53
Atmospheric Optics
Craig F. Bohren Pennsylvania State University, Department of Meteorology, University Park, Pennsylvania, USA Phone: (814) 466-6264; Fax: (814) 865-3663; e-mail: [email protected]
Abstract Colors of the sky and colored displays in the sky are mostly a consequence of selective scattering by molecules or particles, absorption usually being irrelevant. Molecular scattering selective by wavelength – incident sunlight of some wavelengths being scattered more than others – but the same in any direction at all wavelengths gives rise to the blue of the sky and the red of sunsets and sunrises. Scattering by particles selective by direction – different in different directions at a given wavelength – gives rise to rainbows, coronas, iridescent clouds, the glory, sun dogs, halos, and other ice-crystal displays. The size distribution of these particles and their shapes determine what is observed, water droplets and ice crystals, for example, resulting in distinct displays. To understand the variation and color and brightness of the sky as well as the brightness of clouds requires coming to grips with multiple scattering: scatterers in an ensemble are illuminated by incident sunlight and by the scattered light from each other. The optical properties of an ensemble are not necessarily those of its individual members. Mirages are a consequence of the spatial variation of coherent scattering (refraction) by air molecules, whereas the green flash owes its existence to both coherent scattering by molecules and incoherent scattering by molecules and particles.
Keywords sky colors; mirages; green flash; coronas; rainbows; the glory; sun dogs; halos; visibility.
1 Introduction 54 2 Color and Brightness of Molecular Atmosphere 55 2.1 A Brief History 55 54 Atmospheric Optics
2.2 Molecular Scattering and the Blue of the Sky 57 2.3 Spectrum and Color of Skylight 58 2.4 Variation of Sky Color and Brightness 59 2.5 Sunrise and Sunset 62 3 Polarization of Light in a Molecular Atmosphere 63 3.1 The Nature of Polarized Light 63 3.2 Polarization by Molecular Scattering 64 4 Scattering by Particles 66 4.1 The Salient Differences between Particles and Molecules: Magnitude of Scattering 66 4.2 The Salient Differences between Particles and Molecules: Wavelength Dependence of Scattering 67 4.3 The Salient Differences between Particles and Molecules: Angular Dependence of Scattering 68 4.4 The Salient Differences between Particles and Molecules: Degree of Polarization of Scattered Light 69 4.5 The Salient Differences between Particles and Molecules: Vertical Distributions 70 5 Atmospheric Visibility 71 6 Atmospheric Refraction 73 6.1 Physical Origins of Refraction 73 6.2 Terrestrial Mirages 73 6.3 Extraterrestrial Mirages 76 6.4 The Green Flash 77 7 Scattering by Single Water Droplets 78 7.1 Coronas and Iridescent Clouds 78 7.2 Rainbows 80 7.3 The Glory 82 8 Scattering by Single Ice Crystals 83 8.1 Sun Dogs and Halos 83 9 Clouds 86 9.1 Cloud Optical Thickness 86 9.2 Givers and Takers of Light 87 Glossary 89 References 90 Further Reading 90
1 atmosphere and the primary source of Introduction their illumination is the sun. Essentially all light we see is scattered light, even that Atmospheric optics is nearly synonymous directly from the sun. When we say that with light scattering, the only restrictions such light is unscattered we really mean being that the scatterers inhabit the that it is scattered in the forward direction; Atmospheric Optics 55 hence it is as if it were unscattered. glories) resulting from illumination of Scattered light is radiation from matter cloud droplets by sunlight. excited by an external source. When the This article begins with the color source vanishes, so does the scattered light, and brightness of a purely molecu- as distinguished from light emitted by lar atmosphere, including their variation matter, which persists in the absence of across the vault of the sky. This nat- external sources. urally leads to the state of polarization Atmospheric scatterers are either mole- of skylight. Because the atmosphere is cules or particles. A particle is an aggrega- rarely, if ever, entirely free of particles, tion of sufficiently many molecules that the general characteristics of scattering it can be ascribed macroscopic proper- by particles follow, setting the stage ties such as temperature and refractive for a discussion of atmospheric visibil- index. There is no canonical number of ity. molecules that must unite to form a Atmospheric refraction usually sits by bona fide particle. Two molecules clearly itself, unjustly isolated from all those at- do not a quorum make, but what about mospheric phenomena embraced by the 10, 100, 1000? The particle size corre- term scattering. Yet refraction is another sponding to the largest of these numbers manifestation of scattering, coherent scat- −3 µ is about 10 m. Particles this small tering in the sense that phase differences of water substance would evaporate so cannot be ignored. rapidly that they could not exist long under Scattering by single water droplets and conditions normally found in the atmo- ice crystals, each discussed in turn, yields sphere. As a practical matter, therefore, feasts for the eye as well as the mind. The we need not worry unduly about scatterers curtain closes on the optical properties in the shadow region between molecule of clouds. and particle. A property of great relevance to scat- tering problems is coherence,bothofthe 2 array of scatterers and of the incident light. Color and Brightness of Molecular At visible wavelengths, air is an array of Atmosphere incoherent scatterers: the radiant power scattered by N molecules is N times that 2.1 scattered by one (except in the forward ABriefHistory direction). But when water vapor in air condenses, an incoherent array is trans- Edward Nichols began his 1908 presiden- formed into a coherent array: uncorrelated tial address to the New York meeting of the water molecules become part of a single American Physical Society as follows: ‘‘In entity. Although a single droplet is a coher- asking your attention to-day, even briefly, ent array, a cloud of droplets taken together to the consideration of the present state of is incoherent. our knowledge concerning the color of the Sunlight is incoherent but not in an sky it may be truly said that I am inviting absolute sense. Its lateral coherence length you to leave the thronged thoroughfares is tens of micrometers, which is why of our science for some quiet side street we can observe what are essentially where little is going on and you may even interference patterns (e.g., coronas and suspect that I am coaxing you into some 56 Atmospheric Optics
blind alley, the inhabitants of which belong fell short of the mark. Yet his hypothesis to the dead past.’’ that ‘‘the blueness we see in the atmo- Despite this depreciatory statement, sphere is not intrinsic color, but is caused hoary with age, correct and complete by warm vapor evaporated in minute and explanations of the color of the sky still insensible atoms on which the solar rays are hard to find. Indeed, all the faulty fall, rendering them luminous against the explanations lead active lives: the blue infinite darkness of the fiery sphere which sky is the reflection of the blue sea; lies beyond and includes it’’ would, with it is caused by water, either vapor or minor changes, stand critical scrutiny to- droplets or both; it is caused by dust. day. If we set aside Leonardo as sui generis, Thetruecauseoftheblueskyisnot scientific attempts to unravel the origins of difficult to understand, requiring only a theblueskymaybesaidtohavebegunwith bit of critical thought stimulated by belief Newton, that towering pioneer of optics, in the inherent fascination of all natural who, in time-honored fashion, reduced it phenomena, even those made familiar by to what he already had considered: inter- everyday occurrence. ference colors in thin films. Almost two Our contemplative prehistoric ancestors centuries elapsed before more pieces in no doubt speculated on the origin of the the puzzle were contributed by the exper- blue sky, their musings having vanished imental investigations of von Brucke¨ and into it. Yet it is curious that Aristotle, the Tyndall on light scattering by suspensions most prolific speculator of early recorded of particles. Around the same time Clau- history, makes no mention of it in his sius added his bit in the form of a theory Meteorologica even though he delivered that scattering by minute bubbles causes pronouncements on rainbows, halos, and the blueness of the sky. A better theory mock suns and realized that ‘‘the sun was not long in coming. It is associated looks red when seen through mist or with a man known to the world as Lord smoke.’’ Historical discussions of the blue Rayleigh even though he was born John sky sometimes cite Leonardo as the first to William Strutt. comment intelligently on the blue of the Rayleigh’s paper of 1871 marks the sky, although this reflects a European bias. beginning of a satisfactory explanation If history were to be written by a supremely of the blue sky. His scattering law, the disinterested observer, Arab philosophers key to the blue sky, is perhaps the would likely be given more credit for most famous result ever obtained by having had profound insights into the dimensional analysis. Rayleigh argued that workings of nature many centuries before the field Es scattered by a particle small their European counterparts descended compared with the light illuminating it from the trees. Indeed, Moller¨ [1] begins is proportional to its volume V and his brief history of the blue sky with to the incident field Ei. Radiant energy Jakub Ibn Ishak Al Kindi (800–870), who conservation requires that the scattered explained it as ‘‘a mixture of the darkness field diminish inversely as the distance of the night with the light of the dust and r from the particle so that the scattered haze particles in the air illuminated by power diminishes as the square of r.To the sun.’’ make this proportionality dimensionally Leonardo was a keen observer of light in homogeneous requires the inverse square nature even if his explanations sometimes of a quantity with the dimensions of Atmospheric Optics 57 length. The only plausible physical variable 2.2 at hand is the wavelength of the incident Molecular Scattering and the Blue of the Sky light, which leads to Our illustrious predecessors all gave ex- EiV planations of the blue sky requiring the Es ∝ .(1) rλ2 presence of water in the atmosphere: Leonardo’s ‘‘evaporated warm vapor,’’ When the field is squared to ob- Newton’s ‘‘Globules of water,’’ Clausius’s tain the scattered power, the result is bubbles. Small wonder, then, that water Rayleigh’s inverse fourth-power law. This still is invoked as the cause of the blue law is really only an often – but not sky. Yet a cause of something is that with- always – very good approximation. Miss- out which it would not occur, and the sky ing from it are dimensionless proper- would be no less blue if the atmosphere ties of the particle such as its refractive were free of water. index, which itself depends on wave- A possible physical reason for attributing length. Because of this dispersion, there- the blue sky to water vapor is that, because fore, nothing scatters exactly as the inverse of selective absorption, liquid water (and fourth power. ice) is blue upon transmission of white Rayleigh’s 1871 paper did not give the light over distances of order meters. Yet complete explanation of the color and if all the water in the atmosphere at any polarization of skylight. What he did that instant were to be compressed into a liquid, was not done by his predecessors was to the result would be a layer about 1 cm thick, give a law of scattering, which could be which is not sufficient to transform white used to test quantitatively the hypothesis light into blue by selective absorption. that selective scattering by atmospheric Water vapor does not compensate for particles could transform white sunlight its hundredfold lower abundance than into blue skylight. But as far as giving nitrogen and oxygen by greater scattering the agent responsible for the blue sky is per molecule. Indeed, scattering of visible concerned, Rayleigh did not go essentially light by a water molecule is slightly less beyond Newton and Tyndall, who invoked than that by either nitrogen or oxygen. particles. Rayleigh was circumspect about Scattering by atmospheric molecules the nature of these particles, settling on does not obey Rayleigh’s inverse fourth- salt as the most likely candidate. It was not power law exactly. A least-squares fit over until 1899 that he published the capstone the visible spectrum from 400 to 700 nm to his work on skylight, arguing that air of the molecular scattering coefficient of sea- molecules themselves were the source of level air tabulated by Penndorf [2] yields an the blue sky. Tyndall cannot be given inverse 4.089th-power scattering law. β thecreditforthisbecauseheconsidered The molecular scattering coefficient , air to be optically empty:whenpurged which plays important roles in following of all particles it scatters no light. This sections, may be written erroneous conclusion was a result of the β = Nσs,(2) small scale of his laboratory experiments. On the scale of the atmosphere, sufficient where N is the number of molecules light is scattered by air molecules to be per unit volume and σs, the scattering readily observable. cross section (an average because air is 58 Atmospheric Optics
a mixture) per molecule, approximately an ideal gas (to good approximation the obeys Rayleigh’s law. The form of this atmosphere is an ideal gas), scattering by expression betrays the incoherence of N molecules is N times scattering by one. scattering by atmospheric molecules. The This is the only sense in which the blue sky inverse of β is interpreted as the scattering can be attributed to scattering by fluctua- mean free path, the average distance a tions. Perfectly homogeneous matter does photon must travel before being scattered. not exist. As stated pithily by Planck, ‘‘a To say that the sky is blue because chemically pure substance may be spo- of Rayleigh scattering, as is sometimes ken of as a vacuum made turbid by the done, is to confuse an agent with a presence of molecules.’’ law. Moreover, as Young [3] pointed out, the term Rayleigh scattering has many 2.3 meanings. Particles small compared with Spectrum and Color of Skylight the wavelength scatter according to the same law as do molecules. Both can What is the spectrum of skylight? What is be said to be Rayleigh scatterers, but its color? These are two different questions. only molecules are necessary for the blue Answering the first answers the second sky. Particles, even small ones, generally but not the reverse. Knowing the color of diminishthevividnessofthebluesky. skylight we cannot uniquely determine its Fluctuations are sometimes trumpeted spectrum because of metamerism:Agiven asthe‘‘real’’causeofthebluesky.Pre- perceived color can in general be obtained sumably, this stems from the fluctuation in an indefinite number of ways. theory of light scattering by media in which Skylight is not blue (itself an impre- the scatterers are separated by distances cise term) in an absolute sense. When small compared with the wavelength. In the visible spectrum of sunlight outside this theory, which is associated with Ein- the earth’s atmosphere is modulated by stein and Smoluchowski, matter is taken Rayleigh’s scattering law, the result is a to be continuous but characterized by a spectrum of scattered light that is nei- refractive index that is a random function ther solely blue nor even peaked in the of position. Einstein [4] stated that ‘‘it is blue (Fig. 1). Although blue does not pre- remarkable that our theory does not make dominate spectrally, it does predominate direct use of the assumption of a discrete perceptually. We perceive the sky to be distribution of matter.’’ That is, he cir- blue even though skylight contains light of cumvented a difficulty but realized it could all wavelengths. have been met head on, as Zimm [5] did Any source of light may be looked upon years later. as a mixture of white light and light of The blue sky is really caused by scat- a single wavelength called the dominant tering by molecules – to be more precise, wavelength.Thepurity of the source is scattering by bound electrons: free elec- the relative amount of the monochromatic trons do not scatter selectively. Because air component in the mixture. The dominant molecules are separated by distances small wavelength of sunlight scattered according compared with the wavelengths of visible to Rayleigh’s law is about 475 nm, which light, it is not obvious that the power scat- lies solidly in the blue if we take this tered by such molecules can be added. Yet to mean light with wavelengths between if they are completely uncorrelated, as in 450 and 490 nm. The purity of this Atmospheric Optics 59
2.4 Variation of Sky Color and Brightness
Notonlyisskylightnotpureblue, but its color and brightness vary across the vault of the sky, with the best blues at zenith. Near the astronomical horizon the sky is brighter than overhead but of considerably lower purity. That this variation can be observed from an airplane flying at 10 km, well above most particles, suggests that the sky is inherently nonuniform in color and Fig. 1 Rayleigh’s scattering law (dots), the brightness (Fig. 2). To understand why spectrum of sunlight outside the Earth’s requires invoking multiple scattering. atmosphere (dashes), and the product of the two (solid curve). The solar spectrum is taken from Multiple scattering gives rise to observ- Thekaekara, M. P., Drummond, A. J. (1971), Nat. able phenomena that cannot be explained Phys. Sci. 229, 6–9 [6] solely by single-scattering arguments. This is easily demonstrated. Fill a blackened pan scattered light, about 42%, is the upper limit for skylight. Blues of real skies are less pure. Another way of conveying the color of a source of light is by its color temperature, the temperature of a blackbody having the same perceived color as the source. Since blackbodies do not span the entire gamut of colors, not all sources of light can be assigned color temperatures. But many natural sources of light can. The color temperature of light scattered according to Rayleigh’s law is infinite. This follows from Planck’s spectral emission function ebλ in the limit of high temperature, 2πckT hc e λ ≈ , kT,(3) b λ4 λ where h is Planck’s constant, k is Boltz- mann’s constant, c is the speed of light in vacuo,andT is absolute temperature. Thus, the emission spectrum of a black- Fig. 2 Even at an altitude of 10 km, well above body with an infinite temperature has the most particles, the sky brightness increases same functional form as Rayleigh’s scat- markedly from the zenith to the tering law. astronomical horizon 60 Atmospheric Optics
with clean water, then add a few drops of path, the optical thickness is about 35 times milk. The resulting dilute suspension il- greater (Fig. 4), which leads to several luminated by sunlight has a bluish cast. observable phenomena. But when more milk is added, the suspen- Even an intrinsically black object is sion turns white. Yet the properties of the luminous to an observer because of scatterers (fat globules) have not changed, airlight, light scattered by all the molecules only their optical thickness: the blue suspen- and particles along the line of sight sion being optically thin, the white being from observer to object. Provided that optically thick. this is uniformly illuminated by sunlight Optical thickness is physical thickness and that ground reflection is negligi- in units of scattering mean free path, and ble, the airlight radiance L is approxi- hence is dimensionless. The optical thick- mately −τ ness τ between any two points connected L = GL0(1 − e ), (5) by an arbitrary path in a medium populated by (incoherent) scatterers is an integral where L0 is the radiance of incident over the path: sunlight along the line of sight with optical thickness τ.ThetermG accounts for 2 geometric reduction of radiance because τ = β ds.(4) 1 of scattering of nearly monodirectional sunlight in all directions. If the line of τ The normal optical thickness n of the sight is uniform in composition, τ = βd, atmosphere is that along a radial path where β is the scattering coefficient and d extending from the surface of the Earth is the physical distance to the black object. to infinity. Figure 3 shows τn over the If τ is small (1), L ≈ GL0τ.Ina visible spectrum for a purely molecular purely molecular atmosphere, τ varies τ atmosphere. Because n is generally small with wavelength according to Rayleigh’s compared with unity, a photon from law; hence the distant black object in the sun traversing a radial path in the such an atmosphere is perceived to be atmosphere is unlikely to be scattered more than once. But along a tangential
Fig. 4 Optical thickness (relative to the normal optical thickness) of a molecular atmosphere Fig. 3 Normal optical thickness of a pure along various paths with zenith angles between ◦ ◦ molecular atmosphere 0 (normal) and 90 (tangential) Atmospheric Optics 61 bluish. As τ increases so does L but not transmitted to the observer without be- proportionally. Its limit is GL0: The airlight ing scattered out of the line of sight. radiance spectrum is that of the source of For a long optical path, these two pro- illumination. Only in the limit d = 0is cesses compensate, resulting in a hori- L = 0 and the black object truly black. zon radiance spectrum which is that of Variation of the brightness and color of the source. dark objects with distance was called aerial Selective scattering by molecules is not perspective by Leonardo. By means of it we sufficient for a blue sky. The atmosphere estimate distances to objects of unknown also must be optically thin, at least for size such as mountains. most zenith angles (Fig. 4) (the black- Aerial perspective belongs to the same ness of space as a backdrop is taken family as the variation of color and for granted but also is necessary, as brightness of the sky with zenith angle. Leonardo recognized). A corollary of this Although the optical thickness along a path is that the blue sky is not inevitable: an tangent to the Earth is not infinite, it is atmosphere composed entirely of nonab- sorbing, selectively scattering molecules sufficiently large (Figs. 3 and 4) that GL0 is a good approximation for the radiance of overlying a nonselectively reflecting earth the horizon sky. For isotropic scattering (a need not be blue. Figure 5 shows calcu- condition almost satisfied by molecules), lated spectra of the zenith sky over black − ground for a molecular atmosphere with G is around 10 5, the ratio of the solid the present normal optical thickness as angle subtended by the sun to the solid well as for hypothetical atmospheres 10 angle of all directions (4π). Thus, the and 40 times thicker. What we take to horizon sky is not nearly so bright as be inevitable is accidental: If our atmo- direct sunlight. sphere were much thicker, but identical Unlike in the milk experiment, what in composition, the color of the sky is observed when looking at the hori- would be quite different from what it zon sky is not multiply scattered light. is now. Both have their origins in multiple scat- tering but manifested in different ways. Milk is white because it is weakly ab- sorbing and optically thick, and hence all components of incident white light are multiply scattered to the observer even though the blue component traverses a shorter average path in the suspension than the red component. White horizon light has escaped being multiply scat- tered, although multiple scattering is why this light is white (strictly, has the spec- trum of the source). More light at the short-wavelength end of the spectrum is Fig. 5 Spectrum of overhead skylight for the scattered toward the observer than at the present molecular atmosphere (solid curve), as long-wavelength end. But long-wavelength well as for hypothetical atmospheres 10 (dashes) light has the greater likelihood of being and 40 (dots) times thicker 62 Atmospheric Optics
2.5 scatterers not lying along the line of Sunrise and Sunset sight to the sun. A striking example of this is a horizon sky tinged with If short-wavelength light is preferentially oranges and pinks in the direction opposite scattered out of direct sunlight, long- the sun. wavelength light is preferentially trans- The color and brightness of the sun mitted in the direction of sunlight. changes as it arcs across the sky because Transmission is described by an expo- the optical thickness along the line of sight nential law (if light multiply scattered changes with solar zenith angle .Ifthe back into the direction of the sunlight Earth were flat (as some still aver), the is negligible): transmitted solar radiance would be −τ τ / L = L0e ,(6) n cos L = L0e .(7) where L is the radiance at the observer This equation is a good approximation in the direction of the sun, L is the 0 except near the horizon. On a flat earth, radiance of sunlight outside the atmo- the optical thickness is infinite for horizon sphere, and τ is the optical thickness along paths. On a spherical earth, optical thick- this path. nesses are finite although much larger for If the wavelength dependence of τ is horizon than for radial paths. given by Rayleigh’s law, sunlight is red- The normal optical thickness of an dened upon transmission: The spectrum atmosphere in which the number density of the transmitted light is comparatively of scatterers decreases exponentially with richer than the incident spectrum in height z above the surface, exp(−z/H), is lightatthelong-wavelengthendofthe the same as that for a uniform atmosphere visible spectrum. But to say that trans- of finite thickness: mitted sunlight is reddened is not the ∞ same as saying it is red. The perceived τn = β dz = β0H,(8) color can be yellow, orange, or red, de- 0 pending on the magnitude of the optical β thickness. In a molecular atmosphere, where H is the scale height and 0 is the optical thickness along a path from the scattering coefficient at sea level. the sun, even on or below the horizon, This equivalence yields a good approx- is not sufficient to give red light upon imation even for the tangential opti- transmission. Although selective scatter- cal thickness. For any zenith angle, ing by molecules yields a blue sky, reds the optical thickness is given approxi- are not possible in a molecular atmo- mately by sphere, only yellows and oranges. This τ R2 2R can be observed on clear days, when the = e 2 + e + 2 cos 1 horizon sky at sunset becomes succes- τn H H sively tinged with yellow, then orange, but Re not red. − cos , (9) H Equation (6) applies to the radiance only in the direction of the sun. Oranges and where Re is the radius of the Earth. reds can be seen in other directions AflatearthisoneforwhichRe is because reddened sunlight illuminates infinite, in which instance Eq. (9) yields Atmospheric Optics 63 the expected relation from direct sunlight by clouds (that do not, of course, occlude the distant cloud τ 1 lim = .(10) of interest). This may reduce the first →∞ τ Re n cos term in Eq. (11) so that the second term For Earth’s atmosphere, the molecular dominates. Thus, under a partly over- scale height is about 8 km. According to cast sky, distant horizon clouds may be the approximate relation Eq. (9), therefore, reddish even when the sun is high in the horizon optical thickness is about the sky. 39 times greater than the normal optical The zenith sky at sunset and twilight thickness. Taking the exponential decrease is the exception to the general rule that of molecular number density into account molecular scattering is sufficient to ac- yields a value about 10% lower. count for the color of the sky. In the Variations on the theme of reds and absence of molecular absorption, the spec- oranges at sunrise and sunset can be trum of the zenith sky would be essentially seen even when the sun is overhead. The that of the zenith sun (although greatly radiance at an observer an optical distance reduced in radiance), hence would not τ from a (horizon) cloud is the sum bethebluethatisobserved.Thiswas of cloudlight transmitted to the observer pointed out by Hulburt [7], who showed and airlight: that absorption by ozone profoundly af- fects the color of the zenith sky when the −τ −τ L = L0G(1 − e ) + L0Gce ,(11) sun is near the horizon. The Chappuis band of ozone extends from about 450 to where Gc is a geometrical factor that 700 nm and peaks at around 600 nm. Pref- accounts for scattering of nearly monodi- erential absorption of sunlight by ozone rectional sunlight into a hemisphere of over long horizon paths gives the zenith directions by the cloud. If the cloud sky its blueness when the sun is near is approximated as an isotropic reflec- the horizon. With the sun more than tor with reflectance R and illuminated ◦ about 10 above the horizon, however, at an angle , the geometrical factor ozone has little effect on the color of G is R cos/π,where is the c s s the sky. solid angle subtended by the sun at the Earth. If Gc > G, the observed ra- diance is redder (i.e., enriched in light of longer wavelengths) than the incident 3 radiance. If Gc < G, the observed radi- Polarization of Light in a Molecular ance is bluer than the incident radiance. Atmosphere Thus, distant horizon clouds can be red- dish if they are bright or bluish if they 3.1 are dark. The Nature of Polarized Light Underlying Eq. (11) is the implicit as- sumption that the line of sight is uniformly Unlike sound, light is a vector wave, an illuminated by sunlight. The first term electromagnetic field lying in a plane nor- in this equation is airlight; the second mal to the propagation direction. The is transmitted cloudlight. Suppose, how- polarization state of such a wave is deter- ever, that the line of sight is shadowed mined by the degree of correlation of any 64 Atmospheric Optics
two orthogonal components into which observed brightness indicates some degree its electric (or magnetic) field is resolved. of partial polarization. Completely polarized light corresponds to In the analysis of any scattering prob- complete correlation; completely unpolar- lem, a plane of reference is required. This ized light corresponds to no correlation; is usually the scattering plane, determined partially polarized light corresponds to par- by the directions of the incident and scat- tial correlation. tered waves, the angle between them being If an electromagnetic wave is completely the scattering angle. Light polarized perpen- polarized, the tip of its oscillating electric dicular (parallel) to the scattering plane is field traces out a definite elliptical curve, sometimes said to be vertically (horizon- the vibration ellipse. Lines and circles are tally) polarized. Vertical and horizontal in special ellipses, the light being said to this context, however, are arbitrary terms be linearly or circularly polarized, respec- indicating orthogonality and bear no rela- tively. The general state of polarization is tion, except by accident, to the direction elliptical. of gravity. Any beam of light can be consid- ThedegreeofpolarizationP of light ered an incoherent superposition of two scattered by a tiny sphere illuminated by collinear beams, one unpolarized, the unpolarized light is (Fig. 6) other completely polarized. The radiance 1 − cos2 θ of the polarized component relative to P = ,(12) the total is defined as the degree of po- 1 + cos2 θ larization (often multiplied by 100 and where the scattering angle θ ranges from ◦ ◦ expressed as a percentage). This can be 0 (forward direction) to 180 (backward measured for a source of light (e.g., light direction); the scattered light is partially from different sky directions) by rotat- linearly polarized perpendicular to the ing a (linear) polarizing filter and noting scattering plane. Although this equation the minimum and maximum radiances transmitted by it. The degree of (linear) polarization is defined as the difference between these two radiances divided by their sum.
3.2 Polarization by Molecular Scattering
Unpolarized light can be transformed into partially polarized light upon interaction with matter because of different changes in amplitude of the two orthogonal field com- ponents. An example of this is the partial polarization of sunlight upon scattering Fig. 6 Degree of polarization of the light by atmospheric molecules, which can be scattered by a small (compared with the wavelength) sphere for incident unpolarized detected by looking at the sky through a po- light (solid curve). The dashed curve is for a larizing filter (e.g., polarizing sunglasses) small spheroid chosen such that the degree of ◦ while rotating it. Waxing and waning of the polarization at 90 is that for air Atmospheric Optics 65 is a first step toward understanding that in the atmosphere, as opposed to polarization of skylight, more often than the laboratory, multiple scattering is not not it also has been a false step, having led negligible. Also, atmospheric air is almost countless authors to assert that skylight is never free of particles and is illuminated ◦ completely polarized at 90 from the sun. by light reflected by the ground. We must ◦ Although P = 1atθ = 90 according to take the atmosphere as it is, whereas Eq. (12), skylight is never 100% polarized in the laboratory we often can eliminate at this or any other angle, and for everything we consider extraneous. several reasons. Because of both multiple scattering Although air molecules are very small and ground reflection, light from any compared with the wavelengths of visible directionintheskyisnot,ingeneral, light, a requirement underlying Eq. (12), made up solely of light scattered in a the dominant constituents of air are not single direction relative to the incident spherically symmetric. sunlight but is a superposition of beams The simplest model of an asymmetric with different scattering histories, hence molecule is a small spheroid. Although different degrees of polarization. As a it is indeed possible to find a direction consequence, even if air molecules were in which the light scattered by such a perfect spheres and the atmosphere were spheroid is 100% polarized, this direction completely free of particles, skylight would ◦ depends on the spheroid’s orientation. not be 100% polarized at 90 to the sun or In an ensemble of randomly oriented at any other angle. spheroids, each contributes its mite to the Reduction of the maximum degree of total radiance in a given direction, but polarization is not the only consequence each contribution is partially polarized to of multiple scattering. According to Fig. 6, varying degrees between 0 and 100%. It there should be two neutral points in the is impossible for beams of light to be sky, directions in which skylight is unpo- incoherently superposed in such a way that larized: directly toward and away from the the degree of polarization of the resultant sun. Because of multiple scattering, how- is greater than the degree of polarization ever, there are three such points. When ◦ of the most highly polarized beam. the sun is higher than about 20 above the ◦ Because air is an ensemble of randomly horizon there are neutral points within 20 oriented asymmetric molecules, sunlight of the sun, the Babinet point above it, the scattered by air never is 100% polarized. Brewster point below. They coincide when The intrinsic departure from perfection is the sun is directly overhead and move about 6%. Figure 6 also includes a curve apart as the sun descends. When the sun ◦ for light scattered by randomly oriented is lower than 20 ,theArago point is about ◦ spheroids chosen to yield 94% polarization 20 above the antisolar point, the direction ◦ at 90 . This angle is so often singled opposite the sun. outthatitmaydeflectattentionfrom One consequence of the partial polar- nearby scattering angles. Yet, the degree of ization of skylight is that the colors of polarization is greater than 50% for a range distant objects may change when viewed ◦ of scattering angles 70 wide centered through a rotated polarizing filter. If the ◦ about 90 . sun is high in the sky, horizontal airlight Equation (12) applies to air, not to will have a fairly high degree of polariza- the atmosphere, the distinction being tion. According to the previous section, 66 Atmospheric Optics
airlight is bluish. But if it also is partially scattered field, which in turn is the sum polarized, its radiance can be diminished of all the fields scattered by the individual with a polarizing filter. Transmitted cloud- molecules. For an incoherent array we may light, however, is unpolarized. Because the ignore the wave nature of light, whereas radiance of airlight can be reduced more for a coherent array we must take it than that of cloudlight, distant clouds may into account. change from white to yellow to orange Water vapor is a good example to ponder when viewed through a rotated polariz- because it is a constituent of air and ing filter. can condense to form cloud droplets. The difference between a sky containing water vapor and the same sky with the same 4 amount of water but in the form of a cloud Scattering by Particles of droplets is dramatic. According to Rayleigh’s law, scattering Up to this point we have considered only an by a particle small compared with the atmosphere free of particles, an idealized wavelength increases as the sixth power state rarely achieved in nature. Particles of its size (volume squared). A droplet of still would inhabit the atmosphere even diameter 0.03 µm, for example, scatters if the human race were to vanish from about 1012 times more light than does one the Earth. They are not simply by- of its constituent molecules. Such a droplet products of the ‘‘dark satanic mills’’ of contains about 107 molecules. Thus, civilization. scattering per molecule as a consequence All molecules of the same substance are of condensation of water vapor into essentially identical. This is not true of a coherent water droplet increases by particles: They vary in shape and size, about 105. and may be composed of one or more Cloud droplets are much larger than homogeneous regions. 0.03 µm, a typical diameter being about 10 µm. Scattering per molecule in such 4.1 a droplet is much greater than scatter- The Salient Differences between Particles ing by an isolated molecule, but not to and Molecules: Magnitude of Scattering the extent given by Rayleigh’s law. Scat- tering increases as the sixth power of The distinction between scattering by droplet diameter only when the molecules molecules when widely separated and scatter coherently in phase. If a droplet when packed together into a droplet is is sufficiently small compared with the that between scattering by incoherent and wavelength, each of its molecules is ex- coherent arrays. Isolated molecules are cited by essentially the same field and excited primarily by incident (external) all the waves scattered by them inter- light, whereas the same molecules forming fere constructively. But when a droplet a droplet are excited by incident light and is comparable to or larger than the wave- by each other’s scattered fields. The total length, interference can be constructive, power scattered by an incoherent array of destructive, and everything in between, molecules is the sum of their scattered and hence scattering does not increase powers. The total power scattered by a as rapidly with droplet size as predicted by coherent array is the square of the total Rayleigh’s law. Atmospheric Optics 67
The figure of merit for comparing different from the water vapor out of which scatterers of different size is their scatter- it was born that the offspring bears no ing cross section per unit volume, which, resemblance to its parents. We can see except for a multiplicative factor, is the scat- through tens of kilometers of air laden tering cross section per molecule. A scat- with water vapor, whereas a cloud a few tering cross section may be looked upon tens of meters thick is enough to occult as an effective area for removing radiant the sun. Yet a rainshaft born out of a energy from a beam: the scattering cross cloud is considerably more translucent section times the beam irradiance is the than its parent. radiant power scattered in all directions. The scattering cross section per unit 4.2 volume for water droplets illuminated The Salient Differences between Particles by visible light and varying in size and Molecules: Wavelength Dependence of from molecules (10−4 µm) to raindrops Scattering (103 µm) is shown in Fig. 7. Scattering by a molecule that belongs to a cloud droplet is Regardless of their size and composi- about 109 times greater than scattering by tion, particles scatter approximately as an isolated molecule, a striking example the inverse fourth power of wavelength of the virtue of cooperation. Yet in if they are small compared with the wave- molecular as in human societies there are length and absorption is negligible, two limits beyond which cooperation becomes important caveats. Failure to recognize dysfunctional: Scattering by a molecule them has led to errors, such as that yel- that belongs to a raindrop is about 100 low light penetrates fog better because times less than scattering by a molecule it is not scattered as much as light of that belongs to a cloud droplet. This shorter wavelengths. Although there may tremendous variation of scattering by be perfectly sound reasons for choosing water molecules depending on their state yellow instead of blue or green as the of aggregation has profound observational color of fog lights, greater transmission consequences. A cloud is optically so much through fog is not one of them: Scat- tering by fog droplets is essentially in- dependent of wavelength over the visible spectrum. Small particles are selective scatterers; large particles are not. Particles nei- ther small nor large give the reverse of what we have come to expect as nor- mal. Figure 8 shows scattering of visible light by oil droplets with diameters 0.1, 0.8, and 10 µm. The smaller droplets scatter according to Rayleigh’s law; the larger droplets (typical cloud droplet size) are nonselective. Between these two ex- µ Fig. 7 Scattering (per molecule) of visible light tremes are droplets (0.8 m) that scatter (arbitrary units) by water droplets varying in size long-wavelength light more than short- from a single molecule to a raindrop wavelength. Sunlight or moonlight seen 68 Atmospheric Optics
the forward and backward hemispheres, scattering becomes markedly asymmetric for particles comparable to or larger than the wavelength. For example, forward scat- tering by a water droplet as small as 0.5 µm is about 100 times greater than backward scattering, and the ratio of forward to back- ward scattering increases more or less monotonically with size (Fig. 9). The reason for this asymmetry is found in the singularity of the forward direc- tion. In this direction, waves scattered Fig. 8 Scattering of visible light by oil droplets of diameter 0.1 µm (solid curve), 0.8 µm by two or more scatterers excited solely (dashes), and 10 µm(dots) by incident light (ignoring mutual ex- citation) are always in phase regardless of the wavelength and the separation through a thin cloud of these intermediate of the scatterers. If we imagine a par- droplets would be bluish or greenish. This ticletobemadeupofN small sub- requires droplets of just the right size, and units, scattering in the forward direc- hence it is a rare event, so rare that it oc- tion increases as N2,theonlydirec- curs once in a blue moon. Astronomers, tion for which this is always true. For for unfathomable reasons, refer to the sec- other directions, the wavelets scattered by ond full moon in a month as a blue moon, the subunits will not necessarily all be but if such a moon were blue it would be in phase. As a consequence, scattering only by coincidence. The last reliably re- in the forward direction increases with ported outbreak of blue and green suns size (i.e., N) more rapidly than in any and moons occurred in 1950 and was other direction. attributed to an oily smoke produced in Canadian forest fires.
4.3 The Salient Differences between Particles and Molecules: Angular Dependence of Scattering
The angular distribution of scattered light changes dramatically with the size of the scatterer. Molecules and particles that are small compared with the wavelength are nearly isotropic scatterers of unpo- larized light, the ratio of maximum (at ◦ ◦ ◦ 0 and 180 ) to minimum (at 90 )scat- Fig. 9 Angular dependence of scattering of visible light (0.55 µm) by water droplets small tered radiance being only 2 for spheres, compared with the wavelength (dashes), and slightly less for other spheroids. Al- diameter 0.5 µm (solid curve), and diameter though small particles scatter the same in 10 µm(dots) Atmospheric Optics 69
Many common observable phenom- ena depend on this forward-backward asymmetry. Viewed toward the illumi- nating sun, glistening fog droplets on a spider’s web warn us of its presence. But when we view the web with our backs to the sun, the web mysteriously disap- pears. A pattern of dew illuminated by the rising sun on a cold morning seems etched on a windowpane. But if we go outside to look at the window, the pattern vanishes. Thin clouds sometimes hover Fig. 10 Degree of polarization of light scattered over warm, moist heaps of dung, but may by water droplets illuminated by unpolarized go unnoticed unless they lie between us visible light (0.55 µm). The dashed curve is for a and the source of illumination. These are droplet small compared with the wavelength; the solid curve is for a droplet of diameter 0.5 µm; but a few examples of the consequences the dotted curve is for a droplet of diameter of strongly asymmetric scattering by sin- 1.0 µm. Negative degrees of polarization gle particles comparable to or larger than indicate that the scattered light is partially the wavelength. polarized parallel to the scattering plane
4.4 The Salient Differences between Particles and Molecules: Degree of Polarization of Scattered Light
All the simple rules about polarization upon scattering are broken when we turn from molecules and small particles to particles comparable to the wavelength. For example, the degree of polarization of light scattered by small particles is a simple function of scattering angle. But simplicity gives way to complexity as particles grow Fig. 11 Degree of polarization at a scattering ◦ (Fig. 10), the scattered light being partially angle of 90 of light scattered by a water droplet polarized parallel to the scattering plane of diameter 0.5 µm illuminated by unpolarized light for some scattering angles, perpendicular for others. ◦ The degree of polarization of light say, 90 may vary considerably over the scattered by molecules or by small particles visible spectrum (Fig. 11). is essentially independent of wavelength. In general, particles can act as polarizers But this is not true for particles comparable or retarders or both. A polarizer transforms to or larger than the wavelength. Scattering unpolarized light into partially polarized by such particles exhibits dispersion of light. A retarder transforms polarized light polarization: The degree of polarization at, of one form into that of another (e.g., 70 Atmospheric Optics
linear into elliptical). Molecules and small 4.5 particles, however, are restricted to roles The Salient Differences between Particles as polarizers. If the atmosphere were and Molecules: Vertical Distributions inhabited solely by such scatterers, skylight could never be other than partially linearly Not only are the scattering properties of polarized. Yet particles comparable to particles quite different, in general, from or larger than the wavelength often those of molecules; the different vertical are present; hence skylight can acquire distributions of particles and molecules by a degree of ellipticity upon multiple themselves affect what is observed. The scattering: Incident unpolarized light is number density of molecules decreases partially linearly polarized in the first more or less exponentially with height scattering event, then transformed into z above the surface: exp(−z/Hm), where partially elliptically polarized light in the molecular scale height Hm is around subsequent events. 8 km. Although the decrease in number Bees can navigate by polarized sky- density of particles with height is also ap- light. This statement, intended to evoke proximately exponential, the scale height great awe for the photopolimetric pow- for particles Hp is about 1–2 km. As a ers of bees, is rarely accompanied by consequence, particles contribute dispro- an important caveat: The sky must be portionately to optical thicknesses along clear. Figures 10 and 11 show two rea- near-horizon paths. Subject to the approxi- sons – there are others – why bees, re- mations underlying Eq. (9), the ratio of the markable though they may be, cannot do tangential (horizon) optical thickness for the impossible. The simple wavelength- particles τtp to that for molecules τtm is independent relation between the posi- tion of the sun and the direction in τtp τnp Hm which skylight is most highly polarized, = ,(13) τ τ H an underlying necessity for navigating tm nm p by means of polarized skylight, is oblit- erated when clouds cover the sky. This where the subscript t indicates a tangential was recognized by the decoder of bee path and n indicates a normal (radial) path. dances himself von Frisch, [8]: ‘‘Some- Because of the incoherence of scattering times a cloud would pass across the area by atmospheric molecules and particles, of sky visible through the tube; when this scattering coefficients are additive, and happened the dances became disoriented, hence so are optical thicknesses. For equal and the bees were unable to indicate the normal optical thicknesses, the tangential direction to the feeding place. Whatever optical thickness for particles is at least phenomenon in the blue sky served to ori- twice that for molecules. Molecules by ent the dances, this experiment showed themselves cannot give red sunrises and that it was seriously disturbed if the sunsets; molecules need the help of blue sky was covered by a cloud.’’ But particles. For a fixed τnp, the tangential von Frisch’s words often have been for- optical thickness for particles is greater gotten by disciples eager to spread the themoretheyareconcentratednear story about bee magic to those just as the ground. eager to believe what is charming even At the horizon the relative rate of change though untrue. of transmission T of sunlight with zenith Atmospheric Optics 71 angle is 1 dT Re = τn ,(14) T d H where the scale height and normal opti- cal thickness may be those for molecules or particles. Not only do particles, be- ing more concentrated near the surface, give disproportionate attenuation of sun- light on the horizon, but they magnify the angular gradient of attenuation there. A perceptible change in color across the ◦ sun’s disk (which subtends about 0.5 ) on the horizon also requires the help of particles. Fig. 12 Because of scattering by molecules and particles along the line of sight, each successive ridge is brighter than the ones in front of it even 5 though all of them are covered with the same Atmospheric Visibility dark vegetation
On a clear day can we really see for- ever? If not, how far can we see? To account the portion of it that stimu- answer this question requires qualifying lates the human eye or by what relative it by restricting viewing to more or less amount it does so at each wavelength. horizontal paths during daylight. Stars Luminance (also sometimes called bright- at staggering distances can be seen at ness) is the corresponding photometric night, partly because there is no sky- quantity. Luminance and radiance are light to reduce contrast, partly because related by an integral over the visi- stars overhead are seen in directions ble spectrum: for which attenuation by the atmosphere is least. B = K(λ)L(λ) dλ, (15) The radiance in the direction of a black object is not zero, because of light scattered where the luminous efficiency of the hu- along the line of sight (see Sec. 2.4). At man eye K peaks at about 550 nm and sufficiently large distances, this airlight is vanishes outside the range 385–760 nm. indistinguishable from the horizon sky. The contrast C between any object and An example is a phalanx of parallel dark the horizon sky is ridges, each ridge less distinct than those in front of it (Fig. 12). The farthest ridges − = B B∞ ,() blend into the horizon sky. Beyond some C 16 B∞ distance we cannot see ridges because of insufficient contrast. where B∞ is the luminance for an infinite Equation (5) gives the airlight radi- horizon optical thickness. For a uniformly ance, a radiometric quantity that de- illuminated line of sight of length d, scribes radiant power without taking into uniform in its scattering properties, and 72 Atmospheric Optics
with a black backdrop, the contrast is sky assuming a contrast threshold of 0.02 and ignoring the curvature of the earth.
KGL0 exp(−βd) dλ We also observe contrast between ele- C =− .(17) ments of the same scene, a hillside mottled with stands of trees and forest clearings, KGL0 dλ for example. The extent to which we can resolve details in such a scene depends on The ratio of integrals in this equation sun angle as well as distance. defines an average optical thickness: The airlight radiance for a nonreflecting = C =−exp(−τ). (18) object is Eq. (5) with G p( ) s,where p() is the probability (per unit solid angle) This expression for contrast reduction that light is scattered in a direction making with (optical) distance is mathematically, an angle with the incident sunlight and s is the solid angle subtended by the sun. but not physically, identical to Eq. (6), ◦ When the sun is overhead, = 90 ;with which perhaps has engendered the mis- ◦ conception that atmospheric visibility is the sun at the observer’s back, = 180 ; for an observer looking directly into the reduced because of attenuation. Yet as ◦ there is no light from a black object to be sun, = 0 . attenuated, its finite visual range cannot The radiance of an object with a finite re- be a consequence of attenuation. flectance R and illuminated at an angle The distance beyond which a dark is given by Eq. (11). Equations (5) and (11) object cannot be distinguished from the can be combined to obtain the contrast be- horizon sky is determined by the contrast tween reflecting and nonreflecting objects: threshold: the smallest contrast detectable Fe−τ by the human observer. Although this = , C + ( − ) −τ depends on the particular observer, the 1 F 1 e angular size of the object observed, R cos F = .(20) the presence of nearby objects, and the πp() absolute luminance, a contrast threshold of 0.02 is often taken as an average. This All else being equal, therefore, contrast value in Eq. (18) gives decreases as p( ) increases. As shown in Fig. 9, p() is more sharply peaked in the − ln |C|=3.9 =τ=βd.(19) forward direction the larger the scatterer. Thus, we expect the details of a distant To convert an optical distance into a scene to be less distinct when looking physical distance requires the scattering toward the sun than away from it if the coefficient. Because K is peaked at around optical thickness of the line of sight has 550 nm, we can obtain an approximate an appreciable component contributed by value of d from the scattering coefficient particles comparable to or larger than the at this wavelength in Eq. (19). At sea wavelength. level, the molecular scattering coefficient On humid, hazy days, visibility is in the middle of the visible spectrum often depressingly poor. Haze, however, corresponds to about 330 km for ‘‘forever’’: is not water vapor but rather water the greatest distance at which a black that has ceased to be vapor. At high object can be seen against the horizon relative humidities, but still well below Atmospheric Optics 73
100%, small soluble particles in the gas and the scattering cross section σs of a atmosphere accrete liquid water to become gas molecule: solution droplets (haze). Although these = + 1 α ,() droplets are much smaller than cloud n 1 2 N 21 droplets, they markedly diminish visual 4 k 2 rangebecauseofthesharpincreasein σs = |α| ,(22) 6π scattering with particle size (Fig. 7). The same number of water molecules when where N is the number density (not mass aggregated in haze scatter vastly more than density) of gas molecules, k = 2π/λ is the when apart. wave number of the incident light, and α is the polarizability of a molecule (induced dipole moment per unit inducing electric 6 field). The appearance of the polarizabil- Atmospheric Refraction ity in Eq. (21) but its square in Eq. (22) is the clue that refraction is associated with 6.1 electric fields whereas lateral scattering Physical Origins of Refraction is associated with electric fields squared (powers). Scattering, without qualification, Atmospheric refraction is a consequence often means incoherent scattering in all of molecular scattering, which is rarely directions. Refraction, in a nutshell, is co- stated given the historical accident that herent scattering in a particular direction. before light and matter were well un- Readers whose appetites have been derstood refraction and scattering were whetted by the preceding brief discussion locked in separate compartments and sub- of the physical origins of refraction are sequently have been sequestered more directed to a beautiful paper by Doyle [9] rigidly than monks and nuns in neigh- in which he shows how the Fresnel boring cloisters. equations can be dissected to reveal the Consider a beam of light propagating in scattering origins of (specular) reflection an optically homogeneous medium. Light and refraction. is scattered (weakly but observably) later- ally to this beam as well as in the direction 6.2 of the beam (the forward direction). The Terrestrial Mirages observed beam is a coherent superposi- tion of incident light and forward-scattered Mirages are not illusions, any more so light, which was excited by the incident than are reflections in a pond. Reflections light. Although refractive indices are of- of plants growing at its edge are not ten defined by ratios of phase velocities, interpreted as plants growing into the we may also look upon a refractive index water. If the water is ruffled by wind, as a parameter that specifies the phase the reflected images may be so distorted shift between an incident beam and the that they are no longer recognizable forward-scattered beam that the incident as those of plants. Yet we still would beam excites. The connection between not call such distorted images illusions. (incoherent) scattering and refraction (co- And so is it with mirages. They are herent scattering) can be divined from the images noticeably different from what they expressions for the refractive index n of a would be in the absence of atmospheric 74 Atmospheric Optics
refraction, creations of the atmosphere, shallow surface layers, in which the pres- not of the mind. sure is nearly constant, the temperature Mirages are vastly more common than gradient determines the refractive index is realized. Look and you shall see them. gradient. It is in such shallow layers that Contrary to popular opinion, they are mirages, which are caused by refractive- not unique to deserts. Mirages can be index gradients, are seen. seen frequently even over ice-covered Cartoonists by their fertile imaginations landscapes and highways flanked by deep unfettered by science, and textbook writers snowbanks. Temperature per se is not by their carelessness, have engendered what gives mirages but rather temperature the notion that atmospheric refraction can gradients. work wonders, lifting images of ships, for Because air is a mixture of gases, the example, from the sea high into the sky. polarizability for air in Eq. (21) is an A back-of-the-envelope calculation dispels average over all its molecular constituents, such notions. The refractive index of air at although their individual polarizabilities sea level is about 1.0003 (Fig. 13). Light are about the same (at visible wavelengths). from empty space incident at glancing The vertical refractive index gradient can incidence onto a uniform slab with this be written so as to show its dependence on refractive index is displaced in angular pressure p and (absolute) temperature T: position from where it would have been intheabsenceofrefractionby d 1 dp 1 dT ln(n − 1) = − .(23) δ = ( − ). ( ) dz p dz T dz 2 n 1 24 This yields an angular displacement of Pressure decreases approximately ex- ◦ about 1.4 , which as we shall see is a rough ponentially with height, where the scale upper limit. height is around 8 km. Thus, the first term Trajectories of light rays in nonuniform on the right-hand side of Eq. (23) is around media can be expressed in different ways. 0.1 km−1. Temperature usually decreases According to Fermat’s principle of least with height in the atmosphere. An average lapse rate of temperature (i.e., its decrease ◦ with height) is around 6 C/km. The aver- age temperature in the troposphere (within about 15 km of the surface) is around 280 K. Thus, the magnitude of the second term in Eq. (23) is around 0.02 km−1.On average, therefore, the refractive-index gra- dient is dominated by the vertical pressure gradient. But within a few meters of the surface, conditions are far from average. On a sun-baked highway your feet may ◦ be touching asphalt at 50 Cwhileyour ◦ nose is breathing air at 35 C, which cor- Fig. 13 Sea-level refractive index versus ◦ ◦ responds to a lapse rate a thousand times wavelength at −15 C(dashes)and15 C(solid the average. Moreover, near the surface, curve). Data from Penndorf, R. (1957), J. Opt. temperature can increase with height. In Soc. Am. 47, 176–182[2] Atmospheric Optics 75 time (which ought to be extreme time), the actual path taken by a ray between two points is such that the path integral 2 n ds (25) 1 is an extremum over all possible paths. This principle has inspired piffle about the alleged efficiency of nature, which directs light over routes that minimize travel time, presumably freeing it to tend to important business at its destination. Fig. 14 Parabolic ray paths in an atmosphere The scale of mirages is such that in with a constant refractive-index gradient (inferior analyzing them we may pretend that the mirage). Note the vastly different horizontal and vertical scales Earthisflat.Onsuchanearth,with an atmosphere in which the refractive index varies only in the vertical, Fermat’s familiar highway mirage, seen over high- principle yields a generalization ways warmer than the air above them. The downward angular displacement is n sin θ = constant (26) 1 dn δ = s ,(28) of Snel’s law, where θ is the angle between 2 dz the ray and the vertical direction. We could, of course, have bypassed Fermat’s where s is the horizontal distance between principle to obtain this result. object and observer (image). Even for If we restrict ourselves to nearly hori- a temperature gradient 1000 times the zontal rays, Eq. (26) yields the following tropospheric average, displacements of differential equation satisfied by a ray: mirages are less than a degree at distances of a few kilometers. d2z dn If temperature increases with height, = ,(27) dy2 dz as it does, for example, in air over a cold sea, the resulting mirage is called where y and z are its horizontal and vertical a superior mirage. Inferior and superior are coordinates, respectively. For a constant not designations of lower and higher caste refractive-index gradient, which to good but rather of displacements downward approximation occurs for a constant tem- and upward. perature gradient, Eq. (27) yields parabolas For a constant temperature gradient, for ray trajectories. One such parabola for one and only one parabolic ray tra- a constant temperature gradient about 100 jectory connects an object point to an times the average is shown in Fig. 14. image point. Multiple images therefore Note the vastly different horizontal and are not possible. But temperature gra- vertical scales. The image is displaced dients close to the ground are rarely downward from what it would be in the linear. The upward transport of energy absence of atmospheric refraction; hence from a hot surface occurs by molecular the designation inferior mirage. This is the conduction through a stagnant boundary 76 Atmospheric Optics
layer of air. Somewhat above the surface, originating from outside it had to have however, energy is transported by air in been incident on it from an angle δ below motion. As a consequence, the tempera- the horizon: ture gradient steepens toward the ground 2H 2H if the energy flux is constant. This vari- δ = − − 2(n − 1), (29) able gradient can lead to two observable R R consequences: magnification and multi- where R is the radius of the Earth. Thus, ple images. when the sun (or moon) is seen to be on According to Eq. (28), all image points the horizon it is actually more than halfway ◦ at a given horizontal distance are dis- below it, δ being about 0.36 , whereas the placed downward by an amount propor- angular width of the sun (or moon) is ◦ tional to the (constant) refractive index about 0.5 . gradient. A corollary is that the closer Extraterrestrial bodies seen near the an object point is to a surface, where horizon also are vertically compressed. The the temperature gradient is greatest, the simplest way to estimate the amount of greater the downward displacement of the compression is from the rate of change of corresponding image point. Thus, non- angle of refraction θr with angle of inci- linear vertical temperature profiles may dence θi for a uniform slab magnify images. dθ cos θ Multiple images are seen frequently r = i ,(30) θ 2 2 on highways. What often appears to d i n − sin θi be water on the highway ahead but evaporates before it is reached is the wheretheangleofincidenceisthatfor inverted secondary image of either the a curved but uniform atmosphere such horizon sky or horizon objects lighter than that the refracted ray is horizontal. The dark asphalt. result is dθ R r = 1 − (n − 1), (31) 6.3 dθ H Extraterrestrial Mirages i according to which the sun near the When we turn from mirages of terrestrial horizon is distorted into an ellipse with objects to those of extraterrestrial bodies, aspect ratio about 0.87. We are unlikely most notably the sun and moon, we to notice this distortion, however, be- can no longer pretend that the Earth causeweexpectthesunandmoonto is flat. But we can pretend that the be circular, and hence we see them atmosphere is uniform and bounded. that way. The total phase shift of a vertical ray The previous conclusions about the from the surface to infinity is the same downward displacement and distortion of in an atmosphere with an exponentially the sun were based on a refractive-index decreasing molecular number density as in profile determined mostly by the verti- a hypothetical atmosphere with a uniform cal pressure gradient. Near the ground, number density equal to the surface value however, the temperature gradient is the up to height H. prime determinant of the refractive-index A ray refracted along a horizon path gradient, as a consequence of which the by this hypothetical atmosphere and sun on the horizon can take on shapes Atmospheric Optics 77
6.4 The Green Flash
Compared to the rainbow, the green flash is not a rare phenomenon. Before you dismiss this assertion as the ravings of a lunatic, consider that rainbows require raindrops as well as sunlight to illuminate them, whereas rainclouds often completely obscure the sun. Moreover, ◦ the sun must be below about 42 .Asa consequence of these conditions, rainbows are not seen often, but often enough that they are taken as the paragon of color variation. Yet tinges of green on the upper Fig. 15 A nearly triangular sun on the horizon. rim of the sun can be seen every day The serrations are a consequence of horizontal at sunrise and sunset given a sufficiently variations in refractive index low horizon and a cloudless sky. Thus, the conditions for seeing a green flash more striking than a mere ellipse. For are more easily met than those for seeing example, Fig. 15 shows a nearly triangu- a rainbow. Why then is the green flash lar sun with serrated edges. Assigning considered to be so rare? The distinction a cause to these serrations provides a here is that between a rarely observed lesson in the perils of jumping to con- phenomenon (the green flash) and a rarely clusions. Obviously, the serrations are the observable one (the rainbow). result of sharp changes in the temper- Thesunmaybeconsideredtobea ature gradient – or so one might think. collection of disks, one for each visible Setting aside how such changes could be wavelength. When the sun is overhead, produced and maintained in a real at- all the disks coincide and we see the mosphere, a theorem of Fraser [10] gives sun as white. But as it descends in the pause for thought. According to this the- sky, atmospheric refraction displaces the orem, ‘‘In a horizontally (spherically) ho- disksbyslightlydifferentamounts,thered mogeneous atmosphere it is impossible less than the violet (see Fig. 13). Most of for more than one image of an extrater- each disk overlaps all the others except restrial object (sun) to be seen above the for the disks at the extremes of the visible astronomical horizon.’’ The serrations on spectrum. As a consequence, the upper the sun in Fig. 15 are multiple images. rim of the sun is violet or blue, its lower But if the refractive index varies only rim red, whereas its interior, the region in vertically (i.e., along a radius), no mat- which all disks overlap, is still white. ter how sharply, multiple images are not This is what would happen in the ab- possible. Thus, the serrations must owe sence of lateral scattering of sunlight. But their existence to horizontal variations of refraction and lateral scattering go hand in the refractive index, a consequence of hand,eveninanatmospherefreeofpar- gravity waves propagating along a tem- ticles. Selective scattering by atmospheric perature inversion. molecules and particles causes the color 78 Atmospheric Optics
of the sun to change. In particular, the theories. For example, coronas are said to violet-bluish upper rim of the low sun can be caused by diffraction and rainbows by be transformed to green. refraction. Yet both the corona and the According to Eq. (29) and Fig. 13, the rainbow can be described quantitatively to angular width of the green upper rim of high accuracy with a theory (the Mie the- ◦ the low sun is about 0.01 , too narrow to ory for scattering by a sphere) in which be resolved with the naked eye or even to diffraction and refraction do not explicitly be seen against its bright backdrop. But appear. No fundamentally impenetrable depending on the temperature profile, the barrier separates scattering from (specu- atmosphere itself can magnify the upper lar) reflection, refraction, and diffraction. rim and yield a second image of it, thereby Becausethesetermscameintogeneral enabling it to be seen without the aid of a use and were entombed in textbooks be- telescope or binoculars. Green rims, which fore the nature of light and matter was well require artificial magnification, can be understood, we are stuck with them. But seen more frequently than green flashes, if we insist that diffraction, for example, is which require natural magnification. Yet somehow different from scattering, we do both can be seen often by those who know so at the expense of shattering the unity what to look for and are willing to look. of the seemingly disparate observable phe- nomena that result when light interacts with matter. What is observed depends 7 on the composition and disposition of the Scattering by Single Water Droplets matter, not on which approximate theory in a hierarchy is used for quantitative de- All the colored atmospheric displays that scription. result when water droplets (or ice crystals) Atmospheric optical phenomena are are illuminated by sunlight have the same best classified by the direction in which underlying cause: light is scattered in they are seen and by the agents respon- different amounts in different directions sible for them. Accordingly, the following by particles larger than the wavelength, sections are arranged in order of scattering and the directions in which scattering is direction, from forward to backward. greatest depends on wavelength. Thus, When a single water droplet is illumi- when particles are illuminated by white nated by white light and the scattered light, the result can be angular separation light projected onto a screen, the result of colors even if scattering integrated over is a set of colored rings. But in the atmo- all directions is independent of wavelength sphere we see a mosaic to which individual (as it essentially is for cloud droplets and droplets contribute. The scattering pattern ice crystals). This description, although of a single droplet is the same as the correct, is too general to be completely mosaic provided that multiple scattering satisfying. We need something more is negligible. specific, more quantitative, which requires theories of scattering. 7.1 Because superficially different theories Coronas and Iridescent Clouds have been used to describe different op- tical phenomena, the notion has become A cloud of droplets narrowly distributed in widespread that they are caused by these size and thinly veiling the sun (or moon) Atmospheric Optics 79 can yield a spectacular series of colored concentric rings around it. This corona is most easily described quantitatively by the Fraunhofer diffraction theory, a sim- ple approximation valid for particles large compared with the wavelength and for scattering angles near the forward direc- tion. According to this approximation, the differential scattering cross section (cross section for scattering into a unit solid angle) of a spherical droplet of radius a illuminated by light of wave number k is Fig. 16 Scattering of light near the forward |S|2 direction (according to Fraunhofer theory) by a ,() µ 2 32 sphere of diameter 10 m illuminated by red and k green light where the scattering amplitude is 1 + cos θ J (x sin θ) S = x2 1 .(33) droplet size, gives sufficient dispersion to 2 x sin θ yield colored coronas. Suppose that the first angular maxi- The term J1 is the Bessel function of µ first order and the size parameter x = mum for blue light (0.47 m) occurs for a µ ka. The quantity (1 + cos θ)/2 is usually droplet of radius a. For red light (0.66 m) approximated by 1 since only near-forward a maximum is obtained at the same an- + scattering angles θ are of interest. gle for a droplet of radius a a.That The differential scattering cross section, is, the two maxima, one for each wave- which determines the angular distribution length, coincide. From this we conclude of the scattered light, has maxima for that coronas require narrow size distri- x sin θ = 5.137, 8.417, 11.62, ... Thus, butions: if cloud droplets are distributed the dispersion in the position of the first in radius with a relative variance a/a maximum is greater than about 0.4, color separation is not possible. dθ 0.817 ≈ ( ) Because of the stringent requirements λ 34 d a for the occurrence of coronas, they are and is greater for higher-order maxima. not observed often. Of greater occur- This dispersion determines the upper limit rence are the corona’s cousins, iridescent on drop size such that a corona can be clouds, which display colors but usually observed. For the total angular dispersion not arranged in any obviously regular ge- over the visible spectrum to be greater ometrical pattern. Iridescent patches in ◦ than the angular width of the sun (0.5 ), clouds can be seen even at the edges of the droplets cannot be larger than about thick clouds that occult the sun. 60 µmindiameter.Dropsinrain,even Coronas are not the unique signatures of in drizzle, are appreciably larger than spherical scatterers. Randomly oriented ice this, which is why coronas are not seen columns and plates give similar patterns through rainshafts. Scattering by a droplet according to Fraunhofer theory [11]. As a of diameter 10 µm (Fig. 16), a typical cloud practical matter, however, most coronas 80 Atmospheric Optics
probably are caused by droplets. Many of the differential scattering cross section clouds at temperatures well below freezing at which the conditions contain subcooled water droplets. Only dθ b if a corona were seen in a cloud at a = 0, = 0 (36) ◦ db sin θ temperature lower than −40 C could one assert with confidence that it must be an are satisfied. Missing from Eq. (35) are ice-crystal corona. various reflection and transmission coeffi- cients (Fresnel coefficients), which display 7.2 no singularities and hence do not deter- Rainbows mine rainbow angles. A rainbow is not associated with rays In contrast with coronas, which are seen externally reflected or transmitted without looking toward the sun, rainbows are internal reflection. The succession of seen looking away from it, and are rainbow angles associated with one, two, caused by water drops much larger than three ... internal reflections are called those that give coronas. To treat the primary, secondary, tertiary ... rainbows. rainbow quantitatively we may pretend Aristotle recognized that ‘‘Three or more that light incident on a transparent sphere rainbows are never seen, because even is composed of individual rays, each of the second is dimmer than the first, and which suffers a different fate determined so the third reflection is altogether too only by the laws of specular reflection and feeble to reach the sun (Aristotle’s view refraction. Theoretical justification for this was that light streams outward from the is provided by van de Hulst’s ([12], p. 208) eye)’’. Although he intuitively grasped that localization principle, according to which each successive ray is associated with terms in the exact solution for scattering by ever-diminishing energy, his statement a transparent sphere correspond to more about the nonexistence of tertiary rainbows or less localized rays. in nature is not quite true. Although Each incident ray splinters into an infi- reliable reports of such rainbows are rare nite number of scattered rays: externally (unreliable reports are as common as dirt), reflected, transmitted without internal re- at least one observer who can be believed flection, transmitted after one, two, and so has seen one [13]. on internal reflections. At any scattering An incident ray undergoes a total angle θ, each splinter contributes to the angular deviation as a consequence of scattered light. Accordingly, the differen- transmission into the drop, one or more tial scattering cross section is an infinite internal reflections, and transmission out series with terms of the form of the drop. Rainbow angles are angles of (θ) minimum deviation. b db .() θ θ 35 Forarainbowofanyordertoexist, sin d 2 The impact parameter b is a sin i,where n − 1 cos i = (37) i is the angle between an incident p(p + 1) ray and the normal to the sphere. Each term in the series corresponds to one must lie between 0 and 1, where i is of the splinters of an incident ray. A the angle of incidence of a ray that gives rainbow angle is a singularity (or caustic) a rainbow after p internal reflections and Atmospheric Optics 81 n is the refractive index of the drop. A to the pencil-and-paper variety, are nec- primary bow therefore requires drops with essarily observed in an atmosphere, the refractive index less than 2; a secondary molecules and particles of which scat- bow requires drops with refractive index ter sunlight that adds to the light from less than 3. If raindrops were composed the rainbow but subtracts from its purity of titanium dioxide (n ≈ 3), a commonly of color. used opacifier for paints, primary rainbows Although geometrical optics yields the would be absent from the sky and we positions, widths, and color separation wouldhavetobecontentwithonly of rainbows, it yields little else. For secondary bows. example, geometrical optics is blind to If we take the refractive index of water to supernumerary bows, a series of narrow be 1.33, the scattering angle for the primary bands sometimes seen below the primary ◦ rainbow is about 138 .Thisismeasured bow. These bows are a consequence from the forward direction (solar point). of interference, and hence fall outside Measured from the antisolar point (the the province of geometrical optics. Since direction toward which one must look supernumerary bows are an interference in order to see rainbows in nature), this phenomenon, they, unlike primary and ◦ scattering angle corresponds to 42 ,the secondary bows (according to geometrical basis for a previous assertion that rainbows optics), depend on drop size. This poses (strictly, primary rainbows) cannot be the question of how supernumerary bows ◦ seen when the sun is above 42 .The can be seen in rain showers, the drops ◦ secondary rainbow is seen at about 51 in which are widely distributed in size. In from the antisolar point. Between these a nice piece of detective work, Fraser [16] two rainbows is Alexander’s dark band,a answered this question. region into which no light is scattered Raindrops falling in a vacuum are spher- according to geometrical optics. ical. Those falling in air are distorted by The colors of rainbows are a conse- aerodynamic forces, not, despite the de- quence of sufficient dispersion of the pictions of countless artists, into teardrops refractive index over the visible spectrum but rather into nearly oblate spheroids with to give a spread of rainbow angles that their axes more or less vertical. Fraser ar- appreciably exceeds the width of the sun. gued that supernumerary bows are caused The width of the primary bow from violet by drops with a diameter of about 0.5 mm, ◦ to red is about 1.7 ; that of the secondary at which diameter the angular position ◦ bow is about 3.1 . of the first (and second) supernumerary Because of its band of colors arcing bow has a minimum: interference causes across the sky, the rainbow has become the position of the supernumerary bow the paragon of color, the standard against to increase with decreasing size whereas which all other colors are compared. Lee drop distortion causes it to increase with and Fraser [14, 15], however, challenged increasing size. Supernumerary patterns this status of the rainbow, pointing out contributed by drops on either side of the that even the most vivid rainbows are minimum cancel, leaving only the contri- colorimetrically far from pure. bution from drops at the minimum. This Rainbows are almost invariably dis- cancellation occurs only near the tops of cussed as if they occurred literally in a rainbows, where supernumerary bows are vacuum. But real rainbows, as opposed seen. In the vertical parts of a rainbow, a 82 Atmospheric Optics
horizontal slice through a distorted drop at the end of our journey: the glory.Because is more or less circular, and hence these it is most easily seen from airplanes it drops do not exhibit a minimum supernu- sometimes is called the pilot’s bow.Another merary angle. name is anticorona, which signals that it According to geometrical optics, all is a corona around the antisolar point. Al- spherical drops, regardless of size, yield though glories and coronas share some the same rainbow. But it is not necessary common characteristics, there are differ- for a drop to be spherical for it to yield ences between them other than direction rainbows independent of its size. This of observation. Unlike coronas, which may merely requires that the plane defined by be caused by nonspherical ice crystals, glo- the incident and scattered rays intersect ries require spherical cloud droplets. And the drop in a circle. Even distorted a greater number of colored rings may be drops satisfy this condition in the vertical seen in glories than in coronas because part of a bow. As a consequence, the the decrease in luminance away from the absence of supernumerary bows there is backward direction is not as steep as that compensated for by more vivid colors away from the forward direction. To see of the primary and secondary bows [17]. a glory from an airplane, look for colored Smaller drops are more likely to be rings around its shadow cast on clouds be- spherical, but the smaller a drop, the low. This shadow is not an essential part less light it scatters. Thus, the dominant of the glory, it merely directs you to the contribution to the luminance of rainbows antisolar point. is from the larger drops. At the top of a Like the rainbow, the glory may be bow, the plane defined by the incident and looked upon as a singularity in the dif- scattered rays intersects the large, distorted ferential scattering cross section Eq. (35). drops in an ellipse, yielding a range of Equation (36) gives one set of conditions rainbow angles varying with the amount of for a singularity; the second set is distortion, and hence a pastel rainbow. To θ = , (θ) = .() the knowledgeable observer, rainbows are sin 0 b 0 38 no more uniform in color and brightness That is, the differential scattering cross than is the sky. section is infinite for nonzero impact Although geometrical optics predicts parameters (corresponding to incident that all rainbows are equal (neglecting rays that do not intersect the center of the ◦ background light), real rainbows do not sphere) that give forward (0 ) or backward ◦ slavishly follow the dictates of this approx- (180 ) scattering. The forward direction imate theory. Rainbows in nature range is excluded because this is the direction from nearly colorless fog bows (or cloud of intense scattering accounted for by the bows) to the vividly colorful vertical por- Fraunhofer theory. tions of rainbows likely to have inspired For one internal reflection, Eq. (38) leads myths about pots of gold. to the condition n 2 7.3 sin i = 4 − n ,(39) The Glory 2 which is satisfied only for refractive indices Continuing our sweep of scattering direc- between 1.414 and 2, the lower refractive tions, from forward to backward, we arrive index corresponding to a grazing-incidence Atmospheric Optics 83 ray. The refractive index of water lies 8 outside this range. Although a condition Scattering by Single Ice Crystals similar to Eq. (39) is satisfied for rays un- dergoing four or more internal reflections, Scattering by spherical water drops in the insufficient energy is associated with such atmosphere gives rise to three distinct dis- rays. Thus, it seems that we have reached plays in the sky: coronas, rainbows, and an impasse: the theoretical condition for glories. Ice particles (crystals) also can in- a glory cannot be met by water droplets. habit the atmosphere, and they introduce Not so, says van de Hulst [18] in a sem- two new variables in addition to size: shape inal paper. He argues that 1.414 is close and orientation, the second a consequence enough to 1.33 given that geometrical op- of the first. Given this increase in the tics is, after all, an approximation. Cloud number of degrees of freedom, it is hardly droplets are large compared with the wave- cause for wonder that ice crystals are the length, but not so large that geometrical source of a greater variety of displays than optics is an infallible guide to their optical are water drops. As with rainbows, the behavior. Support for the van de Hulstian gross features of ice-crystal phenomena interpretation of glories was provided by can be described simply with geometrical Bryant and Cox [19], who showed that the optics, various phenomena arising from dominant contribution to the glory is from the various fates of rays incident on crys- the last terms in the exact series for scat- tals. Colorless displays (e.g., sun pillars) tering by a sphere. Each successive term are generally associated with reflected rays, in this series is associated with ever larger colored displays (e.g., sun dogs and halos) impact parameters. Thus, the terms that with refracted rays. Because of the wealth give the glory are indeed those correspond- of ice-crystal displays, it is not possible to ing to grazing rays. Further unraveling of treat all of them here, but one example the glory and vindication of van de Hulst’s should point the way toward understand- conjectures about the glory were provided ing many of them. by Nussenzveig [20]. It sometimes is asserted that geometrical 8.1 optics is incapable of treating the glory. Sun Dogs and Halos Yet the same can be said for the rainbow. Geometrical optics explains rainbows only Because of the hexagonal crystalline struc- in the sense that it predicts singularities for ture of ice it can form as hexagonal plates scattering in certain directions (rainbow in the atmosphere. The stable position of angles). But it can predict only the angles a plate falling in air is with the normal of intense scattering, not the amount. toitsfacemoreorlessvertical,whichis Indeed, the error is infinite. Geometrical easy to demonstrate with an ordinary busi- optics also predicts a singularity in the ness card. When the card is dropped with backward direction. Again, this simple its edge facing downward (the supposedly theory is powerless to predict more. aerodynamic position that many people Results from geometrical optics for both instinctively choose), the card somersaults rainbows and glories are not the end in a helter-skelter path to the ground. But but rather the beginning, an invitation when the card is dropped with its face par- to take a closer look with more powerful allel to the ground, it rocks back and forth magnifying glasses. gently in descent. 84 Atmospheric Optics
A hexagonal ice plate falling through air and illuminated by a low sun is ◦ like a 60 prism illuminated normally to its sides (Fig. 17). Because there is no mechanism for orienting a plate within the horizontal plane, all plate orientations in this plane are equally probable. Stated another way, all angles of incidence for a fixed plate are equally probable. Yet all scattering angles (deviation angles) of rays refracted into and out of the plate are not equally probable. Figure 18 shows the range of scattering Fig. 18 Scattering by a hexagonal ice plate (see angles corresponding to a range of rays Fig. 17) in various orientations (angles of ◦ incident on a 60 ice prism that is part of a incidence). The solid curve is for red light, the dashed for blue light hexagonal plate. For angles of incidence ◦ less than about 13 , the transmitted (θ) θ ray is totally internally reflected in the P of scattering angles by prism. For angles of incidence greater p(θ ) ◦ (θ) = i .() than about 70 , the transmittance plunges. P 40 dθ/dθi Thus, the only rays of consequence are ◦ those incident between about 13 and At the incidence angle for which ◦ θ/ θ = (θ) 70 . d d i 0, P is infinite and scat- All scattering angles are not equally tered rays are intensely concentrated probable. The (uniform) probability near the corresponding angle of mini- mum deviation. distribution p(θi) of incidence angles θi is related to the probability distribution The physical manifestation of this singu- larity (or caustic) at the angle of minimum ◦ deviation for a 60 hexagonal ice plate is ◦ abrightspotabout22 from either or both sides of a sun low in the sky. These bright spots are called sun dogs (because they accompany the sun) or parhelia or mock suns. The angle of minimum deviation θm, hence the angular position of sun dogs, ◦ depends on the prism angle (60 for the plates considered) and refrac- tive index:
−1 Fig. 17 Scattering by a hexagonal ice plate θm = 2sin n sin − . (41) illuminated by light parallel to its basal plane. 2 The particular scattering angle θ shown is an angle of minimum deviation. The scattered light Because ice is dispersive, the separation between the angles of minimum deviation is that associated with two refractions by ◦ the plate for red and blue light is about 0.7 (Fig. 18), Atmospheric Optics 85 somewhat greater than the angular width to be randomly oriented by Brownian of the sun. As a consequence, sun dogs motion, they are too small to yield sharp may be tinged with color, most noticeably scattering patterns. toward the sun. Because the refractive But completely randomly oriented plates index of ice is least at the red end of the are not necessary to give halos, especially spectrum, the red component of a sun dog ones of nonuniform brightness. Each part is closest to the sun. Moreover, light of any of a halo is contributed to by plates with two wavelengths has the same scattering a different tip angle (angle between the angle for different angles of incidence normal to the plate and the vertical). if one of the wavelengths does not The transition from oriented plates (zero correspond to red. Thus, red is the purest tip angle) to randomly oriented plates color seen in a sun dog. Away from its red occurs over a narrow range of sizes. In inner edge a sun dog fades into whiteness. the transition region, plates can be small With increasing solar elevation, sun enough to be partially oriented yet large dogs move away from the sun. A falling ice enough to give a distinct contribution to plate is roughly equivalent to a prism, the the halo. Moreover, the mapping between prism angle of which increases with solar tip angles and azimuthal angles on the elevation. From Eq. (41) it follows that the halo depends on solar elevation. When angle of minimum deviation, hence the the sun is near the horizon, plates can sun dog position, also increases. give a distinct halo over much of its At this point you may be wondering why azimuth. ◦ only the 60 prism portion of a hexagonal plate was singled out for attention. As evident from Fig. 17, a hexagonal plate ◦ couldbeconsideredtobemadeupof120 prisms. For a ray to be refracted twice, its angle of incidence at the second interface must be less than the critical angle. This imposes limitations on the prism angle. For a refractive index 1.31, all incident rays are totally internally reflected by prisms ◦ with angles greater than about 99.5 . A close relative of the sun dog is the ◦ ◦ 22 halo, a ring of light approximately 22 from the sun (Fig. 19). Lunar halos are also possible and are observed frequently (although less frequently than solar halos); even moon dogs are possible. Until Fraser [21] analyzed halos in detail, the conventional wisdom had been that they obviously were the result of randomly oriented crystals, yet another example of jumping to conclusions. By combining ◦ optics and aerodynamics, Fraser showed Fig. 19 A22 solar halo. The hand is not for that if ice crystals are small enough artistic effect but rather to occlude the bright sun 86 Atmospheric Optics
When the sun is high in the sky, days a year over a 16-year period, with hexagonal plates cannot give a sharp halo extremes of 29 and 152 halos a year. Al- ◦ but hexagonal columns – another possible though the 22 halo was by far the most form of atmospheric ice particles – can. frequently seen display, ice-crystal displays The stable position of a falling column is of all kinds were seen, on average, more with its long axis horizontal. When the often than once every four days at a loca- sun is directly overhead, such columns tion not especially blessed with clear skies. can give a uniform halo even if they all lie Although thin clouds are necessary for ice- in the horizontal plane. When the sun is crystal displays, clouds thick enough to not overhead but well above the horizon, obscure the sun are their bane. columns also can give halos. A corollary of Fraser’s analysis is that halos are caused by crystals with a range of 9 sizes between about 12 and 40 µm. Larger Clouds crystals are oriented; smaller particles are too small to yield distinct scatter- Although scattering by isolated particles ing patterns. can be studied in the laboratory, parti- More or less uniformly bright halos with cles in the atmosphere occur in crowds the sun neither high nor low in the sky (sometimes called clouds). Implicit in the could be caused by mixtures of hexagonal previous two sections is the assumption plates and columns or by clusters of bullets that each particle is illuminated solely (rosettes). Fraser opines that the latter is by incident sunlight; the particles do not more likely. illuminate each other to an appreciable de- One of the by-products of his analysis is gree. That is, clouds of water droplets or an understanding of the relative rarity of ice grains were assumed to be optically ◦ the 46 halo.Aswehaveseen,theangleof thin, and hence multiple scattering was minimum deviation depends on the prism negligible. Yet the term cloud evokes fluffy angle. Light can be incident on a hexagonal white objects in the sky, or perhaps an ◦ column such that the prism angle is 60 overcast sky on a gloomy day. For such ◦ for rays incident on its side or 90 for clouds, multiple scattering is not negligi- rays incident on its end. For n = 1.31, ble, it is the major determinant of their Eq. (41) yields a minimum deviation angle appearance. And the quantity that deter- ◦ ◦ ◦ of about 46 for = 90 .Yet,although46 mines the degree of multiple scattering is halos are possible, they are seen much less optical thickness (see Sec. 2.4). ◦ frequently than 22 halos. Plates cannot ◦ give distinct 46 halos although columns 9.1 can.Yettheymustbesolidandmost Cloud Optical Thickness columns have hollow ends. Moreover, the range of sun elevations is restricted. Despite their sometimes solid appearance, Like the green flash, ice-crystal phenom- clouds are so flimsy as to be almost ena are not intrinsically rare. Halos and nonexistent – except optically. The fraction sun dogs can be seen frequently – once of the total cloud volume occupied by you know what to look for. Neuberger [22] water substance (liquid or solid) is about reports that halos were observed in State 10−6 or less. Yet although the mass College, Pennsylvania, an average of 74 density of clouds is that of air to within Atmospheric Optics 87 a small fraction of a percent, their optical usually translucent, not transparent, yet thickness (per unit physical thickness) is not completely opaque. much greater. The number density of air The scattering coefficient of cloud molecules is vastly greater than that of droplets, in contrast with that of air water droplets in clouds, but scattering per molecules, is more or less independent molecule of a cloud droplet is also much of wavelength. This is often invoked as greater than scattering per air molecule the cause of the colorlessness of clouds. (see Fig. 7). Yet wavelength independence of scatter- Because a typical cloud droplet is much ing by a single particle is only sufficient, larger than the wavelengths of visible light, not necessary, for wavelength indepen- its scattering cross section is to good dence of scattering by a cloud of particles approximation proportional to the square (see Sec. 2.4). Any cloud that is optically of its diameter. As a consequence, the thick and composed of particles for which scattering coefficient [see Eq. (2)] of a cloud absorption is negligible is white upon having a volume fraction f of droplets is illumination by white light. Although ab- approximately sorption by water (liquid and solid) is not identically zero at visible wavelengths, and d2 selective absorption by water can lead to β = 3f ,(42) d3 observable consequences (e.g., colors of the sea and glaciers), the appearance of all where the brackets indicate an average but the thickest clouds is not determined over the distribution of droplet diameters by this selective absorption. d. Unlike molecules, cloud droplets are Equation (42) is the key to the vastly distributed in size. Although cloud parti- different optical characteristics of clouds cles can be ice particles as well as water and of the rain for which they are the droplets, none of the results in this and the progenitors. For a fixed amount of water following section hinge on the assumption (as specified by the quantity fh), optical of spherical particles. thickness is inversely proportional to mean The optical thickness along a cloud diameter. Rain drops are about 100 times path of physical thickness h is βh for larger on average than cloud droplets, and a cloud with uniform properties. The hence optical thicknesses of rain shafts are ratio d3/d2 defines a mean droplet correspondingly smaller. We often can see diameter, a typical value for which is through many kilometers of intense rain − 10 µm. For this diameter and f = 10 6,the whereas a small patch of fog on a well- optical thickness per unit meter of physical traveled highway can result in carnage. thicknessisaboutthesameasthenormal optical thickness of the atmosphere in the 9.2 middle of the visible spectrum (see Fig. 3). Givers and Takers of Light Thus, a cloud only 1 m thick is equivalent optically to the entire gaseous atmosphere. Scattering of visible light by a single A cloud with (normal) optical thickness water droplet is vastly greater in the ◦ about 10 (i.e., a physical thickness of about forward (θ<90 ) hemisphere than in the ◦ 100 m) is sufficient to obscure the disk of backward (θ>90 ) hemisphere (Fig. 9). the sun. But even the thickest cloud does But water droplets in a thick cloud not transform day into night. Clouds are illuminated by sunlight collectively scatter 88 Atmospheric Optics
much more in the backward hemisphere their surroundings. As so often happens, (reflected light) than in the forward more is not always better. Beyond a hemisphere (transmitted light). In each certain cloud optical thickness, the diffuse scattering event, incident photons are irradiance decreases. For a sufficiently deviated, on average, only slightly, but thick cloud, the sky overhead can be darker in many scattering events most photons than the clear sky. are deviated enough to escape from the Why are clouds bright? Why are they upper boundary of the cloud. Here is an dark? No inclusive one-line answers can be example in which the properties of an given to these questions. Better to ask, Why ensemble are different from those of its is that particular cloud bright? Why is that individual members. particular cloud dark? Each observation Clouds seen by passengers in an airplane must be treated individually; generaliza- can be dazzling, but if the airplane were tions are risky. Moreover, we must keep to descend through the cloud these same in mind the difference between bright- passengers might describe the cloudy sky ness and radiance when addressing the overhead as gloomy. Clouds are both givers queries of human observers. Brightness is and takers of light. This dual role is a sensation that is a property not only of exemplified in Fig. 20, which shows the the object observed but of its surround- calculated diffuse downward irradiance ings as well. If the luminance of an object below clouds of varying optical thickness. is appreciably greater than that of its sur- On an airless planet the sky would be black roundings, we call the object bright. If the in all directions (except directly toward luminance is appreciably less, we call the thesun).Butiftheskyweretobefilled object dark. But these are relative rather from horizon to horizon with a thin cloud, than absolute terms. the brightness overhead would markedly Two clouds, identical in all respects, increase. This can be observed in a partly including illumination, may still appear overcast sky, where gaps between clouds different because they are seen against (blue sky) often are noticeably darker than different backgrounds, a cloud against the horizon sky appearing darker than when seen against the zenith sky. Of two clouds under identical illumi- nation, the smaller (optically) will be less bright. If an even larger cloud were to ap- pear, the cloud that formerly had been de- scribed as white might be demoted to gray. With the sun below the horizon, two identical clouds at markedly different elevations might appear quite different in brightness, the lower cloud being shadowed from direct illumination by sunlight. A striking example of dark clouds can Fig. 20 Computed diffuse downward irradiance below a cloud relative to the incident solar sometimes be seen well after the sun irradiance as a function of cloud has set. Low-lying clouds that are not optical thickness illuminated by direct sunlight but are Atmospheric Optics 89 seen against the faint twilight sky may spectral response of the human ob- be relatively so dark as to seem like server. Also sometimes called photometric ink blotches. brightness. Because dark objects of our everyday .5.5 lives usually owe their darkness to absorp- tion, nonsense about dark clouds is rife: Mirage: An image appreciably different they are caused by pollution or soot. Yet of from what it would be in the absence of all the reasons that clouds are sometimes atmospheric refraction. seen to be dark or even black, absorption is not among them. Neutral Point: A direction in the sky for which the light is unpolarized.
Normal Optical Thickness: Optical thick- Glossary ness along a radial path from the surface of the earth to infinity. Airlight: Light resulting from scattering by all atmospheric molecules and particles Optical Thickness: The thickness of a scat- along a line of sight. tering medium measured in units of photon mean free paths. Optical thick- Antisolar Point: Direction opposite the nesses are dimensionless. sun.
Astronomical Horizon: Horizontal direc- Radiance: Radiant power crossing a unit tion determined by a bubble level. area and confined to a unit solid angle about a particular direction. Brightness: The attribute of sensation by which an observer is aware of differences Scale Height: The vertical distance over of luminance (definition recommended which a physical property of the at- by the 1922 Optical Society of America mosphere is reduced to 1/e of its Committee on Colorimetry). value.
Contrast Threshold: The minimum rela- Scattering Angle: Angle between incident tive luminance difference that can be and scattered waves. perceived by the human observer. Scattering Coefficient: The product of scat- Inferior Mirage: A mirage in which images tering cross section and number density of are displaced downward. scatterers.
Irradiance: Radiant power crossing unit Scattering Cross Section: Effective area of a area in a hemisphere of directions. scatterer for removal of light from a beam by scattering. Lapse Rate: Therateatwhichaphysical property of the atmosphere (usually tem- Scattering Plane: Plane determined by perature) decreases with height. incident and scattered waves.
Luminance: Radiance integrated over the Solar Point: The direction toward the visible spectrum and weighted by the sun. 90 Atmospheric Optics
Superior Mirage: A mirage in which im- [15] Lee, R. (1991), Appl. Opt. 30, 3401–3407. ∗ ages are displaced upward. [16] Fraser, A. B. (1983), J. Opt. Soc. Am. 73, 1626–1628. 1972 29 Tangential Optical Thickness: Optical [17] Fraser, A. B. ( ), J. Atmos. Sci. , 211, 212. thickness through the atmosphere along [18]∗ van de Hulst, H. C. (1947), J. Opt. Soc. a horizon path. Am. 37, 16–22. [19]∗ Bryant,H.C.,Cox,A.J.(1966), J. Opt. Soc. Am. 56, 1529–1532. References [20]∗ Nussenzveig, H. M. (1979), J. Opt. Soc. Am. 69, 1068–1079. [21]∗ Fraser, A. B. (1979), J. Opt. Soc. Am. 69, Many of the seminal papers in atmospheric 1112–1118. optics, including those by Lord Rayleigh, are [22] Neuberger, H. (1951), Introduction to Phys- bound together in Bohren, C. F. (Ed.) (1989), ical Meteorology. University Park, PA: Col- Selected Papers on Scattering in the Atmosphere, lege of Mineral Industries, Pennsylvania Bellingham, WA: SPIE Optical Engineering State University. Press. Papers marked with an asterisk are in this collection.
[1] Moller,¨ F. (1972), Radiation in the at- Further Reading mosphere, in D. P. McIntyre (Ed.), Me- teorological Challenges: A History. Ottawa: Minnaert, M. (1954), TheNatureofLightand Information Canada, pp. 43–71. Colour in the Open Air. New York: Dover [2]∗ Penndorf, R. (1957), J. Opt. Soc. Am. 47, Publications, is the bible for those interested 176–182. ∗ in atmospheric optics. Like accounts of natural [3] Young, A. T. (1982), Phys. Today 35(1), phenomena in the Bible, those in Minnaert’s 2–8. book are not always correct, despite which, [4] Einstein, A. (1910), Ann. Phys. (Leipzig) 33, 175; English translation in Alexan- again like the Bible, it has been and will der, J. (Ed.) (1926), Colloid Chemistry, continue to be a source of inspiration. Vol. I. New York: The Chemical Catalog A book in the spirit of Minnaert’s but with a Company, pp. 323–339. wealth of color plates is by Lynch, D. K., Liv- [5] Zimm, B. H. (1945), J. Chem. Phys. 13, ingston, W. (1995), Color and Light in Nature. 141–145. Cambridge, UK: Cambridge University Press. [6] Thekaekara, M. P., Drummond, A. J. A history of light scattering, From Leonardo to (1971), Nat. Phys. Sci. 229, 6–9. the Graser: Light Scattering in Historical Per- [7]∗ Hulburt, E. O. (1953), J. Opt. Soc. Am. 43, spective, was published serially by Hey, J. D. 1983 79 113–118. ( ), S. Afr. J. Sci. , 11–27, 310–324; 1985 81 [8] von Frisch, K. (1971), Bees: Their Vision, Hey, J. D. ( ), S. Afr. J. Sci. , 77–91, 1986 82 Chemical Senses, and Language, revised 601–613; Hey, J. D., ( ), S. Afr. J. Sci. , edition, Ithaca, NY: Cornell University 356–360. The history of the rainbow is re- Press, p. 116. counted by Boyer, C. B. (1987), The Rainbow. [9] Doyle, W. T. (1985), Am. J. Phys. 53, Princeton, NJ: Princeton University Press. 463–468. A unique, beautifully written and illustrated [10]∗ Fraser,A.B.(1975), Atmosphere 13, 1–10. treatise on rainbows in science and art, [11] Takano, Y., Asano, S. (1983), J. Meteor. both sacred and profane, is by Lee, R. L., Soc. Jpn. 61, 289–300. Fraser, A. B. (2001), The Rainbow Bridge, [12] van de Hulst, H. C. (1957), Light Scattering University Park, PA: Penn State University by Small Particles. New York: Wiley- Press. Interscience. Special issues of Journal of the Optical Society [13] Pledgley, E. (1986), Weather 41, 401. of America (August 1979 and December 1983) [14] Lee, R., Fraser, A. (1990), New Scientist and Applied Optics (20 August 1991 and 20 July 127(September), 40–42. 1994) are devoted to atmospheric optics. Atmospheric Optics 91
Several monographs on light scattering by a few relevant chapters. Two popular science particles are relevant to and contain exam- books on simple experiments in atmospheric ples drawn from atmospheric optics: van physics are heavily weighted toward atmo- de Hulst, H. C. (1957), Light Scattering by spheric optics: Bohren, C. F. (1987), Clouds in Small Particles. New York: Wiley-Interscience; a Glass of Beer. New York: Wiley; Bohren, C. F. reprint (1981), New York: Dover Publica- (1991), What Light Through Yonder Window tions; Deirmendjian, D. (1969), Electromag- Breaks? New York: Wiley. netic Scattering on Polydispersions.NewYork: For an expository article on colors of the sky Elsevier; Kerker, M. (1969), The Scattering see Bohren, C. F., Fraser, A. B. (1985), Phys. of Light and Other Electromagnetic Radiation. Teacher 23, 267–272. New York: Academic Press; Bohren, C. F., An elementary treatment of the coherence Huffman, D. R. (1983), Light Scattering by properties of light waves was given by For- Small Particles. New York: Wiley-Interscience; rester, A. T. (1956), Am.J.Phys.24, 192–196. Nussenzveig, H. M. (1992), Diffraction Effects This journal also published an expository arti- in Semiclassical Scattering.Cambridge,UK: cle on the observable consequences of multiple Cambridge University Press. scattering of light: Bohren, C. F. (1987), Am. J. The following books are devoted to a wide Phys. 55, 524–533. range of topics in atmospheric optics: Although a book devoted exclusively to atmo- Tricker, R. A. R. (1970), Introduction to Mete- spheric refraction has yet to be published, orological Optics. New York: Elsevier; McCart- an elementary yet thorough treatment of mi- ney, E. J. (1976), Optics of the Atmosphere.New rages was given by Fraser, A. B., Mach, W. H. York: Wiley; Greenler, R. (1980), Rainbows, (1976), Sci. Am. 234(1), 102–111. Halos, and Glories.Cambridge,UK:Cambridge Colorimetry, the often (and unjustly) neglected University Press. Monographs of more limited component of atmospheric optics, is treated scope are those by Middleton, W. E. K. (1952), in, for example, Optical Society of Amer- Vision Through the Atmosphere. Toronto: Uni- ica Committee on Colorimetry (1963), The versity of Toronto Press; O’Connell, D. J. K. Science of Color. Washington, DC: Optical (1958), The Green Flash and Other Low Society of America. Billmeyer, F. W., Saltz- Sun Phenomena. Amsterdam: North Holland; man, M. (1981), Principles of Color Technol- Rozenberg, G. V. (1966), Twilight: A Study in ogy, (2nd ed.), New York: Wiley-Interscience. Atmospheric Optics. New York: Plenum; Hen- MacAdam, D. L. (1985), Color Measurement, derson, S. T. (1977), Daylight and its Spectrum, (2nd ed.), Berlin: Springer. (2nd ed.), New York: Wiley; Tricker, R. A. R. Understanding atmospheric optical phenom- (1979), Ice Crystal Haloes. Washington, DC: ena is not possible without acquiring at Optical Society of America; Konnen,¨ G. P. least some knowledge of the properties (1985), Polarized Light in Nature.Cambridge, of the particles responsible for them. To UK: Cambridge University Press; Tape, W. this end, the following are recommended: (1994), Atmosphere Halos. Washington, DC: Pruppacher, H. R., Klett, J. D. (1980), Micro- American Geophysical Union. physics of Clouds and Precipitation.Dor- Although not devoted exclusively to atmospheric drecht, Holland: D. Reidel. Twomey, S. A. optics, Humphreys, W. J. (1964), Physics of the (1977), Atmospheric Aerosols.NewYork: Air. New York: Dover Publications, contains Elsevier. 937
Interferometry
Parameswaran Hariharan School of Physics, University of Sydney, Australia Phone: (612) 9413 7159; Fax: (612) 9413 7200; e-mail: hariharan [email protected]
Katherine Creath Optineering, Tucson, Arizona, USA Phone: (520) 882-2950; Fax: (520) 882-6976; e-mail: [email protected]
Abstract This article reviews the field of interferometry. It begins by outlining the fundamentals of two-beam and multiple-beam interference. The rest of the article discusses the applications of interferometry for measurement of length, optical testing, fringe analysis, interference microscopy, interferometric sensors, interference spectroscopy, nonlinear interferometers, interferometric imaging, space-time and gravitation, holographic interferometry, Moire´ techniques, and speckle interferometry. Each section provides a referenced (and cross-referenced) overview of the application area.
Keywords interferometry; interference; interferometers; optical testing; optical metrology; nondestructive testing.
1 Introduction 939 2 Interference and Coherence 940 2.1 Localization of Fringes 941 2.2 Coherence 941 3 Two-beam Interferometers 942 3.1 The Michelson Interferometer 942 3.2 The Mach–Zehnder Interferometer 943 3.3 The Sagnac Interferometer 943 938 Interferometry
4 Multiple-beam Interference 943 4.1 Fringes of Equal Chromatic Order 944 5 Measurement of Length 944 5.1 Electronic Fringe Counting 944 5.2 Heterodyne Interferometry 944 5.3 Two-wavelength Interferometry 945 5.4 Frequency-modulation Interferometry 946 5.5 Laser-feedback Interferometry 946 6 Optical Testing 946 6.1 Flat Surfaces 946 6.2 Homogeneity 947 6.3 Concave and Convex Surfaces 947 6.4 Prisms 947 6.5 Aspheric Surfaces 948 6.6 Optically Rough Surfaces 948 6.7 Shearing Interferometers 948 6.8 The Point-diffraction Interferometer 949 7 Fringe Analysis 949 7.1 Fringe Tracking and Fourier Analysis 949 7.2 Phase-shifting Interferometry 950 7.3 Determining Aberrations 951 8 Interference Microscopy 951 8.1 The Mirau Interferometer 951 8.2 The Nomarski Interferometer 951 8.3 White-light Interferometry 952 9 Interferometric Sensors 953 9.1 Laser–Doppler Interferometry 953 9.2 Fiber Interferometers 954 9.3 Rotation Sensing 955 10 Interference Spectroscopy 955 10.1 Etendue of an Interferometer 955 10.2 The Fabry–Perot Interferometer 955 10.3 Wavelength Measurements 957 10.4 Laser Linewidth 958 10.5 Fourier-transform Spectroscopy 958 11 Nonlinear Interferometers 959 11.1 Second-harmonic Interferometers 959 11.2 Phase-conjugate Interferometers 960 11.3 Measurement of Nonlinear Susceptibilities 961 12 Interferometric Imaging 961 12.1 The Intensity Interferometer 961 12.2 Heterodyne Stellar Interferometers 962 12.3 Stellar Speckle Interferometry 963 12.4 Telescope Arrays 963 Interferometry 939
13 Space-time and Gravitation 963 13.1 Gravitational Waves 963 13.2 LIGO 965 13.3 Limits to Measurement 965 14 Holographic Interferometry 965 14.1 Strain Analysis 965 14.2 Vibration Analysis 966 14.3 Contouring 966 15 Moire´ Techniques 967 15.1 Grating Interferometry 967 16 Speckle Interferometry 968 16.1 Electronic Speckle Pattern Interferometry (ESPI) 968 16.2 Phase-shifting Speckle Interferometry 968 16.3 Vibrating Objects 969 Glossary 969 References 970 Further Reading 973
1 The most important of these was the Introduction development of the laser, which made available, for the first time, an intense Optical interferometry uses the phe- source of light with a remarkably high nomenon of interference between light degree of spatial and temporal coher- waves to make extremely accurate mea- ence. Lasers have removed most of the surements. The interference pattern con- limitations imposed by conventional light tains, in addition to information on the sources and have made possible many new optical paths traversed by the waves, infor- techniques, including nonlinear interfer- mation on the spectral content of the light ometry. and its spatial distribution over the source. Another development that has revo- Young was the first to state the principle lutionized interferometry has been the of interference and demonstrate that the application of electronic techniques. The summation of two rays of light could give use of photoelectric detector arrays and rise to darkness, but the father of optical in- digital computers has made possible direct terferometry was undoubtedly Michelson. measurements of the optical path differ- Michelson’s contributions to interferom- ence at an array of points covering an etry, from 1880 to 1930, dominated the interference pattern, with very high accu- field to such an extent that optical inter- racy, in a very short time. ferometry was regarded for many years as Light scattered from a moving particle a closed chapter. However, the last four has its frequency shifted by an amount decades have seen an explosive growth of proportional to the component of its interest in interferometry due to several velocity in a direction determined by the new developments. directions of illumination and viewing. 940 Interferometry
Lasers have made it possible to measure 2 this frequency shift and, hence, the velocity Interference and Coherence of the particles, by detecting the beats produced by mixing the scattered light and When two light waves are superimposed, the original laser beam. the resultant intensity depends on whether Another major advance has been the they reinforce or cancel each other. use of single-mode optical fibers to This is the well-known phenomenon of build analogs of conventional two-beam interference (see WAVE OPTICS). interferometers. Since very long optical If, at any point, the complex ampli- paths can be accommodated in a small tudes of two light waves, derived from the space, fiber interferometers are now used same monochromatic point source and = widely as rotation sensors. In addition, polarized in the same plane, are A1 (− φ ) = (− φ ) since the length of the optical path in a1 exp i 1 and A2 a2 exp i 2 ,the −2 such a fiber changes with pressure or intensity (or the irradiance, units W m ) temperature, fiber interferometers have at this point is found many applications as sensors for =| + |2 a number of physical quantities. I A1 A2 1/2 In the field of stellar interferometry, it = I1 + I2 + 2(I1I2) cos(φ1 − φ2), is now possible to combine images from (1) widely spaced arrays of large telescopes where I1 and I2 are the intensities due to obtain extremely high resolution. In- to the two waves acting separately and terferometry is also being applied to the φ1 − φ2 is the difference in their phases. detection of gravitational waves from black The visibility V of the interference holes and supernovae. fringes is defined by the relation Holography (see HOLOGRAPHY)isacom- pletely new method of imaging based on I − I V = max min optical interference. Holographic interfer- Imax + Imin ometry has made it possible to map the 2(I I )1/2 displacements of a rough surface with an = 1 2 .() + 2 accuracy of a few nanometers, and even to I1 I2 make interferometric comparisons of two Interference effects can be observed stored wavefronts that existed at different quite easily by viewing a transparent plate times. Holographic interferometry and a illuminated by a point source of monochro- related technique, speckle interferometry matic light. In this case, interference takes (see SPECKLE AND SPECKLE METROLOGY), are place between the waves reflected from the now used widely in industry for nonde- front and back surfaces of the plate. structive testing and structural analysis. For a ray incident at an angle θ1 on a The applications outlined above provide plane-parallel plate (thickness d, refractive a glimpse of the many areas of optics that index n) and refracted within the plate at use interferometry. This article is meant an angle θ2,theopticalpathdifference to provide an overview. More detail is between the two reflected rays is available in the cross-referenced articles as well as in the lists of works cited and λ p = 2nd cos θ2 + ,(3) further reading. 2 Interferometry 941 since an additional phase shift of π is introduced by reflection at one of the ∆x surfaces. The interference fringes are circles centered on the normal to the plate (fringes of equal inclination, or Haidinger fringes). x With a collimated beam, the interfer- ence fringes are contours of equal optical thickness (Fizeau fringes). The variations in the phase difference observed can rep- Fig. 1 Evaluation of the errors of a polished flat resent variations in the thickness or the test surface by interference refractive index of the plate. A polished flat surface can be compared with a reference With a plane-parallel plate, the inter- flat surface, by placing them in contact ference fringes are localized at infinity. and observing the fringes of equal thick- With a wedged thin film, and near-normal ness formed in the air film between them. incidence, the interference fringes are lo- Introduction of a small tilt between the calized in the wedge. test and reference surfaces produces a set of almost straight and parallel fringes. Any 2.2 deviations of the test surface from a plane Coherence are seen as a departure of the fringes from straight lines. The errors of the test sur- A more detailed analysis [1] shows that the interference effects observed depend face can then be evaluated, as shown in on the degree of correlation between the Fig. 1, by measuring the maximum devia- wavefieldsatthepointofobservation. tion (x) of a fringe from a straight line The intensity in the interference pattern is as well as the spacing between successive given by the relation fringes (x). Each fringe corresponds to a 1/2 change in the optical path difference of I = I1 + I2 + 2(I1I2) |γ | half a wavelength. × cos(arg γ + 2πντ), (4) 2.1 where I1 and I2 are the intensities of Localization of Fringes the two beams, ν is the frequency of the radiation, τ is the mean time delay between An extended monochromatic source can the arrival of the two beams and γ is the be considered as an array of independent (complex) degree of coherence between the point sources. Since the light waves from wave fields. these sources take different paths to the With two beams of equal intensity, the point where interference is observed, the visibility of the interference fringes is equal elementary interference patterns produced to |γ |, with a maximum value of 1 when by any two of them will not, in general, the correlation between the wave fields coincide. Interference fringes are then is complete. observed with maximum contrast only The correlation between the fields at any in a particular region (the region of two points, when the difference between localization). the optical paths to the source is small 942 Interferometry
enough for effects due to the spectral Young’s experiment and in the Rayleigh bandwidth of the light to be neglected, interferometer. is a measure of the spatial coherence of More commonly, two beams are derived thelight.Ifthesizeofthesourceand from the same portion of the primary wave- the separation of the two points are very front (amplitude division) using a beam small compared to their distance from splitter (a transparent plate coated with the source, it can be shown that the a partially reflecting film), a diffraction complex degree of coherence is given by grating or a polarizing prism. the normalized two-dimensional Fourier transform of the intensity distribution 3.1 over the source (see FOURIER AND OTHER The Michelson Interferometer TRANSFORM METHODS). Similarly, the correlation between the In the Michelson interferometer, a single fieldsatthesamepointatdifferent beam splitter is used, as shown in Fig. 2, times is a measure of the temporal to divide and recombine the beams. coherence of the light and is related to its However, to obtain interference fringes with white light, the two optical paths spectral bandwidth. With a point source must contain the same thickness of glass. (or when interference takes place between Accordingly, a compensating plate (of the corresponding elements of the original same thickness and the same material wavefront), the visibility of the fringes as the beam splitter) is introduced in as a function of the delay is the Fourier one beam. transform of the source spectrum. The interference pattern observed is To make this analysis complete, we must similar to that produced in a plate (n = 1) also take into account the polarization ef- bounded by one mirror and the image of fects. In general, for maximum visibility, the other mirror produced by reflection the beams must start in the same state from the beam splitter. With an extended of polarization (see POLARIZED LIGHT, BA- source, the interference fringes are circles SIC CONCEPTS OF) and interfere in the same localized at infinity (fringes of equal state of polarization. For natural (unpolar- ized) light, the optical path difference must be the same for all polarizations. The ef- fects of deviations from these conditions, which can be quite complex, have been discussed by [2].
3 Two-beam Interferometers BS Two methods are used to obtain two beams from a common source. In wavefront division, two beams are isolated from separate areas of the primary wavefront. This technique was used in Fig. 2 The Michelson interferometer Interferometry 943 inclination). With collimated light (the Twyman–Green interferometer), straight, parallel fringes of equal thickness (Fizeau fringes) are obtained.
3.2 The Mach–Zehnder Interferometer
As shown in Fig. 3, the Mach–Zehnder interferometer (MZI) uses two beam split- ters to divide and recombine the beams. BS The MZI has the advantage that each optical path is traversed only once. In addition, with an extended source, the region of localization of the fringes can be made to coincide with the test section. Fig. 4 The Sagnac interferometer The MZI has been used widely to map local variations of the refractive index in exactly the same path in opposite direc- wind tunnels, flames, and plasmas. tions. However, with an odd number of re- A variant, the Jamin interferometer, flections in the path, the wavefronts are lat- along with the Rayleigh interferometer, is erally inverted with respect to each other. commonly used to measure the refractive Since the optical paths traversed by index of gases and mixtures of gases. Accu- the two beams are always very nearly rate measurements of the refractive index equal, fringes can be obtained easily with of air are a prerequisite for interferometric an extended, white-light source. Modified measurements of length (see Sect. 5). forms of the Sagnac interferometer are used for rotation sensing, since the 3.3 rotation of the interferometer with an The Sagnac Interferometer angular velocity about an axis making an angle θ with the normal to the In one form of the Sagnac interferometer, plane of the beams introduces an optical asshowninFig.4,thetwobeamstraverse path difference 4 A p = cos θ, (5) c between the two beams, where A is the BS area enclosed by the beams and c is the speed of light.
4 BS Multiple-beam Interference
With two highly reflecting surfaces, we Fig. 3 The Mach–Zehnder interferometer have to take into account the effects 944 Interferometry
of multiply reflected beams (see WAVE and a spectrograph. FECO fringes permit OPTICS). The intensity in the interference measurements with a precision of λ/500. pattern formed by the transmission is A major application of FECO fringes has been to study the microstructure of sur- T2 I (φ) = ,(6) faces [3, 4]. However, the test surface must T + 2 − φ 1 R 2R cos be coated with a highly reflective coating. where R and T are, respectively, the reflectance and transmittance of the sur- faces and φ = (4π/λ)nd cos θ2.Asthe 5 reflectance R increases, the intensity at the Measurement of Length minima decreases, and the bright fringes become sharper. One of the earliest applications of interfer- The finesse, defined as the ratio of the ometry was in measurements of lengths. separation of adjacent fringes to the full Because of the limited distance over which width at half maximum (FWHM) of the interference fringes could be observed fringes (the separation of points at which with conventional light sources, Michel- the intensity is equal to half its maximum son had to perform a laborious series value), is of comparisons to measure the number πR1/2 of wavelengths of a spectral line in the F = .(7) 1 − R standard meter. The extremely narrow spectral bandwidth of light from a laser The interference fringes formed by the has led to the development of a number of reflected beams are complementary to interferometric techniques for direct mea- those obtained by transmission. surements of large distances. The values obtained for the optical path length are di- 4.1 vided by the value of the refractive index of Fringes of Equal Chromatic Order air, under the conditions of measurement, to obtain the true length. With a white-light source, interference fringes cannot be seen for optical path dif- 5.1 ferences greater than a few micrometers. Electronic Fringe Counting However, if the reflected light is exam- ined with a spectroscope, the spectrum If an additional phase difference of π/2 will be crossed by dark bands correspond- is introduced between the beams in ing to interference minima. With a thin one half of the field, two detectors can ) λ film (thickness d, refractive index n ,if 1 provide signals in quadrature to drive a λ and 2 are the wavelengths corresponding bidirectional counter. These signals can to adjacent dark bands, we have also be processed to obtain an estimate of λ λ the fractional interference order [5]. d = 1 2 .(8) 2n|λ2 − λ1| 5.2 With two highly reflecting surfaces en- Heterodyne Interferometry closing a thin film, very sharp fringes of equal chromatic order (FECO fringes) can In the Hewlett–Packard interferome- be obtained, using a white-light source ter [6], a He-Ne laser is forced to oscillate Interferometry 945
Beam C1 Solenoid expander
Q 1 C Q2 2
± ∆ Q2 Q Laser l/4 plate Polarizers Detectors D D R S Counter − ± ∆ Q2 Q1 Q
Subtractor Display
Reference − Q2 Q1 signal Counter
Fig. 5 Fringe-counting interferometer using a two-frequency laser (after Dukes, J. N., Gordon, G. B. (1970), Hewlett-Packard J. 21, 2–8 [6]. Hewlett–Packard Company. Reproduced with permission) simultaneously at two frequencies sep- viewing the interference pattern contains arated by about 2 MHz by applying a a component at the difference frequency, longitudinal magnetic field. As shown in and the phase of this heterodyne signal cor- Fig. 5, these two frequencies that have op- responds to the phase difference between posite circular polarizations pass through a the interfering beams. λ/4 plate that converts them to orthogonal In another technique, the two mirrors of linear polarizations. a Fabry–Perot interferometer are attached A polarizing beam splitter reflects one to the two ends of the sample, and the frequency to a fixed cube-corner, while the wavelength of a laser is locked to a other is transmitted to a movable cube- transmission peak [8]. A change in the corner. Both frequencies return along a separation of the mirrors results in a common axis and, after passing through change in the wavelength of the laser and, ◦ apolarizersetat45, are incident on a hence, in its frequency. These changes can photodetector. The beat frequencies from be measured with high precision by mixing this detector and a reference detector go to the beam from the laser with the beam a differential counter. If one of the cube- from a reference laser, and measuring the cornersismoved,thenetcountgivesthe beat frequency. change in the optical path in wavelengths. Very small changes in length can be 5.3 measured by heterodyne interferometry. Two-wavelength Interferometry In one technique [7], a small frequency shift is introduced between the two beams, If an interferometer is illuminated simul- typically by means of a pair of acousto-optic taneously with two wavelengths λ1 and λ2, modulators operated at slightly different the envelope of the fringes yields the inter- frequencies. The output from a detector ference pattern that can be obtained with 946 Interferometry
a synthetic wavelength by half a wavelength corresponds to one λ λ cycle of modulation. λ = 1 2 .() A very simple laser-feedback interfer- s |λ − λ | 9 1 2 ometer can be set up with a single-mode One way to implement this technique is semiconductor laser. An increased mea- with a CO2 laser, which is switched rapidly surement range and higher accuracy can between two wavelengths, as one of the be obtained by mounting the mirror on a mirrors of an interferometer is moved piezoelectric translator, and using an ac- over the distance to be measured. The tive feedback loop to hold the optical path output signal from a photodetector is then constant [12]. processed to obtain the phase difference at any point [9]. 6 5.4 Optical Testing Frequency-modulation Interferometry A major application of interferometry is Absolute measurements of distance can be in testing optical components and optical made with a semiconductor laser by sweep- systems (see OPTICAL METROLOGY). ing its frequency linearly with time [10]. If the optical path difference between the two 6.1 beams in the interferometer is L,onebeam Flat Surfaces reaches the detector with a time delay L/c, and they interfere to yield a beat signal The Fizeau interferometer (see Fig. 7) is with a frequency used widely to compare a polished flat surface with a standard flat surface without L df f = ,(10) placing them in contact and risking c dt damage to the surfaces. Measurements where df /dt istherateatwhichthelaser are made on the fringes of equal thickness frequency varies with time. formed with collimated light in the air space separating the two surfaces. 5.5 To determine absolute flatness, it is Laser-feedback Interferometry possible to use a liquid surface as a reference [13]; however, a more often-used If, as shown in Fig. 6, a small fraction of method is to test a set of three nominally the output of a laser is fed back to it by flat surfaces in pairs. Errors of each of the an external mirror, the output of the laser three surfaces can be evaluated using this varies cyclically with the position of the technique without the need for a known mirror [11]. A displacement of the mirror standard flat surface [14–16].
M
Detector Laser Fig. 6 Laser-feedback interferometer Interferometry 947
Reference flat
Laser
Test surface Fig. 7 Fizeau interferometer used to test flat surfaces
6.2 6.3 Homogeneity Concave and Convex Surfaces
The homogeneity of a material can be The Fizeau and Twyman–Green interfer- checked by preparing a plane-parallel sam- ometers can be used to test concave and ple and placing it in the test path of convex surfaces [18, 19]. Typical test con- the interferometer. The effects of sur- figurations for curved optical surfaces are face imperfections and systematic er- shown in Fig. 8. rors can be minimized by submerging 6.4 the sample in a refractive-index match- Prisms ing oil and making measurements with and without the sample in the test Figure 9 shows a test configuration for a ◦ path [17]. 60 prism. With a prism having a roof
Reference flat Converging lens
Concave surface test
Convex surface test Fig. 8 Fizeau interferometer used to test concave and convex surfaces 948 Interferometry
is to use oblique incidence [26]. An- other means of measuring these sur- faces uses scanning white-light tech- niques similar to those described in Sect. 8.3. Prism 6.7 Shearing Interferometers
Shearing interferometers, in which inter- ference takes place between two images of the test wavefront, have the advan- tage that they eliminate the need for a reference surface of the same dimen- Fig. 9 Twyman–Green interferometer used to sions as the test surface [23, 24]. With test a prism a lateral shear, as shown in Fig. 10(a), the two images undergo a mutual lateral ◦ angle of 90 , the beam is retro-reflected displacement. If the shear is small, the back through the system. wavefront aberrations can be obtained by integrating the phase data from two in- 6.5 terferograms with orthogonal directions Aspheric Surfaces of shear. With a radial shear, as shown in Fig. 10(b), one of the images is con- Problems can arise in testing an aspheric tracted or expanded with respect to the surface against a spherical reference wave- front because the fringes in some parts x of the resulting interferogram may be too closely spaced to be resolved. One way to solve this problem is to use a compen- s sating null-lens [20]; another is to use a computer-generated hologram to produce y a reference wavefront matching the de- sired aspheric wavefront [21, 22]. Shearing interferometry is yet another way to reduce the number of fringes in the interfero- (a) gram [23, 24].
6.6 Optically Rough Surfaces
d1 d2 One way to test fine ground surfaces, before they are polished, is by infrared interferometry with a CO2 laser at a (b) wavelength of 10.6 µm [25]. A simpler al- Fig. 10 Images of the test wavefront in ternative, with nominally flat surfaces, (a) lateral and (b) radial shearing interferometers Interferometry 949 other. If the diameter of one image is 7 less than (say) 0.3 of the other, the inter- Fringe Analysis ferogram obtained is very similar to that obtained with a Fizeau or Twyman–Green High-accuracy information can be ex- interferometer. tracted from the fringe pattern, including Other forms of shear, such as rotational, the calculation of aberration coefficients inverting or folding shears, can also be (see OPTICAL ABERRATIONS; [28]), by using used for specific applications. an electronic camera interfaced with a computer to measure and process the 6.8 intensity distribution in the interference The Point-diffraction Interferometer pattern. Several methods are available for this purpose (see [29]). As shown in Fig. 11, the point-diffraction interferometer [27] consists of a pinhole in 7.1 a partially transmitting film placed at the Fringe Tracking and Fourier Analysis focus of the test wavefront. The interference pattern formed by Early approaches to fringe analysis were the test wavefront, which is transmitted based on fringe tracking [30]. In order by the film, and a spherical reference to analyze a single fringe pattern, it is wave produced by diffraction at the pin- desirable to introduce a tilt between the hole corresponds to a contour map of interfering wavefronts so that a large the wavefront aberrations. Both wave- number of nominally straight fringes are fronts traverse the same path, making obtained. The shape of the fringe will this compact interferometer insensitive to be modified by the errors of the test vibrations. wavefront. Fourier analysis of the fringes
Transmitted Image of wave point source Pinhole
Diffracted spherical Partially reference wave Test wavefront transmitting film Fig. 11 The point-diffraction interferometer [27] 950 Interferometry
can then determine the test wavefront tilting a glass plate, moving a grating, deviation [31–33]. frequency-shifting, or rotating a half-wave plate or analyzer. Typically, intensity infor- 7.2 mation from four or five interferograms Phase-shifting Interferometry are used to calculate the original phase difference between the wavefronts on a Direct measurement of the phase differ- point-by-point basis [34, 35]. The repeata- ence between the beams at a uniformly bility of measurements is around λ/1000. spaced array of points offers many advan- Because the phase-calculation algorithm tages. In order to determine the phase utilizes an arctangent function, which does of the wavefront at each data point, at not yield any information on the integral least three interferograms are required. interference order, it is necessary to use The phase difference between the inter- a phase unwrapping procedure to detect fering beams is usually varied linearly changes in the integral interference order with time, and the intensity signal is and remove discontinuities in the retrieved integrated at each point over a num- phase [29, 36]. ber of equal phase segments covering Figure 12 shows a three-dimensional one period of the sinusoidal output sig- plot of the errors of a flat surface pro- nal. This technique is often simplified by duced by an interferometer using a digital adjusting the phase difference in equal phase-measurement system. steps. Normally, to implement such a phase The most common way to accomplish unwrapping procedure, it is necessary to the phase shift between the object and have at least two measurements per fringe reference beams is by changing the op- spacing; this constraint, limits the phase tical path difference between the beams gradients that can be measured. However, through a shift of the reference mirror techniques are available that can be used along the optical axis. Other ways include in special situations, with some apriori
105.7
109.9 mm
Fig. 12 Three-dimensional plot of the errors of a flat surface (326 nm Peak-to-Valley) obtained with a phase-measurement interferometer (Courtesy of Veeco Instruments Inc.) Interferometry 951 knowledge about the test surface, to work 8.1 around this limitation [37]. The Mirau Interferometer
The Mirau interferometer permits a very 7.3 compact optical arrangement. As shown Determining Aberrations schematically in Fig. 13, light from an illuminator is incident through the mi- With most optical systems, it is then croscope objective on a beam splitter. The convenient to express the deviations of transmitted beam falls on the test surface, the test wavefront as a linear combina- while the reflected beam falls on an alu- tion of Zernike circular polynomials in minized spot on a reference surface. The the form two reflected beams are recombined at the n k same beam splitter and return through W(ρ, θ) = ρk the objective. k=0 l=0 AsshowninFig.14,veryaccurate measurements of surface profiles can × (A cos lθ + B sin lθ), (11) kl kl be made using phase shifting. With a rough surface, the data can be processed ρ θ where and are polar coordinates to obtain the rms surface roughness ( − ) over the pupil, and k l is an even and the autocovariance function of the number. If the optical path differences surface [41]. at a suitably chosen array of points are known, the coefficients A and B kl kl 8.2 can be calculated from a set of linear The Nomarski Interferometer equations [38, 39]. Common-path interference microscopes use polarizing elements to split and re- 8 combine the beams [42]. In the Nomarski Interference Microscopy interferometer (see Fig. 15), two polarizing
Interference microscopy provides a non- contact method for studies of sur- facesaswellasamethodforstudy- Microscope objective ing living cells without the need to Reference surface stain them. Two-beam interference microscopes have been described using optical systems similar to the Fizeau and Michelson interferometers. For high magnifications, Beam splitter a suitable configuration is that described by Linnik [40] in which a beam splitter directs the light onto two identical objectives; one beam is incident on the test surface, while the other is directed to Sample the reference mirror. Fig. 13 The Mirau interferometer 952 Interferometry
Fig. 14 Pits (approximately 90 nm deep) on the surface of a mass-replicated CD-ROM. 11 µm × 13 µm field of view (Courtesy of Veeco Instruments Inc.)
With small isolated objects, two im- ages are seen covered with fringes that map the phase changes introduced by the object. With larger objects, the in- terference pattern is a measure of the phase gradients, revealing edges and local defects. Objective The use of phase-shifting techniques to extract quantitative information from the interference pattern has been described Object by [43].
8.3 Condenser White-light Interferometry
With monochromatic light, ambiguities can arise at discontinuities and steps producing a change in the optical path difference greater than a wavelength. This problem can be overcome by using a broadband (white-light) source. When the surface is scanned in height and Fig. 15 The Nomarski interferometer the corresponding variations in intensity at each point are recorded, the height prisms (see MICROSCOPY) introduce a lat- position corresponding to equal optical eral shear between the two beams. paths at which the visibility of the fringes Interferometry 953
2q Detector Laser Flow Fig. 16 Laser–Doppler interferometer for measurements of flow velocities is a maximum yields the height of the a moving particle has its frequency surface at that point. shifted. The method most commonly used to re- In the arrangement shown in Fig. 16, cover the fringe visibility function from two intersecting laser beams making the fringe intensity is by digital filter- angles ±θ with the direction of observation ing [44]. More recent techniques combine are used to illuminate the test field. phase-shifting techniques with signal de- The frequency of the beat signal ob- modulation [45, 46]. More detail on these served is techniques is available in OPTICAL METROL- θ ν = 2v sin ,() OGY. λ 12 Another technique that can be used with white light is spectrally resolved where v is the component of the velocity interferometry [47, 48]. A spectroscope is oftheparticleintheplaneofthebeamsat used to analyze the light from each point right angles to the direction of observation. on the interferogram. The optical path Simultaneous measurements of the ve- difference between the beams at this point locity components along orthogonal direc- can then be obtained from the intensity tions can be made by using two pairs of distribution in the resulting channeled illuminating beams (with different wave- spectrum (see Sect. 4.1). White-light interferometry techniques lengths) in orthogonal planes. used for biomedical applications are gen- It is also possible to use a self-mixing erally referred to as optical coherence configuration for velocimetry, in which the tomography or coherence radar (BIOMEDI- light reflected from the moving object is CAL IMAGING TECHNIQUES; [49, 50]). mixed with the light in the laser cavity. A very compact system has been described by [52] using a laser diode operated near 9 threshold with an external cavity to ensure Interferometric Sensors single-frequency operation. Very small vibration amplitudes can be Interferometers can be used as sensors for measured by attaching one of the mirrors several physical quantities. in an interferometer to the vibrating object. If the reflected beam is made to interfere 9.1 with a reference beam with a fixed- Laser–Doppler Interferometry frequency offset, the time-varying output contains, in addition to a component at Laser–Doppler interferometry [51] makes the offset frequency (the carrier), two use of the fact that light scattered by sidebands [53]. The vibration amplitude is 954 Interferometry
given by the relation used to shift and modulate the phase of the reference beam. The output goes to a λ I = s ,() pair of photodetectors, and measurements a π 13 2 Ic are made with a heterodyne system or a phase-tracking system [55]. where Ic and Is are, respectively, the power in the carrier and the sidebands. It is also possible, as shown in Fig. 18, to use a length of a birefringent single-mode 9.2 fiber, in a configuration similar to a Fizeau Fiber Interferometers interferometer, as a temperature-sensing element [56]. Since the length of the optical path in an The outputs from the two detectors are optical fiber changes when it is stretched, processed to give the phase retardation or when its temperature changes, inter- between the waves reflected from the front ferometers in which the beams propagate and rear ends of the fiber. Changes in ◦ in single-mode optical fibers (see FIBER OP- temperature of 0.0005 Ccanbedetected TICS and SENSORS, OPTICAL) can be used as with a 1-cm-long sensing element. sensors for a number of physical quanti- Measurements of electric and magnetic ties [54]. High sensitivity can be obtained, fields can also be made with fiber interfer- because it is possible to have very long, ometers by bonding the fiber sensor to a noise-free paths in a very small space. piezoelectric or magnetostrictive element. In the interferometer shown in Fig. 17, In addition, it is possible to multiplex sev- light from a laser diode is focused on the eral optical fiber sensors in a single system end of a single-mode fiber and optical to make measurements of various quanti- fiber couplers are used to divide and ties at a single location or even at different recombine the beams. Fiber stretchers are locations (see [57, 58]).
Optical fiber sensing element
Splitter Combiner Detectors Laser Phase modulator
Optical fiber reference path
Phase shifter
Oscillator Detector system Fig. 17 Interferometer using a single-mode fiber as a sensing element. (Giallorenzi,T.G.,Bucaro,J.A.,Dandridge,A.,Sigel,JrG.H.,Cole,J.H., Rashleigh, S. C., Priest, R. G. (1982), IEEE J. Quantum Electron. QE-18, 626–665 [55] IEEE, 1982. Reproduced with permission.) Interferometry 955
Beam Laser splitter Monomode high birefringence fiber
l/2 plate
Detector Signal processing Polarizing beam splitter Detector
Fig. 18 Interferometer using a birefringent single-mode fiber as a sensing element [56]
9.3 etendue (see OPTICAL RADIATION SOURCES Rotation Sensing AND STANDARDS). In the optical system shown in Fig. 20, Another application of fiber interferom- the effective areas AS and AD of the eters is in rotation sensing [59, 60]. The source and detector are images of each configuration used, in which the two waves other. traverse a closed multiturn loop in oppo- The etendue of the system is site directions, is the equivalent of a Sagnac interferometer (see Sect. 3.3). A typical sys- E = AS S = AD D,(14) tem is shown in Fig. 19 [61].
where S is the solid angle subtended by the lens L at the source and is the 10 S D solid angle subtended by the lens L at Interference Spectroscopy D the detector. Interferometric techniques are now used Since the etendue of a conventional spec- widely in high-resolution spectroscopy troscope is limited by the entrance slit, a (see SPECTROSCOPY, LASER) because they much higher etendue can be obtained with offer, in addition to higher resolution, a an interferometer. higher throughput. 10.2 10.1 The Fabry–Perot Interferometer Etendue of an Interferometer The Fabry–Perot interferometer (FPI) [62] The throughput of an optical system is uses multiple-beam interference between proportional to a quantity known as its two flat, parallel surfaces coated with 956 Interferometry
Polarization Polarization controller Polarizer controller
Semiconductor Fiber laser loop
Coupler Coupler
Detector Dead ends
Phase Modulated signal Reference signal modulator
Chart recorder Lock-in amplifier AC generator
Rotation signal
Fig. 19 Fiber interferometer for rotation sensing [61]
Ω Ω S LS D LD
Source Detector
AS AD
Interferometer Fig. 20 Etendue of an interferometer
semitransparent, highly reflecting coat- wavelength difference λ2/2nd.Thiswave- ings. With a fixed spacing d, any single length difference, known as the free wavelength produces a system of sharp, spectral range (FSR), is the range of bright rings (fringes of equal inclination), wavelengths that can be handled by the defined by Eq. (6), centered on the normal FPI without successive interference orders to the surfaces. overlapping. For any angle of incidence, with a The resolving power of the FPI is broadband source, it also follows from obtained by dividing the FSR by the Eq. (6) that the separation of succes- finesse (see Eq. 7) and is given by the sive intensity maxima corresponds to a relation Interferometry 957 λ = R λ π 1/2 = 2nd R ,() MM λ − 15 1 R Fig. 21 Confocal Fabry–Perot interferometer where λ is the half-width of the peaks, and R is the reflectance of the surfaces. two spherical mirrors separated by a One way to overcome the limited FSR of distance equal to their radius of curvature, the FPI is by imaging the fringes onto the so that their foci coincide. Since the optical slit of a spectrometer, but this procedure path difference is independent of the angle limits the etendue of the system. A better of incidence, a uniform field is obtained. way is to use two or more FPIs, with An extended source can be used, and the different values of d,inseries. transmitted intensity is recorded as the Another important characteristic of an separation of the plates is varied [64]. FPI is the contrast factor, defined by the ratio of the intensities of the maxima and 10.3 minima, which is Wavelength Measurements 1 + R 2 C = .(16) Accurate measurements of the wavelength 1 − R of the output from a tunable laser, such For applications such as Brillouin spec- as a dye laser, can be made with an troscopy, in which a weak spectrum line interferometric wavelength meter. maybemaskedbythebackgrounddueto In the dynamic wavelength meter shown a neighboring strong spectral line, a much in Fig. 22, a beam from the dye laser as higher contrast factor can be obtained by well as a beam from a reference laser, passing the light several times through the whose wavelength is known, traverse the same FPI [63]. same two paths. The wavelength of the dye A much higher throughput can be laser is determined by counting fringes obtained with the confocal Fabry–Perot simultaneously at both wavelengths, as the interferometer shown in Fig. 21 that uses end reflector is moved [65].
l Detector D1 ( 1)
l Reference laser ( 2)
l Dye laser ( 1) Movable cube-corner l Detector D2 ( 2) Fig. 22 Dynamic wavelength meter [65] 958 Interferometry
In a simpler arrangement [66], the AND VISIBLE LIGHT and SPECTROMETERS, IN- fringes of equal thickness formed in a FRARED). wedged air film are imaged on a linear de- With a scanning spectrometer, the total tector array. The spacing of the fringes is time of observation T is divided between, used to evaluate the integral interference say, m elements of the spectrum. Since order, and their position to determine the in the infrared, the main source of noise fractional interference order. is the detector, the signal-to-noise (S/N) ratio is reduced by a factor m1/2.This 10.4 reduction in the S/N ratio can be avoided Laser Linewidth by varying the optical path difference in an interferometer linearly with time, in The extremely small spectral bandwidth of which case each element of the spectrum the output from a laser can be measured generates an output modulated at a by mixing, as shown in Fig. 23, light from frequency that is inversely proportional the laser with a reference beam from the to its wavelength. It is then possible to same laser that has undergone a frequency record all these signals simultaneously shift and a delay [67]. (or, in other words, to multiplex them) and then, by taking the Fourier transform 10.5 of the recording (see FOURIER AND OTHER Fourier-transform Spectroscopy TRANSFORM METHODS), to recover the spectrum [68–70]. Major applications of Fourier-transform AsshowninFig.24,aFourier- spectroscopy include measurements of transform spectrometer is basically a infrared absorption spectra as well as Michelson interferometer illuminated emission spectra from faint astronomical with an approximately collimated beam objects (see SPECTROMETERS, ULTRAVIOLET from the source whose spectrum is to be
Acousto- Optical Laser optic isolator modulator B1
B2 Detector Single-mode optical fiber
Spectrum Subtractor Detector analyzer
Fig. 23 Measurement of laser linewidth by heterodyne interferometry [67] Interferometry 959
Movable reflector
Source
Detector
Fig. 24 Fourier-transform spectrometer recorded. The mirrors are often replaced new areas of interferometry based on by ‘‘cat’s eye’’ reflectors to minimize the use of nonlinear optical materials problems due to tilting as they are moved. (see NONLINEAR OPTICS). The interferogram is then sampled at a number of equally spaced points. To 11.1 avoid ambiguities (aliasing) the change Second-harmonic Interferometers in the optical path difference between samples must be less than half the short- Second-harmonic interferometers produce est wavelength in the spectrum. Finally, a fringe pattern corresponding to the the spectrum is computed using the fast phase difference between two second- Fourier-transform algorithm [71]. Errors harmonic waves generated at differ- in the computed spectrum can be reduced ent points in the optical path from by a process called apodization, where the original wave at the fundamental the interference signal is multiplied by frequency. a symmetrical weighting function whose Figure 25 is a schematic of an inter- value decreases gradually with the optical ferometer using two frequency-doubling path difference. crystals that can be considered as an analog of the Mach–Zehnder interferom- 11 eter [72]. Nonlinear Interferometers In this interferometer, the infrared beam from a Q-switched Nd:YAG laser The high light intensity available with (λ1 = 1.06 µm) is incident on a frequency- pulsed lasers has opened up completely doubling crystal, and the green (λ2 = 960 Interferometry
Green
IR laser
DoublerTest piece Doubler Fig. 25 Second-harmonic interferometer using two frequency-doubling crystals
0.53 µm) and infrared beams emerging 11.2 from this crystal pass through the test Phase-conjugate Interferometers piece. At the second crystal, the infrared beam undergoes frequency doubling to In a phase-conjugate interferometer, the produce a second green beam that in- test wavefront is made to interfere with its terferes with the one produced at the conjugate, eliminating the need for a ref- first crystal. The interference order at any erence wave and doubling the sensitivity. point is In the phase-conjugate interferometer shown in Fig. 26, which can be regarded as an analog of the Fizeau interferometer, (n − n ) d(x, y) ( , ) = 2 1 ,() a partially reflecting mirror is placed in N x y λ 17 2 front of a single crystal of barium titanate, which functions as a self-pumped phase- where d(x, y) is the thickness of the test conjugate mirror [75, 76]. ( , ) specimen at any point x y and n1 and n2 An interferometer in which both mirrors are its refractive indices for infrared and have been replaced by a single-phase green light, respectively. conjugator has the unique property that Other types of interferometers that are the field of view is normally completely analogs of the Fizeau, Twyman, and point- dark and is unaffected by misalignment diffraction interferometers have also been or by air currents. However, because of described [73, 74]. thedelayintheresponseofthephase
Phase-conjugate Input wavefront mirror
Beam splitter
Reference surface
Output wavefronts Fig. 26 Phase-conjugate interferometer [76] Interferometry 961 conjugator, any sudden local change in the optical path produces a bright spot that slowly fades away [77].
11.3 Measurement of Nonlinear Susceptibilities
A modified Twyman–Green interferom- eter can be used for measurements of nonlinear susceptibilities. A system that can be used to measure the relative phase shift between two-phase conjugators, as well as the ratio of their susceptibilities, and yields high sensitivity, even with weak Telescope signals, has been described by [78].
12 Fig. 27 Michelson’s stellar interferometer Interferometric Imaging to zero when Interferometric imaging started with the 1.22λ development of techniques to measure = .() D α 19 the diameters of stars that could not 2 be resolved with conventional telescopes Measurements with Michelson’s stellar in- (see ASTRONOMICAL TELESCOPES AND IN- terferometer over baselines longer than STRUMENTATION). 6 m presented serious difficulties because Michelson’s stellar interferometer [79] of the difficulty of maintaining the optical used the fact that the angular diameter of a path difference between the beams stable star can be calculated from observations and small enough not to affect the visibility of the visibility of the fringes in an of the fringes. However, modern detection, interferometer using light from the star control, and data handling techniques have reaching the surface of the earth at two made possible a new version of Michel- points separated by a known distance. son’s stellar interferometer [80] designed If we assume the star to be a uniform to make measurements over baselines up circular source with an angular diameter to 640 m. 2α,andD is the separation of two mirrors receiving the light from the star and 12.1 feeding it to a telescope, as shown in The Intensity Interferometer Fig. 27, the visibility of the fringes is The problem of maintaining the equality 2J (u) of the two paths was minimized in V = 1 ,(18) u the intensity interferometer, which used measurements of the degree of correlation where u = 2παD/λ and J1 is a first-order between the fluctuations in the outputs Bessel function. The fringe visibility drops of two photodetectors at the foci of two 962 Interferometry
large light collectors separated by a variable 12.2 distance [81, 82]. Heterodyne Stellar Interferometers The actual instrument used light collec- tors operated with separations up to 188 m. In these instruments, as shown in Fig. 28, With a bandwidth of 100 MHz it was only light from the star is received by two necessary to equalize the two optical paths telescopes and mixed with light from a to within 30 cm, but, because of the narrow laser at two photodiodes. The resulting bandwidth, measurements could only be heterodyne signals are multiplied in a made on 32 of the brightest stars. correlator. The output signal is a measure
Telescope Telescope
5 MHz offset
Laser Laser
Detector Detector Heterodyne signals (< 1500 MHz)
Delay lines
Computer Multiplier
5 MHz signal
Processor
Fringe amplitude Fig. 28 Schematic of a heterodyne stellar interferometer [83] Interferometry 963 of the degree of coherence of the wave stars. Unfortunately, with a two-element fields at the two photo detectors [83]. interferometer, it is only possible to ob- As with the intensity interferometer, tain information on the fringe amplitude this technique only requires the two op- becausethevalueofthephaseisaffected tical paths to be equalized to within a by instrumental and atmospheric effects. few centimeters; however, the sensitivity However, with a triangular array, the clo- is higher, since it is proportional to the sure phase is determined only by the product of the intensities of the laser coherence function. As the number of ele- and the star. An infrared interferome- ments increases, the image becomes better ter comprising two telescopes with an constrained [87]. aperture of 1.65 m has been constructed, Some images have already been ob- capable of yielding an angular resolu- tained from a large, multielement inter- tion of 0.001 second of arc [84]. Larger ferometer (the Cambridge Optical Aper- telescopes are planned (see ASTRONOMICAL ture Synthesis Telescope) [88] and several TELESCOPES AND INSTRUMENTATION). other telescope arrays are nearing comple- tion (see ASTRONOMICAL TELESCOPES AND 12.3 INSTRUMENTATION). Stellar Speckle Interferometry
Stellar speckle interferometry makes use 13 of the fact that, due to local inhomo- Space-time and Gravitation geneities in the earth’s atmosphere, the image of a star produced by a large Michelson’s classical experiment to test telescope, when observed under high mag- the hypothesis of a stationary ether showed nification, has a speckle structure [85] an effect that was less than one-tenth of that expected. This experiment was re- (see also ASTRONOMICAL TELESCOPES AND peated by [89], with a much higher degree INSTRUMENTATION). However, individual speckles have dimensions correspond- of accuracy, by locking the frequency of ing to a diffraction-limited image of the a He-Ne laser to a resonance of a ther- star. Reference [86] showed that a high- mally isolated Fabry–Perot interferometer resolution image could be extracted from a mounted along with it on a rotating hor- number of such speckled images recorded izontal granite slab. When the frequency with sufficiently short exposures to freeze of this laser was compared to that of a the speckles. stationary, frequency-stabilized laser, the While the angular resolution that can frequency shifts were found to be less 6 be obtained by speckle interferometry is than 1 part in 10 of those expected with a limited by the aperture of the telescope, it stationary ether. has been applied successfully to a number 13.1 of problems, including the study of close Gravitational Waves double stars. 12.4 It follows from the general theory of Telescope Arrays relativity that binary systems of neutron stars, collapsing supernovae, and black Theultimateobjectivewouldbetheabil- holes should be the sources of gravita- ity to produce high-resolution images of tional waves. 964 Interferometry
Because of the transverse quadrupolar the interferometer. The frequency of the nature of a gravitational wave, the local laser is locked to a transmission peak of distortion of space-time due to it stretches one interferometer, while the optical path spaceinadirectionnormaltothedirection length in the other is continually adjusted of propagation of the wave, and shrinks so that its peak transmittance is also at this it along the orthogonal direction. This frequency [92]. local strain could, therefore, be measured A further increase in sensitivity can by a Michelson interferometer in which be obtained by recycling the available the beam splitter and the end mirrors light. Since the interferometer is normally are attached to separate, freely suspended adjusted so that observations are made masses [90, 91]. on a dark fringe, to avoid overloading Theoretical estimates of the intensity the detector, most of the light is returned of gravitational radiation due to various toward the source and is lost. If this light possible events, suggest that a sensitivity is reflected back into the interferometer in to strain of the order of 10−21 over a the right phase, by an extra mirror placed bandwidth of a kilohertz would be needed. in the input beam, the amount of light This would require an interferometer with traversing the arms of the interferometer unrealistically long arms. can be increased substantially [93]. One way to obtain a substantial increase Two other techniques that can be com- in sensitivity is, as shown in Fig. 29, by bined with these techniques for obtaining using two identical Fabry–Perot cavities, even higher sensitivity are signal recy- with their mirrors mounted on freely cling [94] and resonant side-band extrac- suspended test masses, as the arms of tion [95].
End mirrors T = 0% T = 0%
= T = 3% T 3% Corner L mirrors 1 L 2
T = 50% Recycling mirror To T = 3% detector From laser Fig. 29 Schematic of an interferometric gravitational wave detector [91] Interferometry 965
13.2 real time, any changes in the shape of the LIGO object. Alternatively, two holograms can be recorded with the object in two different The laser interferometer gravitational ob- states and the wavefronts reconstructed servatory (LIGO) project [96, 97] involves by these two holograms can be made the construction of three laser interferom- to interfere. eters with arms up to 4-km long, two at Since holographic interferometry makes onesiteandthethirdatanothersitesep- it possible to measure, with very high pre- arated from the first by almost 3000 km. cision, changes in the shape of objects with The test masses and the optical paths in rough surfaces, it has found many applica- these interferometers are housed in a vac- tions including nondestructive testing and uum. Correlating the outputs of the three vibration analysis [99–101]. interferometers should make it possible to distinguish the signals due to gravitational 14.1 waves from the bursts of instrumental and Strain Analysis environmental noise. The phase difference at any point (x, y) in 13.3 the interferogram is given by the relation Limits to Measurement (see Fig. 30)
The limit to measurements of such small φ = L(x, y) · (k1 − k2) = L(x, y) · K, displacements is ultimately related to the (21) number of photons n that pass through where L(x, y) is the vector displacement of the interferometer in the measurement the corresponding point on the surface of time. The resulting uncertainty in mea- the object, k1 and k2 are the propagation surements of the phase difference between vectors of the incident and scattered light, the beams has been shown to be [98] and K = k1 − k2 is known as the sensitivity 1 vector [102]. φ ≥ √ ,(20) To evaluate the vector displacements 2 n (out-of-plane and in-plane), it is convenient and is known as the standard quantum to use a single direction of observation and limit (SQL).
Image Incident light 14 Holographic Interferometry k P′ 1 Holography (see HOLOGRAPHY and L OPTICAL TECHNIQUES FOR MECHANICAL K P MEASUREMENT) makes it possible to k2 store and reconstruct a perfect three- dimensional image of an object. The Deformed To reconstructed wave can then be made to object observer interfere with the wave generated by the Fig. 30 Phase difference produced by a object to produce fringes that contour, in displacement of the object 966 Interferometry
record four holograms with the object illu- fringes corresponding to the zeros of the minated from two different angles in the Bessel function. vertical plane and two different angles in Alternatively, a hologram can be recor- the horizontal plane. Phase shifting is used ded of the stationary object, and the real- to obtain the phase differences at a network time interference pattern obtained with the of points [103]. These values can then be vibrating object can be viewed using stro- used, along with information on the shape boscopic illumination. A brighter image of the object, to obtain the strains. can be obtained by recording the hologram with stroboscopic illumination, synchro- 14.2 nized with the vibration cycle, and viewing Vibration Analysis the interference fringes formed with the stationary object, using continuous illu- One way to study the vibrating objects is to mination. Phase-shifting techniques can record a hologram of the vibrating object then be used to map the instantaneous with an exposure time that is much longer displacement of the vibrating object (see than the period of the vibration [104]. The Fig. 31) [105]. intensity at any point (x, y) in the image is then given by the relation 14.3 Contouring ( , ) = ( , ) K · L( , ) ,() I x y I0 x y J0[ x y ] 22 The simplest method of contouring an object is by recording two holograms where I0(x, y) is the intensity with the with the object illuminated from slightly stationary object, J0 is a zero-order Bessel different angles. More commonly used function and L(x, y) is the amplitude of techniques are two-wavelength contouring vibration of the object. The fringes ob- and two-refractive-index contouring. served (time-average fringes) are contours In two-wavelength contouring [106], a of equal vibration amplitude, with the dark telecentric lens system is used to image
1.0
m) 0.5 µ (
25 50 75 (mm) Fig. 31 Three-dimensional plot of the instantaneous displacement of a metal plate vibrating at 231 Hz obtained by stroboscopic holographic interferometry using phase shifting[105] Interferometry 967 the object on the hologram plane and shifting, using the two-refractive-index exposures are made with two different technique [108]. wavelengths, λ1 and λ2. When the holo- gram is illuminated with one of the wavelengths (say λ2), fringes are seen con- 15 touring the reconstructed image, separated Moire´ Techniques by an increment of height λ λ Moire´ techniques complement hologra- |δz|= 1 2 (23) phic interferometry and can be used where 2(λ − λ ) 1 2 a contour interval greater than 10 µmisre- In two-refractive-index contouring [107], quired [109] (see also OPTICAL TECHNIQUES the object is placed in a cell with a FOR MECHANICAL MEASUREMENT). glass window and imaged by a telecen- A simple way to obtain Moirefringes´ tric system. is to project interference fringes (or a Two holograms are recorded on a plate grating) onto the object and view it placed near the stop of the telecentric through a grating of approximately the system with the cell filled with liquids same spacing. The contour interval is having refractive indices n1 and n2,re- determined by the fringe (grating) spacing spectively. Contours are obtained with a and the angle between the illumination spacing and viewing directions. Phase shifting is λ possible by shifting one grating or the |δz|= .(24) projected fringes. 2(n1 − n2) Digital phase-shifting techniques can 15.1 be used with both these methods of Grating Interferometry holographic contouring. Figure 32 shows a three-dimensional plot of a wear mark The in-plane displacements of nearly on a flat surface obtained by phase flat objects can be measured with
200
m) 100 µ (
0
Fig. 32 Three-dimensional plot of a wear mark on a flat surface obtained by phase shifting, using the two-refractive-index technique [108] 968 Interferometry
submicrometer sensitivity by grating displacements using an optical system in interferometry [109, 110]. which the surface is illuminated by two A reflection grating is attached to the beams making equal but opposite angles object under test with a suitable adhesive to the normal. and illuminated by two coherent beams symmetrical to the grating normal. The 16.1 interference pattern produced by the two Electronic Speckle Pattern Interferometry diffracted beams reflected from the grating (ESPI) yields a map of the in-plane displacements. Polarization techniques can be used for Measurements can be made at video rates phase shifting. using a TV camera interfaced to a com- puter [111, 113]. As shown in Fig. 33, the object is imaged on a CCD array along with a coaxial reference beam. This technique 16 Speckle Interferometry was originally known as electronic speckle pattern interferometry (ESPI). The image of an object illuminated by a If an image of the object in its original laser is covered with a stationary gran- state is subtracted from an image of ular pattern known as a speckle pat- the object at a later stage, regions in which the speckle pattern has not changed, tern (see SPECKLE AND SPECKLE METROL- corresponding to the condition OGY). Speckle interferometry [111] utilizes interference between the speckled image K · L(x, y) = 2mπ, (25) of an object illuminated by a laser and a reference beam derived from the same where m is an integer, appear dark, while laser. Any change in the shape of the regions where the pattern has changed are object results in local changes in the in- covered with bright speckles [114, 115]. tensity distribution in the speckle pattern. This technique is also known as If two photographs of the speckled image TV holography. are superimposed, fringes are obtained, corresponding to the degree of correlation 16.2 of the two speckle patterns that contour Phase-shifting Speckle Interferometry the changes in shape of the surface [112]. Speckle interferometry can be a very Each speckle can be regarded as an simple way of measuring the in-plane individual interference pattern and the
Single-mode fiber Reference beam
TV Object camera Fig. 33 System for ESPI Interferometry 969 phase difference between the two beams Coherence: A complex quantity whose at this point can be measured by phase magnitude denotes the correlation be- shifting, with the object before and after tween two wave fields; its phase de- an applied stress. The result of subtracting notes the effective phase difference be- the second set of values from the first is tween them. then a contour map of the deformation of Degree of Coherence: The value of the the object [116, 117]. coherence expressed as a fraction of Further developments in speckle that for complete correlation between the interferometry with phase-measurement wave fields. techniques, high-resolution CCD arrays, and real-time processing have created Fringes of Equal Inclination: Interference many techniques encompassing what is fringes created from two collinear in- now most commonly known as digital terfering beams having wavefronts with holography (see OPTICAL TECHNIQUES different radii of curvature. FOR MECHANICAL MEASUREMENT and HOLOGRAPHY). Fringes of Equal Thickness: Interference fringes created with collimated beams 16.3 when the optical path difference depends Vibrating Objects only on the thickness and refractive index.
Interference Order: The number of wave- If the period of the vibration is small lengths in the optical path difference compared to the exposure time or the between two interfering beams. scan time of the camera, the contrast of the speckles at any point is given by Interferogram: The varying part of an the expression interference pattern, after subtracting any uniform background. {1 + 2αJ2[K · L(x, y)]}1/2 C = 0 ,(26) 1 + α Moire´ Fringes: Relatively coarse fringes produced by the superposition of two where α is the ratio of the intensities of fine fringe patterns with slightly different the reference beam and the object beam, spacings or orientations. K is the sensitivity vector and L(x, y) is the vibration amplitude at that point. Regions Optical Path Difference: The difference corresponding to the zeros of the J0 Bessel in the optical path length between two function appear as dark fringes. interfering beams. Phase-shifting techniques can also be applied to the analysis of vibrations [118]. Optical Path Length: The product of the refractive index of the medium traversed by a beam and the length of the path in the medium. Glossary Optical Phase: Theresultantphaseofa Beam splitter: An optical element that light beam after allowing for changes due divides a single beam of light into two to the optical path traversed and reflection beams of the same wave form. at any surfaces. 970 Interferometry
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Laser Cooling and Trapping of Neutral Atoms
Harold J. Metcalf Department of Physics, State University of New York, Stony Brook, N.Y. 11794-3800, USA Phone: (631) 632-8185; Fax: (631) 632-8176; e-mail: [email protected]
PetervanderStraten Debye Institute, Department of Atomic and Interface Physics, Utrecht University, 3508 TA Utrecht, The Netherlands Phone: 31-30-2532846; Fax: 31-30-2537468; e-mail: [email protected]
Abstract This article presents a review of some of the principal techniques of laser cooling and trapping that have been developed during the past 20 years. Its approach is primarily experimental, but its quantitative descriptions are consistent in notation with most of the theoretical literature.
Keywords laser cooling; atom trapping; optical lattice; Bose–Einstein condensation.
1 Introduction 977 1.1 Temperature and Entropy 977 1.2 Phase Space Density 978 2 Optical Forces on Neutral Atoms 978 2.1 Radiative Optical Forces 979 2.2 Dipole Optical Forces 979 2.3 Density Matrix Description of Optical Forces 980 2.3.1 Introduction 980 2.3.2 Open Systems and the Dissipative Force 980 2.3.3 Solution of the OBEs in Steady State 981 976 Laser Cooling and Trapping of Neutral Atoms
2.3.4 Radiative and Dipole Forces 982 2.3.5 Force on Moving Atoms 982 3 Laser Cooling 982 3.1 Slowing Atomic Beams 982 3.2 Optical Molasses 984 3.2.1 Doppler Cooling 984 3.2.2 Doppler Cooling Limit 985 3.2.3 Atomic Beam Collimation – One-dimensional Optical Molasses – Beam Brightening 986 3.2.4 Experiments in Three-dimensional Optical Molasses 987 3.3 Cooling Below the Doppler Limit 988 3.3.1 Introduction 988 3.3.2 Linear ⊥ Linear Polarization Gradient Cooling 988 3.3.3 Origin of the Damping Force 990 3.3.4 The Limits of Sisyphus Laser Cooling 991 4 Traps for Neutral Atoms 991 4.1 Dipole Force Optical Traps 992 4.1.1 Single-beam Optical Traps for Two-level Atoms 992 4.1.2 Blue-detuned Optical Traps 993 4.2 Magnetic Traps 994 4.2.1 Introduction 994 4.2.2 Magnetic Confinement 994 4.2.3 Classical Motion of Atoms in a Quadrupole Trap 996 4.2.4 Quantum Motion in a Trap 997 4.3 Magneto-optical Traps 997 4.3.1 Introduction 997 4.3.2 Cooling and Compressing Atoms in an MOT 999 4.3.3 Capturing Atoms in an MOT 999 4.3.4 Variations on the MOT Technique 1000 5 Optical Lattices 1000 5.1 Quantum States of Motion 1000 5.2 Properties of 3-D Lattices 1002 5.3 Spectroscopy in 3-D Lattices 1003 5.4 Quantum Transport in Optical Lattices 1004 6 Bose–Einstein Condensation 1005 6.1 Introduction 1005 6.2 Evaporative Cooling 1005 6.2.1 Simple Model 1006 6.2.2 Application of the Simple Model 1007 6.2.3 Speed of Evaporation 1008 6.2.4 Limiting Temperature 1008 6.3 Forced Evaporative Cooling 1009 7 Conclusion 1010 Glossary 1010 References 1012 Laser Cooling and Trapping of Neutral Atoms 977
1 assignment of a thermodynamic ‘‘temper- Introduction ature’’ is completely inappropriate. Nevertheless, it is convenient to use the The combination of laser cooling and label of temperature to describe an atomic atom trapping has produced astounding sample whose average kinetic energy Ek new tools for atomic physicists [1]. These has been reduced by the laser light, and experiments require the exchange of mo- this is written simply as kBT/2 =Ek, mentum between atoms and an optical where kB is the Boltzmann’s constant (for field, usually at a nearly resonant fre- the case of one dimension, 1-D). It must quency. The energy of light ¯hω changes be remembered that this temperature the internal energy of the atom, and the assignment is absolutely inadequate for angular momentum ¯h changes the orbital atomic samples that do not have a angular momentum of the atom, as de- Maxwell–Boltzmann velocity distribution, scribed by the well-known selection rule whether or not they are in thermal contact =±1. By contrast, the linear momen- with the environment; there are infinitely tum of light p = ¯hω/c = ¯hk cannot change many velocity distributions that have the the internal atomic degrees of freedom, same value of Ek but are so different and therefore must change the momen- from one another that characterizing them tum of the atoms in the laboratory frame. by the same ‘‘temperature’’ is a severe The force resulting from this momentum error. (In the special case where there exchange between the light field and the is a true damping force, F ∝−v,and atoms can be used in many ways to control where the diffusion in momentum space atomic motion, and is the subject of this is a constant independent of momentum, article. solutions of the Fokker–Planck equation can be found analytically and can lead 1.1 to a Maxwell–Boltzmann distribution that Temperature and Entropy does indeed have a temperature.) Since laser cooling decreases the tem- The idea of ‘‘temperature’’ in laser cool- perature of a sample of atoms, there is ing requires some careful discussion and less disorder and therefore less entropy. disclaimers. In thermodynamics, temper- This seems to conflict with the second law ature is carefully defined as a parameter of thermodynamics, which requires the of the state of a closed system in thermal entropy of a closed system to always in- equilibrium with its surroundings. This, crease with time. The explanation lies in of course, requires that there be thermal the consideration of the fact that in laser contact, that is, heat exchange, with the en- cooling, the atoms do not form a closed vironment. In laser cooling, this is clearly system. Instead, there is always a flow of not the case because a sample of atoms laser light with low entropy into the sys- is always absorbing and scattering light. tem and fluorescence with high entropy Furthermore, there is essentially no heat outofit.Thedecreaseofentropyofthe exchange (the light cannot be considered atoms is accompanied by a much larger as heat even though it is indeed a form increase in entropy of the light field. En- of energy). Thus, the system may very tropy considerations for a laser beam are well be in a steady state situation, but cer- far from trivial, but recently it has been tainly not in thermal equilibrium, so the shown that the entropy lost by the atoms 978 Laser Cooling and Trapping of Neutral Atoms
is many orders of magnitude smaller than to use a force that is not conservative, the entropy gained by the light field. such as a velocity-dependent force. In laser cooling, the force on the atoms can be 1.2 a damping force, that is, always directed Phase Space Density opposite to the atomic velocity, so that the momentum part of ρφ increases. This The phase space density ρ(r, p, t) is process arises from the irreversible nature defined as the probability that a single of spontaneous emission. particle is in a region dr around r and has momentum dp around p at time t. In classical mechanics, ρ(r, p, t) is just the 2 sum of the ρ(r, p, t) values of each of the N Optical Forces on Neutral Atoms particles in the system divided by N.Since the phase space density is a probability, it The usual form of electromagnetic forces is is always positive and can be normalized given by F = q(E + v × B),butforneutral over the six-dimensional volume spanned atoms, q = 0. The next order of force is the by position r and momentum p.Foragas dipole term, but this also vanishes because of cold atoms, it is convenient to choose the neutral atoms have no inherent dipole 3 elementary volume for ρ(r, p, t) to be ¯h , moment. However, a dipole moment can so it becomes the dimensionless quantity be induced by a field, and this is most efficient if the field is alternating near the ρφ = nλ3 ,(1) deB atomic resonance frequency. Since these λ where deB is the deBroglie wavelength of frequencies are typically in the optical the atoms in the sample and n is their range, dipole moments are efficiently spatial density. induced by shining nearly resonant light The Liouville theorem requires that ρφ on the atoms. cannot be increased by using conservative If the light is absorbed, the atom forces. For instance, in light optics one makes a transition to the excited state, can focus a parallel beam of light with a and the subsequent return to the ground lens to a small spot. However, that simply state can be either by spontaneous or by produces a high density of light rays in the stimulated emission. The nature of the focus in exchange for the momentum part optical force that arises from these two of ρφ because the beam entering the lens different processes is quite different and is parallel but the light rays are divergent will be described separately. at the focus. The spontaneous emission case is For classical particles, the same principle different from the familiar quantum- applies. By increasing the strength of the mechanical calculations using state vec- trapping potential of particles in a trap, one tors to describe the system. Spontaneous can increase the density of the atoms in the emission causes the state of the sys- trap but at the same time, the compression tem to evolve from a pure state into of the sample results in a temperature a mixed state and so the density ma- increase, leaving the phase space density trix is needed to describe it. Sponta- unchanged. neous emission is an essential ingredient In order to increase the phase space for the dissipative nature of the optical density of an atomic sample, it is necessary forces. Laser Cooling and Trapping of Neutral Atoms 979
2.1 the atoms). The force is thus velocity- Radiative Optical Forces dependent and the experimenters’ task is to exploit this dependence to the desired In the simplest case–the absorption of goal, for example, optical friction for laser well-directed light from a laser beam–the cooling. momentum exchange between the light The maximum attainable deceleration field and the atoms results in a force is obtained for high intensities of light. High-intensity light can produce faster ab- dp F = = ¯hkγp,(2) sorption, but it also causes equally fast dt stimulated emission; the combination pro- duces neither deceleration nor cooling. where γp is the excitation rate of the atoms. The absorption leaves the atoms in their The momentum transfer to the atoms excited state, and if the light intensity by stimulated emission is in the oppo- is low enough that they are much more site direction to what it was in absorption, likely to return to the ground state by resulting in a net transfer of zero mo- spontaneous emission than by stimulated mentum. At high intensity, Eq. (3) shows γ γ/ emission, the resulting fluorescent light saturation of p at 2, and since the force is given by Eq. (2), the deceleration satu- carries off momentum ¯hk in a random = γ direction. The momentum exchange from rates at a value amax ¯hk /2M. the fluorescence averages zero, so the net total force is given by Eq. (2). 2.2 Dipole Optical Forces The excitation rate γp depends on the laser detuning from atomic resonance δ ≡ While detuning |δ|γ ,spontaneous ωl − ωa,whereωl is the laser frequency emission may be much less frequent than and ωa is the atomic resonance frequency. This detuning is measured in the atomic stimulated emission, unlike the case of the rest frame, and it is necessary that the dissipative radiative force that is necessary Doppler-shifted laser frequency in the rest for laser cooling, given by Eqs. (2) and (3). frame of the moving atoms be used to In this case, absorption is most often fol- lowed by stimulated emission, and seems calculate γp. In Sect. 2.3.3, we find that to produce zero momentum transfer be- γp for a two-level atom is given by the cause the stimulated light has the same Lorentzian momentum as the exciting light. How- γ/ γ = s0 2 ,() ever, if the optical field has beams with at p 2 3 1 + s0 + [2(δ + ωD)/γ ] least two different k-vectors present, such as in counterpropagating beams, absorp- where γ ≡ 1/τ is an angular frequency tion from one beam followed by stimulated corresponding to the natural decay rate of emission into the other indeed produces a the excited state. Here, s0 = I/Is is the ratio nonzero momentum exchange. The result of the light intensity I to the saturation is called the dipole force, and is reversible 3 intensity Is ≡ πhc/3λ τ,whichisafew and hence conservative, so it cannot be − mW cm 2 for typical atomic transitions. used for laser cooling. The Doppler shift seen by the moving The dipole force is more easily calculated atoms is ωD =−k · v (note that k opposite from an energy picture than from a to v produces a positive Doppler shift for momentum picture. The force then derives 980 Laser Cooling and Trapping of Neutral Atoms
from the gradient of the potential of an intensity Imax at the antinodes is four times atom in an inhomogeneous light field, that of the single traveling wave. which is appropriate because the force is conservative. The potential arises from the 2.3 shift of the atomic energy levels in the Density Matrix Description of Optical light field, appropriately called the ‘‘light Forces shift”, and is found by direct solution of the Schrodinger¨ equation for a two-level atom 2.3.1 Introduction in a monochromatic plane wave. After Use of the density matrix ρ for pure states making both the dipole and rotating wave provides an alternative description to the approximations, the Hamiltonian can be more familiar one that uses wave functions written as and operators but adds nothing new. Its equation of motion is i¯h(dρ/dt) = [H,ρ], h − δ H = ¯ 2 ( ) and can be derived directly from the ∗ 4 2 0 Schrodinger¨ equation. Moreover, it is a √ straightforward exercise to show that the where the Rabi frequency is | |=γ s /2 0 expectation value of any operator A that for a single traveling laser beam. Solution represents an observable is A=tr (ρA). of Eq. (4) for its eigenvalues provides Application of the Ehrenfest theorem the dressed state energies that are light- gives the expectation value of the force as shifted by F=−tr (ρ∇H). Beginning with the two- level atom Hamiltonian of Eq. (4), we find | |2 + δ2 − δ ω = .() the force in 1-D to be ls 5 2 ∗ ∂ ∗ ∂ F=¯h ρ + ρeg .(7) For sufficiently large detuning |δ|| |, ∂z eg ∂z approximation of Eq. (5) leads to ω ≈ ls Thus, F depends only on the off-diagonal | |2/4δ = γ 2s /8δ. 0 elements ρ = ρ∗ , terms that are called In a standing wave in 1-D with |δ|| |, eg ge ω the optical coherences. the light shift ls varies sinusoidally from node to antinode. When δ is sufficiently large, the spontaneous emission rate 2.3.2 Open Systems and the Dissipative may be negligible compared with that of Force ω The real value of the density matrix stimulated emission, so that ¯h ls may be treated as a potential U.Theresulting formalism for atom-light interactions is dipole force is its ability to deal with open systems. By not including the fluorescent light that is γ 2 lost from an atom-laser system undergoing =−∇ =−¯h ∇ ,() F U I 6 cooling, a serious omission is being made 8δIs in the discussion above. That is, the closed where I is the total intensity distribution system of atom plus laser light that can be of the standing-wave light field of period described by Schrodinger¨ wave functions λ/2. For such a standing wave, the optical and is thus in a pure state, undergoes electric field (and the Rabi frequency) at the evolution to a ‘‘mixed’’ state by virtue of antinodesisdoublethatofeachtraveling the spontaneous emission. This omission wave that composes it, and so the total canberectifiedbysimpleadhocadditions Laser Cooling and Trapping of Neutral Atoms 981 to the equation of motion, and the result is γρee. These equations have to be solved in called the optical Bloch equations (OBE). order to evaluate the optical force on the These are written explicitly as atoms.
dρgg i ∗ =+γρee + ( ρ˜eg − ρ˜ge) 2.3.3 Solution of the OBEs in Steady State dt 2 In most cases, the laser light is applied dρee i ∗ =−γρee + ( ρ˜ge − ρ˜eg ) for a period long compared to the typical dt 2 evolution times of atom-light interaction, dρ˜ge γ i ∗ that is, the lifetime of the excited state τ = =− + iδ ρ˜ge + (ρee − ρgg ) dt 2 2 1/γ . Thus, only the steady state solution of theOBEshavetobeconsidered,andthese dρ˜eg γ i =− − iδ ρ˜eg + (ρgg − ρee), are found by setting the time derivatives in dt 2 2 ρ (8) Eq. (8) to zero. Then the probability ee to −iδt where ρ˜eg ≡ ρeg e for the coherences. be in the excited state is found to be In these equations, the terms propor- γ / / ρ = p = s0 2 = s 2 , tional to the spontaneous decay rate γ have ee 2 γ 1 + s0 + (2δ/γ ) 1 + s been put in ‘‘by hand”, that is, they have (9) 2 been introduced into the OBEs to account where s ≡ s0/[1 + (2δ/γ ) ]istheoff- for the effects of spontaneous emission. resonance saturation parameter. The The spontaneous emission is irreversible excited-state population ρee increases lin- and accounts for the dissipation of the early with the saturation parameter s for cooling process. For the ground state, the small values of s,butfors of the order of decay of the excited state leads to an in- unity, the probability starts to saturate to a crease of its population ρgg proportional to value of 1/2. The detuning dependence of γρee, whereas for the excited state, it leads γp (see Eq. 3) showing this saturation for to a decrease of ρee, also proportional to variousvaluesofs0 is depicted in Fig. 1.
0.50 = s0 100.0 = s0 10.0 0.40 = s0 1.0 )
g = s0 0.1 (
p g 0.30
0.20 Scattering rate 0.10
0.00 –10 –5 0 5 10 Detuning d (g)
Fig. 1 Excitation rate γp as a function of the detuning δ for several values of the saturation parameter s0. Note that for s0 > 1, the line profiles start to broaden substantially from power broadening 982 Laser Cooling and Trapping of Neutral Atoms
2.3.4 Radiative and Dipole Forces atomic velocities that are small compared Some insight into these forces emerges with γ/k, the motion can be treated as a by expressing the gradient of the Rabi small perturbation in the atomic evolution frequency of Eq. (7) in terms of a real that occurs on the time scale 1/γ . Then the and imaginary part so that (∂ /∂z) = first-order result is given by (qr + iqi) . Then Eq. (7) becomes d ∂ ∂ ∂ ∗ ∗ = + v = + v(qr + iqi) . F = ¯hqr( ρ + ρeg) dt ∂t ∂z ∂t eg (12) + ( ρ∗ − ∗ρ )() i¯hqi eg eg 10 For the case of atoms moving in a standing wave, this results in the same damping Thus, the first term of the force is related force as Eq. (13) below. to the dispersive part of the atom-light interaction, whereas the second term is related to the absorptive part of the atom- 3 light interaction. Laser Cooling To appreciate the utility of the separation of ∇ into real and imaginary parts, 3.1 consider the interaction of atoms with a Slowing Atomic Beams i(kz−ωt) traveling plane wave E(z) = E0(e + c.c.)/2. In this case, qr = 0andqi = k,and Among the earliest laser cooling exper- so the force is caused only by absorption. iments was the deceleration of atoms The force is given by Fsp = ¯hkγρee and is in a beam [2]. The authors exploited the the radiative force of Eqs. (2) and (3). Doppler shift to make the momentum For the case of counterpropagating plane exchange (hence the force) velocity de- waves, there is a standing wave whose pendent. It worked by directing a laser −iωt electric field is E(z) = E0 cos(kz)(e + beam opposite an atomic beam so that the c.c.).Thus,qr =−k tan(kz) and qi = 0, so atoms could absorb light, and hence mo- there is only the dispersive part of the mentum ¯hk,verymanytimesalongtheir force, given by paths through the apparatus as shown in Fig. 2 [2, 3]. Of course, excited-state atoms δ = 2¯hk s0 sin 2kz .() cannot absorb light efficiently from the Fdip 2 2 11 1 + 4s0 cos kz + (2δ/γ ) laser that excited them, so between ab- sorptions they must return to the ground This replaces Eq. (6) for the dipole force state by spontaneous decay, accompanied |δ|| | and removes the restriction , by the emission of fluorescent light. The thereby including saturation effects. Even spatial symmetry of the emitted fluores- though the average of this force over a cence results in an average of zero net wavelength vanishes, it can be used to momentum transfer from many such fluo- trap atoms in a region smaller than the rescence events. Thus, the net force on the wavelength of the light. atoms is in the direction of the laser beam, and the maximum deceleration is limited 2.3.5 Force on Moving Atoms by the spontaneous emission rate γ . In order to show how these forces can be Since the maximum deceleration amax = used to cool atoms, one has to consider ¯hkγ/2M is fixed by atomic parameters, it is the force on moving atoms. For the case of straightforward to calculate the minimum Laser Cooling and Trapping of Neutral Atoms 983
Extraction Laser 1 Profile magnets magnet Bias PMT magnet Sodium 80-MHz source AOM l/4-plate
Laser 2
C.M. 80-MHz AOM
4 cm Pump Probe beam beam
zs zc zp
40 cm 125 cm 25 cm 40 cm Fig. 2 Schematic diagram of apparatus for beam slowing. The tapered magnetic field is produced by layers of varying length on the solenoid stopping length Lmin and timetmin for Tab. 1 Parameters of interest for slowing various atoms. The stopping length L and the rms velocity of atoms v¯ = 2 kBT/M min time t are minimum values. The oven at the source temperature. The result is min temperature T that determines the peak = 2/ = / oven Lmin v 2amax and tmin v amax.In velocity is chosen to give a vapor pressure of Table 1 are some of the parameters for 1 Torr. Special cases are H at 1000 K for slowing a few atomic species of interest dissociation of H2 into atoms, and He in the from the peak of the thermal velocity metastable triplet state, for which two rows are shown: one for a 4-K source and another for the distribution. typical discharge temperature Maximizing the scattering rate γp re- quires δ =−ωD in Eq. (3). If δ is chosen Atom Toven v Lmin tmin for a particular atomic velocity in the [K] [m s−1] [m] [ms] beam, then as the atoms slow down, their changing Doppler shift will take them H 1000 5000 0.012 0.005 out of resonance. They will eventually He* 4 158 0.03 0.34 cease deceleration after their Doppler shift He* 650 2013 4.4 4.4 Li 1017 2051 1.15 1.12 has been decreased by a few times√ the γ = γ + Na 712 876 0.42 0.96 power-broadened width 1 s0, K 617 626 0.77 2.45 corresponding to v of a few times Rb 568 402 0.75 3.72 vc = γ /k. Although this v of a few Cs 544 319 0.93 5.82 ms−1 is considerably larger than the typ- ical atomic recoil velocity vr = ¯hk/M of a few cm s−1, it is still only a small frac- In order to achieve deceleration that tion of the average thermal velocity v of changes the atomic speeds by hundreds the atoms, such that significant further of m s−1, it is necessary to maintain cooling or deceleration cannot be accom- (δ + ωD) γ by compensating such large plished. changes of the Doppler shift. This can be 984 Laser Cooling and Trapping of Neutral Atoms
done by changing ωD through the angular t = 10 − 50 µs with an acoustic optical dependence of k · v,orchangingδ either modulator (AOM). A pulse of atoms in via ωl or ωa. The two most common the selected hfs passes the pump region methods for maintaining this resonance and travels to the probe beam. The time ω are sweeping the laser frequency l dependence of the fluorescence induced along with the changing ωD of the by the probe laser, tuned to excite the decelerating atoms [4–6], or by spatially selected hfs, gives the time of arrival, and varying the atomic resonance frequency this signal is readily converted to a velocity with an inhomogeneous d.c magnetic distribution. Figure 3 shows the measured field to keep the decelerating atoms velocity distribution of the atoms slowed in resonance with the fixed frequency by Laser 2. laser [2, 3, 7]. The use of a spatially varying magnetic 3.2 field to tune the atomic levels along the Optical Molasses beam path was the first method to succeed in slowing atoms [2, 3]. It works as long 3.2.1 Doppler Cooling as the Zeeman shifts of the ground and A different kind of radiative force arises excited states are different so that the in low intensity, counterpropagating light resonant frequency is shifted. The field beams that form a weak standing wave. It can be tailored to provide the appropriate is straightforward to calculate the radiative Doppler shift along the moving atom’s force on atoms moving in such a standing path. A solenoid that can produce such wave using Eq. (3). In the low intensity a spatially varying field has layers of case where stimulated emission is not decreasing lengths. The technical problem important, the forces from the two light of extracting the beam of slow atoms from beams are simply added to give FOM = the end of the solenoid can be simplified by
reversing the field gradient and choosing a 6 transition whose frequency decreases with Instrumental resolution increasing field [9]. For alkali atoms such as sodium, a 4 time-of-flight (TOF) method can be used –1 to measure the velocity distribution of 2.97 m s 2 (FWHM) atoms in the beam [8]. It employs two Atom flux (arb. units) additional beams labeled pump and probe from Laser 1 as shown in Fig. 2. Because 0 ◦ these beams cross the atomic beam at 90 , ω =− · = 125 130 135 140 145 150 155 D k v 0, and they excite atoms at Velocity (m s–1) all velocities. The pump beam is tuned Fig. 3 The velocity distribution measured with to excite and empty a selected ground the TOF method. The experimental width of 1 (γ / ) hyperfine state (hfs), and it transfers approximately 6 k is shown by the dashed more than 98% of the population as the vertical lines between the arrows. The Gaussian atomspassthroughits0.5mmwidth. fit through the data yields an FWHM (full width at half maximum) of 2.97 m s−1 (figure taken To measure the velocity distribution of from Molenaar, P. A., vander Straten, P., atoms in the selected hfs, this pump Heideman,H.G.M.,Metcalf,H.(1997), Phys. laser beam is interrupted for a period of Rev. A 55, 605–614) Laser Cooling and Trapping of Neutral Atoms 985 F+ + F−,whereF± are found from Eqs. (2) laser cooling and related aspects of optical and (3). Then the sum of the two forces is control of atomic motion, the forces arise because of the exchange of momentum 2δ ∼ 8¯hk s0v between the atoms and the laser field. FOM = ≡−βv, γ(1 + s + (2δ/γ )2)2 These necessarily discrete steps of size ¯hk 0 ( ) 13 constitute a heating mechanism that must ( /γ )4 where terms of order kv and higher be considered. have been neglected. The force is pro- Since the atomic momentum changes portional to velocity for small enough by ¯hk, their kinetic energy changes on velocities, resulting in viscous damping an average by at least the recoil energy for δ<0 [10, 11] that gives this technique 2 2 Er = ¯h k /2M = ¯hωr. This means that the the name ‘‘optical molasses’’ (OM). average frequency of each absorption is These forces are plotted in Fig. 4. For ω = ω + ω at least abs a r. Similarly, the en- δ<0, this force opposes the velocity ergy ¯hωa available from each spontaneous and therefore viscously damps the atomic decay must be shared between the out- motion.√ The force FOM has maxima near going light and the kinetic energy of ≈±γ + / v s0 1 2k and decreases rapidly the atom recoiling with momentum ¯hk. for larger velocities. Thus, the average frequency of each emis- sion is ωemit = ωa − ωr. Therefore, the 3.2.2 Doppler Cooling Limit light field loses an average energy of If there were no other influence on the ¯h(ωabs − ωemit) = 2¯hωr for each scatter- atomic motion, all atoms would quickly ingevent.Thislossoccursatarateof2γp decelerate to v = 0andthesamplewould (two beams), and the energy is converted reach T = 0, a clearly unphysical result. In to atomic kinetic energy because the atoms
0.4
0.2 ) g
hk 0.0 Force (
– 0.2
– 0.4 –4 –2 0 2 4
Velocity (g /k) Fig. 4 Velocity dependence of the optical damping forces for 1-D optical molasses. The two dotted traces show the force from each beam, and the solid curve is their sum. The straight line shows how this force mimics a pure damping force over a restricted velocity range. These are calculated for s0 = 2andδ =−γ ,so there is some power broadening evident 986 Laser Cooling and Trapping of Neutral Atoms
recoil from each event. The atomic sample and step frequency 2γp, where the factor is thereby heated because these recoils are of 2 arises because of the two beams. The in random directions. random walk results in an evolution of The competition between this heating the momentum distribution as described and the damping force of Eq. (13) results by the Fokker–Planck equation, and can in a nonzero kinetic energy in steady state, be used for a more formal treatment of where the rates of heating and cooling are laser cooling. It results in diffusion in equal. Equating the cooling rate, FOM · v,to momentum space with diffusion coeffi- 2 2 the heating rate, 4¯hωrγp, we find the steady cient D0 ≡ 2(p) /t = 4γp(¯hk) .Then state kinetic energy to be (¯hγ/8)(2|δ|/γ + the steady state temperature is given by γ/2|δ|). This result is dependent on kBT = D0/β.Thisturnsouttobe¯hγ/2as |δ|, and has a minimum at 2|δ|/γ = 1, above for the case s0 1whenδ =−γ/2. whence δ =−γ/2. The temperature found This remarkable result predicts that the from the kinetic energy is then TD = final temperature of atoms in OM is inde- ¯hγ/2kB,whereTD is called the Doppler pendent of the optical wavelength, atomic temperature or the Doppler cooling limit. mass, and laser intensity (as long as it is For ordinary atomic transitions, TD is not too large). typically below 1 mK. Another instructive way to determine 3.2.3 Atomic Beam TD is to note that the average momentum Collimation – One-dimensional Optical transfer of many spontaneous emissions Molasses – Beam Brightening is zero, but the rms scatter of these When an atomic beam crosses a 1-D OM about zero is finite. One can imagine as shown in Fig. 5, the transverse motion these decays as causing a random walk of the atoms is quickly damped while in momentum space, similar to Brownian the longitudinal component is essentially motion in real space, with step size ¯hk unchanged. This transverse cooling of
Optical Optical molasses molasses
Oven
Very bright atomic beam
Laser or magnetic lens
Fig. 5 Scheme for optical brightening of an atomic beam. First, the transverse velocity components of the atoms are damped out by an optical molasses, then the atoms are focused to a spot, and finally the atoms are recollimated in a second optical molasses (figure taken from Sheehy, B., Shang, S. Q., van der Straten, P., Metcalf, H. (1990), Chem. Phys. 145, 317–325) Laser Cooling and Trapping of Neutral Atoms 987 an atomic beam is an example of a wander away from the center experience method that can actually increase its no force directing them back. They are brightness (atoms/s-sr-cm2) because such allowed to diffuse freely and even escape, active collimation uses dissipative forces as long as there is enough time for to compress the phase space volume their very slow diffusive movement to occupied by the atoms. By contrast, the allowthemtoreachtheedgeofthe usual focusing or collimation techniques region of intersection of the laser beams. for light beams and most particle beams Since the atomic velocities are randomized is restricted to selection by apertures or during the damping time M/β = 2/ωr, conservative forces that preserve the phase atoms execute a random walk in position space density of atoms in the beam. space√ with a step size of 2vD/ωr = This velocity compression at low in- λ/(π 2ε) =∼ few µm. To diffuse a distance tensity in one dimension can be easily of 1 cm requires about 107 steps or about estimated√ for two-level√ atoms to be about 30 s [13, 14]. vc/vD = γ/ωr ≡ 1/ε.HerevD is the In 1985, the group at Bell Labs was the velocity associated with the Doppler limit first to observe 3-D OM [11]. Preliminary for laser cooling discussed above: vD = measurements of the average kinetic en- −1 ¯hγ/2M.ForRb,vD = 12cms , vc = ergy of the atoms were done by blinking −1 γ/k 4.6ms , ωr 2π × 3.8kHz,and off the laser beams for a fixed interval. 1/ε 1600. (The parameter ε character- Comparison of the brightness of the fluo- izes optical forces on atoms.) Includ- rescence before and after the turnoff was ing two transverse directions along with used to calculate the fraction of atoms that the longitudinal slowing and cooling left the region while it was in the dark. discussed above, the decrease in three- The dependence of this fraction on the dimensional 3-D phase space volume for duration of the dark interval was used to es- laser cooling of an Rb atomic beam from timate the velocity distribution and hence the momentum contribution alone can the temperature. This method, which is exceed 106. Clearly optical techniques usually referred to as release and recapture, can create atomic beams enormously is specifically designed to measure the tem- more times intense than ordinary thermal perature of the atoms, since the usual way beams and also many orders of magnitude of measuring temperatures cannot be ap- brighter. plied to an atomic cloud of a few million atoms. The result was consistent with TD 3.2.4 Experiments in Three-dimensional as calculated from the Doppler theory, as Optical Molasses described in Sect. 3.2.2. By using three intersecting orthogonal Later a more sensitive ballistic technique pairs of oppositely directed beams, the was devised at NIST that showed the movement of atoms in the intersection astounding result that the temperature of region can be severely restricted in all 3-D, the atoms in OM was very much lower than and many atoms can thereby be collected TD [15]. These experiments also found that and cooled in a small volume. OM was less sensitive to perturbations Even though atoms can be collected and more tolerant of alignment errors and cooled in the intersection region, it than was predicted by Doppler theory. is important to stress that this is not a For example, if the intensities of the two trap (see Sect. 4 below), that is, atoms that counterpropagating laser beams forming 988 Laser Cooling and Trapping of Neutral Atoms
an OM were unequal, then the force steady state have ground-state orientations on the atoms at rest would not vanish, caused by optical pumping processes that but the force on the atoms with some distribute the populations over the differ- nonzero drift velocity would vanish. This ent ground-state sublevels. In the presence drift velocity can be easily calculated by of polarization gradients, these orienta- using unequal intensities s0+ and s0−, tions reflect the local light field. In the to derive an analog of Eq. (13). Thus, low-light-intensity regime, the orientation atoms would drift out of an OM, and the of stationary atoms is completely deter- calculated rate would be much faster than mined by the ground-state distribution; observed by deliberately unbalancing the the optical coherences and the excited- beams in the experiments [16]. state population follow the ground-state distribution adiabatically. 3.3 For atoms moving in a light field Cooling Below the Doppler Limit that varies in space, optical pumping acts to adjust the atomic orientation 3.3.1 Introduction to the changing conditions of the light Itwasanenormoussurprisetoobserve field. In a weak pumping process, the that the ballistically measured temperature orientation of moving atoms always lags of the Na atoms was as much as 10 behind the orientation that would exist for times lower than TD = 240 µK [15], the stationary atoms. It is this phenomenon of temperature minimum calculated from nonadiabatic following that is the essential theory. This breaching of the Doppler limit featureofthenewcoolingprocess. forced the development of an entirely new Production of spatially dependent optical picture of OM that accounts for the fact pumping processes can be achieved in that in 3-D, a two-level picture of atomic several different ways. As an example, structure is inadequate. The multilevel consider two counterpropagating laser structure of atomic states, and optical beams that have orthogonal polarizations, pumping among these sublevels, must be as discussed below. The superposition of considered in the description of 3-D OM. the two beams results in a light field In response to these surprising mea- having a polarization that varies on the surements of temperatures below TD,two wavelength scale along the direction of the groups developed a model of laser cooling laser beams. Laser cooling by such a light that could explain the lower tempera- field is called polarization gradient cooling. tures [17, 18]. The key feature of this model In a 3-D OM, the transverse wave character that distinguishes it from the earlier pic- of light requires that the light field always ture is the inclusion of the multiplicity of has polarization gradients. sublevels that make up an atomic state (e.g., Zeeman and hfs). The dynamics 3.3.2 Linear ⊥ Linear Polarization of optically pumping the moving atoms Gradient Cooling among these sublevels provides the new One of the most instructive models for mechanism for producing ultralow tem- discussion of sub-Doppler laser cooling peratures [19]. was introduced in Ref. [17] and very well The dominant feature of these models described in Ref. [19]. If the polarizations is the nonadiabatic response of moving of two counterpropagating laser beams atomstothelightfield.Atomsatrestina are identical, the two beams interfere Laser Cooling and Trapping of Neutral Atoms 989
and produce a standing wave. When px the two beams have orthogonal linear − ω s polarizations (same frequency l)with → k their Eˆ vectors perpendicular (e.g., xˆ and l/2 2 ˆ ⊥ → y), the configuration is called lin lin k1 0 + or lin-perp-lin. Then the total field is the s p sum of the two counterpropagating beams y given by 0 l/4 l/2 ⊥ E = E xˆ cos(ω t − kz) + E yˆ cos(ω t + kz) Fig. 6 Polarization gradient field for the lin lin 0 l 0 l configuration = E (ˆ + ˆ) ω 0[ x y cos lt cos kz ˆ ˆ + (x − y) sin ωlt sin kz].(14) Since the coupling of the different states of multilevel atoms to the light field de- At the origin, where z = 0, this becomes pends on its polarization, atoms moving in a polarization gradient will be coupled E = E (ˆ + ˆ) ω ,() 0 x y cos lt 15 differently at different positions, and this will have important consequences for laser which corresponds to linearly polarized cooling. For the Jg = 1/2 → Je = 3/2tran- +π/ light at an angle 4tothe√ x-axis. The sition (one of the simplest transitions that E amplitude of this field is 2 0. Similarly, show sub-Doppler cooling [20]), the opti- = λ/ = π/ + for z 4, where kz 2, the field is cal pumping process in purely σ light −π/ also linearly polarized but at an angle 4 drives the ground-state population to the to the x-axis. Mg =+1/2 sublevel. This optical pump- = λ/ Between these two points, at z 8, ing occurs because absorption always = π/ where kz 4, the total field is produces M =+1 transitions, whereas π the subsequent spontaneous emission pro- E = E ˆ ω + 0 x sin lt duces M =±1, 0. Thus, the average 4 − π M ≥ 0 for each scattering event. For σ - − yˆ cos ω t + .(16) l 4 light, the population is pumped toward the Mg =−1/2 sublevel. Thus, atoms trav- Since the xˆ and yˆ components have eling through only a half wavelength in sine and cosine temporal dependence, the light field, need to readjust their pop- they are π/2 out of phase, and so ulation completely from Mg =+1/2to Eq. (16) represents circularly polarized Mg =−1/2 and back again. light rotating about the z-axis in the The interaction between nearly resonant negative sense. Similarly, at z = 3λ/8 light and atoms not only drives transitions where kz = 3π/4, the polarization is between atomic energy levels but also circular but in the positive sense. Thus, shifts their energies. This light shift of in this lin ⊥ lin scheme the polarization the atomic energy levels, discussed in cycles from linear to circular to orthogonal Sect. 2.2, plays a crucial role in this scheme linear to opposite circular in the space of of sub-Doppler cooling, and the changing only half a wavelength of light, as shown in polarization has a strong influence on the Fig. 6. It truly has a very strong polarization light shifts. In the low-intensity limit of gradient. two laser beams, each of intensity s0Is,the 990 Laser Cooling and Trapping of Neutral Atoms
light shifts Eg of the ground magnetic substates are given by a slight variation of the approximation to Eq. (5) that accounts for the multilevel structure of the atoms. 0 We write M = +1/2 2 γ 2 Energy ¯hs0Cge Eg = ,(17) 8δ
M = −1/2 where Cge is the Clebsch–Gordan coeffi- cient that describes the coupling between 0 λ/4 λ/2 3λ/4 the particular levels of the atom and the Position (z) light field. Fig. 7 The spatial dependence of the light shifts of the ground-state sublevels of the In the present case of orthogonal linear = / ⇔ / ⊥ = / → / J 1 2 3 2 transition for the case of lin lin polarizations and J 1 2 3 2, the light polarization configuration. The arrows show the shift for the magnetic substate Mg = 1/2 path followed by atoms being cooled in this is three times larger than that of the arrangement. Atoms starting at z = 0inthe M =+1/2 sublevel must climb the potential Mg =−1/2 substate when the light field g + hill as they approach the z = λ/4pointwherethe is completely σ . On the other hand, − light becomes σ polarized, and they are when an atom moves to a place where =− / − optically pumped to the Mg 1 2 sublevel. the light field is σ ,theshiftofMg = Then they must begin climbing another hill −1/2 is three times larger. So, in this toward the z = λ/2pointwherethelightisσ + case, the optical pumping discussed above polarized and they are optically pumped back to =+ / causes a larger population to be there the Mg 1 2 sublevel. The process repeats until the atomic kinetic energy is too small to in the state with the larger light shift. climb the next hill. Each optical pumping event This is generally true for any transition results in the absorption of light at a frequency Jg → Je = Jg + 1. A schematic diagram lower than emission, thus dissipating energy to showing the populations and light shifts the radiation field for this particular case of negative detuning is illustrated in Fig. 7. a position where the light field is σ −- polarized, and are optically pumped to 3.3.3 Origin of the Damping Force Mg =−1/2, which is now lower than the To discuss the origin of the cooling process Mg = 1/2 state. Again, the moving atoms in this polarization gradient scheme, are at the bottom of a hill and start to consider atoms with a velocity v at a climb. In climbing the hills, the kinetic position where the light is σ +-polarized, as energy is converted to potential energy, shownatthelowerleftofFig.7.Thelight and in the optical pumping process, the optically pumps such atoms to the strongly potential energy is radiated away because negative light-shifted Mg =+1/2 state. In the spontaneous emission is at a higher moving through the light field, atoms must frequency than the absorption (see Fig. 7). increase their potential energy (climb a Thus, atoms seem to be always climbing hill) because the polarization of the light is hills and losing energy in the process. changing and the state Mg = 1/2 becomes This process brings to mind a Greek less strongly coupled to the light field. After myth, and is thus called ‘‘Sisyphus laser traveling a distance λ/4, atoms arrive at cooling’’. Laser Cooling and Trapping of Neutral Atoms 991
≡ / = 2/ The cooling process described above is afewtimesEr kBTr 2 Mvr 2isintu- effective over a limited range of atomic itively comforting for two reasons. First, velocities. The force is maximum for one might expect that the last spontaneous atoms that undergo one optical pumping emission in a cooling process would leave process while traveling over a distance λ/4. atoms with a residual momentum of the Slower atoms will not reach the hilltop order of ¯hk, since there is no control over before the pumping process occurs and its direction. Thus, the randomness asso- faster atoms will travel a longer distance ciated with this would put a lower limit before being pumped toward the other of vmin ∼ vr on such cooling. Second, the sublevel, so E/z is smaller. In both polarization gradient cooling mechanism cases, the energy loss is smaller and described above requires that atoms be therefore the cooling process less efficient. localizable within the scale of ∼ λ/2π in Nevertheless, the damping constant β order to be subject to only a single polariza- for this process is much larger than for tion in the spatially inhomogeneous light Doppler cooling, and therefore the final field. The uncertainty principle then re- steady state temperature is lower [17, 19]. quires that these atoms have a momentum In the experiments of Ref. [21], the spread of at least ¯hk. temperature was measured in a 3-D The recoil limit discussed here has molasses under various configurations been surpassed by evaporative cooling of the polarization. Temperatures were of trapped atoms [22] and two different measured by a ballistic technique, in which optical cooling methods, neither of which the flight time of the released atoms was can be described in simple notions. One of measured as they fell through a probe these uses optical pumping into a velocity- a few centimeters below the molasses selective dark state [23]. The other one region. The lowest temperature obtained uses carefully chosen, counterpropagating, was 3 µK, which is a factor 40 below the laser pulses to induce velocity-selective Doppler temperature and a factor 15 above Raman transitions, and is called Raman the recoil temperature of Cs. cooling [24].
3.3.4 The Limits of Sisyphus Laser Cooling The extension of the kind of thinking 4 about cooling limits in the case of Doppler Traps for Neutral Atoms cooling to the case of the sub-Doppler processes must be done with some care, In order to confine any object, it is because a naive application of similar necessary to exchange kinetic for poten- ideas would lead to an arbitrarily low tial energy in the trapping field, and in final temperature. In the derivation in neutral atom traps, the potential energy Sect. 3.2.2, it is explicitly assumed that must be stored as internal atomic energy. each scattering event changes the atomic Thus, practical traps for ground-state neu- momentum p by an amount that is a small tral atoms are necessarily very shallow fraction of p and this clearly fails when compared with thermal energy because the velocity is reduced to the region of the energy-level shifts that result from vr ≡ ¯hk/M. convenient size fields are typically con- This limitation of the minimum steady siderably smaller than kBT for T = 1K. state value of the average kinetic energy to Neutral atom trapping therefore depends 992 Laser Cooling and Trapping of Neutral Atoms
on substantial cooling of a thermal atomic where w0 is the beam waist size. Such sample, and is often connected with the a trap has a well-studied and important cooling process. In most practical cases, macroscopic classical analog in a phe- atoms are loaded from magneto-optical nomenon called optical tweezers [27–29]. traps (MOTs) in which they have been With the laser light tuned below reso- efficiently accumulated and cooled to mK nance (δ<0), the ground-state light shift temperatures (see Sect. 4.3), or from op- is negative everywhere, but largest at tical molasses, in which they have been the center of the Gaussian beam waist. optically cooled to µK temperatures (see Ground-state atoms, therefore, experience Sect. 3.2). a force attracting them toward this cen- The small depth of typical neutral ter, given by the gradient of the light atom traps dictates stringent vacuum shift, which is found from Eq. (5), and for requirements because an atom cannot δ/γ s0 is given by Eq. (6). For the Gaus- remain trapped after a collision with a sian beam, this transverse force at the waist thermal energy background gas molecule. is harmonic for small r and is given by Since these atoms are vulnerable targets γ 2 ¯h I0 r −2r2/w2 for thermal energy background gas, the F e 0 .(19) 4δ I 2 mean free time between collisions must s w0 exceed the desired trapping time. The In the longitudinal direction, there is cross section for destructive collisions is also an attractive force but it is more quite large because even a gentle collision complicated and depends on the details (i.e., large impact parameter) can impart of the focusing. Thus, this trap produces enough energy to eject an atom from a an attractive force on the atoms in three trap. At pressure P sufficiently low to be dimensions. ∼ of practical interest, the trapping time is Although it may appear that the trap does −8 (10 /P)s,whereP is in Torr. not confine atoms longitudinally because of the radiation pressure along the laser 4.1 beam direction, careful choice of the laser Dipole Force Optical Traps parameters can indeed produce trapping in 3-D. This can be accomplished because 4.1.1 Single-beam Optical Traps for the radiation pressure force decreases as Two-level Atoms 1/δ2 (see Eqs. 2 and 3), but by contrast, The simplest imaginable optical trap con- the light shift and hence the dipole force sists of a single, strongly focused Gaussian decreases only as 1/δ for δ (see laser beam (see Fig. 8) [25, 26] whose in- Eq. 5). If |δ| is chosen to be sufficiently tensity at the focus varies transversely with large, atoms spend very little time in r as − 2/ 2 the untrapped (actually repelled), excited ( ) = 2r w0 ,() I r I0e 18 state because its population is proportional to 1/δ2. Thus, a sufficiently large value of |δ| produces longitudinal confinement Laser and also maintains the atomic population primarily in the trapped ground state. 2w0 The first optical trap was demonstrated Fig. 8 A single focused laser beam produces in Na with light detuned below the D- the simplest type of optical trap lines [26]. With 220 mW of dye laser light Laser Cooling and Trapping of Neutral Atoms 993 tuned about 650 GHz below the atomic the optical frequency such as the Nd : YAG transition and focused to an ∼10 µm laser trap [31] or the CO2 laser trap [32]. waist, the trap depth was about 15¯hγ More important in some cases is that the corresponding to 7 mK. trap relies on Stark shifting of the atomic Single-beam dipole force traps can energy levels by an amount equal to the be made with the light detuned by a trap depth, and this severely compromises significant fraction of its frequency from the capabilities for precision spectroscopy the atomic transition. Such a far-off- in a trap [33]. resonance trap (FORT) has been developed Attracting atoms to the region of low- for Rb atoms using light detuned by nearly est intensity would ameliorate both these 10% to the red of the D1 transition at concerns, but such a trap requires positive λ = 795 nm [30]. Between 0.5 and 1 W of detuning (blue), and an optical configu- power was focused to a spot about 10 µmin ration having a dark central region. One size, resulting in a trap 6 mK deep where of the first experimental efforts at a blue the light-scattering rate was only a few detuned trap used the repulsive dipole hundreds per second. The trap lifetime force to support Na atoms that were oth- was more than half a second. erwise confined by gravity in an optical There is a qualitative difference when the ‘‘cup’’ [34]. Two rather flat, parallel beams trapping light is detuned by a large frac- detuned by 25% of the atomic resonance tion of the optical frequency. In one such frequency were directed horizontally and case, Nd : YAG light at λ = 1064 nm was oriented to form a V-shaped trough. Their used to trap Na whose nearest transition Gaussian beam waists formed a region is at λ = 596 nm [31]. In a more extreme 1 mm long where the potential was deep- case, a trap using λ = 10.6 µmlightfrom est, and hence provided confinement along aCO2 laser has been used to trap Cs their propagation direction as shown in whose optical transition is at a frequency Fig. 9. The beams were the λ = 514 nm ∼12 times higher (λ = 852 nm) [32]. For and λ = 488 nm from an argon laser, and such large values of |δ|,calculationsof the trapping force cannot exploit the rotat- ing wave approximations as was done for 0.6 Eqs. (4) and (5), and the atomic behavior is 0.4
similar to that in a DC field. It is impor- Intensity 0.2 tant to remember that for an electrostatic 0 trap Earnshaw’s theorem precludes a field 1000 maximum, but that in this case there is in- 0 deed a local 3-D intensity maximum of the y focused light because it is not a static field. – 1000 0 50 – 50 x 4.1.2 Blue-detuned Optical Traps Fig. 9 The light intensity experienced by an One of the principal disadvantages of the atom located in a plane 30 µmabovethebeam optical traps discussed above is that the waists of two quasi-focused sheets of light negative detuning attracts atoms to the traveling parallel and arranged to form a V-shaped trough. The x and y dimensions are in region of highest light intensity. This may µm (figure taken from Davidson, N., Lee, H. J., result in significant spontaneous emission Adams, C. S., Kasevich, M., Chu, S. (1995), Phys. unless the detuning is a large fraction of Rev. Lett. 74, 1311–1314) 994 Laser Cooling and Trapping of Neutral Atoms
the choice of the two frequencies was not trap cooled Cs atoms to 3 µKatadensity simply to exploit the full power of the mul- of 3 × 1010/cm3 in a sample whose 1/e tiline Ar laser, but also to avoid the spatial height in the gravitational field was only interference that would result from the use 19 µm. Simple ballistics gives a frequency of a single frequency. of 450 bounces per second, and the 6-s Obviously, a hollow laser beam would lifetime (limited only by background gas also satisfy the requirement for a blue- collisions) corresponds to several thousand detuned trap, but conventional textbook bounces. However, at such low energies, wisdom shows that such a beam is not the deBroglie wavelength of the atoms is an eigenmode of a laser resonator [35]. 1/4 µm, and the atomic motion is no Some lasers can make hollow beams, longer accurately described classically, but but these are illusions because they requires deBroglie wave methods. consist of rapid oscillations between the TEM01 and TEM10 modes of the cavity. 4.2 Nevertheless, Maxwell’s equations permit Magnetic Traps the propagation of such beams, and in the recent past there have been 4.2.1 Introduction studies of the LaGuerre–Gaussian modes Magnetic trapping of neutral atoms is that constitute them [36–38]. The several well suited for use in very many areas, ways of generating such hollow beams including high-resolution spectroscopy, have been tried by many experimental collision studies, Bose–Einstein conden- groups and include phase and amplitude sation (BEC), and atom optics. Although holograms, hollow waveguides, axicons or ion trapping, laser cooling of trapped ions, related cylindrical prisms, stressing fibers, and trapped ion spectroscopy were known and simply mixing the TEM01 and TEM10 for many years [42], it was only in 1985 modes with appropriate cylindrical lenses. that neutral atoms were first trapped [43]. An interesting experiment has been Such experiments offer the capability of performed using the ideas of Sisyphus the spectroscopic ideal of an isolated atom cooling (see Sect. 3.3) with evanescent at rest, in the dark, available for interaction waves combined with a hollow beam with electromagnetic field probes. formed with an axicon [39]. In the pre- Because trapping requires the exchange viously reported experiments with atoms of kinetic energy for potential energy, the bouncing under gravity from an evanes- atomic energy levels will necessarily shift cent wave field [40, 41], they were usually as the atoms move in the trap. These lost to horizontal motion for several rea- shifts can severely affect the precision of sons, including slight tilting of the surface, spectroscopic measurements. Since one surface roughness, horizontal motion as- of the potential applications of trapped sociated with their residual motion, and atoms is in high-resolution spectroscopy, horizontal ejection by the Gaussian pro- such inevitable shifts must be carefully file of the evanescent wave laser beam. considered. The authors of Ref. [39] simply confined their atoms in the horizontal direction by 4.2.2 Magnetic Confinement surrounding them with a wall of blue- The Stern–Gerlach experiment in 1924 detuned light in the form of a vertical first demonstrated the mechanical action hollow beam. Their gravito-optical surface of inhomogeneous magnetic fields on Laser Cooling and Trapping of Neutral Atoms 995 neutral atoms having magnetic moments, directions [44]. Its experimental simplicity and the basic phenomenon was subse- makes it most attractive, both because of quently developed and refined. An atom ease of construction and of optical access with a magnetic moment µ can be con- to the interior. Such a trap was used in the fined by an inhomogeneous magnetic field first neutral atom trapping experiments at because of an interaction between the mo- NIST on laser-cooled Na atoms for times ment and the field. This produces a force exceeding 1 s, and that time was limited given by only by background gas pressure [43]. F = ∇ (µ · B)(20) The magnitude of the field is zero at the center of this trap, and increases in all since E =−µ · B. Several different mag- directions as netic traps with varying geometries that exploit the force of Eq. (20) have been B =∇B ρ2 + 4z2,(21) studied in some detail in the literature. where ρ2 ≡ x2 + y2 and the field gradient The general features of the magnetic fields is constant (see Ref. [44]). The field gradi- of a large class of possible traps has been entisconstantalonganylinethroughthe presented [44]. origin, but has different values in differ- W. Paul originally suggested a quadru- ent polar directions because of the ‘4’ in pole trap composed of two identical coils Eq. (21). Therefore, the force of Eq. (20) carrying opposite currents (see Fig. 10). that confines the atoms in the trap is This trap clearly has a single center in neither harmonic nor central, and orbital which the field is zero, and is the simplest angular momentum is not conserved. of all possible magnetic traps. When the The requisite field for the quadrupole coils are separated by 1.25 times their trapcanalsobeprovidedintwodimen- radius, such a trap has equal depth in the sions by four straight currents as indicated radial (x-y plane) and longitudinal (z-axis) in Fig. 11. The field is translationally in- variant along the direction parallel to the currents, so a trap cannot be made this way without additional fields. These are provided by end coils that close the trap, as I shown. Although there are very many different kinds of magnetic traps for neutral parti- cles, this particular one has played a special
− + I I I + −
Fig. 11 The Ioffe trap has four straight current Fig. 10 Schematic diagram of the coil elements that form a linear quadrupole field. The configuration used in the quadrupole trap and axial confinement is accomplished with end coils the resultant magnetic field lines. Because the as shown. These fields can be achieved with currentsinthetwocoilsareinopposite many different current configurations as long as directions, there is a |B|=0 point at the center the geometry is preserved 996 Laser Cooling and Trapping of Neutral Atoms
role. There are certain conditions required hundred Gauss depth, a ∼ 250 m s−2,and for trapped atoms not to be ejected in a the fastest trappable atoms in circular −1 region of zero field such as occurs at the orbits have vmax ∼ 1ms so ωT /2π ∼ center of a quadrupole trap (see Sects. 4.2.3 20 Hz. Because of the anharmonicity and 4.2.4). This problem is not easily cured; of the potential, the orbital frequencies the Ioffe trap has been used in many of the depend on the orbit size, but in general, BEC experiments because it has |B| = 0 atoms in lower-energy orbits have higher everywhere. frequencies. For the quadrupole trap to work, the 4.2.3 Classical Motion of Atoms in a atomic magnetic moments must be ori- Quadrupole Trap ented with µ · B < 0 so that they are Because of the dependence of the trapping repelled from regions of strong fields. force on the angle between the field This orientation must be preserved while and the atomic moment (see Eq. 20), the the atoms move around in the trap even orientation of the magnetic moment with though the trap fields change directions respect to the field must be preserved in a very complicated way. The condi- as the atoms move about in the trap. tion for adiabatic motion can be written Otherwise, the atoms may be ejected as ωZ |dB/dt|/B,whereωZ = µB/¯h is instead of being confined by the fields the Larmor precession rate in the field. of the trap. This requires velocities low Since v/ρ1 = v∇B/B =|dB/dt|/B for a enough to ensure that the interaction uniform field gradient, the adiabaticity between the atomic moment µ and the condition is field B is adiabatic especially when the atom’s path passes through a region ωZ ωT .(22) where the field magnitude is small and therefore the energy separation between More general orbits must satisfy a simi- the trapping and nontrapping states is lar condition. For the two-coil quadrupole small. This is especially critical at the trap, the adiabaticity condition√ can be eas- low temperatures of the BEC experiments. ily calculated. Using v = ρ1a for circular Therefore energy considerations that focus orbits in the z = 0plane,theadiabaticcon- only on the trap depth are not sufficient dition for a practical trap (∇B ∼ 1T/m)re- 2 2 1/3 to determine the stability of a neutral quires ρ1 (¯h /M a) ∼ 1 µmaswell / − atom trap; orbit and/or quantum state as v (¯ha/M)1 3 ∼ 1cms 1.Notethat calculations and their consequences must violation of these conditions (i.e., v ∼ − also be considered. 1cms 1 in a trap with ∇B ∼ 1T/m)re- For the two-coil quadrupole magnetic sults in the onset of quantum dynamics for trap of Fig. 10, stable circular orbits of the motion (deBroglie wavelength ≈ orbit radius ρ1 in the z = 0 plane can be found size). µ∇ = 2/ρ classically√ by setting B Mv 1,so Since the nonadiabatic region of the trap −18 3 that v = ρ1a,wherea ≡ µ∇B/M is is so small (less than 10 m compared the centripetal acceleration supplied by with typical sizes of ∼2 cm, corresponding − the field gradient (cylindrical coordinates to 10 5 m3), nearly all the orbits of most are appropriate). Such orbits√ have an atoms are restricted to regions where angular frequency of ωT = a/ρ1.For they are adiabatic. Therefore, most of traps of a few centimeter size and a few such laser-cooled atoms stay trapped for Laser Cooling and Trapping of Neutral Atoms 997 many thousands of orbits corresponding separation of the rapid precession from to several minutes. At laboratory vacuum the slower orbital motion is reminiscent chamber pressures of typically 10−10 torr, of the Born–Oppenheimer approximation the mean free time between collisions that for molecules. can eject trapped atoms is ∼2min,sothe On the other hand, the small-n states, transitions caused by nonadiabatic motion whose orbits are confined to a region near are not likely to be observable in atoms the origin where the field is small, have that are optically cooled. much larger coupling to the continuum states. Thus, they are rapidly ejected from 4.2.4 Quantum Motion in a Trap the trap. The transitions to unbound states Since laser and evaporative cooling have resulting from the coupling of the motion the capability to cool atoms to energies with the trapping fields are called Majorana where their deBroglie wavelengths are on spin flips, and effectively constitute a ‘‘hole’’ the scale of the orbit size, the motional at the bottom of the trap. The evaporative dynamics must be described in terms cooling process used to produce very cold, of quantum mechanical variables and dense samples reduces the average total suitable wave functions. Quantization of energy of the trapped atoms sufficiently the motion leads to discrete bound states that the orbits are confined to regions within the trap having µ · B < 0, and also a near the origin and so, such losses continuum of unbound states having µ · B dominate [44, 45]. with opposite sign. There have been different solutions Studying the behavior of extremely slow to this problem of Majorana losses for (cold) atoms in the two-coil quadrupole confinement of ultracold atoms for the trap begins with a heuristic quantization BEC experiments. In the JILA-experiment, of the orbital angular momentum using the hole was rotated by rotating the 2 magnetic field, and thus, the atoms do Mr ωT = n¯h forcircularorbits.Theen- ergy levels are then given by not spend sufficient time in the hole to make a spin flip. In the MIT experiment, 3 2/3 theholewaspluggedbyusingafocused En = E1n , where 2 laser beam tuned to the blue side of atomic 2 2 1/3 resonance, which expelled the atoms from E1 = (Ma ¯h ) ∼ h × 5kHz,(23) the center of the magnetic trap. In the For velocities of optically cooled atoms of Rice experiment, the atoms were trapped afewcms−1, n ∼ 10 − 100. It is readily in an Ioffe trap, which has a nonzero found that ωZ = nωT , so that the adiabatic field minimum. Most BEC experiments condition of Eq. (22) is satisfied only for arenowusingtheIoffetrapsolution. n 1. These large-n bound states have small 4.3 matrix elements coupling them to the Magneto-optical Traps unbound continuum states [45]. This can be understood classically since they spend 4.3.1 Introduction most of their time in a stronger field, and The most widely used trap for neutral thus satisfy the condition of adiabaticity atoms is a hybrid employing both optical of the orbital motion relative to the and magnetic fields to make a magneto- Larmor precession. In this case, the optical trap (MOT), first demonstrated in 998 Laser Cooling and Trapping of Neutral Atoms
1987 [46]. The operation of an MOT de- Energy = + pends on both inhomogeneous magnetic Me 1 fields and radiative selection rules to ex- ploit both optical pumping and the strong M = 0 radiative force [46, 47]. The radiative in- e d+ teraction provides cooling that helps in d d− loading the trap and enables very easy op- M = −1 eration. MOT is a very robust trap that e does not depend on precise balancing of wl the counterpropagating laser beams or on s+ beam s− beam
a very high degree of polarization. = Mg 0 The magnetic field gradients are modest z ′ Position and have the convenient feature that Fig. 12 Arrangement for an MOT in 1-D. The the configuration is the same as the horizontal dashed line represents the laser quadrupole magnetic traps discussed in frequency seen by an atom at rest in the center of Sect. 4.2.2. Appropriate fields can readily the trap. Because of the Zeeman shifts of the atomic transition frequencies in the be achieved with simple, air-cooled coils. inhomogeneous magnetic field, atoms at z = z The trap is easy to construct because it can are closer to resonance with the σ − laser beam be operated with a room-temperature cell than with the σ + beam, and are therefore driven in which alkali atoms are captured from toward the center of the trap the vapor. Furthermore, low-cost diode lasers can be used to produce the light the magnetic field, therefore, tunes the appropriate for many atoms, so the MOT M =−1 transition closer to resonance has become one of the least expensive ways and the M =+1 transition further out of to make atomic samples with temperatures resonance. If the polarization of the laser below 1 mK. beam incident from the right is chosen Trapping in an MOT works by opti- to be σ − and correspondingly σ + for the cal pumping of slowly moving atoms in other beam, then more light is scattered a linearly inhomogeneous magnetic field from the σ − beam than from the σ + beam. B = B(z) (see Eq. 21), such as that formed Thus, the atoms are driven toward the by a magnetic quadrupole field. Atomic center of the trap where the magnetic field transitions with the simple scheme of is zero. On the other side of the center of Jg = 0 → Je = 1 have three Zeeman com- the trap, the roles of the Me =±1 states are ponents in a magnetic field, excited by each reversed and now more light is scattered of three polarizations, whose frequencies from the σ + beam, again driving the atoms tune with the field (and therefore with po- toward the center. sition) as shown in Fig. 12 for 1-D. Two So far, the discussion has been limited counterpropagating laser beams of oppo- to the motion of atoms in 1-D. However, site circular polarization, each detuned the MOT scheme can easily be extended below the zero-field atomic resonance by to 3-D by using six instead of two laser δ, are incident as shown. beams. Furthermore, even though very Because of the Zeeman shift, the excited few atomic species have transitions as state Me =+1 is shifted up for B > simple as Jg = 0 → Je = 1, the scheme 0, whereas the state with Me =−1is works for any Jg → Je = Jg + 1 transition. shifted down. At position z in Fig. 12, Atoms that scatter mainly from the σ + Laser Cooling and Trapping of Neutral Atoms 999 √ laser beam will be optically pumped toward κ/M. For magnetic field gradients ∇B ≈ −1 the Mg =+Jg substate, which forms a 0.1Tm , the oscillation frequency is typ- closed system with the Me =+Je substate. ically a few kHz, and this is much smaller than the damping rate that is typically a 4.3.2 Cooling and Compressing Atoms in few hundred kHz. Thus, the motion is an MOT overdamped, with a characteristic restor- For a description of the motion of atoms ing time to the center of the trap of /ω2 ≈ in an MOT, consider the radiative force in 2 MOT MOT several milliseconds for the low intensity limit (see Eqs. 2 and 3). typical values of detuning and intensity of The total force on the atoms is given by the lasers. F = F+ + F−,whereF± can be found from Eqs. (2) and (3), and the detuning δ± for 4.3.3 Capturing Atoms in an MOT each laser beam is given by δ± = δ ∓ k · Although the approximations that lead to
v ± µ B/¯h. Here, µ ≡ (geMe − ggMg)µB Eq. (24) for force hold for slow atoms near is the effective magnetic moment for the the origin, they do not apply for the capture transition used. Note that the Doppler of fast atoms far from the origin. In the shift ωD ≡−k · v and the Zeeman shift capture process, the Doppler and Zeeman
ωZ = µ B/¯h both have opposite signs for shifts are no longer small compared to opposite beams. the detuning, so the effects of the position The situation is analogous to the velocity and velocity can no longer be disentangled. damping in an OM from the Doppler effect However, the full expression for the force as discussed in Sec. 3.2, but here the effect still applies and the trajectories of the also operates in position space, whereas atoms can be calculated by numerical for molasses it operates only in velocity integration of the equation of motion [48]. space. Since the laser light is detuned The capture velocity of an MOT is below the atomic resonance in both cases, serendipitously enhanced because atoms compression and cooling of the atoms is traveling across it experience a decreasing obtained simultaneously in an MOT. magnetic field just as in beam deceleration When both the Doppler and Zeeman described in Sect. 3.1. This enables reso- shifts are small compared to the detuning nance over an extended distance and ve- δ, the denominator of the force can be locity range because the changing Doppler expanded as for Eq. (13) and the result shift of decelerating atoms can be com- becomes pensated by the changing Zeeman shift as atoms move in the inhomogeneous mag- F =−βv − κr,(24) netic field. Of course, it will work this wayonlyifthefieldgradient∇B does not where the damping coefficient β is defined demand an acceleration larger than the in Eq. (13). The spring constant κ arises maximum acceleration a . Thus, atoms max from the similar dependence of F on the aresubjecttotheopticalforceoveradis- Doppler and Zeeman shifts, and is given tance that can be as long as the trap size, by κ = µ β∇B/¯hk and can therefore be slowed considerably. The force of Eq. (24) leads to damped The very large velocity capture range harmonic motion of the atoms, where the vcap of an MOT can be estimated by using damping rate is given by MOT = β/M Fmax = ¯hkγ/2 and choosing a maximum and the oscillation frequency ωMOT = size of a few centimeters for the beam 1000 Laser Cooling and Trapping of Neutral Atoms
diameters. Thus, the energy change can low speed through the center of the MOT be as large as a few K, corresponding until they emerge on the other side, into −1 to vcap ∼ 100ms [47]. The number of the region where light of both frequen- atoms in a vapor with velocities below vcap cies is present and begin absorbing again. in the Boltzmann distribution scales as Then they feel the trapping force and are 4 vcap, and there are enough slow atoms to driven back into the ‘‘dark’’ center of the fall within the large MOT capture range trap. Such an MOT has been operated at even at room temperature, because a few MIT [57] with densities close to 1012/cm3, − Kincludes10 4 of the atoms. and the limitations are now from colli- sions in the ground state rather than from 4.3.4 Variations on the MOT Technique multiple light scattering and excited-state Because of the wide range of applications collisions. of this most versatile kind of atom trap, a number of careful studies of its properties have been made [47, 49–56], and several 5 variations have been developed. One of Optical Lattices these is designed to overcome the density 5.1 limits achievable in an MOT. In the Quantum States of Motion simplest picture, loading additional atoms into an MOT produces a higher atomic As the techniques of laser cooling ad- density because the size of the trapped vanced from a laboratory curiosity to a tool sample is fixed. However, the density for new problems, the emphasis shifted cannot increase without limit as more from attaining the lowest possible steady atoms are added. The atomic density − state temperatures to the study of ele- is limited to ∼1011 cm 3 because the mentary processes, especially the quantum fluorescent light emitted by some trapped atoms is absorbed by others. mechanical description of the atomic mo- One way to overcome this limit is to tion. In the completely classical description have much less light in the center of the of laser cooling, atoms were assumed to MOT than at the sides. Simply lowering have a well-defined position and momen- the laser power is not effective in reducing tum that could be known simultaneously the fluorescence because it will also reduce with arbitrary precision. However, when the capture rate and trap depth. But those atoms are moving sufficiently slowly that advantageous properties can be preserved their deBroglie wavelength precludes their λ/ π while reducing fluorescence from atoms at localization to less than 2 ,thesede- the center if the light intensity is low only scriptions fail and a quantum mechanical in the center. description is required. Such exotic be- The repumping process for the alkali havior for the motion of whole atoms, as atomsprovidesanidealwayofimple- opposed to electrons in the atoms, had not menting this idea [57]. If the repumping been considered before the advent of laser light is tailored to have zero intensity at cooling simply because it was too far out of the center, then atoms trapped near the the range of ordinary experiments. A series center of the MOT are optically pumped of experiments in the early 1990s provided into the ‘‘wrong’’ hfs state and stop fluo- dramatic evidence for these new quantum rescing. They drift freely in the ‘‘dark’’ at states of motion of neutral atoms, and Laser Cooling and Trapping of Neutral Atoms 1001 led to the debut of deBroglie wave atom means that their deBroglie wavelengths are optics. equal to the optical wavelengths divided by The quantum description of atomic a ‘‘few’’. If the depth of the optical potential motion requires that the energy of such wells is high enough to contain such very motion be included in the Hamiltonian. slow atoms, then their motion in potential The total Hamiltonian for atoms moving wells of size λ/2 must be described in a light field would then be given by quantum mechanically, since they are confined to a space of size comparable H = H + H + H + H ,( ) atom rad int kin 25 to their deBroglie wavelengths. Thus, they do not oscillate in the sinusoidal wells as where Hatom describes the motion of the atomic electrons and gives the internal classical localizable particles, but instead occupy discrete, quantum mechanical atomic energy levels, Hrad is the energy of the radiation field and is of no concern here bound states [60], as shown in the lower part of Fig. 13. because the field is not quantized, Hint describes the excitation of atoms by the The basic ideas of the quantum me- light field and the concomitant light shifts, chanical motion of particles in a periodic H potential were laid out in the 1930s with the and kin is the kinetic energy operator of the motion of the center of mass of the Kronig–Penney model and Bloch’s theo- atoms. This Hamiltonian has eigenstates rem, and optical lattices offer important of not only the internal energy levels and opportunities for their study. For example, the atom-laser interaction that connects them, but also that of the kinetic energy H ≡ 2/ operator kin P 2M. These eigenstates will therefore be labeled by quantum numbers of the atomic states as well as the center of mass momentum p.Anatom in the ground state, |g; p, has an energy 2 Eg + p /2M, that can take on a range of values. In 1968, V.S. Letokhov [58] suggested that it is possible to confine atoms in the wavelength-size regions of a standing wave Energy by means of the dipole force that arises from the light shift. This was first accom- plished in 1987 in 1-D with an atomic beam traversing an intense standing wave [59]. 0 l/4 l/2 3l/4 l Since then, the study of atoms confined Position to wavelength-size potential wells has be- Fig. 13 Energy levels of atoms moving in the come an important topic in optical control periodic potential of the light shift in a standing of atomic motion because it opens up con- wave. There are discrete bound states deep in figurations previously accessible only in the wells that broaden at higher energy, and condensed matter physics using crystals. become bands separated by forbidden energies above the tops of the wells. Under conditions The limits of laser cooling discussed in appropriate to laser cooling, optical pumping Sect. 3.3.4 suggest that atomic momenta among these states favors populating the lowest can be reduced to a ‘‘few’’ times ¯hk.This ones as indicated schematically by the arrows 1002 Laser Cooling and Trapping of Neutral Atoms
these lattices can be made essentially free of defects with only moderate care in spatially filtering the laser beams to as- sure a single transverse mode structure. Furthermore, the shape of the potential is exactly known and does not depend on the effect of the crystal field or the ionic energy level scheme. Finally, the laser parameters can be varied to mod- ify the depth of the potential wells without changing the lattice vectors, and the lat- tice vectors can be changed independently Fig. 14 The ‘‘egg-crate’’ potential of an optical by redirecting the laser beams. The sim- lattice shown in two dimensions. The potential λ/ plest optical lattice to consider is a 1-D wells are separated by 2 pair of counterpropagating beams of the same polarization, as was used in the first motion can also be observed by stimu- experiment [59]. lated emission [62, 63] and by direct RF Of course, such tiny traps are usually spectroscopy [64, 65]. very shallow, so loading them requires cooling to the µK regime. Even atoms 5.2 Properties of 3-D Lattices whose energy exceeds the trap depth must be described as quantum mechanical The name ‘‘optical lattice’’ is used rather particles moving in a periodic potential than optical crystal because the filling that display energy band structure [60]. fraction of the lattice sites is typically Such effects have been observed in very only a few percent (as of 1999). The limit careful experiments. arises because the loading of atoms into Because of the transverse nature of light, the lattice is typically done from a sample any mixture of beams with different k- of trapped and cooled atoms, such as an vectors necessarily produces a spatially MOT for atom collection, followed by an periodic, inhomogeneous light field. The OM for laser cooling. The atomic density in importance of the ‘‘egg-crate’’ array of such experiments is limited by collisions potential wells arises because the asso- and multiple light scattering to a few times ciated atomic light shifts can easily be 1011 cm−3. Since the density of lattice sites comparable to the very low average atomic of size λ/2isafewtimes1013 cm−3,the kinetic energy of laser-cooled atoms. A filling fraction is necessarily small. With typical example projected against two di- the advent of experiments that load atoms mensions is shown in Fig. 14. directly into a lattice from a BEC, the filling Atoms trapped in wavelength-sized factor can be increased to 100%, and in spaces occupy vibrational levels similar some cases it may be possible to load more to those of molecules. The optical spec- than one atom per lattice site [66, 67]. trum can show Raman-like sidebands that In 1993 a very clever scheme was result from transitions among the quan- described [68]. It was realized that an n- tized vibrational levels [61, 62] as shown in dimensional lattice could be created by Fig. 15. These quantum states of atomic only n + 1 traveling waves rather than Laser Cooling and Trapping of Neutral Atoms 1003
0.6 0.5
0.4 0.4 0.3 ×10 0.2 ×20 0.2
0.1 0 Fluorescence (arb. units) Fluorescence (arb. units) 0 −200 −100 0 100 200 −200 −100 0 100 200 (a)Frequency (kHz) (b) Frequency (kHz) Fig. 15 (a) Fluorescence spectrum in a 1-D lin ⊥ lin optical molasses. Atoms are first captured and cooled in an MOT, and then the MOT light beams are switched off leaving a pair of lin ⊥ lin because of spontaneous emission of the atoms to the same vibrational state from which they are excited, whereas the sideband on the left (right) is due to spontaneous emission to a vibrational state with one vibrational quantum number lower (higher) (see Fig. 13). The presence of these sidebands is a direct proof of the existence of the band structure. (b) Same as (a) except that the 1-D molasses is σ + − σ −, which has no spatially dependent light shift and hence no vibrational states (figure taken from Jessen, P. S., Gerz, C., Lett,P.D.,Phillips,W.D.,Rolston,S.L.,Spreeuw,R.J.C.,Westbrook,C.I.(1992), Phys. Rev. Lett. 69, 49–52)
2n. The real benefit of this scheme is 5.3 that in case of phase instabilities in the Spectroscopy in 3-D Lattices laser beams, the interference pattern is only shifted in space, but the interference The group at NIST developed a new pattern itself is not changed. Instead method that superposed a weak probe of producing optical wells in 2-D with beam of light directly from the laser upon four beams (two standing waves), these some of the fluorescent light from the authors used only three. The k-vectors atoms in a 3-D OM, and directed the of the coplanar beams were separated by light from these combined sources onto 2π/3, and they were all linearly polarized on a fast photodetector [70]. The resulting in their common plane (not parallel beat signal carried information about the to one another). The same immunity Doppler shifts of the atoms in the optical to vibrations was established for a 3- lattices [61]. These Doppler shifts were D optical lattice by using only four expected to be in the sub-MHz range for beams arranged in a quasi-tetrahedral atoms with the previously measured 50- configuration. The three linearly polarized µK temperatures. The observed features beams of the 2-D arrangement described confirmed the quantum nature of the above were directed out of the plane motion of atoms in the wavelength-size toward a common vertex, and a fourth potential wells (see Fig. 15) [15]. circularly polarized beam was added. The NIST group also studied atoms All four beams were polarized in the loaded into an optical lattice using of same plane [68]. The authors showed that laser light from the spatially ordered this configuration produced the desired array [71]. They cut off the laser beams potential wells in 3-D. that formed the lattice, and before the 1004 Laser Cooling and Trapping of Neutral Atoms
atoms had time to move away from their to a speed faster than that corresponding positions, they pulsed on a probe laser to the edge of a Brillouin zone, and beam at the Bragg angle appropriate for that at longer times the particles would one of the sets of lattice planes. The execute oscillatory motion. Ever since then, Bragg diffraction not only enhanced the experimentalists have tried to observe reflection of the probe beam by a factor these Bloch oscillations in increasingly of 105, but by varying the time between pure and/or defect-free crystals. the shut-off of the lattice and turn-on of Atoms moving in optical lattices are ide- the probe, they could also measure the ally suited for such an experiment, as was ‘‘temperature’’ of the atoms in the lattice. beautifully demonstrated in 1996 [69]. The The reduction of the amplitude of the authors loaded a 1-D lattice with atoms Bragg-scatteredbeamwithtimeprovided from a 3-D molasses, further narrowed the some measure of the diffusion of the atoms velocity distribution, and then instead of away from the lattice sites, much like the applying a constant force, simply changed Debye–Waller factorDebye–Waller factor the frequency of one of the beams of the in X-ray diffraction. 1-D lattice with respect to the other in a controlled way, thereby creating an ac- 5.4 Quantum Transport in Optical Lattices celerating lattice. Seen from the atomic reference frame, this was the equivalent of In the 1930s, Bloch realized that applying a constant force trying to accelerate them. a uniform force to a particle in a periodic After a variable time ta, the 1-D lattice potential would not accelerate it beyond a beams were shut off and the measured certain speed, but instead would result atomic velocity distribution showed beau- in Bragg reflection when its deBroglie tiful Bloch oscillations as a function of ta. wavelength became equal to the lattice The centroid of the very narrow velocity period. Thus, an electric field applied to distribution was seen to shift in velocity a conductor could not accelerate electrons space at a constant rate until it reached
= ta t
= ta 3t/4 0.5 = ta t/2
= 0 ta t/4 − Atom number (arb. units) = Mean velocity 0.5 ta 0 −101 −2 −1 0 1 2 (a)Atomic momentum (hk) (b) Quasi-momentum (k) Fig. 16 Plot of the measured velocity distribution verses. time in the accelerated 1-D lattice. (a) Atoms in a 1-D lattice are accelerated for a fixed potential depth for a certain time ta and the momentum of the atoms after the acceleration is measured. The atoms accelerate only to the edge of the Brillouin zone where the velocity is +vr, and then the velocity distribution appears at −vr. (b) Mean velocity of the atoms as a function of the quasi-momentum, that is, the force times the acceleration time (figure taken from Ben Dahan, M., Peik, E., Reichel, J., Castin, Y., Salomon, C. (1996), Phys.Rev.Lett.76, 4508–4511) Laser Cooling and Trapping of Neutral Atoms 1005 vr = ¯hk/M, and then it vanished and reap- Temperatures lower than the recoil limit peared at −vr as shown in Fig. 16. The are readily achieved by evaporative cooling, shape of the ‘‘dispersion curve’’ allowed and so all BEC experiments employ it in measurement of the ‘‘effective mass’’ of their final phase. Evaporative cooling is the atoms bound in the lattice. inherently different from the other cooling processes discussed in Sect. 3, and hence discussed here separately. 6 Since the first observations in 1995, BEC Bose–Einstein Condensation has been the subject of intense investiga- tion, both theoretical and experimental. No 6.1 attempt is made in this article to even ad- Introduction dress, much less cover, the very rich range of physical phenomena that have been un- In the 1920s, Bose and Einstein predicted veiled by these studies. Instead, we focus that for sufficiently high phase space on the methods to achieve ρφ ∼ 1and density, ρφ ∼ 1 (see Sect. 1.2), a gas of BEC. atoms undergoes a phase transition that is now called Bose–Einstein condensation. 6.2 It took 70 years before BEC could be Evaporative Cooling unambiguously observed in a dilute gas. From the advent of laser cooling and Evaporative cooling is based on the trapping, it became clear that this method preferential removal of those atoms from could be instrumental in achieving BEC. a confined sample with an energy higher BEC is another manifestation of quanti- than the average energy followed by a zation of atomic motion. It occurs in the rethermalization of the remaining gas by absence of resonant light, and its onset is elastic collisions. Although evaporation is characterized by cooling to the point where a process that occurs in nature, it was the atomic deBroglie wavelengths are com- applied to atom cooling for the first time parable to the interatomic spacing. This is in 1988 [72]. in contrast with the topics discussed in One way to think about evaporative Sect. 5 where the atoms were in an optical cooling is to consider cooling of a con- field and their deBroglie wavelengths were tainer of hot liquid. Since the most comparable to the optical wavelength λ. energetic molecules evaporate from the Laser cooling alone is inherently inca- liquid and leave the container, the remain- pable of achieving ρφ ∼ 1. This is easily ing molecules obtain a lower temperature seen from the recoil limit of Sect. 3.3.4 and are cooled. Furthermore, it requires that limits λdeB to λ/‘‘few’’. Since the cross the evaporation of only a small fraction section for optical absorption near reso- of the liquid to cool it by a considerable nance is σ ∼ λ2 near ρφ ∼ 1, this limit of amount. λdeB ∼ λ/‘‘few’’ results in the penetration Evaporative cooling works by remov- depth of the cooling light into the sample ing the higher-energy atoms as sug- being smaller than λ. Thus, the sample gested schematically in Fig. 17. Those would have to be extremely small and con- that remain have much lower average en- tain only a few atoms, hardly a system ergy (temperature) and so they occupy a suitable for investigation. smallervolumenearthebottomofthe 1006 Laser Cooling and Trapping of Neutral Atoms
Several models have been developed to describe this process, but we present here the simplest one [77] because of its pedagogical value [22]. In this model, the trap depth is lowered in one single step and the effect on the thermodynamic quantities, such as temperature, density, and volume, is calculated. The process can be repeated and the effects of multiple steps added up cumulatively. In such models of evaporative cooling, Fig. 17 Principle of the evaporation technique. the following assumptions are made: Once the trap depth is lowered, atoms with energy above the trap depth can escape and the remaining atoms reach a lower temperature 1. The gas behaves sufficiently ergodically, that is, the distribution of atoms trap, thereby increasing their density. For in phase space (both position and trapped atoms, it can be achieved by lower- momentum) depends only on the ing the depth of the trap, thereby allowing energy of the atoms and the nature the atoms with energies higher than the of the trap. trap depth to escape, as discussed first by 2. The gas is assumed to begin the Hess [73]. Elastic collisions in the trap then process with ρφ 1(farfromtheBEC lead to a rethermalization of the gas. This transition point), and so it is described technique was first employed for evapora- by classical statistics. tive cooling of hydrogen [72, 74–76]. Since 3. Even though ρφ 1, the gas is cold both the temperature and the volume de- enough that the atomic scattering is ρ crease, φ increases. pure (s-wave) quantum mechanical, Recently, more refined techniques have that is, the temperature is sufficiently been developed. For example, to sustain low that all higher partial waves do the cooling process the trap depth can not contribute to the cross section. be lowered continuously, achieving a Furthermore, the cross section for continuous decrease in temperature. Such elastic scattering is energy-independent a process is called forced evaporation and is and is given by σ = 8πa2,wherea is the discussed in Sect. 6.3 below. scattering length. It is also assumed that the ratio of elastic to inelastic collision 6.2.1 Simple Model rates is sufficiently large that the elastic This section describes a simple model of collisions dominate. evaporative cooling. Since such cooling is 4. Evaporation preserves the thermal na- not achieved for single atoms but for the ture of the distribution, that is, the whole ensemble, an atomic description of thermalization is much faster than the the cooling process must be replaced by rate of cooling. thermodynamic methods. These methods 5. Atoms that escape from the trap neither are completely different from the rest of the collide with the remaining atoms nor material in this article, and will therefore exchange energy with them. This is remain rather elementary. called full evaporation. Laser Cooling and Trapping of Neutral Atoms 1007
6.2.2 Application of the Simple Model quantities for this process (the values af- The first step in applying this simple model ter the process are denoted by a prime). is to characterize the trap by calculating One of these is ν ≡ N /N, the fraction of how the volume of a trapped sample atoms remaining in the trap after the cool- of atoms changes with temperature T. ing. The other is γ (This γ is not to be Consider a trapping potential that can be confused with the natural width of the ex- expressed as a power law given by cited state.), a measure of the decrease in temperature caused by the release of hot s1 s2 x y atoms and subsequent cooling, modified U(x, y, z) = ε1 + ε2 a1 a2 by ν,anddefinedas
s3 + ε z ,() log(T /T) log(T /T) 3 26 γ ≡ = .(28) a3 log(N /N) log ν where aj is a characteristic length and sj the This yields a power-law dependence for power, for a certain direction j. Then the the decrease in temperature caused by volume occupied by trapped atoms scales the loss of the evaporated particles, that γ as V ∝ Tξ [78], where is, T = Tν .Thedependenceofthe other thermodynamic quantities on the ξ ≡ 1 + 1 + 1 .() parameters ν and γ can then be calculated. 27 γ s1 s2 s3 The scaling of N = Nν, T = Tν ,and V = Vνγξ can provide the scaling of Thus, the effect of the potential on the all the other thermodynamic quantities volume of the trapped sample for a of interest by using the definitions for given temperature can be reduced to a the density n = N/V, the phase space ξ − / single parameter . This parameter is ρφ = λ3 ∝ 3 2 density n deB nT ,andthe independent of how the occupied volume / elastic collision rate k ≡ nσv ∝ nT1 2. is defined, since many different definitions el TheresultsaregiveninTable2.Fora lead to the same scaling. When a gas is held given value of η, the scaling of all quantities in a 3-D box with infinitely high walls, then depends only on γ . Note that for successive = = =∞ ξ = s1 s2 s3 and 0, which means steps j, ν has to be replaced with νj. that V is independent of T,asexpected. For a harmonic potential in 3-D, ξ = 3/2; Tab. 2 Exponent q for the scaling of the for a linear potential in 2-D, ξ = 2; and for thermodynamic quantities X = Xνq with the a linear potential in 3-D, ξ = 3. reduction ν of the number of atoms in the trap The evaporative cooling model itself [77] starts with a sample of N atoms in volume Thermodynamic Symbol Exponent q V having a temperature T held in an in- variable finitely deep trap. The strategy for using η Number of N 1 the model is to choose a finite quantity , atoms and then (1) lower the trap depth to a value Temperature T γ γξ ηkBT, (2) allow for a thermalization of the Volume V sample by collisions, and (3) determine the Density n 1 − γξ Phase space ρ 1 − γ(ξ + 3/2) change in ρφ. density Only two parameters are needed to com- Collision rate k 1 − γ(ξ − 1/2) pletely determine all the thermodynamic 1008 Laser Cooling and Trapping of Neutral Atoms
The fraction of atoms remaining is fully 6.2.4 Limiting Temperature determined by the final trap depth η for In the models discussed so far, only a given potential characterized by the trap elastic collisions have been considered, parameter ξ. In order to determine the that is, collisions where kinetic energy change of the temperature in the cooling is redistributed between the partners. process, it is necessary to consider in However, if part of the internal energy detail the distribution of the atoms in the of the colliding partners is exchanged with trap, and this is discussed more fully in their kinetic energy in the collision, then Refs. [1, 22, 77]. it is inelastic. Inelastic collisions can cause problems for two reasons: (1) the internal 6.2.3 Speed of Evaporation energy released can cause the atoms to So far, the speed of the evaporative cooling heat up and (2) the atoms can change their process has not been considered. If the internal states, and the new states may trap depth is ramped down too quickly, the no longer be trapped. In each case, such thermalization process does not have time collisions can lead to trap loss and are to run its course and the process becomes therefore not desirable. less efficient. On the other hand, if the Apart from collisions with the back- trap depth is ramped down too slowly, ground gas and three-body recombination, the loss of particles by inelastic collisions there are two inelastic processes that are becomes important, thereby making the important for evaporative cooling of alkali evaporation inefficient. atoms: dipolar relaxation and spin relax- The speed of evaporation can be found ation. The collision rate nkdip for them from the principle of detailed balance [22]. at low energies is independent of veloc- Its application shows that the ratio of the ity [82]. Since the elastic collision rate is = σ evaporation time and the elastic collision given by kel n vrel, the ratio of good time is √ (= elastic) to bad (= relaxation) collisions τ η goes down when the temperature does. ev = 2e .() 29 This limits the temperature to a value T τel η e near which the ratio between good and Note that this ratio increases exponentially bad collisions becomes unity, and Te is with η. given by Experimental results show that ∼2.7 π 2 elastic collisions are necessary to rether- Mkdip k T = .(30) malize the gas [79]. In order to model the B e 16σ 2 rethermalization process, Luiten et al. [80] have discussed a model based on the Boltz- The limiting temperature for the alkalis mann equation where the evolution of the is of the order of 1 nK, depending on the phase space density ρ(r, p, t) is calculated. values of σ and kdip. This evolution is not only caused by the In practice, however, this ratio has to trapping potential, but also by collisions be considerably larger than unity, and between the particles. Only elastic col- so the practical limit for evaporative lisions, whose cross section is given by cooling occurs when the ratio is ∼103 [22]. σ = 8πa2 with a as the scattering length, In the model of Ref. [80], the authors are considered. This leads to the Boltz- discuss different strategies for evaporative mann equation [81]. cooling. Even for the strategy of the Laser Cooling and Trapping of Neutral Atoms 1009 lowest temperature, the final temperature the cutoff are expelled from the trap. By is higher than Te. using RF-evaporation, one can expel the The collision rate between atoms with atoms in all three dimensions equally and one in the excited state (S + P collisions) is thus obtain a true 3-D evaporation. In the also much larger at low temperatures than case of the time orbiting potential (TOP) the rate for such collisions with both atoms trap, the atoms are evaporated along the in the ground state (S + S collisions). Since outer side of the cloud that is exposed to S + P collisions are generally inelastic and the highest magnetic field on the average. the inelastic energy exchange generally This is a cylinder along the direction of leads to a heating of the atoms, increasing rotation axis of the magnetic field and thus the density increases the loss of cold the evaporation takes place in 2-D. atoms. To achieve BEC, resonant light Once the energy of the atoms becomes should therefore be avoided, and thus laser very small, the atoms sag because of gravity cooling is not suitable for achieving BEC. and the outer shell of the cloud is no longer at a constant magnetic field. Atoms at the 6.3 bottom of the trap have the highest energy Forced Evaporative Cooling and thus the evaporation becomes 1-D. In case of harmonic confinement, Utrap = In all the earliest experiments that achieved 2 U z/2, the equipotential surface is at BEC, the evaporative cooling was ‘‘forced’’ z ≈ 2ηkBT/U . Now, the gravitational by inducing rf transitions to magnetic energy is given by Ugrav = mgz and sublevels that are not bound in the thus the limiting temperature for 1-D magnetic trap. Atoms with the highest evaporation to take place is given by [22] energies can access regions of the trap where the magnetic field is stronger, and 2η(mg)2 thus their Zeeman shifts would be larger. < ( ) kBT µ 31 A correspondingly high-frequency rf field B would cause only these most energetic = −2 atoms to undergo transitions to states For a curvature of B 500 T m ,the µ 7 that are not trapped, and in doing so, the limiting temperature becomes 1 Kfor Li, µ 23 µ 87 departing atoms carry away more than the 10 Kfor Na, and 150 Kfor Rb. Below average energy. Thus, a slow sweep of this temperature, evaporation becomes the rf frequency from high to low would less efficient. continuously shave off the high-energy tail In the three experiments that obtained of the energy distribution, and thereby BEC for the first time in 1995, the problem continuously drive the temperature lower of this ‘‘gravitational sag’’ was not known, and the phase space density higher. The butitdidnotpreventtheexperimentalist results of evaporative cooling from the first from observing BEC. The solution used three groups that have obtained BEC have in those experiments was because of the shown that using this rf shaving technique, light mass (7Li), tight confinement (23Na), it is much easier to select high-energy and TOP trap (87Rb). In the last case, the atoms and waste them than it is to cool axis of rotation is in the z-direction and them. thus the evaporation always remains 2- For the evaporation of the atoms, it is D. Table 3 shows typical values of ρφ for important that atoms with an energy above various situations. 1010 Laser Cooling and Trapping of Neutral Atoms
Tab. 3 Typical numbers for the phase space density as obtained in the experiments aimed at achieving BEC. The different stages of cooling and trapping the atoms are discussed in the text
Stages T λdeB n ρφ
◦ Oven 300 C0.02nm1010 cm−3 10−16 Slowing 30 mK 2 nm 108 cm−3 10−12 Pre-cooling 1 mK 10 nm 109 cm−3 10−9 Trapping 1 mK 10 nm 1012 cm−3 10−6 Cooling 1 µK0.3µm1011 cm−3 3 × 10−3 Evaporation 70 nK 1 µm1012 cm−3 2.612
7 potential, predicted in the 1930s by Bloch Conclusion for electrons in solid state.
In this article, we have reviewed some Bose–Einstein Condensation (BEC): New of the fundamentals of optical control of state of matter theoretically described in atomic motion. The reader is cautioned the 1920s by Bose and Einstein and that this is by no means an exhaustive experimentally first observed in dilute review of the field, and that many impor- gases in 1996, where the interatomic tant and current topics have been omitted. spacing is smaller than the deBroglie Much of the material here was taken from wavelength of the atoms. our recent textbook [1], and the reader is encouraged to consult that source for the Brightness: The number of atoms emitted origin of many of the formulas presented from a source or in a beam per unit of in the present text, as well as for further time, per unit of solid angle, and per unit reading and more detailed references to of area of the source. the literature. Capture Range: The range in velocity, where the optical forces are effective.
Glossary Damping Force: Theopticalforceonatoms that leads to damping of their velocity. Atomic Beam Slowing: Using laser light to slow down the velocities of atoms in a Dark-spot Magneto-optical Trap: Magneto- beam. optical trap, where in the center of the Bad Collisions: Inelastic scattering of trap the atoms are not kept in the cycling atoms leading to loss of atoms from the transition, which reduces the loss of atoms trap during evaporation. due to inelastic collisions.
Beam Brightening: Increasing the bright- Dipole Force Trap: Trap using the dipole ness of an atomic beam by using laser optical force to trap atoms. cooling. Dipole Forces: Theopticalforceonthe Bloch Oscillations: The oscillatory motion atom caused by the gradient of the light of particles moving through a periodic shift of the atom. Laser Cooling and Trapping of Neutral Atoms 1011
Doppler Cooling Limit: Limit of cooling Magnetic Traps: Traps where atoms are atoms given by the Doppler theory. trapped due to their magnetic moment by an inhomogeneous magnetic field. Doppler Cooling Techniques: Using the Doppler shift and the detuning of the light Magneto-optical Trap (MOT): Combination from atomic resonance in order to cool of light forces and inhomogeneous mag- atoms. netic field, leading to cooling and trapping of atoms. Doppler Theory: Theory describing the cooling of two-level atoms by exploiting the Majorana Losses: Loss of atoms from Doppler shift of the atoms in combination magnetic traps, where the atoms in the with the detuning of the laser light from center of the trap undergo a motion- resonance. induced transition to an untrapped state. Ehrenfest Theorem: Theorem by Ehrenfest Multilevel Atoms: Atoms that have degen- making the correspondence between clas- erate magnetic substates in ground and sical relations, like F =−gradV,andtheir excited state, which are coupled by laser quantum-mechanical counterpart, in this light. case F=−gradH. Optical Bloch Equations (OBE): Relations Entropy: Thermodynamic measure of the describing the evolution of the state vector disorder in the system. of the internal state of the atom due to the interaction with laser light. Evaporative Cooling: The preferential re- moval of high-energy atoms from a gas Optical Forces: The force induced on the sample, thereby reducing the temperature atom by laser light. of the remaining atoms. Optical Lattice: The periodic trapping po- Far-off-resonance Traps: Dipole force trap, tential for atoms created by the interfer- where the light is far detuned from reso- ence of laser beams. nance, which minimizes the scattering of Optical Molasses: Viscous damping of light due to spontaneous emission. the atomic velocities by the use of Fokker-Planck Equation: Equation describ- counterpropagating laser beams. ing the evolution of the atomic momentum Phase Space Density: Probability of find- distribution under the combined influence ing a particle in a certain region of space, of force and diffusion. with a certain range of velocities. Good Collisions: Elastic scattering of atoms leading to the thermalization of the atoms Polarization Gradient Cooling: The use of during evaluation. polarization gradients to cool atoms below the Doppler limit. Laser Cooling: Using the interaction be- tween laser light and atoms to reduce the Polarization Gradient: The interference average kinetic energy of the atoms. pattern that occurs when two counter- propagating laser beams have different lin ⊥ lin Cooling: Polarization gradient polarization. The resulting polarization of cooling, where the two beams have the light field is not constant in space, that perpendicular, linear polarization. is, shows a gradient. 1012 Laser Cooling and Trapping of Neutral Atoms
Quadrupole Trap: The simplest magnetic s-wave Scattering: Scattering of atoms at trap, where the magnetic field is generated low temperatures, where only the lowest- by two separated identical coils carrying order partial wave is effective. opposite currents. Temperature: A measure of the average Radiative Forces: Theopticalforceonthe kinetic energy of the atoms, that is, atom caused by the scattering of light. kBT/2 =Ek. Recoil Limit: Limit in laser cooling deter- Time-of-flight (TOF) Method: Technique to mined by the recoil by one photon on the determine the flight time of atoms over a atom. well-defined flight path in order to detect the atomic velocity. Saturation Intensity: Intensity of the light where the rate for spontaneous emission Time-orbiting Potential (TOP) trap: Mag- and the rate for stimulated emission are netic quadrupole trap, where the zero of equal. the magnetic field oscillates in space in order to reduce trap loss due to Majorana Saturation Parameter: Ratio between the flips. intensity of the laser light and the saturation intensity. Two-level Atoms: Hypothetical atoms that only have one nondegenerate ground σ +-σ − Cooling: Polarization gradient cool- state and one nondegenerate excited state, ing, where the two beams have opposite, which are resonant with the laser light. circular polarization.
Sisyphus Cooling: The continuous trans- References fer of kinetic energy to potential en- ergy in laser cooling, where the poten- tial energy is radiated away by sponta- [1] Metcalf, H. J., vander Straten, P. (1999), neously emitted photons. A special case Laser Cooling and Trapping.NewYork: Springer-Verlag. of Sisyphus cooling is polarization gra- [2] Phillips, W., Metcalf, H. (1982), Phys. Rev. dient cooling to sub-Doppler tempera- Lett. 48, 596–599. ture. [3] Prodan, J., Phillips, W., Metcalf, H. (1982), Phys. Rev. Lett. 49, 1149–1153. Spontaneous Emission: The return of the [4] Prodan,J.,Phillips,W.(1984), Prog. Quant. atom from the excited state to the ground Electron. 8, 231–235. state by the emission of one photon in a [5] Ertmer, W., Blatt, R., Hall, J. L., Zhu, M. random direction. (1985), Phys.Rev.Lett.54, 996–999. [6] Watts, R., Wieman, C. (1986), Opt. Lett. 11, Stimulated Emission: The return of the 291–293. atom from the excited state to the ground [7] Bagnato, V., Lafyatis, G., Martin, A., state by the emission of one photon in the Raab, E., Ahmad-Bitar, R., Pritchard, D. 1987 58 direction of the laser light. ( ), Phys.Rev.Lett. , 2194–2197. [8] Molenaar, P. A., vander Straten, P., Heide- 1997 Sub-Doppler Cooling: Using laser cool- man, H. G. M., Metcalf, H. ( ), Phys. Rev. A 55, 605–614. ing techniques beyond the Doppler tech- [9] Barrett, T. E., Dapore-Schwartz, S. W., Ray, niques to cool atoms below the Doppler M. D., Lafyatis, G. P. (1991), Phys.Rev.Lett. limit. 67, 3483–3487. Laser Cooling and Trapping of Neutral Atoms 1013
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Lasers, Semiconductor
Nagaatsu Ogasawara University of Electro-Communications, Department of Electronics Engineering, Tokyo, Japan
Ryoichi Ito Meiji University, Department of Physics, Kawasaki, Japan Phone/Fax: +81-44-934-7425; e-mail: [email protected]
Abstract Following a brief history of semiconductor lasers, this article describes their opera- tion principles in relation to their device structures and materials. Their fundamental operation characteristics, including the static, spectral, and dynamic aspects, are then presented. Finally, the current state of the art in semiconductor laser technology is de- scribed by reviewing up-to-date devices incorporating new structures and new functions.
Keywords semiconductor laser; laser diode; photonics; compact light source; heterostructure; fiber-optic telecommunications; optical disk.
1 Introduction 1252 2History1253 3 Structures, Materials, and Operation Principles 1254 3.1 Double Heterostructure 1254 3.2 Fabry–Perot´ Cavity 1257 3.3 Lasing Threshold Condition 1257 3.4 Stripe Geometry 1258 3.5 Materials and Emission Wavelengths 1259 3.5.1 III–V Compounds 1259 3.5.2 IV–VI Compounds 1260 3.5.3 II–VI Compounds 1260 1252 Lasers, Semiconductor
4 Fundamental Characteristics 1260 4.1 Light–Current Characteristics 1260 4.2 Beam Profile and Polarization 1262 4.3 Spectral Characteristics 1263 4.4 Dynamic Characteristics 1265 5 New Structures and Functions 1267 5.1 Quantum-well Lasers 1267 5.2 Distributed-feedback and Distributed Bragg-reflector Lasers 1268 5.3 Semiconductor-laser Arrays 1269 5.4 Surface-emitting Lasers 1270 6 Summary 1270 Glossary 1271 References 1272 Further Reading 1273
1 efficiency of converting electrical power Introduction into optical power is several tens of percent, which should be compared with Semiconductor lasers have now grown to that of 0.1% or less for gas and other be key components in modern photonics lasers. technology, most notably as light sources 3. Capability for high-speed direct modula- in fiber-optic telecommunications systems tion: Light output can be modulated at and optical disk systems. They have frequencies of 10 GHz or more simply by also found an increasing number of modulating the pumping current. applications ranging from instruments 4. Wide emission spectrum: The emission such as bar code scanners and computer wavelength is determined by the band- printers to solid-state laser pumps and gap energy of the active-layer material. measuring and sensoring equipment for The blue-near-infrared region is covered engineering use. by III–V compounds. The infrared re- Semiconductor lasers have the following gion (3–30 µm) is covered by IV–VI features that distinguish them from other compounds. lasers. 5. High reliability: Semiconductor lasers 1. Compactness:Thetypicalsizeofalaser have become highly reliable devices be- chip is 300 × 200 × 100 µm3. The small cause of the remarkable improvement in chip contains all the ingredients of a laser crystal quality, although rapid degradation structure: a resonator, a waveguide, an was a great problem at the early stage of active medium, and a p-n junction to pump their development. the active medium. 6. Sensitivity to temperature: The threshold 2. High efficiency: Semiconductor lasers current and the lasing wavelength strongly can be driven by low electrical power depend on temperature. This is a disadvan- [(several tens of mA) × (1–2 V)]. The tage for most applications. On the other Lasers, Semiconductor 1253 hand, wavelength tunability is useful in the exploration of many other III–V spectroscopic research. and IV–VI compound semi-conductors. Unfortunately, these early homojunction This article is intended to serve as a devices, consisting of a single semicon- useful reference for those who wish to ductor, were not considered for serious grasp the basic concepts of semiconductor applications since continuous operation lasers and utilize them in engineering and at room temperature was not feasible be- research activities. Following this intro- cause of their high threshold current den- duction is a brief history of semiconductor sities for lasing (several tens of kA cm−2). lasers. Next, the operation principles of A breakthrough was brought about in semiconductor lasers are described in re- 1970 by employing a double heterostruc- lation to their structures and materials. ture grown by liquid-phase epitaxy [4, 5]. The fourth section presents a general view In the double heterostructure, stimulated of fundamental operation characteristics emission occurred only within a thin ac- covering the static, spectral, and dynamic tive layer of GaAs sandwiched between aspects. The fifth section gives the cur- p-andn-doped AlGaAs layers that have rent state of the art in semiconductor a wider band gap. The threshold current laser technology by reviewing up-to-date density was dramatically reduced to several devices incorporating new structures and − kA cm 2 or less, and continuous opera- new functions. Because of the limited tion at room temperature was eventually space, citation of published papers has accomplished. The invention of the dou- been kept to a minimum. Several books ble heterostructure was obviously the most listed in the (Further Reading) section can important step in the history of the devel- be consulted both for further study and for opment of semiconductor lasers toward access to the relevant published papers. practical utility. For the invention of het- erostructure, Alferov and Kroemer were honored with a Nobel Prize in physics in 2 History 2000, while Alferov, Hayashi and Panish shared the Kyoto Prize in 2001. The advent of semiconductor injection There still remained, however, a trou- lasers dates back to 1962, only two years blesome problem of degradation; the reli- later than the first achievement of laser ability of the early double-heterostructure action in ruby, when stimulated emis- devices was very poor and many of them sion from forward-biased GaAs diodes stopped their continuous operation within was demonstrated [1–3]. The stimulated a few or several tens of minutes. For- emission resulted from the radiative re- tunately, intensive investigations of the combination of electrons and holes in degradation mechanism led to the identi- the direct-gap semiconductor GaAs. The fication of the causes of major failure: the electrons and holes were injected by for- recombination-enhanced growth of dis- ward bias into the depletion region in locations, and facet erosion [6 Part B, the vicinity of the p-n junction. Optical Chapter 8]. These studies have revealed feedback was provided by polished facets that a long operating life is obtained by perpendicular to the junction plane. The eliminating dislocation generation during demonstration of GaAs lasers prompted material growth and device processing and 1254 Lasers, Semiconductor
by coating the mirror facets with dielectric the stable single–longitudinal-mode oscil- thin films. lation they exhibit even under high-speed Another task undertaken in parallel modulation. with the improvement of reliability was Recent advances in growth technol- the stabilization of the optical mode in ogy of semiconductor ultrathin layers by the lateral transverse direction (parallel molecular-beam epitaxy and metal-organic to the junction plane). It had been vapor-phase epitaxy have also created a recognized by the late 1970s that the new class of laser structures. Quantum- lateral-mode instability often observed at well lasers incorporating such ultrathin higher output powers was detrimental active layers (∼20 nm or less) have been to most applications and that such an shown to display a variety of superior laser instability could be effectively suppressed characteristics, such as low-threshold cur- by incorporating a built-in refractive-index rents and wide modulation bandwidths, profile in the lateral direction [7]. A large which are attributed to the size quanti- number of laser structures have been zation of electrons in the ultrathin active proposed and demonstrated to introduce layers [9]. Quantum-well structures as well lateral variations of the refractive index. as multiple quantum-well structures are As a result of these research and develop- now commonly employed in most semi- ment efforts during the first two decades, conductor lasers. the maturity of AlGaAs lasers reached the In the visible spectrum, GaInP lasers stage of practical use in printers and disk were first commercialized as red-emitting players early in the 1980s. Meanwhile, the lasers (0.63–0.69 µm) in 1988 by Sony, rapid reduction of transmission loss in Toshiba, and NEC. They are now exten- optical fibers achieved during the 1970s sively used in DVD systems. Blue–green stimulated intensive development efforts laser emission was achieved first in on GaInPAs lasers, which have emission ZnCdSe lasers [10] and then in InGaN µ wavelengths(1.3–1.55 m) in the low-loss lasers [11, 12]. InGaN blue lasers are be- and dispersion-free region of silica fibers. ing employed in second-generation DVD During the first half of the 1980s, GaIn- players (see DATA STORAGE, OPTICAL). PAs lasers emerged as indispensable light sources in telecommunications systems using silica fibers (see OPTICAL COMMUNI- 3 CATIONS). Structures, Materials, and Operation Progress has continued to date in both Principles device structures and materials. In the following, let us look at some major topics. 3.1 The idea of utilizing periodic gratings Double Heterostructure incorporated in a laser medium as a means of optical feedback was proposed A semiconductor laser is a diode, as by Kogelnik and Shank in [8]. Distributed- shown schematically in Fig. 1. When a feedback and distributed-Bragg-reflector forward current is passed through the p-n semiconductor lasers utilizing this prin- junction, electrons and holes injected into ciple have since become the major device theactiveregionfromthen and p regions structures used in long-haul, high bit-rate respectively, recombine to emit radiation. telecommunications systems because of The wavelength λ of the radiation is Lasers, Semiconductor 1255
y x Electrode Current Hole z Laser µm emission 200
p-type
Active layer 100 µm n -type
300 µm Electron
Cleaved facet
Fig. 1 Schematic illustration of a double-heterostructure semiconductor laser basically determined by the relation λ ∼ the DH structure facilitates the interaction hc/Eg,whereh is the Planck constant, c is needed for laser action between the optical the light velocity, and Eg is the band-gap field and the injected carriers. energy of the active-region material. Let us use a GaAs(active)/AlxGa1−xAs For low injection currents, light is emit- (cladding) DH structure as an example. ted through spontaneous emission. In The dependence of the band-gap energy of order for lasing to take place, a suf- AlxGa1−xAs on the AlAs mole fraction x ficiently high concentration of carriers can be approximated for x < 0.42 (direct- must be accumulated within the active re- gap region) by [6 Part A, p. 193]. gion to induce population inversion. This E = 1.424 + 1.247x eV.(1) is effectively accomplished by adopting g a double-heterojunction (DH) structure, The energy-gap difference between the ∼ . µ where a thin active layer, typically 0 1 m active and cladding layers Eg is divided thick,butasthinas10nminquantum- into the band discontinuities Ec in the well lasers, is sandwiched between n-and conduction band and Ev in the valence p-type cladding layers, which have wider band according to [13]. band gaps than the active layer. Electrons ∼ . ( ) and holes injected into the active layer Ec 0 62 Eg 2 through the heterojunctions are confined and within the thin active layer by the poten- Ev ∼ 0.38Eg.(3) tial barriers at the heteroboundaries, as illustrated in Fig. 2(a). The DH structure The efficiency of the carrier confinement forms an efficient optical waveguide as depends strongly on the magnitude of well, because of the refractive-index dif- the band discontinuity. The AlAs mole ference between the active and cladding fraction x of the cladding layers is usually layers shown in Figs. 2(b) and 2(c). Thus, chosen to be greater than that of the active 1256 Lasers, Semiconductor
n -cladding Active p -cladding layer layer layer
Electron Ec
eV Ev
Electron energy d Hole (a) Refractive index (b)
Γ Light intensity y (c) Fig. 2 Diagram illustrating carrier confinement and waveguiding in a double heterostructure (a) Energy-band diagram at high forward bias; (b) refractive-index distribution; (c) light intensity distribution. Ec and Ev: The edges of the conduction and valence bands. V: Applied voltage. d: Active-layer thickness (∼0.1 µm). : Confinement factor
layer by ∼0.3. This gives rise to a Ec supports only the lowest-order fundamen- approximately 10 times as large as the tal mode is expressed as room-temperature thermal energy, which is sufficient to suppress electron diffusion D <π. (5) over the heterobarriers. The hole leakage current is less important because of the Meanwhile, the refractive index of smaller diffusion constant for holes. AlxGa1−xAs, η(x), can be approximated The characteristics of a three-layer slab for a light wavelength of ∼0.9 µmby[6, waveguide are conveniently described in Part A, p. 45] terms of the normalized waveguide thick- ness D,definedas η(x) = 3.590 − 0.710x + 0.091x2.(6) 2π = η2 − η2,() D λ d a c 4 Thus, for a waveguide with a GaAs active layer surrounded by Al0.3Ga0.7As where ηa and ηc are the refractive indices of layers and a wavelength λ of 0.87 µm, the the active and cladding layers respectively condition of Eq. (5) simply becomes and d is the active-layer thickness. For example, the condition that a waveguide d < 0.36 µm.(7) Lasers, Semiconductor 1257
Since the thickness d is typically 0.1 a full round trip in the cavity without to 0.2 µm, the single, fundamental-mode attenuation; that is, the optical gain should condition is almost always satisfied in equal the losses both inside the cavity and practical devices. through the partially reflecting end facets. The confinement factor ,definedasthe Thus, the gain coefficient at threshold gth fraction of the electromagnetic energy of is given by the relation the guided mode that exists within the active layer, is an important parameter 1 1 g = αa + (1 − )αc + αs + ln . representing the extent to which the th L R α waveguide mode is confined to the active i ( ) layer. for a fundamental mode is 9 α α approximately given by [14] Here, a and c denote the losses in the active and cladding layers respectively, 2 D due to free-carrier absorption. αs accounts ∼ .(8) 2 + D2 for scattering loss due to heterointerfacial imperfections. The first three loss terms For a GaAs/Al . Ga . As waveguide 0 3 0 7 on the right-hand side combined are with d = 0.1 µm, ∼ 0.27. termed internal loss αi and add up to 10 to 20 cm−1 [6, Part A, p. 174–176]. The 3.2 − − reflection loss L 1 ln R 1 due to output Fabry–Perot´ Cavity coupling (∼40 cm−1 for L ∼ 300 µm, R ∼ 0.3) is normally the largest among the In addition to the optical gain, optical loss terms. Despite the fact that both feedback is the other ingredient for laser oscillation. This is provided by a pair the internal and the reflection losses are of mirror facets at both ends of the exceptionally large as compared to other devices. These facets are normally formed lasers, lasing is brought about by virtue of simply by cleaving the crystal. Since the the high optical gain in semiconductors. refractive index of major semiconductor It is not an easy task to analyze precisely laser materials is ∼3.6, the reflectivity the optical gain spectrum in semiconduc- of the mirror is ∼0.3. This is very low tor lasers. Fortunately, however, we can compared to other types of lasers but is utilize a phenomenological linear relation- still sufficient to provide optical feedback ship between the maximum gain g (the in semiconductor lasers. peak value of the gain spectrum for a given carrier density) and the injected carrier 3.3 density n, Lasing Threshold Condition ∂g g(n) = (n − n ), (10) When a sufficient number of electrons ∂n t and holes is accumulated to form an inverted population, the active region to a good approximation [15–17]. Here, exhibits optical gain and can amplify ∂g/∂n is termed differential gain,andnt light passing through it since stimulated denotes the carrier density required to emission overcomes interband absorption. achieve transparency where stimulated The condition for self-sustained laser emission balances against interband ab- oscillation to occur is that the light makes sorption corresponding to the onset of 1258 Lasers, Semiconductor
population inversion. For GaAs lasers, Electrode
∂g − ∼ 3.5 × 10 16 cm2 (11) w ∂n Insulator and p -cladding layer 18 −3 Active layer nt ∼ 1.5 × 10 cm .(12) n -cladding layer Substituting Eqs. (10) to (12) into n -substrate Eq. (9) and assuming that = 0.27, αi = 10 cm−1,andL−1 ln R−1 = 40 cm−1,we Electrode y get a threshold carrier density nth of ∼2 × 1018 cm−3. x The threshold current density Jth is z expressed as (a)
= ednth ,() Electrode Jth 13 Insulator τs w p -cladding layer τ where s is the carrier lifetime due to Active layer spontaneous emission. Assuming that n -burying layer τ = = . µ s 2–4 ns and d 0 1 m, we ob- p -burying layer tain a threshold current density Jth or − ∼1kAcm 2. n -cladding layer n -substrate 3.4 Electrode Stripe Geometry (b) Fig. 3 Two representative examples of Most modern semiconductor lasers adopt stripe-geometry lasers (front view of the end a stripe geometry, where current is injected facet) (a) Stripe-contact laser; (b) Buried only within a narrow region beneath a heterostructure (BH) laser. W: Stripe width stripe contact several µm wide, in order (2–10 µm) to keep the threshold current low and to control the optical field distribution in the direction) is determined by the gain profile lateral direction. As compared with broad- produced by carrier density distribution. area lasers, where the entire laser chip Devices incorporating a built-in refr- is excited, the threshold current of the active-index variation in the lateral di- stripe-geometry lasers is reduced roughly rection are termed index-guided lasers. proportional to the area of the contact. Figure 3(b) shows the structure of a buried Figure 3(a) shows the simplest form of heterostructure (BH) laser [19] as a repre- the stripe geometry [18]. Lasing occurs in sentative example of index-guided lasers. a limited region of the active layer beneath The active region is surrounded by materi- the stripe contact where a high density of als with lower refractive indices in both the current flows. Such lasers are termed gain- vertical (y) and lateral (x) transverse direc- guided lasers because the optical intensity tions, thus forming a waveguide structure distribution in the lateral direction (x in both directions. In the BH laser, the Lasers, Semiconductor 1259 widthofcurrentflowisdelineatedbythe laser printers. GaInPAs/InP/InP lasers p-n junction formed by the burying layers, emitting at 1.2 to 1.6 µm are used which is reverse biased when the active in fiber-optic communications systems. region is forward biased. GaInN/AlGaN/sapphire lasers emitting at 0.38 to 0.45 µm are starting to be 3.5 used in second-generation DVD players. Materials and Emission Wavelengths All of them exhibit low-threshold, room- temperature continuous wave (cw) opera- Semiconductor-laser materials that have tion with high reliability. been studied range over III–V, IV–VI, The common features among them are and II–VI compounds as listed in Table 1. as follows:
1. The active layer consists of direct-gap 3.5.1 III–V Compounds materials. III–V compounds are the most impor- 2. Binary compounds (GaAs, InP) are tant and popular laser materials [6, Part used as the substrate except for B, Chapter 5]. In particular, AlGaAs (ac- GaInN/AlGaN/sapphire lasers, for tive layer)/AlGaAs (cladding layer)/GaAs which GaN substrate will be used in (substrate) lasers emitting at 0.7 to the future. 0.9 µm and GaInP/AlGaInP/GaAs lasers 3. The active and cladding layers have emitting at 0.63 to 0.69 µmareexten- nearly the same lattice constant as the sively used in optical disk systems and substrate.
Tab. 1 Major semiconductor-laser materials and their emission wavelengths
Materials (active/cladding/substrate) Emission wavelengths [µm]
III–V compounds AlGaAs/AlGaAs/GaAs 0.7–0.9 GaInPAs/InP/InP 1.2–1.6 GaInP/AlGaInP/GaAs 0.66–0.69 GaInPAs/GaInP/GaAs or GaPAs 0.65–0.9 GaInPAs/AlGaAs/GaAs 0.62–0.9 AlGaAsSb/AlGaAsSb/GaSb 1.1–1.7 GaInAsSb/AlGaAsSb/GaSb or InAs 2–4 InPAsSb/InPAsSb/GaSb or InAs 2–4 GaInAs(strained)/AlGaAs/GaAs 0.9–1.1 GaInPAs(strained)/GaInPAs/InP ∼1.55 GaInN/AlGaN/sapphire 0.38–0.45 IV–VI compounds PbSnTe/PbSnSeTe/PbTe 6–30 PbSSe/PbS/PbS 4–7 PbEuTe/PbEuTe/PbTe 3–6 II–VI compounds ZnCdSe/ZnSSe/GaAs ∼0.5 1260 Lasers, Semiconductor
The use of direct-gap semiconductors, 3.5.3 II–VI Compounds featuring efficient radiative recombina- ZnCdSe/ZnSSe/ZnMgSSe/GaAs quan- tion, as the active-region material (1)is tum-well lasers emitting in the blue–green essential in achieving laser action. Fea- spectral region emerged around 1991 as tures 2 and 3 are related to the crystal a result of success in overcoming the quality of the hetero-interfaces. To min- difficulty of p-type doping into these ma- imize the generation of lattice defects that terials. [10, 21] Despite extensive efforts, may impair the device reliability, the lat- however, these lasers have never attained tice parameter of the active and cladding device life times exceeding several hun- layers should be matched to that of the dred hours and hence have not been put high-quality binary substrate (the lattice- into practical use. matching condition). The lattice parameter of AlxGa1−xAs is essentially independent of alloy concentration x. This is the only 4 exception in ternary alloys, arising from Fundamental Characteristics the fortuity that GaAs and AlAs have In this section, several aspects of fun- virtually the same lattice parameter, the damental characteristics are described. difference being as small as ∼0.1%. In Numerical examples are given based on quaternary alloys like Ga In − P As − x 1 x y 1 y AlGaAs (λ ∼ 0.8 µm) and GaInPAs (λ ∼ and Al Ga In − − P, the band-gap en- x y 1 x y 1.3 µm) lasers. ergy can be tuned in a certain range for a fixed lattice parameter by choos- 4.1 ing appropriate pairs of x and y values. Light–Current Characteristics The design of lattice-matched DH struc- tures is facilitated by taking advantage Figure 4 shows an example of output of this degree of freedom in quater- power (P) versus DC injection current (I) nary alloys. In some quantum-well struc- characteristics. The ordinate is the light tures, a certain degree of strain in the power emitted from one end facet, and active material caused by inevitably or essentially the same power is emitted from deliberately introduced slight lattice mis- the other end facet. matching is used to provide better laser The threshold current at room temper- performance. ature is typically several tens of mA. The dependence of the threshold current Ith on device temperature T is phenomenologi- 3.5.2 IV–VI Compounds cally expressed as PbSnTe and PbSSe lasers emitting in the infrared region (3–30 µm) [20] are used in ∝ T ,() high-resolution gas spectroscopy and in air Ith exp 14 T0 pollution monitoring. These lasers operate only at cryogenic temperatures. However, where T0 is a constant referred to as an appreciable spectral tuning is feasible the characteristic temperature and takes by simply changing temperature and injec- a value of 100 to 150 and 50 to tion current, which is extremely desirable 80 K for AlGaAs and GaInPAs lasers, for spectroscopy. respectively. Lasers, Semiconductor 1261
The differential external quantum effi- ciency ηext, defined as the ratio of the 15 number of photons coupled out to the number of carriers injected, is given by
/ ω 1 L−1 ln(1/R) η = P ¯h l = η 2 . ext / i α + −1 ( / ) 10 I e i L ln 1 R (mW) (17) P Above threshold, the stimulated emis- ∆P sion predominates over the nonradia- tive processes and ηi ∼ 1. Typical val- = µ = . α = Light output ues of L 300 m, R 0 3, and i 5 20 cm−1 give η ∼ 0.3. Since hω ∼ E , ∆I ext ¯ l g ηD ∼ (Eg/e)ηext. For GaAs lasers (Eg = Threshold 1.424 eV), we get ηD ∼ 0.4W/A. current The maximum rating for output power istypically5to30mW.Figure5shows 0 0 20 40 60 schematically the three phenomena that Current (mA) determine the maximum power rating. Fig. 4 Light output versus injection current curveinanAlGaAslaser 1. Kink: The nonlinearity in the curve of light versus current caused by lateral- mode instability is referred to as a kink. The light power emitted by injection Since a kink is often associated with a above threshold is expressed as beam-profile shift and the generation of intensity noise, lasers cannot be used − 1 L−1 ln(1/R) = ω I Ith η 2 .( ) P ¯h l i −1 15 e αi + L ln(1/R) Thermally Here, (I − Ith)/e is the rate of excess induced saturation carrier injection beyond the threshold. ηi is the internal quantum efficiency repre- senting the fraction of injected carriers Kink that recombine radiatively and gener- ate photons of energy ¯hωl.Thefac- tor 1/2L−1 ln R−1/(α + L−1 ln R−1) repre- i Catastrophic sents the fraction of the generated photons failure that are coupled out of the cavity through Light output one of the end facets. From Eq. (15), the differential efficiency ηD,definedasthe slope of output power versus current curve above threshold, is written as
− ω 1 L 1 ln(1/R) Current η = P = ¯h l η 2 . D i −1 I e αi + L ln(1/R) Fig. 5 Schematic illustration of the three (16) phenomena occurring at high injection levels 1262 Lasers, Semiconductor
in the kink region for most applica- 4.2 tions. The lateral-mode instability arises Beam Profile and Polarization from the optical power dependence of the refractive index. By incorporating a Typical radiation intensity patterns of laser refractive-index profile in the lateral di- diodes are shown in Figs. 6 and 7. Shown rection to form a waveguide, the power in Fig. 6 are the near-field patterns, that level for kink generation can be apprecia- is, the spatial distributions of optical bly increased. intensity on the end facet in the direc- 2. Catastrophic optical damage: For opti- tions (a) perpendicular and (b) parallel to cal power densities higher than several the junction plane. The far-field pattern MW cm−2, the end facets of semiconduc- (Fig. 7) is the angular distribution of radi- tor lasers may be melted. This results in ant intensity measured at distances several a sudden decrease of output power and mm or more away from the facet, which failure of the device. The catastrophic dam- is mathematically a Fourier transform of ageisconsideredtobebroughtaboutby the near-field pattern. The single-lobed the enhanced light absorption associated radiation patterns show that the trans- with surface states at the end facets. The verse mode of the device is single and importance of this process seems to de- fundamental. Transverse-mode-stabilized pend on materials and photon energies. lasers incorporating appropriate waveg- In GaInPAs lasers (λ ∼ 1.3 µm), the catas- uide structures exhibit such beam profiles trophic optical damage practically does not stably up to reasonable output power occur. levels. 3. Thermally induced saturation:Heating Normally, the angular width or beam of the junction under high current in- divergence (the full width at half maxi- jections raises the threshold current. mum) perpendicular to the junction, θ⊥ ◦ ◦ This may lead to saturation of output (20 –60 ), is larger than that parallel to ◦ ◦ power with an increase in injection cur- the junction θ⊥ (10 –30 )sincethenear- rent, especially in lasers with lower T0 field spot is an ellipsoid because of the values. largedifferencebetweenthestripewidth Intensity Intensity (arb. unit) (arb. (arb. unit) (arb.
−20+2 −20+2 (a)Distance (µm) (b) Distance (µm) Fig. 6 Near-field optical intensity patterns (a) perpendicular; (b) parallel to the junction plane Lasers, Semiconductor 1263 (arb. unit) (arb. (arb. unit) (arb. Radiant intensity Radiant intensity
−20 0 +20 −20 0 +20 (a) Angle (deg) (b) Angle (deg) Fig. 7 Far-field radiant intensity patterns (a) perpendicular; (b) parallel to the junction plane
(several µm) and the active-layer thickness oscillation in DH lasers because of the (∼0.1 µm). θ⊥ can be reduced by adopting higher reflectivity. thinner active layers; then, most of the op- tical energy penetrates into the cladding 4.3 layers, increasing the spot size perpendic- Spectral Characteristics ular to the junction. The output beam from gain-guided Figure 8 shows the variations of lasing lasers can be considerably astigmatic; spectrum with output power in an index- the beam waist perpendicular to the guided AlGaAs laser. At lower powers, junction plane is located essentially on the laser oscillates in several longitudinal the end facet, while the virtual beam modes. Here, the longitudinal-mode wave- waist along the junction plane is formed length λN is determined by the relation in the cavity at a position several tens of µm behind the end facet. This is 1 λN brought about by the difference of the N = L (N : positive integers), 2 η waveguiding mechanism between the two (18) directions. In contrast, in index-guided where η is the effective refractive index of lasers, the beam waist, both parallel and the waveguide and L is the cavity length. perpendicular to the junction plane, is Themodespacingλ is given by located within several µmofthefacet. The astigmatism should be taken into λ2 consideration when the beam is coupled λ = ,() η 19 into lenses. 2 gL The laser beam is linearly polarized η = η − (∂η ∂λ)λ along the junction plane. This is because, where g / is the group in- in a slab waveguide, the facet reflectivity for dex. λ is typically 3 A(˚ λ = 0.8 µm)–8 A˚ the TE mode is higher than that for the TM (λ = 1.3 µm) for L = 300 µm. mode [22]. Since the gain needed to reach As injection is increased, the laser threshold depends on the facet reflectivity power tends to concentrate in a single- (see Eq. (9)), the TE mode is selected for longitudinal mode, while the power in the 1264 Lasers, Semiconductor
Light output P (mW) 15 10 837 m)
835 µ ( λ 836 (nm) l 834 l Wavelength 835 20 25 30 Temperature T (°C)
Wavelength Wavelength Fig. 9 Temperature dependence of the lasing 833 l wavelength
At ranges of temperature and bias cur- 30 40 50 60 rent where the oscillation mode jumps to Current I (mA) a neighboring mode, it is often observed Fig. 8 Lasing wavelength versus injection that the lasing is randomly switched be- current characteristics in an AlGaAs laser tween the two longitudinal modes. The random switching is associated with a large-intensity noise since the output remaining modes saturates. This is com- power is different between modes by mon among index-guided lasers. Gain- (0.1–1%). Some devices exhibit hysteresis guided lasers, in contrast, tend to exhibit in the lasing wavelength versus tem- multiple–longitudinal-mode spectra even perature and/or injection-current charac- at higher powers. teristics. These phenomena have been It can be seen in Fig. 8 that the lasing interpreted in terms of nonlinear mode wavelength shifts toward longer wave- coupling among longitudinal modes [17]. lengths as the output power is increased. The spectral linewidth of a single- This is induced by the temperature rise longitudinal mode is inversely propor- in the active region with the increase tional to the output power to a good ap- in injection current. Shown in Fig. 9 is proximation. The product of the linewidth an example of the wavelength shift due f and the output power P is typically in to a heat sink temperature change at a the range of 1 to 100 MHz mW. This value fixed injection current. Each longitudinal- is 10 to 50 times larger than fST obtained − mode shifts at a rate of 0.5 AK˚ 1 (λ ∼ from the well-known Schawlow–Townes − 0.8 µm)–0.8 AK˚ 1 (λ ∼ 1.3 µm) because formula. An analysis shows that the en- of the temperature dependence of the hancement is expressed as [23] refractive index. In addition, the lasing = ( + α2), ( ) wavelength jumps toward longer wave- f fST 1 20 lengths, as temperature is raised, at − − where α is the linewidth enhancement arateof2AK˚ 1 (λ ∼ 0.8 µm)–5 AK˚ 1 factor defined as (λ ∼ 1.3 µm). This is caused by the tem- perature dependence of the band-gap 4π (∂η/dn) α =− (21) energy. λ (∂g/dn) Lasers, Semiconductor 1265 by the derivatives of the refractive index η and the gain g with respect to the car- rier density n. α represents the magnitude of the amplitude-phase coupling, inher- ent in semiconductor lasers, originating Ith
from the strong dependence of the re- fractive index on the carrier density. The 0 linewidth is determined not only by the t 1/fr direct phase fluctuation caused by spon- d taneous emission but there also exists an additional contribution from the ampli- tude fluctuation since it is coupled into the phase fluctuation through the carrier Light output Current density fluctuation. α takes a value of 2 to 6 depending
on the active-region material, the injected 0 carrier density, and the lasing photon Time energy [24], while in most gas and solid- Fig. 10 Laser response to a staircase injection current. τ :Turn-ondelaytime.f : Relaxation state lasers, α can be virtually taken to d r oscillation frequency bezero.Thisisbecausethegainspec- trum in semiconductor lasers based on a band-to-band transition is asymmetric shown in Fig. 10. There is a delay time τd with respect to the gain-peak frequency for the turn-on of lasing because a finite (corresponding to the laser frequency) and, time is required before the injection car- as a consequence, the associated refractive riers are accumulated to form population index in the vicinity of the lasing fre- inversion. τd is approximated by [25] quency varies appreciably with the injected carrier density. In contrast, in ordinary τ = τ Ith −1 Ith ,() lasers where the lasing transition takes d s tanh 22 I I place between two discrete levels, the gain spectrum is symmetric and the associated where τs is the carrier lifetime at threshold ∼ refractive-index dispersion crosses zero at ( 2ns),Ith is the threshold current, and the lasing frequency. The nonzero value of I is the current pulse height. τd can be τ this parameter affects a number of semi- reduced by prebiasing the laser. Then d conductor laser characteristics including is expressed by the DC bias current I0 linewidth broadening, lateral-mode insta- ( time interval of several nanoseconds be- represents the damping of the relax- fore the steady state is attained. The ation oscillation and is given by relaxation oscillation occurs because of 1 c ∂g the resonant interaction between pho- = + .() τ S0 ∂ 26 tons and carriers; the energy stored in s ng n the cavity is transferred back and forth τp is the photon lifetime of the cavity between the photon and carrier subsys- (∼2 ps) expressed as tems. Therefore, the periodic oscillation of the laser output is accompanied by a τ −1 = c α + 1 1 .() p i ln 27 modulation of carrier density. The tem- ηg L R poral variation of gain spectrum caused by the carrier density modulation leads Va denotes the active-region volume. The to a multiple–longitudinal-mode oscilla- magnitude of the modulated component of output power P(ω) is proportional to tion. Furthermore, the frequency of each S(ω);thatis, individual mode is modulated because of the carrier-density–dependent refrac- 1 c 1 1 Va P(ω) = ¯hωl ln S(ω) . tive index, leading to a broadening of 2 ηg L R the time-averaged spectral linewidth. This (28) phenomenon is referred to as frequency Figure 11 shows an example of relative chirping. The magnitude of the chirping is modulation response |P(ω)/P(0)|= ω2 (ω2 − ω2)2 − ω2 2 1/2 governed by the linewidth enhancement r /[ r ] . A flat re- factor α. sponse at modulation frequencies less When a laser is biased above threshold, than fr is followed by a peak at fr and a the bandwidth of the dynamic response sharp drop at modulation frequencies ex- of the laser is essentially determined by ceeding fr. Therefore fr is the uppermost the relaxation oscillation frequency fr.Let useful modulation frequency. fr is propor- I(ω)eiωt be a small-amplitude sinusoidal tional to the square root of output power current superimposed on a DC bias I0 (>Ith). Then the photon density inside the active region is made up of a DC component S0 and an AC component 10 S(ψ)eiωt. The magnitude of the mod- ulated component S(ω) is analyzed to be [26] 0 2 τpω I(ω) S(ω) = r .(24) −ω2 + ω + ω2 i r eVa Modulation response (dB) –10 Here, the angular relaxation oscillation fr frequency ωr is given by 108 109 1010 Modulation frequency (Hz) S c ∂g ω = π = 0 .() Fig. 11 Calculated small-signal modulation r 2 fr τ η ∂ 25 p g n response. fr is chosen to be 1.3 GHz Lasers, Semiconductor 1267 P and typically takes a value of 3 to 5 GHz GRIN (graded index)–SCH (separate con- at P = 10 mW. In practice, however, the finement heterostructure)–SQW (single- modulation bandwidth may be limited by quantum-well) laser [27, 28]. In the latter the electrical parasitics associated with a structure, the injected carriers are confined specific device structure that leads to a in the SQW, while the laser light is guided rolloff in the modulation response. The by the GRIN waveguide, thus ensuring cutoff frequency due to the parasitic rolloff an effective interaction between carriers varies 3 to 30 GHz depending on the device and light without resort to multiple-well structure. structure. The quantization of electronic states in the well gives rise to a variety of superior 5 laser characteristics over the conventional New Structures and Functions lasers with bulk active region. A major 5.1 benefit is the reduction of the threshold Quantum-well Lasers current. Figure 13 helps us understand the low-threshold characteristics of QW lasers, Quantum-well (QW) lasers have an ac- where the energy distributions of the tive region composed of ultrathin lay- density of states and the injected carriers ers (∼20 nm thick or less) forming are compared between a bulk material and narrow potential wells for injected car- a QW structure. As the parabolic density riers. Illustrated in Fig. 12 are the of states in the bulk material changes into two representative examples of the the staircase density of states in the QW active-region structures in AlGaAs QW structure, the energy distribution of the lasers. Lasers comprising multiple wells injected carriers narrows. The narrower (Fig. 12a) are termed multiple-quantum- energy distribution of the injected carriers well (MQW) lasers [27, 28]. Shown in leads to a narrower gain spectrum, with a Fig. 12(b) is the active region of a higher peak gain value for a given carrier n-cladding Active p-cladding n-cladding Active p-cladding layer layer layer layer layer layer Aly Ga1-y As barriers Alz Ga1-z As Alz Ga1-z As cladding layer cladding layer Aly Ga1-y As Al mole fraction Al mole fraction Alx Ga1-x As well yy (a) Alx Ga1-x As wells (b) GRIN waveguide Fig. 12 Two representative examples of active-region structures in AlGaAs quantum-well (QW) lasers (a) Multiple-quantum well (MQW) laser; (b) Graded index (GRIN)–separate confinement heterostructure (SCH)–single-quantum-well (SQW) laser 1268 Lasers, Semiconductor − Bulk comprising a strained GaxIn1 xAs well and AlGaAs cladding layers grown on Density QW of states a GaAs substrate, laser emission at 0.9 to 1.1µm can be realized by varying the Electron population composition x and the well width. This wavelength region corresponds to the gap Bulk Electron population QW between AlGaAs and GaInPAs laser wave- lengths and is fit for such applications Electron energy as the excitation of Er-doped fibers and Fig. 13 Comparison of energy distribution of blue–green light generation by the second- injected carriers between bulk and quantum-well active regions harmonic generation technique. Furthermore, it is predicted theoret- ically that the strain is beneficial in density. Because of the staircase density obtaining such good laser properties as of states, however, optical gain in QW low-threshold currents, high-relaxation os- lasers saturates for higher injection levels. cillation frequencies, and low linewidth- Since the saturated gain value depends enhancement factors [31, 32]. This is re- on the number of QW layers, SQW is lated to the strain-induced alleviation of preferable for high-Q cavities while MQW the asymmetry in the density of states be- structure should be employed for lossy tween the conduction and valence bands. cavities. A threshold current as low as The strain effects on high-speed modula- 0.55 mA has been demonstrated in a tion characteristics in GaInPAs lasers are high-reflectivity coated (R ∼ 0.8) GRIN- now being studied experimentally. SCH-SQW laser [29]. The narrowing of the gain spectrum in QW lasers as compared 5.2 to bulk lasers is accompanied by an Distributed-feedback and Distributed enhancement of ∂g/∂n (nearly doubled) Bragg-reflector Lasers and a reduction of α (see Eq. (21)). This leads to an enhancement of modulation In place of mirror facets in Fabry–Perot´ bandwidths (see Eq. (25)) and a reduction cavities, periodic gratings incorporated of linewidths (see Eqs. (20) and (21)) under within laser waveguides can be uti- both DC and pulsed excitation [30]. lizedasameansofopticalfeedback. The DH structure is generally formed Integrated optical feedback from the by successive epitaxial growth of materi- periodic grating provides strong wave- als lattice-matched to the substrate. The length selectivity. Devices incorporating lattice-matching condition imposes severe the grating in the pumped region are limitations on the combination of DH termed distributed-feedback (DFB) lasers structure materials. However, the limita- (Fig. 14a), while those incorporating the tion is alleviated to a certain extent if the grating in the passive region are termed grown layer is thin enough to be elasti- distributed Bragg-reflector (DBR) lasers cally strained without any generation of (Fig. 14b) [33]. By virtue of the strong wave- dislocations. Applying this principle to the length selectivity, DFB and DBR lasers active region, we can obtain lasers whose oscillate in a single-longitudinal mode emission wavelengths cannot be realized even under high-speed modulation, in by lattice-matched systems. In QW lasers contrast to Fabry–Perot´ lasers, which Lasers, Semiconductor 1269 Current laser utilizing this principle, which can be tuned continuously over a wavelength p-type range of several nanometers. Here, the Active layer Bragg wavelength of the DBR structure, at which the reflection loss is minimized, n-type is tuned by the injection current I3.The current I2 is adjusted so that one of the res- (a) onant wavelengths of the cavity is brought Current into coincidence to the Bragg wavelength. Active layer Thus, by choosing appropriate pairs of I2 and I3, the oscillation wavelength can be p-type Waveguide varied continuously without mode jump- ing. Tunable lasers are expected to be used DBRn-type DBR in future wavelength-division multiplexing communications systems. (b) 5.3 Fig. 14 (a) Distributed-feedback (DFB); (b) distributed Bragg-reflector (DBR) lasers Semiconductor-laser Arrays The major drawbacks of semiconductor exhibit multiple–longitudinal-mode oscil- lasers, the relatively low output powers, lation when pulsed rapidly. This is an and the conspicuous beam divergence advantageous feature for optical data trans- can be alleviated by integrating multiple mission using fibers because the spec- laser stripes into an array as shown in tral width determines the maximum bit Fig. 16. The stripes are closely spaced so rate transmitted in the presence of fiber- that the radiation from neighboring stripes chromatic dispersion. is coupled to form coherent modes of The carrier-density dependence of the the entire array. The array modes, often refractive index can be exploited to provide referred to as supermodes, are phase- wavelength tunability to the periodic grat- locked combinations of the individual ings [34]. Illustrated in Fig. 15 is a tunable stripe modes and are characterized by the phase relationship between the optical fields supported by adjacent stripes. If Active region Tuning region an array is properly designed so that I1 I2 I3 Active layer Waveguide Fig. 15 Wavelength-tunable distributed Bragg-reflector (DBR) laser Fig. 16 Semiconductor-laser array 1270 Lasers, Semiconductor one particular supermode, which is a Light output uniphase superposition of the individual ◦ stripe modes (so called 0 shift mode), Electrode is excited, a single-lobed, low-divergence beam is attained; the divergence angle θ is approximated by the relation θ ∼ λ/Ns, where s is the center-to-center separation DBR of adjacent stripes and N is the number of Active region stripes. A serious problem in the performance of laser arrays is their liability to multisuper- mode oscillation; the difference among the modal gains of the individual supermodes is so small that the lateral-mode behavior of DBR an array is susceptible to spatial hole burn- ing and temperature distribution in the lateral direction induced at high-excitation Electrode levels. The multisupermode oscillation is generally accompanied by a multilobed output beam, which greatly reduces the utility of arrays for most potential applica- Fig. 17 Vertical-cavity Surface-emitting Laser tions. The maximum output power for the single-lobed beam operation is typically 300 to 500 mW. through a pair of electrodes in the upper and lower surfaces to excite a cylindrical µ 5.4 active region typically of 10- mdiameter, µ Surface-emitting Lasers 1to3 mthick. The surface-emitting laser has sev- Conventional semiconductor lasers utiliz- eral advantages over the conventional ing cleaved end facets as a means of optical edge-emitting laser, such as the capabil- feedback are not fit for monolithic inte- ity of being integrated on a wafer to gration. For monolithic integration, it is form a two-dimensional densely packed desirable to have laser output normal to the laser array, the feasibility of stable wafer surface. Illustrated in Fig. 17 is the single–longitudinal-mode operation be- basic configuration of a surface-emitting causeofthelargemodespacingdue laser incorporating two mirrors parallel to to the short cavity length, and the at- the surface to form a vertical cavity typically tainability of a narrow circular output 5to10µm long [35]. It is crucially impor- beam. tant to increase the reflectivity of mirrors, since the reflection loss otherwise becomes very high because of the very short gain 6 region. Bragg reflectors composed of mul- Summary tilayered semiconductors or dielectrics can be exploited to provide reflectivities exceed- We have reviewed the history and the state ing 95%. Carrier injection is performed of the art of semiconductor lasers. We note Lasers, Semiconductor 1271 here that their successful development is Cladding Layers: Layers that sandwich an the result of contributions of scientists active layer made of materials that have and engineers from a number of fields, wider band gaps than the active layer, p including semiconductor physics, quan- and n doped to facilitate carrier injection tum electronics, crystal growth, processing into the active layer. technologies, and system applications, as is most notably represented by the appli- Distributed Bragg Reflector Laser: Alaser cation to the fiber-optic communications incorporating a periodic grating at both systems. ends of the cavity as a means of opti- Ongoing research and development ac- cal feedback. tivities indicate that the same will be true in the future. There is growing inter- Distributed-feedback Laser: Alaserincor- est in improving laser characteristics by porating a periodic grating in the vicinity miniaturizing device structures. The nat- of the active layer as a means of opti- cal feedback. ural extension of the QW lasers will be quantum wire and box lasers, where we Double Heterojunction Structure: An can take advantage of the two- and three- active-region structure in which an ac- dimensional quantization of electronic tive layer of one material is sandwiched systems. Researchers in quantum optics between two cladding layers of an- are interested in the physics of microcavity other material. semiconductor lasers, which have cavity lengths comparable to light wavelength. Gain-guided Laser: A laser in which the Such a microcavity will drastically alter optical field distribution in the transverse the nature of spontaneous emission and direction is determined by the gain profile can lead to ultralow-threshold (∼1 µAor produced by carrier density distribution. less), low-noise, and high-speed lasers. In order to convert the microcavity lasers as Index-guided Laser: A laser in which the well as the quantum wire and box lasers optical field distribution in the transverse from a laboratory curiosity to a practical direction is determined by a built-in light source, further progress in growth refractive-index profile. and processing technology of semicon- ductor microstructures is indispensable. Linewidth-enhancement Factor: The ratio Integration of these superior devices will between the carrier-induced variations of find new application fields such as optical the real and imaginary parts of susceptibil- interconnection in large-scale-integration ity χ(n), i.e., circuits and two-dimensional information ∂[Reχ(n)]/∂n processing. α = ,(29) ∂[Imχ(n)]/∂n where n is the carrier density. An equiva- Glossary lentexpressionisgiveninEq.(21). 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(1977), Semiconductor Lasers and Heterojunction LEDs.NewYork: Academic Press.