ELEGANT CONNECTIONS IN PHYSICS

Diffraction, Par t 3 CURVED WAVEFRONTS AND FRESNEL DIFFRACTION Fig. 1. The geometry of the Fresnel diffraction analysis. Fig. 1. The geometry of the Fresnel diffraction analysis. When Thomas Young presented his wave theory of Please note the distances ρ = , r = , ro , the angles by DwightdiffractionWhen E.Thomas Neuenschwander, before Young Great presented Brita Southernin’s his Royal wave Nazarene Society theory on ofUniversity 16 January φPlease and θ note, and the the distances annular ring ρ = of area, r = da =, (2ro πρ sinφ, )(theρ d anglesφ). 1800,diffraction it was before cooly Great received. Brita Onein’s Royalhistorian Society of physics on 16 January φ and θ, and the annular ring !"of area da!" = (2πρ= sin!"φ)(ρ dφ). !" !" = !" writes1800, [1]it was cooly received. One historian of physics writes[1]

When“…the Thomas wave Youngtheory presentedmight have hissuffered wave sterility theory and of diffraction oblivion before Great Britain’s Royal Society on January 16,1800, it had“…the not wave sounder theory critics might revived have sufferedit in France. sterility In 1815and oblivion was coolly received. One historian of physics writes[1] Aughad unotstin sounder Frensel, critics a brilliant revived young it in militaryFrance. engineer In 1815 and mathematiciaAugustin Frensel,n, submitted a brilliant to theyoung Acad militaryemy a engineerpaper on and “…thediffraction,mathematicia wave which,theoryn, submitted asmight was have theto the custo suffered Acadm ofemy sterility that a papelearned andr on oblivion body, had not sounder critics revived it in France. In 1815 Augustin Fresnel,wasdiffraction, reported a brilliant which, upon youngasby wastwo military themembers custo engineerm— ofFrançois that and learned Aragomathematician, body, and submitted to the Academy a paper on diffraction, which, as was reported upon by two members—François Arago and wasLouis the Poinsot. custom ofThe that former learned took body, up the was matter reported with upon great by two members—François Arago and Louis Poinsot. The former Louis Poinsot. The former took up the matter with great tookenthusiasm up the matter and drew with Fresnel’s great enthusiasm attention and to the drew almost Fresnel’s attention to the almost identical views of Young published identicalenthusiasm views and of drew Young Fresnel’s published attention fifteen to years the almost previously. fifteen years previously. Although this was the year of the battle of Waterloo, Fresnel paid a generous tribute to Young and Althougidenticalh viewsthis was of Youngthe year published of the battle fifteen of yearsWaterloo, previously. Fresnel theypaidAlthoug corresponded a generoush this was tribute frequentlythe year to Young of until the battle andthe yearthey of Waterloo,corresponded1827 when Fresnel the death of the former put an end to a career full of great promise.” frequentlypaid a generous until the tribute year to 1827 Young when and the they death corresponded of the former Partsputfrequently an 1 andend until2to of a careerthisthe yearseries[2,3] full 1827 of great when considered promise.” the death diffraction of the former produced by plane wave fronts—Fraunhofer diffraction. Its signature phenomena put an end include to a Young’scareer full famous of great double-slit promise.” experiment, which demonstrated the wave nature of light. But of course there is more Parts 1 and 2 of this series[2,3] considered diffraction to the story.Parts 1 An and unobstructed 2 of this series[2,3] wave front considered radiating diffraction from a point source forms an expanding spherical surface. Fresnel diffraction takes into producedaccount this by sphericalplane wave shape, fronts, and Fraunhofer I would be diffraction. remiss to not Its discuss it, and show how to derive it, in this summary of the basic ele- signatureproduced phenomenby plane wavea include fronts,s Young’s Fraunhofer famous diffraction. double- slitIts mentsexperimentsignature of diffraction phenomen that theory.demonstrateda include s theYoung’s wave naturefamous of double light. - slitBut of Ascourseexperiment in the there plane that is morewave demonstrated toparadigm, the story. the Huygens’ Anwave unobstructed nature principle of light. wformsave But front the of working tool for understanding Fresnel diffraction. The principle holdsradiatingcourse that each there from infinitesimal is amore point to source the patch story. forms of Ansurface an unobstructed expanding on a primary spherical wave wave front front may be considered the source of a secondary wave. The signal subsequentlysurface.radiating Fresneldetectedfrom a pointdiff at araction pointsource Ptakes formsbeyond into an the accountexpanding surface this isspherical spherical the superposition of those secondary waves that reach P. surface. Fresnel diffraction takes into account this spherical Considershape, and a pointI would source be remiss S that toradiates not discuss a monochromatic it in this summary wave of angular frequency ω, so that the signal leaving S is proportional shape, and I would be remiss to not discuss it in this summary to cos(of ωtthe). basic It is sufficientelements of to diffraction consider a theory.monochromatic harmonic wave because, so long as the wave equation is linear, any wave can of As the i nbasic the plane elements wave of paradigm, diffraction Huygens theory. ’ Principle forms the As working in the plane tool inwave Fresnel paradigm, diffraction Huygens. The’ Principleprinciple holdsforms thatthe working each infinitesimal tool in Fresnel patch diffraction of surface. on The a primaryprinciple wave holds frontthat each may infinitesimal be considered patch the sourceof surface of a on secondary a primary wave. wave The signalfront may subsequently beAS considered IN THEdetected the PLANE source at a point of a PWAVE secondary beyond thePARADIGM, wave. surface The is HUYGENS’ PRINCIPLE FORMS thesignal superposition subsequently of detectedthose secondary at a point waves P beyond that reach the surface P. is the Consider superposition a point of source those Ssecondary that radiates waves a monochromatic that reach P. wave Considerthe of angular a point workingfrequenc sourcey S ω that, so radiatesthat the asignal monochromatictool leaving S isin Fresnel diffraction proportionalwave of angular to frequenc. yIt ω is, sosufficient that the to signal consider leaving a S is monochromaticproportional to harmonic. Itwave is sufficient because, to so consider long as thea wave equationmonochromatic is linear,cos harmonic any(!" wave) wave can because,be written so as long a superposition as the wave equation is linear,cos any(!" wave) can be written as a superposition be writtenof harmonics. as a superposition Picture the of spherical harmonics. wave Picture front whenthe spherical it has of harmonics. Picture the spherical wave front when it has waveexpanded front when to radiusit has expandedρ, to form to a radiusspheri calρ, andsurface formed σ. Becausea spheri - ofexpanded energy conservation,to radius ρ, to theform amplitude a spheri calof asurface spherical σ. Becausewave cal surface σ. Because of energy conservation, the amplitude of dropsof energy off asconservation, the inverse distancethe amplitude from theof a source. spherical The wave a sphericaldrops off wave as thedrops inverse off as distance the inverse from of the the source. distance The from the spherical wave front at σ carries time-dependence source.spherical The spherical wave front wave at σfront carries at σ time carries-dependence time-dependence cos [ω(t-ρ/c)] with with amplitude amplitude ~ ~1/ 1/ρ,ρ, as as described described by by the the wave wavecos funct function [!ion! − ! with amplitude ~ 1/ρ, as described by the wave function cos [! ! − ! ] ! ψ](ρ,t) = (1) (1) ψ(ρ,t) = ℇ! (1) ℇ! ! cos !" − !" wherewhere k = 2π/λ k = ! 2denotesπcos/λ denotes!" the− !"wavenumber, the wavenumber, and λwith the λwavelength. the The wavelength.wherecoefficient k = 2 π /,Theλ determined denotes coefficient the by wavenumber, the, determinedsource’s withluminosity by λ thethe source’s L, will wavelength. o The coefficient , determined by the source’s be consideredluminosity known; L, will inbe particular, considered for known a light; in wave, particular, for a luminosity ℇL, will be consideredℇ! known; in particular, for a = √(lightL/2 wave,πϵ c) .[4] .[!4] o light wave,o ℇ.[4] On On another another spherical spherical surface surface σ’ σcentered’ centered on onS butS but having having On anotherℇ !spherical= !/2 surface!!!! σ’ centered on S but having ℇ radius ρ + ro (see! Fig. 1), i.e.! a surface that includes point P, radiusradius ρ + rρo (see+ rℇ (seeFig.= Fig.1), ! i.e./ 1),2! a !i.e. surface! a surface that includesthat includes point point P, Eq. P, (1) Eq. (1) sayso that for the wave front passing over P, Fig.FIG. 1.1: The geometry of the Fresnel diffraction analysis. says Eq.that (1) for says the wavethat for front the passing wave front over passing P, over P, When Thomas Young presented his wave theory of Please note the distances ρ = , r = , ro , the angles diffraction ψ(ρ + ro ,tbefore) = Great Britain’s Royal Society. on(2) 16 January φ and θ, and the annular ring of area da = (2πρ sinφ)(ρ dφ). 1800, ψ(ρ it+ wasro ,t) cooly= ℇ! received. One historian .of physics(2) (2) !" !" = !" ℇ! ! ! writes [1] !!! cos [!(! + ! ) − !"] !!!! cos [!(! + !!) − !"] The SPS Observer / Fall 2013 25 “…the wave theory might have suffered sterility and oblivion had not sounder critics revived it in France. In 1815 Aug ustin Frensel, a brilliant young military engineer and mathematician, submitted to the Academy a paper on diffraction, which, as was the custom of that learned body, was reported upon by two members—François Arago and Louis Poinsot. The former took up the matter with great enthusiasm and drew Fresnel’s attention to the almost identical views of Young published fifteen years previously. Although this was the year of the battle of Waterloo, Fresnel paid a generous tribute to Young and they corresponded frequently until the year 1827 when the death of the former put an end to a career full of great promise.”

Parts 1 and 2 of this series[2,3] considered diffraction produced by plane wave fronts, Fraunhofer diffraction. Its signature phenomena includes Young’s famous double-slit experiment that demonstrated the wave nature of light. But of course there is more to the story. An unobstructed wave front radiating from a point source forms an expanding spherical surface. Fresnel diffraction takes into account this spherical shape, and I would be remiss to not discuss it in this summary of the basic elements of diffraction theory. As in the plane wave paradigm, Huygens’ Principle forms the working tool in Fresnel diffraction. The principle holds that each infinitesimal patch of surface on a primary wave front may be considered the source of a secondary wave. The signal subsequently detected at a point P beyond the surface is the superposition of those secondary waves that reach P. Consider a point source S that radiates a monochromatic wave of angular frequency ω, so that the signal leaving S is proportional to . It is sufficient to consider a monochromatic harmonic wave because, so long as the wave equation is linear,cos any(!" wave) can be written as a superposition of harmonics. Picture the spherical wave front when it has expanded to radius ρ, to form a spherical surface σ. Because of energy conservation, the amplitude of a spherical wave drops off as the inverse distance from the source. The spherical wave front at σ carries time-dependence

with amplitude ~ 1/ρ, as described by the wavecos funct [!ion! − ! ! ] ψ(ρ,t) = (1) ℇ!

! cos !" − !" where k = 2π/λ denotes the wavenumber, with λ the wavelength. The coefficient , determined by the source’s luminosity L, will be considered known; in particular, for a ! light wave, ℇ.[4] On another spherical surface σ’ centered on S but having ℇ! = !/2!!!! radius ρ + ro (see Fig. 1), i.e. a surface that includes point P, Eq. (1) says that for the wave front passing over P,

ψ(ρ + ro ,t) = . (2) ℇ!

!!!! cos [!(! + !!) − !"]

ELEGANT CONNECTIONS IN PHYSICS

Now return to Huygens’ principle. Each little patch of area “back side” of the sphere makes it to P (signal from O at θ = on the spherical surface σ, such as the one at a typical point H π never reaches P). How do we account for this time delay as shown in Fig. 1, serves as a source of a secondary wave. when S emits a sinusoidal wave continuously? Here the′ The signal increment that leaves H at time zero arrives at Fresnel zones fall readily to hand. P at time t > 0 with an amplitude modified from that of Eq. The Huygens analysis of spherical wave fronts with (1), is !" strengths that oscillate harmonically proceeds by partitioning σ into a set of Fresnel zones centered on the “north pole” at O. dψ(H→P) = (3) By definition, the vibrations from the inner and outer ! boundaries of a zone are half a cycle out of phase when they ℇ !(!) !" ! arrive at P. To map them in terms of the r of Fig. 1 and the where depends on cos; relating [!(! +them!) − forms !"] one of our tasks. H o wavelength λ, let r = r + λ/2, r = r + λ, r = r + 3λ/2, etc. The factor K(θ) (see θ in Fig. 1), called the “obliquity factor,” 1 o 2 o 3 o All points on σ for which r lies between r and r are within was originallyℇ includedℇ ad hoc by Fresnel because of his o 1 the first Fresnel zone, those for which r ≤ r ≤ r lie in the intuition that the amplitude of the signal that makes it from H 1 2 second Fresnel zone, and so on. to P depends on θ. To see the need for this, look at the

extremes on the surface σ. The da at O will assuredly send

signal towards P, but signal radiated from the da located at O’

never arrives at P. Evidently, K(0) = 1 and K(π) = 0. A

function that exhibits this behavior is

. (4) ! ! !But! until= demonstrated1 + cos ! rigorously, this expression merely offers a plausible candidate for describing Fresnel diffraction. We may assume, however, that K(θ) will change very slowly especially for small θ because . Although ! Fresnel’s intuition was sound, an obliquity factor! derived with ! Fig. 2. Fresnel zones. rigor had to await some 60 yearscos, until! ≈ 1Gustav− ! Kirchhoff FIG. 2: Fresnel zones.

published in 1883 a diffraction theory based on the Consider the situation when S continuously emits a differential equations of waves. sinusoidal wave. Suppose that at time t = r /c a wave crest The conceptual distinction between Eqs. (2) and (3) raises o from O sweeps over P. At the same instant, a wave trough basic questions: Why bother with the Huygens surface σ in from r will be passing through P, and another wave crest the first place? Why not merely go with Eq. (2) and be done 1 from r will be at P, and so on. At any time, the signals with it? After all, integrating Eq. (3) over σ must reproduce 2 arriving at P from boundaries of adjacent Fresnel zones will Eq. (2). be half a cycle out of phase. Their amplitudes at P diminish That’s a good question. If spherical waves always traveled with larger r , thanks to the varying r and the obliquity factor. unobstructed from S to P then Huygens’ principle would n Let us prepare to integrate Eq. (3). For an infinitesimal indeed be a redundant complication. But neither would there area da consider an annular ring on Fig. 1 for which da = be much interest in diffraction, where the interesting results . For fixed ρ, da may be written terms of r arise in the interactions of waves and apertures. The with the help of the law of cosines applied to triangle SHP, superposition of Huygens secondary waves that get through ( 2!" sin !)(! !") an aperture makes possible the calculation of diffraction for (5) patterns. The aperture problem is what we intend to solve, ! ! ! taking into account the curvature of the wave front. Thus the ! = ! + (! + !!) − 2! ! + !! cos !, spherical wave front of interest will be the one with a radius ρ from which that places its surface at the aperture. . (6) !!"# !"

!" = !!!! FRESNELFresnel Zones ZONES Now consider an aperture of area Γ, located at O and oriented with the aperture plane perpendicular to the SP axis. At first glance, our task may appear rather simple. To The number of Fresnel zones N (assumed for simplicity to be predict the diffraction pattern produced by an aperture, we an integer) that pass through the aperture follows from could merely integrate Eq. (3) over limits defined by the aperture boundary, it would seem. However, the finite speed (7) of the wave (e.g., c for light) offers a subtle complication. Let’s approach it this way: If the source S emits a flash of ! ! ! !where+ !An +denotes⋯ ! the= Γarea of the nth Fresnel zone. Its value light of infinitesimal duration, in the reference frame of S that follows by integrating Eq. (6) from rn−1 to rn, so that flash arrives at all points on σ at a time we may set to zero. What does the observer at P (assumed to be at rest with πρλ (8) ρ respect to S) detect? Nothing at all for 0 < t < ro/c. Then at !! !

the time ro/c the first light arrives at P, the light from point O. !! = !!! 1 + !!! 2! + 1 . After the signal from O sweeps by, subsequent signals arrive Whenever λ << ro (typical for visible light and macroscopic later at P from other locations on σ. A secondary signal apertures), then An becomes approximately independent of n: leaving a point on σ at the distance r from P (see Fig. 1) arrives at t = r/c. The last light to arrive at P from σ occurs as (9) ! t → (ro + 2ρ)/c, when some fraction of the signal from the !"#! !! ≈ !!!! .

26 Fall 2013 / The SPS Observer For instance, with a circular aperture of radius a, Eq. (7) gives nature of K(θ) and consider averages under various alternative grouping scenarios. Requiring consistency offers as an estimate the median value between the first and last obliquity ! ! (10) !! ! ! ! factors[5]

! = !! = ! (! + !!). Numerically, if ρ = r = 1 m, λ = 600 nm, and a = 1 mm then o (18) N ≈ 3; but if a = 1 cm then N = 333. Seen another way, if ρ = ! ro then the distance ro that allows only the first Fresnel zone to ! ! ! ! T he ≈ intensity! ! + !(or “.irradiance”) at P is proportional to |ψ|2. pass is which, for a ~ 1 mm and λ ~ 500 nm gives r ! o Slide the detector (move point P) along the SP axis, and a !! ~ 4 m! At larger values for ro and ρ, only a fraction of the first series of intensity maxima and minima appear as passes ! ! Fresnel! zone= , approximately, flat, passes through the aperture. through successively odd and even values of N. (!) The situation reduces to the Fraunhofer diffraction of plane Since adjacent Fresnel zones tend to cancel one! another, waves for which ρ → ∞. Fresnel zone plates can be constructed, typically by Before continuing discussion with this or that aperture, let photographic reduction. These are masks consisting of us investigate the contribution at P of just one Fresnel zone. concentric opaque rings that block out either the even Fresnel The signal arriving at P from the nth Fresnel zone follows zones or the odd ones. For example, should all the even zones by integrating Eq. (3) over that zone only: ! be blocked and an aperture allow no more than 10 zones to ! pass, then the total signal arriving at P would be that of the (11) odd-numbered zones 1 through 9. Assuming all their obliquity !!!ℇ! !! factors to be near unity, !!! !! = !!!! ! !! ! cos ! ! + ! − !" !" where in general θ is a function of r. But K(θ) varies slowly with θ, so it may be safe to assume that, within any single (19) Fresnel zone, its obliquity factor K is approximately constant. n ! ≈ !! + !! + !! + !! + !!. Then Eq. (11) can be integrated at once: Since each for small n, in this example , giving an intensity 25 times that of the first Fresnel zone by ! ! ! itself![6] ! ≈ ! ! ≈ 5!

A special case of an aperture would be no aperture at all! !!!ℇ! !! ! Then N becomes very large. The number of Fresnel zones !! = ! !!! [sin ! + !" − sin ! + ! − 1 !) = (12) across the entire sphere is Nall = 4ρ/λ, which follows by !!! counting the Fresnel zones between O (r = ro) and O’ (r = ro ! ! 5 where k! = (2−π/1λ), α = !k(ρ + ro) – ωt, and + 2ρ). For instance, N ~ 10 for a sphere of 1 cm radius with λ = 400 nm. At least for waves in the optical portion of the . (13) spectrum, including all the Fresnel zones across the entire !!"ℇ! sphere suggests taking the limit as N → ∞. Because K∞ ≈ K(π) ! ! ! ! = !!! sin[! ! + ! − !"] = 0, it follows from Eq. (18) that Z∞ ≈ ½K1. As we anticipated, the signal that passes through P at So far this result has been shown to hold only for odd N. time t, coming from Fresnel zones 1 through N, forms an The case of even N must also be worked out. For large N one !(!) alternating series: finds the same result[5], which might be expected because the difference between one more or one fewer Fresnel zone makes no difference as N → ∞. Therefore all Fresnel zones ! !!! (!) ! !!! ! contributing to the signal at P (no aperture) yields the elegant ! = ! (−1) ! ! ] result !!! ! ! ! ! ! =≡ ! [ ! − ! + ! − … + ( − 1 ) ! (14) (20) ! ! ! Because ! K ! varies. slowly with θ, we may assume that K ≈ !"" ! ! n n+1 !where ≈ ! has been used. About one-quarter of the K . In that case, should N be an even number, then ≈ 0. n intensity of the unobstructed wave comes from the first ! But for odd N one obliquity factor survives. The Kn are not all ! ≈ 1 !(!) Fresnel zone. exactly equal, because K(θ) does diminish with increasing θ. It is interesting to note from Eq. (14), that by separating the This ambiguity presents us with a technical problem. first Fresnel zone’s contribution from all others, Consider, for example, the case N = 5, for which (21) . (15)

(!) ! (! !!) ! ! ! ! ! ! !then for= large! + N ! and from Eq. (20) it follows that !To =make ! use− ! of +the ! near−- cancellation! + ! of adjacent obliquity . Blocking out the first zone of an otherwise factors, should Z5 be grouped as ! unobstructed wave produces a wave function inverted relative !(! !!) ≈ − ! !! , (16) to and carrying half its amplitude. This is the Fresnel diffraction version of the famous Poisson spot, mentioned ! ! ! ! ! ! ! ! !or as= ! − ! + ! − ! + ! ≈ ! earlier! [3] in the context of its Fraunhofer analog that applied Babinet’s principle. (17) Now we can relate the of Eq. (3) to the of Eq. (2). Using Eqs. (13) and (20) and K1 ≈ 1, the unobstructed wave ! ! ! ! ! ! ! ! ! !or in= some ! − other ! −way !? In− the ! absence− ! of≈ an ! explicit, obliquity function arriving at P, dueℇ to all the Huygensℇ secondary factor the sum is ambiguous. Pending a justification of a sources on σ, is result like Eq. (4), we have to make use of the decreasing

The SPS Observer / Fall 2013 27 ELEGANT CONNECTIONS IN PHYSICS

(22) differences in distances. From Fig. 3, the Pythagorean ℇ! !" theorem followed by the binomial expansion gives

!!"" = !!!! sin ! ! + !! − !" . Comparing this amplitude to that of Eq. (2) requires ! ! ! (26) ! ! ! ! and ! ! ! (23) ! = (!! + ! + ! ) ≈ !! + !!! ℇ! ! ! ! !. (27) ℇ = !" ! ! ! so that, for the unobstructed wave, ! ! ! ! ! = (!! + ! + ! ) ≈ !! + !!! If only a small number of Fresnel zones are allowed to pass . (24) through the aperture, the approximation K(θ) ≈ 1 may be used ℇ! for each contributing zone. !!"" ≈ !!!! sin[! ! + !! − !"] There remains the difficulty of the π/2 phase shift between the Now we are ready to integrate Eq. (25) over the aperture. In cosine in Eq. (2) and the sine in Eq. (24). We leave its terms using rectangular coordinates, resolution as a question for the Kirchhoff theory (see Appendix) and turn to diffraction with specific apertures. (28) ! ! ℇ! ! ! ! ! ! !! ! !" !"# ! ! ! /! ! ≈ !!!!! ! ! ! !" where FRESNEL DIFFRACTION WITH AN Fresnel Diffraction with an Aperture APERTURE . (29) !(!!! !!) Interference between the Huygens sources emanating from ! ≡ !!!!! the spherical surface σ produces the wave function that From Eq. (1), the wave that would have arrived as a spherical arrives at P. Apply Eqs. (3) and (23), switch to complex wave front at P in the absence of an aperture would have been notation for the harmonic dependence, and we obtain described by

(25) . (30) ! ! ℇ !(!)!" ![! !!! ! !"] ℇ ![! !!!!! !!"] ! ! !" = !"# ! !! ≈ ! !! ! which will be integrated to find the contribution to the signal Writing and using Eq. (29) at P that comes from the Fresnel zones allowed pass through turns Eq. (28) into !![! !!!!! !!"] ! ! ! ! the aperture. In Eq. (25) the area da is a patch of area on the ℇ = ! (! + ! )! sphere σ, but the aperture is presumably a plane. However, (31) where the sphere σ intersects the aperture in the plane of the ! ! ! ! ! ! ! !"/! aperture, the aperture boundary forms the limits of ! = !! ! ! ! !". integration. Towards this end we construct an xy coordinate The integral modifies the amplitude and phase of thanks to system in the aperture plane, with the origin lying at the the superposition of the portion of σ not blocked by the !! intersection of that plane and the SP axis. Note in Fig. 3 the aperture. Consider a rectangular aperture. The limits on Eq. (31) distinction between ρ and ρo, the latter being the distance from S to the origin of the aperture plane. become

. (32) ! ! ! !! !! !! !"/! !! !"/! ! = !! ! !! ! !" !! ! !" Let and . Then

! = ! ! ! = ! ! (33) ! ! !! !! ! !"! /! !"! /! ! = ! !! !! ! !" !! ! !". Introduce the Fresnel integral[7]

! !"! ! !

! ! ≡ ! ! !" (34)

where ≡ ! ! + !"(!)

! (35) ! !!

! ! ≡ ! cos ! !" and

FIG.Fig. 3.3: AA spherical spherical wave wave front front passing passing through through the the aperture aperture of of ! (36) area Γ,Γ, andand thethe areaarea da.da. ! !! ! ! !Now! Eq. ≡ ( 33)sin may be written !". In Eq. (25) we may approximate ρ ≈ ρo and r ≈ ro in the amplitude. However, the phase is far more sensitive to small (37) ! ! = ! !! ! !! − ! !! ! !! − ! !! .

28 Fall 2013 / The SPS Observer is a complex number. Starting at w = 0, compute produces a result analogous to the thin lens equation of successive values of and , then plot them on the geometrical optics, complex! = ! + plane.!" This procedure generates the Cornu spiral, a phasor diagram (Fig.! 4()! named) !after(!) the French physicist (40) Alfred Cornu (1841-1902). ! ! ! ! ! ! ! + ! = ! 2 where fm = a /2mλ. This exhibits serious chromatic aberration!

There remains the task to confirm with a derivation the

behavior of the obliquity factor, K(θ), and put the whole

Huygens-Fresnel approach on a less cobbled-together and

more rigorous basis. The main ideas are described in the

Appendix, highlighting the principal elements of Kirchhoff’s

diffraction theory.

APPENDIX: KIRCHHOFF’S Appendix: Kirchoff’s Diffraction Theory

DIFFRACTION THEORY

The German physicist Gustav Kirchhoff (1824-1887) derived

a theory of diffraction based on the linear wave equation.[9] It

uses a standard tool of field theory, Gauss’s divergence

theorem, which says the integral of the divergence of a vector field A over a volume V equals the flux of A through the closed surface Σ that forms the boundary of V:

(A1) ! ! ! where! ∙ ! denotes ! ! = the !outward∙ ! !"-pointing unit vector normal to Σ on the patch of area da. Consider the case for some FIG.Fig. 4:4. The The Cornu Cornu spiral spiral (schematic). (schematic). scalar fields! ζ and η. The divergence theorem gives ! = ! !η To graphically determine draw a (A2) straight line connecting and . The length!" of this ! ! !" line is the magnitude ! ! − ! ! ≡ and!! δ is the angle ! [ζ∇ ! + !ζ ∙ !η ] ! ! = ! ! !" !" the line makes with the real! ! axis. ! ! where denotes the directional derivative. From the definitions !of ! − ! and! = !, one may easily Interchange!" ζ and η and subtract the two versions to obtain !" ≡ !ζ ∙ ! show that and and that “Green’s identity,”[10] .[8] For! (a! consistency) !(!) check consider no ! −! = −!(!) ! −! = −! ! , aperture at all: The! limits on both x and y go from −∞ to +∞, . (A3) ! ∞ = ! ∞ = ! 2 2 !" !" and Eq. (35) gives the expected intensity, |ψ| /|ψo| = 1. ! ! ! ! ! To map the wave function and intensity pattern off the SP Now[ζ∇ c!onnect− !∇ thisζ] ! fancy! = mathematics[! !" − ! !"to] some !" physics. For axis—a main objective of diffraction theory— make an x y wave propagation let η be a solution to the inhomogeneous plane parallel to the aperture plane and passing through P, and wave equation, locate the origin at P. To examine the wave function at (x′,y′ ) = (a ,b ), instead of moving P move the aperture in the reverse (A4) direction: Merely translate the origin of the aperture ′ ′ ! ! ! ! ! !,! ′ ′ ! ! coordinates by the amount and . Adjust ∇ !(!, !) − ! !! = −4!"(!, !) the limits on the Fresnel integrals, use Eq. (37), and ψ(a ,b ) where J denotes a source density. Maxwell’s equations lead to results. ∆! = −!′ ∆! = −!′ such a wave equation. The total signal is a superposition of A circular aperture of radius a suggests mapping the plane′ ′ harmonics, where each harmonic carries some angular of the aperture in polar coordinates, so that da = 2πrdr. Now frequency ω. Writing and similarly for Eq. (31) becomes J, the wave equation becomes the inhomogeneous!!"# Helmholtz equation for , ! !, ! = ! ! ! (38) ! !(!) (A5) ! !"#! /! ! ! ! ! Let! = !"! ! , so that ! !". ∇where! ! k += ω!/c!. (The!) = Helmholtz−4!"(!) equation can be solved by the ! ! = !"! /2 method of Green’s functions, for which a function G(r – r ) is ! found that solves the Helmholtz equation when J gets !"! /! !" replaced by a point source, ′ ! ! = ! ! ! !" (39) ! ! (A6) !"#! /! !! ! = 2!!! sin ! . ! ! ! ! Minima (maxima) occur when where m T∇he! Dirac+ ! ! delta= − function4!! ! − ! . is the density of a point ! denotes a positive integer (half an odd integer), which source (think of a point mass):! It vanishes! everywhere except !! !/4 = !" ! ! − !

The SPS Observer / Fall 2013 29 ELEGANT CONNECTIONS IN PHYSICS

at the source itself, where it blows up. Yet its integral is finite Now Eq. (A10) becomes (e.g., the integral over all the particle’s density equals its mass, whatever its distribution). Thus the Dirac delta may be operationally defined as ! ! = !" !!! ℇ! ! !" ! ! ! (A7) !! ! ! −!" ! !!! ! ∙ ! − (A14) ! !" ! ! ! ! ! ! ! ! ! − ! ! ! = !(!) ! ! − ! ! ∙ ! !"2 . 2 provided a lies within V; otherwise the integral vanishes. If λ << r and ρ, then we may neglect the 1/r and 1/ρ terms. Once G is found, then η follows from the original source Eq. (A14) becomes density J, by Green’s theorem,[11]

!!"# !" !!! (A15) (A8) !!ℇ!! ! !∙! ! !∙!

! ! ! ! ! ≈ !! ! !" ! !". where! ! = the integral! ! !( !is− over! ) !all! space.′ As shown in The differential form of Eq. (A15) says that electrodynamics textbooks, the Green’s function for the Helmholtz equation, with outward-traveling spherical waves, ! ![! !!! ! !" ! ] (A16) ! is a damped oscillation:[12] ℇ! ! !∙! ! !∙!

!" ! ≈ ! !" ! where has been used. Eq. (3), when rewritten as a !"# (A9) complex harmonic!!"/!, appears as ! ! – ! = ! where! ! − ! = ! . In Eq. (A3) let η = G, and let ζ be the “optical disturbance,” e.g., the electric potential ψ of the K( (A17) ℇ! electromagnetic! = ! − field.!′ With the help of Eq. (A5) in source- ![! !!! !!"] !" = !"# !) ! !" free regions, and Eqs. (A6) and (A7), Eq. (A3) becomes where Eq. (23) has also been used. Comparing Eqs. (A16) and (A17), the former has the correct phase, and comparing the (A10) coefficients shows the obliquity factor to be

! !" !" !! ! !" !" !For! the= surface Σ! consider− ! the !"arrangement. of Fig. 5. Σ does . (A18) ! not enclose the original wave source S, but it does enclose the ! ! = ! ! ∙ ! − ! ∙ ! point P, and the point H lies on the surface of Σ. It remains to evaluate the dot products between the unit vectors, expressing them in terms of observables. Consider for From Eq. (2), on Σ at H, this application the doubly-connected surface Σ (Fig. 6), whose outer surface Σ2 encloses point P, and whose inner (A11) surface Σ1 excludes the source point S. A study of Fig. 6 ℇ! shows that Eq. (A18) reduces to Eq. (4). // !(!"! !") ! = ! ! so that

Fig. 6. The doubly-connected closed surface Σ made by inner Fig. 5. The closed surface Σ used in Eq. (A10). FIG. 5: The closed surface Σ used in Eq. (A10). FIG.surface 6: The Σ1 and doubly-connected outer surface Σ 2closed. surface Σ made by inner

surface Σ and outer surface Σ . 1 2 . (A12) !" !" ! !(!"!!") ! ! !" ! = (! ∙ !)[ ! − ! ]ℇ ! Similarly, by Eq. (A9),

ACKNOWLEDGEMENTAcknowledgment . (A13) Thanks to Toni Sauncy for carefully reading a draft of this !" !" ! !"# ! manuscript and making useful suggestions. !" ! = (! ∙ !)[ ! − ! ]!

30 Fall 2013 / The SPS Observer REFERENCES AND NOTES [1] W.P.D. Wightman, The Growth of Scientific Ideas (Yale Univ. Press, New Haven, CT, 1953), pp. 146–150. [2] “Elegant Connections in Physics, Diffraction Part 1: Huygens’ Principle and Young’s Interference,” SPS Observer, Winter 2012–13, pp. 20–23. [3] “Elegant Connections in Physics, Diffraction, Part 2: Multiple Point Sources, Apertures, and Diffraction Limits,” SPS Observer, Spring 2013, pp. 18–23. [4] At a distance r from a point source of luminosity (power output) L, the intensity I (power per unit area received) relates to L according to I = L/4πr2. For light waves, the intensity is also given by the time-average of the Poynting’s vector S = 2 Experimenting (E×B)/µo. Since E and B are perpendicular, |E| = c|B| and 1/c = µo εo, the result follows. [5] Eugene Hecht, Optics, 4th ed. (Addison–Wesley, San with your Francisco, CA, 2002), pp. 485–489; Francis A. Jenkins and Harvey E. White, Fundamentals of Optics (McGraw–Hill, New job search? York, 1950), p. 352. The arguments go like this: For odd N the series may be grouped as

! ! ! ! ! ! ! ! ! ! !! = ! + ! − !! + ! + ! − !! + ! + …

!!! ! ! ! ! ! + ! − !!!! + ! + !

! ! ! ! ! ! ! ! ! ! ! ! ! ! = ! + ! − !! + ! − !! + … + ! (a) ! ! where! R! !denotes a “remainder.” If the nth obliquity factor is greater≡ !than+ the! average of its two neighboring ones, then R < 0 and ZN < ½(K1 + KN). A different grouping gives

(b) !! ! !!!! with !! remainder= !! + !T!. A−gain, !if the −nth ! obliquity factor is larger than the mean of its neighbors, then T < 0. Because Kn+1 − Kn is small, for N not very large inequality Eq. (b) gives ZN > ½(K1 + KN). Consistency between the two inequalities requires ZN ½(K1 + KN). The same conclusion holds when the nth obliquity factor is less than the mean of its neighbors. [6] Another ≈way to describe the effect of the obliquity factor is through the “vibration curve,” a phasor diagram. See Hecht (Ref. 5), p 489; Jenkins and White (Ref. 5), p. 286; Bruno Rossi, Optics (Addison-Wesley, Reading, MA, 1965), p. 162. [7] Fresnel integrals and the Cornu spiral are well tabulated numerically and graphically, e.g., see Hecht (Ref. 5), pp. 497– 509; Jenkins and White (Ref. 5), pp. 358–366; Rossi (Ref. 6), pp. 189–198. [8] This result requires the Gaussian integral,

!! ! !!! ! ! !" = . [9] Gustav Kirchhoff, !“Z!ür Theorie der Lichtstrahlen”! (“On the Theory of Light Rays”), Annalen der Physik-Berlin 254 (4), 663–695 (1883). See also Hecht (Ref. 5), p. 510. [10] George Green (1793–1841) was an amazing fellow, brilliant and self-taught in mathematics. Despite being stuck in the family’s bakery and grain milling business, Green laid down some fundamental results in vector calculus and differential equations that would be used later by James Maxwell in his theory of the electromagnetic field. [11] “Discrete Sources and the Continuum, and the Functions of Dirac and Green,” SPS Newsletter (October 1996), pp.10–12. [12] E.g., see J.D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), p. 224.

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