SIMG-455 Physical Optics SPSP-455 Optical Physics

Roger L. Easton, Jr. Chester F. Carlson Center for Imaging Science Rochester Institute of Technology 54 Lomb Memorial Drive Rochester, NY 14623-5604 585-475-5969 585-475-5988 (FAX) [email protected]

29 April 2008 Contents

Preface ix 0.1 Sources: ...... 1 0.1.1 Optics Texts: ...... 2 0.1.2 Physics Texts: ...... 2

1 Wave Optics = Physical Optics 1 1.1 The Wave Equation ...... 2 1.1.1 Traveling Waves ...... 2 1.1.2 Notation and Dimensions for Traveling Sinusoidal Waves in a Medium . . . . 4 1.2 Electric and Magnetic Fields ...... 6 1.2.1 A Note on Units ...... 6 1.2.2 Magnetic Fields ...... 7 1.3 Maxwell’sEquations ...... 9 1.3.1 Vector Operations ...... 9 1.3.2 Vector Calculus ...... 15 1.3.3 Curl of Curl ...... 22 1.4 1. Gauss’Law for Electric Fields ...... 22 1.5 2. Gauss’Law for Magnetic Fields ...... 23 1.6 3. Faraday’sLaw of Magnetic Induction ...... 23 1.7 4. Ampère’sLaw ...... 23 1.8 Solution of Maxwell’sEquations ...... 24 1.8.1 Wave Equation for EM Waves ...... 25 1.8.2 Electromagnetic Waves from Maxwell’sEquations ...... 28 1.8.3 Phase Velocity of Electromagnetic Waves ...... 30 1.9 Consequences of Maxwell’sEquations ...... 31 1.9.1 “Shape”of the Wavefronts ...... 32 1.10 Optical Frequencies –Detector Response ...... 33 1.11 Dual Nature of Light: Photons ...... 34 1.11.1 Momentum of Photons ...... 36

2 Index of Refraction 37 2.0.2 Refractive Indices of Di¤erent Materials: ...... 39 2.0.3 Optical Path Length ...... 39 2.1 Lorentz’sModel of Refractive Index ...... 40 2.1.1 Refractive Index in terms of Phase ...... 41 2.1.2 Refractive Index in Terms of Forces ...... 43 2.1.3 Damping Forces and Complex Refractive Index ...... 47 2.1.4 Dispersion Curves ...... 51 2.1.5 Empirical Expressions for Dispersion ...... 53 2.1.6 Negative Refractive Index ...... 56

v vi CONTENTS

3 Polarization 57 3.1 Plane Polarization = Linear Polarization ...... 57 3.2 Circular Polarization ...... 58 3.2.1 Nomenclature for Circular Polarization ...... 59 3.2.2 Elliptical Polarization, Re‡ections ...... 59 3.2.3 Change of Handedness on Re‡ection ...... 59 3.2.4 Natural Light ...... 60 3.3 Description of Polarization States ...... 60 3.3.1 Jones Vector ...... 60 3.3.2 Actions of Optical Elements on Jones Vectors ...... 61 3.3.3 Coherency Matrix J ...... 64 3.4 Generation of Polarized Light ...... 69 3.4.1 Selective Emission: ...... 69 3.4.2 Selective Transmission or Absorption ...... 69 3.4.3 Generating Polarized Light by Re‡ection –Brewster’sAngle ...... 70 3.4.4 Polarization by Scattering ...... 70 3.5 Birefringence –Double Refraction ...... 71 3.5.1 Examples: ...... 71 3.5.2 Phase Delays in Birefringent Materials; Wave Plates ...... 71 3.5.3 Critical Angle –Total Internal Re‡ection ...... 73

4 Electromagnetic Waves at an Interface 75 4.1 Fresnel Equations ...... 75 4.1.1 De…nitions of Vectors ...... 75 4.1.2 Snell’sLaw for Re‡ection and Refraction of Waves ...... 77 4.1.3 Boundary Conditions for Electric and Magnetic Fields ...... 78 4.1.4 Transverse Electric (TE) Waves = “s”= “ ”Polarization ...... 81 4.1.5 Transverse Magnetic (TM) Waves = “p”=? “ ”polarization ...... 84 4.1.6 Comparison of Coe¢ cients for TE and TM Wavesk ...... 85 4.1.7 Angular Dependence of Amplitude Coe¢ cients at “Rare-to-Dense”Interface 87 4.1.8 Re‡ectance and Transmittance at “Rare-to-Dense”Interface ...... 88 4.1.9 Re‡ection and Transmittance at “Dense-to-Rare”Interface, Critical Angle . . 91 4.1.10 Practical Applications for Fresnel’sEquations ...... 92

5 Optical Interference and Coherence 93 5.1 Addition of Oscillations ...... 93 5.1.1 Addition of 1-D Traveling Waves ...... 94 5.2 Division-of-Wavefront Interference ...... 97 5.2.1 Addition of 3-D Traveling Waves with same ! ...... 97 5.2.2 Superposition of Two Plane Waves with !1 = !2 ...... 103 5.3 Fringe Visibility –Temporal Coherence ...... 6 ...... 106 5.3.1 Coherence of Light ...... 111 5.3.2 Coherence Time and Coherence Length ...... 112 5.3.3 Polarization and Fringe Visibility ...... 114 5.4 Van Cittert - Zernike Theorem ...... 115 5.5 Division-of-Amplitude Interferometers ...... 115 5.5.1 Fizeau Interferometer ...... 116 5.5.2 Michelson Interferometer ...... 118 5.5.3 Mach-Zehnder Interferometer ...... 125 5.5.4 Sagnac Interferometer ...... 126 5.6 Interference by Multiple Re‡ections ...... 127 5.6.1 Thin Films ...... 127 5.6.2 Fabry-Perot Interferometer ...... 135 CONTENTS vii

6 Optical Di¤raction and Imaging 139 6.1 Huygens’Principle ...... 140 6.2 Di¤raction Integrals ...... 141 6.3 Fresnel Di¤raction; the Near Field ...... 143 6.3.1 Fresnel Di¤raction as a Convolution Integral ...... 145 6.3.2 Characteristics of Fresnel Di¤raction ...... 154 6.3.3 Transfer Function of Fresnel Di¤raction ...... 155 6.4 Fraunhofer Di¤raction ...... 156 6.4.1 Examples of Fraunhofer Di¤raction ...... 158 6.5 Coherence of Light and Imaging ...... 160 6.6 Transmissive Optical Elements ...... 161 6.6.1 Optical Elements with Constant or Linear Phase ...... 162 6.6.2 Lenses with Spherical Surfaces ...... 163 6.7 Fraunhofer Di¤raction and Imaging ...... 166 6.7.1 “Di¤raction-Limited”and “Detector-Limited”Imaging Systems ...... 171 6.8 Depth of Field and Depth of Focus ...... 171 Preface

ix

0.1 SOURCES: 1

The science of optics is often divided into three classi…cations based on the scale of the phenomena considered.

I. Geometrical Optics (Ray Optics): Macroscopic-scale Phenomena

(a) considers light to be a RAY that travels in a straight line until it encounters an interface between media. The wavelength and temporal frequency of the light are assumed to be zero and in…nity, respectively: 0; ; ! !1 (b) explains re‡ection and refraction; (c) calculations are simple; (d) useful for designing imaging systems (locate the images and their magni…cations) (e) more di¢ cult to assess the quality of the resulting image

II. Physical Optics (Wave Optics): Microscopic-scale Phenomena

(a) considers light (electromagnetic radiation) to be a WAVE; (b) action of light is described by Maxwell’s equations; (c) “complicated”mathematical calculations; (d) light has wavelength , frequency , velocity c; (e) leads to explanations of re‡ection, refraction, di¤raction, interference, polarization, dis- persion; (f) Useful for assessing the quality of the images.

III. Quantum Optics: Atomic-scale Phenomena

(a) light is a PHOTON with both wave-like and particle-like characteristics; (b) used to analyze the interaction of light and matter on a sub-microscopic level; (c) explains the photoelectric e¤ect, lasers.

Phenomena in the …rst two categories are most relevant to imaging. You may have already explored the concepts of ray optics in an earlier course. This course considers the principles within the second category. The …rst several chapters are reviews of physical principles that are most relevant to wave optics, starting with oscillations, complex numbers, traveling waves, and the Doppler e¤ect. From there, we will consider electromagnetic waves and Maxwell’s equations, leading to the interaction of light within matter as demonstrated by the phenomenon of dispersion. The interaction of light at an interface between two media is considered next, including some discussion of Fresnel’s equations for re‡ectivity and transmittance. Then follows a discussion of polarization of light and optical birefringence. Finally we will consider the aspects of wave optics that are most relevant to optical imaging systems: interference and di¤raction. As we shall see, these are really just two aspects of the same phenomenon that arises from interactions of light from di¤erent locations as the light propagates. This will lead to the discussion of the fundamental limitation to the image created by an optical system; the constraint to the ability of the image to distinguish point sources that are close together.

0.1 Sources:

Many references exist for the subject of wave optics, some from the point of view of physics and many others from the subdiscipline of optics. Unfortunately, relatively few from either camp concentrate on the aspects that are most relevant to imaging. 2 Preface

0.1.1 Optics Texts:

[H] Eugene Hecht, Optics, 4th Edition, Addison-Wesley, 2002.

[PON] Reynolds, DeVelis, Parrent, Thompson, The New Physical Optics Notebook, SPIE, 1989.

[BW] Max Born and Emil Wolf, Principles of Optics, 7th Expanded Edition, Cambridge University Press, 2005.

[GF] Grant R. Fowles, Introduction to Modern Optics (Second Edition), Dover Publications, 1975.

[RHW] Robert H. Webb, Elementary Wave Optics, Dover Publications, 1997.

[FLS] R. Feynman, R. Leighton, M. Sands, The Feynman Lectures on Physics, Addison- Wesley, 1964.

[KF] M.V. Klein and T.E. Furtak, Optics, Second Edition, Wiley, 1986.

[JW] F. Jenkins and H. White, Fundamentals of Optics, 4th Edition, McGraw-Hill, 1976.

[NP] A. Nussbaum and R. Phillips, Contemporary Optics for Scientists and Engineers, Prentice-Hall, 1976.

[I] K. Iizuka, Engineering Optics, Springer-Verlag, 1985.

[FBS] D. Falk, D. Brill, and D. Stork, Seeing the Light, Harper and Row, 1986.

Lawrence Mertz, Transformations in Optics, John Wiley & Sons, 1965.

0.1.2 Physics Texts:

[HR] D. Halliday and R. Resnick, Physics, 3rd Edition, Wiley, 1978.

[C] F. Crawford, Waves, Berkeley Physics Series Vol. III, McGraw-Hill, 1968.

John D. Jackson, Classical Electrodynamics, Third Edition, Wiley, 1998 §6 Chapter 1

Wave Optics = Physical Optics

Ray (geometrical) optics allows determination of locations and magni…cations of images created by optical systems. The study of “wave optics”(also called physical optics) uses the more sophisticated model of electromagnetic radiation as a wave “disturbance” that can exert a force on a charged particle (typically an electron because of its small mass). The wave model allows explanation and modeling of microscopic-scale phenomena, including refractive index and dispersion, polarization, re‡ectance and transmittance at interfaces (Fresnel equations), di¤raction, and interference. The last two items (which arguably are the same phenomenon) are very useful in imaging, because they allow assessment of the quality of images created by imaging systems, which is only somewhat feasible in geometrical optics. The discussion of wave optics begins with brief reviews of the wave equation, Maxwell’sequations (which demonstrate the simultaneous propagation of electric and magnetic …elds), and refractive index. We then use Maxwell’s equation to derive the Fresnel equations for the re‡ectance and transmittance of light at an interface between two media. This naturally leads to the study of polarization. The …nal subject to consider is optical di¤raction, which begins with the discussion of interference and coherence and derives the e¤ect of di¤raction on imaging system performance. The entire discussion will depend on the previous study of linear systems and Fourier methods, especially the discussions of convolution, impulse responses, and transfer functions. The need for wave optics in imaging may be illustrated by considering the observation of the “size” of the spot of light projected through an aperture, as shown in the …gure. If the diameter d of the aperture is “large” (based on some criterion to be determined), then the diameter D of the “image” satis…es D d; this is the domain where ray (or geometric) optics applies. If d is / 1 decreased below some limit, then D d and thus D increases as d decreases. This phenomenon of di¤raction results in the ultimate/ limit to the ability of an imaging system to “resolve”detail in the object.

1 2 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS

Light from a point source through aperture in opaque screen: (a) diameter d1 of aperture is “large” = diameter of projection is D1 d1 (larger hole = larger “image”); (b) if aperture diameter ) / ) 1 decreased, eventually diameter of projection is D2 (d2) (smaller hole = larger “image”). / ) 1.1 The Wave Equation

A wave is a “self-replicating pattern”of energy that travels through space and time (and therefore is a function of both r = [x; y; z] and t). In our usual experience with sound, some attribute of a system (e.g., the air pressure) simultaneously oscillates and propagates. The reciprocating nature of an oscillation clearly requires the joint presence of two forces: 1. A force of motion (inertia) that displaces the physical attribute (e.g., the position angle of the pendulum or the voltage in the LC oscillator) from its equilibrium value; and 2. A static force (the restoring or return force) that acts in opposition to the force of motion so to return the attribute to equilibrium. The restoring force acts as negative feedback; the greater the deviation from equilibrium, the larger the restoring force. In many cases in our experience (acoustic or water waves), the restoring force is supplied by a characteristic of a medium, such as air or water that exerts a pressure on the oscillating attribute. This observation led to the belief that a stationary medium had to exist for light to propagate through a vacuum. Waves generated by a moving oscillation in a stationary medium that travel to an observer exhibit changes in the observed oscillation frequency as Doppler shifts. However, it is easy to show that the sizes of Doppler shifts depend on whether the source or the observer is moving relative to the medium. If a medium for light propagation exists (the “aether”), then it should be possible to use the Doppler shift to determine whether the source or the observer is moving. This led to a quandary that lasted until 1864 when James Clerk Maxwell explained how the “aether”is not necessary because light acts as its “carries its medium with it” and so is its own propagation medium. Maxwell showed that light consists of two wave …elds that are both necessary for light to propagate. The …elds are electric and magnetic, and thus light is called an electromagnetic wave. This observation leads to the interpretation that the magnetic …eld is the “medium”for the electric …eld to propagate and vice versa. The …nal blows against the aether theory of light propagation came in the 1880s when Michelson and Morley demonstrated that the velocity of light propagation was the same regardless of the velocity of the measuring device and in 1905 with the publication of Einstein’sspecial theory of relativity. The simplest oscillations, and thus the simplest waves, are harmonic, which means that the oscillation of the attribute is composed of a single sinusoidal frequency (usually de…ned as a cosine rather than a sine because cos [] is a symmetric function and is more compatible with complex notation). The wave equation was …rst considered by d’Alembert in 1747.

1.1.1 Traveling Waves Recall the di¤erential equation for the motion of a 1-D simple harmonic oscillator:

d2y m = k y [t] dt2 which says that the motion y [t] has the same “form”(“shape”)as its second derivative. The motion of a traveling wave also satis…es a particular di¤erential equation, but the position y is a function of both spatial and temporal coordinates. To avoid confusion between the displacement and the y-coordinate, we will rename the displacement as , so in the case of a single spatial dimension, the position is [z; t] and in the 3-D case, the position is [x; y; z; t] = [r; t]. For simplicity, we derive the 1-D spatial case in this section and then extend the interpretation to 3-D. Because the waveform [z; t] “travels,”it must evaluate to the same value at di¤erent locations and at di¤erent times: [z0; t0] = [z1; t1] 1.1 THE WAVE EQUATION 3

In other words, the traveling wave must have the same “shape” at a di¤erent location and at speci…c di¤erent times, and thus that the derivative of the function [z; t] must evaluate to zero at appropriate pairs [zn; tn]. The derivation makes use of the chain rule of di¤erentiation: d = 0 dt @ dz @ = + @z dt @t @ @ (z vt) dz @ @ (z vt) = + @ (z vt) @z dt @ (z vt) @t @ @ (z vt) dz @ (z vt) = + @ (z vt) @z dt @t @ dz = 1 + ( v) @ (z vt) dt @ dz dz = 1 v = 0 = = v @ (z vt) dt ) dt

The amplitude is identical at coordinate pairs [z1; t1] and [z2; t2] that satisfy z2 = z1 v (t2 t1), where v is the velocity of the wave. This means that we can generate a traveling wave from a sinusoidal function of time by substituting t t z . We can use any function for , but we ! v also already know that sinusoidal functions are convenient because we can decompose any into sinusoidal component functions. The decomposition is discrete or continuous depending on whether is periodic or not.

A sinusoidal wave traveling toward z = + is: 1 z z [z; t] = A0 cos !0 t + 0 = A0 cos !0 t + 0 v v !0 = A0 cos z !0t + 0 v where the quantity !0 has dimensions of radians s = radians , which is an angular spatial frequency, v s mm mm i.e., the rate of change of the phase along the direction of propagation z. Designate this by the scalar k0: [z; t] = A0 cos [k0z !0t + 0]

The wave equation for [z; t] may be con…rmed by evaluating its second partial derivatives with respect to the two coordinates z and t: @ = k0A0 sin [k0z !0t + 0] @z = k0A0 cos (k0z !0t) + 0 2 h i = +k0A0 cos (k0z !0t) + + 0 2 2 h i @ 2 2 = (k0) A0 cos [k0z !0t + 0] = k [z; t] @z2 0 1 @2 = [z; t] = 2 2 ) k0 @z 4 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS

@ = ( !0) A0 sin [k0z !0t + 0] = +!0A0 sin [k0z !0t + 0] @t = +!0A0 cos (k0z !0t) + 0 2 2 h i @ 2 2 = (!0) A0 cos [k0z !0t + 0] = ! [z; t] @t2 0 1 @2 = [z; t] = 2 2 ) !0 @t

The 1-D wave equation is obtained by equating these two expressions:

1 @2 1 @2 2 2 = 2 2 k0 @z !0 @t 2 2 2 2 @ k0 @ 1 @ = 2 = 2 = 2 2 ) @z !0 @t !0 @t k0 The dimensions of the scale factor may be evaluated to see that:

! radians mm 0 = s = k radians s 0 mm This is the velocity of the same point on the wave or the velocity of a point of constant phase on the !0 traveling wave which is commonly called the phase velocity v . This con…rms the 1-D wave k0 equation: @2 1 @2 1-D [z; t] : 2 = 2 2 @z v @t The corresponding equations for waves de…ned over two and three spatial dimensions are:

@2 @2 1 @2 2-D [x; y; t] : 2 + 2 = 2 2 @x @y v @t @2 @2 @2 1 @2 3-D [x; y; z; t] : 2 + 2 + 2 = 2 2 @x @y @z v @t

As an aside, the sums of the second-order partial derivatives with respect to the space coordinates in the 2-D and 3-D cases are both called the Laplacian, commonly denoted 2: r 2 2 2 @ @ 2 1 @ 2-D: 2 + 2 = = 2 2 @x @y r v @t 2 2 2 2 @ @ @ 2 1 @ 3-D: 2 + 2 + 2 = = 2 2 @x @y @z r v @t

The Laplacian appears in many contexts in imaging science, including image processing (in its discrete form).

1.1.2 Notation and Dimensions for Traveling Sinusoidal Waves in a Medium

For a 1-D oscillation in the form of a sinusoid that travels down the z-axis. It may be written using either trigonometric or complex notation; the former is perhaps more obvious, while the latter is easier to use in calculations. 1.1 THE WAVE EQUATION 5

Trigonometric Notation: y [z; t] y0 = A0 cos [z; t] = A0 cos [k0z !0t + 0] f g +i[z;t] Complex Notation: y [z; t] = A0e = Re A0 exp [+i (k0z !0t + 0)] f g The variables and parameters in the equations are:

1. y is the coordinate (or position) of the oscillating attribute of the system, (e.g., angle,voltage, etc.).

2. y0 is the equilibrium value of the attribute.

3. A0 is the amplitude of the oscillation, i.e., the maximum displacement from equilibrium. The units of A0 are the same as those of y, [A] = [y]. 4. z; t are the spatial and temporal coordinates, [z] = length (e.g., mm), [t] = s.

radians 5. !0 is the angular temporal frequency of the oscillation, units of [!0] = second .

cycles !0 6. 0 is the temporal frequency of the oscillation, units of [0] = second = Hertz (Hz) ; 0 = 2 . 1 2 7. T0 = = is the temporal period of the oscillation, units of [T0] = s. !0

8. 0 is the wavelength measured in mm. 2 2 radians 9. k0 = is the angular spatial frequency of the wave, k0 = , [k0] = . 0 0 mm 10. [z; t] is the phase angle of the oscillation (the argument of the sinusoid), measured in radians at position z and time t.

11. 0 = initial phase angle of the wave, i.e., phase angle evaluated at t = 0 AND z = 0; [0] = radians.

1 12. 0 = is the wavenumber, which is a “spatial frequency” (the number of wavelengths per 0 1 unit length), [0] = mm .

The temporal frequencies 0 and !0 = 20 and the spatial frequencies 0 and k0 = 20 of the oscillation are partial derivatives of the phase [z; t]:

@ !0 = @t !0 1 @ 0 = = 2 2 @t @ k = + 0 @z 1 @ = + 0 2 @z

The extension to a 3-D wave requires substitution of a wavevector k0 = [kx; ky; kz] that speci…es the direction of propagation and the spatial wavelength 0 of the wave. The equation for the 3-D wave is:

[x; y; z; t] = A0 cos [kxx + kyy + kzz !0t + 0] = [r; t] = A0 cos [k r !0t + 0] 0 = A0 cos [k r !0t + 0] 0 2 2 2 2 where: k0 = kx + ky + kz = j j 0 q where “ ”denotes the scalar product of the two vectors k and r. 0 6 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS

3-D wavevector k de…nes the direction of propagation of the wave; length of k is proportional to 0 j 0j the reciprocal of the wavelength 0.

1.2 Electric and Magnetic Fields

By 1864, much was known about electric and magnetic e¤ects on materials. Faraday had discovered that a time-varying magnetic …eld (as produced by a moving magnet) can generate an electric …eld, and Ampère demonstrated the corresponding e¤ect that a time-varying electric …eld (as from a moving electric charge) produces a magnetic …eld. Both electric and magnetic …elds were known to be vector quantities that could vary in time and space: the amplitudes of the electric and magnetic …elds as functions of position. Both quantities are spatial 3-D vectors that vary over time, and may be denoted by E [x; y; z; t] and B [x; y; z; t], respectively. The electric …eld E is measured by the force it exerts on a “test”electric charge Q (measured in coulombs). The force is determined by:

F kg m F Q E = E measured in / ) Q / s2 C kg m where the force is measured in newtons s2 as the product of the charge Q and the electric …eld E, which has units of volts per meter (equivalenth i to joules per coulomb).

1.2.1 A Note on Units If you consult other books, you will likely see many di¤erences in the details of these equations due to the di¤erent systems of units used in electromagnetics (and thus in optics); many students (including the author!) …nd it more di¢ cult than probably necessary to cut through the seeming morass of di¤erences. For example, two of the well-known physics texts on the subject, by Lorrain and Corson and by Jackson, use di¤erent systems; the former uses the rationalized MKS system (meter, kilogram, second, which engineers tend to use), while the latter uses CGS units (centimeter, gram, second, more useful for physicists). The MKS system includes many factors of 4. The systems evolved from Coulomb’slaw that evaluates the force between two electrical charges q1 and q2: q1q2 F 2 ^r12 / r12 The constant of proportionality may be called b:

q1q2 F = b 2 ^r12 r12

If the charges are measured in electrostatic units (esu) (also called statcoulombs), the distance r12 2 2 in cm, and the force in dynes (g cm s ), then k = 1. This means that two charges of 1 esu separated by 1 cm produces a force of 1 dyn. But what if the charges are measured in coulombs 1.2 ELECTRIC AND MAGNETIC FIELDS 7

2 [C], the distance in meters [m], and the force in newtons (1 N = 1 kg m s )? The value of k is determined from the knowledge that there are 105 dyn per N, 2:998 109 electrostatic units (esu) per Coulomb, and 100 cm per m:

9 esu 2 2 k 2:998 10 C 9 N m = 2 = 8:988 10 2 1 102 cm 105 dyn C m N 10 10 In words, the force between two charges of 1 C separated by 1 m is nearly 10 N = 4:5 10 pounds of force [lbf], or about 1; 100; 000 tons! The constant k generally is normalized by a factor of 4:

1 1 q1q2 k = F = 2 ^r12 40 ) 40 r12 2 1 12 C 12 F where 0 = = 8:854 10 = 8:854 10 = 0 4k N m2 m where 1 Farad [F] is the unit of capacitance equivalent to:

C2 C 1 F = 1 = 1 N m V (one coulomb per volt). This last expression is a restatement of the equation for electrical capaci- tance: Q [coulombs] = C [farads] V [volts] which also means that one volt is equivalent to one newton-meter per coulomb: N m 1 V = 1 C

This “new” normalization constant 0 is called the “dielectric constant” or “permittivity” of free space. A similar procedure for the magnetic force between two current-carrying wires leads to the exact value of a proportionality constant 0:

7 N 0 4 10 A2 where 1 ampere is one coulomb per second. The magnetic …eld in free space B (the so-called magnetic induction, measured in tesla) is then related to the magnetic …eld intensity H (also called the auxiliary …eld) in free space (measured in amperes per meter) by

B = 0H

1.2.2 Magnetic Fields

The concept of a magnetic …eld is seemingly somewhat less intuitive than that of an electric …eld, so we’llconsider it in somewhat more detail. The magnetic …eld is measured in terms of the “‡ux” (often labelled by ), which is a term arising from the original concept of “lines” of magnetic ‡ux emanating from the magnetic “poles.” In fact, the original CGS unit for magnetic ‡ux was called the “line” (now called the “maxwell,” Mx). The ‡ux emanating from a unit …eld of 1 gauss is 4 lines because the area of the sphere is 4r2. The MKS unit of magnetic ‡ux is the “weber” (1 Wb = 108 Mx), which is the ‡ux which, when changing uniformly in one second, induces 1 volt in 1 turn of a conductor. In electromagnetism, the more important quantity is the magnetic ‡ux ‡ux line 2 line density ( unit area ) labeled B and measured in gauss (CGS, 1 G = 1 cm2 = 10 mm2 ) or tesla (MKS, 8 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS

Wb 1 T = 1 m2 ): Mx 1 G = 1 cm2 Wb N 1 T = 1 = 1 = 104 G m2 A m

Just like electric …elds, magnetic ‡ux will “‡ow”only under the in‡uence of a force or “pressure.” The pressure of an electric …eld is the electromotive force (EMF) measured in volts. The magnetic “pressure” is the magnetomotive force (MMF), which is measured in “ampere-turns,” the force radiated when one ampere of current is pushed through one loop of a conductor. The resulting force A -turn 1000 A -turn …eld is measured in m (MKS) or oersteds (CGS), where 1 oersted = 4 m : The ‡ux within a material varies with the property that the material can “resist”the ‡ux, which is called reluctance and is measured as the induced magnetic “pressure” per unit of magnetic ‡ux, MMF A -turn i.e. as ‡ux (CGS) or Wb (MKS). This property of a material is more commonly expressed as the reciprocal, which is called the permeability . The permeability is analogous to electrical conductivity, the ability of a material to conduct electricity, so it would be appropriate to think of the permeability as the magnetic conductivity. The electric …eld E and the magnetic ‡uxdensity B refer to the “strength”of the force that exists in a vacuum. Two other vector …elds are required when describing propagation of electromagnetism through matter: the electric displacement D and the magnetic …eld intensity H (also called the “magnetizing force” or the “auxiliary …eld”). In this discussion, we assume that any material is linear, isotropic, and homogeneous. “Linearity” means that the response of the medium to an incident …eld varies in proportion to the …eld. The response of “isotropic” media does not change with orientation of the …eld, while the characteristics of a “homogeneous”medium do not vary with position in the medium. The electric displacement D de…nes the total electric …eld within a material that is due to the external …eld E. D is the sum of E and any local …eld P generated within the matter due to the changes in positions of positive and negative electric charges within the atoms (and thus within the material) due to the presence of the E …eld; this induced …eld P is called the “polarization”of the material (which may be called the “polarizability”to avoid confusion with the “polarization”of the electric …eld vector, which we will mention later). H is a similar construct for the magnetic …eld in a medium generated by the incident magnetic …eld (“induction”) B. E and D, and B and H are related by the so-called constitutive equations that are determined by constants of the medium:

D = E B = H where and are the electric permittivity and magnetic permeability of the material, respectively, which we have already seen to be measures of the ability of the electric and magnetic …elds to “permeate” the medium. If is increased, then a larger percentage of the external electric …eld exists within the material. Similarly, for a larger permeability , more of the magnetic …eld is present in the material. Since we will consider propagation of light only in vacuum, D = E and B = H. In vacuum, and are denoted 0 and 0 and both are set to unity in CGS units. In MKS units, the quantities are:

7 N 2 0 = 4 10 (newton per ampere ) A2 12 F 12 C 0 = 8:85 10 (farads per meter) = 8:85 10 : m V m

The permittivity and permeability are larger in matter than in vacuum: > 0, and > 0. In fact (though we won’tdiscuss it in detail), and determine the phase velocity v and the refractive 1.3 MAXWELL’SEQUATIONS 9 index n via: 1 v = p c p n = = v p00

Particularly note the latter equation, which de…nes the index of refraction in terms of the permittivity and permeability of the medium. In many (if not most) cases of interest in optics, 0 = , and therefore: n = in most optical materials r 0 1.3 Maxwell’sEquations

In 1864, James Clerk Maxwell published a paper on the dynamics of electromagnetic …elds in which he collected four previously described equations that relate electric and magnetic forces, modi…ed one (by adding a term to remove an inconsistency), and combined them to demonstrate the true nature of light waves. He demonstrated that the amplitudes of the electric and magnetic …elds would vary as the reciprocal of the distance (rather than the square of the reciprocal of the distance, as is true for static electric …elds). In this way, an electric current in one location has a much larger e¤ect on a distant electric charge than a static electric charge at the same location as the current. As we shall see, Maxwell’s equations demonstrate that energy can propagate from a source through vacuum to a sensor as traveling waves formed of two component vector …elds that are oriented transverse (perpendicular) to the direction of propagation. Both of these …elds, the electric and magnetic …elds, must be present for propagation. Maxwell’s interpretation is a revolutionary concept, as it negated the need for a “medium”(the so-called aether) that had been hypothesized to provide the restoring force for oscillations to propagate. Maxwell’sequations show that the electric …eld provides the restoring force for the magnetic …eld, and vice versa, so it is reasonable to interpret that energy is conveyed by the copropagation of the two …elds.

1.3.1 Vector Operations Any physical or mathematical quantity whose amplitude may be decomposed into “directional” components often is represented conveniently as a vector. In this discussion, vectors are denoted by bold-faced underscored lower-case letters, e.g., x. The usual notation for a vector with N elements is a column of N individual numerical scalars, where N is the dimensionality of the vector. For example, the 3-D vector x is speci…ed by a vertical column of the three ordered numerical components:

x1 x 2 x 3 2 6 7 6 x3 7 6 7 4 5 Both real- and complex-valued scalars will be used as the components xn with the same notation. If the xn are real, then the vector x speci…es a location in 3-D Cartesian space. The individual scalar components x1, x2, and x3 are equivalent to the distances along the three axial directions (commonly labeled x, y, and z, respectively, in the space domain). In common situations, the components of the vector x have dimensions of length, but other representations are possible. For example, we shall often use a convenient representation of a sinusoid in the x y plane that is speci…ed by a vector whose components have the dimensions of spatial frequency (e.g., cycles per mm). To minimize any confusion resulting from the use of the symbol “x”to represent both a vector and a particular component of a vector, a normal-faced “xi”with a subscript will be used to indicate the th th i component of the vector x, while the bold-faced subscripted symbol “xi”denotes the i member 10 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS of a set of vectors. Other notations also will be employed during certain aspects of the discussion, but these cases will be explicitly noted.

De…nitions of the algebraic operations of vectors will be essential to this discussion. For example, the sum of two N-D vectors x and y is generated by summing the pairs of corresponding components:

x1 y1 x1 + y1

2 x2 3 2 y2 3 2 x2 + y2 3 x + y = + = 6 . 7 6 . 7 6 . 7 6 . 7 6 . 7 6 . 7 6 7 6 7 6 7 6 7 6 7 6 7 6 xN 7 6 yN 7 6 xN + yN 7 6 7 6 7 6 7 4 5 4 5 4 5 The notation “x” and “y” used here merely distinguish between the two vectors and their com- ponents; they are not references to the x- and y-coordinates of 2-D or 3-D space. Note that this de…nition implies that two vectors must have the same dimension for their sum to exist.

The de…nition of the di¤erence of two vectors is evident from the equation for the sum:

x1 y1 x1 y1 2 x2 3 2 y2 3 2 x2 y2 3 x y = = 6 . 7 6 . 7 6 . 7 6 . 7 6 . 7 6 . 7 6 7 6 7 6 7 6 7 6 7 6 7 6 xN 7 6 yN 7 6 xN yN 7 6 7 6 7 6 7 4 5 4 5 4 5

Obviously, if the number of dimensions N of the vector is 1, 2, or 3, then the corresponding vector x speci…es a location on a line, on a plane, or within a volume, respectively. This interpretation of a vector as the location of a point in space is so pervasive and intuitive that it may obscure other useful and perhaps more general interpretations of vectors and vector components. For example, we can use the vector notation to represent a two-dimensional (2-D) sampled object. Such an object formed from an N N array of samples or by “stacking” the N columns to create a 1-D vector with N 2 components. This stacking process is known as lexicographic ordering of the matrix. Such a representation often is used when constructing computer algorithms for processing digital images, but will not be considered further here.

The transpose of the column vector x is the same set of scalar components arrayed as a hori- zontal row, and is denoted in this discussion by a superscript T ; another common notation uses an overscored tilde: T x = x1 x2 x3 = ~x By analogy with the usual interpretationh of a vector in Cartesiani space, the length of a vector with real-valued components is a real-valued scalar computed from the 2-D or 3-D “Pythagorean” sum of the components:

N 2 2 (xn) x n=1 j j X The result is the squared magnitude of the vector. The vector’slength, or norm, is the square root of Eq.(3.5), as shown in the …gure and thus also is real valued. 1.3 MAXWELL’SEQUATIONS 11

Length, or “norm”, of 2-D vector with real-valued components.

N 2 x = (xn) j j v un=1 uX From this de…nition, it is evident that the normt of a vector must be nonnegative ( x 0) and that it is zero only if all scalar components of the vector are zero. j j

Vectors with unit length will be essential in the discussion of transformations into alternate representations. Such a unit vector often is indicated by an overscored caret. The unit vector pointing in the direction of any vector x may be generated by dividing each component of x by the scalar length x of the vector: j j x1 x j j 2 x2 3 x x ^x = = 6 j . j 7 x 6 . 7 j j 6 . 7 6 7 6 xN 7 6 x 7 6 j j 7 4 5

The squared-magnitude operation is the …rst example of the vector scalar product (also called the dot product), which de…nes a “product”of two vectors of the same dimension that generates a scalar. Following common mathematical notation, the scalar-product operation will be denoted by a “dot”( ) between the symbols for the vectors. The process also may be written as the transpose of x multiplied from the right by x. Therefore, the scalar product of a vector x with itself may be written in equivalent ways. N 2 T 2 x = (x x) x x = xn j j n=1 X 12 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS

Scalar Product of Two Vectors

It is easy to generalize the squared magnitude operation to apply to distinct vectors a and x that have real-valued components and that have the same dimension N:

x1

2 x2 3 a x aT x = a a a 1 2 N 6 . 7 6 . 7 h i 6 7 6 7 6 xN 7 6 7 N 4 5

= a1x1 + a2x2 + + aN xN = anxn n=1 X In words, the scalar product of two vectors is obtained by multiplying pairs of vector components with the same indices and summing these products. Note that the scalar product of two distinct vectors may be positive, negative, or zero, whereas that the squared magnitude of a vector must be nonnegative. From these equivalent mathematical expressions, it is apparent that the scalar product of vectors with real-valued components in either order are identical:

a x = x a Any process that performs an action between two entities and that may be performed in either order is commutative. The simple concept of the scalar product is the basis (future pun intended) for some very powerful tools for describing vectors and, after appropriate generalization, for functions of continuous variables. The features of the various forms of scalar product are the subject of much of the remainder of this chapter.

The scalar product of an arbitrary “input”vector x with a “reference”vector a has the form of an operator acting on x to produce a scalar g: The appropriate process was just de…ned:

N

x = a x = an xn = g O f g n=1 X It is apparent that a multiplicative scale factor k applied to each component of the real-valued input vector x results in the same scaling of the output scalar:

N N

k x = an (k xn) = k an xn = k g O f g n=1 n=1 X X which demonstrates that the scalar product “operator”satis…es the linearity condition.

The geometrical interpretation of a 2-D vector as the endpoint of a line drawn from the origin on the 2-D plane leads to an alternate expression for the scalar product of two vectors. It is convenient to use 2-D vectors denoted by f n with Cartesian components [xn; yn], or represented in polar coordinates by the length f n and the azimuth angles n. The geometric picture of the vector establishes the relationship betweenj j the polar and Cartesian representations to be:

f = [xn; yn] = [ f cos [n] ; f sin [n]] n j nj j nj where, in this case, xn and yn represent x- and y-coordinates of the vector fn. The scalar product of two such vectors f 1 and f 2 is obtained by applying the de…nition and casting into a di¤erent form 1.3 MAXWELL’SEQUATIONS 13 by using the well-known trigonometric identity for the cosine of the di¤erence of two angles:

f f = x1x2 + y1y2 1 2 = ( f cos [1]) ( f cos [2]) + ( f sin [1]) ( f cos [2]) j 1j j 2j j 1j j 2j = f f (cos [1] cos [2] + sin [1] sin [2]) j 1j j 2j = f f cos [1 2] = f f cos 1 2 j 1j j 2j j 1j j 2j j j where the symmetry of the cosine function has been used in the last step. In words, the scalar product of two 2-D vectors is equal to the product of the lengths of the vectors and the cosine of the included angle 1 2. The knowledgeable reader is aware that this result has been obtained by circular reasoning; we are de…ning the scalar product form by using the Cartesian components of polar vectors, which were themselves determined by scalar products with the Cartesian basis vectors. This quandary is due in part to the familiarity of these concepts. Rather than resolve the issue from …rst principles, we will instead “sweep it under the rug”while continuing to use our existing intuition as a springboard to generalize these concepts to other applications. For example, it is easy now to generalize the scalar product to real-valued vectors a and x with arbitrary dimension N:

a x = a x cos [a x] = a x cos [] j j j j j j j j where represents the “included”angle between the two N-D vectors. This last de…nition for the scalar product may be used to derive the Schwarz inequality for vectors by recognizing that cos [] 1: a x a x j j j j The equality is satis…ed only for vectors a and x that “point” in the same direction, which means that the ratios of the corresponding components of a and x are equal, and that the included angle = 0 radians, which means that the vectors are scaled replicas. Note both the similarity and di¤erence between the Schwarz inequality and triangle inequality for vectors:

a + x a + x j j j j j j In words, the Schwarz inequality says that the scalar product of two vectors can be no larger than the product of their lengths, while the triangle inequality establishes that one side of a triangle can be no longer than the sum of the other two sides. Both relations are illustrated in the …gure.

Graphical comparison of Schwarz’and the triangle inequalities for the same pair of 2-D vectors x and a.

The Schwarz inequality may be combined with the de…nition of the unit vector to obtain an expres- 14 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS sion for the included angle between two unit vectors: a x a x = = ^a ^x = cos [] 1 a x a x j j j j j j j j Cross Product Consider the area of the parallelogram formed by two vectors a and b, as shown:

The cross product of the two vectors a and b yields a third vector orthogonal to those two and with length equal to a b sin [], which is equal to the area of the parallelogram formed by those two j j j j vectors. The area of the parallelogram is a b sin [], as shown, and may be computed as a 3-D vector that points perpendicular to the twoj componentj j j vectors with length equal to the area; the calculation is the “cross product”of the two 3-D vectors. Given the two vectors with components:

a = ^xax + ^yay + ^zaz

b = ^xbx + ^yby + ^zbz the cross product may be de…ned as the determinant of the speci…c 3 3 matrix: ^x ^y ^z a b = det 2 a a a 3 x y z 6 7 6 bx by bz 7 6 7 = ^x (a4ybz azby) +5 ^y (azbx axbz) + ^z (axby aybx) In the example shown, the two vectors are a = ^x a and b = ^x ( b cos []) + ^y ( b sin []), so that j j j j j j az = bz = 0 ^x ^y ^z det 2 a 0 0 3 = ^z ( a b sin []) j j j j j j 6 7 6 b cos [] b sin [] 0 7 6 j j j j 7 It is easy to see that: 4 5 b a = a b Note that the cross product is de…ned for 3-D vectors ONLY!

Triple Vector Product The “triple vector product” is the cross product of two 3-D vectors a and b crossed with a third vector c. The result may be evaluated by straightforward (yet tedious!) calculation and produces the result:

a b c = b (c a) a (c b) = b (a c) a (b c) 1.3 MAXWELL’SEQUATIONS 15 where the fact that the scalar product commutes for vectors with real-valued components. The result is the di¤erence of two scaled replicas of the …rst two vectors, where the scaling factors are the scalar products of c with a and b. The “output”is a vector, as it must be.

1.3.2 Vector Calculus

The four equations are now collected into a group that bears his name. To interpret the four Maxwell equations, we must …rst understand some concepts of di¤erential vector calculus, which may seem intimidating but is really just an extension of normal di¤erentiation applied to scalar and vector …elds. For our purposes, a scalar …eld is a description of scalar values in space (one or more spatial dimensions). One example of a scalar …eld is the temperature distribution in the air throughout the atmosphere. Obviously, a single number is assigned to each point in the space. On the other hand, a vector …eld de…nes the values of a vector quantity throughout a volume. For example, the vector …eld of wind velocity in the atmosphere assigns a three-dimensional vector to each point in the 3-D space. Scalar quantities are denoted by normal-face type and vectors (usually) by underscored bold-face characters, e.g., f [x; y; z] and g [x; y; z] describe scalar and vector …elds, respectively. Unit vectors (vectors with unit magnitude, also called unit length) are indicated by bold-faced characters topped by a caret, e.g., ^x, ^y, and ^z. In preparation of the discussion of vector calculus, we’ll review a few concepts of classical me- chanics. Consider a force descibed by the vector F = ^xFx + ^yFy + ^zFz. The force performs “work” if it acts to create a displacement (described by the vector s).

W = F s If the displacement is the di¤erential element ds = ^x dx+ ^y dy +^z dz, then the scalar product yields a di¤erential element of work dW = F ds and the work resulting by the action of the force from point a to point b is:

b W = F ds Za No work is performed if the force acts at right angles to the displacement so that F ds = 0; the work is “positive” if the force acts in the direction of the displacement (e.g., a weight dropping in a gravitational …eld); the work is “negative” if the force acts in opposition to the displacement (a spring resisting motion). If the work is independent of the path from a to b, then the force is conservative. The work can be evaluated via:

b b W = F ds = ^x Fx + ^y Fx + ^z Fz ^x dx + ^y dy + ^z dz Za Za b b b = Fx dx + Fy dy + Fz dz = T + c Za Za Za where T is the kinetic energy and c is a constant determined from boundary conditions. It the vector force is a function only of the distance from some reference point, it may be written in terms of a scalar function of that distance, called the 3-D “potential”(or “potential energy”) V 16 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS that satis…es the conditions: @V Fx = @x @V Fy = @y @V Fz = @z We can substitute these di¤erential expressions into the integral equation for the work:

F ds = ^x Fx + ^y Fx + ^z Fz ^x dx + ^y dy + ^z dz Z Z @V @V @V = dx + dy + dz @x @y @z Z Z Z = dV = V = T + c Z = T + V E = constant ) The sum of the potential and kinetic energies is the total energy, a constant under these conditions of a “conservative system.”

For a simple illustration, consider the force of gravity near the earth’s surface; the vector force is:

F = ^x Fx + ^y Fx + ^z Fz = 0^x + 0^y + ^z ( mg) so that: @V = 0 = V = c1 @x ) @V = 0 = V = c2 @y ) @V = mg = V = mg dz = mgz + c3 @z ) Z = V [x; y; z] = mgz + (c1 + c2 + c3) = mgz + constant ) 1 E = mgz + mv2 2 Under the conditions of a conservative force, we can write di¤erentiate the …rst two expressions with respect to the “other”variable and equate them:

@ @ @V @2V Fx = = @y @y @x @[email protected] @ @ @V @2V @ Fy = = = Fx @x @x @y @[email protected] @y @ @ = Fx = Fy ) @y @x 1.3 MAXWELL’SEQUATIONS 17

The same pattern of operations leads to two other relations: @ @ F = F @z x @x z @ @ F = F @z y @y z These three are necessary and su¢ cient conditions that a force is conservative. We can then write: @V @V @V F = ^x + ^y + ^z @x @y @z which can be written in a shorthand form by de…ning the …rst-order di¤erential vector operator (called “del”) with three components: r

@ @ @ = ; ; r @x @y @z It also may be written in explicit vector form as: @ @ @ = ^x + ^y + ^z r @x @y @z where ^x; ^y; and ^z are the unit vectors along the x; y, and z axes respectively. Thus we can write that the force is proportional to the gradient of a potential:

F = V r It is easy to show that satis…es the requirements for a linear operator: r (A + B) = A + B r r r ( A) = A r r where A; B are scalar functions and is a numerical constant. The del operator may be applied in the same manner as a vector, though the result is a description of the rater of change of the entity to which it is applied. The operator may be applied to a 3-D “…eld” of scalars (such as f [x; y; z], where f is a scalar “weight”); an example is the measurement of temperature at each point in [x; y; z]. The result f [x; y; z] assigns a 3-D vector to each point in space. (the gradient). The operator may be applied tor a …eld of vectors (e.g. g [x; y; z]) via a scalar product to create a scalar …eld g [x; y; z]; this is the divergence of the vector …eld. Finally, it may be applied to a …eld of 3-D vectorsr to create a di¤erent 3-D vector …eld g [x; y; z] (the curl of the vector …eld). The …rst two operations may be generalized to operate onr or generate 2-D vectors, whereas the curl is de…ned only for 3-D vector …elds.

Gradient of a Scalar Field

Derives a Vector Field f from a Scalar Field f r Application of the del operator to a scalar …eld f [x; y; z] with three dimensions (such as the temperature of air at all points in ther atmosphere) generates a …eld of 3-D vectors which describes the spatial rate-of-change of the scalar …eld, i.e., the gradient of the temperature at each point in the atmosphere is a vector that describes the direction and magnitude of the change in air temperature. In the 2-D case where the scalar …eld describes the altitude of landform topography, the gradient vector is the size and direction of the maximum slope of the landform.

@f @f @f @f @f @f f [x; y; z] ; ; = ^x + ^y + ^z = a vector r @x @y @z @x @y @z 18 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS

As implied by its name, the gradient vector at [x; y; z] points “uphill”in the direction of maximum rate-of-change of the …eld; the magnitude of the gradient f is the slope of the scalar …eld.x: jr j

Scalar …eld represented as contour map and as 3-D display.

The “gradient” of the scalar …eld f [x; y] speci…es the vector f at each point in the …eld. The vector points in the direction of maximum increase in amplituder and its length is equal to this “slope.”

Divergence of a Vector …eld

Derives a Scalar Field g from a Vector Field g: r

@ @ @ g [x; y; z] ; ; [gx; gy; gz] r @x @y @z @g @g @g = x + y + z = a scalar @x @y @z The divergence at each point in a vector …eld is a number that describes the total spatial rate-of- change, such as the total outgoing vector ‡ux per unit volume (‡ux = net outward ‡ow), and thus is equal to: ‡ux = (average normal vector component) (surface area) For a vector …eld g [x; y; z] and an in…nitesmal surface “element”described by its normal di¤erential vector element da directed outward from the volume, the di¤erential element of the “‡ux” z (a scalar) of g through the surface element da is the scalar (“dot”) product of the vector that describes the …eld with the vector normal to the surface. Thus the total ‡ux is the integral over the surface:

dz = g da = z = g da ) ZZsurface area S 1.3 MAXWELL’SEQUATIONS 19

The divergence of the vector …eld describes the total ‡ux through the macroscopic surface area composed of di¤erential surface elements da enclosing the volume. Unless the volume encloses a net “source”or “sink”of the vector …eld (a point from which the vector …eld “diverges”or “converges”), then the divergence over that surface must be zero:

g = g da r ZZsurface area S = 0 if no “source” or “sink” of vector …eld within volume enclosed by S

This is Gauss’theorem (also called the divergence theorem).

The divergence of the scalar …eld f [x; y] is f [x; y] and calculates a scalar at each point in space. The divergence of vectors in a volume is zeror if there are no sources or sinks of the vector …eld in that volume.

Of course, the ‡ux of an electric …eld is not made up of a substance that “moves” through the surface, since the electric …eld is not the “velocity of anything”(in Feynman’swords)

Curl of a Vector Field Derives a 3-D VectorField from a 3-D Vector Field g: The curl of a vector …eld describes the state of spatial nonuniformity of the 3-D vector …eld g [x; y; z]. If the …eld describes the ‡ow of a liquid (matter moving with a velocity), the curl deter- mines whether the liquid is “circulating,”i.e. whether there is a net rotational motion about some location. The word de…nition of “circulation”is:

circulation = (average tangential component) (circumference) Rather than develop the measure from this equation, we again de…ne an operator (the “curl”) and show that it measures the quantity in question. The “curl”of a 3-D vector …eld is the cross product of the di¤erential operator with the …eld: r ^x ^y ^z g [x; y; z] = det 2 @ @ @ 3 r @x @y @z 6 7 6 gx gy gz 7 6 7 [email protected] @g 5 @g @g @g @g = ^x z y + ^y x z + ^z y x = a 3-D vector @y @z @z @x @x @y To visualize curl, imagine a vector …eld that describes the motion of a ‡uid (e.g., water or wind). If a paddle-wheel placed in the ‡uid does not revolve, then the …eld has no curl. If the wheel does revolve, the curl is nonzero. The direction of the curl vector is that of the axis of the paddlewheel when the rotation is maximized and its magnitude is that rotation rate. The algebraic sign of the curl is determined by the direction of rotation (clockwise = positive curl). The paddle rotates only if the vector …eld is spatially nonuniform. Note that the) curl of the …eld typically varies from 20 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS

Figure 1.1: Two vector …elds with nonzero curl. The “paddlewheel” rotates in both cases.

point to point; some points in the …eld can have zero curl while others have nonvanishing curl. Both vector …elds shown in the examples of divergence have zero curl, since a paddle wheel placed at any point in either …eld will not rotate. The vector …eld shown in the next …gure have nonzero curl; the paddlewheel rotates. Two of Maxwell’s four equations relate the curl of one of the two …elds (electric and magnetic) to the time derivative of the other …eld.

Example of Function with Large Curl Consider the 3-D …eld composed of vectors de…ned by the equation: g [x; y; z] = ( y) ^x + (+x) ^y + 0^z The vectors in this …eld lie in the x y plane and those located on the x or y axes are oriented perpendicular to the axes and get longer with increasing distance from the origin.

Vector …eld g [x; y; z] = [ y; x; 0] in the plane z = 0. The curl of this …eld is the vector g = [0; 0; +2], which points out of the plane of the paper towards the reader. r The vectors in this …eld de…ne a “‡ow” in the counterclockwise direction whose velocity increases with radial distance from the origin. This is a 2-D analogue of the complement of the well-known “bathtub drain vortex,” in which the vectors converge on the center of the drain and the velocity increases with decreasing distance from the center. The magnitudes and azimuth angles of the vectors in this …eld may be evaluated easily:

g [x; y; z] = ( y)2 + (+x)2 + 02 = x2 + y2 q 1 x p [x; y; 0] = tan = y Now evaluate the partial derivatives of the vectors: 1.3 MAXWELL’SEQUATIONS 21

@g @g @g x = 0; x = 1; x = 0 @x @y @z @g @g @g y = +1; y = 0; y = 0 @x @y @z @g @g @g z = z = z = 0 @x @y @z The curl of the …eld is obtained by direct substitution:

@g @g @g @g @g @g g [x; y; z] = ^x z y + ^y x z + ^z y x r @y @z @z @x @x @y = ^x (0 0) + ^y (0 0) + ^z (+1 ( 1)) 0 = 0 ^x + 0 ^y + 2 ^z = 2 0 3 6 7 6 +2 7 6 7 4 5 The curl vector points in the direction of the +z axis, i.e., out of the plane of the ‡ow towards the viewer. The direction of the curl determines that the ‡ow is in the x y plane, and the magnitude of the curl is related to the “speed”of the ‡ow, if the vectors describe a motion.

Laplacian of Scalar and Vector Fields

The divergence of the gradient of a scalar function often appears in problems in electromagnetic theory and in imaging; it is a measure of the “curvature” or the local variation of the function f [x; y; z]:

@ @ @ @f @f @f f 2f = ^x + ^y + ^z ^x + ^y + ^z r r r @x @y @z @x @y @z @ @f @ @f @ @f = (^x ^x) + ^x ^y + (^x ^z) @x @x @x @y @x @z @ @f @ @f @ @f + ^y ^x + ^y ^y + ^y ^z @y @x @y @y @y @z @ @f @ @f @ @f + (^z ^x) + ^z ^y + (^z ^z) @z @x @z @y @z @z @2f @2f @2f = 1 + 0 + 0 @x2 @[email protected] @[email protected] @2f @2f @2f @2f + 0 + 1 + 0 @[email protected] @y2 @[email protected] @y2 @2f @2f @2f + 0 + 0 + 1 @[email protected] @[email protected] @z2 @2f @2f @2f = + + 2f @x2 @y2 @z2 r

Because it is a derivative, the Laplacian of a vector …eld is the sum of the Laplacians of the three 22 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS component functions:

( ) g = 2g r r r @2g @2g @2g = ^x x + ^y y + ^z z g @x2 @y2 @z2 The Laplacian is the spatial derivative in the 3-D wave equation:

@2 @2 @2 @ + + = @x2 @y2 @z2 @t2 1 @2 = 2 = ) r v2 @t2 The discrete analogue of the Laplacian is a very useful operation in digital image processing.

1.3.3 Curl of Curl

Since the curl of a vector is a vector, it is possible to compute its curl (the “curl of the curl”). The shorthand solution uses the vector triple product that was presented earlier:

a b c = b (a c) a (b c) = g = g 2g ) r r r r r = g ( ) g r r r r In words, it is the di¤erence of the gradient of the divergence and the Laplacian of the vector …eld.

1.4 1. Gauss’Law for Electric Fields

Gauss’law relates the ‡uxof the electric …eld through a closed surface to the total charge enclosed by the surface. In its simplest terms, Gauss’law states that electrical charges within a volume produce an electric …eld that passes through the surface of the volume and that the ‡ux of the …eld through the surface is proportional to the “amount”of charge within the volume. If the volume is enlarged, then so is the surface area, so the ‡ux density through the surface must decrease at the same rate that the surface area increases. There still can be ‡ux through the surface of a volume even if no charge is enclosed, but the ingoing and outgoing parts of the ‡ux due to external charges will cancel. A di¤erential element of the closed surface is de…ned by its normal vector da. The ‡ux of the electric …eld E through this surface element is its scalar product with da:

d = E da where the symbol “ ”denotes the scalar product of the two vector quantities. According to Gauss’ law, the integral of the ‡ux over the entire enclosing surface is proportional to the volume integral of the charge density [x; y; z] (measured coulombs per unit volume).

Q 1 E da = = [x; y; z] dV ZZsurface ZZZvolume If the surface encloses no charges, then this integral evaluates to zero. The surface integral may be equated to the divergence of the electric …eld via Gauss’integral law: 1.5 2. GAUSS’LAW FOR MAGNETIC FIELDS 23

@ @ @ E da = E [x; y; z; t] + E [x; y; z; t] + E [x; y; z; t] dV @x @y @z ZZsurface ZZZvolume = ( E [x; y; z; t]) dV r ZZZvolume We can equate the volume integrals of the charge density to the volume integral of the divergence, which means that the integrands must be equal: 1 ( E [x; y; z; t]) dV = [x; y; z] dV r ZZZvolume ZZZvolume [x; y; z] = E [x; y; z; t] = ) r

1.5 2. Gauss’Law for Magnetic Fields

Since there are no magnetic analogues for “charges”,the volume cannot enclose a magnetic analogue of . which leads to the particularly simple equivalent forms for Gauss’s law for the magnetic ‡ux density:

B da = 0 ZZsurface = B [x; y; z; t] = 0 ) r In other words, the ‡ux of the magnetic …eld through any enclosed surface ALWAYS is zero. This is often interpreted by the statement that there are no magnetic “monopoles.”

1.6 3. Faraday’sLaw of Magnetic Induction

Michael Faraday observed in 1831 the phenomenon that he called “electromagnetic induction,” that generates (“induces”) electricity in a wire by means of the electromagnetic e¤ect of a current in another wire. In other words, he discovered the basis for the electric transformer. Shortly thereafter, Faraday discovered magneto-electric induction: the production of a steady electric current by mechanical manipulation of a magnet. He attached two wires to a copper disc through a sliding contact. He rotated the disc between the poles of a horseshoe magnet and generated a continuous direct current; in short, this was the …rst generator. The mathematical formulation of Faraday’s magneto-electric induction is called Faraday’s law, which states that the rate of change of a magnetic …eld through a surface is equivalent to the circulation of the electric …eld around the perimeter of the surface. In mathematical terms, the time derivative of the magnetic …eld is proportional to the particular spatial derivative (the curl) of the electric …eld: @B = E @t r Thus a time-varying magnetic …eld produces a spatially varying electric …eld, and vice versa.

1.7 4. Ampère’sLaw

The analogue of Faraday’s law relates the rate of change of the ‡ux of an electric …eld through a surface to the circulation of the magnetic …eld around the perimeter of the surface. Maxwell added a “correction term”due to the ‡ux of electric current (due to moving electric charges) through the 24 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS surface. The corrected form of Ampère’slaw is:

@E B + + J = @t r where the additional source term J is the “current density”of the electric …eld (measured in amperes per unit volume, or coulombs per second per unit volume). Note the change of sign in the two analogues, Faraday’slaw and Ampère’slaw.

We have already seen that: 1 = c2 8 1 where c is the velocity of light, c = 2:99792458 10 m s , which shows that the e¤ect of the spatial variation of the magnetic …eld produces a much smaller temporal change in the electric …eld than vice versa.

There are two “source” terms in Maxwell’s equations: the “static” charge density and the “dynamic” current density J. These can only be nonzero within media (such as copper wire) and thus both vanish in vacuum without sources. If we consider the propagation of light only in a vacuum, neither electric charges nor conductors are present and both source terms vanish.

1.8 Solution of Maxwell’sEquations

(Jackson, Classical Electrodynamics, §6)

In 1864, James Clerk Maxwell collected these four equations and derived the form of the …elds that simultaneously satisfy them in some simple cases. In rationalized MKS units, the di¤erential forms of the equations in the presence of charges and currents are:

E = Gauss’Law for Electric Fields, Coulomb’s Law (A) r B = 0 Gauss’Law for Magnetic Fields (B) r @B = E Faraday’s Law of Magnetic Induction (C) @t r @E B + = J Ampère’s Law (including correction for moving charges) (D) @t r The de…nition of curl may be used to rewrite the four vector equations of Maxwell as eight scalar equations: 1.8 SOLUTION OF MAXWELL’SEQUATIONS 25

@E @E @E x + y + z = Gauss’Law for E (1) @x @y @z @B @B @B x + y + z = 0 Gauss’Law for B (2) @x @y @z @B @E @E x = z y Faraday’s Law (3) @t @y @z @B @E @E y = x z (4) @t @z @x @B @E @E z = y x (5) @t @x @y @Ex @Bz @By + = Jx Ampère’s Law (6) @t @y @z @Ey @Bx @Bz + = Jy (7) @t @z @x @Ez @By @Bx + = Jz (8) @t @x @y These four coupled …rst-order di¤erential equations can be solved directly in many cases. If we assume that the volume surrounding the waves that we analyze includes no sources or sinks (as would be the case for waves emitted by sources at a large distance away), then the charge density = 0 and the current density J = 0 = Jx = Jy = Jz = 0. The corresponding equations are: )

@E @E @E x + y + z = 0 (1) @x @y @z @B @B @B x + y + z = 0 (2) @x @y @z @B @E @E x = z y (3) @t @y @z @B @E @E y = x z (4) @t @z @x @B @E @E z = y x (5) @t @x @y @E @B @B + x = z y (6) @t @y @z @E @B @B + y = x z (7) @t @z @x @E @B @B + z = y x (8) @t @x @y

1.8.1 Wave Equation for EM Waves

Take the curl of both sides of Faraday’s law (Eq.(B) above) and apply the expression for the “curl of the curl”previously mentioned (though not derived). In a region containing no electric charges, then Gauss’law for the electric …eld simpli…es to E = 0 and the result for the left-hand side of r 26 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS the equation is:

( E) = ( E) ( ) E r r r r r r = ( E) E r r r r = 0 2E = 2E r r The right side of the equation may be rewritten by applying Ampère’slaw:

@B @ = ( B) r @t @t r @ @E = @t @t 1 @2E = c2 @t2 Equate these two equations to obtain:

1 @2E 2E = r c2 @t2 which relates the spatial and temporal second derivatives of the electric …eld; this again is the wave equation, but for 3-D functions. It assumes that no energy of the wave is lost to friction or damping forces.

We already inferred that the wave equation for 3-D spatial waves is written:

2 2 1 @ [x; y; z; t] = 2 2 [x; y; z; t] r v @t where v is the velocity of a point of constant phase: this is our old friend, the phase velocity.

The wave equation for electric …elds con…rms our earlier observation:

1 1 = = c = c2 ) r Think about this result for a second; it says that the phase velocity of the wave in a medium is a simple function of two properties of the medium; the electrical permittivity and the magnetic permeability.

The 1-D equation may be written in the form of a “second-order homogeneous” di¤erential equation: @2 1 @2 2 2 2 [z; t] = 0 @z v @t !

Any di¤erential equation is linear, so that if 1 [z; t] and 2 [z; t] are solutions to the equation, so is a 1 [z; t] + b 2 [z; t], where a and b are arbitrary constants. The linearity property means that light beams can pass “through”each other and that waves can constructively or destructively interfere.

This wave equation has the simple solution:

[z; t] = f [z vt] where f [u] is any function that may be di¤erentiated twice. 1.8 SOLUTION OF MAXWELL’SEQUATIONS 27

Proof. @u @u De…ne u z vt = = 1; = v ) @z @t @f @f @u @f @f @2f @2f Apply chain rule: = = = 1 = = @z @u @z ) @z @u ) @z2 @u2 2 2 @f @f @u @f @f @ f @ f 2 Apply chain rule: = = = ( v) = = v @t @u @t ) @t @u ) @t2 @u2 @2f 1 @2f 1 @2f @2f Substitute into wave equation: = = v2 = @z2 v2 @t2 v2 @u2 @u2

The expressions for sinusoidal waves derived in the last section satisfy the wave equation:

2 @ 2 (^x E0 cos [kz !t]) = ^x E0 k cos [kz !t] @z2 2 1 @ 1 2 2 2 (^x E0 cos [kz !t]) = 2 ^x E0 ! cos [kz !t] v @t v !2 = ^x E0 2 cos [kz !t] v ! !2 = v2 = ) k2

If the general solution to the wave equation has the form:

[z; t] = f [z vt] where the shape (“form”) of the function f is arbitrary, then the argument of the function [z vt] (the “phase”)remains constant if the location z increases with increasing time t, so that the “shape” f moves towards z = + with increasing time without changing its shape (i.e., without “dispersion”). 1

A second solution to the wave equation is:

[z; t] = g [z + vt] which moves towards z = . 1

The spatial derivative of the corresponding 3-D wave equation is the sum of the three second partial derivatives:

1 @2 @2 @2 @2 [x; y; z; t] = [x; y; z; t] + [x; y; z; t] + [x; y; z; t] v2 @t2 @x2 @y2 @z2 = 2 [x; y; z; t] r The 3-D wave may still be a sinusoid with argument in radians, so we must be more careful about how the 3-D function becomes a 1-D function. The x, y, and z dependencies all have associated “wavelengths”that may be de…ned by the three corresponding “wavenumbers”kx; ky; kz, which may 28 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS

be written as a “wavevector”k0:

[x; y; z; t] = A cos [ [x; y; z; t]]

= A cos [kxx + kyy + kzz !0t 0] kx x

= A cos 20 k 1 0 y 1 !0t 03 y 6B C B C 7 6B kz C B z C 7 6B C B C 7 = A cos [[email protected] r [email protected] A0] 5 0 The length of the wavevector is proportional to the reciprocal of the wavelength:

2 2 2 2 k0 k0 = kx + ky + kz = j j 0 q The components of the electric and magnetic …elds (Ex;Ey;Ez;Bx;By; and Bz) all satisfy the wave equation.

1.8.2 Electromagnetic Waves from Maxwell’sEquations In the general case, the electric …eld and magnetic …elds can have the form:

E [x; y; z; t] = ^xEx [x; y; z; t] + ^yEy [x; y; z; t] + ^zEz [x; y; z; t]

B[x; y; z; t] = ^xBx[x; y; z; t] + ^yBy[x; y; z; t] + ^zBz[x; y; z; t]

We will now solve these equations for a single speci…c case: an in…nite plane electric …eld wave propagating in vacuum toward z = + . The locus of points of constant phase (often called a wavefront) of a plane wave is (obviously)1 a plane. The electric …eld E has no variation along x or y at a particular value of z, but can vary with z; this variation will be shown to be sinusoidal. This constraint a¤ects the derivatives of the components of the electric …eld: @E @E @E @E @E @E x = x = y = y = z = z = 0 (9) @x @y @x @y @x @y and the 4-D vector …eld E [x; y; z; t] can be written as:

E [x; y; z; t] = E [z; t] = ^xEx [z; t] + ^yEy [z; t] + ^zEz [z; t] (10)

From (9) and Gauss’law for electric …elds (1), we …nd that:

@E @E @E @E x + y + z = 0 = z = 0 (11) @x @y @z ) @z

Since the derivative of Ez with respect to z vanishes, then the z-component of the electric …eld Ez is an arbitrary constant, which we select to be 0:

Ez [x; y; z] = constant 0 (12) ! Therefore, the electric …eld is now expressable in a much simpler form:

E [x; y; z] = Ex [z; t] + Ey [z; t] (13) i.e., the only existing electric …eld is perpendicular (transverse) to z! This alone is a signi…cant result. We can simplify eq.(13) by rotating the coordinate system about the z axis such that E is aligned with the x-axis so that Ey[z; t] = 0 by assumption (14) 1.8 SOLUTION OF MAXWELL’SEQUATIONS 29

The expression for the electric …eld is:

E [x; y; z] = ^x Ex [z; t] (15)

Given the expression for E [x; y; z; t], we can substitute these results into Faraday’sLaw (eqs. 3,4,5) to …nd the magnetic …eld:

@Bx @Ez @Ey @Ey @Bx @Ey = = 0 = = = 0 = Bx [t] is constant (3) @t @y @z @z ) @t @z ) @B @E @E @E @B @E y = x z = x = y = x (4) @t @z @x @z ) @t @z @Bz @Ey @Ex = = 0 = Bz [t] is constant (5) @t @x @y )

We can arbitrarily set the constant term Bz = 0, so the only remaining equation is:

@B @E y = x (4) @t @z which says that the time derivative of the magnetic …eld By is equal to to the negative of the space derivative of Ex. We can now …nd a relation between By and Ex by standard solution techniques of di¤erential equations. Assume that: E is a vector …eld that varies sinusoidally with z:

E [x; y; z; t] = ^x Ex [z; t] = ^x E0 cos [k0z !0t] @By @Ex = = = k0E0 sin [k0z !0t] ) @t @z @Ex = By [z; t] = dt = ( k0E0) sin [k0z !0t] dt ) @z Z Z cos [k0z !0t] By [z; t] = +k0E0 !0 k0 = E0 cos [k0z !0t] !0 E0 = cos [k0z !0t] !0 k0

E0 = cos [k0z !0t] v

E0 = B [z; t] = ^y cos [k0z !0t] ) v where v is the phase velocity of the electromagnetic wave

E0 Ex By = cos [k0z !0t] = = Ex = vBy v v )

Note that the only existing component of B is By, which is perpendicular to the component Ex of E. Also note that the sinusoidal variations of E and B have the same arguments, which means that they oscillate “in phase”. The amplitude of the magnetic …eld is smaller by the factor of the phase velocity v = c, so the e¤ect of the magnetic …eld on observations is generally much smaller and often ignored. 30 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS

1.8.3 Phase Velocity of Electromagnetic Waves

Given the form for the plane electromagnetic wave in a vacuum, we can now use the three Ampère relations to …nd something else useful:

@E @B @B + x = z y (6) @t @y @z @E @B @B + y = x z (7) @t @z @x @E @B @B + z = y x (8) @t @x @y

Because E = ^xE0, only (6) does not vanish:

@ @ E0 (E0 cos [k0z !0t]) = cos [k0z !0t] @t @z v E0 = E0 (!0 sin [k0z !0t]) = ( k0 sin [k0z !0t]) ) v E0k0 = !0E0 = ) v k 1 ! 2 1 = = 0 = = v2 = 0 = ) ! v v2 ) k 0 0 1 v = p 1 which we already knew from the wave equation. In vacuum, 0; 0; v c, and c = . 00 The permittivity and permeability of free space (vacuum) can be measured in laboratoryq experi- ments, thus allowing a calculation of the phase velocity of electromagnetic waves. The permeability and permittivity in vacuum are respectively:

J 7 newton 6 N 6 m 4 10 = 1:26 10 = 1:26 10 0 ampere2 2 2 A A 12 F 12 C 1 0 = 8:85 10 = 8:85 10 m V m Their product is:

12 C 6 J 00 = 8:85 10 1:26 10 V m A2 m 17 C J = 1:11 10 V m C 2 sec m 2 17 1 J sec = 1:11 10 V m C m 2 17 sec = 1:11 10 m2 1 m = c = = 2:99 108 , which agrees with experiment: ) sec r 0 0 In media (i.e., not in vacuum), the phase velocity is slower than c. The same relation may be written using the permittivity and permeability of the medium. The permeability of most optical materials 1.9 CONSEQUENCES OF MAXWELL’SEQUATIONS 31 is close to that of vacuum ( = 0), while the permittivity of optical materials is larger than in vacuum: 12 F > 0 = 8:85 10 m So therefore 1 1 1 1 v = = = = c r r r 0 r 0 0

1.9 Consequences of Maxwell’sEquations

1. Copropagation of E and B: The wave travels in a direction mutually perpendicular to both E and B, and in fact the propagation direction S^ is de…ned by the direction:

S = c2 (E B) –the Poynting vector

2. The force induced by the electric …eld on an electric charge q0 (measured in Coulombs) is speci…ed by the Lorentz relation: v F = q0 E + B c which indicates that a force is induced on the charge by the magnetic …eld only if the charge is moving with velocity v, and that the force is scaled by the velocity of light c. In short, the force due to the electric …eld dominates under most conditions. This observation, and the fact that the waves are transverse to the direction of propagation leads to the concept of of the electric …eld, which is merely its orientation measured from some radial reference line transverse to the direction of propagation.

3. Both electric and magnetic …elds are required for energy and force to propagate. In other words, the two vector …elds copropagate. I like to interpret this result as meaning that the magnetic …eld provides the medium for propagation of the electric …eld and vice versa.

4. In vacuum, E and B are in phase, which means that the phases of the sinusoidal variation of E and B are identical (the phases of the …elds often are out of phase in some types of matter).

1 5. Both E and B travel at c = p00 , the phase velocity of the wave.

6. Energy is carried by both the electric and magnetic …elds, and the magnitude of the energy E E2. / 0

7. There is no limitation on the possible frequencies of the waves, i.e., [0 ! ], which implies the allowed wavelengths are in the interval [ 0] 1 1

8. The average power of the light wave per unit area is the “irradiance,” which is a measure of the average energy through one unit of area in one unit of time. The irradiance is determined 32 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS

from the Poynting vector. For a wave propagating down the z-axis, the irradiance is:

t+ T 1 2 I [x; y; z; T ] = S [x; y; z; t] S [x; y; z; t0] dt0 h i T t T Z 2 T t+ 2 1 2 E0 = c 0 (^xE0 cos [kzz !0t]) ^y cos [kzz !0t] dt0 T t T c Z 2 T t+ 2 2 1 2 E0 2 = ^x ^y c 0 cos [kzz !0t] dt0 T t T c Z 2 T t+ 2 2 1 2 = ^z c0E0 cos [kzz !0t] dt0 T t T Z 2 2 2 1 c0E0 = ^z c0E = ^z 1 [x; y] 0 2 2 In words, the unit vector ^z indicates that the irradiance is propagating in this direction; 1 [x; y] indicates that the irradiance is uniform in the plane orthogonal to this direction, and 2 c0E0 the iradiance “value”is 2 , which has dimensions of:

2 2 2 m F V FV c 0 (E0) = = s m m m2 s h i But the relationship of capacitance Q = C V demonstrates that one farad is one coulomb per volt, which allows restatement:

2 C 2 C 2 FV V V s V VA c 0 (E0) = = = = m2 s m2 s m2 m2 h i because one ampere is one coulomb per volt. Now recall that the product of volts and amperes is power measured in watts, so we can rewrite this one more time

2 VA W c 0 (E0) = = m2 m2 h i This measure of power through one square meter of area is an appropriate measure of irradi- ance.

Relationship between E and B for a linearly (plane) polarized wave traveling from left to right. The two …elds are in phase and each oscillates in its own plane; the two planes are orthogonal.

1.9.1 “Shape” of the Wavefronts Maxwell’s equations state that the electromagnetic …eld propagates away from the source, but the resulting description of the waves does not describe the “shape” of the wavefront, i.e, the locus of 1.10 OPTICAL FREQUENCIES –DETECTOR RESPONSE 33 points in the …eld that have the same phase because they were emitted by the source at the same instant. The shape of the wavefront depends on the shape of the source: a point source emits spherical wavefronts; a line source emits cylindrical wavefronts, and a planar source emits planar wavefronts.

1.10 Optical Frequencies –Detector Response

The general equation for a traveling electromagnetic wave is:

y [z; t] = A0 cos [k0z !0t + 0] z = A0 cos 2 0t + 0 0 = A0 Re exp [+i (k0z !0t + 0)] f g

We “see”electromagnetic radiation with sensors that respond in some form to incident light by conversion to some other, measurable, attribute, e.g., to heat (bolometer), to a change in chem- ical form (photochemical reaction), or to a change in an electrical property (e.g., resistance in a photoconductor). The type of conversion determines the speed of the response of the sensor.

The human eye is sensitive only to light with wavelengths in the range 400 nm / / 700 nm. However, this is not the case for all lifeforms. The pit viper can see radiation emitted by humans at = 10 m with special receptors on the sides of its head. Clearly, the frequencies of visible wavelengths are quite large: c = 400 nm = = = 7:5 1014 Hz ) = 550 nm = = 5:5 1014 Hz ) = 700 nm = = 4:3 1014 Hz ) The frequency bandwidth of visible light is:

14 = max min = 3:2 10 Hz = 320 THz and the temporal period of an optical wave is therefore

1 15 T = = 1:8 10 s

Sensors, including the human eye, cannot respond at this speed and therefore cannot detect the periodic oscillation of wave amplitude: the optical phase. Human eyes and other sensors of visible light “see”a constant response to incident light. As asides, we mention that it is possible to measure relative phase of light using indirect techniques, such as optical interference used in holography, and that human ears also cannot sense phase in sound waves at frequencies above a few Hz. It is possible to use the relative time delay in sound arrival at your two ears to locate sound sources. For other types of waves, e.g., waves in media (such as water waves) and lower-frequency elec- tromagnetic waves (e.g., radio waves or microwaves), the amplitude and phase is measurable. This allows many more detection techniques to be applied, such as modulation (mixing) for phase mea- surement. Now consider the e¤ect of time averaging; the average amplitude of a sinusoidal wave measured 34 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS

over the time interval Td is:

1 Td y [z; t] = y [z; t] dt h i T d Z0 1 Td = A0 cos [k0z !0t + 0] dt T d Z0 A0 t=Td = sin [k0z !0t + 0] t=0 !0Td j

1 2 If Td is large, then y [z; t] 0. If Td = , then y [z; t] may be …nite. However, the power / 0 !0 of the sinusoidal waveh (itsi squared ! magnitude) does noth averagei to zero:

1 Td I [z] y2 [z; t] = y2 [z; t] dt / T d Z0 Td 1 2 2 = A cos [k0z !0t] dt T 0 d Z0 2 Td A0 2 = cos [k0z !0t] dt T d Z0 A2 T A2 = 0 d = 0 Td 2 2 2 2 A0 1 = I [z] y [z; t] = if Td >> ) / ) 2

2 1 because the average value of cos [x] = 2 .

2 1 x (a) y1 [x] = cos [2x] (red dashed line); (b) y2 [x] = cos [2x] = 2 1 + cos 2 1 (solid black ( 2 ) 2 1 line); (c) cos [2x] = 2 (solid blue thick line).

Detectors of visible light are sensitive to time-averaged power (called irradiance).

1.11 Dual Nature of Light: Photons

Though the wave picture of light is essential in analyzing the resolution of imaging systems, the “particle” picture of light is often more appropriate in the study of imaging sensors. Consider images created of light with di¤erent exposure times. Photographs taken in with shorter exposures are made with fewer photons and generally look “grainier”because of statistical uncertainty in the number of counted photons is more apparent: 1.11 DUAL NATURE OF LIGHT: PHOTONS 35

Images created from increasing numbers of photons, showing increase in signal-to-noise ratio.

The “particle of light” is the photon, which conveys energy in discrete “packets” or “quanta.” Einstein demonstrated that the energy carried by a photon is proportional to its temporal oscillation frequency: c E = h = h = ! ~ where h and ~ are the various ‡avors of Planck’s constant; ~ (called “h-bar”) is normalized by a factor of 2:

34 27 h = 6:625 10 J s = 6:625 10 erg s h 34 ~ = = 1:054 10 J s 2 For example, the energy per photon at = 550 nm is:

8 m 27 3 10 s 12 E = 6:625 10 erg s = 3:6 10 erg @ = 550 nm 550 nm

Large swarms of photons may be analyzed only by their statistics, which requires the study of statistical mechanics and is beyond our scope. For our purposes, we will make only the briefest mention of the physics. Classical mechanics with distinguishable particles may be analyzed by Maxwell-Boltzmann statistics, where the number of particles with energy state En in an ensemble of many particles (N0) held at temperature T is a decaying exponential distribution:

En Nn = N0 exp kT 23 1 k = Boltzmann’s constant = 1:38 10 JK which clearly indicates that Nn as En . Quantum mechanics involves indistinguishable particles whose statistics depend on the “spin”characteristics# " of the particles; particles with integer spins are called “bosons”and obey Bose-Einstein statistics, including photons, which have unit spin. Bosons tend to cluster in an energy state, which means that the electromagnetic …eld created by an ensemble of photons clustered in one energy (i.e., they have the same corresponding frequency) appears as a continuous electromagnetic wave. Electrons are fermions, which must each have a distinctive energy and thus cannot cluster in one state. The “photon ‡ux” is the number of photons per second in a light beam, which may be evaluated from the total power of the beam (measured in watts)

W 2 P [W] I 2 A m = = m h [J s] [Hz] h [J s] [Hz] 36 CHAPTER 1 WAVE OPTICS = PHYSICAL OPTICS where I is the irradiance of the light and A is the cross-sectional area. Typical ‡uxes per unit area for some sources are shown in the table:

photons Light Source A sec mm2 h i focused laser 1020 unfocused laser 1015 bright sunlight 1012 indoor light 1010 twilight 108 moonlight 106 starlight 104

The “pattern”of photon arrivals measured by a sensor tells something about the source. Random (incoherent) light sources (such as light bulbs) emit photons with a Bose-Einstein distribution of random arrival times. Coherent light sources, on the other hand, emit photons with a Poisson distribution, which is more uniform but still random.

1.11.1 Momentum of Photons Atoms that emit photons “recoil” in the opposite direction, and surfaces that absorb photons also recoil. The momentum of a single photon is h p = = k ~ The pressure due to radiation is the force per unit area, which is equal to the energy per unit volume, or the energy density. Radiation pressures are often neglected, but cannot be if the mass is small or the ‡ux is large, e.g., in the motion of comet tails or spacecraft, in stellar interiors, and in the light of lasers. Chapter 2

Index of Refraction

The idea that the refractive index relates the velocity of light in a medium to that in vacuum is ubiquitous, but does not by itself address the reason “why”light slows down within a medium. This section attempts to address this issue.

The refractive index n is the dimensionless ratio of the velocity of light in vacuum to that in the medium: c n 1 v 1 n = r 0 0 = = 1 r 0 0 r 0 r = n2 = ) 0 in dielectric materials that have no free electrons to conduct electric current. The ratio is the 0 relative permittivity of the medium. In a dielectric medium with > 0, the refractive index n > 1. We can relate refractive index to the wavelength of light in the medium after recognizing that E = h and energy conservation requires that not change in di¤erent media. This means that the wavelength must change. If the wavelength in vacuum is 0 the wavelength in a medium with index n is 0: 0 = 0 1 n 2 2n 2n0 n!0 = k0 = = = = ) 0 0 c c In words, the wavenumber (length of the wavevector) in a medium is scaled by the index n of the medium.

In nondielectric materials such as metals or at wavelengths near dielectric absorption features, the index of refraction is complex valued and often is denoted by n~ and its imaginary part by : n~2 (n + i)2 = 00 (some authors write n~ = n (1 + i), where is the “attenuation index”). The length of the 37 38 CHAPTER 2 INDEX OF REFRACTION propagation wavevector for a complex-valued index is: ! ! k =n ~ 0 = (n + i) 0 0 c c !0 Re k0 = n f g c !0 Im k0 = f g c This means that the wavevector k also is complex valued. If ^s is the unit vector in the direction of k, we can write the electric …eld:

E [x; y; z; t] = E exp [+i (k r !0t)] 0 0 = E exp [+i ( k (^s r) !0t)] 0 j 0j = E exp [+i (k0 (^s r) !0t)] 0 !0 = E exp +i (n + i) (^s r) !0t 0 c h n !i0 = E exp +i!0 ^s r t exp (^s r) 0 c c h i h i Note the distinct multiplicative exponential factors: the …rst has a complex-valued argument and represents a …eld that propagates sinusoidally, while the second has a real-valued negative argument that decays with increasing distance r = r. j j If we assume that the direction of propagation ^s is in the direction of r (as would be the case for a plane wave), then ^s r = ^s r cos (0) = r = r j j j j j j and the electric …eld may be written in a simpli…ed form:

n !0 n r E0 exp i!0 r t exp r E0 exp i!0 r t exp c c c 0 h i h i h i where the parameter 0 is de…ned in terms of the wavelength 0 measured in vacuum and the imaginary part of the refractive index:

1 !0 c 0 0 = = c !0 2 m s The quantity 0 has dimensions s 1 = m and is the distance over which the …eld amplitude decreases 1 by a factor of e = 0:368; it is called the skin depth of the medium. The amplitude decreases exponentially with increasing r, so that the electric …eld is attenuated as it propagates. Consider the e¤ect in a conductive medium where is large. This means that 0 is small and the electric …eld decreases rapidly with distance “into” the conductor. This observation demonstrates that electric current is propagated along the surfaces of conductors and not within the “bulk.”

If the imaginary part of the refractive index is zero, there is no attenuation and the electric …eld is the simple oscillating term:

n r E0 exp i!0 r t = E0 exp i!0 t c v h i which con…rms that the velocity in the medium is the ratio of the velocity in vacuum and n: c v = n The magnetic …eld is obtained from the electric …eld via the cross product with the electric-…eld 39 vector: k E n + i B = = (^s E) ! c 0 which indicates that it also decays with with depth within a conductor.

2.0.2 Refractive Indices of Di¤erent Materials: Values of the refractive index for common nonconductive materials include:

Medium Refractive Index n

vacuum = 1:0 (by de…nition)

oxygen = 1:000271 at S.T.P. ( 20 C, 1 atm) air = 1:0002929 at S.T.P., 1:01241 at 42 atm water 1:33 4 = = 3 “crown”glass 1:5 = 3 = 2 “‡int”glass = 1:7 diamond = 2:417 germanium = 4:0 (though only transparent at wavelengths ' 2 m)

The entries for air at di¤erent pressures demonstrate that n increases with increasing pressure, which sati…es the intuition that the refractive index should increase with increasing number density of atoms. This also means that n decreases with increasing temperature. This variation in n with pressure and temperature is the source of “atmospheric image blur,” where an image through the atmosphere changes from moment to moment due to variations in temperature and pressure that change the refractive index. The image moves both transversely (around the image plane) and longitudinally (to go “in and out of focus”). These changes happen over time intervals of the order 1 of 100 sec, which means that longer exposures are blurred due to image motion in exactly the same fashion as a long-exposure image of a moving object taken with a …xed camera is blurred. Observations of atmospheric turbulence indicate that images may be modeled as created through isolated areas of the atmosphere within which the refractive index is approximately uniform (“isopla- 2 natic patches”) if measured over time intervals = 10 sec. The apparent diameter of these regions is of the order of 100 mm, which sets the ultimate resolution limit for images of astronomical objects taken through the atmosphere. The complete picture of the e¤ect of the atmosphere on images also depends on the location of the medium relative to the object and imaging system. Think of the two complementary vertical imaging modes through the atmosphere: from the ground up and from orbits down. In the former, the imaging system is “close to” (indeed, it is immersed in the atmosphere), whereas in the latter the object is close to the atmosphere and distant from the imaging system. The transverse motion of the atmosphere may be characterized as an angular motion, which is relatively much larger in the “ground up”situation. Thus it is easier to get good images of the ground from spacecraft than of astronomical objects (or spacecraft) from the ground, which is fortunate for people working in “adaptive optics,”which tries to compensate for the atmospheric blur in real time.

2.0.3 Optical Path Length We have already seen that the phase velocity of light in a medium is slower than in vacuum, which means that light takes longer to travel through a given thickness of material than through the same 40 CHAPTER 2 INDEX OF REFRACTION

“thickness”of vacuum. For a …xed medium thickness d, we know that:

d = v t (distance = velocity time) = c t1 (in vacuum) c = t2 (in medium of index n) n t2 = t1 = = t2 > t1 ) n )

In the time t2 required for light to travel the distance d in a material of index n, light would travel a longer distance in the same time in vacuum: nd = ct2. The distance

nd d which is the distance light travels in vacuum in the same time as it would traverse the distance d in the medium with index n is the optical path length within the medium:

OPL = nd [mm]

2.1 Lorentz’sModel of Refractive Index

(Feynman Lectures on Physics Vol. I §31, §32 and II §32).

Electric waves interacting with a layer of transparent glass: some of the …eld is re‡ected and the original …eld plus a “correction” term are transmitted.

H.A. Lorentz proposed a simpli…ed explanation of the refractive index that is based on the “solar system”model of atomic structure, where positively charged massive nucleus of protons and neutrons surrounded by a cloud of negatively charged electrons. To ensure stability of the atomic structure, the electrostatic force between electrons and protons must be balanced by a force of motion, which we assume applies only to electrons since they are so much less massive than protons (mp = 1836 me). The forces on the electrons produce an equation of motion that has an oscillatory solution whose temporal frequency depends upon the parameters of the atom or molecule. An electron oscillates at a “natural” frequency !0 that depends on its mass and the force. An incident electric …eld can stimulate the electron; it absorbs and re-emits the light, which is called scattering. Under the in‡uence of the incident electric …eld, the electron acts as a forced harmonic oscillator. The 2.1 LORENTZ’SMODEL OF REFRACTIVE INDEX 41 scattering from the electron “system”varies with the temporal frequency of the incident radiation, and it is this variation in response that is called dispersion. The con…guration of the system is shown in the …gure; a point source is located a large distance away to the left, so that the incident electric …eld is a plane wave. The observation point is a large distance to the right (the …gure is “not to scale!”). The observed electric …eld is the sum of the original electric source …eld Es and a “correction term” E2 of electric …elds generated by moving electrical charges in the plate; this is the “scattered light.”We assume that only electrons can absorb (and therefore emit) light, since the massive protons are much more di¢ cult to move. The index of refraction is due to this correction term E2, which makes the electric …eld inside the glass “appear” to be moving at a di¤erent (slower) phase velocity. In reality, a single electron in the glass is a¤ected by the incident …eld and by the …elds generated by all other electrons in the glass, just as the motions of all other charges are in‡uenced by the single observed electron. To simplify the problem, we assume that the in‡uences of the other electrons are small compared to the e¤ect of the incident …eld, so that the total …eld at the observation point is little a¤ected by the motions of the other charges. In e¤ect, we are assuming that the index of refraction of the glass is very close to unity. The calculation will produce a …eld E2 that travels in the same direction as the incident …eld and a …eld E1 that travels in the opposite direction (the “re‡ected”light), but E1 and E2 are small because n = 1.

2.1.1 Refractive Index in terms of Phase

Because its source is far away, the incident electric …eld is a traveling plane wave that may be written in 1-D complex notation: z Es = E0 exp [+i (k0z !t)] = E0 exp +i! t c h i We avoid use of the notation “!0” for the temporal frequency of the variable incident (“driving”) …eld in this derivation because this frequency will be varied and because the subscripted notation is reserved for the “natural oscillation”frequency of the electron due to the forces from the nucleus. Assume that z = 0 at the “front”(input) side of the thin plate of glass and z = z at the back side. The phase of the electric …eld at the front of the plate is a function only of the observation time: [0; t] = k0 0 !t = !t If there were no glass, then the phase at the location of the back of the plate would be incremented by the additional propagation to z = z:

z z [z; t0; vacuum] = ! t = !t + ! c c Insertion of the glass “slows down” the light by an additional factor because the phase velocity in the glass is v < c. The phase of light at the rear of the glass includes an additional factor due to the slower phase velocity v:

z z [z; t; glass] = ! t > ! t v c If we substitute the index of refraction n, then the parts of the phase due to the distance z di¤er by its scale factor: c n = v n z z = [z; t; glass] = ! t > ! t ) c c 42 CHAPTER 2 INDEX OF REFRACTION

The “additional”phase due to the extra time to travel through the glass is:

n z z [z; t; glass] [z; t0; vacuum] = ! t ! t c c (n 1) z = ! c

The electric …eld at the back of the glass is the product of the incident …eld and an additional phase delay: z E [z; t] = E0 exp +i! t exp [ i ] c h z i (n 1) z = E0 exp +i! t exp i! c c h i This allows the contribution of the glass plate to the electric …eld to be interpreted as a multiplica- tive contribution to the optical phase instead of as a multiplicative contribution to the amplitude. The phase interpretation is much easier to analyze because it separates into additive terms. The incremental phase may be expanded into a Taylor series:

(n 1) z exp [ i ] = exp i! c (n 1) z ( i)2 (n 1) z 2 = 1 i! + ! + c 2! c 2 If we assume that the glass is su¢ ciently thin (so that 1 >> z ' 0 = z >> (z) ), then we can ignore all terms of order two or larger without much error and the) exponential reduces to the sum of two terms: (n 1) z exp [ i ] = 1 i! c which, after substitution, leaves:

z (n 1) z Eafter = E0 exp +i! t 1 i! c c h z i (n 1) z z = E0 exp +i! t i! E0 exp +i! t c c c h i h i The …rst term is just the source …eld at the front of the plate: z Es = E0 exp +i! t c h i and the second term is identi…ed as the approximate contribution from the charges within the glass, which is labeled in the Figure as E2:

(n 1) z z E2 = i! E0 exp +i! t c c h i (n 1) z z = exp i ! E0 exp +i! t 2 c c h i h i The leading factor of i = exp i may be interpreted as a phase di¤erence of the electric …eld 2 due to the charges in the glass; in words, the electric …eld E2 is in quadrature to the original electric …eld. The vector (phasor) contributions of the two …elds are shown in the …gure. 2.1 LORENTZ’SMODEL OF REFRACTIVE INDEX 43

Argand diagram of phasor contributions from incident …eld E0 and …eld due to charges in the glass E2, which is oriented approximately perpendicular to E0 and “delays”the phase of the electric …eld.

If the …eld E2 thus evaluated is interpreted as due to the oscillating charges in the glass, then we have explained the concept of refractive index, which is quanti…ed in terms of the physical forces in the next section.

2.1.2 Refractive Index in Terms of Forces

The task now is to evaluate the …elds in terms of the forces on the electrons in the system. Again, we assume that the incident …eld has the form of a plane wave that oscillates sinusoidally at angular temporal frequency !: Es [z; t] = E0 exp [+i (k0z !t)] The source …eld at the “front edge”of the glass may be evaluated by setting z = 0:

Es [0; t] = E0 exp [ i!t] The electrons at this part of the glass “feel”this …eld and are driven to move in the same direction by the force due to the electric …eld Es:

F = eEs [0; t] = eE0 exp [ i!t] (we assume no speci…c orientation –polarization –of the incident …eld) These electrons have mass m and act as though bound to the atomic nuclei (protons) by little “springs” that exert restoring forces proportional to the distance of the electron from its equilibrium position:

F = k (x x0) k Note that the dimensions of m are

kg m N 2 k N s2 radians = m = = = s 2 = m kg kg m kg m s which is the square of an angular frequency: the resonant frequency !0, which is the “normal oscillating frequency”of the system composed of electron + spring in classical mechanics:

k !0 rm The equation of motion of an electron under the in‡uence of the driving …eld is that of an undamped driven harmonic oscillator:

2 d x 2 m + m! x = F = eE0 exp [ i!t] dt2 0 44 CHAPTER 2 INDEX OF REFRACTION where the last term is the “driving force”due to the electric …eld. This equation is solved by standard methods of di¤erential equations, by assuming that the position x also oscillates at the same rate. The position x and its time derivatives are:

x = x0 exp [ i!t] dx x_ = i!x0 exp [ i!t] = i!x dt 2 d x 2 2 x• = ( i!) x0 exp [ i!t] = ! x dt2 These are substituted into the equation of motion simpli…es to obtain an equation for the amplitude x0 of the oscillation:

2 2 m ! x0 exp [ i!t] + m! x0 exp [ i!t] = eE0 exp [ i!t] 0 2 2 = m! + m! x0 = eE0 ) 0 eE0 = x0 = ) m (!2 !2) 0 We assume that the motions of ALL individual electrons in the glass plate due to the incident electric …eld are identically described by the same simple expression:

eE0 x [t] = x0 exp [ i!t] = exp [ i!t] m (!2 !2) 0 Note that this does not include the initial positions of the charges, and thus the phase of the position x [t], which are (obviously) di¤erent for each.

We now must calculate the …eld at the observation point (well beyond the plate) due to a “thin plane” of charges that all move with the same equation of motion x [t]. We …nd the …eld at the observation point by adding the contributions from each of the charges in the glass. The electric …eld radiated by each electron in the glass is proportional to the acceleration just evaluated:

2 d x 2 2 = ( i!) x0 exp [ i!t] = ! x dt2 The electric …eld at large distances from the oscillating charge falls o¤ approximately as the reciprocal of the distance and includes the time delay for the …eld to arrive:

e 2 r E [r; t] = ! x0 exp i! t r c h i We assume that the observation point is so far away that the …eld oscillates approximately perpen- dicular to the “line of sight.”

The total …eld generated by the charges at the observation point is the vector sum of the con- tributions from the individual electrons, which may be integrated in polar coordinates because the contributions along all azimuth angles are assumed to be identical. If is the “area number density” of electrons in the glass (number of electrons per unit area, which may be multiplied by z to …nd the number density per unit volume), then the electric …eld is:

=+ 1 e 2 r Eall = E2 = ! x0 exp i! t 2 d r c Z=0 =+h i 2 1 r = 2e ! x0 exp [ i!t] exp +i! d c r Z=0 h i where the length of the 3-D vector r is the Pythagorean sum of the lengths of the 2-D vector and 2.1 LORENTZ’SMODEL OF REFRACTIVE INDEX 45 the longitudinal distance z:

2 r 2 = r2 = + z2 = 2 + z2 j j = r dr = d )

The integral in the expression for the total electric …eld Eall becomes:

=+ r=+ 1 r 1 1 r 1 exp +i! d = exp +i! r dr c r c r Z=0 Zr=z h i r=+ h i 1 r = exp +i! dr r=z c Zc h i z = exp [+i ] exp +i! i! 1 c c z h i = exp +i! i! c ic h z i = + exp +i! ! c h i The …rst term oscillates “in…nitely rapidly”so that the integral of its contributions must be negligible. The electric …eld due to all charges is:

2 ic z E2 = 2e ! x0 exp [ i!t] exp +i! ! c z h i = (2ce) ( i!) x0 exp +i! t c h i which shows that the …eld measured at the observation point due to all of the oscillating charges is out of phase by 2 radians and delayed due to the …nite propagation time.

All that is left to do is to substitute the formula for x0 derived above for the driven harmonic oscillator:

eE0 z E2 = (2ce) ( i!) exp [ i!t] exp +i! m (!2 !2) c 0 2 h i 2ce E0 ! z = i 2 2 exp +i! t m ! !0 ! c h i which has the form of a sinusoidal traveling wave. In words, the electrons in the glass that oscillate due to the incident …eld emit a wave that travels in the same direction (towards z = + ). The amplitude of the wave is proportional to the area density of atoms and to the strength1 of the source …eld E0. This …eld resembles that of E2 that was evaluated in the last section:

(n 1) z z E2 = i! E0 exp +i! t c c 2 h i 2ce E0 z = i! 2 2 exp +i! t m (!0 ! ) c h i If we equate the two and cancel the common factors, we obtain an expression for the index of 46 CHAPTER 2 INDEX OF REFRACTION refraction in terms of the parameters of the system and the driving frequency !:

(n 1) z 2ce2 = c m (!2 !2) 0 2c2e2 = n = 1 + ) m (!2 !2) z 0 We now de…ne N to be the number density of electrons per unit volume in the glass, which is the ratio of the area density and the thickness z:

mm 2 = N mm 3 z [mm] to obtain the …nal expression for the frequency-dependent refractive index in the simple model of oscillating bound (but undamped) electrons:

2Nc2e2 n = 1 + (no damping) m (!2 !2) 0 The magnitude of the refractive index is approximately unity if evaluated at angular temporal frequencies ! distant from the natural frequency !0: ! << !0 or ! >> !0. If ! / !0, n >> 1; if ! ' !0 then n << 1 and may be negative. This last behavior is a clue that the picture is incomplete.In fact, the electron response actually is “damped”by other forces in the medium, which is the subject of the next section.

2 2 (2Nc e )E0 Model of refractive index n = 1 + in vicinity of a resonance, plotted as functions of m(!2 !2) 0 frequency and of wavelength. The resonance is assumed !0 = 1 and 0 = 1. The constants are arbitrary so that only the shapes of the curves are signi…cant. Note the discontinuity in n at the 2.1 LORENTZ’SMODEL OF REFRACTIVE INDEX 47

resonance due to the lack of damping force. The index decreases with increasing (normal dispersion) except at the discontinuity.

2.1.3 Damping Forces and Complex Refractive Index

The oscillator will absorb energy from the electric …eld if the electron motion is damped, as by frictional forces. This leads to a complex-valued refractive index:

n~ = n + i

The equation of motion includes a term proportional to the velocity:

d2x dx m + + kx = eE dt2 dt

Again, the incident …eld E0 oscillates sinusoidally at the driving frequency !:

E = E0 exp [ i!t] = x = x0 exp [ i!t] )

The derivatives are substituted into the equation of motion to evaluate the amplitude x0 of the oscillation: dx = i!x0 exp [ i!t] dt 2 d x 2 = ! x0 exp [ i!t] dt2

2 d d 2 m + + k x0 exp [ i!t] = m! i! + k x0 exp [ i!t] dt2 dt 2 = m! i! + k x0 exp [ i!t] = eE0 exp [ i!t] ) eE0 eE0 = x0 = = ) ( m!2 i! + k) m !2 + i! k m m where we have already identi…ed: k !2 m 0 We now also rename the term:

[Hz] m which is the reciprocal of a time constant, which may be interpreted as the time interval over which the amplitude of the oscillation decays to a speci…c value. The expression for the amplitude of the oscillation may be written in two equivalent forms:

e E0 x [!] = m 0 (!2 !2) + i! 0 e 2 2 E0 ! ! i! = m 0 (!2 !2) + i! (!2 !2) i! 0 0 ! E !2 !2 i! = e 0 0 m (!2 !2)2 + 2!2 0 which is clearly complex valued. Note that if ! = !0, the amplitude x0 no longer is unde…ned, but rather is imaginary: eE0 1 x0 [! = !0] = i m !0 48 CHAPTER 2 INDEX OF REFRACTION

We can insert the new expression for the amplitude into the formula for polarizability:

P = Nex0 e E0 = Ne m (!2 !2) + i! 0 Ne2 E P = 0 (!2 !2) i! m 0 First, we should test to see that this reverts to the static value for P if ! = 0 (no incident light):

Ne2 E lim P [!] = 0 ! 0 !2 m ! f g 0

The refractive index n may be evaluated exactly as before:

(n 1) z 2ce2 = c m (!2 !2 i! ) 0 2Nc2e2 = n = 1 + ) m !2 !2 i! 0 m Note that n now is complex valued; it often is denoted n~ n + i:

2Nc2e2 n~ = 1 + m (!2 !2 i! ) 0 This expression is rewritten into its real and imaginary parts:

2 2 2 2 2Nc e E0 ! ! + i! n~ = 1 + 0 m (!2 !2) i! ((!2 !2) + i! ) 0 m 0 2 2 2 2 2Nc e E0 ! ! + i! = 1 + 0 m (!2 !2)2 + (! )2 0 2 2 2 2 2 2 2Nc e E0 !0 ! 2Nc e E0 ! = 1 + 2 2 + i 2 2 m (!2 !2) + (! ) ! m (!2 !2) + (! ) ! 0 0

2 2 2 2 2Nc e E0 ! ! n = 1 + 0 m 2 2 2 2 (!0 ! ) + (! ) 2 2 2Nc e E0 ! = m (!2 !2)2 + (! )2 0 2.1 LORENTZ’SMODEL OF REFRACTIVE INDEX 49

Attenuation due to Im [n]

The complex-valued refractive index may be inserted into the propagation phase:

n~ z E [z; t] = E0 exp +i! t c (n + i) z = E0 exp +i! t c n z i z = E0 exp +i! t exp +i! c c h n z i z = E0 exp +i! t exp c c=! h i where the last term may be rewritten in the form:

z z exp = exp c=! h i c where = ! has dimensions of length. This last term is the source of the decaying amplitude of the scattered sinusoidal waves emitted by the charges. The distance is the decay length over which 1 the amplitude decays to the factor e = 0:367 of its original maximum.

Refraction index in the limit of no damping

Note that the real part of the refractive index di¤ers from the undamped index, but reverts to the 1 undamped formula if = 0, which implies that its reciprocal with units of time goes to in…nity:

2 2 2 2 2Nc e !0 ! lim (Re [~n]) = lim (n) = lim 1 + 2 2 0 0 0 m (!2 !2) + (! ) ! ! ! ! 0 2Nc2e2 = 1 + = n m (!2 !2) undamped 0 2Nc2e2a ! lim (Im [~n]) = lim () = lim 2 2 = 0 0 0 0 m (!2 !2) + (! ) ! ! ! ! 0

The real part of the refractive index is graphed as functions of the driving frequency ! and driving wavelength , along with the undamped case (dashed line, for comparison). Both !0 and 0 were selected to be unity for convenience. The damping “smooths”the dispersion curve so that n varies more slowly with and the real part decreases

with increasing (normal dispersion) except in the vicinity of the resonance where the index increases with (anomalous dispersion). 50 CHAPTER 2 INDEX OF REFRACTION

Model of the real part of the refractive index in the damped case in vicinity of a resonance, plotted as function of both frequency ! and wavelength and compared to the undamped case (dashed line). Again, the constants are arbitrary. The complex refractive index also is shown as functions of both ! and . Note the Lorentzian fom of the imaginary part, which is the source of the spectral absorption. 2.1 LORENTZ’SMODEL OF REFRACTIVE INDEX 51

The complex refractive index in the vicinity of a resonance located at !0 = 1 and 0 = 1, plotted as real part (solid) and imaginary part (dashed). The imaginary part attenuates the transmission.

Attenuation due to Im [n] Recall the formula for the electric …eld traveling down the z-axis in a medium with complex refractive index n~ = n + i

n~ n + i E exp +i!0 z t = E exp +i!0 z t 0 c 0 c n!0z 2 = E exp +i i!0t + i z 0 c c h nz i = E exp +i!0 t z 0 c c h nz i = E exp +i!0 t exp z 0 c c h i h i nz z = E0 exp +i!0 t exp c c " # h nz i z E exp +i!0 t exp 0 c h i h i The amplitude of the last term decreases exponentially with increasing z at a rate determined by c , the attenuation depth. This represents the attenuation (or absorption) due to the imaginary part of n~.

2.1.4 Dispersion Curves Though the behaviors of di¤erent optical materials di¤er in details, they have many properties in common. We have already stated that the index of refraction n relates the phase velocity of light in vacuum with that in matter: c n = 1: v In a transparent dispersive medium over the visible spectrum, the index n decreases with increasing ! , which ensures that the phase velocity k (of the average wave) is larger than the group velocity d! dk (of the modulation wave).

Dispersion curves for various optical glasses in the visible region of the spectrum.

In addition, the slope of the dispersion curve decreases with increasing : 52 CHAPTER 2 INDEX OF REFRACTION

Typical dispersion curve for glass at visible wavelengths, showing the decrease in n with increasing and the three spectral wavelengths speci…ed by Fraunhofer and used to specify the “refractivity”, “mean dispersion”, and “partial dispersion” of a material.

Dispersion curve of a material from very short to very long wavelengths. The index tends to increase with increasing as additional resonances are passed, but the index of refraction for visible wavelengths (in bold face) decreases with increasing wavelength..

The dispersion curves for optically transparent materials, such as glass and air, exhibit some very similar features, though the details may be very di¤erent. Starting at very short wavelengths ( ' 0), the refractive index n = 1. In words, the wavelength is so short, the frequency so large, and the energy so large that the photons pass through the material without interacting with the atoms. For nonzero, but still very short, wavelengths (“hard”X rays), the refractive index actually is slightly less than unity, which means that X rays incident on a prism are refracted away from the prism’s base, rather than towards the base in the manner of visible light. This is the reason why X rays can be totally re‡ected at grazing incidence, which is the focusing mechanism in X-ray telescopes (such as Chandra). As the wavelength of the incident light increases further, though still within the X-ray region, the radiation incident on the material will be heavily absorbed. This is the “K-absorption edge”where the energy of the incident X rays is just su¢ cient to ionize an electron in the innermost atomic “shell”–the “K shell.”The wavelength of this absorption is K = 0:67 nm 2.1 LORENTZ’SMODEL OF REFRACTIVE INDEX 53 for silicon. Other absorptions occur at yet longer wavelengths (smaller incident photon energies), where electrons in the L and M shells, etc., of the atom are ionized. The spectrum of a material with a large atomic number (and thus several …lled electron shells) will exhibit several such resonant absorptions.

Ionization of a K-shell electron by an incoming X ray of su¢ cient energy. This is the reason for the large absorptions of “hard” X rays by materials. Lower-energy (longer-wavelength) X rays will ionize electrons in the L or M shells, thus producing other absorption “edges.” As the wavelength of the incident radiation increases further, into the “far ultraviolet”region of the spectrum, the real part of the refractive index decreases to a value much less than unity within a wide band of anomalous dispersion. The fact that n < 1 in this region may be confusing because it seems that the velocity of light exceeds c, but these waves do not propagate in the material due to the strong absorption (large value of ). The wavelength of maximum absorption corresponds to the largest of the several “natural oscillation frequencies”of bound electrons in the material. In the visible region of the spectrum, the dispersion curve exhibits the familiar decrease in n with that was shown above. For example, the index of air is n = 1:000279 at = 486:1 nm (Fraunhofer’s“F”line) and n = 1:000276 at = 656:3 nm (“C”line). The corresponding values for diamond are nF = 2:4354 and nC = 2:4100. The closer the nearest ultraviolet absorption to the dn visible spectrum, the steeper will be the slope d in the visible region and thus the larger the visible dispersion (de…ned below). The dispersion curve descends yet more steeply somewhere in the near infrared region and then rises due to anomalous dispersion in the vicinity of an infrared absorption band (labeled “2” on the graph). For quartz (crystalline SiO2), the center of this band is located at = 8:5 m, but the absorption already is quite strong for wavelengths as short as = 4 m. Most optical materials have several such infrared absorption bands and the “base level”of the index of refraction is larger after each such band. This behavior is con…rmed by far-infrared measurements of the refractive index of quartz (crystalline SiO2), which varies over the interval 2:40 n 2:14 for 51 m 63 m. The large values of n ensure that the focal length of a convex quartz lens is much shorter at far-infrared wavelengths than at visible wavelengths. As the wavelength is increased still further into the radio region of the spectrum, eventually all of the absorption bands will be passed. The refractive index decreases slowly and approaches a limiting value of . 0 q 2.1.5 Empirical Expressions for Dispersion To a …rst approximation, the indices of refraction of optical materials in the visible region of the 1 spectrum vary as , which allows us to write an empirical expression for the refractivity of the medium n 1: b n [] 1 = a + where a and b are parameters determined from measurements. Clearly the refractive index n = a = as + . The observation that the index decreases with increasing ensures that b > 0. 0 ! 1 q 54 CHAPTER 2 INDEX OF REFRACTION

Cauchy’sEquation

Cauchy came up with a better empirical relation for the refractivity by adding some additional parameters A, B, C, :::: B C n [] 1 = A + + + 2 4 The parameters are determined by measurements of n for the corresponding material. Again, A = is the refractive index for long wavelengths ( + ). To determine three empirical constants, 0 ! 1 itq obviously is necessary to measure n at three wavelengths. Again, the behavior of normal dispersion ensures that A and B are both positive. The dispersion (variation of n with wavelength ) may be expressed as the derivative of n with respect to wavelength : dn B C B = 2 4 + = 2 if C B d 3 5 3 / which again agrees with the observation that the variation in refractive index becomes “‡atter”for longer wavelengths.

Sellmeier’sEquation

These two approximations for n [] both follow “smooth”curves that decrease with increasing wave- length, and thus do not model anomalous dispersion. Sellmeier proposed an equation for dispersion in 1871 that includes an empirically …xed wavelength where the electrons in the material oscillate at their “natural”or “resonant”frequency !0:

A2 A n [] = 1 + 2 2 = 1 + 2 0 s 0 s 1

2v where the two constants A > 1 and 0 = are determined empirically. The denominator !0 2 0 1 < 0 and n [] < 1 for wavelengths less than 0; the computed n [] > 1 for > 0. This expression exhibits the behavior of the refractive index in regions of anomalous dispersion, as shown in the …gure. 2.1 LORENTZ’SMODEL OF REFRACTIVE INDEX 55

Index of refraction model by Sellmeier, showing anomalous dispersion in the vicinity of the resonance at = 0. Multiple resonances may be accommodated by adding terms that resonate at their speci…c wave- lengths:

N 2 N Ai Ai n [] = 1 + 2 2 = 1 + 2 v v i u i=1 i u i=1 1 u X u X t t In the limit + , this evaluates to: ! 1

N n [ ] = 1 + Ai = ! 1 v u i=1 0 u X r t which, in words, shows that the index of refraction at “DC”(! = 0) includes the sum of the weighting constants for each resonance.

Refractive Constants for Glasses The refractive properties of glass are approximately speci…ed by the refractivity and the measured di¤erences in refractive index at the three Fraunhofer wavelengths F, D, and C:

Refractivity nD 1 1:75 nD 1:5 Mean Dispersion nF nC > 0 di¤erences between blue and red indices Partial Dispersion nD nC > 0 di¤erences between yellow and red indices nD 1 Abbé Number ratio of refractivity and mean dispersion, 25 65 nF nC Glasses are speci…ed by six-digit numbers abcdef, where nD = 1:abc, to three decimal places, and = de:f. Note that larger values of the refractivity mean that the refractive index is larger and thus so is the deviation angle in Snell’slaw. A larger Abbé number means that the mean dispersion is smaller and thus there will be a smaller di¤erence in the angles of refraction. Such glasses with larger Abbé numbers and smaller indices and less dispersion are crown glasses, while glasses with smaller Abbé numbers are ‡int glasses, which are “denser”. Examples of glass speci…cations include Borosilicate crown glass (BSC), which has a speci…cation number of 517645, so its refractive index in the D line is 1:517 and its Abbé number is = 64:5. The speci…cation number for a common ‡int glass is 619364, so nD = 1:619 (relatively large) and = 36:4 (smallish). Now consider the refractive indices in the three lines for two di¤erent glasses: “crown”(with a smaller n) and “‡int:”

Line [nm] n for Crown n for Flint

C 656:28 1:51418 1:69427 D 589:59 1:51666 1:70100 F 486:13 1:52225 1:71748

The glass speci…cation numbers for the two glasses are evaluated to be:

For the crown glass:

refractivity: nD 1 = 0:51666 = 0:517 1:51666 1 Abbé number: = = 64:0 1:52225 1:51418 Glass number = 517640 56 CHAPTER 2 INDEX OF REFRACTION

For the ‡int glass:

refractivity:L nD 1 = 0:70100 = 0:701 0:70100 1 Abbé number: = = 30:2 1:71748 1:69427 Glass number = 701302

2.1.6 Negative Refractive Index In 1968, a Russian physicist named Victor Veselago presented theoretical arguments that the in- dex of refraction need not be positive (V.G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Sov. Phys. Uspekhi 10, 509-514, 1968). We have al- ready demonstrated that the refractive index is related to the electrical permittivity and magnetic permeability:

1 c = r 0 0 1 v = r c n = = v r 0 0 The presence of the square root has always been taken to imply that n > 0 (and, in fact, that n 1 in real materials). Veselago showed that if both and are negative, then the negative sign must be taken: n = if < 0 and < 0 r 0 0 In the case n < 0, the direction of propagation is reversed so that light travels backwards towards the source. This also implies that the laws of geometrical optics are “‡ipped;”light emerging from a “rare”medium to a “dense”medium is refracted away from the normal, rather than towards it. Materials whose electromagnetic properties arise from features of the internal structure, rather than from the constituent elements, are metamaterials. An analogy is a material that exhibits apparent color due to interference from re‡ective coatings or structures within the material. One example of a metamaterial with a negative index was recently announced for which n = 0:6 at = 780 nm. These metamaterials are said by some to be the key technology for making “cloaking devices”that would make an object invisible by forcing light from behind to propagate “around”it. http://www.newscientisttech.com/article/dn10816.html Chapter 3

Polarization

Maxwell’sequations demonstrated that light is a transverse wave (as opposed to longitudinal waves, e.g., sound). Both the E and B vectors are perpendicular to the direction of propagation of the radiation. Even before Maxwell, Thomas Young inferred the transverse character of light in 1817 when he passed light through a calcite crystal (calcium carbonate, CaCO3). Two beams emerged from the crystal, which Young brilliantly deduced were orthogonally polarized, i.e., the E vectors of the two beams oscillate in orthogonal directions. The polarization of electromagnetic radiation is de…ned by the plane of vibration of the electric vector E because the e¤ect of the magnetic …eld on a free charge (electron) is much smaller. This is 1 seen from the factor of c applied to the magnetic …eld Lorentz force law: v F q0 E + B / c

q0 = charge [coulombs]

kg m F = force on the charge [measured in newtons: 1 N = 1 s2 m h i v = velocity of the charge q0 [ s ] c = velocity of light [3 108 m ] s

3.1 Plane Polarization = Linear Polarization

In the most familiar type of polarization, the E-vector oscillates in the same plane at all points on the wave. This is called planar or linear polarization. Any state of linear polarization can be expressed as a linear combination (sum) of two orthogonal states (basis states), e.g., the x- and y-components of the E-vector for a wave traveling toward z = : 1

E = E [r; t] = [^xEx + ^yEy] cos [k0z !0t + ] ^x, ^y = unit vectors along x and y

Ex;Ey = amplitudes of the x- and y-components of E

For a wave of amplitude E0 polarized at an angle relative to the x-axis:

57 58 CHAPTER 3 POLARIZATION

Ex = E0 cos []

Ey = E0 sin []

2 2 E0 = Ex + Ey q E = tan 1 y E x Linearly polarized radiation oscillates in the same plane at all times and at all points in space. Especially note that Ex and Ey are in phase for linearly polarized light, i.e., both components have zero-crossings and have extrema at the same points in time and space

Electric …eld vector E and magnetic …eld vector H of a plane-polarized wave.

3.2 Circular Polarization

If the E-vector describes a helical motion in space, the projection of the E-vector onto a plane normal to the propagation direction k exhibits circular motion over time, hence the name for this polarization:

Circular polarization occurs when the electric …elds along orthogonal axes have the same amplitude by their phases di¤er by radians. 2 3.2 CIRCULAR POLARIZATION 59

To an observer sitting at a …xed point in space (z = z0), the motion of the E-vector is the sum of two orthogonal linearly polarized states with one component delayed in phase by 90 = 2 radians. The math is identical to that used to describe oscillator motion as the projection of rotary motion: motion = ^x cos [k0z0 !0t] + ^y cos k0z0 !0t = ^x cos [!0t] ^y sin [!0t] 2 h i We can consider the constant k0z0 to be an initial phase value 0:

motion = ^x cos [!0t 0] ^y sin [!0t 0] where the upper sign applies to right-handed circular polarization (angular momentum convention)

3.2.1 Nomenclature for Circular Polarization Like linearly polarized light, circularly polarized light has two orthogonal states, i.e., clockwise and counterclockwise rotation of the E-vector. These are termed right-handed (RHCP) and left-handed (LHCP). There are two conventions for the nomenclature: 1. Angular Momentum Convention (my preference because easier to visualize): Point the thumb right of the hand in the direction of propagation. If the …ngers point in the direction of 8 left 9 < = RHCP rotation: of the;E-vector, then the light is : 8 LHCP 9 < = 2. Optics (also called screw-shape or screwy): Convention:; The path traveled by the E-vector of RHCP light is the same path described by a right-hand screw. Of course, the natural laws de…ned by Murphy ensure that the two conventions are opposite: RHCP light by the angular momentum convention is LHCP by the screw convention.

3.2.2 Elliptical Polarization, Re‡ections If the amplitudes of the x- and y-components of the E-vector are not equal, or if the phase di¤erence is not 2 = 90, then the projection of the path of the E-vector is an ellipse and the polarization (obviously) is elliptical. Note that elliptical polarization may be either right- or left-handed, as de…ned above.

3.2.3 Change of Handedness on Re‡ection By conservation of angular momentum, the direction of rotation of the E-vector does not change on re‡ection. Since the direction of propagation reverses, the handedness of the circular or elliptical polarization changes:

Change in “handedness” of a circularly (or elliptically) polarized wave upon re‡ection by a mirror. 60 CHAPTER 3 POLARIZATION

3.2.4 Natural Light

The superposition of emissions from a large number of thermal source elements (as in a light bulb) has a random orientation of polarizations. The state of polarization of the resulting light changes 14 direction randomly over very short time intervals (= 10 s). The radiation is unpolarized, even though the electric …eld oscillates in one plane if measured over this short time period. Natural light is neither totally polarized nor totally unpolarized, but rather partially polarized.

3.3 Description of Polarization States

3.3.1 Jones Vector

The components of the electric …eld in the two orthogonal directions may be used to represent a vector with complex components. This is called a Jones vector, which is useful only for completely polarized light. Assume that light propagates down the z-axis (or rotate the coordinate system to ensure this situation). The electric …eld is:

E = Re E exp [+i (k0z !0t)] = [Re Ex exp [+i (k0z !0t)] ; Re Ey exp [+i (k0z !0t )] ] f 0 g f g f g i = Re [Ex;Eye ] exp [+i (k0z !0t)] The Jones vector that describes this state has components equal to the amplitudes in the orthogonal directions, plus any phase delay in one of the components (generally taken to be the y-component):

Ex Jones Vector = E 2 i 3 Ey e 4 5 The total irradiance is the sum of the irradiances in orthogonal polarizations:

Itotal = Ix + Iy = ExE + EyE = E E = y h xi y h i E E D E where the “dagger”is used to indicate the Hermitian conjugate vector, which is the complex conju- gate of the transpose: T T y ( ) = E E E In words, the irradiance is the time average of the scalar product of the Jones vector with its complex conjugate.

Examples of Jones Vectors:

1. Plane-polarized light along x-axis

E0 = E 2 0 3 4 5 2. Plane-polarized light along y-axis: 0 = 2 3 E E0 4 5 3.3 DESCRIPTION OF POLARIZATION STATES 61

3. Plane-polarized light at angle to x-axis:

E0 cos [] = 2 3 E E0 sin [] 4 5 4. RHCP

E = ^xE0 cos [k0z !0t] + ^yE0 sin [k0z !0t] = ^xE0 cos [k0z !0t] + ^yE0 cos k0z !0t = = + 2 ) 2 h i ( = + because of negative sign in phase of Jones vector) 2

E0 = Re exp [+i (k0z !0t)] 82 E exp + i 3 9 < 0 2 = 4 5 : 1 1 ; = = E0 = E0 for RHCP )E 2 i 3 2 3 exp + 2 +i 4 5 4 5 so the Jones vector for LHCP obviously is:

1 1 = E0 = E0 for LHCP E 2 exp i 3 2 i 3 2 4 5 4 5 The Jones vectors of two orthogonally polarized states satis…es the condition:

= 0 E1 E2 Other representations of the state of polarization are available (e.g., Stokes’ parameters, co- herency matrix, Mueller matrix, Poincare sphere). They are more complicated, and hence more useful, i.e., they can describe partially polarized states. For more information, see (for example), Polarized Light by Shurcli¤.

3.3.2 Actions of Optical Elements on Jones Vectors We can now imagine the action of an optic on a state of polarization expressed by the corresponding Jones vector. The optic is expressed as a 2 2 matrix M, where again we recall that Jones vectors may only represent completely polarized light. The form of the operation is:

a b Ex M = +i E 2 c d 3 2 Ey e 3 4 5 4 5 We can infer the form of the matrix for some di¤erent cases. If the incident light is linearly polarized along x and the optic passes only light that is linearly polarized along x, then all of the light should pass to the output:

a b Ex Ex = 2 c d 3 2 0 3 2 0 3 4 5 4 5 4 5 1 b = a = 1; c = 0 = M=0 = ) ) 2 0 d 3 4 5 62 CHAPTER 3 POLARIZATION

If light that is linearly polarized along y is incident on the same polarizer, the ouptut should be zero:

1 b 0 0 = 2 0 d 3 2 Ey 3 2 0 3 4 5 4 5 4 5 1 0 = b = 0; d = 0 = M=0 = ) ) 2 0 0 3 4 5 If the linear polarizer is oriented along the y direction, it is easy to show that:

0 0 M= = 2 2 0 1 3 4 5 Note that det M = det M = 0 =0 = 2

If the x-matrix is applied to light that is linearly polarized at = , the output is: 4

1 0 I0 I0 p2 p2 M = = =0 4 E 2 0 0 3 2 I0 3 2 0 3 p2 4 5 4 5 4 5 A linear polarizer oriented at measured relative to the x-axis may be derived in similar fashion:

cos [] M = = E0 E E 2 sin [] 3 4 5 a b cos [] cos [] = 2 c d 3 2 sin [] 3 2 sin [] 3 4 5 4 5 4 5 M = 0 E 2 cos 2 0 ME0 = 2 sin 3 2 0 3 2 4 5 4 5 a b sin [] 0 = 2 c d 3 2 cos [] 3 2 0 3 4 5 4 5 4 5

These two relations may be solved simultaneously:

a b cos [] sin [] cos [] 0 = 2 c d 3 2 sin [] cos [] 3 2 sin [] 0 3 4 5 4 5 4 5 3.3 DESCRIPTION OF POLARIZATION STATES 63 which means that:

1 a b cos [] 0 cos [] sin [] M = = 2 c d 3 2 sin [] 0 3 2 sin [] cos [] 3 4 5 4 5 4 5 cos [] 0 cos sin = 2 sin [] 0 3 2 sin cos 3 4 5 4 5 cos2 sin cos = 2 sin cos sin2 3 4 5 Note that det M = 0, as was true for M and M . =0 = 2 A polarization rotator will convert light that is linearly polarized at angle and rotate it to + , so that:

a b cos [] cos [ + ] M = = 2 c d 3 2 sin [] 3 2 sin [ + ] 3 4 5 4 5 4 5 a cos [] + b sin [] cos [] cos [ ] sin [] sin [ ] = = ) 2 c cos [] + d sin [] 3 2 sin [] cos [ ] + cos [] sin [ ] 3 = 4a = cos [ ] , b = sin5 [ ] 4, c = + sin [ ] , d = cos [ ] 5 ) cos [ ] sin [ ] M = 2 + sin [ ] cos [ ] 3 4 5 det [M ] = 1

We can also retard the phase of one component more or less than the other:

+ix Ex Exe M = +iy 2 Ey 3 2 Eye 3 4 5 4 5 e+ix 0 = M = ) 2 0 e+iy 3 4 5 +ix +iy so that det M = e e = exp [+i (x + )]. A quarter-wave retarder changes the phase of one axis by radians: 2 1 0 1 0 M = = 2 i 3 2 3 0 e 2 0 i 4 5 4 5 A quarter-wave plate acting on light that is linearly polarized at = 45 yields:

1 0 1 1 = 2 0 i 3 2 1 3 2 i 3 4 5 4 5 4 5 which is circularly polarized. 64 CHAPTER 3 POLARIZATION

3.3.3 Coherency Matrix J

For light with amplitudes Ex and Ey in the x and y directions, respectively, the coherency matrix J is the 2 2 array formed from the outer product of time averages of the Jones vector: T Ex Ex J y = 0 1 EE *2 E e+i 3 2 E e+i 3 + D E y y B C 4 5 @4 5 A Ex +i = E Ey e +i x *2 Ey e 3 + h i 4 5 i 2 i Ex E Ex Ee E Ex Ey e = h xi y = x 2 +i 3 2 +i 2 3 Ey Exe Ey Ey Ex Ey e Ey 4 5 4 5 Note that the coherency matrix is identical to the complex conjugate of its transpose: it is a Hermitian matrix: T J = J Jy The irradiance I is the sum of the diagonal terms, which is called the trace of the matrix:

2 2 I Ex + Ey = Ex E + Ey E = tr [J] j j j j h xi y D E D E

Coherency Matrix of Completely Linearly Polarized Light

The coherency matrix of completely polarized light has the form:

2 E Ex E J = x y 2 2 3 E E y E h x i y 4 5 Note that

2 2 det J = E E E Ey Ex E x y h x i y = 0 for linearly polarized light. We can rotate the coordinate system of the matrix J to …nd the equivalent diagonal form (call it ), where the diagonal elements are the eigenvalues. Since J is only 2 2, the eigenvalues are easily derived using the well-known “brute force” method. We set the determinant of the “secular equation”to zero and solve the quadratic equation to …nd two values for :

det [J I] = 0 Jxx Jxy = det 2 Jyx Jyy 3 = (Jxx4 )(Jyy ) J5xyJyx 2 = (Jxx + Jyy) + (Jxx Jyy Jxy Jyx) = 2 (tr [J]) + det [J] = 0 where tr [J] is the sum of the diagonal elements of J. The eigenvalues are the solutions of this 3.3 DESCRIPTION OF POLARIZATION STATES 65 equation:

tr [J] 4 det [J] + 1 + 1 2 s tr [J] ! tr [J] 4 det [J] 1 1 2 s tr [J] ! so that + and the diagonal form is:

+ 0

2 0 3 4 5 For completely linearly polarized light, + = Itotal and = 0, so the diagonal form of the coherency matrix of linearly polarized light is:

Itotal 0 = 2 0 0 3 4 5

Coherency Matrix of Unpolarized Light

+i The two electric …eld amplitudes of unpolarized light are uncorrelated, so that ExEye = E E e i = 0 and the coherency matrix reduces to: x y

1 1 0 J = Itotal 2 2 0 1 3 4 5 which is diagonal, so the eigenvalues of the coherency matrix for unpolarized light are equal: 1 + = = Itotal 2 and det J = Itotal.

Examples:

LP at angle

E0 cos [] J = y = E0 cos [] E0 sin [] EE *2 E sin [] 3 + D E 0 h i 4 2 5 2 cos [] cos [] sin [] = E0 j j 2 cos [] sin [] sin2 [] 3 D E 4 5 cos2 [] cos [] sin [] JLP = I0 2 cos [] sin [] sin2 [] 3 4 5 det J = cos2 [] sin2 [] (cos [] sin [])2 = 0 66 CHAPTER 3 POLARIZATION

Horizontal LP Light (along x) 1 0 JLP x = y = I0 EE 2 0 0 3 D E 4 5 1 0 det = 0 2 0 0 3 4 5

Vertical LP (along y) 0 0 JLP y = I0 2 0 1 3 4 5 0 0 det = 0 2 0 1 3 4 5

Diagonal LP at = + 4 cos2 [] cos [] sin [] JLP + = I0 4 2 cos [] sin [] sin2 [] 3 = 4 4 5 1 1 2 2 I0 1 1 = I0 = 2 1 1 3 2 2 3 2 2 1 1

4 15 1 4 5 det 2 2 = 0 2 1 1 3 2 2 4 5

Diagonal LP at = 4

JLP = y 4 EE D E cos2 [] cos [] sin [] = I0 2 cos [] sin [] sin2 [] 3 = 4 4 5 1 1 2 2 I0 1 1 = I0 = 2 1 1 3 2 2 1 1 3 2 2 4 5 4 5 1 1 det 2 2 = 0 2 1 1 3 2 2 4 5 3.3 DESCRIPTION OF POLARIZATION STATES 67

RHCP

J = y EE D E2 E0 1 = 1 exp i 2 2 i 3 2 * exp + 2 + h i 4 5 E 2 1 i = 0 2 2 3 * + i 1

4 5 I0 1 i JRHCP = 2 2 i 1 3 4 5 1 i det = 0 2 i 1 3 4 5 LHCP I0 1 +i JLHCP = y = EE 2 2 i 1 3 D E 4 5 Coherency Matrix of Partially Polarized Light In this case we decompose the matrix into a weighted sum of polarized and unpolarized pieces. This is easy to see in the diagonal forms:

2 +i E Ex Eye J = x 2 i 2 3 Ex Eye Ey 4 5 + 0 + 0 0 = = + 2 0 3 2 0 0 3 2 0 3 4 5 4 5 4 5 1 0 1 0 = (+ ) + 2 0 0 3 2 0 1 3 4 5 4 5 The degree of polarization is the ratio of the polarized irradiance to the total irradiance as obtained from the eigenvalues: P

Ipolarized + = P Itotal + + 4 det [J] = + 1 2 s (tr [J])

If either + = 0 or = 0, then = 1 and the light is completely polarized; if + = , then P = 0 and the light is completely unpolarized. If + = , the light is partially polarized. P 6 Action of Optic on Coherency Matrix The action of the optical element on a coherency matrix J is:

J y = J0 L L 68 CHAPTER 3 POLARIZATION

where the output irradiance is again the invariant trace of the matrix, tr [J] = tr J0 :

Example: Rotation of Plane of Polarization:

cos [] sin [] = = L R 2 sin [] cos [] 3 4 5 Ex cos [] + Ey sin [] 0 = = E LE 2 Ex cos [] + Ey sin [] 3 4 5

2 2 Itotal = Ex + Ey

q 2 2 I0 = (Ex cos [] + Ey sin []) + ( Ex cos [] + Ey sin []) total D 2 2 E = (Ex cos [] + Ey sin []) + ( Ex cos [] + Ey sin []) 3.4 GENERATION OF POLARIZED LIGHT 69

3.4 Generation of Polarized Light 3.4.1 Selective Emission: If all emitting elements of a source (e.g., electrons in a bulb …lament) vibrate in the same direction, the radiated light will be polarized in that direction. This is di¢ cult to achieve at optical frequencies 14 14 8 (t / 10 s = ' 10 Hz), but is easy at radio or microwave frequencies ( / 10 Hz) by proper design of) the antenna that radiates the energy. For example, a radio-frequency oscillator attached to a simple antenna forces the free electrons in the antenna to oscillate along the long (vertical) dimension of the antenna. The emitted radiation is therefore mostly oscillating in the vertical direction; it is vertically polarized.

Emission of electromagnetic radiation (“light”) by a “dipole” radiator is polarized in the direction of motion of the emitting electrons (vertical, in this case). Rather than generating polarized light at the source, we can obtain light of a selected polarization from natural light by removing unwanted states of polarization. This is the mechanism used in the next section.

3.4.2 Selective Transmission or Absorption A man-made device for selecting a state of polarization by selective absorption is Polaroid. This operates like the microwave-polarizing skein of wires. The wires are parallel to the y-axis in the …gure. Radiation incident on the wires drives the free electrons in the wires in the direction of polarization of the radiation. The electrons driven in the y-direction along the surface of the wire and strike other such electrons, thus dissipating the energy in thermal collisions. What energy that is reradiated by such electrons is mostly directed back toward the source (re‡ected). The x-component of the polarization is not so a¤ected, since the electrons in the wire are constrained against movement in that direction. The x-component of the radiation therefore passes nearly una¤ected. Common Polaroid sheet acts as a skein of wires for optical radiation. It is made from clear polyvinyl acetate which has been stretched in one direction to produce long chains of hydrocarbon molecules. The sheet is then immersed in iodine to supply lots of free electrons.

Polarization by “skein of wires” –the radiation polarized parallel to the direction of the wires in the skein is absorbed, so the radiation polarized perpendicular to the wires is transmitted. In other words, the “picket fence” model of polarization is not appropriate in this case. 70 CHAPTER 3 POLARIZATION

3.4.3 Generating Polarized Light by Re‡ection –Brewster’sAngle

1 n2 We have already observed that rTM = 0 if the incident angle is Brewster’s angle B = tan . n1 One polarization of obliquely incident light “sees”the bound electrons in the dense medium “di¤er-h i ently”and therefore is re‡ected “di¤erently.”The re‡ected wave is polarized to some extent; the de- gree of polarization depends on the angle of incidence and the index of refraction n2. The polarization mechanism is simply pictured as a forced electron oscillator. The bound electrons in the dielectric material are driven by the incident oscillating electric …eld of the radiation E exp [+i (k0z0 !0t)], !0 and hence vibrate at frequency 0 = 2 . Due to its acceleration, the vibrating electron reradiates radiation at the same frequency 0 to produce the re‡ected wave. The state of polarization of the re‡ected radiation is a function of the polarization state of the incident wave, the angle of incidence, and the indices of refraction on either side of the interface. If the re‡ected wave and the refracted wave are orthogonal (i.e., 0 + t = 90 = t = 2 0), then the re‡ected wave is completely plane polarized parallel to the surface (and thus) polarized perpendicular to the plane of incidence). This angle appeared in the discussion of the re‡ectance coe¢ cients in the previous section. In this case, the electrons driven in the plane of the incidence will not emit radiation at the angle required by the law of re‡ection. This angle of complete polarization is called Brewster’s Angle B, which we mentioned earlier during the discussion of the Fresnel equations.

Brewster’s angle: the incident beam at 0 = B is unpolarized. The re‡ectance coe¢ cient for light polarized in the plane (TM waves) is 0, and the sum of the incident and refracted angle is 90 = 2 . Thus B + t = = t = B. 2 ) 2

3.4.4 Polarization by Scattering

Light impinging on an air molecule drives the electrons of the molecule in the direction of vibration of the electric …eld vector. This motion causes light to be reradiated in a dipole pattern; i.e., no light is emitted along the direction of electron vibration. If we look at scattered light (e.g., blue sky) at 90 from the source, the light is completely linearly polarized. Note that if the light is multiply scattered, as in fog, each scattering disturbs the state of polarization and the overall linear state is perturbed into unpolarized radiation. 3.5 BIREFRINGENCE –DOUBLE REFRACTION 71

Scattering of sunlight by atmospheric molecules. The light scattered into the eye at an angle of 90 is completely linearly polarized perpendicular to the line from the sun to the point in the sky.

3.5 Birefringence –Double Refraction

H§8.4 Many natural crystals and man-made materials interact with the two orthogonal polarizations di¤erently. This is often due to an anisotropy (nonuniformity) in the crystalline structure; such materials are called dichroic or birefringent. Many crystals (e.g., calcite) divide a nonpolarized light wave into two components with orthogonal polarizations. The two indices of refraction are sometimes denoted nf and ns for fast and slow axes, where nf < ns. They are also denoted no and ne for ordinary and extraordinary axes.By dividing the incoming natural light into two beams in such a crystal, we can select one of the two polarizations.

3.5.1 Examples: Refractive indices along the fast and slow axes at = 589:3 nm

Material ns nf

Calcite (CaCO3) 1:6584 1:4864

Crystalline Quartz (SiO2) 1:5534 1:5443

Ice (crystalline H2O) 1:313 1:309

Rutile (TiO2) 2:903 2:616

Sodium Nitrate (SiNO3) 1:5854 1:3369

The wavelength of light in a medium is 0 = n , so light along the two polarization directions have di¤erent wavelengths: s0 = < f0 = ns nf

3.5.2 Phase Delays in Birefringent Materials; Wave Plates Consider light incident on a birefringent material of thickness d. The electric …eld as a function of distance z and time t is:

E [z; t] = ^xEx + ^yEy exp [+i (k0z !0t)] : 72 CHAPTER 3 POLARIZATION

At the input face of the material (z = 0) and the output face (z = d), the …elds are:

E [z = 0; t] = ^xEx + ^yEy exp [ i!0t] E [z = d; t] = ^xEx + ^yEy exp [+i (k0d !0t)]

If nx = ns > ny = nf , then f > s and : 2n 2n k = k = s > k = k = f s x f y The …eld at the output face (z = d) is therefore:

i (2d n ) 2d n s f i!0t E [d; t] = ^xE exp + + ^yE exp +i e x y 2i 2dns = ^xEx + ^yEy exp d (nf ns) exp +i 2 By de…ning a constant phase term d (nf ns), the electric …eld at the output face of the birefringent material can be expressed as:

2dn E [d; t] = ^xE + ^yE e+i exp +i s x y On emergence from the material, the y-component of the polarization has a di¤erent phase than the x-component; the phase di¤erence is .

Quarter-Wave Plate: = + 2 = (nf ns) d = 4 , and there is a phase di¤erence of one quarter wavelength between the polarizations) of the x- and the y-components of the wave. This is a quarter-wave plate. The required thickness d of the material is: d = 4 (ns nf ) And the emerging …eld is:

+ i E [d; t] = ^xEx + ^yEy e 2 exp [+i (ksd !0t)] h i If Ex = Ey, (i.e., the incident wave is linearly polarized @ 45 to the x-axis), then the emerging wave is circularly polarized. This is the principle of the circular polarizer.

Half-Wave Plate: If = + = d = , and the relative phase delay is 180. Such a device is a half-wave ) 2(ns nf ) plate. If the incident light is linearly polarized along the orientation midway between the fast and slow axes, the plane of polarization of the exiting linearly polarized light is rotated by 90.

Circular Polarizer: A circular polarizer is a sandwich of a linear polarizer and a 4 plate, where the polarizing axis is oriented midway between the fast and slow axes of the quarter-wave plate. The LP ensures that equal amplitudes exist along both axes of the quarter-wave plate, which delays one of the components to create circularly polarized light. Light incident from the back side of a circular polarizer is not circularly polarized on exit; rather it is linearly polarized. A circular polarizer can be recognized and properly oriented by placing it on a re‡ecting object (e.g.,.a dime). If the image of the coin is 3.5 BIREFRINGENCE –DOUBLE REFRACTION 73 dark, the polarizer has the linear polarizer on top. This is because the handedness of the light is changed on re‡ection; the light emerging from the 4 plate is now linearly polarized perpendicular to the axis of the LP and no light escapes.

A circular polarizer is a sandwich of a linear polarizer and a quarter-wave plate.

3.5.3 Critical Angle –Total Internal Re‡ection

We mentioned the critical angle during the discussion of the Fresnel equations. From Snell, we have the relation: n1 sin [1] = n2 sin [2]

If n1 > n2 then a speci…c angle C exists that satis…es the condition where its sine is equal to the ratio of the two indices:

n1 n2 sin [C ] = 1 = sin [C ] = < 1 = 2 = n2 ) n1 ) 2 The outgoing ray is refracted parallel to the interface (the “surface”):

n Critical Angle: = sin 1 2 C n 1

Re‡ection at a “dense-to-rare” interface for light incident at the critical angle c.

For crown glass with n = 1:52, = sin 1 1 0:718 radians 41 . For a common ‡int glass d C 1:52 = = with nd = 1:70, then C = 0:629 radians = 36. If the incident angle 1 > C and n1 > n2 (e.g., the …rst medium is glass and the second is air), then no real-valued solution for Snell’s law exists, and there is no refracted light. This is the well-known phenomenon of total internal re‡ection where all incident light is re‡ected at the interface. 74 CHAPTER 3 POLARIZATION

Schematic of total internal re‡ection at a “dense-to-rare” interface.

This may be analyzed rigorously by applying Maxwell’sequations to show that the refracted angle 2 is complex valued instead of real valued, so that the electromagnetic …eld is attenuated exponentially as it crosses the interface. In other words, the electric …eld decays so rapidly across the interface that no energy can ‡ow across the boundary, and hence no light escapes. However, we can “frustrate” the total internal re‡ection by placing another medium (such as another piece of glass) within a few light wavelengths of the interface. If close enough to the boundary, then some electric …eld can get into the second glass and a refracted wave “escapes”.

Schematic of “frustrated total internal re‡ection”: some energy can “jump” across a small gap between two pieces of glass even though the incident angle exceeds the critical angle. As the width of the gap increases, then the quantity of energy coupled across the gap decreases very quickly. Chapter 4

Electromagnetic Waves at an Interface

In optics, and particularly in imaging, we are interested in evaluating electromagnetic …elds on both sides of an interface between two media, e.g., air and glass. We use these properties to design particular shapes of the interfaces to make the light “do what we want,”so that we can redirect the …elds to converge at images.

4.1 Fresnel Equations

A beam of light (implicitly a plane wave) in vacuum or in an isotropic medium propagates in the particular …xed direction speci…ed by its Poynting vector until it encounters the interface with a di¤erent medium. The light causes the charges (electrons, atoms, or molecules) in the medium to oscillate and thus emit additional light waves that can travel in any direction (over the sphere of 4 steradians of solid angle). The oscillating particles vibrate at the frequency of the incident light and re-emit energy as light of that frequency (this is the mechanism of light “scattering”). If the emitted light is “out of phase”with the incident light (phase di¤erence = radians), then the two waves interfere destructively and the original beam is attenuated. If the attenuation is nearly complete, the incident light is said to be “absorbed.”Scattered light may interfere constructively with the incident light in certain directions, forming beams that have been re‡ected and/or transmitted. The constructive interference of the transmitted beam occurs at the angle that satis…es Snell’slaw; while that after re‡ection occurs for re‡ected = incident. The mathematics are based on Maxwell’s equations for the three waves and the continuity conditions that must be satis…ed at the boundary. The equations for these three electromagnetic waves are not di¢ cult to derive, though the process is somewhat tedious. The equations determine the properties of light on either side of the interface and lead to the phenomena of:

1. equal angles of incidence and re‡ection;

2. Snell’sLaw that relates the incident and refracted wave;

3. relative “strengths”and phases of the three light waves, and;

4. orientations of the electric …elds of the three waves (the states of polarization of the three waves).

4.1.1 De…nitions of Vectors For simplicity, we consider only plane waves, so that all beams (incident, re‡ected, and transmitted) are speci…ed by single wavevectors kn that are valid at all points in a medium and that point in the

75 76 CHAPTER 4 ELECTROMAGNETIC WAVES AT AN INTERFACE direction of propagation. The length of the wavevector in a medium with index n is: 2 2 n kn = = = 2 0 j j n n 0

0 where 0 is the wavelength in vacuum and n = n is the wavelength in the medium. The interface between the media is assumed to be the x y plane located at z = 0. The incident wavevector k , 0 the re‡ected vector kr, the transmitted (refracted) vector kt and the unit vector ^n normal to the interface are shown:

The k vectors of the incident, re‡ected, and “transmitted” (refracted) wave at the interface between two media of index n1 and n2 (where n2 > n1 in the example shown).

All angles 0, r, and t are measured from the normal, so that 0 and t are positive and r < 0 as drawn. The incident and re‡ected beams are in the same medium (where n = n1) and so have the same wavelength and their k vectors have the same magnitude: 2n 2n = 1 = 1 1 k k j 0j j rj !0 2n1 k0 = kr = = j j j j v1 0 The wavelength of the transmitted (refracted) beam is di¤erent because of the di¤erent index of refraction: 2n = 2 2 k j tj As drawn, the normal to the surface is speci…ed by the unit vector perpendicular to the interface; in this case, it points in the direction of the positive z-axis:

0 ^n = 2 0 3 6 7 6 1 7 6 7 4 5 (n.b., we could have de…ned ^n in the opposite direction, which would have changed the signs of the angles but would have had no e¤ect on the physics). The incident electric …eld is a sinusoidal oscillation that may be written in complex notation:

E = E exp [+i (k r !0t)] incident 0 0 4.1 FRESNEL EQUATIONS 77

where r = [x; y; z] is the position vector of the location where the phase k0 r !0t is measured; note that the phases measured at all positions in a plane perpendicular to the incident wavevector k0 are identical because this is a plane wave. The re‡ected and transmitted waves have the general forms:

E = E exp [+i (k r !rt + r)] reflected r r E = E exp [+i (k r !tt + t)] transmitted t t where we have yet to demonstrate that !r = !t = !0 and that k = k . The constants r and j rj j 0j t are the (perhaps di¤erent) initial phases of the re‡ected and transmitted waves, i.e., the phases measured at r = 0 and t = 0.

4.1.2 Snell’sLaw for Re‡ection and Refraction of Waves One boundary condition that must be satis…ed is that the phases of all three waves must match at the interface (speci…ed by z = 0) at all times t:

(k r !0t) = (k r !t + r) = (k r !t + t) 0 jz=0 r jz=0 t jz=0 This equivalence at all times immediately implies that the temporal frequencies ! of the three waves must be identical (! = !0) because otherwise the phases would di¤er at di¤erent times. In words, this demonstrates that the temporal frequency is invariant with medium, which is equivalent to saying that the “color”of the light does not change if the light propagates into a di¤erent medium. By cancelling the temporal pars of the phases, we see that the spatial vectors must satisfy the conditions: (k r) = (k r + r) = (k r + t) 0 jz=0 r jz=0 t jz=0 Since the scalar products of the three wavevectors with the same position vector r must be equal, then the three vectors k0; kr and kt must all lie in the same plane (call it the x z plane, as shown in the drawing). The number of waves per unit length at any instant of time must be equal at the boundary for all three waves, as shown, which means that the x-components of the three wavevectors must be equal:

(k0)x = (kr)x = (kt)x

The x-components of the three wavevectors (for the incident, re‡ected, and transmitted refracted waves) must be equal at the interface to ensure that each produces the same number of waves per unit length along the interface, so that the three wavefronts “match” despite the di¤erence in wavelengths in the two media. 78 CHAPTER 4 ELECTROMAGNETIC WAVES AT AN INTERFACE

From the de…nitions of the wavevectors, we can also see that: (k0) = k cos 0 = k sin [0] x j 0j 2 j 0j h i (kr) = k cos r = k sin [ r] x j rj 2 j rj h i where the factor of 1 applied to the re‡ected angle arises from the fact that this angle is clockwise from the normal, and hence negative. The equality of the lengths of the incident and re‡ected wavevectors immediately demonstrates that:

(k0) = (kr) = k sin [0] = k sin [ r] x x j 0j j rj = k sin [0] = k sin [ r] ) j 0j j 0j = sin [0] = sin [ r] ) = r = 0 ) In words, the angle of re‡ection is equal to the negative of the angle of incidence. The sign of the angle often is ignored, leading to the statement that the angles of incidence and re‡ection are equal. Now make the same observation for the transmitted wave; its angle is measured counter-clockwise from the normal and hence is positive:

2n1 (k0)x = k0 sin [0] = sin [0] j j 0 2n2 (kt)x = kt cos t = kt sin [t] = sin [t] j j 2 j j 0 h i We equate these to derive the relationship of the angles of the incident and transmitted wavevectors:

2n1 2n2 sin [0] = sin [t] 0 0

= n1 sin [0] = n2 sin [t] ) We recognize this to be (of course) Snell’slaw for refraction. The re‡ection law may be cast into the form of Snell’srefraction law by assuming that the index of refraction is negative for the re‡ected beam:

n1 sin [0] = n1 sin [r] = sin [r] = sin [0] ) = r = 0 ) Note that these laws were derived without having to consider the vector nature of the electric and magnetic …elds, but rather just the spatial frequencies of the waves at the boundaries. The next task is not quite this simple.....

4.1.3 Boundary Conditions for Electric and Magnetic Fields

We’ve determined the angles of the re‡ected and transmitted (refracted) plane waves in the form of Snell’s law(s). We also need to evaluate the “quantity” of light re‡ected and refracted due to the boundary. Since the geometries of the …elds will depend on the directions of the electric …eld vectors, we will have to consider this aspect in the derivations. In short, this discussion will depend on the “orientation”of the electric …eld relative to the interface, which is what is called the “polarization” of the electric …eld (n.b., this is di¤erent from the “polarizability” of the medium). We will again have to match appropriate boundary conditions at the boundary, but these conditions apply to the vector components of the electric and magnetic …elds on each side of the boundary. We use the same 4.1 FRESNEL EQUATIONS 79 notation as before for amplitudes of the electric …elds of the incident, re‡ected, and transmitted (refracted) waves. Faraday’s and Ampère’s laws (the Maxwell equations involving curl) for plane waves can be recast into forms that are more useful for the current task: @B E r / @t @E B + r / @t We need the constants of proportionality in this derivation. Recall that they depend on the system of units. In the MKS system, the laws have the form: @B E = r @t @E B = + r @t where and are the permittivity and permeability of the medium, respectively and the derivation of the wave equation ensures that phase velocity of light in the medium is:

1 v = r The incident …eld is assumed to be a plane wave of the form already used:

E [x; y; z; t] = E exp [+i (k r !0t)] incident 0 0 E exp +i [k ] x + [k ] y + [k ] z !0t 0 0 x 0 y 0 z E exp [+h i (k0xx + k0yy + k0zz !0t)] i 0 = ^xE0x + ^yE0y + ^zE0z exp [+i (k0xx + k0yy + k0zz !0t)] where the usual convention for naming the scalar components of the vector along the three axes has been used, and we know that E k . In our coordinate system, the incident wave vector lies in the 0? 0 x z plane (the plane de…ned by k and ^n), so that k0y = 0: 0

E [x; y; z; t] = E exp [+i (k r !0t)] incident 0 0 = ^xE0x + ^yE0y + ^zE0z exp [+i (k0xx + k0zz !0t)] The boundary conditions that must be satis…ed by the electric …elds and by the magnetic …elds at the boundary are perhaps not obvious, but result from Maxwell’s equations in two di¤erent geometries. Consider the …gure on the left:

The boundary conditions on the electric and magnetic …elds at the boundary are established from these situations.

We assume that there is no charge or current on the surface and within the cylinder that straddles 80 CHAPTER 4 ELECTROMAGNETIC WAVES AT AN INTERFACE the boundary. If the height dh of the cylinder is decreased towards zero, then the two expressions for Gauss’ law demonstrate that the ‡ux of the electric and magnetic …elds through the top and bottom of the cylinder (the z components in this geometry) must cancel:

1E ^n 2E ^n = 0 1 2 = 1E1z = 2E2z )

B ^n B ^n = 0 1 2 = B1z = B2z ) The ‡ux of the electric …eld in a medium is the so-called “displacement”…eld D = E and the ‡ux of the magnetic …eld is the …eld B. In short, Gauss’laws determine that the normal components of D and of B must be continuous across the boundary of the medium. The …gure on the right is a rectangular path (a “loop”) that also straddles the boundary. The unit vector ^t ^n points along the interface surface. If the “height” of the loop dh 0, then the circulations of? the electric and magnetic …elds around the loop must cancel: !

E ^t E ^t = 0 1 2 = E1x = E2x )

B B 1 ^t 2 ^t = 0 1 2 B B = 1x = 2x ) 1 2

We now need to solve Maxwell’sequations for an incident plane wave, which will depend on the incident angle 0 and on the vector direction of the electric …eld. It is convenient to evaluate these conditions in two cases of linearly polarized waves: (1) where the polarization is perpendicular to the plane of incidence de…ned by ^n and k0 and (2) the polarization is parallel to the plane of incidence de…ned by ^n and k0. The traditional jargon for the …rst state is “s polarization,”though Born and Wolf call it the “perpendicular”or “ ”polarization, while other authors (including Jackson) call it the transverse electric (TE) polarization? because the electric …eld vector is “perpendicular” to the interface. The second case is called “p” or “ ” or the transverse magnetic (TM) polarization. We shall follow Jackson’snaming convention of TEjj and TM. The two cases are depicted below:

The electric …eld perpendicular to the plane of incidence; this is the TRANSVERSE ELECTRIC …eld (TE, also called the “s” or “ ”polarization). ? 4.1 FRESNEL EQUATIONS 81

The electric …eld is parallel to the plane of incidence; this is the TRANSVERSE MAGNETIC …eld (TM, also called the “p” or “ ” polarization). jj

4.1.4 Transverse Electric (TE) Waves = “s” = “ ” Polarization ? In the TE case in our geometry, the incident electric …eld Eincident is oriented along the y direction and the wavevector k0 has components in the x and z directions:

E [x; y; z; t] = ^x 0 + ^y E + ^z 0 exp [+i (k0xx + k0zz !0t)] incident j 0j = ^yE0 exp [+i (k0xx + k0zz !0t)] The magnetic …eld is derived from the relation: n B = k E c 0

Bincident [x; y; z; t]

E0 E0 = cos [0] n1 j j ^x + 0 ^y + + sin [0] n1 j j ^z exp [+i (k0xx + k0zz !0t)] c c The re‡ected …elds are:

E [x; y; z; t] = ^y E exp [+i (krxx + krzz !0t)] reflected j 0j

Breflected [x; y; z; t]

E0 E0 = + cos [ 0] n1 j j ^x + sin [ 0] n1 j j ^z exp [+i (k0xx + k0zz !0t)] c c E0 E0 = + cos [0] n1 j j ^x + sin [0] n1 j j ^z exp [+i (k0xx + k0zz !0t)] c c and the transmitted (refracted) …elds are:

E [x; y; z; t] = ^y E exp [+i (ktxx + ktzz !0t)] transmitted j tj

Btransmitted [x; y; z; t]

Et Et = cos [t] n2 j j ^x + sin [t] n2 j j ^z exp [+i (k0xx + k0zz !0t)] c c 82 CHAPTER 4 ELECTROMAGNETIC WAVES AT AN INTERFACE

The only components of the electric …eld at the interface are transverse, so the only boundary conditions to be satis…ed are the tangential electric …eld:

Er Et E0 + Er = Et = 1 + = ) E0 E0 This is typically expressed in terms of the re‡ection and transmission coe¢ cients for the amplitude of the waves (not the power of the waves; these are the re‡ectance R and transmittance T of the interface, which will be considered very soon):

Er rTE E0 Et tTE E0 where the subscripts denote the transverse electric polarization. The boundary condition for the normal magnetic …eld yields the expression: n n 1 sin [ ](E + E ) = 2 sin [ ] E c 0 0 r c t t while that for the tangential magnetic …eld:

n1 n2 cos [0](E0 Er) = cos [t] Et 1c 2c These may be solved simultaneously for r and t to yield expressions in terms of the indices, perme- abilities, and angles:

Re‡ectance Coe¢ cient for TE Waves

n1 n2 E cos [0] cos [t] r = r = 1 2 TE n1 n2 E0 cos [0] + cos [t] 1 2

n1 cos [0] n2 cos [t] rTE = if 1 = 2 (usual case) n1 cos [0] + n2 cos [t] This equation for the amplitude re‡ectance (amplitude of the re‡ected wave) of the transverse electric polarization is one of the Fresnel Equations that calculates the e¤ect of an interface on incident electromagnetic waves. The corresponding Fresnel equation for the amplitude transmittance is:

Transmission Coe¢ cient for TE Waves

n1 E +2 cos [0] t = t = 1 TE n1 n2 E0 cos [0] + cos [t] 1 2

+2n1 cos [0] tTE = if 1 = 2 n1 cos [0] + n2 cos [t] Again, these are the amplitude coe¢ cients; the re‡ectance and transmittance of light at the surface relate the energies or powers. These measure the ratios of the re‡ected or transmitted power to the incident power. The power is proportional to the product of the magnitude of the Poynting vector and the area of the beam. The areas of the beams before and after re‡ection are identical, which means that the re‡ectance is just the ratio of the magnitudes of the Poynting vectors. This reduces to the square of the amplitude re‡ection coe¢ cient:

R = r2 4.1 FRESNEL EQUATIONS 83 which reduces to this expression for the TE case:

2 n1 cos [0] n2 cos [t] R = TE n cos [ ] + n cos [ ] 1 0 2 t

The transmittance T is a bit more complicated to compute, because the refraction at the interface changes the “width”of the beam (and thus its the cross-sectional area in one direction). The example in the …gure shows a case with n1 > n2, where the width of the transmitted (“refracted”) beam along the x-axis is larger, and thus the area of the transmitted beam is larger, in the medium with the larger index:

Demonstration that the cross-sectional area of the beam is changed by refraction at the interface between two media with di¤erent refractive indices. The area is larger in the medium with the larger index. This di¤erence must be accounted for in the calculation of the power transmission T across the interface.

The magnitude of the Poynting vector is proportional to the product of the index of refraction and the squared magnitude of the electric …eld:

2 s n1 E0 j 1j / j j 2 s n2 Et j 2j / j j The ratio of the transmitted to incident power is:

2 n2 Et A2 s A2 n2 A2 T = j 2j = j j = t2 s A1 2 n1 A1 1 n1 E0 A1 j j j j The area of the transmitted beam changes in proportion to the dimension along the x-axis in this case, which allows us to see that:

A2 w2 sin 2 t cos [t] = = = A1 w1 sin 0 cos [0] 2 which is larger than unity if t < 0 , as is the case ifn2 > n1. This leads to the …nal expression j j j j 84 CHAPTER 4 ELECTROMAGNETIC WAVES AT AN INTERFACE for the transmission at the interface: n cos [ ] T = 2 t2 t n cos [ ] 1 0 n cos [ ] T = 2 t t2 n cos [ ] 1 0 Snell’slaw is used to derive the relationship between the incident and transmitted angles:

n1 sin [0] = n2 sin [t] n1 = sin [t] = sin [0] ) n2 2 2 n1 1 2 2 2 = cos [t] = 1 sin [t] = 1 sin [0] = n2 n1 sin [0] ) s n2 n2 q q Thus we can write down the transmittance T in terms of the refractive indices and the incident angle: 2 2 2 n cos [ ] n2 n1 sin [0] T = 2 t t2 = t2 n1 cos [0] 0q n1 cos [0] 1 where the appropriate expression for t for the [email protected] is inserted.A For the TE case, the trans- mission is: 2 2 n2 cos [t] +2n1 cos [0] n cos [ ] n cos [ ] + n cos [ ] 1 0 1 0 2 t n2 n2 sin2 [ ] 2 2 1 0 +2n1 cos [0] TTE = 0q n cos [ ] 1 n cos [ ] + n cos [ ] 1 0 1 0 2 t @ A These will be plotted for some speci…c cases after we evaluate the coe¢ cients for TM waves.

4.1.5 Transverse Magnetic (TM) Waves = “p” = “ ” polarization k

In the TM case in our geometry, the electric …eld is in the x-z plane and the wavevector has components in the x and z directions:

E [x; y; z; t] = ^x E cos [0] + ^y 0 + ^z E sin [ 0] exp [+i (k0xx + k0zz !0t)] incident j 0j j 0j = (^x E cos [0] ^z E sin [0]) exp [+i (k0xx + k0zz !0t)] j 0j j 0j The magnetic …eld is in the y-direction:

E0 B [x; y; z; t] = n1 j j ^y exp [+i (k0xx + k0zz !0t)] incident c The re‡ected …elds are:

E [x; y; z; t] = (^x E cos [0] ^z E sin [0]) exp [+i (k0xx + k0zz !0t)] reflected j 0j j 0j Er B [x; y; z; t] = n1 j j ^y exp [+i (k0xx + k0zz !0t)] reflected c 4.1 FRESNEL EQUATIONS 85 and the transmitted (refracted) …elds are:

E [x; y; z; t] = (^x E cos [t] ^z E sin [t]) exp [+i (k0xx + k0zz !0t)] transmitted j 0j j 0j Et B [x; y; z; t] = n2 j j ^y exp [+i (ktxx + ktzz !0t)] transmitted c In the case, the boundary condition on the normal component of B is trivial, but the other compo- nents are:

1 sin [0](E0 + Er) = 2 sin [2] Et

cos [0](E0 Er) = cos [2] Et n1 n2 (E0 + Er) = Et 1c 1c These are solved for the re‡ection and transmission coe¢ cients:

Transverse Magnetic Waves

n2 n1 + cos [0] cos [t] r = 2 1 TM n2 n1 + cos [0] + cos [t] 2 1 which simpli…es if the permeabilities n are equal (as they usually are):

+n2 cos [0] n1 cos [t] rTM = if 1 = 2 +n2 cos [0] + n1 cos [t]

The corresponding re‡ectance is:

2 +n2 cos [0] n1 cos [t] R = TM +n cos [ ] + n cos [ ] 2 0 1 t The amplitude transmission coe¢ cient evaluates to:

n1 2 cos [0] t = 1 TM n2 n1 + cos [0] + cos [t] 2 1 again, if the permeabilities are equal, this simpli…es to:

2n1 cos [0] tTM = if 1 = 2 +n2 cos [0] + n1 cos [t]

The corresponding transmittance function is:

n2 n2 sin2 [ ] 2 2 1 0 2n1 cos [0] TTM = 0q n cos [ ] 1 +n cos [ ] + n cos [ ] 1 0 2 0 1 t @ A

4.1.6 Comparison of Coe¢ cients for TE and TM Waves

The re‡ectance coe¢ cients for the two cases of TE and TM waves are: 86 CHAPTER 4 ELECTROMAGNETIC WAVES AT AN INTERFACE

n1 cos [0] n2 cos [t] rTE = n1 cos [0] + n2 cos [t] +n2 cos [0] n1 cos [t] rTM = +n2 cos [0] + n1 cos [t] where the angles are also determined by Snell’slaw:

n1 sin [0] = n2 sin [t] 2 n1 = cos [t] = 1 sin [0] ) n s 2

Note that angles and the indices for the TE case are “in”the same media, i.e., the index n1 multiplies the cosine of 0, which is in the same medium, and the same holds for n2 and t. However, the opposite situation applies to the TM case: n1 is applied to cos [t] and n2 to cos [0]. These same observations also apply to the corresponding transmission coe¢ cients:

+2n cos [ ] +2 cos [ ] t = 1 0 = 0 TE n2 n1 cos [0] + n2 cos [t] cos [0] + cos [t] n1 +2n cos [ ] +2 cos [ ] t = 1 0 = 0 TM n2 +n2 cos [0] + n1 cos [t] + cos [0] + cos [t] n1

Normal Incidence (0 = 0)

In the case of normal incidence, where 0 = r = t = 0, then the TE and TM equations evaluate to: n n r = 1 2 TE 0=0 j n1 + n2 +n n r = 2 1 = r TM 0=0 TE 0=0 j +n2 + n1 j +2n t = 1 TE 0=0 j n1 + n2 +2n t = 1 = t TM 0=0 TE 0=0 j n1 + n2 j Note that the re‡ectance coe¢ cients are not identical, even though the two polarizations are indis- tinguishable at normal incidence. Since the areas of the incident and transmitted waves are identical, there is no area factor in the amplitude transmittance. The resulting formulas for the observable re‡ectance and transmittance are identical:

normal incidence (0 = 0)

2 n1 n2 RTE (0 = 0) = RTM (0 = 0) R = n + n 1 2

4n1n2 T = 2 (n1 + n2)

“Rare-to-Dense” Re‡ection at Normal Incidence If the input medium has a smaller re- fractive index n (a rarer medium) than the second (denser) medium (e.g., “air to glass” with n1 = 1:0 < n2 = 1:5, then the Fresnel coe¢ cients are: 4.1 FRESNEL EQUATIONS 87

1:0 1:5 +i rTE = = 0:2 = 0:2e 1:0 + 1:5 1:5 1:0 r = = +0:2 TM 1:5 + 1:0 2 1:0 t = t = = +0:8 TE TM 1:0 + 1:5

for “rare-to-dense” re‡ection with n1 = 1:0 and n2 = 1:5

Note the value of rTE; in words, the phase of the re‡ected light is changed by radians = 180 if re‡ected at a “rare-to-dense”interface such as the usual air-to-glass case. The re‡ectivity R and the transmittance T are:

1 1:5 2 R = R = = 0:04 TE TM 1 + 1:5 4 1 1:5 TTE = TTM = = 0:96 (1 + 1:5)2 which means that: R + T = 1

“Dense-to-Rare”Re‡ection at Normal Incidence If the input medium is “denser”(n1 = 1:5 > n2 = 1:0), then the amplitude re‡ection coe¢ cients at normal incidence are:

1:5 1:0 r = = +0:2 TE 1:5 + 1:0 1:0 1:5 +i rTM = = 0:2 = 0:2e 1:0 + 1:5 The re‡ectivity is the same as in the “rare-to-dense”re‡ection:

R = ( 0:2)2 = 0:04 The amplitude transmission coe¢ cients at normal incidence in a “dense-to-rare”re‡ection are iden- tical: 2 1:5 t = t = = +1:2 > 1:0 TE TM 1:0 + 1:5 The fact that these values are larger than unity may seem strange, at least until we recall that the transmittance requires the additional geometrical factor:

n cos [ ] T = 2 t t2 n cos [ ] 1 0 1 = (+1:2)2 = 0:96 1:5 Again, we see that energy is conserved: R + T = 1

4.1.7 Angular Dependence of Amplitude Coe¢ cients at “Rare-to-Dense” Interface The graphs of the Fresnel coe¢ cients using the equations just derived for the cases of the “rare-to- dense” interface (n1 = 1 < n2 = 1:5) are plotted vs. incident angle measured in degrees from 0 88 CHAPTER 4 ELECTROMAGNETIC WAVES AT AN INTERFACE

(normal incidence) to 90 (grazing incidence). Note that the amplitude re‡ectance coe¢ cient for the TM state (“ ” or p polarization) goes to zero at a speci…c angle B = 60 (Brewster’s Angle), jj which means that RTM = 0 and TTM = 1 there. This phenomenon was discovered in 1815 by David Brewster.

Amplitude re‡ectance and transmittance coe¢ cients for “rare-to-dense” re‡ection: n1 = 1:0 (air) and n2 = 1:5 (glass) for both TE and TM waves, plotted as functions of the incident angle from 0 = 0 (normal incidence) to 0 = 90 (grazing incidence). The re‡ectance coe¢ cient rTE < 0 for all , which means that there is a phase shift upon re‡ection, whereas rTM > 0 for 0 < B (Brewster’s angle). Also note that the transmittance coe¢ cients are very similar functions.

4.1.8 Re‡ectance and Transmittance at “Rare-to-Dense” Interface

The re‡ectance and transmittance the two polarizations with n1 = 1:0 and n2 = 1:5 as functions of the incident angle 0 show that the re‡ectance of the TM wave at Brewster’sangle is zero. 4.1 FRESNEL EQUATIONS 89

Re‡ectance and transmittance for n1 = 1:0 and n2 = 1:5 for TE and TM waves. Note that RTM = 0 and TTM = 1 at “Brewster’s angle.”

Brewster’sAngle for Complete Polarization of Re‡ected Wave

In a “rare-to-dense” re‡ection at Brewster’s Angle 0 = B, we see that rTM = 0 and therefore the re‡ected amplitude of this wave is zero. In other words, the only light re‡ected at this angle must have the TE polarization only. This is Brewster’s angle where the electrons driven in the plane of the incidence will not emit radiation at the angle required by the law of re‡ection; this is sometimes called the angle of complete polarization. The incident angle B may be evaluated by setting rTM = 0:

+n2 cos [B] n1 cos [t] rTM = = 0 +n2 cos [B] + n1 cos [t]

= +n2 cos [B] = n1 cos [t] ) 2 2 n2 2 = cos [t] = cos [B] ) n 1 2 We also have an expression for sin [t] from Snell’slaw:

2 2 n1 2 n1 sin [B] = n2 sin [t] = sin [t] = sin [B] ) n 2 2 Square and add to the expression for cos [t]:

n 2 n 2 cos2 [ ] + sin2 [ ] = 1 = 2 cos2 [ ] + 1 sin2 [ ] t t n B n B 1 2 90 CHAPTER 4 ELECTROMAGNETIC WAVES AT AN INTERFACE

2 2 But we also know that cos [B] + sin [B] = 1, so the two expressions may be equated:

n 2 n 2 cos2 [ ] + sin2 [ ] = 2 cos2 [ ] + 1 sin2 [ ] B B n B n B 1 2 2 2 2 2 n2 n1 2 n1 n2 2 = 2 cos [B] + 2 sin [B] = 0 ) n1 n2 2 2 sin [B] 2 n2 = 2 = tan [B] = 2 ) cos [B] n1 So the expression that is satis…ed by Brewster’sangle at a “rare-to-dense”re‡ection is:

n = tan 1 2 B n 1 If n1 = 1 (air) and n2 = 1:5 (glass), then B = 56:3. For incident angles larger than about 56, the re‡ected light is plane polarized parallel to the plane of incidence. We can also …nd t by applying Snell’slaw:

n1 sin [B] = n2 sin [t]

1 1:5 1 sin tan = 1:5 sin [t] 1 1 1 1 1:5 = sin sin tan = 33:7 t 1:5 1 Note that + = 56:3 + 33:7 = 90 = radians B t 2 If the dense medium is water (n = 1:33), then 52:4 and = 37:6 ; again, + = 2 B = t B t 2 radians. This means that light incident at Brewster’sangle B is refracted at t such that: t + B = = sin [t] = sin B = cos [B] 2 ) 2 h i

Polarization of re‡ected light at Brewster’s angle. The incident beam at 0 = B is unpolarized. The re‡ectance coe¢ cient for light polarized in the plane (TM waves) is 0, and the sum of the incident and refracted angle is 90 = . Thus B + t = = t = B. 2 2 ) 2 The re‡ection at Brewster’s angle provides a handy means to determine the polarization axis of a linear polarizer – just look through a linear polarizer at light re‡ected at a shallow angle relative 4.1 FRESNEL EQUATIONS 91 to the surface (e.g., a waxed ‡oor). Note that the transmitted (refracted) light contains both polarizations, though not in equal amounts.

4.1.9 Re‡ection and Transmittance at “Dense-to-Rare” Interface, Criti- cal Angle

At a “dense-to-rare”interface (n1 > n2), the re‡ectance of the TM wave (s polarization) is:

n2 cos [0] + n1 cos [t] r = +n2 cos [0] + n1 cos [t]

The numerator evaluates to zero for a particular incident angle C that satis…es:

n2 cos [C ] = n1 cos [t] n cos [ ] 1 = C n2 cos [t] This corresponds to the situation where Snell’slaw requires that:

n1 n2 sin t = = 1 = sin [C ] = sin [C ] = 2 n2 ) n1 h i If the incident angle exceeds this critical angle, then the amplitude re‡ectance coe¢ cients rTE and rTM are both unity, and thus so are the re‡ectances RTE and RTM : This means that light incident at 0 C is totally re‡ected at a “dense-to-rare”interface. This phenomenon is the source of total internal re‡ectance (TIR); the re‡ection is “internal” because it within the dense medium, e.g., glass. TIR is why optical …bers are useful in communications. If n1 = 1:5 and n2 = 1:0, then 2 sin [C ] = = 0 = 0:73 radians = 41:8 C 3 ) Compare this to Brewster’sangle in the same case satis…es:

1 n2 1 2 B = tan = tan = B = 0:59 radians = 33:7 < C n 3 ) 1

Amplitude re‡ectance coe¢ cients for TE and TM waves at a “dense-to-rare” interface with 92 CHAPTER 4 ELECTROMAGNETIC WAVES AT AN INTERFACE

n1 = 1:5 (glass) and n2 = 1:0 (air). Both polarizations rise to r = +1:0 at the “critical angle” c, for which t = 90 = 2 . Also noted is Brewster’s angle, where rTM = 0. The coe¢ cients for 0 > c may be interpreted as being complex-valued.

4.1.10 Practical Applications for Fresnel’sEquations The 4% re‡ectivity at normal incidence for one surface of glass is the reason why windows resemble mirrors at night for persons in brightly lit rooms. The windows at the ends of the gas tube in a He:Ne laser are often oriented at Brewster’s angle to eliminate re‡ective losses; the emitted laser light is linearly polarized. The basic principle of optical …ber transmission is total internal re‡ection to propagate the beam. Hollow optical …bers use high-incidence-angle near-unity re‡ections. Chapter 5

Optical Interference and Coherence

We now return to the more accurate model of light as a wave with the goal of understanding the performance limitations of imaging systems. We …rst consider “interference,” the result of adding waves from a small number of sources which may “cancel out” or reinforce, depending on the amplitudes and relative phase angles. This study leads to the concept of “coherence,” which is a measure of the statistical behavior of the optical phase over time, space, or both. Finally, we consider the subject of di¤raction, which is really just an extension of the concept of interference to large numbers of sources. We shall see that the e¤ects of di¤raction lead to the ultimate limit on the performance of an optical imaging system. The basic concept of interference (and of di¤raction) arises from the di¤erence in the phase of light waves that are superposed at one point in space; the two waves may have been generated from two locations on the same wavefront or they may have been derived from the same point on the wavefront and traveled two di¤erent paths to recombination. In either event, the signi…cant aspect is the optical phase di¤erence between the recombined waves. The importance of the optical phase di¤erence is illustrated by a simple addition of two oscillations and then will be generalized to the sum of traveling waves.

5.1 Addition of Oscillations

The simple mathematical sum of two sinusoids with the same amplitude and di¤erent frequencies and phases may be evaluated via the identity for the sum of two cosines:

'1 + '2 '1 '2 cos ['1] + cos ['2] = 2 cos cos 2 2 2 cos ['avg ] cos ['mod] This relationship may be derived easily using complex analysis. This result may then be generalized to the sum of two sinusoidal oscillations with the same amplitude A but di¤erent angular temporal frequencies !1 and !2:

y1 [t] = A cos [!1t + 1]

y2 [t] = A cos [!2t + 2]

!1 + !2 1 + 2 !1 !2 1 2 y1 [t] + y2 [t] = 2A cos t+ cos t+ 2 2 2 2 2A cos !avg t + cos [!modt + ] avg mod

!1+!2 !1 !2 1+2 1 2 where !avg 2 , !mod 2 , avg 2 , and mod 2 . In words, the sum of two oscillations with di¤erent frequencies is identical to the product of two oscillations: one at the

93 94 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

slower varying modulation !mod and the other is the more rapidly oscillating average frequency !avg . The latter is called the carrier frequency in some applications. A perhaps familiar example of the modulation results from the excitation of two mistuned piano strings, where a low-frequency oscillation (the beat) is heard. As one string is tuned to the other, the frequency of the beat decreases until reaching zero when the string frequencies match. Acoustic beats may be thought of as interference of the summed oscillations in time.

Sidebar: Addition of Di¤erent-Ampitude Oscillations

As an aside, the corresponding result for waves with di¤erent amplitudes may be evaluated by decomposing the larger term into two pieces, of which one has the same amplitude:

A1 cos [!1t + 1] + A2 cos [!2t + 2]

= (A1 A2) cos [!1t + 1] + A2 (cos [!1t + 1] + cos [!2t + 2]) = (A1 A2) cos [!1t + 1] + 2A2 cos !avg t + cos [!modt + ] avg mod Sidebar: Multiplication of Oscillations

We have just seen that the sum of two sinusoidal oscillations with the same amplitude may be written as the product of oscillations with the average and modulation frequencies. In passing, we note the (also useful) complementary result that the product of two oscillations with the sum and di¤erence frequencies:

cos [!1t + 1] cos [!2t + 2] 1 1 = cos [(!1 + !2) t + (1 + 2)] + cos [(!1 !2) t + (1 2)] 2 2 In words, the product of two oscillations is equivalent to the sum of two oscillations: one at the sum frequency (with the sum of the initial phases) and one at the di¤erence frequency. This is the principle of signal mixing or modulation, as in AM radio where the transmitted radio-frequency signal (RF) is multiplied by an intermediate frequency (IF) generated within the receiver to produce the sum of high-frequency and low-frequency signals. The sum-frequency signal is stripped away by a lowpass …lter, leaving the desired audio signal.

“Heterodyne” detection by multiplying two oscillations. The di¤erence frequency is passed by the lowpass …lter.

5.1.1 Addition of 1-D Traveling Waves

The expression for the summation of two oscillations may be extended to 1-D traveling waves with the same amplitude A:

f1 [z; t] = A cos [(k1z !1t) + 1] f2 [z; t] = A cos [(k2z !2t) + 2] f1 [z; t] + f2 [z; t] = 2A cos [(kmodz !modt) + mod] cos [(kavg z !avg t) + avg ] 5.1 ADDITION OF OSCILLATIONS 95

k1 k2 k = mod 2 !mod !1 !2 ! vmod = = = kmod k1 k2 k 1 2 = mod 2

k + k k = 1 2 avg 2 !avg !1 + !2 vavg = = kavg k1 + k2 + = 1 2 avg 2 In words, the superposition of two traveling waves with di¤erent temporal frequencies (and thus di¤erent wavelengths) generates the product of two component traveling waves, one oscillating more slowly in both time and space, i.e., a “traveling modulation.” Note that both the average and modulation waves move along the z-axis. In this case, k1, k2, !1, and !2 are all positive, and so kavg and !avg must be also. However, the modulation wavenumber and frequency may be negative. In fact, the algebraic sign of kmod may be negative even if !mod is positive (or vice versa). In either of these cases, the modulation wave moves in the opposite direction to the average wave.

Example: 1-D Waves Traveling in Opposite Directions (k2 = k1)

Note that if the two 1-D waves traveling in the same direction along the z-axis have the same 2 frequency !, they must have the same wavelength 0 and the same wavenumber k = . The 0 modulation terms kmod and !mod must be zero, and the summation wave exhibits no modulation. Recall also such waves traveling in opposite directions generate a waveform that “moves”but does not “travel:”

f1 [z; t] = A cos [(k1z !1t) + 1] f2 [z; t] = A cos [( k1z !1t) + 2] = A cos [(k1z + !1t) 2] f1 [z; t] + f2 [z; t] = A cos [(k1z !1t) + 1] + A cos [(k1z + !1t) 2] k1 + k1 !1 !1 1 2 = 2A cos z t + 2 2 2 k1 k1 !1 + !1 1 + 2 cos z t + 2 2 2 1 + 2 = 2A cos k1z + cos !1t + 2 2 = 2A cos [k1z + mod] cos [ !1t + avg ] = 2A cos [k1z + mod] cos [!1t avg ] where the symmetry of cos[] was used in the last step. The result is the product of two terms: one each that oscillates in space and time. The spatial variation is “…xed,”but the amplitude oscillates in time at !1 radians per second. This is commonly called a “standing wave.” 96 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

The behavior of a standing wave over time: the spatial variation does not move but its amplitude radians does oscillate at an angular frequency of !0 sec . Note that the amplitude at a value of z is negative half of the time. Though not apparent at this point, the goal of optical interference is to produce a standing wave that may be recorded and measured.

Traveling Waves with “Similar” Frequencies !1 ' !2

If !1 ' !2 (which means that k1 = k1 / k2), then the propagation velocities of the two waves are approximately equal j j ! + ! ! = 1 2 = ! avg 2 1 kavg = k1 = k2 !1 !2 ! = 0 mod 2 ' k1 k2 k = 0 mod 2 / The propagation velocity of the “average wave”is:

!avg !1 vavg = = = v1 = v2 kavg k1 The velocity of the “modulation wave”is:

! !mod 2 ! d! vmod = = = in the limit k2 k1 k k k ! dk ! mod 2

Another common name for vmod is the group velocity, which is a measure of the velocity of propa- gation of information carried by the modulation. The “game” of interference is to combine two (or more) sinusoidal traveling waves that di¤er in some attribute: di¤erent propagation directions, di¤erent phases, di¤erent path lengths, from di¤erent locations on a source, etc. The two waves may have been extracted from di¤erent points on the same source wavefront (“division of wavefront”interference) or obtained by dividing the light amplitude from the same source wavefront with some type of beamsplitter (“division of amplitude” interference). The next chapter generalizes the concept of interference to include a “large”number of waves (usually an in…nite number). The resulting concept of “di¤raction” is very important in 5.2 DIVISION-OF-WAVEFRONT INTERFERENCE 97

Figure 5.1: Two classes of interference used to measure statistical variation in the phase: (a) light from two locations on the same wavefront is added in division-of-wavefront interference, which mea- sures the spatial coherence; (b) light from two di¤erent wavefronts is combined using a beamsplitter in division-of-amplitude interference, which measures the temporal coherence..

imaging because it is the fundamental limit to the capability of a system to distinguish di¤erent objects.

5.2 Division-of-Wavefront Interference

References: Hecht, Optics §8 The …rst example of optical interference involves summing two wavefronts that were generated by one source and traveled di¤erent paths before being recombined. This was the …rst type of interference observed, having been noted by Thomas Young in the early 1800s. The mathematical derivation is a simple extension of the addition of 1-D traveling waves to three dimensions.

5.2.1 Addition of 3-D Traveling Waves with same !

Traveling waves also may be de…ned over two or three spatial dimensions; the waves have the form f[x; y; t] and f[x; y; z; t], respectively. The direction of propagation of such a wave in a multidi- mensional space is determined by a vector analogous to k0; a 3-D wavevector k0 has components [(k0)x ; (k0)y ; (k0)z]. The vector may be written:

(k0)x 2 3 k0 = (k0)x ^x + (k0)y ^y + (k0)z ^z = (k0)y 6 7 6 (k0)z 7 6 7 4 5 2 The corresponding wave travels in the direction of the wavevector k and has wavelength 0 = k . In other words, the length of k is the magnitude of the wavevector: j j

2 2 2 2 k0 = k0 k0 = (k0)x + (k0)y + (k0)z = : j j 0 p q The angular temporal frequency !0 is determined from the magnitude of the wavevector through the dispersion relation: v !0 = v k0 = 0 = j j ) 0 98 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

Example: (k1)y = (k2)y = 0 For illustration, consider a simple 2-D analogue of the 1-D traveling plane wave. The wave travels in the direction of a 2-D wavevector k that lies the x z plane, so that ky = 0:

(k0)x

k0 = 2 0 3 6 7 6 (k0)z 7 6 7 4 5 The points of constant phase with phase angle = C radians is the set of points in the 2-D space r = [x; y; z] = (r; ) such that the scalar product k r = C: 0

k r = r k = k r cos [] 0 0 j 0j j j = (k0) x + 0 y + (k0) z = C radians, for point of constant phase x z

Therefore, the equation of a 2-D wave traveling in the direction of k0 with linear wavefronts is:

f [x; y; t] = A cos [(k0) x + (k0) z !0t] x z = A cos [k r !0t] 0 In three dimensions, the set of points with the same phase lie on a planar surface so that the equation of the traveling wave is:

f [x; y; z; t] = f [r; t]

= A cos (k0) x + (k0) y + (k0) z !0t x y z = A cos [hk r !0t] i 0

Plane wave traveling in direction k This plane wave could have been created by a point source at a large distance to the left and below the z-axis. 5.2 DIVISION-OF-WAVEFRONT INTERFERENCE 99

Now, we will apply the equation derived when adding oscillations with di¤erent temporal fre- quencies. In general, the form of the sum of two traveling waves is:

f1 [x; y; z; t] + f2 [x; y; z; t] = A cos [k1 r !1t] + A cos [k2 r !2t] = 2A cos k r !avg t cos [k r !modt] avg mod where the average and modulation wavevectors are:

k + k (k ) + (k ) (k1) + (k2) (k ) + (k ) k = 1 2 = 1 x 2 x ^x + x y ^y + 1 z 2 z ^z avg 2 2 2 2 k k (k1) (k2) (k1)x (k2)y (k1) (k2) k = 1 2 = x x ^x + ^y + z z ^z mod 2 2 2 2 and the average and modulation angular temporal frequencies are: ! + ! ! = 0 0 = ! avg 2 0 !0 !0 ! = = 0 mod 2

The average and modulation wavevectors kavg and kmod generally point in di¤erent directions and thus the corresponding waves move in di¤erent directions at velocities determined from:

!avg vavg = kavg ! v = mod mod k j modj Because the phase of the multidimensional traveling wave is a function of two parameters (its wavevector kn and angular temporal frequency !n), the phases of two traveling waves usually di¤er even if the temporal frequencies are equal. Since the temporal frequencies are equal, so must be the wavelengths: 1 = 2 = k = k k k ! j 1j j 2j j j In this case, the component waves travel in di¤erent directions so the components of the wavevectors di¤er: k = (k1) ; (k1) ; (k1) = k = (k2) ; (k2) ; (k2) 1 x y z 6 2 x y z The summation of two travelingh waves with identicali magnitudesh and temporali frequencies is:

f1[x; y; z; t] + f2[x; y; z; t] = A cos(k r !0t) + A cos(k r !0t) 1 2 = 2A cos(k r !avg t) cos(k r) avg mod because !mod = 0. The superposition of two 2-D wavefronts with the same temporal frequency but traveling in di¤erent directions results in two multiplicative components: a traveling wave in the direction of kavg , and a stationary wave in space along the direction of kmod. 100 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

The wavevectors of two interfering plane waves with the same wavelength.

These two waves could have been generated at point sources located above and below the z-axis at a large distance to the left. This is the classic “Young’sdouble slit”experiment, where light from a single source is split into two waves (spherical waves in this case) that propagate large distances to the observation plane:

How two “tilted” plane waves are generated in the Young double-aperture experiment. The two apertures in the opaque screen on the left divide the incoming wave into two expanding spherical waves. After propagating a long distance, the spherical waves approximate plane waves that are d tilted relative to the axis by = 2 L : 5.2 DIVISION-OF-WAVEFRONT INTERFERENCE 101

The “tilts”of the two waves are evaluated from the two (equal) distances:

(d=2) d tan [] = = L 2L If L >> d, then d tan [] = sin [] = = 2L

The superposition of the two electric …elds is:

f[x; y; z; t] = f1[x; y; z; t] + f2[x; y; z; t]

= 2A cos k r !avg t cos [k r] avg mod z x = 2A cos 2 cos [] !0t cos 2 sin [] 0 0 The …rst term (with the time dependence) is a traveling wave in the direction speci…ed by k = 2 [0; 0; kz] = cos [] ^z, while the second term (with no dependence on time) is a spatial wave along 0 the x direction. The amplitude variation of this modulation wave is:

x x 2A cos 2 sin [] = 2A cos 2 0 2 0 3 sin[] 4 5 which is a sinusoidal oscillation with period sin[] .

The irradiance (measurable intensity) of the superposition is:

2 2 2 z 2 x f [x; y; z; t] = 4A cos 2 cos [] !0t cos 2 j j 0 2 0 3 sin[] 4 5 Both cosine terms can be rewritten using: 1 cos2 [] = (1 + cos [2]) 2

2 z 1 z cos 2 cos [] !0t = 1 + cos 2 2!0t 0 2 0 2 0 31 2 cos[] @ 4 5A 2x sin [] 1 x cos2 = 1 + cos 2 0 2 0 2 0 31 2 sin[] @ 4 5A 14 radians As before, the argument of the …rst term varies rapidly because !0 = 10 sec . The result of the 1 measurement is the average value of 2 :

1 1 x f [x; y; z; t] 2 = 4A2 1 + cos 2 j j 2 2 0 2 0 31 2 sin[] D E @ 4 5A x = A2 1 + cos 2 0 2 0 31 2 sin[] @ 4 5A 102 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

This derivation may also be applied to …nd the irradiance of the individual component traveling waves:

2 2 2 2 2 1 I1 = f1 [x; y; z; t] = A cos [k r !0t] = A cos [k r !0t] = A j j j 1 j 1 2 D 2E D 2 1 E I2 = f2 [x; y; z; t] = A = I1 I0 j j 2 D E So the irradiance of the sum of the two waves can be rewritten in terms of the irradiance of a single wave:

2 x f [x; y; z; t] = 2I0 1 + cos 2 j j 0 2 0 31 D E 2 sin[] @ 4 5A This is a “biased sinusoid” that is positive or zero everywhere. In other words, the modulation of 0 the irradiance is sinusoidal between 0 and 2I0 (2) = 4I0 with period X0 = , so that the 2 sin[] average irradiance is 2I0. The period varies directly with and inversely with sin []; for small , the period of the sinusoid is large, for = 0 (so that the both waves travel “straight down”the optical axis); there is no modulation of the irradiance. The alternating bright and dark regions of this time-stationary sinusoidal intensity pattern are called interference fringes. The shape, separation, and orientation of the interference fringes are determined by the form of and direction traveled by the incident wavefronts, and the pattern provides information about the incident waves. In fact, if one of the two waves is known, then the parameters of the other wave can be inferred from the interference pattern; this is the principle behind holography. The argument of the cosine function is the optical phase di¤erence of the two waves, which is a function of the wavelength 0 and the angle of each beam from the axis of symmetry. The phase di¤erence does not vary with time, which means that the two waves are temporally coherent. At locations where the optical phase di¤erence is an even multiple of , the cosine evaluates to unity and a maximum of the interference pattern results; this is constructive interference. If the optical phase di¤erence is an odd multiple of , the cosine evaluates to -1 and the irradiance is zero; this is destructive interference.

Interference of two “tilted plane waves” with the same wavelength. The two component traveling waves are shown as “snapshots” at one instant of time on the left (white = 1, black = -1); the sum of the two is shown in the center (white = 2, black = -2), and the squared magnitude on the right (white = 4, blace = 0). The modulation in the vertical direction is constant, while that in the 1 horizontal direction is a traveling wave and “averages” out to a constant value of 2 . 5.2 DIVISION-OF-WAVEFRONT INTERFERENCE 103

The amplitude and irradiance observed at one instant of time when the irradiance at the origin (“on axis”) is a maximum is shown:

Interference patterns observed along the x-axis at one value of z: (a) bipolar amplitude fringes, with period equal to sin[] ; nonnegative irradiance (intensity) fringes, with period equal to 2 sin[] . This 1 pattern is averaged over time and scales by a factor of 2 :

Again, the traveling wave in the images of the amplitude and intensity of the superposed images moves in the z-direction (to the right), thus blurring out the oscillations in the z-direction. The oscillations in the x-direction are preserved as the interference pattern, which is plotted as a function of x below. Note that the spatial frequency of the intensity fringes is twice as large as that of the amplitude fringes.

Irradiance patterns observed at the output plane at several instants of time, showing that the spatial variation of the irradiance is preserved but the averaging reduces the maximum value by half.

5.2.2 Superposition of Two Plane Waves with !1 = !2 6 For further illustration, consider the case the two waves travel in the same directions but with di¤erent temporal frequencies !1 = !2, which means that k^ = k^ and k = k The average and 6 1 2 j 1j 6 j 2j modulation wavevectors are found as before, but the fact that !mod = 0 means that modulation wave is no longer stationary along the x-axis. 6 104 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

1 = 8 units 1 ! = radians per second 1 8 1 = radians = 30 6

2 = 12 units 1 ! = radians per second 2 12 = + radians = +30 2 6 average and modulation frequencies: 3 ! = ! 1 2 2 1 + 1 5 ! = 8 12 = radians per second avg 2 48 1 1 1 ! = 8 12 = radians per second mod 2 48

Length of Wavevectors:

– 1 = 8 units = ) 2 radians k = = = 0:785 j 1j 8 4 unit length ! 1 radians per second 1 units v = 1 = 8 = 1 k radians 2 second j 1j 4 unit length

– 1 = 12 units = ) 2 radians k = = = 0:785 j 2j 8 4 unit length ! 1 radians per second 1 units v = 2 = 8 = 2 k radians 2 second j 2j 4 unit length

– v1 = v2 = nondispersive waves )

Derive k 1 2 2 1 radians (k1) = sin [1] = = = 0:393 x 8 2 8 unit length 1 2 2 p3 radians (k1)z = cos [1] = = 0:680 1 8 2 unit length

2 2 p3 – Check: k1 = 8 + 8 = 4 (agrees with result from wavelength j j r 5.2 DIVISION-OF-WAVEFRONT INTERFERENCE 105

– Components of k1: 0:393 8 2 3 2 3 k1 = 0 = 0 6 p3 7 6 7 6 + 8 7 6 +0:680 7 6 7 6 7 4 5 4 5 – Derive k2 2 1 radians (k2) = sin [ 1] = + = + = +0:262 x 6 2 12 unit length 2 2 2 p3 p3 radians (k2)z = cos [ 1] = = = 0:453 2 12 2 12 unit length

2 2 p3 – Check: k2 = 12 + 12 = 6 (also agrees) j j r – Components of k2: + 12 +0:262 2 3 2 3 k2 = 0 = 0 6 p3 7 6 7 6 + 12 7 6 +0:453 7 6 7 6 7 4 5 4 5 Average and modulation wavevectors: 8 + 12 48 k + k 1 k = 1 2 = 02 0 3 + 2 0 31 = 2 0 3 avg 2 2 B6 p3 7 6 p3 7C 6 5p3 7 B6 + 8 7 6 + 12 7C 6 48 7 B6 7 6 7C 6 7 p19 [email protected] 5 radians4 5A 4 5 k = = 0:571 avg 24 unit length unit length 1 48 avg = tan = 0:115 radians = 6:6 5p3 " 48 #

5 8 + 12 48 k k 1 1 2 = 02 0 3 2 0 31 = 2 0 3 2 2 B6 p3 7 6 p3 7C 6 p3 7 B6 + 8 7 6 + 12 7C 6 48 7 B6 7 6 7C 6 7 [email protected]4 radians5 4 5radiansA 4 5 k = = 0:346 j mod j 24 unit length unit length 5 1 48 mod = tan = 1:237 radians = 71 p3 " 48 #

Note that both the average and modulation waves travel; they are headed in di¤erent directions 5 with di¤erent frequencies and di¤erent velocities. The temporal frequencies are avg = 48 Hz and 2 mod = 48 Hz. In this case, the time-averaged irradiance of the modulation wave is also “averaged out”to create a uniform irradiance pattern and no stationary fringe pattern is visible from interfer- ence of beams with two di¤erent wavelengths. Put another way, the phase di¤erence encapsulated in the phase of the modulation oscillation is a function of time and space; this variation means that the light is not coherent. 106 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

3 Sum of two sinusoidal traveling waves where the periods are related by 2 = 2 1. The two waves travel in the directions 30, respectively. The resulting amplitude sum and power are depicted as “snapshots” at one instant of time. Since the modulation wave now travels too, both waves are averaged to constant values and no fringes are visible.

Intensity patterns observed at the output plane at several instants of time. The velocity of the modulation wave makes this pattern “migrate” towards x, and thus the time-averaged pattern is a constant; no interference is seen. The same principles just discussed may be used to determine the form of interference fringes from wavefronts with other shapes. Some examples will be considered in the following sections.

5.3 Fringe Visibility –Temporal Coherence

The visibility of a sinusoidal fringe pattern is a quality that corresponds quite closely to the concept of modulation of sinusoids. Given a nonnegative sinusoidal irradiance (intensity) distribution with maximum Imax and minimum Imin (so that Imin 0), the visibility of the sinusoidal fringe pattern is: Imax Imin V Imax + Imin 5.3 FRINGE VISIBILITY –TEMPORAL COHERENCE 107

Note that if Imin = 0, then V = 1 regardless of the value of Imax . The visibility of the fringe pattern is determined by the relative irradiances of the individual wavefronts and by the coherence of the light source, which determines the distance over which the phase of the light from the source is well de…ned.

To introduce the concept of coherence, consider …rst the Young’stwo-aperture experiment where the source is composed of equal-amplitude emission at two distinct wavelengths 1 and 2 incident on the observation screen at . Possible pairs of wavelengths could be those of the sodium doublet ( = 589:0 nm and 589:6 nm), or the pair of lines emitted by a “greenie”He:Ne laser (1 = 543 nm (green), 2 = 594 nm (yellow)). In air or vacuum, the corresponding angular frequencies obviously 2c 2c are !1 = c k1 = and !2 = . j j 1 2 To …nd the irradiance pattern created by the interference of the four beams, we must compute the superposition of the amplitude of the electromagnetic …eld, …nd its squared-magnitude, and then determine the average over time. The sum of the four component terms is straightforward to compute by recognizing that it is the sum of the amplitude patterns from the pairs of waves with the same wavelength. We have already shown that the sum of the two terms with = 1 is: 2z 2x f1 [x; z; 1] + f2 [x; z; 1] = 2A cos cos [] !1t cos sin [] 1 1 a traveling wave in the +z-direction and a stationary sinusoid along the x-direction. Now add a second pair of plane waves with same “tilts” but the light has di¤erent wavelength 2 = 1. The interference pattern from this pair of waves alone is easy to evaluate: 6

2z 2x f3 [x; z; 2] + f4 [x; z; 2] = 2A cos cos [] !2t cos sin [] 2 2 so the period of the interference pattern di¤ers only by the wavelength. If both wavelengths are used at the same time, the amplitudes of the four waves may still be added in pairs:

4 fn [x; z; ] = (f1 [x; z; 1] + f2 [x; z; 1]) + (f3 [x; z; 2] + f4 [x; z; 2]) n=1 X x x = 2A cos 2 cos (kavg )1 z !1t + 2A cos 2 cos (kavg )2 z !2t 2 1 3 2 2 3 sin[] sin[] 4 5 4 5 where: 2 k (k ) = avg 1 avg 1 1 cos [] 2 k (k ) = avg 2 avg 2 2 cos []

The velocities of the average waves are:

!1 21 11 c (vavg ) = = = cos [] = cos [] 1 2 z (kavg )1 z z 1 cos[]

!2 22 22 c v2 = = 2z = cos [] = cos [] = v1 (kavg ) z z 2 2 cos[] = v2 = v1 ) which means that both “average”waves travel down the z-axis with the same velocity, as they should 108 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE because there is no dispersion. The …eld at the observation plane is now the sum of two terms:

4 fn [x; z; ] n=1 X x x = 2A cos 2 cos (kavg )1 (z v1t) + 2A cos 2 cos (kavg )2 (z v1t) 2 1 3 2 2 3 sin[] sin[] 4 5 4 5 The irradiance is the time average of the squared magnitude of the summation, which has three terms:

4 2 fn [x; z; ]

n=1 X 2

2 x x = 2A cos 2 cos (kavg )1 z !1t + cos 2 cos (kavg )2 z !2t j j 2 1 3 2 2 3

sin[] sin[] 4 5 4 5 2 2 x 2 2 x 2 = 4A cos 2 cos (kavg )1 z !1t + cos 2 cos (kavg )2 z !2t 0 2 1 3 2 2 3 1 sin[] sin[] @ 4 5 4 5 A 2 x x + 4A 2 cos 2 cos 2 cos (kavg )1 z !1t cos (kavg )2 z !2t 0 2 1 3 2 2 3 1 sin[] sin[] @ 4 5 4 5 A The time averages of each of the …rst two terms (squares of the traveling wave cosines) are identically 1 2 , but the time average of the product of the two cosine traveling waves may be written as the summation of two bipolar cosines:

cos (kavg ) z !1t cos (kavg ) z !2t 1 2 1 1 = cos (kavg ) + (kavg ) z (!1 + !2) t + cos (kavg ) (kavg ) z (!1 !2) t 2 1 2 2 1 2

1 The time average evaluates to 0 if the time period T >> mod , so the irradiance of the summation is:

4 2 2 2 x 1 2 x 1 fn [x; z; t; ] = 4A cos 2 + cos 2 + 0 * + 0 2 1 3 2 2 2 3 2 1 n=1 sin[] sin[] X @ 4 5 4 5 A

x x = I [x; z] = 2A2 0cos2 22 3 + cos2 22 31 1 2 B 6 7 6 7C B 6 sin [] 7 6 sin [] 7C @ 4 5 4 5A This is a very simple and signi…cant result; it shows that the irradiance from a two-aperture system from two wavelengths is the sum of irradiances from the individual wavelengths. 5.3 FRINGE VISIBILITY –TEMPORAL COHERENCE 109

The sum of four tilted plane waves can be calculated by summing the pair due to one wavelength and that due to the other.

This expression as the sum of two squared cosines may be rewritten yet again by using the identity: 1 1 cos2 [] = + cos [2] 2 2 to obtain:

1 1 x 1 1 x I [x; z] = 2A2 + cos 2 + + cos 2 02 2 2 1 3 2 2 2 2 31 2 sin[] 2 sin[] @ 4 5 4 5A 1 x x = 2A2 1 + cos 2 + cos 2 0 2 0 2 1 3 2 2 311 2 sin[] 2 sin[] @ @ 4 5 4 5AA The sum of the two stationary cosine waves also may be recast yet one more time as the product of cosines with the average and modulation frequencies:

1 + 2 1 2 cos [21x] + cos [22x] = 2 cos 2 x cos 2 x 2 2 where: 2 sin [] 1 = 1 2 sin [] 2 = 2 1 2 1 1 2 1 = = sin [] = sin [] ) 2 1 2 1 2 110 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

Substitute these expressions into the summation:

x x x x cos 2 + cos 2 = cos 2 + cos 2 2 1 3 2 2 3 2 c 3 2 c 3 2 sin[] 2 sin[] 21 sin[] 22 sin[] 4 2 sin5 [] 4 52 sin [4] 5 4 5 = cos 2 1 x + cos 2 2 x c c 1 + 2 sin [] 1 2 sin [] = 2 cos 2 2 x cos 2 2 x 2 c 2 c sin [] sin [] = 2 cos 2 2avg x cos 2 2mod x c c x x = 2 cos 2 cos 2 x 2 c 3 2 c 3 2a v g sin[] 2m o d sin[] 4 5 4 5

1+2 1 2 where avg 2 and mod 2 . These may be expressed in terms of the average and modulation wavelengths :

+ c + c c 1 1 = 1 2 = 1 2 = + avg 2 2 2 1 2 c + c + c = 1 2 = 1 2 = 2 2 avg 1 2 1 2 1 2

c c 1 2 1 2 c 1 1 mod = = = 2 2 2 1 2 c 2 1 c 2 1 c = = = 2 2 mod 1 2 1 2 1 2

So the expression for the irradiance becomes:

1 x x I [x; z] = 2A2 1 + 2 cos 2 cos 2 0 2 0 2 c 3 2 c 311 2a v g sin[] 2m o d sin[] @ @ 4 5 4 5AA

x x I [x; z] = 2A2 1 + cos 2 cos 2 0 2 c 3 2 c 31 2a v g sin[] 2m o d sin[] @ 42x 2x5 4 5A 2A2 1 + cos cos D D avg mod where the periods are:

c 12 1 Davg = (avg ) 2avg sin [] 2 sin [] avg / c 12 1 Dmod = () 2mod sin [] 2 sin [] mod / If the two wavelengths are close together, so that 1 = 2 = avg >> mod, then the expressions for 5.3 FRINGE VISIBILITY –TEMPORAL COHERENCE 111 the periods of the two component oscillations may be simpli…ed:

avg Dmod = Davg >> Davg

In words, the period of the modulation due to mod is much longer than that due to avg if the emitted wavelengths are approximately equal. Where the amplitude with the longer period Dmod decays to zero, the short-period fringes are no longer visible. In fact, the sinusoidal fringes due to avg are visible over a range of x equal to range of x between the zeros of Dmod , i.e., half the period of Dmod . The pattern resulting from the example considered above is shown. Note that the amplitude of maxima of the irradiance fringes decreases away from the center of the observation screen, where the optical path lengths are equal for all four beams, and thus where they will add constructively.

Interference patterns from two wavelengths with same input amplitude: (a) the two amplitude patterns di¤er in period in proportion to the wavelength; (b) the sum of the two amplitude patterns at one instant of time; (c) the squared magnitude at the same instant, showing that the amplitude of the fringe varies with x.

In this case where << avg , the fringes are visible over a large interval of x. We speak of such a light source as temporally coherent; the phase di¤erence of light emitted at 1 and at 2 changes slowly with time, and thus with the position along the x-axis. Therefore, fringes are visible over a large range of x. On the other hand, if is of the same order as avg , the wavelengths are widely separated. The phase di¤erence of light emitted at the two extreme wavelengths changes rapidly with time, and thus with position along the x-axis. The fringes are visible only if the phase di¤erence remains approximately constant (for x = 0) over the averaged time interval. Such a source is said to be temporally incoherent. It is di¢ cult (though not impossible) to see fringes generated by a temporally incoherent source.

5.3.1 Coherence of Light The coherence of an electromagnetic …eld is a description of the correlation properties (i.e., the statistics) of the optical phase, which determines the interference properties of the …eld, i.e., how radiation from di¤erent points on the object or that has travelled di¤erent optical paths will interact during the imaging process. If the radiation is coherent, then it will interfere constructively and destructively when it produces the image. Incoherent radiation cannot interfere, and thus the image has signi…cantly di¤erent properties. Imaging systems that act in coherent light are linear in the amplitude of the …eld, while systems acting in incoherent light are linear in irradiance (time average of the squared magnitude of the amplitude). The measure of the coherence of light is based on the statistics of the phase di¤erence measured at two points that di¤er in location and/or time:

[R ; t1; R ; t2] [R ; t1] [R ; t2] 1 2 1 2 112 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE which may be written in the equivalent form:

[R ; t1; R; t] [R ; t1] [R + R; t1 + t] 1 1 1 where

R2 = R1 + R

t2 = t2 + t

A …eld that is spatially (or laterally) coherent exhibits a …xed phase di¤erence measured at any pair of points in space separated by the same vector R and at the same time (t = 0). A radiation …eld is temporally (or longitudinally) coherent if phase di¤erences evaluated at any point in space at two di¤erent times separated by t (i.e., such that R = 0) are invariant. In other words, two beams are combined after one has been delayed in time to determine whether the phase di¤erence is preserved; this leads to the concept of coherence length. Spatial coherence is tested by a division- of-wavefront interferometer that can “shear”a part of the wavefront to one side to superpose it on another part. Light from two di¤erent points on a wavefront can interfere if the source is “small.” Temporal coherence is tested using a division-of-amplitude interferometer that can delay one part of the wavefront relative to another; it is a measure of the frequency bandwidth of a point source; a source with a narrow bandwidth is temporally coherent over a longer distance. The phase di¤erence of a …eld of light that is both spatially and temporally coherent is identical for all pairs of points separated by the same increments [R; t], regardless of the initial location [R1; t1], which means that the light comes from a point source of purely monochromatic light. Pure coherence is an idealized concept, because neither do ideal point sources nor truly monochromatic waves exist. A true monochromatic wave must have in…nite length. If the two superposed plane waves have the same wavelength, then !1 = !2 = !avg and !mod = 0, which means that the modulation wave remains …xed in space. The traveling wave is averaged out in the irradiance, leaving only the …xed modulation wave.

T0 + 2 1 2 2 I [x; y; z; t] = E [x; y; z; t] dt = 4 cos [kmod r] T T0 j j Z 2 This stationary pattern of irradiance demonstrates the interference between the two plane waves. If the two waves have di¤erent wavelengths, then the modulation pattern also moves and “smears out” the interference pattern. Maxima occur at values of the position vector r such that kmod r = n, where n is any integer. The “strength” of each maximum is 4 units, which is twice as large as the irradiance of the two waves added independently. Minima with “strength”0 occur halfway between the maxima. This phenomenon of “ampli…cation” and “cancellation” in coherent light does not occur for incoherent light, and is the reason why the transfer functions di¤er for the two cases. It is straightforward to superpose wavefronts from other sources, such as cylinders or spheres, to generate modulation waves that have di¤erent “shapes.” Strictly speaking, no radiation …eld is perfectly coherent or incoherent, but the psf and OTF of optical imaging systems are easily computed in these two limiting cases and we will assume that one of them is applicable.

5.3.2 Coherence Time and Coherence Length If two wavelengths emitted by the source are separated by , the corresponding frequency di¤erence often is called the bandwidth of the source:

c c 2 1 1 2 = = c j j = c j j 1 2 1 2 1 2

Now consider that the spectrum of the source includes a third wavelength 3 with the same “strength” located midway between the extrema 1 and 2 (i.e., 3 = avg ), the factors and 5.3 FRINGE VISIBILITY –TEMPORAL COHERENCE 113

avg are unchanged, but the irradiance pattern must be di¤erent in some way. This pattern is more di¢ cult to calculate directly as before, but the result may be modeled easily by recognizing that the wavefronts generated by 3 through the two apertures combine in amplitude to create a third pattern of sinusoidal fringes with a spatial period between those due to 1 and 2. The three such patterns may be summed and squared to model the irradiance fringes. Consider …rst the individual fringe patterns due to the extrema 1 and 2 as shown in (a):

(a) Irradiance pattern resulting from two wavelengths with equal “powers,” showing the long-period fringes due to and the short-period fringes due to avg; (b) Irradiance pattern after adding a third wavelength at avg with the same “power” –the distance between peaks of the fringe pattern has increased; (c) pattern obtained from light over the full bandwidth between maximum and minimum –amplitudes of neighboring maxima are smaller.

The irradiance pattern generated from the superposition of these fringe patterns exhibits the same “short-period”fringes due to avg and the same long-period fringes due to , but the amplitudes of the adjacent maxima of the long period fringes varies; the distance between the “maxima of the maxima”is longer. Now generalize to the case where the spectrum contains equal amplitudes of all wavelengths between min and max; the distance between the maxima gets larger yet so that the most visible fringes are present only in the region centered about x = 0: Because the region where the short-period fringes created by the three-line source are visible is similar in size to that from the two-line source, but (in…nitely) much smaller than the region of interference from the single-line source, we say that light from the two wavelengths max and min and light from max, min, and avg are equally coherent, the light from one wavelength is completely coherent. In short, coherence is quanti…ed from the temporal bandwidth, which may be evaluated from the wavelength passband: c c = max min = min max max min = c = c max min max min 1 where max min. Note that the dimensions of are (sec) . The time delay over which the phase di¤erence of light emitted from one source point is deterministic (and thus over which 1 fringes may be generated) is the reciprocal () :

1 max min = = ; c which is called the coherence time. Obviously, if is large, then so is and the corresponding coherence time is small. The coherence length ` is the distance traveled by an electromagnetic 114 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE wave during the coherence time:

c max min c ` = = This is a measure of the length of the electromagnetic wave “packet”over which the phase di¤erence is predictable. Recall that for interference of waves from a source with wavelength range , the range of coordinate x over which irradiance fringes are visible is half of the modulation period:

1 Dmod 1 1 1 1 max min = 2 2 sin [] 1 2 2 sin [] 1 1 1 = = ` 2 sin [] c 2 sin [] In words, the domain of x over which fringes are visible is proportional to the coherence length; if the bandwidth is “wide,” then the coherence length ` is “short” and the fringes are visible over only a small region. Lasers supply the most coherent light available because, to a very good approximation, many lasers emit a single wavelength 0, so that = 0 = ` = = Dmod = and fringes are visible at all x. A coherent source should be employed) when an1 optical) interference1 pattern is to be used to measure a parameter of the system, such as the optical image quality (as was used to test the Hubble space telescope). Thus the range of x over which fringes are visible is determined by the coherence length (and thus the bandwidth) of the source. Therefore, the visibility of the interference fringes may be used as a measure the source coherence.

“White-Light” Fringes The range of visible wavelengths of light is generally taken to be:

= red blue = 700 nm 400 nm = 300 nm The corresponding frequency bandwidth is:

1 1 m 1 1 = c = 3 108 = 3:21 1014 Hz s 400 nm 700 nm blue red The width of the modulation region is:

12 Dmod = 2 sin [] mod 700 nm 400 nm = sin [] 300 nm 1 m =

If the angle 2 between the interfering beams is small, then Dmod may be large. For example, we can expect that Dmod would have to be of the order of millimeters to be visible, say Dmod = 5 mm, which implies that the angle would have to be: 1 2 = radians = 80 arc seconds = 1:3 arc minutes 2500

5.3.3 Polarization and Fringe Visibility Up to this point, we have ignored the e¤ect on the orientation of the electric …eld vectors on the sum of the …elds. In fact, this is an essential consideration; two orthogonal electric …eld vectors cannot add to generate a time-invariant modulation in the irradiance. Consider the sum of two electric …eld 5.4 VAN CITTERT - ZERNIKE THEOREM 115

vectors E1 and E2 to generate a …eld E. The resulting irradiance is:

I = E E = E 2 = E + E 2 h i j j j 1 2j = (E + E ) (E + E ) 1 2 1 2 = (E E ) + (E E ) + (E E ) + (E E ) 1 1 2 2 1 2 2 1 = (E E ) + (E E ) + 2(E E ) 1 1 2 2 1 2 = I1 + I2 + 2(E E ) 1 2 where I1 and I2 are the irradiances due to E1 and E2, respectively. Consider the irradiance in the case where the incident …elds are plane waves with polarizations ^e1 and ^e2, respectively:

E = ^e E1 cos [k r !1t] 1 1 1 E = ^e E2 cos [k r !2t] 2 2 2 I = I1 + I2 + 2 ^e E1 cos [k r !1t] ^e E2 cos [k r !2t] h 1 1 2 2 i = I1 + I2 + 2E1E2(^e ^e ) cos [k r !1t] cos [k r !2t] 1 2 h 1 2 i = I1 + I2 + 2E1E2(^e ^e ) cos [(k k ) r (!1 !2) t] 1 2 h 1 2 i = I1 + I2 + 2E1E2(^e ^e ) cos [2k r 2!modt] 1 2 h mod i

In the case where the two components are polarized orthogonally so that ^e1 ^e2 = 0, then the irradiance is the sum of the component irradiances and no interference is seen.

In the case where !1 = !2 so that !mod = 0, and I1 = I2, then the output irradiance is:

I = 2I1 (1 + cos [2k r]) mod which again says that the irradiance includes a stationary sinusoidal fringe pattern with spatial period = 2 . mod k j modj

5.4 Van Cittert - Zernike Theorem

The size of the region over which interference fringes are visible is related to the “degree”of coherence of the sources through the van Cittert - Zernike theorem, which says that the (normalized) Fourier transform of the irradiance distribution across the source is the degree of coherence, also called the spatial coherence function.

2 I [x; y] = E [r1] E [r2] [r1; r2] F f g h i

5.5 Division-of-Amplitude Interferometers

We have seen several times now that optical interference results when two (or more) waves are su- perposed in such a way to produce a time-stationary spatial modulation of the superposed electric …eld, which may be observed by eye or a photosensitive detector. Interferometers use this result to measure di¤erent parameters of the light (e.g., wavelength , bandwidth or coherence length `, angle and sphericity of the wavefront, etc.), or of the system (path length, traveling distance, index of refraction, etc.). We have seen that the interference pattern in a division-of-wavefront interferometer is generated from the kmod portion of the sum of the wavefronts. Division-of-amplitude interferom- eters use a partially re‡ecting mirror –a beamsplitter –to divide the wavefront into two beams that then travel down di¤erent paths and are recombined by the original or another beamsplitter. The optical interference is generated by the phase di¤erence between the recombined wavefronts. The beamsplitter re‡ects part of the wave and transmits the rest. The amplitude re‡ection 116 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE coe¢ cient for normal incidence is:

n1 n2 r = at normal incidence n1 + n2 which shows that the amplitude is multiplied by a negative number if n1 < n2, meaning that the phase is changed by radians if re‡ected at a rare-to-dense interface (second surface has larger n). The re‡ection at a dense-to-rare interface exhibits no phase shift.Irradiance depends on the relative phases of the light due to the re‡ections, which depend on type of re‡ection. A re‡ective dielectric coating has an electrically “resistive”surface and a “rare-to-dense”(“external”) re‡ection induces a phase change = radians.

Optical schematic of the Michelson interferometer where the incident wavefront is planar.

5.5.1 Fizeau Interferometer The Fizeau interferometer uses a single beamsplitter and may be used to measure the di¤erence in shape between a test optical surface and a reference surface. In the drawing, the physical length di¤erence between the path re‡ected from of the bottom of the test optic and from the top of the reference surface is d.

Part of the incident beam is re‡ected from the glass-air interface of the test object. This dense-to- rare re‡ection has no phase shift. The re‡ection from the glass reference surface is rare-to-dense, and the phase of the light is changed by radians. The two waves are recombined when they emerge from the top of the test surface and the irradiance exhibits fringes from the optical path di¤erence. Because the beams traverse the same path in each direction, the optical path di¤erence is doubled, 5.5 DIVISION-OF-AMPLITUDE INTERFEROMETERS 117

so an increment in the physical path of 2 changes the optical path by and one fringe cycle is seen. The fringe that appears where the two surfaces are in contact corresponds to a phase di¤erence of radians and thus is dark. If the test optic is spherical, then the physical path di¤erence d may be expressed in terms of the radius of curvature R and the radial distance r:

Pythagoras says that:

R2 = (R d)2 + r2 = r2 = 2Rd d2 = 2Rd for d << R ) r2 d = for d << R (Sag Formula) 2R Sag square of radial distance / If the interstice between the optical elements in …lled with air, and if m fringes are counted between two points at radial distances r1 and r2, then the corresponding thickness change is: m = OPD = nd d = m in air 2 ! 2 This is the same paraboloidal approximation for a sphere that is used in Fresnel di¤raction (quadratic dependence on radial distance). If the interstice between optical elements …lled with air and m fringes counted between two points at radial distances r1 and r2. The corresponding thickness change is: m = OPD = nd d = m in air 2 ! 2 The test of a spherical surface against an optical “‡at”produces “Newton’sRings”

Interference patterns observed using Fizeau interferometer of spherical lens compared to planar test surface: (a) large radius of curvature R; (b) small R. 118 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

5.5.2 Michelson Interferometer

A Michelson interferometer uses mirrors to change the optical path di¤erence of the two beams. The two wavefronts are directed down di¤erent paths before recombining to create interference. For example, consider the Michelson interferometer:

Michelson interferometer: beam in input arm is partially re‡ected and transmitted by beamsplitter. Beams in two arms travel di¤erent distances before being recombined. The compensator is a piece of glass with plane-parallel sides that ensures that the path lengths in glass in the two arms are equal.

In the example shown, the two beams out of one output port have experienced one external and one internal re‡ection at beamsplitter, so the inherent phase di¤erence is radians. The beams returning out the “input port”have experienced 0 re‡ections and 2 external re‡ections from the beamsplitter, which means that the inherent phase shift = 2 = 0 radians, so the phase di¤erence is due only to the optical path di¤erence. 5.5 DIVISION-OF-AMPLITUDE INTERFEROMETERS 119

Re‡ections from beamsplitter in Michelson interferometer: the two components of the normal “output” port have experienced 1 external re‡ection and 1 internal re‡ection at the beamsplitter, while light that re‡ects back out the “input” port have experienced 0 and 2 external re‡ections. The internal re‡ection (“dense-to-rare”) adds a phase shift of radians.

If the beamsplitter both re‡ects and transmits 50% of the irradiance (NOT 50% of the amplitude), then equal portions of the energy are directed toward the mirrors M1 and M2. The amplitude of the electric …eld of the re‡ected beam is:

2 I0 E0 E1 = I1 = = r 2 r 2 p 1 = E0 = 0:707 E0 p2 Because each beam is re‡ected once and transmitted once before being recombined, the amplitude of each component when recombined is:

1 1 1 (E1)out = (E2)out = E0 = E0 r2 r2 2 Each beam experiences a phase delay proportional to the optical distance traveled in its arm of the interferometer: 2 2 1 = d1 = n1d1 = k n1d1 1 1 where d1 is the distance traveled by beam #1 and n1 is the refractive index in that path (n = 1 in vacuum or air). The beam directed at mirror M1 travels distance L1 from the beamsplitter to the mirror and again on the return, so the total physical path length d1 = 2L1. Similarly the physical length of the second path is d2 = 2L2 and the optical path is n2d2 = 2n2L2. After recombination, 120 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE the relative phase delay is:

= 1 2 = k (n1d1 n2d2) = 2k (n1L1 n2L2) 4 = (n1L1 n2L2) 1

Note that the phase delay is proportional to 1 , i.e., longer wavelengths (red light) experience 1 smaller phase di¤erences than shorter wavelengths (blue light) because the ratio of the physical path di¤erence to the wavelength is smaller. The amplitude at the detector is the sum of the amplitudes:

E0 2 E0 2 E [t] = cos (n1 2L1) !1t + cos (n2 2L2) !1t p2 0 p2 E0 2 2 = cos n1 2L1 !1t + cos n2 2L2 !1t p2 0 0 which has the form: A + B A B cos [A] + cos [B] = 2 cos cos 2 2 So the amplitude may be rewritten as the productd of the average and modulation frequencies:

E0 2 (2n1L1 + 2n2L2) 2 (2n1L1 2n2L2) E [t] = 2cos !1t cos + 0 t p2 20 20 2 (n1L1 + n2L2) 2 (n1L1 n2L2) = p2E0 cos 21t cos 0 0 Note that the …rst term is rapidly oscillating function of time and the second term is the stationary fringe pattern. The time average of squared magnitude is the irradiance:

2 2 2 (n1L1 + n2L2) 2 2 (n1L1 n2L2) I = E(t) = 2E0 cos 2 1t cos j j 0 0 D E 2 1 1 (n1L1 + n2L2) 1 1 4 (n1L1 n2L2) = 2E + cos 4 1t + cos 0 2 2 2 2 0 0 1 1 1 4 (n1L1 n2L2) = 2E2 + cos 0 2 2 2 0 2 2E0 2 = 1 + cos 2 (n1L1 n2L2) 4 0 2 = I0 1 + cos 2 (n1L1 n2L2) 0 2 I [L1;L2; 0; n1; n2] = I0 1 + cos 2 (n1L1 n2L2) 0 If the two arms are in air, then n1 = n2 = 1 and the irradiance pattern is: 2 I [L1;L2; 0] = I0 1 + cos 2 (L1 L2) if n1 = n2 = 1 0 The Michelson interferometer with monochromatic plane-wave inputs and “parallel” images of mirrors generates uniform irradiance that depends on ONLY of lengths of arms and on 0; not on time or position, so the irradiance is uniform; the Michelson interferometer with monochromatic plane-wave inputs generates a uniform irradiance related to the path di¤erence. If input wavefronts with other shapes are used, then fringes are generated whose period is a function of the local optical 5.5 DIVISION-OF-AMPLITUDE INTERFEROMETERS 121 path di¤erence, which is directly related to the shape of the wavefronts.

Example: Tilted Mirror If the incident light is a tilted plane wave, then the resulting pattern is analogous to that from a division-of-wavefront interferometer (Young’sdouble-slit). I

Example: Spherical Waves If a point source emitting spherical waves is used, then the interferometer may be modeled as two point sources at di¤erent distances from the observation plane:

Optical schematic of the Michelson interferometer where the incident wavefront from the source S is spherical. The interferometer may be “unfolded” and modeled as two point sources emitting the same wavefronts but at di¤erent distances from the observation plane.

Images of the point source are formed at S1 (due to mirror M1) and at S2 (due to M2). The wavefronts superpose and form interference fringes. The positions of the fringes may be determined from the optical path di¤erence (OPD).

“Unwrapped” schematic of the Michelson interferometer where the wavefronts are spherical. The optical path di¤erence is 2 (L1 L2) cos []. The OPD is the excess distance that one wavefront has to travel relative to the other before being recombined. For a ray oriented at angle measured relative to the axis of symmetry, the ray re‡ected 122 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

from mirror M2 travels an extra distance:

OPD = 2 (L1 L2) cos [] The symmetry about the central axis ensures that the fringes are circular. The phase di¤erence of the waves is the extra number of radians of phase that the wave must travel in that OPD: 2 = k OPD = 2 [L1 L2] cos [] 1 4 (L1 L2) cos [] = [radians] 1

If the phase di¤erence is a multiple of 2 radians, then the waves recombine in phase and a maximum of the amplitude results. If the phase di¤erence is an odd multiple of radians, then the waves recombine out of phase and a minimum of the irradiance results due to destructive interference. The locations of constructive interference (irradiance maxima) may be speci…ed by:

4 (L1 L2) cos [] = 2m = m1 = 2 (L1 L2) cos [] 1 ! The corresponding angles are speci…ed by:

1 m1 = cos 2 (L1 L2)

m As the physical path di¤erence L1 L2 decreases, then increases and decreases. In 2(L1 L2) other words, if the physical path di¤erence is decreased, the angular size of a circular fringe decreases and the fringes disappear into the center of the pattern. Since a particular fringe occurs at the same angle relative to the optical axis, these are called fringes of equal inclination.

Spherical waves from point source recombined after traveling through arms with di¤erent lengths. The curvatures of the two wavefronts are di¤erent, which means that the local optical phase di¤erence varies with location in the observation plane. 5.5 DIVISION-OF-AMPLITUDE INTERFEROMETERS 123

The OPD is approximately equal to di¤erence in sags between the two spherical wavefronts:

r2 d = for d << R (Sag Formula) 2R r2 d1 = 2R1 r2 d2 = 2R2 2 2 1 1 r OPD [m] = 2 (d1 d2) = r = (R2 R1) R R R R 1 2 1 2 2 OPD [m] = r R [m] R1R2

The OPD may be rewritten as a pure number of wavelengths by dividing by 0

r2 R OPD [number of wavelengths] = R1R2 0 The resulting optical phase di¤erence is obtained by multiplying by 2 radians per wavelength, which is a quadratic function of the radial distance r from the axis of symmetry:

2 r2 R r2 r r 2 = 2 = R R = R R 0 1 2 R R 1 2 0 2R 0 0 1 2 1 2R @ A where is the radius of the circle measured from the originq where the phase di¤erence is radians. The optical phase di¤erence is substituted into the irradiance formula

r 2 R R I [r] 1 + cos , where = 0 1 2 [m] , chirp function / 2R r

0R1R2 The irradiance changes from a bright fringe where r = 0 to a dark fringe where r = = 2R . Bright fringes are located at values of r where the phase di¤erence is an integer multipleq of 2 radians, i.e., for r = p2 , r = p4 = 2a, r = p6 , etc. If the projected separation between the two mirrors in the Michelson is 0, there is no phase di¤erence and so no fringesincreased, R increases and decreases. In words, increasing the mirror separation decreases the radial distance of the phase change of radians; the fringe spacing decreases. 124 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

x 2 The observed pattern is a “chirp” function cos , which means that the fring spacing at decreases the observation plane with increasing radialh distance i from center. If one of the wavefronts is not spherical, then the fringes are not circular. In fact, the shape of fringes is determined by the shape of wavefront. If one mirror is tilted relative to the other, then the output beams travel in di¤erent directions when recombined. The fringes thus obtained are straight and have a constant spacing just like those from a Young’sdouble-slit experiment.

Applications of Michelson Interferometers 1. Measure refractive index n: Insert a plane-parallel plate of known thickness t and unknown index n into one arm of a Michelson interferometer illuminated with light of wavelength . Count the number of fringes due to the plate:

OPD = (n 1) t = m 1 2. Measure the wavelength of an unknown source or between two spectral lines of a single source. Count the number of fringes that pass a single point as one mirror is moved a known distance.

3. Measure lengths of objects: The standard m now is de…ned in terms of a particular spectral line:

1 m = 1; 650; 763:73 wavelengths of emission at = 605:8 nm from Kr-86 measured in vacuum 5.5 DIVISION-OF-AMPLITUDE INTERFEROMETERS 125

This standard has the advantage of being transportable.

4. Measure de‡ections of objects: The optical phase di¤erence will be signi…cant for very small physical path di¤erences. Place one mirror on an object and count fringes to measure the de‡ection.

5. Measure the velocity of light (Michelson-Morley experiment)

Twyman-Green Interferometer This is a version of the Michelson interferometer with a point source and a collimating lens that ensures that the incoming light is composed of plane waves

Fourier-Transform Spectrometry (FTS) Reference: Transformations in Optics, Lawrence Mertz, John Wiley & Sons, 1965. The Michelson interferometer is the basis for an elegant (though indirect) technique for measuring the wavelength spectrum of a source. Its primary advantage is that it does not “waste”(i.e., discard) light, but the data collection takes time and the spectrum must be reconstructed indirectly. One traditional method for measuring a spectrum is to interpose a set of bandpass …lters between the object and the measurement system, so that the output is a set of images of the source in di¤erent passbands. The spectrum of the object is constructed from the di¤erent images. Consider a Michelson interferometer that is con…gured so that the two arms are exactly equal in length so that the OPD = 0. In theory, the waves emitted by the source are split and reunited exactly in phase and appear as though the interferometer is not present. Now the path length of one arm of the interferometer is changed slowly at a uniform rate, which means that the waves from the source are combined with a slowly increasing OPD. Consider the case if the source emits only a single wavelength (or, equivalently, a single optical frequency). At the time when OPD = 0, the two waves reunite in phase. As the OPD is increased, the phase di¤erence between the two waves changes from = 0 to = to = 2, and the irradiance of the combined signals changes from I = Imax at = 0 to I = 0 at = and back to I = Imax at = 2. The last measurement is again a maximum because the source emits only a single wavelength. If the source emits two wavelengths, the interfering beams at both frequencies are in phase if OPD = 0, so that the resulting irradiance I [OPD = 0] = Imax. If OPD is increased from 0, the phase of the shorter wavelength will change more quickly with distance so that the phase di¤erence at this wavelength will be shorter = when longer < . The two beams at the shorter wavelength cancel exactly, but those at the longer wavelength do not quite do so. As the OPD increases further, eventually shorter = 2 when longer < 2, so the resulting irradiance I < Imax. Now consider the case if the source is completely incoherent, which means that there is the optical phase is perfectly correlated if the beams are reunited after traveling exactly the same path, but the phase is completely uncorrelated for anly OPD = 0. In other words, the correlation of the optical phase of completely incoherent light is a Dirac delta6 function. In the Michelson, the output irradiance is a maximum value if OPD = 0 and is If the source emits two wavelengths

5.5.3 Mach-Zehnder Interferometer The M-Z interferometer is very similar to the Michelson, except that a second beamsplitter is used to recombine the beams so that the light does not traverse the same path twice. Therefore there is no factor of 2 in the OPD for the M-Z. Mach-Zehnder interferometers often are used to measure the refractive index of liquids or gases. The container C1 (or C2) is …lled with a gas while examining 126 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE the fringe pattern. The length of the gas path in the cell is . As the container …lls, the refractive index n increases and so does the optical path length in that arm. The optical path di¤erence is:

OPD = (n 1)

The fringes move and each new cycle of the pattern corresponds to an increase in the OPD of . After m fringes are counted, the index of refraction is found via:

2 m1 (n1 1) = m 2 n1 = 1 + 1 !

Optical schematic of the Mach-Zehnder interferometer, where the amplitudes divided by the beamsplitter travel two di¤erent paths and are recombined by a second beamsplitter.

5.5.4 Sagnac Interferometer

The Sagnac interferometer is a single-beamsplitter version of the M-Z; the output beamsplitter is exchanged for a mirror which is reversed to create a loop path. Light travels around the loop in both directions so that the optical path di¤erence is zero for a stable con…guration. However, if the interferometer (including the illuminator) is rotated as on a turntable, then light in one path will experience a Doppler shift with increasing frequency (blue-shift), while light in the reverse direction will experience a red shift. The phase of the two beams will change in proportion to the frequency shift, and the superposed light will exhibit a sinusoidal variation in the detected signal over time:

cos [!1t] + cos [!2t] = 2 cos [!avg t] cos [!modt]

The slower-varying modulation frequency is detectable and linearly proportional to the rotation rate. This device may be used as a gyroscope with no moving parts, and in fact may be constructed from a single optical …ber that forms a loop with counterrotating beams. 5.6 INTERFERENCE BY MULTIPLE REFLECTIONS 127

Optical schematic of the Sagnac interferometer, where the two beams travel the same path in opposite directions.

5.6 Interference by Multiple Re‡ections

Division-of-amplitude interference also may occur within a medium by multiple re‡ections between two surfaces. This is the basis for thin-…lm interference and the Fabry-Perot interferometer. Thin- …lm interference is the basis for optical coating design, e.g., for “anti-re‡ection” (“AR”) coatings applied to lenses and for coatings that exhibit near total re‡ection. The coating is designed so that the multiple re‡ections within the coating either “cancel out”by destructive interference (in an AR coat) or contructively interfere in phase (in a mirror coating). Fabry-Perot interferometers are used to measure the spectrum of a source or to select a narrow band of wavelengths from a source, as to decrease the bandwidth and extend the coherence length of a laser used to make optical holograms.

5.6.1 Thin Films

Parameters in thin-…lm interference. The ray with amplitude a (not irradiance) is incident from air onto the medium with index n at angle . The Fresnel equations determine the percentage re‡ected and transmitted at each surface. The front-surface re‡ection exhibits a phase shift of radians, while there is no phase shift at the “dense-to-rare” interface. 128 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

Simple geometry determines the optical path di¤erence (OPD) between the …rst two re‡ected rays is: OPD = 2 nd cos [0] Since the …rst ray is re‡ected at a “rare-to-dense”interface, it includes an intrinsic phase di¤erence of radians that is not present in the second ray that re‡ected from the rear surface (“dense-to-rare”). The total optical phase di¤erence is: 2 4 = OPD + = nd cos [0] + 0 0 If = 2`, where ` is any integer, the two beams superpose out of phase (due to the front- surface re‡ection) to produce an interference maximum, which produces high-re‡ectance coating. The thickness d is selected for a re‡ectance coating tuned to a particular wavelength 0: 4 = 2` = nd cos [0] 0 2 1 = ` = nd cos [0] ) 0 2 1 = d = ` + 0 for high-re‡ectance coating ) 2 2n cos [ ] 0 The complementary condition for minimizing the re‡ection is: 4 = (2` + 1) = nd cos [0] + 0 ` = d = 0 for antire‡ectance (AR) coating ) 2n cos [0]

All re‡ections of “transmitted” rays are “dense to rare” with no intrinsic phase change. The OPD is the same as for the re‡ected rays, so the optical phase di¤erence is: 2 4 = OPD = (nd cos [0]) 0 0 (2` + 1) = d = 0 ) 4n cos [0] so the irradiance of a “transmitted” fringe is a maximum if = 2` and a minimum if = (2` + 1). In other words, the re‡ected and transmissive fringes are complementary; the “bright” and “dark”bands interchange.

Visibility of Thin-Film Fringes

The visibility of the fringes depends on the relative re‡ectance and transmittance coe¢ cients at the surface, which may be calculated is an elegant way put forth by Sir George Stokes. First consider part (a) of the …gure. The re‡ectance and transmission coe¢ cients are determined by the Fresnel equations for the particular polarization used. The amplitude of the electric …eld in the incident ray is a and of the re‡ected ray is ar < 0 because of the “rare-to-dense”re‡ection. The amplitude of the electric …eld of the transmitted ray is at > 0, where t is the transmission coe¢ cient at a “rare-to-dense” interface. Time has been reversed In part (b) so that there are two incident rays with amplitudes ar and at. The “output” ray in the time-reversed case that had amplitude a in 2 part (a) has amplitude ar + att0, where t0 is the transmission coe¢ cient in the “dense-to-rare” interaction. The incident wave with amplitude ar transmits to produce an amplitude art, while the incident amplitude at produces a re‡ection with amplitude atr0, where r0 is the “dense-to-rare” 5.6 INTERFERENCE BY MULTIPLE REFLECTIONS 129 re‡ection coe¢ cient. Since all four terms must be the same, this establishes two conditions:

2 2 a = ar + att0 = r + tt0 = 1 ) atr0 + art = 0 = r0 = r (which we knew already) )

Re‡ectance and transmission coe¢ cients at the surface of a thin …lm: (a) the incident ray has amplitude r, the amplitude of the re‡ected ray is ar < 0, and the transmitted ray is at; (b) the “time-reversed” situation, where the rays with amplitudes ar and at are incident on the surface.

We can apply these results to determine the amplitudes of multiply re‡ected rays. All rays in the following system arise from the single incident ray with …eld amplitude a, so all rays are mutually coherent. Only the …rst re‡ected ray includes a phase shift of radians, so the negative sign has been included explicitly. We already know that the optical phase di¤erence between adjacent pairs of rays is: 2 = 2nd cos [0] 0 130 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

Now consider the amplitude of the sum of all re‡ected rays:

2 4 ar = ar + atrt0 exp [+i ] + atrt0 r exp [+i 2 ] + atrt0 r exp [+i 3 ] + 2 4 = ar + artt0 exp [+i ] 1 + r exp [+i ] + r exp [+i 2 ] + 1 2 +i n = ar + artt0 exp [+i ] r e n=0 X Now use the identity: 1 1 tn = where t < 1 1 t n=0 j j X 5.6 INTERFERENCE BY MULTIPLE REFLECTIONS 131

+i 1 ar = ar + artt0e 1 r2e+i tt e+i = ar 1 0 1 r2e+i 2+i +i 1 r e tt0e = ar 1 r2e+i 2 +i 1 r + tt0 e = ar 1 r2e+i ! +i 1 e ar = ar 1 r2e+i The ratio of the irradiances is:

2 +i 2 Ir ar 2 1 e = j j2 = r 2 +i Iin a 1 r e j j + i +i 1 e 1 e = r2 1 r2e+i 1 r2e+i +i i 1 e 1 e = r2 1 r2e+i 1 r2e i 2 2 cos [] = r2 1 + r4 2r2 cos [] 2 (1 cos []) = r2 1 + r4 2r2 cos [] Now use the identities: 1 cos [] = 2 sin2 2 2 1 + r4 = 1 r2 + 2r2 4 sin2 Ir 2 2 = r 2 Iin 2(1 r2) + 2r2h 2ri2 cos []3 4 2 2 5 4r sin 2 = (1 r2)2 + 2r2 (1h icos []) 2 2 4r sin 2 = (1 r2)2 + 2r2 h2 sini2 2 2 2 h i 4r sin 2 = (1 r2)2 + 4rh2 sini2 2 2 2r h i (1 r2) sin 2 = 2 2r h i 1 + (1 r2) sin 2 h i 132 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

We also know that I I t = 1 r Iin Iin 2 2r (1 r2) sin 2 = 1 2 2r h i 1 + (1 r2) sin 2 1 h i = 2 2r 1 + (1 r2) sin 2 h i 5.6 INTERFERENCE BY MULTIPLE REFLECTIONS 133

Example: Small Re‡ectance Coe¢ cient r

Now consider the case of r2 0, e.g., r = 0:2 = r2 = 0:04 ' ) 2r 2 :4 2 = = 0:1736 << 1 1 r2 :96 2 I 0:1736 sin 2 = r = ) Iin h2 i 1 + 0:1736 sin 2 I 1 h i = t = ) Iin 2 1 + 0:1736 sin 2 h i

Ir (black) and It (red) for r = 0:2 Iin Iin 134 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE

Figure 5.2: Ir (black) and It (red) for r = 0:95 Iin Iin

Example: Large Re‡ectance Coe¢ cient r

If the surfaces have re‡ective coatings with r = 0:95 = r2 = 0:952 = 0:9 ) 2r 2 1:9 2 = = 380 >> 1 1 r2 1 0:952 2 I 380 sin 2 = r = ) Iin h2 i 1 + 380 sin 2 I 1 h i = t = ) Iin 2 1 + 380 sin 2 h i Again, the re‡ective and transmissive patterns are complementary. As r2 increases, the transmitted maxima become “sharper” while the re‡ected maxima become “broader.” This observation is the basis for the Fabry-Perot etalon, which is a division-of-amplitude instrument that acts by multiple re‡ections from two highly (though not totally) re‡ecting surfaces many times before recombining. 5.6 INTERFERENCE BY MULTIPLE REFLECTIONS 135

It for r:= 0:2 (black), r = 0:4 (red), r = 0:7 (brown), r = 0:84 (green), and r = 0:97 (blue), which Iin shows the “sharpening”of the transmission peaks with increasing r:

5.6.2 Fabry-Perot Interferometer

Fabry-Perot interferometer

The Fabry-Perot interferometer uses two highly-re‡ecting surfaces that amplify the internal re- ‡ections and “narrow” the fringes in transmitted light. The width of the transmissive peaks are characterized by the “…nesse,”which is de…ned:

period of It 2 Iin = F FWHM W 1 2 where “FWHM” is the “full width at half maximum.” As r increases, the widths of the peaks decrease and the …nesse increases. The FWHM is determined from the increment " of the optical 136 CHAPTER 5 OPTICAL INTERFERENCE AND COHERENCE phase di¤erence that satis…es the conditions:

I [ = 2`] t = 1 Iin I [ = 2 (` + ")] 1 t = Iin 2 2 = ) F 2" 2 r = = F 2" " 1 r2 If the source included a second wavelength, the spacing between the fringe maxima would be scaled and the fringes from the two wavelengths with the same order ` would be positioned at di¤erent spots. The two wavelengths can be distinguished if is large. For quasimonochromatic F light so that << 0, the maximum of order ` is spread over the

Optical Path Di¤erence:

OPD = 2nd cos [t] = m for constructive interference

2d cos [t] in air (n = 1) !

All rays that are incident at same angle t arrive at P

Add collimating lens on input side (distance from lens to interferometer = f 5.6 INTERFERENCE BY MULTIPLE REFLECTIONS 137

Finesse of Fabry-Perot interferometer

E¤ect of varying amplitude re‡ectance coe¢ cient r on fringes from a Fabry-Perot interferometer: (a) r ' 0 so that the maxima are “wide;” (b) r / 1 producing “narrow” maxima.

Resolving power of Fabry-Perot etalon: Two fringes from di¤erent wavelengths will overlap 2m signi…cantly if the FWHM W 1 is large. The resolving power = = m 2 W 1 R 2 F

Chapter 6

Optical Di¤raction and Imaging

In geometrical (ray) optics, light is assumed to propagate in straight lines from the source (rectilinear propagation), and cannot penetrate into the shadow of an opaque obstruction. However, Grimaldi observed in the 1600s that light is able to “bend”into the shadow region and named this phenomenon di¤raction (from Latin, dis + frangere, “to break apart”). This spreading of a bundle of rays a¤ects the sharpness of shadows cast by opaque objects by making edges become “fuzzy.”A second observation of di¤raction is the comparison of diameters D of the light beams through an aperture of diameter d. In the regime where geometrical explanations apply, the observed diameter D d, if 1 / d is made su¢ ciently small, then D d . /

Incoming light from a source at an in…nite distance away encounters an opaque obstruction. Some light penetrates into the geometrical shadow, indicating the existence of di¤raction.

E¤ect of di¤raction: light through an aperture in an opaque screen: (a) if the diameter d1 of the aperture is “large,” the diameter of the projection is D1 d1 (larger hole = larger “image”); / ) 1 (b) if the aperture diameter is decreased, eventually the diameter of the projection is D2 (d2) (smaller hole = larger “image”). / ) Di¤raction is really a generalization of interference that di¤ers only in the number of sources of the light that is added; in interference, light from a “few”sources (typically 2-5, or so) is recombined

139 140 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING by deliberate design to create the e¤ects, while light from many sources (up to an in…nite number) is combined in di¤raction, typically by natural propagation. Just as in interference, the observed e¤ects in di¤raction depend on the state of coherence of the light source. In both cases, light from a source is combined after traveling di¤erent paths (few paths for interference, many for di¤raction). The amplitude of the superposition depends on the relative phases of the component beams; the amplitude is increased if the beams are in phase (or nearly so) at some point in space so that their vector sum is large. If the phases produce a small vector sum, the power is small. In other words, the wave character of light creates stationary regions of constructive and destructive interference that may be observed as bright and dark regions. In the simplest case of two sources of in…nitesimal size, the superposition wave may be determined by summing spherical wave contributions from the sources. If the number of sources generating beams to be added, the e¤ect is considered to be interference. In di¤raction, the sources are large compared to the wavelength 0, and light with the same wavelength emerging from the large number of “subsources” at di¤erent locations on the source will interfere constructively or destructively depending on the relative phases of the light waves. The spherical-wave contributions from the large number of subsources are summed (by integrating over the area of the aperture) to determine the total electric …eld. The superposition electric …eld vector (magnitude and phase) is the vector sum of the …elds due to these spherical-wave subsources. Interference is often considered …rst in courses in optics because it is arguably easier to analyze and understand the interactions of small numbers of sources. However, this tactic often shortchanges the study of di¤raction, even though it is more applicable to imaging. You already have developed the necessary mathematical tools for the analysis of di¤raction in your study of Fourier methods, so we can transfer directly into this study; there is no need to pass “Go”beforehand. The mathematical model for di¤raction is straightforward to develop, though computations may be tedious or even unfeasible. The model for di¤raction is based on Huygen’sprinciple:

6.1 Huygens’Principle

In 1678, Christiaan Huygens theorized a model for light propagation that claimed that each point on a propagating wavefront (regardless of the “shape” of the wavefront) acted as the source of a new spherical wave. The sum of these secondary spherical “wavelets” produced the subsequent wavefronts. Huygens’principle had the glaring problem that these secondary spherical wavefronts propagated “backwards”as well as forwards, but this problem was addressed by Fresnel and Kirchho¤ in the 19th century by introducing an “obliquity”factor that attenuates the parts of the wavefront that propagate “o¤ axis.”With this correction, the Huygens’principle provides a very useful model for light propagation that naturally leads to expressions for “di¤racted”light. Consider The mathematical expression for a spherical wave emitted by a source located at the origin of coordinates; energy conservation requires that the energy density of the electric …eld of a spherical wave decrease as the square of the distance from the source. Correspondingly, the electric …eld decreases as the distance from the source. The electric …eld observed at location [x1; y1; z1] due to a spherical wave emitted from the origin is: E s [x ; y ; z ; t] = 0 cos [k x + k y + k z ! t] 1 1 1 2 2 2 x y z 0 x1 + y1 + z1

2 2 2 p where x1 + y1 + z1 = r is the distance from the source. This observation that light from a point source generates a spherical wave is the …rst step towards Huygen’sprinciple, which states that every p point on a wavefront may be modeled as a “secondary source”of spherical waves. The summation of the waves from the secondary sources (sometimes called “wavelets”) produces a new wavefront that is “farther downstream”in the optical path. In the more general case of a spherical wave emitted from a source located at coordinates 6.2 DIFFRACTION INTEGRALS 141

[x0; y0; z0] and observed at [x1; y1; z1] has the form:

E0 E0 s [r; t] = cos [k r !0t] exp [k r !0t] r 0 ! r 0 j j j j where

2 2 2 r = (x1 x0) + (y1 y0) + (z1 z0) j j p2 and k0 = j j 0

If the propagation distance r is large, then the spherical wave may be approximated accurately as a paraboloidal wave, and forj VERYj large r the sphere becomes a plane wave. The region where the …rst approximation is acceptable de…nes Fresnelj j di¤raction, while the more distant region where the second approximation is valid is the Fraunhofer di¤raction region.

6.2 Di¤raction Integrals

Consider the electric …eld emitted from a point source located at [x0; y0; z0 = 0]. The wave propagates in all directions. The electric …eld of that wave is observed on an observation plane centered about coordinate z1. The location in the observation plane is described by the two coordinates [x1; y1]. The electric …eld at [x1:y1] at this distance z1 from a source located in the plane [x0; y0] centered about z = z0 = 0 is:

E0 E [x1; y1; z1; x0; y0; z0 = 0] = cos [k r !0t] , r 0 j j 2 2 2 where r = (x1 x0) + (y1 y0) + z j j 1 q Though this may LOOK complicated, it is just an expression of the electric …eld propagated as a spherical wave from a source in one plane to the observation point in another plane; the ampli- tude decreases as the reciprocal of the distance and the phase is proportional to the distance and time. Di¤raction calculations based on the superposition of spherical waves is Rayleigh-Sommerfeld di¤raction.

Now, observe the electric …eld at that same location [x1; y1; z1] that is generated from many point sources located in the x y plane located at z = 0. The summation of the …elds is computed as an integral of the electric …elds due to each point source. The integral is over the area of the source plane. If all sources emit the same amplitude E0, then the integral is simpli…ed somewhat:

+ 2 1 Etotal[x1; y1; z1] = E[x1; y1; z1; x0; y0; 0] dx0 dy0 ZZ1 E = 0 2 2 2 aperture (x1 x0) + (y1 y0) + z ZZ 1 2 p 2 2 2 cos (x1 x0) + (y1 y0) + z1 !0t dx0 dy0 0 q where is a constant that is evaluated from physical considerations including dimensional analysis. 1 2 1 It can be shown that = pi0 = = , which includes a constant phase factor i0 1 ) i = exp i . This integral may be recast into a di¤erent form by de…ning the shape of the 2 142 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING

aperture to be a 2-D function f[x0; y0] in the source plane z0 = 0:

E0 Etotal [x1; y1; z1] = i0 + 1 f [x0; y0] 2 2 2 2 cos (x1 x0) + (y1 y0) + z 20t dx0 dy0 2 2 2 1 (x1 x0) + (y1 y0) + z 0 ZZ1 1 q This expressionp is the di¤raction integral. Again, this expression may LOOK complicated, but really represents just the summation of the electric …elds due to all point sources. Virtually the entire study of optical di¤raction is the application of various schemes to simplify and apply this equation. We will approximate the di¤raction integral for two distinct cases, as shown in the …gure:

1. Observation plane z1 is located “close to” the source plane z0 = 0; this is “near-…eld” or “Fresnel di¤raction”(the s in “Fresnel”is silent –the name is pronounced Frey0 nel).

2. Observation plane z1 located a large distance the source plane at z0 = 0 so that z1 = . This is called “far-…eld,” or “Fraunhofer di¤raction,” which is particularly interesting (and1 easier to compute results) because the di¤raction integral is proportional to the Fourier transform of the object distribution (the shape of the planar aperture).

The di¤raction regions for spherical waves emitted by a point source. Rayleigh-Sommerfeld di¤raction is based on the spherical waves emitted by the source. Fresnel di¤raction is an approximation based on the assumption that the wavefronts are parabolic and with unit amplitude for distances from some limit outward to . The “width” of the quadratic phase is indicated by 1 n; this is the o¤-axis distance from the origin where the phase change is radians. Fraunhofer di¤raction assumes that the spherical wave has traveled a large distance and the wavefronts may be approximated by planes.

Also note that the di¤raction integral depends on the wavelength 0, which means that it strictly applies ONLY to monochromatic light and thus cannot be used to describe the e¤ect in temporally incoherent. 6.3 FRESNEL DIFFRACTION; THE NEAR FIELD 143

6.3 Fresnel Di¤raction; the Near Field

Consider the …rst case of the di¤raction integral where the observation plane is near to the source plane, where the concept of near must be de…ned. Note that the distance r appears twice in the expression for the electric …eld due to a point source –once in the denominatorj j and once in the phase of the cosine. The …rst term a¤ects the size (magnitude) of the electric …eld, and the scalar product of the second with the wavevector k is computed to determine the rapidly changing phase angle of 15 the sinusoid. The optical phase changes very quickly with time (because !0 is very large, !0 = 10 7 radians/second) and with distance (because k is very small, k = 10 m), so the phase di¤erence of light observed at one point in the observationj j plane but generatedj j from two points in the source plane may di¤er by MANY radians. Simply put, small changes in the propagation distance r are very signi…cant in the computation of the phase, but much less so when computing the amplitudej j of the electric …eld. Therefore, the distance may be approximated more crudely in the denominator than in the phase.

Now consider the approximation of the distance r . The exact expression is: j j

2 2 2 r = (x1 x0) + (y1 y0) + z j j 1 q 2 2 (x1 x0) + (y1 y0) = z2 1 + 1 z2 s 1 2 2 (x1 x0) + (y1 y0) = z1 1 + 2 s z1 1 2 2 2 (x1 x0) + (y1 y0) = z1 1 + z2 1 This expression may be expanded into a power series by applying the binomial theorem. Recall that the general binomial expansion is:

n (n 1) n (n 1) (n 2) n! (1 + )n = 1 + n + 2 + 3 + + r + 2! 3! (n r)!r! 2 1 which converges to the correct value if < 1. In the current example of n = 2 (square root), the series is: 1 1 1 (1 + ) 2 = 1 + 2 + 3 2 8 16 which may be applied to …nd an approximation for the distance r : j j

1 2 2 2 (x1 x0) + (y1 y0) r = z1 1 + j j z2 1 2 2 2 1 (x x )2 + (y y )2 1 (x1 x0) + (y1 y0) = z 1 + 1 0 1 0 + 1 0 2 4 1 2 z1 8 z1 B C @ A If z1 is “su¢ ciently”large, terms of second and larger order are assumed to be su¢ ciently close to zero to be ignored. In other words, we assume that:

2 2 2 (x1 x0) + (y1 y0) << z 1 so that third term is MUCH smaller than the second, which is much smaller than the …rst. The 144 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING resulting approximation for the propagation distance is:

1 (x x )2 + (y y )2 r z 1 + 1 0 1 0 = 1 2 j j 2 z1 ! 2 2 1 (x1 x0) + (y1 y0) = z1 + 2 z1 In words, the propagation distance from the source to the observation point is the sum of the “longitudinal” distance z1 (down the optical axis) and a quadratic “correction” arising from the “o¤-axis”distance of the observation point. The expression for the electric …eld may be simpli…ed further by recasting into complex notation, so that: cos [] = Re exp [+i] f g The electric …eld from the single point source located at coordinates [x0; y0; z0 = 0] and observed at 2 2 [x1; y1; z1] where z1 >> x1 + y1 is:

p 2 2 E0 2i (x1 x0) + (y1 y0) E [x1; y1; z1; x0; y0; 0] = Re exp + z1 + 0t i z 2z 0 1 0 1 E0 2i i 2 2 = Re exp + z1 exp [ 2i0t] exp + (x1 x0) + (y1 y0) i z z 0 1 0 0 1 The phase of this approximation of the spherical wave includes three terms: a constant phase 2z1 0 due to the distance propagated down the optical axis from the source plane to the observation plane, a phase term that is a linear function of time (and that may be combined with the constant phase), and the last term with phase proportional to the square of the distance o¤-axis from the source point at [x0; y0] to the observation point [x1; y1]. This last spatial quadratic-phase term is the important one; it approximates the wavefront emitted by a point source as a paraboloidal shape instead of as the sphere in Huygens’principle. Note that one part of the assumption of Fresnel di¤raction is clearly unreasonable: that the wavefront is assumed to have constant magnitude regardless of the o¤-axis location [x1; y1] where the …eld is measured. In other words, the paraboloidal wave in Fresnel di¤raction has the same “brightness”regardless of where it is measured in the observation plane. This electric …eld emitted from a point source may be substituted into the di¤raction integral to obtain the approximate expression in the near …eld:

+ 1 Etotal [x1; y1; z1; 0] = E [x1; y1; z1; x0; y0; 0] dx0 dy0 ZZ1 + 1 z1 1 i 2 2 = exp +2i 0t f [x0; y0] exp + (x1 x0) + (y1 y0) dx0 dy0 i0z1 0 0z1 ZZ1 Again, this expression LOOKS complicated, but really is just a collection of the few parts that we have considered already. In words, the integral says that the electric …eld downstream but near to the source function is the summation of paraboloidal …elds from the individual sources. The paraboloidal approximation signi…cantly simpli…es the computation of the di¤racted light.

For still larger values of z1 (observation plane still farther from the source), the radius of curvature of the approximate paraboloidal waves increases, so the change in phase measured for nearby points in the observation plane decreases. In other words, as z1 increases still further, the paraboloid “‡attens out” and approaches a plane wave; this is the criterion for the Fraunhofer di¤raction region. The mathematical expression for light from an object observed in the Fraunhofer di¤raction region is derived shortly. 6.3 FRESNEL DIFFRACTION; THE NEAR FIELD 145

6.3.1 Fresnel Di¤raction as a Convolution Integral

The Fresnel di¤raction integral

+ 1 z1 1 i 2 2 g [x1; y1; z1; 0] = exp 2i 0t f [x0; y0] exp + (x1 x0) + (y1 y0) dx0 dy0 i0z1 0 0z1 ZZ1 may be rewritten by de…ning the exponential term (in the parentheses) as a function h that depends on the four variables in a particular way:

1 z1 i 2 2 h [x1 x0; y1 y0; z1; 0] exp 2i 0t exp + (x1 x0) + (y1 y0) i0z1 0 0z1 which leads to a simpli…ed expression for the Fresnel di¤raction integral:

+ 1 g [x1; y1; z1; 0] = f [x0; y0] h [x1 x0; y1 y0; z1; 0] dx0 dy0 ZZ1 This is a convolution integral of the form that abounds in all areas of physical science, and particu- larly in imaging. The function h is the impulse response of the integral operator, and often is called the point-spread function in imaging and particularly in optics. It has other names in other disci- plines (e.g., Green’sfunction in physics). The integral operator often is given a shorthand notation, such as the asterisk “ .” The variables of integration also often are renamed as dummy variables, such as , :

+ 1 g [x; y; z1; 0] = f [ ; ] h [x ; y ; z1; 0] d d ZZ1 f [x; y] h [x; y; z1; 0] The impulse response for Fresnel di¤raction is:

2 2 1 z1 x + y h [x; y; z1; 0] = exp 2i 0t exp +i i z z 0 1 0 " 0 1 # which is the product of a constant and a quadratic-phase “chirp”function, whose real and imaginary parts are sinusoids with varying spatial frequency that depends on the propagation distance z1. The parameters of the chirp often are combined into the chirp rate 0 p0z1 that has dimensions of length:

2 2 1 z1 x + y h [x; y; z1; 0] = exp 2i 0t exp +i i z 2 0 1 0 0 2 2 1 z1 x y = exp 2i 0t exp +i exp +i i z 2 2 0 1 0 0 0 2 2 The chirp rate is the radial distance r = x + y = 0 measured from the origin of coordinates over which the phase changes by radians, which means that the sign of the impulse response p “‡ips.” Again note that the magnitude of the impulse response is the unit constant: 1 h [x; y; z1; 0] = 1 [x; y] j j 0z1 which indicates that the assumed illumination from a point source in the Fresnel di¤raction region is constant o¤ axis; there is no “inverse square law.”This obviously unphysical assumption limits the usefulness of calculated di¤raction patterns to the immediate vicinity of the optical axis of symmetry. Pro…les of the impulse response along a radial axis are shown for 0 = p0z1 = 1 and 0 = 2. 146 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING

In the second case, pz1 is larger by a factor of two, so z1 is larger by a factor of four.

1-D pro…les of the impulse response of Fresnel di¤raction for (a) pz1 = 1 and (b) pz1 = 2, so that z1 is four times larger in (b). Note that the phase increases less rapidly with x for increasing distances from the source.

1-D pro…les of the impulse response of Fresnel di¤raction for 0z1 = 1. Note that the magnitude of the impulse response is 1 and the phase is a quadratically increasing function of x.

The convolution integral is straightforward to implement.

Computed Examples of Fresnel Di¤raction

The convolution formulation of di¤raction allows easy calculation of pro…les of di¤raction patterns. 6.3 FRESNEL DIFFRACTION; THE NEAR FIELD 147

Fresnel Di¤raction from a Point Source

f [x; y] = [x x0; y y0] 2 2 1 z1 x y exp 2i 0t exp +i exp +i i z z z 0 1 0 0 1 0 1 2 2 1 z1 x y = exp 2i 0t [x x0] exp +i [y y0] exp +i i z z z 0 1 0 0 1 0 1 2 2 1 z1 (x x0) (y y0) = exp 2i 0t exp +i exp +i i z z z 0 1 0 " 0 1 # " 0 1 # 2 2 1 z1 (x x0) + (y y0) = exp 2i 0t exp +i i z z 0 1 0 " 0 1 # The irradiance is proportional to the squared magnitude:

2 2 2 1 z1 (x x0) + (y y0) f [x; y] = exp 2i 0t exp +i j j i0z1 0 " 0z1 # 2 2 2 2 2 1 z1 (x x0) + (y y 0) = exp 2i 0t exp +i i0z1 0 " 0z1 # 2 1 = z 0 1 The impulse response is translated and now centered about the location of the point source. 148 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING

Fresnel di¤raction from single on-axis point source.

Fresnel Di¤raction from Two Point Sources

f [x; y] = [x + x0; y] + [x x0; y] = ( [x + x0] + [x x0]) [y] The separability of the quadratic-phase kernel allows the convolution to be evaluated easily because the “input function”f [x; y] also is separable:

g [x; y] = f [x; y] h [x; y] = ( [x + x0] + [x x0]) 2 2 1 z1 x y exp 2i 0t exp +i exp +i i z z z 0 1 0 0 1 0 1 1 z1 = exp 2i 0t i z 0 1 0 x2 y2 ( [x + x0] + [x x0]) exp +i [y] exp +i z z 0 1 0 1 1 z1 = exp 2i 0t i z 0 1 0 2 2 2 (x + x0) (x x0) y exp +i + exp +i exp +i z z z " 0 1 # " 0 1 #! 0 1 6.3 FRESNEL DIFFRACTION; THE NEAR FIELD 149

1 z1 g [x; y] = exp 2i 0t i z 0 1 0 2 2 2 2 x + x + 2xx0 x + x 2xx0 exp +i 0 + exp +i 0 z z 0 1 0 1 y2 exp +i z 0 1 1 z1 = exp 2i 0t i z 0 1 0 +2xx0 2xx0 exp +i + exp +i z z 0 1 0 1 x2 + x2 y2 exp +i 0 exp +i z z 0 1 0 1 1 z1 = exp 2i 0t i z 0 1 0 xx x2 x2 + y2 2 cos +2 0 exp +i 0 exp +i z z z 0 1 0 1 " 0 1 # 2 2 2 2 z1 xx0 x0 x + y = exp 2i 0t cos +2 exp +i exp +i i z z z z 0 1 0 0 1 0 1 " 0 1 # The irradiance is proportional to squared magnitude:

2 2 2 z1 2 xx0 g [x; y] = exp 2i 0t cos +2 j j i z z 0 1 0 0 1 2 x2 2 x2 + y2 exp +i 0 exp +i z z 0 1 " 0 1 #

4 x = cos2 +2 2 z (0z1) 2 0 1 3 x0 4 5 4 1 x = 1 + cos 2 +2 2 z (0z1) 02 0 2 0 0 1 1311 x0 @ @ 4 @ A5AA 2 x = 1 + cos +2 2 z (0z1) 0 2 0 1 31 2x0 2 @ 4 x 5A = 2 1 + cos +2 (0z1) D h i where the period of the sinusoidal fringe pattern is: z D = 0 1 = 0 2x0 2 sin [] where is the angle of each aperture measured from the optical axis at the observation plane. This is the same pattern from interference of two plane waves that are tilted relative to the z-axis by angle . The length of the period increases with increasing propagation distance z1, increasing wavelength 0, or decreasing separation of the point sources x0. 150 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING

Fresnel di¤raction from two point sources

Fresnel Di¤raction from a Knife Edge that would be generated from a knife edge at the same distances from the source as shown above. Note the “ringing” at the edges and that the fringes are farther apart when observed farther from the source. Compare these images to actual Fresnel di¤raction patterns in Hecht.

One feature to note is that the irradiance at the projected location of the edge is 0:25 of the “average”maximum value, regardless of the propagation distance. 6.3 FRESNEL DIFFRACTION; THE NEAR FIELD 151

1-D pro…les of the irradiance (squared magnitude) of di¤raction patterns from a sharp “knife edge” (modeled as the STEP function shown) for the same distances from the origin: (a) pz1 = 1; (b) pz1 = 2 = z is four times larger. Note that the “period” of the oscillation has increased with ) 1 increasing distance from the source and that the irradiance at the origin is not zero but rather 4 .

Fresnel Di¤raction from a Rectangular Aperture (“slit”) Recall that a 1-D rectangle func- tion may be constructed by subtracting appropriately translated STEP functions:

RECT [x] generated from STEP x + 1 STEP x 1 . 2 2

Since convolution is linear and shift invariant, the Fresnel di¤raction pattern of a rectangular aperture may be calculated by convolving the impulse response for the appropriate propagation distance with appropriately translated step functions, subtracting the amplitude disttributions, and calculating the squared magnitudes. 152 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING

Fresnel di¤raction of rectangle as di¤erence of two edges: (a) f1 [x] = STEP [x + 4] and x 2 x 2 f2 [x] = STEP [x 4]; (b) Re f1 [x] exp +i and Re f2 [x] exp +i ; (c) 1 1 x 2 x 2 Im f1 [x] exp +i andn Im f2 [x]h exp +iio ; (d)n real and imaginaryh partsio of 1 1 2 n h io n h io x 2 x di¤erence; (e) squared magnitude (f1 [x] f2 [x]) exp + i compared to RECT . 1 8 h i

6.3 FRESNEL DIFFRACTION; THE NEAR FIELD 153

Fresnel di¤raction of “narrower” rectangle as di¤erence of two edges: (a) f1 [x] = STEP [x + 2] x 2 x 2 and f2 [x] = STEP [x 2]; (b) Re f1 [x] exp +i and Re f2 [x] exp +i ; (c) 1 1 x 2 x 2 Im f1 [x] exp +i andnIm f2 [x] hexp + i io ; (d)n real and imaginaryh parts io of 1 1 2 n h io n h io x 2 x di¤erence; (e) squared magnitude (f1 [x] f2 [x]) exp + i compared to RECT . 1 4 h i

154 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING

Fresnel di¤raction of rectangle farther “downstream:” (a) f1 [x] = STEP [x + 4] and x 2 x 2 f2 [x] = STEP [x 4]; (b) Re f1 [x] exp +i and Re f2 [x] exp +i ; (c) 2 2 x 2 x 2 Im f1 [x] exp +i andn Im f2 [x]h exp +iio ; (d)n real and imaginaryh partsio of 2 2 2 n h io n h io x 2 x di¤erence; (e) squared magnitude (f1 [x] f2 [x]) exp + i compared to RECT . 1 8 h i

6.3.2 Characteristics of Fresnel Di¤raction

The parabolic approximation to the spherical impulse response of light propagation produces “im- ages”of the original object that have “fuzzy edges”and oscillating amplitude on the bright side of an edge. At a …xed distance from the object, the width of the di¤raction pattern is proportional to the width of the original object; if the object becomes wider, so does the “image”in the Fresnel 6.3 FRESNEL DIFFRACTION; THE NEAR FIELD 155 di¤raction pattern.

The Fresnel di¤raction patterns of at the same distance from the origin for two rectangles with di¤erent widths, showing that the “width” of the Fresnel pattern is proportional to the width of the object. Note that the “period” of the “ringing” is larger for larger observation distance z.

One of the more important aspects of imaging using the Fresnel approximation to di¤raction is that the “imaging system” is linear and shift invariant; the action of light propagation may be described by an impulse response and a corresponding transfer function. Note that this is di¤erent from reality where light propagates as spherical waves, which is linear but not shift invariant because the amplitude generated at an observation point from individual identical sources changes due to the di¤erent propagation distances.

6.3.3 Transfer Function of Fresnel Di¤raction

Fresnel di¤raction is described by a convolution with a quadratic-phase impulse response, which is the product of a spatial constant and a separable quadratic-phase factor:

2 2 1 z1 x + y h [x; y; z1; 0] = exp 2i 0t exp +i i z 2 0 1 0 " p0z1 # x2 y2 A0 exp +i 2 exp +i 2 " p0z1 # " p0z1 # where the leading constants were collapsed into A0:

1 z1 A0 exp 2i 0t i z 0 1 0 We may describe the same system by an equivalent representation in the frequency domain via the known 1-D Fourier transform of the quadratic-phase factor:

2 2 1 exp ix = exp i exp i F 4 h i and the scaling theorem: x 1 f = F [ ] = F 1 F j j j j n h io 156 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING

Therefore, we have:

x2 y2 H [; ] = 2 h [x; y] = 2 A0 exp +i 2 exp +i 2 F f g F ( " p0z1 # " p0z1 #) x2 y2 = A0 1 exp +i 2 1 exp +i 2 F ( " p0z1 #)! F ( " p0z1 #)! 2 2 = A0 0z1 exp +i exp i 0z1 4 p h i p 2 2 0z1 exp +i exp i 0z1 4 h i p 2 2 p = A0 0z1 exp +i exp i0z1 + j j 2 h i 1 z1 2 2 = exp 2i 0t i0z1 exp i0z1 + i z 0 1 0 z1 2 2 = exp 2i 0t exp i0z1 + 0 The transfer function of light propagation in the Fresnel region is circularly symmetric:

z1 2 H [; ] = exp 2i 0t exp i0z1 0 h i 2 z1 = exp 2i 0t exp i 2 2 3 0 1 6 0z1 7 4 2 5 z1 q = exp 2i 0t exp i 2 0 1 where 2 + 2. The leading factor has “disappeared”into the complex factor that carries the linear phase dependence on propagation distance z1 and on time t. This modulates a “downchirp” p 1 1 factor (leading negative sign) whose chirp rate is 1 = . 0z1 0 Some authors call H [; ] the transfer function of freeq space, but to me this name does not convey the idea that it applies to propagation by the speci…c distance z1.

6.4 Fraunhofer Di¤raction

The di¤raction integral may be further simpli…ed for the case where the distance from the source to the observation plane is su¢ ciently large to allow the electric …eld from an individual source to be approximated by a plane wave. The process may be considered for a paraboloidal wave emitted from a single source:

i 2 2 exp + (x1 x0) + (y1 y0) z 0 1 2 2 2 2 i(x0 + y0) i(x1 + y1) 2i = exp + exp + exp (x0x1 + y0y1) z z z 0 1 0 1 0 1

If the source is restricted to be near to the optical axis so that x0; y0 = 0 (or, more rigorously, 2 2 x0 + y0 << 0z1), then (x2 + y2) 0 0 = 0 0z1 6.4 FRAUNHOFER DIFFRACTION 157 and: i(x2 + y2) exp + 0 0 = 1: z 0 1 This resulting di¤raction integral simpli…es to:

+ 1 1 i 2 2 g [x1; y1] = f [x0; y0] exp + (x1 x0) + (y1 y0) dx0 dy0 i0z1 0z1 ZZ1 2 2 + 1 i(x1 + y1) 1 2i = exp + f [x0; y0] exp (x0x1 + y0y1) dx0 dy0 i0z1 0z1 0z1 ZZ1 2 2 Similarly, if the observation point is near to the optic axis so that x1 + y1 << 0z1, then: i(x2 + y2) exp + 1 1 = 1: z 0 1

Though x0 and x1 are su¢ ciently small for these approximations, the third exponential term is retained because its contribution may be signi…cant, since the phase is not a quadratic function of any coordinate:

i 2 2 2i (x0x1 + y0y1) exp + (x1 x0) + (y1 y0) = exp 0z1 0z1 Considered in the observation plane as functions of x1 and y1, the phase of the wavefront is propor- tional to the source variables [x0; y0]; since the phase is a linear function of the source coordinates, the wavefront due to light emitted from these coordinates is a plane. The corresponding approximation for the di¤raction integral is:

+ 1 z1 1 2i (x0x1 + y0y1) Etotal [x1; y1; z1] = exp +2i 0t f [x0; y0] exp dx0 dy0 i0z1 0 0z1 ZZ1 In words, the di¤racted light measured far from the source is approximately equal to the summation of plane waves generated by each source point. This is called the Fraunhofer di¤raction formula, and the resulting di¤raction patterns are VERY di¤erent from those from the same aperture measured in the Fresnel region. In fact, the Fraunhofer di¤raction formula may be interpreted as a Fourier transform by mapping the frequency coordinates back to the space domain via the proportional x1 x1 y1 y1 relationships = 2 = , = 2 = : 0 0z1 0 0z1

1 z1 Etotal [x1; y1; z1] = exp +2i 0t i z 0 1 0 + 1 x1 y1 f [x0; y0] exp 2i x0 + y0 dx0 dy0 0z1 0z1 ZZ1 1 z1 = exp +2i 0t 2 f [x; y] = x1 ;= y1 z F f gj 0z1 0z1 1 0 The irradiance pattern in the Fraunhofer region again is proportional to the time-averaged squared magnitudes of the …eld:

2 2 1 Etotal [x1; y1; z1] 2 f [x; y] x1 y1 2 2 = z ;= z j j / 0z1 F f gj 0 1 0 1

1 x y 2 = F 1 ; 1 2z2 z z 0 1 0 1 0 1

The amplitude of the …eld evaluated in the Fraunhofer di¤raction region is proportional to the 158 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING appropriately scaled Fourier transform of the original amplitude, which therefore demonstrates that the process is linear and shift variant, rather than the shift-invariant result in the Fresnel region. Propagation into the Fraunhofer region in this model cannot be described by an impulse response or transfer function. However, also note that the Fresnel region includes the Fraunhofer region, so it is possible to calculate the amplitude by that means. The fact that the two calculations will not agree is a byproduct of the additional approximation in the Fraunhofer region. All of the Fourier transform relationships apply to Fraunhofer di¤raction patterns. The scaling theorem of the Fourier transform shows that the patterns (the “images”) scale in inverse proportion to the original functions, so that larger input functions f [x; y] produce smaller (and brighter) dif- fraction patterns. The shift theorem indicates that movement of the input object “o¤ axis”produces a linear phase in the di¤raction pattern; since the irradiance is (generally) viewed, the translation o¤ axis does not generally have a visible e¤ect on the di¤raction pattern, but its e¤ect is visible when two such patterns are added.

6.4.1 Examples of Fraunhofer Di¤raction

No Aperture

We can now easily evaluate the Fraunhofer di¤raction patterns produced by simple apertures. First, consider the case if the aperture does not exist, so that:

f1 [x; y] = 1 [x; y] = 1 [x] 1 [y] = F1 [; ] = [; ] = [] [] ) Apply the Fourier transform to evaluate the spectrum in the irradiance in the Fraunhofer di¤raction region: 2 2 1 x1 y1 Etotal [x1; y1; z1] ; j j / 2z2 z z 0 1 0 1 0 1

This does not make sense! The reason is actually quite simple; the size of the (nonexistent) aperture does not obey the assumed constraint for Fraunhofer di¤raction, that x0; y0 = 0, which means that the aperture is small.

Rectangular Aperture

The pro…les of 2-D square apertures and computed simulations of the resulting pro…les of the am- plitude di¤raction pattern in the Fraunhofer di¤raction region are easy to calculate: x y f [x; y; z = 0] = RECT ; a b ab h i x1 y1 Etotal [x1; y1; z1] j j SINC ; / 2z2 z z 0 1 0 1 0 1 The irradiance is the squared magnitude of the plotted amplitude. The original point source located at z1 = so that the light “…lls” the aperture with zero phase. Two examples are shown for the same …xed1 (large) distance from the object; the “images”“narrow”and get “taller”as the aperture width increases. 6.4 FRAUNHOFER DIFFRACTION 159

1-D pro…les of Fraunhofer di¤raction patterns: (a) input objects are two rectangles that di¤er in width; (b) amplitude (NOT irradiance) of the corresponding Fraunhofer di¤raction patterns, showing that the wider aperture produces a “brighter” and “narrower” amplitude distributions.

A useful measure of the “width” of the Fraunhofer pattern is labeled for both cases; this is the distance from the center of symmetry to the …rst zero. This “width” provides a measure of the ability of the system to resolve …ne detail, because the individual di¤raction patterns generated by two point sources may overlap and become di¢ cult to distinguish as individual sources as the angular separation of the point sources decreases. 160 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING

Illustrations of resolution in Fraunhofer di¤raction: (a) individual images of two point sources in the Fraunhofer domain; (b) sum of the two images, showing that they may be distinguished easily; (c) images of two sources that are closer together; (d) sum showing that it is much more di¢ cult to distinguish the sources.

6.5 Coherence of Light and Imaging

We should also emphasize that the analysis and the “images”just considered assumed illumination with a single wavelength 0, which means that light can “interfere”constructively and destructively after traveling di¤erent paths (distances and/or times) from the source. Of course, this constraint does not apply in realistic imaging systems; the range of wavelengths typically is large (for visible light, = 300 nm). It often is convenient to specify the bandwidth in frequency:

max min c c max min c = = c = 2 min max max min 0 Therefore the frequency bandwidth of white visible light is:

8 1 700 nm 400 nm 14 = 3 10 m s = 3:2 10 Hz 400 nm 700 nm 6.6 TRANSMISSIVE OPTICAL ELEMENTS 161 which is of the same order as a wavelength of light, because

8 1 8 1 3 10 m s 3 10 m s 14 red = = = 4:3 10 Hz 700 nm 700 10 9 m The bandwidth of the light source has a profound e¤ect on the “quality”of the image produced by the system. If, as we have assumed thus far, the source generates the single wavelength 0, then the bandwidth = 0 Hz and the light is “perfectly coherent,”which means that the phase of the light at all points in space is deterministic (this is an idealization, because a traveling wave with = 0 must be in…nitely long). When the amplitude of light is combined from di¤erent points on the source or after traveling di¤erent paths, the deterministic phase means that the interference e¤ects are important at the image. For example, the light can destructively interfere to create a null amplitude, or it may even produce a “negative” amplitude. In imaging systems used in coherent light, the amplitudes of the light waves add and phase e¤ects are paramount. In short, an optical imaging system acting in coherent light is linear in the complex-valued amplitude. At the opposite extreme of a very wide bandwidth, the light is “incoherent,”which means that the phase of the light measured over a long time scale is random. Incoherent light does not interfere in the conventional sense to create negative amplitudes, though it is still possible to construct meaningful interferograms (as demonstrated by Brown and Twiss). The quantity of importance when incoherent light waves are added is the irradiance. The irradiance cannot be negative and the importance of the relative phase of the light is buried in the calculation. An optical imaging system acting in incoherent light is linear in irradiance, which is the real-valued squared magnitude of the complex amplitude. The quality of the light source cannot be either perfectly coherent or perfectly incoherent, but the action of the imaging system may be calculated rather easily in these two extreme cases, whereas the calculation is di¢ cult or impossible in intermediate cases.

6.6 Transmissive Optical Elements

The imaging “systems” that we have considered so far is as simple as you can get; the light in the …rst cases merely propagates from one plane to another by Huygens’principle, and the amplitude is calculated using one of the two approximations: the Fresnel transform in the near …eld and the Fourier transform in the far …eld. The shift-variant expression for the di¤raction integral simpli…ed to a convolution with a quadratic-phase factor in the Fresnel region, and to a scaled Fourier transform in the Fraunhofer region. In the second case, light propagated from the source to the aperture to the observation plane. At this point, we consider methods for manipulating or modifying the propagating light waves by inserting optical elements at the pupil plane that are based on the physical interaction of refraction; elements may also be constructed based on the mechanisms of re‡ection and di¤raction. The physical mechanism of refraction is the result of the fact that light waves (or rays) within a medium slow down in inverse proportion to a parameter known as the refractive index of the medium, i.e.,

c 8 m vglass = ; c = 3 10 n s If the glass has thickness (measured in mm, say), then the “arrival time” of a …xed point (point of constant phase) on a light wave that travels through the glass will be delayed when compared to that for a wave traveling the same distance through vacuum. The time delay is:

(n 1) t = tglass tvacuum = = seconds vglass c c This time delay produces a di¤erence in the phase of the light that passes through the glass when 162 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING compared to that traveling through vacuum: 2 = glass vacuum = (c t) = 2 (n 1) radians. 0 0 An ideal nonabsorptive transmissive optical element of refractive index n and circularly symmetric physical thickness (r) may be described by a complex-valued transmittance function:

(r) (r) (r) +i[(r)] +2i n +2i(n 1) +2i t (r) = 1 e = e 0 = e 0 e 0 Our task now is to evaluate this transmittance function for di¤erent phase functions.

6.6.1 Optical Elements with Constant or Linear Phase

If the optical element is a slab of glass with uniform thickness , then the phase incident on the glass measured at all points after the glass additional constant factor of exp +2i n . Since 0 the phase change is constant, the e¤ect of the glass plate on the observed irradianceh is undetectable.i An example is shown in the …gure.

Phase shift induced by a ‡at piece of glass compared to the phase of light propagating in vacuum. The locus of points of constant phase on the entering wavefront is a vertical plane. The relative phase shift of light traveling through the glass is constant (not a function of x or y and the locus of points of constant phase is now two discontinuous planes.

In a slightly more complicated case where the glass thickness varies with position [x; y], so will the phase delay. For example, if the thickness is a linear function of x, then the transmittance function is: +2i(n 1) x [x; y] = x 1 [y] t [x; y] = e 0 1 [y] ! (n 1) Such an optical element is called a prism, as shown in the …gure. The term has dimensions 0 of reciprocal length and thus may be identi…ed as a spatial frequency: 6.6 TRANSMISSIVE OPTICAL ELEMENTS 163

Schematic of the phase shift induced by a prism with thickness function [x; y] = x. The locus of the points with constant phase on the entering wavefront is a vertical plane. Distances between several points where the optical path di¤erence is 6 (phase di¤erence of 12 radians) are shown, which con…rm that the wavefront was tilted “down” by the action of the prism.

t [x; y] = exp [+2i0x] 1 [y] where 0 = (n 1) 0

Assume that the prism located at z1 >> 0 (in the Fraunhofer region) is illuminated by a point source at z = 0. The light emerging from the prism propagates to a plane at z = z1 + z2 that is in the Fraunhofer region measured from z1. The “image” generated at the observation plane is proportional to the Fourier transform of the unit-magnitude “aperture”function t [x1; y1]:

E [x; y; z + z ] = t [x ; y ] 1 2 2 1 1 x ; y F f gj ! 0z2 ! 0z2 = 2 exp [+2i0x] 1 [y] F f g = [ ] [] 0 x ; y j ! 0z2 ! 0z2 x y = (n 1) z z 0 2 0 0 2 2 = 0z2 [x (n 1) z2] [y] j j The action of the prism is a change in position of the displayed “image” from the origin to the o¤-axis location [x2; y2] = [(n 1) z2; 0].

6.6.2 Lenses with Spherical Surfaces

Optical systems typically are used to form images of the source distribution by redirecting the radiation from point sources to point images. For example, consider a slab of glass with maximum thickness and with a convex spherical face with radius of curvature R (i.e., a plano-convex lens). The thickness function of the glass is found by using the Pythagorean formula in the following picture: (R s (r))2 + r2 = R2 = r2 2R s (r) + s2 (r) = 0 ) If both R and r are much larger than s (r), then the factor s2 (r) may be ignored. The result is a simple formula, the sag formula for a spherical lens:

r2 s (r) = 2R 164 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING as shown in the …gure. In words, the sag formula approximates the spherical surface by a paraboloid. This process is analogous to the approximation of a spherical wavefront as a paraboloid in the Fresnel di¤raction region.

2 Parameters in the sag formula for a spherical lens, showing that s [r] r ' 2R

The thickness of glass as a function of radial distance from the optical axis is the di¤erence between the maximum thickness and the parabolic sag:

r2 (r) = s (r) = 2R which measures the thickness of glass as a function of radial distance r from the optical axis. The phase delay as a function of radial distance from the optic axis is the sum of the phase delay through the glass and the phase delay through air in the sag region. The delay due to the glass is the number of wavelengths multiplied by the refractive index

2 2 (r) = distance traveled in glass + distance traveled in vacuum 0 n ! 0 2 2 = (r) + s (r) 0 n ! 0 n r2 2 r2 = 2 + 2R 2R 0 0 2 r2 = + n (n 1) 0 0R The …rst (constant) term is the …xed phase delay of light rays traveling through the maximum thickness of glass, while the second term is a quadratic function that decreases as a function of the radial distance r measured from the optical axis due to the more rapid velocity of light in the sag region. The quadratic approximation to the spherical surface is the source of the quadratic phase contribution of a spherical lens. The electric …eld exiting the lens as a function of radial position r from the optical axis is the product of the incident …eld and the phase delays, leading to constant- and quadratic-phase terms:

n r2 E [r; t] = E0 [r] exp +2i exp i (n 1) R 0 0

In the common case of a lens has two convex spherical surfaces with radii R1 and R2, the radius of curvature of the one surface must have the opposite sign because it is oriented in the opposite direction. By convention, a surface with center of curvature to the right of the surface has positive 6.6 TRANSMISSIVE OPTICAL ELEMENTS 165 radius. The phase transmittance of the lens has contributions from each sag:

n r2 r2 t [r] = exp +2i exp i (n 1) exp +i (n 1) 0 0R1 0 ( R2) n r2 1 1 = exp +2i exp i (n 1) R R 0 0 1 2 The expression may be simpli…ed by combining terms to de…ne the factor f that has dimensions of length: 1 1 1 (n 1) f R R 1 2 This possibly is a familiar equation; it is the lens-maker’s formula that relates the focal length f of the lens to the index of refraction n and the radii of curvature R1 and R2 of the two surfaces. The e¤ect of the lens in on the electric …eld is to change the phase of the transmitted light due to the optical transmission function t (r):

n r2 t (r) = exp +2i exp i f 0 0 n r2 = exp +2i exp i if f > 0 0 0 f j j Note that if f is negative, the algebraic sign of the quadratic phase due to the lens is positive:

n r2 t (r) = exp +2i exp +i if f < 0 0 0 f j j which means that the phase of the transmitted light increases with increasing radial distance from the optical axis. At this point, the transmittance is a pure phase function and thus has in…nite support. The …nite diameter of a realistic lens may be introduced as a real-valued function p [x; y] (which may be circularly symmetric or not) that de…nes the pupil of the lens. The pupil function usually is binary (p [x; y] = 0; 1) but we can (and will) use pupil functions whose transmission varies with radial distance. A circularly symmetric spherical lens with positive focal length (converging lens) will be modeled by the transmittance function:

n r2 t (r) = p (r) exp +2i exp i f 0 0 as shown in the …gure. In words, a lens changes the radius of curvature of the incident parabolic wave by adding or subtracting a phase factor that is a quadratic function of the distance from the optical axis. The variable part of the phase contribution from a spherical lens is quadratic with a negative sign, while the (approximate) impulse response of light propagation in the Fresnel di¤raction region is a quadratic-phase factor with a positive sign. 166 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING

The negative quadratic phase induced by a positive lens changes the radius of curvature of incident plane waves to converging spherical waves that focus at the distance f.

6.7 Fraunhofer Di¤raction and Imaging

A monochromatic point object located a long distance away from an imaging system produces a set of wavefronts that are approximately planar. The entrance pupil of the optical system (the image of the aperture stop in object space) collects a section of each plane wavefront and the optical elements convert it to a spherical (actually, only approximately spherical) wavefront that converges to an image “point.”We can use the concept of Fraunhofer di¤raction to de…ne the “angular resolution” of the imaging system. As an introduction, consider an optical system that consists of only an aperture, which we can think of as the entrance pupil or the aperture stop, because they coincide if no other optics are involved. The pieces of the wavefront from the object source that are collected by the stop would continue to propagate “downstream.”If observed a long distance from the stop/pupil, the irradiance pattern from each point source is the Fraunhofer di¤raction pattern of the stop, and thus proportional to the appropriately scaled Fourier transform of the stop. Recall that the Fourier transform has a “reciprocal”relationship: the smaller the stop, the larger the di¤raction pattern and vice versa. Of course, the observed irradiance is the time average of the squared magnitude of the amplitude, and thus is nonnegative. Now consider the situation if the object instead consists of two monochromatic point sources that appear to be displaced by the small angle 0 when viewed from the aperture/pupil. The light from each source propagates from the aperture at the same angle 0 to create the Fraunhofer di¤raction pattern of the aperture at the observation plane. The light from the “o¤-axis”source forms a replica of the same Fraunhofer di¤raction pattern but at the same angular separation 0. The observed irradiance again is the time average of the squared magnitude of this amplitude. If the aperture stop is “wide”in some sense, then the di¤raction patterns will be “narrow”because of the reciprocal relation between the widths of the aperture and its Fraunhofer di¤raction pattern. If the di¤raction patterns are “narrow”, the fact that the object consisted of two point sources may be apparent. Conversely, if the stop is “narrow” and the di¤raction patterns are “wide,” the patterns from the two sources may overlap and be di¢ cult to distinguish. Thus the ability of the system to resolve two point sources separated by the angle depends on the size of the entrance pupil. 6.7 FRAUNHOFER DIFFRACTION AND IMAGING 167

Fraunhofer di¤raction of aperture stop with no optic: the distance from the monochromatic point object to the stop is z1 and the light di¤racted by the stop forms a Fraunhofer irradiance / 1 pattern on the observation screen at z2 . If the object consists of two point sources at the same / 1 distance z1 separated in angle by 0, the di¤racted amplitude is the sum of the individual amplitudes separated by the same angle 0.

Of course, real optical imaging systems consist of more than just an aperture; they also include lenses and/or mirrors that change the curvature of that part of the wavefront that passes through the stop/pupil. The section of the (approximately) spherical wave emerging from the optical system converges approximately to a point in the image plane. Because the distance z1 from the object to the system is large, the image plane is approximately located at the focal plane, i.e., the distance z2 = f, the focal length of the lens. We can interpret the action of the lens as “bringing in…nity up close (and personal?).”In other words, the lens creates a scaled replica of the light pattern that would have been generated at a large distance from the stop if the lens were not present; the image of a point object is created by the imaging system is a scaled replica of the Fraunhofer di¤raction pattern of the entrance pupil (and thus of the aperture stop); the scale factor is the ratio of the original propagation distance z1 to the focal length f (or to the “e¤ective focal length feff if the optic is composed of multiple elements). This scaled replica of the Fraunhofer di¤raction pattern of the stop is the impulse response of the optical imaging system in monochromatic (coherent) light. Note that the lens also adds a phase factor to the wavefront due to the curvature of the converging wave. This phase factor may be approximated by an additional quadratic-phase factor, but this e¤ect is not generally signi…cant in imaging because the sensor measures the irradiance of the light, which is proportional to the squared magnitude of the amplitude. eliminates this factor. If we ignore this additional quadratic-phase factor, the amplitude produced by a single on-axis point source in 168 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING monochromatic light is proportional to the Fourier transform of the entrance pupil p [x; y]:

x1 y1 Etotal [x1; y1; f] = h [x1; y1; f] P ; / 0 f 0 f So, again, a large-diameter entrance pupil produces a “small”impulse response and “better”angu- lar resolution, if the performance of the system is limited by di¤raction (which often is not true; performance is frequently limited by aberrations in the optical system ).

If the object consists of two point sources separated by the apparent angle 0 as before, the amplitude pattern at the image plane is the sum of the two Fraunhofer di¤raction patterns displaced by same angle but that have been “mini…ed” by a constant factor equal to the ratio of the image f distances in the two cases: . z2

An optical system imaging a point source creates a scaled replica the Fraunhofer di¤raction pattern of the stop on the image plane. In other words, the impulse response of the imaging system is the Fraunhofer di¤raction pattern of the aperture stop.

Lord Rayleigh derived a metric for the angular resolution of an imaging system if the wavelength is 0 and the two objects are separated by the distance d in the object plane, the distance from the object to the imaging system is z1 = L, and the diameter of the entrance pupil is D, then the Rayleigh criterion states that the two sources are just resolved in the image if the separation in the image plane is equal to the radius of the di¤raction spot. For square apertures, the “half-width”of the di¤raction spot is: L d = 0 D For circular apertures, the “half-width”(radius) of the di¤raction spot is:

L d 1:22 0 ' D (note the similarity to the formula for the separation between interference fringes in Young’sexper- 6.7 FRAUNHOFER DIFFRACTION AND IMAGING 169 iment). The angular radius of the image spot is a good measure of the angular resolution:

d = 0 , for square apertures L ' D

To …nd the image pattern from a point source through a circular entrance pupil of diameter D, we may use Gaskill’snotation for the so-called cylinder function: r f [x; y] = CYL D Its Fourier transform is:

D 2 F [; ] = SOMB (D) 2 2 D 2J1 (D) D = = J1 (D) 4 D 2 We make the substitution that takes the equation from the frequency domain to the space domain by dividing by the product of the wavelength and the e¤ective focal length:

x2 + y2 r = 2 + 2 = f f ! p0 eff 0 eff p r = D D ) ! 0 feff The …rst zero of J1 (u) occurs at u0 = 1:22, which corresponds to a radial distance r0 that satis…es the condition: Dr 0 = 1:22 0 feff feff = r0 = 1:22 0 ) D

The ratio of the e¤ective focal length feff and the diameter of the entrance pupil D is the “f-number” (often abbreviated “f/#”) of the imaging system, which is a convenient measure of the angle of the cone of rays from the optical system to the image. The radius of the di¤raction spot measured to the …rst zero may be written in concise form:

r0 = 1:22 0 f/# It often is “easier”to think of the diameter of the resolution “spot:”

d0 = 2:44 0 f/#

This leads to a useful mnemonic that applies for visible light (0 = 0:5 m). The product 2:44 0 = 2:44 0:5 m = 1:22 = 1:0, so that: d0 [m] = f/# In words, the diameter (in micrometers) of the resolution spot of an imaging system operating in visible light is approximately equal to the f-number. This is the di¤raction limit of the imaging system; it is a measure of the smallest distance on the image plane where two point objects may be resolved. The angular diameter of the image spot produced by a system with a circular aperture is: 2 = 2:44 0 : D 170 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING

Check this for the most accessible imaging system you have: your eyes. At night, the diameter of the pupil of a dark-adapted eye is D = 7 mm, so the Rayleigh limit of angular resolution near the middle of its sensitivity In green light (0 = 550 nm) is approximately:

0:55 m 4 2 = 2:44 = 1:92 10 radians = 0:192 milliradians 7 mm To convert this to seconds of arc, we need the to know the number of arc-seconds in a radian: 360 60 arc-minutes 60 arc-seconds 1 radian = 206; 265 arc-seconds 2 radians arc-minute = 1 radian = 2 105 arc-seconds ) 6 = 1 arc-second = 5 10 radians = 5 microradians ) which is a useful factor to remember. The Rayleigh limit to the resolution of the dark-adapted human eye is approximately:

0 3 arc-seconds 2:44 = 0:192 10 radians 206; 265 D radians 2 = 40 arc-seconds = arc-minute 3

1 which is approximately 40 of the diameter of the moon. Since the eye is not “perfect,” we more 1 often assume that the resolution of the eye is about 1 minute of arc, or 30 of the moon’sdiameter.

human eye = 1 arc-minute

The diameter of the primary mirror of the Hale telescope on Palomar Mountain is D = 5 m, so the theoretical minimum angular separation is = 0:24 0, for 0 measured in meters. The angular separation is: 7 = 1:32 10 radians = 0:132 radians = 0:027 arc-seconds Again, this is the theoretical limit that would apply to a perfectly …gured mirror with no other “optical elements”in the system. The …rst condition is approximately true, but the second clearly is not. The ground-based Hale telescope views astronomical objects through the atmosphere, which is composed of volumes of gases with di¤erent temperatures (and thus di¤erent refractive indices) at di¤erent locations that move rapidly in time. The variation in refractive index over time has the e¤ect of reducing the resolution of the telescope from its theoretical limit of about 0:3 arc- seconds to angles of the order of arc-seconds, i.e., a factor of 30 or so. This approximate angular resolution of the Hale telescope is that of an optical system with D = 100 mm = 4 in. The additional aperture of a large telescope does have the notable bene…t of collecting more light to allow imaging of fainter objects, but increasing the ability of the telescope to resolve smaller separations requires other technology, e.g., interferometric methods (stellar spectral interferometry) to correct the images after collection, or adaptive optics (“rubber mirrors”)to correct the images in “real time.”Adaptive optical systems have to respond to phase changes within a few hundredths of a second. The diameter of the primary mirror of the Hubble space telescope is approximately half that of the Hale telescope (D = 2:4 m), so the angular resolution of the HST optics at 550 nm is approximately twice as large (0:26radian = 0:054 arc-seconds). Of course, there is no atmosphere to mess up the Hubble images, though readers may well recall some issues (to put it mildly) with mirror fabrication. That the dimension of the resolution “spot”is directly proportional to the wavelength means that a system with …xed focal length and …xed entrance pupil diameter can resolve smaller separations at shorter wavelengths. Also, systems with smaller focal ratios have better resolution (“faster systems have better resolution”). In real life, it is very di¢ cult (meaning very expensive) to fabricate an optical imaging system with a large entrance pupil and whose performance is limited by di¤raction. The performance of the 6.8 DEPTH OF FIELD AND DEPTH OF FOCUS 171 system is much more often constrained by the limitations of the optical design and fabrication so that the outgoing wavefront is not spherical, but has another shape. This behavior is characterized by various optical aberrations (e.g., spherical aberration, astigmatism, distortion) that quantify the di¤erence from the ideal spherical wave. Generally speaking, aberrations become more di¢ cult and expensive to correct as the size of the entrance pupil increases. Of course, the ultimate resolution of the Hale telescope actually is limited by atmospheric tur- bulence, which creates random variations in local air temperature and thus in the refractive index of “patches” of the atmosphere. These variations are often decomposed into the aberrations intro- duced into the wavefront by the phase errors. The constant phase (“piston”) error has no e¤ect on the irradiance (the squared magnitude of the amplitude). Linear phase errors move the image from side to side and or top to bottom (“tip-tilt”). Quadratic phase errors (“defocus”) act like additional power in the lens that moves the image plane backwards or forwards along the optical axis. In general, tip-tilt error is the most signi…cant, which means that correcting this aberration signi…cantly improves the image quality. The …eld of correcting atmospheric aberrations is called “adaptive optics,”and is an active research area.

6.7.1 “Di¤raction-Limited” and “Detector-Limited” Imaging Systems The “blur”of an imaging and/or sensor system discussion just concluded on the e¤ect of Fraunhofer di¤raction in imaging shows that images generated by ideal optical imaging systems exhibit “blur” due to unavoidable di¤raction of the light focused by the optics. The linear dimension of this di¤raction spot de…nes one possible measure of the spatial “resolution” of the images, i.e., the smallest separation between point objects (such as stars) that may be “just” distinguished by the system. It is important to note a couple of points about resolution. First, the range of wavelengths in the light beam being imaged has an important impact on resolution. In other words, the resolution of an imaging system used in “natural” (incoherent) light that has a wide range of wavelengths is di¤erent from the resolution of the same imaging system used in “coherent”light (as from a laser) where the range of wavelengths is small. Also note that several metrics of resolution are commonly used, including the Rayleigh and Dawes limits, so it is important to understand which is being used to specify the quality of the system. In most optical systems of interest, the image is focused onto an imaging sensor with its own resolution properties. This concept perhaps has become more familiar with the widespread availabil- ity of discrete electronic imaging sensors, such as the charge-coupled device (CCD). The concept of resolution of a discrete sensor is quite straightforward, because we usually assume that the individual sensor elements have identical uniform response over their entire area. We should note that some these assumptions are not su¢ cient for some critical applications. For our purposes, the simplify- ing assumptions are su¢ cient. The transverse magni…cation of the imaging system and the linear dimension of the sensor elements determines the smallest separation between two point sources that may be resolved by this sensor; the resolution limit reqires that the image …eld measured by the sensor of two equally bright point objects must be separated by a sensor element that responds less.

6.8 Depth of Field and Depth of Focus

In a system consisting of both optics and sensor, the system is “di¤raction-limited” if the pixel size is smaller than the linear dimension of the di¤raction spot. The system is “detector-limited”or “sensor-limited”if the linear dimension of the individual sensor elements is larger than the di¤raction spot. The discussion of the limiting “blur”of an imaging system may be extended to characterize the range of “distances”(or “depths”)over which images of point objects exhibit the “same”(or at least “similar”) blur dimensions. If speci…ed in object space, the distance range is called the “depth of …eld;”the same metric in image space is the “depth of focus.” There is no one way to de…ne the depths of …eld and focus, but we can rather easily derive a “hybrid” metric (based on both ray and wave optics). The measurement is based on the linear dimension of the blur spot, which is the di¤raction spot in a di¤raction-limited system or on the size 172 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING of the sensor element in a detector-limited system. Until the arrival of digital imaging systems, the former case of di¤raction-limited optics has been most often considered because di¤raction provided the fundamental limit to resolution. As we shall see, the depth of …eld is determined by the f-number of the optical system.

Assume that the linear dimension of image blur has been measured for a particular imaging system at the speci…c pair of object and image distances (z and z0 respectively) of interest:

Blur in a di¤raction-limited system with aperture diameter D. The image of the point source is a di¤raction pattern at the image plane whose linear dimension (using some criterion) is B : 0

For example, the image of a point source located a distance z from the system could be measured to …nd this limiting “blur diameter” B0, where the prime indicates that the measurement is made in image space. In a di¤raction-limited system, the discussion of Fraunhofer di¤raction in imaging shows that one possible measure for B0 is the diameter of the central lobe of the di¤raction spot:

z0 feff B0 = 2:44 0 = 2:44 0 = 2:44 0 f/# D D

From B0, it is easy to evaluate the corresponding dimension B in object space from the transverse z0 B0 magni…cation MT = = . z B 6.8 DEPTH OF FIELD AND DEPTH OF FOCUS 173

The calculation of depth of …eld: B0 is the linear dimension of the blur for the system, either the diameter of the di¤raction spot in a di¤raction-limited system or the dimension of the sensor element in a detector-limited system. The locations z0 0 specify locations in image space where the geometrical blur has the same size. The corresponding locations in object space are the limits of the “depth of …eld.” As shown in the …gure, the “blur” due to geometrical rays from the point object has the same dimension B0 as the blur spot at two locations equidistant from the “in-focus” image. We assign the name 0 to the distance between the “in-focus”image and the geometrically blurred images, so these two planes are located at z0 0. The depth of focus in this model is twice 0:

z0 = 2 0 In the ray model, the drawing shows that:

D B0 z0 = = 0 = B0 = B0 f/# z0 0 ) D

If B0 is small, so must be 0; if the f/# is large, so must be 0. The corresponding locations in object space (and the depth of …eld) may be determined by substituting these distances and the focal length feff into the imaging equation. For example 1 1 1 1 1 = = z1 feff z feff z 10 0 0

The distances z1 and z2 may be evaluated from the imaging equation for the corresponding image distances z0 = z0 0 and z0 = z0 + 0. It is easy to see that the absolute magni…cation MT is 1 2 j j smaller for the smaller image distance, i.e., MT for z0 = z0 0 is smaller than MT for the larger 1 object distance z20 = z0 + 0. The nonlinearity of the imaging equation ensures that the distances between the in-focus object distance z and the extrema are not equal, i.e., z1 z = z z2, thus 6 174 CHAPTER 6 OPTICAL DIFFRACTION AND IMAGING

requiring labels for both: z1 = z + 1 and z2 = z 2. However, if 0 is small, then the concept of longitudinal magni…cation allows simple approximate expressions for the object distances:

0 0 z1 = z + 1 = z + = z + 2 ML M j j T 0 0 z2 = z 2 = z = z 2 ML M j j T The depth of …eld may be written as:

B f/# z = + 2 0 = 2 0 1 2 = 2 2 MT MT

In the di¤raction-limited case, we can substitute the known width of the di¤raction pattern for B0 to obtain: f/# ( f/#)2 z 2 (2:44 f/#) = 4:88 0 = 0 2 2 MT MT In the detector-limited case with pixel dimension b, the depth of …eld is:

b f/# z 2 = 2 MT In words, the depth of …eld in image space is proportional to the wavelength and the square of the f-number of the system, which means that “slower”systems have longer depths of …eld. This is rather obvious from the drawing, because the angle of the extreme rays is smaller for a slow system. Another important metric in imaging is the hyperfocal distance, which is the shortest object distance where the depth of …eld extends to in…nity. The corresponding image distance z0 is the sum of the focal length and 0:

z1 = z + 1 = = z0 0 = feff 1 ) = z0 = feff + 0 ) The hyperfocal object distance satis…es the imaging equation: 1 1 1 + = z z0 feff where z0 = feff + 0, so that:

1 1 1 z = f f + eff eff 0 1 feff + 0 feff = (feff ) (feff + ) 0 (f + ) f = eff 0 eff 0 2 2 feff feff = = feff D feff 2:44 0 f/# 2:44 0 / D