3.3. FoundationsFoundations ofof ScalarScalar DiffractionDiffraction TheoryTheory

Introduction to Fourier Optics, Chapter 3, J. Goodman

Maxwell’s equations Scalar wave equation in the absence of free charge, in a linear, isotropic, homogeneous, G G and nondispersive medium, ∇⋅εE =0 nuPt22∂ (,) GG ∇−2uPt(,) = 0, ∇⋅μH =0 ct22∂ G GG ∂H where, ∇×E =− ∂t ε 1 μ nc== , G ε GG ∂E o oo ∇×H = ∂t με ε If the medium is inhomogeneous with ε(P) (See Appendix.) that depends on position P, G G G G G n2 ∂ 2 E ∇2 E + 2∇ ()E ⋅∇ ln n − = 0 c2 ∂ t 2 HelmholtzHelmholtz EquationEquation

For a monochromatic wave, u (P, t )()= A P cos [2πν t + φ (P )] u (P, t )= {}Re U (P )exp (− j2πν t ) Complex notation

The complex function of position (called a phasor)

U (P )()= A P exp [− jφ (P )]

(∇ 2 + k 2 )U = 0

where, Helmholtz Equation π ν 2π kn== 2 c λ HelmholtzHelmholtz equation equation

Helmholtz sought to synthesize Maxwell's electromagnetic theory of light with the central force theorem. To accomplish this, he formulated an electrodynamic theory of action at a distance in which electric and magnetic forces were propagated instantaneously.

Helmholtz, Hermann von (1821-1894) In 3 dimension,

Cartesian

Cylindrical

Spherical GreenGreen’’ss TheoremTheorem

Let U(P) and G(P) be any two complex-valued functions of position, and let S be a closed surface surrounding a volume V.

If U, G, and their first and second partial derivatives are single- valued and continuous within and on S, then we have G V n ⎛ ∂G ∂U ⎞ S ()U∇2G − G∇2U dυ = ⎜U − G ⎟ ds ∫∫∫ ∫∫⎝ ∂n ∂n ⎠ P ν s

Where ∂ signifies a partial derivative in the outward normal ∂n direction at each point on S. GreenGreen’’ss FunctionFunction

Consider an inhomogeneous linear differential equation of, d 2U dU a ()x + a ()x + a ()x U = V x () 2 dx2 1 dx 0 where V(x) is a driving function and U(x) satisfies a known set of boundary conditions.

If G(x) is the solution when V(x) is replaced by the δ(x-x’), such that

d 2G()x − x' dG(x − x' ) a ()x + a ()x + a ()x G()()x − x' = δ x − x' 2 dx2 1 dx 0

Then, the general solution U(x) can be expressed through a convolution integral,

' ' ' U ()x = ∫ G()x − x V (x )dx Since the system is linear shift-invariant.

“Green’s function” : impulse response of the system KirchhoffKirchhoff’’ss GreenGreen’’ss functionfunction

P 1 Kirchhoff’s Green’s function : r01 A unit amplitude spherical wave expanding about

the point Po (called “free-space Green’s function”)

exp(jkr01 ) V’ : the volume lying between S and S G()P1 = ε r S’ = S + Sε 01

Within the volume V’, the disturbance G satisfies

22 (∇+kG) =0

IntegralIntegral TheoremTheorem ofof HelmholtzHelmholtz andand KirchhoffKirchhoff

⎛ ∂G ∂U ⎞ Green’s Theorem (U∇2G − G∇2U )dυ = ⎜U − G ⎟ ds ∫∫∫ ∫∫⎝ ∂n ∂n ⎠ ν s

P1 Kirchhoff’s Green’s function r01 exp(jkr ) G()P = 01 1 r V’ : the volume lying between S and Sε 01

S’ = S + Sε

22 22 ⎛⎞∂∂GU ∫∫∫()()UGGUd∇−∇υυ =− ∫∫∫ UkGGkUd − ==0 ∫∫ ⎜⎟ U − G ds νν''s '⎝⎠∂∂nn

⎛⎞⎛⎞⎛⎞∂∂GU ∂∂ GU ∂∂ GU ∫∫⎜⎟⎜⎟⎜⎟UGds−=⇒−−=−0 ∫∫ UGdsUGds ∫∫ sSS' ⎝⎠⎝⎠⎝⎠∂∂nnε ∂∂ nn ∂∂ nn Note that for a general point P1 on S’, we have P1 exp jkr()01 r G()P1 = 01 r01 GG ∂GP()1 1 exp( jkr01 ) =−cos(nr ,01 )( jk ) ∂nrr01 01 GG G G where cos( nr , 01 ) represents the cosine of the angle between n and r 01 . G G For a particular case of P1 on Sε , cos( n,r 01 ) = − 1 and

exp( jkε ) ∂GP() exp( jkε ) 1 GP()= 1 =−()jk 1 ε ∂n εε Letting ε -> 0, π πε

⎛⎞∂∂GU2 ⎡⎤exp( jkεε ) ⎛⎞ 1∂UP()o exp( jk ) limU−= G ds lim 4 UP (o ) −− jk = 4 UP (o ) εε→∞∫∫ ⎜⎟ →∞ ⎢⎥εε ⎜⎟ ε S ⎝⎠∂∂nn⎣⎦ ⎝⎠ ∂ n ε 1 ⎧ ∂U ⎡ exp ( jkr )⎤ ∂ ⎡ exp ( jkr )⎤ ⎫ U P = 01 − U 01 ds ()0 ∫∫ ⎨ ⎢ ⎥ ⎢ ⎥ ⎬ 4π S ⎩ ∂n ⎣ r01 ⎦ ∂n ⎣ r01 ⎦ ⎭ Integral theorem of Helmholz and Kirchhoff KirchhoffKirchhoff’’ss FormulationFormulation ofof DiffractionDiffraction (A(A planarplanar screenscreen withwith aa singlesingle aperture)aperture)

Have infinite screen with aperture A nˆ Let the hemisphere (radius R) and screen Σ with aperture comprise the surface (Σ) enclosing P. S r P r’ θ’ θ Radiation from nˆ source, S, arrives Since R →∞ at aperture with amplitude E=0 on Σ. R eikr' Σ E = Eo Also, E = 0 on r' side of screen facing V. KirchhoffKirchhoff’’ss FormulationFormulation ofof DiffractionDiffraction

From the integral theorem of Helmholz and Kirchhoff

1 ⎛⎞∂∂UG exp(jkr01 ) U() P0 =−⎜⎟ G U ds, where G =. 4π ∫∫ ∂∂nn r SS12+ ⎝⎠ 01

Kirchhoff boundary conditions S1 - On ∑ , U and ∂U/∂n are as if there were no screen. S = S1 +S2 - On S1 except Σ, U = 0 and ∂U/∂n = 0.

Σ : aperture FresnelFresnel--KirchhoffKirchhoff’’ss DiffractionDiffraction FormulaFormula (I)(I)

∂G (P ) G G ⎛ 1 ⎞ exp ( jkr ) 1 ⎜ ⎟ 01 = cos ()n, r01 ⎜ jk − ⎟ ∂n ⎝ r01 ⎠ r01

G G exp jkr ⎛ 1 ⎞ ()01 ⎜ ⎟ ≈ jk cos ()n , r01 ⎜ for r01 >> λ → k >> ⎟ r01 ⎝ r01 ⎠

For an illumination consisting of a single point source at P2 :

Aexp(jkr21 ) U ()P1 = r21 G G G G A exp[jk(r + r )]⎡cos(n,r )− cos(n,r )⎤ U ()P = 21 01 01 21 ds 0 ∫∫ ⎢ ⎥ jλ ∑ r21r01 ⎣ 2 ⎦ FresnelFresnel--KirchhoffKirchhoff’’ss DiffractionDiffraction FormulaFormula (II)(II)

Huygens-Fresnel exp(jkr ) U ()P = U′ P ()01 ds 0 ∫∫ 1 ∑ r01 G G G G 1 ⎡ A exp ( jkr )⎤ ⎡ cos (n, r )− cos (n, r )⎤ U ′()P = 21 01 21 1 ⎢ ⎥ ⎢ ⎥ jλ ⎣ r21 ⎦ ⎣ 2 ⎦

restricted to the case of an aperture illumination consisting of a single expanding spherical wave. Kirchhoff’s boundary conditions are inconsistent! : Potential theory says that “If 2-D potential function and it normal derivative vanish together along any finite curve segment, then the potential function must vanish over the entire plane”. Rayleigh-Sommerfeld theory ‹ the scalar theory holds. ‹ Both U and G satisfy the homogeneous scalar wave equation. ‹The Sommerfeld radiation condition is satisfied. SommerfeldSommerfeld radiationradiation conditioncondition

1 ⎛⎞∂∂UG UP()0 = ∫∫ ⎜⎟ G−=→∞ U ds 0 as R 4π S 2 ⎝⎠∂∂nn

On the surface of S2, for large R, exp(jkR ) G = R ∂GjkR⎛⎞1 exp( ) S =−⎜⎟jk ≈ jkG 1 ∂nRR⎝⎠ ⎡⎤⎛⎞∂∂UU GjkGUdsGjkURd−=−() 2 ω ∫∫⎢⎥ ∫Ω ⎜⎟ S 2 ⎣⎦⎝⎠∂∂nn ⎛⎞∂U ∴ limRjkU⎜⎟−= 0 R→∞ ⎝⎠∂n

It is satisfied if U vanishes at least as fast as a diverging spherical wave. FirstFirst RayleighRayleigh--SommerfeldSommerfeld SolutionSolution (When(When UU atat ΣΣ isis known)known)

1 ⎛ ∂U ∂G ⎞ U ()P = ⎜ G −U ⎟ds 0 4π ∫∫ ∂n ∂n S1 ⎝ ⎠ ~ Suppose two point sources at P0 and P0 (mirror image): ~ exp(jkr01 ) exp(jkr01 ) G− ()P1 = − ~ r01 r01

−1 ∂G ∂G (P ) ∂G(P ) U ()P = U − ds − 1 1 I 0 ∫∫ = 2 4π ∑ ∂n ∂n ∂n

−1 ∂G U ()P = U ds I 0 ∫∫ 2π ∑ ∂n SecondSecond RayleighRayleigh--SommerfeldSommerfeld SolutionSolution

(When(When UU’’n atat ΣΣ isis known)known)

~ exp ( jkr 01 ) exp ( jk r01 ) G + ()P1 = + ~ r01 r01 1 ∂ U U P = G ds II ()0 ∫∫ + 4π ∑ ∂ n

G + = 2G

1 ∂ U U P = Gds II ()0 ∫∫ 2π ∑ ∂ n

RayleighRayleigh--SommerfeldSommerfeld DiffractionDiffraction FormulaFormula

The 1st and 2nd R_S solutions are,

1 exp(jkr ) G G U ()P = U ()P 01 cos()n,r ds I 0 ∫∫ 1 01 jλ ∑ r01 1 ∂U (P )exp(jkr ) U ()P = 1 01 ds II 0 ∫∫ 2π ∑ ∂n r01

exp(jkr21 ) For the case of a spherical wave illumination, U ()P1 = A r21 A exp[jk(r + r )] G G U ()P = 21 01 cos()n,r ds I 0 ∫∫ 01 jλ ∑ r21r01 A exp[jk(r + r )] G G U ()P = − 21 01 cos()n,r ds II 0 ∫∫ 21 jλ ∑ r21r01 ComparisonComparison (I)(I)

1 ⎛ ∂U ∂G ⎞ U ()P = G −U K ds F – K : 0 ∫∫⎜ K ⎟ 4π ∑ ⎝ ∂n ∂n ⎠

1 ∂G st U ()P = − U K ds 1 R-S : 1 0 ∫∫ 2π ∑ ∂n

1 ∂U nd U ()P = G ds 2 R-S : II 0 ∫∫ K 2π ∑ ∂n

Wow! Surprising! The Kirchhoff solution is the arithmetic average of the two Rayleigh-Sommerfeld solutions! ComparisonComparison (II)(II) λ For a spherical wave illumination, ψ A exp [jk (r21 + r01 )] U ()P0 = ∫∫ ds , ψ : obliquity factor j ∑ r21 r01

⎧ 1 G G G G []cos ()n , r − cos ()n , r Kirchhoff theory ⎪ 2 01 21 ⎪ ⎪ G G First Rayleigh-Sommerfeld solution ψ = ⎨ cos ()n , r 01 ⎪ G G ⎪ − cos n , r Second Rayleigh-Sommerfeld solution ⎪ ()21 ⎩

For a normal plane wave illumination, (that means r21 Æ infinite) ⎧ 1 []1 + cos Kirchhoff theory ψ ⎪ 2 ⎪ θ ⎪ = ⎨ cos θ First Rayleigh-Sommerfeld solution ⎪ ⎪1 Second Rayleigh-Sommerfeld solution ⎪ ⎩ Again, remember …… After the Huygens-Fresnel principle ……

Fresnel’s shortcomings : He did not mention the existence of backward secondary wavelets, however, there also would be a reverse wave traveling back toward the source. He introduce a quantity of the obliquity factor, but he did little more than conjecture about this kind.

Gustav Kirchhoff : Fresnel-Kirhhoff diffraction theory A more rigorous theory based directly on the solution of the differential wave equation. He, although a contemporary of Maxwell, employed the older elastic-solid theory of light. He found K(χ) = (1 + cosθ )/2. K(0) = 1 in the forward direction, K(π) = 0 with the back wave.

Arnold Johannes Wilhelm Sommerfeld : Rayleigh-Sommerfeld diffraction theory A very rigorous solution of partial differential wave equation. The first solution utilizing the electromagnetic theory of light. Again, let’s compare the Kirchhof’s and Sommerfeld …… HuygensHuygens--FresnelFresnel PrinciplePrinciple revisedrevised byby 11st RR –– SS solutionsolution λ 1 exp(jkr ) U ()P = U ()P 01 cosθ ds 0 ∫∫ 1 j ∑ r01 1 amplitude ∝ λ ∝ ν 1 phase : (lead the incident phase by 90o ) j

U (P )()= h P , P U (P )ds 0 ∫∫ 0 1 1 ∑ λ 1 exp(jkr01 ) h()P0 , P1 = cosθ j r01 Each secondary wavelet has a directivity pattern cosθ GeneralizationGeneralization toto NonmonochromaticNonmonochromatic WavesWaves

At the observation point ν πν ()∞ ( ) ( ) u P0 ,t = ∫−∞U P0 , exp j2 t dν

At an aperture point ν πν ()∞ ( ) ( ) u P1,t = ∫−∞U P1, exp j2 t dν πυ G G cos(n,r ) d ⎛ r ⎞ u()P ,t = 01 u P ,t − 01 ds 0 ∫∫ ⎜ 1 ⎟ ∑ 2 r01 dt ⎝ υ ⎠ AngularAngular SpectrumSpectrum

∞ A()f X , fY ;0 = ∫∫U ()x, y,0 exp[]− j2π ()f X x + fY y dxdy −∞ π α λ β α 2 2π G G j ()x+ y j γz λ P()x, y, z = exp()jk ⋅ r = e e λ β λ G 2π α whereγ , k = ( , β , γ ) α β λ λ λ λ 2 2 = f X = f Y = 1 − ()()f X − λ f Y

π α ⎛ ⎞ ∞ ⎡ ⎛ β ⎞⎤ A⎜ , ;0⎟ = ∫∫U ()x, y,0 exp⎢− j2 ⎜ λ x + y⎟⎥dxdy ⎝ ⎠ −∞ ⎣ ⎝ λ ⎠⎦

Angular spectrum α β λ λPropagationPropagation ofof thethe AngularAngular SpectrumSpectrum α β π α ⎛ ⎞ ∞ λ ⎡ ⎛ β ⎞⎤ A , ; z = U ()xλ, y, z exp − j2 λ x + y dxdy ⎜ ⎟ ∫∫ ⎢ π α ⎜ ⎟⎥ ⎝ ⎠ −∞ ⎣ ⎝ β λ ⎠⎦ λ α ∞ ⎛ ⎞ ⎡ ⎛ λ ⎞⎤ β U ()x, y, z = ∫∫A⎜ , ; z⎟exp⎢ j2 ⎜ x + y⎟⎥ d λ d −∞ ⎝ ⎠ ⎣ ⎝ ⎠⎦ λ α β αλ U must satisfyλβ theπ Helmholtz equation, ∇ 2U + k 2U = 0 λ λ α 2 λ 2 α β α d ⎛ ⎞ ⎛ 2 ⎞ 2 β 2 ⎛ β ⎞ 2 A⎜ , ; z⎟ + ⎜ ⎟ []1− − A⎜ λ , ; z⎟ = 0 dz ⎝ ⎠ ⎝ ⎠ λ ⎝ λ ⎠ λ π ⎛ ⎞ ⎛ ⎞ ⎛ 2λ α ⎞ ∴ A⎜ , ; z⎟ = A⎜ , ;0⎟ exp⎜ j 1− 2 − β 2 z⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ α β λ PropagationPropagation ofof thethe AngularAngular SpectrumSpectrum λ α β λ λ π ⎛ ⎞ ⎛ ⎞ ⎛ 2λ α ⎞ A⎜ , ; z⎟ = A⎜ , ;0⎟exp⎜ j 1− 2 − β 2 z⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ α 2 + β 2 < 0 : each plane - wave component propagates at a different angle α α 2 + β 2 > 0 : evanescentβ waves λ λ π α ∞ ⎛ β ⎞ ⎛ 2λ α ⎞ () 2 2 U x, y, z = ∫∫A⎜ , ;0⎟exp⎜ jπ α 1− − β z⎟ −∞ ⎝ ⎠ ⎝ β ⎠ λ α λ 2 2 ⎡ ⎛ ⎞⎤ β × circ( + )exp⎢ j2 ⎜ x + y⎟⎥d λ d ⎣ ⎝ ⎠⎦ λ EffectEffect ofof ApertureAperture α U x,β()y;0 t ()x, y = t : amplitude transmittance function of the aperture A λ ()α Ui x, yλ;0 β λ α ⎛ ⎞ ⎛ λ ⎞ ⎛ β ⎞ At ⎜ , ⎟ = Ai ⎜ , ⎟ ⊗T⎜ λ , ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ λ ⎠ α α β • For the caseβ of a unit amplitude plane wave incidence, λ δ α λ λλ δ α β ⎛ ⎞ ⎛ λ ⎞ α Ai ⎜ , ⎟ = ⎜ λ , λ ⎟ ⎝ ⎠ ⎝ λ ⎠ β λ α ⎛ ⎞ ⎛ ⎞ ⎛ λ ⎞ ⎛ β ⎞ At ⎜ , ⎟ = ⎜ , ⎟ ⊗T⎜ , ⎟ = T⎜ λ , ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ λ ⎠ PropagationPropagation asas aa LinearLinear SpatialSpatial FilterFilter

λ A()()()()f , f ; z = A f , πf ;0 circ⎜⎛ f 2 + λf 2 ⎟⎞ X Y X Y ⎝ X Y ⎠ λ λ ⎡ z 2 2 ⎤ × exp⎢ j2 1− ()()f X − λfY ⎥ ⎣ ⎦

Transfer function of waveπ propagation phenomenon in free space λ λ ⎧ z 1 ⎡ 2 2 ⎤ f 2 + f 2 〈 ⎪exp⎢ j2 1− ()()f X − λfY ⎥ X Y ⎪ ⎣ ⎦ λ H ()f X , f y = ⎨ ⎪ ⎩⎪0 otherwise α β λ λ δ BPMBPM (Beam(Beamα βPropagationPropagation Method)Method) λ λ δ δ n(z) = average[n(x, y, z)] on (x,y) plane α β πα λ λ δ β ⎛ ⎞ ⎛ ⎞ ⎛ 2 2 λ 2 π 2 ⎞ Ah ⎜ , ; z + z⎟ = A⎜ , ; z ⎟exp⎜ j z n(πz) ko − (2 ) − (2 ) ⎟ ⎝ ⎠ ⎝ ⎠ ⎜ λ ⎟ ⎝ λ α ⎠ β α ∞ ⎛ ⎞ ⎛ 2 ⎞ ⎛ ⎞ ⎛β ⎞ U ()x, y, z + z = A , ; z + z exp j ( x + y) d λ d h ∫∫ h ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ −∞ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ λ ⎠

U (x, y, z +δz)= U h (x, y, z +δz)exp[− jδn(z)koδz]

Invalid for wide angle propagation -> WPM (wave propagation method) AppendixAppendix From Maxiwell’s equations to wave equations