Diffraction theory Scalar diffraction theory History of diffraction theory From vector to a scalar theory Some mathematical Preliminaries Hemholtz equation Green’s theorem The Kirchhoff formulation of diffraction by a planar screen The integral theorem of Helmholtz and Kirchhoff The Kirchhoff boundary conditions Fresnel-Kirchhoff diffraction formula The Rayleigh-Sommerfeld (R-S) formulation of diffraction Alternative Green’s function The R-S diffraction Formula Comparison between Kirchhoff and R-S theories GlititGeneralization to nonmonoc htihromatic waves Angular spectrum of a plane wave Physical interpretation of the angular spectrum Propagation of the angular spectrum Effects of a diffraction appgperture on the angular spectrum Propagation phenomenon as a linear spatial filter

1 Diffraction

• Diffraction is a phenomena in the realm of physical optics. Applicable to all waves such as acoustic and EM waves. • Diffraction is a limiting factor on data processing and imaging system performance. So we need to understand it to design better systems. • Diffraction should not be mistaken with – Refraction: change of direction of propagation of light due to a change in index of refraction of the environment – Penumbra: finite extend of a source causes the light transmitted from an aperture to spread away from it. There is no bending of light involved in Penumbra effect. • Diffraction (Sommerfeld): any deviation of light rays from rectilinear that cannot be interpreted as reflection or refraction. • Diffraction is caused by confinement of the lateral extend of a wave (obstruction of the wavefront) and its effects are most pronounced when size of the confinement is comparable to the wavelength of the light.

2 History of diffraction theory • 1665 Grimaldi reported diffraction for the first time. • 1678 Huygens attempted to explain the phenomenon – Each point on the wavefront of a disturbance is considered to be a new source of a “secondary” spherical disturbance. Then the wavefront at later instances can be found by constructing the “envelope” of the secondary wavelet. • 1700s P rogress on wave th eory was suppressed b y th e f act th at N ewt on f avored th e corpuscular theory of light (geometrical optics). • 1804 Thomas Young introduced the concept of interference to the wave theory of light (production of darkness from light). • 1818 Augustin Jean Fresnel used the wavelets from Huygens theory and Young’ s interference theory letting the wavelets interfere mutually to calculate distribution of light in diffraction patterns with excellent accuracy. • 1860 Maxwell identified light as electromagnetic field. • 1882 Gustav Kirchhoff put the Fresnel and Maxwell ’ s ideas together . He made two assumptions about the boundary values of the light incident on surface of an obstacle that were not absolutely correct but an approximation and constructed a theory that exhibited excellent agreement with experimental results. He concluded – The amplitudes and phases ascribed to the secondary sources of Huygens wavelets are log ica l consequences o f the wave na ture o f the lig ht. • 1892 Poincare; 1894 Sommerfeld proved that the boundary values set by Kirchhoff are inconsistent with one another. So Kirchhoff’s formulation of Huygens-Fresnel principle is regarded as the first approximation although under most conditions it yields excellent results. • 1896 Sommerfeld modified the Kirchhoff’s theory using theory of Green’s function. The result is Rayleigh-Sommerfeld diffraction theory. • 1923 Kottler: first satisfactory generalization of the vectorial diffraction theory. 3 From vector to a scalar theory I

• In all of these theories light is treated as a scalar phenomenon. • At boundaries the various components of the electric and magnetic fie lds are coup le d throug h Maxwe ll equa tions an d canno t be trea te d independently. • We stay away from those situations when using scalar theory. • Scalar theory yields correct values under two conditions: – The diffracting aperture must be large compared with a wavelength. – The diffracting fields must not be observed too close to the aperture. • Our treatment is not ggpyood for some optical systems such as diffraction from – high-resolution gratings – Small pits on optical recording media • Read Goodman 3.2 From a Vector to a Scalar Theory

4 From vector to a scalar theory II FltFor electromagneti c waves propagati tiing inmedia w ith the following properties, an scalar wave equation is obeyed by all components of the field vectors EEEHHHxyz,,, x , y , z . nuPt22∂ (,) ∇−2uPt(,) = 0 ct22∂ Depth of uPt(,),,. is any of the scalar field components xyz at time t penetration in the Properties: media is few wavelengths. Not much effect on linear; if uPt12(,) and uPt (,) are solutions to the wave euation, then total wavefront αβuPt12(,)+ uPt (,) is a solution, passing through the aperture isotropic; properties are independent of direction of polarization of the wave , homogeneous; permitivity is constant throughout the region of propagation, nondispersive; permitivity is independent of frequency over the region of propagation, nonmagnetic; magneti c permeabilit y is equaltl to μ0 At the boundaries, the above criteria are not met and coupling betwen electric and magnetic components of the EM wave happens. We can use the scalar theory if the boundaries are small portion of the total area through which the wave is passing.

5 The Helmholtzφ equation For a monochromatic wave uPt(,) the scalar field can be written as uPt(,)=−= AP ()cos[2πν φ t ()]Re{() P UPe− jt2πν } where Up()= Ape ()jP() space dependent part of the field e− jt2πν time dependent part of the field Pxyz can be any of the space coordinates ,, or Substituting the scalar field in scalar wave equation nuPt22∂ (,) ∇−2uPt(,) = 0 ct22∂ nUPe22∂ ()− jt 2πν ∇−22UPe()− jtπν = 0 ct22∂ 22 22 ⎛⎞2 nj(2)− πν − jt2πν n (2πνπ ) 2 ⎜⎟∇−UP()2 UP () e ===0 with wavenumber k 2 ⎝⎠c c λ ()()0∇+22kUP = time-independent Helmholtz equation . Complex amplitude of any monochromatic w ave propagating in vacuum or in homogeneous dielectric media has to obey the Helmholtz equation.

6 Gauss’s Theorem

Gauss's theorem: connecting surface integral and volume integral of a vector total outflow of flux from the volume V ∇=iiUUsdV d ∫∫VS net outflow of flux per total outflow of flux unit volume from the surface S Green' s theorem is a corollary of Gauss's theorem

7 Green’s Theorem: a mathematical tool If UG and are two scalar functions ∇∇=∇∇+∇∇iii()UG U G()() U G - ∇∇=∇∇+∇∇iii()GU G U( G) ( U) ∇i()UGGU ∇−∇ =∇ U22 GG −∇ U Assuming UG,, their first and second deriva tives are continuous over the volume V and on the surface SV enclosing the ∇∇−∇i()()U G G U dV = U ∇−∇22 G G U dV ∫∫∫VV ∫∫∫ Using the Gauss's theorem convert the volume integral on LHS ()()UGGUd∇∇∇− ∇ i s = U∇ 22 GG− ∇ UdV ∫∫SV ∫∫∫ If we take the gradiant in the outward normal direction n the LHS can be written in a scalar form since ns|| at everyyp point. ∂∂GU ()()U−=∇−∇← G ds U22 G G U dV Green's theorem ∫∫SV∂∂nn ∫∫∫ 8 Physical meaning of the Green’s function I

Imagine an inhomogeneous linear differential euation dU2 () x dUx () aaaUxVx++=() () 210dx2 dx Vx() is a driving force Ux() is the solution for a known set of boundary conditions (BC) Gx() is a solution to the equation with Vx()→−δ ( x x ') the impulse driving force andthesameBCsand the same BCs. Gx() is an impulse response and we can expand Ux () in terms of Gs Ux()= ∫ VxGxxdx (' ))( (-' ) ' Gx() known as the Green's function of the problem. Gx() may be regarded as an auxiliary function chosen cleverly to solve our problem.

9 Physical meaning of the Green’s function II

Imagine an oscillator dU2 () x aaUxVx+=() () 20dx2 Vx() is a driving force Ux() is the solution for a known set of boun dary conditions (BC) Gx() is impulse response Ux()= VxGxxdx (')( ) (-') ' ∫x' The solution is convolution of the driving force with the impulse respons of the system. In case of diffraction application of Green's theorem will yeild different variations of the diffraction theory based on the choice of Green's function.

10 Application of Green’s Theorem in scalar diffraction theory

Goal: calculation of the complex disterbance U at an observation point

iiG'Thin space, P0 , using Green's Theorem. ∂∂GG ()()U−=∇−∇ G ds U22 G G U dV ∫∫SV∂∂nn ∫∫∫ Green's theorem is the prime foundation of the scalar diffraction theory. To apply it to the diffraction problem we need to have a proper choice of 1) an aux iliary func tion G (Green'f's functi ti)on) P1 2) a close surface S V n

P0 is an arbitraryyp point of observation PS is an arbitrary point on the surface ro1 1 . S P0 We want solution of the wave equation UP()0

at P0 in terms of the value of the solution and its derivatives on the surface S . 11 The integral theorem of Helmholtz & Kirchhoff 1) Choice of Green ' s function: a unit-am plitude spherical wave expanding e jkr01 about point PG0 (impulse). has to be a solution of the wave equation. At PGP11 : ()= r01

2))g Treating the discontinuit y at P0 byyg isolating it with Sε P1 4 3) New surface & volume: SSSVV''=+ ; =−πε 3 ε 3 V’ 4) Use Green's theorm with Green's function of r01 e jkr01 GP()=∇+= and Helmholtz equation ( 2 k2 ) U 0 ε 1 r S 01 P . n 0 Note both GU and are the solutions of the same wave euation. After all G is the impulse response and Sε n

UUP is the disturbance. We want to find ()0 or field after the aperture. At the linit of ε → 0 we get (follow from Goodman page 41):

1 ⎧⎫∂∂Uejkr01 e jkr 01 UP()= - U ds 0 ∫∫S ⎨⎬ 4π ⎩⎭∂∂nr01 nr 01 This result is known as the integral thorem of Helmholtz and Kirchhoff. It has important role in

development of the scaler theory of diffraction. UP()0 , the field at point P0 is expressed in terms of the "boundary values" of the wave on any closed surface surrounding that point. 12 Fresnel-Kirchhoff diffraction formula I Problem: diffraction of light by an aperture in an infinite opaque screen.

The field UP at 0 behind the aperture is to be calculated.

r01 is the distance from aperture to observ ation point .

S2 Assumptions: rkr01>>λ and >>1/ 01 Wave Choice of S : a plane surface plus a impinging S1 spherical cap SSS=+ R 12 ∑ . jkr e 01 n P0 Choice of Green's function: G = r01 P1 r01 We apply the Helmholtz-Kirchhoff integral theorem 1 ⎛⎞∂∂UG UP( 0 )=−⎜⎟ G U ds to find UP()0 4π ∫∫ ∂∂nn SS12+ ⎝⎠ Somefeld radiation condition: if the disturbance U vanishes at least as fast as a diverging spherical wave then:

as RSUP→∞ contribution of 20 to () vanishes.

13 Kirchhoff’s Boundary conditions

The screen is opaque and the aperture is shown by Σ 1) Across the surface Σ, the field and its derivatives are exactly the same as they would be in the absence of the screen.

2) Over the portion of S1 that lies in the geometrical shadow of the screen the filed and its derivative are identically zero. 1 ⎛⎞∂∂UG UP=− G U ds ()0 ∫∫ ⎜⎟ 4π Σ ⎝⎠∂nn∂ S Wave 2 Condition 1 is not exactly true since close impinging S1 to the boundaries the field is disturbed. R ∑ Condition 2 is not true since the . n P0 r01 shadow is never perfect and some P1 field will extend behind the walls . On these portions of S : For Σ>>λ this is an OK approximation. 1 ∂∂UU2 U ==2 =0 ∂∂nn 14 Fresnel-Kirchhoff diffraction formula II S2 S Withe above assumptions and kr>>1/ we arrive at 1 01 R 1 ⎧⎫eUjkr01 ⎡⎤∂ ∑ . UP( 001)=−⎨⎬jkUcos(, n r ) ds n P ∫∫∑ ⎢⎥ r 0 4π ⎩⎭rn01 ⎣⎦∂ 01 P1 If the aperture is illuminated by a single spherical wave Ae jkr21 UP()1 = located at point P2 , at a distance r21 from P 1 . r21

If r21 >> λ we can show that (problem 3.3)

jkr()21+ r 01 Ae⎧⎫⎡cos(nr ,01 )− cos( nr , 21 ) ⎤ UP()0 = ⎨⎬⎢⎥ ds jrrλ ∫∫∑ ⎣⎦2 ⎩⎭21 01 The Fresnel-Kirchhoff diffraction formula that hldholds on ly for a s inglitle point source illitiillumination . r21 The Fresnel-Kirchhoff diffraction formula P2 P0 r01 n P1 is symmetrical with respect to P02 and P .

A point source at P0 will produce the same effect at P2. This result is known as: reciprocity theorem of Helmholtz 15 Huygens’ wavelets

Huygens- Fresnel theory: the light distur bance at a point P0 arises from the superposition of secondary waves that produced from a surface situated between this point and the light source. If we rewrite Ae⎧⎫jk() r21+ r 01 ⎡⎤cos(nr , )− cos( nr , ) UP()= 01 21 ds 0 ∫∫∑ ⎨⎬⎢⎥ jrrλ ⎩⎭21 01 ⎣⎦2 e jkr01 UP()01= UP'( ) ds where ∫∫∑ r01 illuminating wavefront ⎡⎤Observation angle Illumination angle jkr21 1 ⎡⎤Ae ⎢⎥cos(nr ,01 )− cos( nr , 21) UP'(1 ) = ⎢⎥⎢⎥ jrλ 2 ⎣⎦21 ⎢⎥ ⎣⎦

Seems like UP()0 arises from sum of infinite fictitious sources with amplitudes and phases expressed by UP'(1 ). We used point source to get this result but it is possible to generalize this result for any illlumination by using Rayleigh-Sommerfeld theory. 16 The Rayleigh-Sommerfeld Formulation of diffraction Potential theory: If a two-dimensional potential function and its normal derivaive vanish together along any finite curve segment , then the potential function must vanish over the entire plane. This is also true for solution of a three-dimensional wave equation.

1) The Kirchhoff boundary conitions suggests that the diffracted field mustbt be zero everywh ere b ehi hidthnd the aper ture. NttNot true. 2) Also close to aperture the theory fails to produce the observed diffraction field. Inconsistencies of the Kirchhoff theory were removed by Sommerfeld. He elliminated the need of imposing boundary values on the disturbance and its derivative simultaneously.

17 An alternative Green’s function I

Observe d fie ld s treng th in terms o f the inc iden t fie ld an d its norma l der iva tives: 1 ⎛⎞∂∂UG UP()=− G U ds (Fresnel-Kirchhoff diffraction formula) 0 ∫∫S ⎜⎟ 2π 1 ⎝⎠∂∂nn

Conditions for validity: 1) The scalar theory holds 2) Both UG and are solutions of the homogeneous scalar wave equation 3) The sommerfeld radiation condition holds i.e. if the disturbance U vanishes at least as fast as a diverging spherical wave then: as RSUP→∞ contribution of 20 to () vanishes.

If the Green's function of Kirchhoff theory was modified so that either G or ∂∂Gn/, vanished over entire surface S1 then there is no need to impose boundary condtions on both UUn and ∂ /.∂

18 An alternative Green’s function II SflddthtG'ftiSomerfeld argued that one Green's function tha t mee ts these crit eri a i s composed of two identical point sourcers at two sides of the aperture, mirror image of each other, oscillating with a 1800 phase difference:

eejkr01 jkr 01 GP()=− − 1 rr 01 01 Now G = 0 on the plane of aperture. Kirchhoff's BC may be applied only on U Σ 1 ⎛⎞∂∂UG 1 UP()=− G U ds 0 ∫∫S ⎜⎟ 2π 1 ⎝⎠∂∂nn S =Σ +Σ+Σ . r . 11 2 01 Σ P P1 P0 1 ⎛⎞∂∂UG 0 r we need UP()=−= G U ds 0 n 01 0 ∫∫Σ +Σ ⎜⎟ 2π 12⎝⎠∂nn∂ GU|0==ΣΣ so if we require only 0 on and Σ S1 12 2 ∂U then does not need to be zero to make UP()| ∂n 0 Σ+Σ12 With the new BC on the U only there is no conflict with the potential theorem. 19 The Rayleigh-Sommerfeld diffraction Formula

With the Green's function G− eejkr01 jkr 01 GP()=− − 1 rr 01 01 the UP( 0 ) takes the form 1 e jkr01 UP()= UP() cos(,) nrds or I 0101∫∫S 1 jrλ 01

Assuming r01 >> λ Now applying the Kirchhoff BC only on UUn and not on ∂∂/ we get 1 e jkr01 UP()= UP() cos(,) nrds I 0101∫∫Σ jrλ 01 And U will not vanish on the other side of the aperture .

20 Rayleigh-Sommerfeld Diffraction formula

With the Green ' s function G+ two sources oscillating in phase wwithith each other eejkr01 jkr 01 GP()=+ the UP() takes the form + 10rr 01 01 1 ∂UP()e jkr01 UP()= 1 ds II 0 ∫∫Σ 2π ∂nr01 Now for the spacial case illumination: . r . 01 P jkr21 P 0 e 0 r01 a diverging spherical wave from point PUP21 : ()= A n P1 r21 We apply the Kirchhoff BC only on UUn and not on ∂∂/ and using G− we get Aejk() r21+ r 01 UPI ()001= cos( nrds , ) ∫∫Σ jrrλ 21 01 and and G+ gives Aejk() r21+ r 01 U II ()Pnrds021= cos( , ) ∫∫Σ jrrλ 21 01 0 Where the angle between nr and 21 is greater than 90 .

This is Rayleigh-Sommerfeld Diffraction formula where we assumed r21 >> λ 21 Comparison of the Kirchhoff and Rayleigh- Sommerfeld (R-S) theorem eejkr01 jkr 01 ee jkr 01 jkr 01 e jkr 01 GP()=− ; GP () =+ , G () P = −+11rr rrK 1 r 0101 01 01 01 GfGreen functions of fS the Sommerf ffeld formulation Green function of the Kirchhoff formulation ∂G On the surface Σ== we can show that GG22 and − G + KK∂n 1 ⎛⎞∂U ∂G For the Kirchhoff theory: UP=− G UK ds ()0 ∫∫ ⎜⎟K 4π Σ ⎝⎠∂∂nn

−1 ∂GK For the R-S theory: UPI ( 0 )= U ds 2π ∫∫Σ ∂n 1 ∂U UPII ()0 = GdsK 2π ∫∫Σ ∂n We can see that UP()+ U () P UP()= III00 0 2 Summary: the Kirchhoff solution is the arithmatic average of the two

Rayleigh-Sommerfeld solutions. 22 Comparison of the Kirchhoff and R-S theory Kirchhoff theory: Obliquity factor ψ jk() r21+ r 01 jk () r21+ r 01 Ae⎡⎤cos(nr ,01 )− cos( nr , 21 ) Ae UP()0 ==⎢⎥ dsψ ds jjjλ ∫∫∑∑rr ⎣⎦2 jλ ∫∫ rr 21 01λλ 21 01 Obliquity factor ψ jk() r+ r jk() r+ r Ae21 01 Aeλ 21 01 R-S theory: UP()= cos( nr , ) ds= ψ ds I 001∫∫ΣΣ ∫∫ jrrλ 21 01 jrr21 01 Obliquity factor ψ jkrr()++ jkrr () Ae21 01 Ae 21 01 U() P==cos( n , r ) dsψ ds II 021∫∫ΣΣ ∫∫ jrr21 01 jrr 21 01

r . 21 . . r . 01 P2 P0 P r P0 0 01 r01 n P1 n P1

23 Comparison of the Kirchhoff and R-S theory Obliquity factor of both Kirchhoff and R-S theory ⎧ 1 ⎫ [cos(nr , )− cos( nr , )] Kirchhof theory ⎪ 2 01 21 ⎪ ⎪ ⎪ ψ =⎨ cos(nr , 01 ) First R-S solution ⎬ ⎪⎪ r −cos(nr ,21 ) Second R-S solution . 21 r . 01 ⎪⎪P P1 P0 ⎩⎭2 θ When a point source is at a very far distance n ⎧⎫1 [1+ cosθ ] Kirchhof theory ⎪⎪2 ⎪⎪ ψ =⎨⎬cos(θ ) First R-S solution ⎪⎪1 Second R -S solutio n ⎪⎪ ⎩⎭ In summary: for small angles all three solutions are identical When observation point or illumination source are r r . 01 P 01 . far away, the angles are small. P0 1 θ P0 R-S solution requires the diffracting screens be planar. n Kichhoff solution is not limitted to planar surfaces. For most applications both are OK. We will use the first R-S solution for simplicity. 24 Huygens-Fresnel Principle

25 Generalization to Non-monochromatic waves I

We generalize the R-S's first solution to nonmonochromatic waves (chromatic?) Monochromatic time function of the disturbance: u()R()(Pt, )R= e{ U() Pe− jt2πν }

Time dependent chromatic functions: uPt(,)1 at the aperture,(,) uPt0 observation point uPt(,),(,)10 uPt in terms of their Fourier transforms. Let's change the variable νν '= - ∞∞ uPt(,)==− UP (,) πννν ejt22'πν d UP ν (, ') ν e− j t d ' 11∫∫ 1 −∞ −∞ ∞ ∞ uPt(,)== UP (,)ν ejt2πν dν UP (,')−ν ed− jt2'πν ν ' 00∫ ∫ 0 −∞ −∞ complex amplitudes monochromatic of the disturbance elementary at frequency ν ' function of freuency '

Where UP()( 11,ν )(==FF{u () P,t)(} and UP () 0,ν )({uν ()P 0,t)}

uPt(,)0 uPt(,)1 r . 01 . We see that the cromatic function is sum P P1 P 0 r01 θ 0 of the monochromatic functions over n different frequencies.

26 Generalization to Non-monochromatic waves II ∞∞ uPt()( , )(== UP ( πν,νν ))( ejt22'πν d UP ν (,− ' ν) e− j t d ' 11∫∫ 1 −∞ −∞ ∞∞ νπν jt22'πν − j t uPt(,)0 uPt(,)==− UP (,)νν e d ν UP (, ν ') e d ' uPt(,)1 r 00∫∫ 0 . 01 . −∞ −∞ ν eltlementary P P1 P complex amplitude 0 r01 θ 0 of the disturbance function of n at frequency ' freuency ν' Now we introduce the R-S first solution for the diffracted field

UP()1 Obliquity factor ψ jkr jkr jkr 11Ae21 e01 e 01 UI () P001101==cos( nr , ) ds UP ( ) cos( nr , ) ds ∫∫ΣΣ ∫∫ jrrλλ21 01 j r 01 ν ' e jr2'πν 01 /V UP(),'−=−νν j UP (,') − cos(nr , ) ds where V= c / n 01∫∫Σ 01 V r01 This is one frequency component. Summing over all of them we get:

Elementary Complex amplitude at each frequency function at that frequency jrV2'/πν ∞ ⎡⎤ν ' e 01 uPt(),=− j UP ( , − ') cos( nr , ) ds ed− jt2'πν ν ' 0101∫−∞ ⎢⎥∫∫Σ ⎣⎦Vr01 ν π r01 cos(nr , ) ∞ −−jt2'()πν uPt(),2'(,')'=−−01 ν j ν πν UP eV d ds 01∫∫Σ−∞ ∫ 2 Vr01 27 Generalization to Non-monochromatic waves III Next we want to relate the disturbance at the observation point

π r01 cos(nr , ) ∞ −−jt2'()πν uPt(),2'(,')'=−−01 ν j ν πν UP eV d ds 01∫∫Σ−∞ ∫ 2 Vr01 u()(P ,t) tthditbto the disturbance at tth the apert ure l ocat ion uPt(,) 0 1 r01 ∞ . P . P0 r 1 θ P0 uPt(,)=− UP (,νν ') e− jt2'πν d ' we use the identity 01 11∫ n −∞ dd∞ uPt(,)=− UP (,νν ') e− jt2'πν d ' 11∫ dt dt −∞ d ∞ uPt(,)=− ν j 2 ν πν '(, UP − ') e− jt2'πν d ' 11∫ dt −∞ cos(nr , ) d r uPt( , ) =−01uPt()( , 01 ) ds 0 ∫∫Σ 1 2πVr01 dt V The wave disturbance Incident wave at at P0 is linearly proportional Over all angles to the time derivative of the the "retarded" time or the time that the disturbance at eah point P1 on the aperture wave was generated In summary the results of the diffration theory for the monochromatic waves is applicable to the more general case of the chromatic waves. 28 The angular spectrum of plane a wave I Next we want to formulate the diffraction theory in a framework of linear, invariant systems. Assume a transverse monochomatic wave traveleing in + z direction incident on a transverse (,xy ) plane Across the zUUxy==0(,,0) plane Across the zz== plane UUxyz(, ,) Objective: to calculate the resulting field UUxyz=>(, ,) down the road z 0 as a function of Ux((),,0y ) ∞ FT of the Uz at ==0(,;0)(, plane: Aff Uxye,0) −+jfxfy2(π XY ) dxdy XY ∫∫−∞ ∞ And Ux(,yyffff ,,)0)(,;= A (ff , ;)0) edjfxfy2(πXY+ ) f df ∫∫−∞ XY XY What is the physical meaning of these components?

So far we have looked at Af(,;0)XY f as the spatial frequency spectrum of the disturbance. What is the direction of propagation of each these components?

29 Physical interpretation of angular spectrum

CidillConsider a simple plane wave propaga ting iditiin direction o fkf k: 2π Pxyz(, ,)==++=++ ejkr. where rk xx yy zz and ( x y z ); λ αβγ 2π αβ,.| , γ and are the direction cosines of kk Also |=

Using 222++=1, between the direction cosines we rewrite αβγ kxx2/πλλ λ 22 αλβλγλλλ== ==fffXYZ;; = = =− 1() ff XY −( ) ||k 2/ x 22ππ πλ λjxyjz()+ jfxfy2(+ ) jff21−−22 Pxy(,xy ,)z ==ee ejk. r eλλαβ γ e = eXy e XY x

Now with fff===αλ/, βλ / 0 we can write λ -1 XYzπ cos α ∞ −+jfxfy2(π XY ) Af(, f;0)= U ( x , y ,0) e dxdy as π XY ∫∫−∞ cos-1γ π⎛⎞αβ ⎛⎞αβ ∞ −+jxy2 ⎜⎟ AUxyedxdy,;0= (,,0) ⎝⎠λλ ⎜⎟∫∫−∞ ⎝⎠λλ z is the anggpular spectrum of the disturbance Ux((),,0y ). cos-1β y In summary this results shows that: each spatial frequency component is propagating at a different angle. 30 Propagation of the angular spectrum I CidthlConsider the angular spec trum o fthf the U across a plllltlane parallel to ()(x, y) at a distance z from it:

π⎛⎞αβ ∞ −+jxy2 ⎜⎟ ⎛⎞αβ ⎝⎠ AzUxyzedxdy⎜⎟, ;();(= ,,) λλ ⎝⎠λ λ ∫∫−∞ Our goal is to find the effects of the wave propagation on the angular spectrum ⎛⎞⎛⎞αβ αβ of the disurbance or the relationship of AAz⎜⎟⎜⎟, ;0 and , ; λλ λλ αβ α β⎝⎠⎝⎠ π⎛⎞αβ ∞ jxy2 ⎜⎟+ ⎛⎞⎝⎠ We start from Uxyz(, ,)= A⎜⎟ , ; zeλλ d d ∫∫−∞ ⎝⎠λ λλλ U must satisfy the Helmholtz equation ∇+22UkU =0 wherever there is no source. The resultλλ λλ is λ that A must satisfy the following differential equation. 2 d 2 ⎛⎞⎛⎞αβ αβ π 2 ⎛⎞ A ,;zAz+−−⎡⎤ 1α 22β ,; = 0 2 ⎜⎟⎜⎟⎣⎦ ⎜⎟ dz ⎝⎠⎝⎠ ⎝⎠ coefficient The solution can have the form:

⎛⎞2π 22 ⎜⎟jz1−− ⎛⎞⎛⎞αβ αβ ⎝⎠λ αβ AzAe⎜⎟⎜⎟,;= ,;0 ⎝⎠⎝⎠λλ λλ 31 Propagation of the angular spectrum II

⎛⎞2π 22 ⎜⎟jzλ 1−−αβ ⎛⎞⎛⎞α βαβ⎝⎠ AZAe⎜⎟⎜⎟,;= ,;0 ⎝⎠⎝⎠λλ λλ For this solution two cases are recognized: 1) When αβ22+<1 (true for all direction cosines) the effect of propagation on the angular spectrum is simply a change in phase of each component. 2) When αβ22+>ββ1 (α an d β are no longer the direc tiifftion cosines effect of aperture is present here) the angular spectrum has the form: Fourier transform of a field distributiion on which BCs of the ⎛⎞A real number aperture is imposed ⎜⎟2π ⎜⎟jz1−−22 ⎜⎟λ αβ ⎜⎟ ⎛⎞⎛⎞αβ αβ ⎝⎠⎛⎞αβ π − z 2 22 AzA⎜⎟⎜⎟,;= ,;0eAe= ⎜⎟,;0; =+− 1 ⎝⎠⎝⎠λλ λλ ⎝⎠λλ λ Since z is a positive real number, the wave components μαβare attenuating as they propagate. They are also called evanescent waves.μ Simm ilar to the case o f m icrowave waveguides th ere is a cu to ff f requency. Below cutoff frequency, these evanescent waves carry no energy away

from the aperture. 32 Propagation of the angular spectrum III

⎛⎞2π 22 ⎜⎟jzλ 1−−αβ ⎛⎞⎛⎞αβ αβ ⎝⎠ Substituting AZAe⎜⎟⎜⎟,;= ,;0αβ in the Uxyz (,,) we get: ⎝⎠⎝⎠λλ λλ ⎛⎞ ∞ jxy2π ⎜⎟+ ⎛⎞αβββ⎝⎠λλ α β Uxyz(, ,)= A⎜⎟ , ; Z e d d ∫∫−∞ ⎝⎠λλ λ λ

⎛⎞⎛⎞2 παβ22 ∞ ⎜⎟⎜⎟jzjxy12−− + ⎛⎞αβββ⎝⎠⎝⎠λλλα β UAU ()(xyz,,) = A⎜⎟,; 0eeαβ π dd ∫∫−∞ λ⎝⎠λλ λ ⎧11 xy22+< ⎪ 22⎪ 22 circxy( +=⎨1/2 ) xy += 1 ⎪0 otherwise ⎪⎩

⎛⎞2παβ22 ⎛⎞ βα∞ β⎛⎞αβ ⎜⎟jz12−− jxy ⎜⎟ + Uxyz(, ,)=+ A , ;0 e⎝⎠λλλ παβ circ22 e ⎝⎠ d d ∫∫−∞ ⎜⎟ λλ λ⎝⎠λλ No angular spectrum contribute to U (,xyz ,) beyond the evanescent

wave cutoff αβ22+<1 . αβ 33 Physical meaning of cutoff frequency

No angular spectrum contribute to Uxyz(, ,) beyond the evanescent

wave cutoff αβ22+<1 . No imaging system can resolve a periodic structure that its period is less than the wavelength of the light used.

2 2 ⎛⎞kx ⎛⎞ky 22 22 ⎜⎟+<→+<⎜⎟1||||kkxy k or xy +>λ . ⎝⎠kk⎝⎠ Near-Field imaging couples to the evanescent waves of a very fine structure and recovers the phase information that would be lost otherwise.

34 Effects of a diffracting aperture on the angular spectrum I Now an infinite opaque screen containing a diffracting structure is placesd in the z = 0 plane. GlfidfftGoal: find effects of fth the screen on th e angultfthditblar spectrum of the disturbance. Amplitude transmitance function:

Uxyt (, ;0) txyA (, )= Uxyi (, ;0)

UxytAi(, ;0)= t (, xyUxy ) (, ;0) take the Fourier transform of the both sides andthfd use the frequency convol ltithution theorem: ⎛⎞αβ ⎛⎞αβ αβ ⎛⎞ At ⎜⎟, =⊗ATi ⎜⎟,, ⎜⎟ ⎝⎠λλ ⎝⎠λλ λλConvolved ⎝⎠ with Angular spectrum Angular spectrum A second angular of the transmitted of the incident spectrum that is a disturbance disturbance result of the diffracting structure

⎛⎞αββ ∞ −+jxy2π ⎜⎟+ ⎛⎞αββ ⎝⎠λλ where Ttxy⎜⎟,(,= A )edxdy ⎝⎠λλ ∫∫−∞ is the Fourier transform of the txy(, ) A 35 Effects of a diffracting aperture on the angular spectrum II Example: for a unit amplitude plane wave illuminating the diffracting structure, the angular spectrum of the input is a delta function: ⎛⎞⎛⎞αβ αβ αβ αβ α ⎛⎞⎛⎞⎛⎞β AAAT,,==⊗ , ,, iti⎜⎟⎜⎟ ⎜⎟⎜⎟⎜⎟λλ ⎝⎠⎝⎠λλ λλ λλδ λλ ⎝⎠⎝⎠⎝⎠ Fourier transform of the Transmitted angular amplitude transmitance spectrum function of the aperture ⎛⎞⎛⎞⎛⎞⎛⎞αββββα β α β α ββ ATTt ⎜⎟⎜⎟⎜⎟⎜⎟,,,= δ ⊗ = , ⎝⎠⎝⎠⎝⎠⎝⎠λλ λλ λλ λλ If the diffracting structure is an aperture that limits the extend of the field distribution,

π⎛⎞αβ ∞ −+jxy2 ⎜⎟ ⎛⎞αβ ⎝⎠λλ Ttx⎜⎟,(,)= A yyyedxdy ⎝⎠λλ ∫∫−∞ the angular spectrum of the disturbance will be broadened.

36 The propagation phenomenon as a linear filter I Consider propagation from plane zzz= 0 to the plane = Across the zUUxy==0(,,0) plane Across the zz== plane UUxyz(, ,) Goals: a) to show that the propagation phenomenon acts like a linear space-invariant suystem. b) find the system's transfer function The system is linear since it is governed by a linear wave equation or considering the superposition integral (R-S first solution). 11eejkr01 jkr 01 U( P )== UP ( ) cos( nr ,01 ) ds UP ( ) cosθ ds I 01∫∫ ∫∫ 1 jrλλΣΣ01 jr 01 1λe jkr01 UP()== hPPUPds (,)() with hPP (,) cosθ 0011∫∫ 01 Σ jr01 To establish the space-invariance we need to derive the transfer function of the system and show that the mapping is space-invariant. 37 Transfer function of the linear invariant systems

g222()(x , ySg)(= S { 111 (x , y } action of a linear operator on a inpu t ∞ gxy(, )= g (,)(,;,)ξη ξη hxy ξ η dd h is the impulse response 222∫∫−∞ 1 22

hx(,22 y ;,)ξη δ ξ η=−− S {( x 1 ; y 1 )}

hx(,22 y ;,)ξη ξ η=− hx ( 2 , y 2 − ) for linear invariant systems ∞ gxy222(, )=−−==⊗ g 1 (,)(ξη ξ η ξ hxη 2 , y 2 ) dd g 1 * h g 1 h ∫∫−∞ Object function Impulse response of the system Take Fourier transform

Gf21(,)XY f= HffGf (,)(,) XY XY f Where H is the Fourier transffform of the impulse response. ∞ −+jf2(πξηXY f ) Hf(,)XY f= h (,) ξξη η e dd ∫∫−∞ And it is called transfer function of the system that indicates effects of the system in the freuency domain. 38 Propagation phenomenon as a linear filter II To establish the space-invariance we need to derive the transfer function of the system and show that the mapping is space-invariant.

Let Af(,;0)XY f be spatial spectrum (Fourier transform) of Uxy (,,0) at z = 0

Let Af(;)( XY, f;) z be spatial spectrum (Fourier transform) of Uxyz ()(,,) at z= z

So following Gf21(,)XY f= HffGf (,)(,) XY XY f and

Af(,;)XY f z = H (,)(,;0) f XY f Af XY f we can see H connects the two frequency spectrums before propagation and after the propagation. We got

⎛⎞2παβ22 ⎛⎞ ∞ ⎜⎟jz12−− jxy ⎜⎟ + βα β⎛⎞αβ ⎝⎠λλλ παβ 22 ⎝⎠ Uxyz(, ,)=+ A⎜⎟ , ;0 e circ e d d ∫∫ λ−∞ λ⎝⎠λλ αβ with ==ffXY, λλ αβ ∞ ⎛⎞2π 2 2 2 ⎜⎟jffz1()−(λλXY) − Uxyz(, ,)=+ A f , f ;0 circλλ f ( f )2 eed⎝⎠λ jfxfy2π ( XY+ ) f df ∫∫ ()XY( X) Y X Y −∞ Imposes the bandwidth limitation associated with evanescent waves ∞ Ux(,y ,)zA= (ff , ;)zejfxfy2(π XY+ ) df df comparing two equations shows: ∫∫−∞ XY XY

⎛⎞2π 2 2 2 ⎜⎟jffz1()−−()XY 2 ⎝⎠λ λλ Af(,;)XY f z= A() f XY ,;0 f circ()λλ fXY+ () f e 39 Propagation phenomenon as linear filter III ⎛⎞2π z 2 2 2 ⎜⎟jff1()−−()XY 2 ⎝⎠λ λλ Af()( XY,; fz) = Af A( f XY,; f 0() 0)ci rc (λλff f X) + () f Ye And the transfer function is then

⎛⎞2π z 2 2 ⎧ ⎜⎟jff1−−(λ XY) ()λ ⎪ ⎝⎠λ 2 2 e (λ ffXY) + ()λλ<1 / Hf(,)XY f = ⎨ ⎩⎪0 otherwise This shows that the propagation phenonmenon can be considered as a linear, dispersive spatial filter with a finite bandwidth. Transmission is zero outside the circular region of radius λ -1 (in spatial frequency space) Its transfer function is exponential. Within the circular frequency bandwidth the modulus (or amplitude) of the function is 1, but frequency dependent phase shifts are introduced. Phase dispersion is largest at high spatial frequencies (below the cutoff)

AdthhdiiihAs ffXY→→00 and then phase dispersion vanishes.

For a fixed spatial frequency pair, ffXY and :

the phase dispersion increases as z increases 40