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Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Chap 3 Foundations of Scalar Diffraction Theory 1 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • • • • • • • • • • Content 3.1 3.2 3.3 3.4 Historical introduction From a vector to a scalar theory Some mathematical preliminaries The Kirchhoff formulation of diffraction by a planar screen 3.5 The Rayleigh-Sommerfeld formulation of diffraction 3.6 Comparison of the Kirchhoff and Rayleigh-Sommerfeld theories 3.7 Further discuss of the Huygens-Fresnel principle 3.8 Generalization to nonmonochromatic waves 3.9 Diffraction at boundaries 3.10 The angular spectrum of plane waves 2 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 3.1 Historical introduction • While the theory discussed here is sufficiently general to be applied in other field, such as acoustic-wave and radiowave propagation, the applications of primary concern will be in the realm of physical optics. • To fully understand the properties of optical imaging and data processing system, it is essential that diffraction and the limitation it imposes on system performance be appreciated. 3 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Area Small part The edge of the aperture is thin enough such that light maybe regarded as unpolarized. In addition the area of the aperture can not be too small. (or be large enough) 4 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • Refraction can be defined as the bending of light rays mat takes place when they pass through a region in which there is a gradient of the local velocity of propagation of the wave. • The most common example occurs when the light wave encounters a sharp boundary between two regions having different refractive indices. 5 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • The Kirchhoff and Rayleigh-Sommerdeld theories share certain major simplifications and approximations. • Most important light is treated as a scalar phenomenon, neglecting the fundamentally vectorial nature of the electromagnetic fields. 6 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 3.2 From a vector to a scalar theory • In the case of diffraction of light by an aperture, the • E and H field modified only at the edges of the aperture where light interacts with the material which the edges are composed of , and the effects extend over only a few wavelengths into the aperture itself. 7 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • 1.Wave eq. (in vectorial form) • (from Maxwell’s eq.) 2 Note : a ay az x 1 E 2 x y z E 2 V 2 t 2 2 x 2 y 2 z 2 • 2. Wave eq. (in scalar forms) • . 1 2 2 u V 2 t 2 u stands for E x , E y , E z or H x , H y , H z • 3. Wave eq. (in phasor forms). 1 2U 2 U V 2 t 2 8 a. b. c. d. e. Linear, Homogeneous, Isotropic, Nondispersive, Nonmagnetic. Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU j ( kr-wt φ) • Where the optical disturbance u(p, t) = A e r • -jwt • = U(p) e A jkr jφ is called phasor • U(p) = e r • and represents position variable (i.e. r) • It follow that ( 2 k 2 ) U 0 (called Helmholtz eq.) 9 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • In general j (kr-wt) ψ (r,t ) A e r Secondary wavelets Huygens (Huygen’s principle) Envelop (new wavefront) (Young) Fresnel Primary wavefront two assumption Kirchhoff (Fresnel-Kirchhoff formula) Rayletgh-Summerfled Fig. 3.1 Huygens’ envelope construction 10 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 3.3 Some mathematical preliminaries • 3.3.1 The Helmholtz equation For a monochromatic wave, the scalar field may be written explicitly u ( P, t ) A( P) cos[2π vt φ(P)] (3-1) where A(P) and (P ) are the amplitude and phase, respectively, of the wave at position P, while v is the optical frequency. If the real disturbance u (P, t ) is to represent an optical wave, it must satisfy the scalar wave equation. 11 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 2 u- n 2 2u 2 c t 2 0 (3-2) • The complex function U(P) serves as an adequate description of the disturbance, since the time dependencies known a priori. If (3-1) is substituted in (3-2), it follows that U must obey the timeindependent equation. ( 2 k 2 ) U 0 (3-3) 12 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • 3.3.2 Green’s theorem Give U(P), G(p), U, G, 2U, and 2G 2 2 Let F UG-GU F (U G U G)-(G U G U ) According Gauss’s divergence thm. v F dv s F ds ( U G-G U ) a s n ds G U 2 2 v (U G-G U )dv s (U -G )ds n n v (U 2 G-G 2U )dv 13 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • Choice of the Green’s func. G(P) v 2 2 (U G-G U )dv s (UG-GU ) a n ds Helmholtz eqs. ( 2 k 2 ) U 0 e jkr G(P)= r (from Huygens-Fresnel principle) , ( 2 k 2 )G 0 14 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • 3.3.3 The integral thm. of Helmholtz and Kirchhoff The Kirchhoff formulation of the diffraction problem is based on a certain integral theorem which expresses the solution of the homogenous wave equation at an arbitrary point in terms of the values of the solution and its first derivative on an arbitrary closed surface surrounding that point. The problem is to express the optical disturbance at Po in terms of its values on the surface S. To solve this problem, we follow Kirchhoff in applying Green’s theorem and in choosing as an auxiliary function a unti-amplitude spherical wave expanding about the point Po (the so-called free space Green’s function). 15 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • Within the volume V ' , the disturbance G, being simply an expanding spherical wave, satisfies the Helmholtz equation ( 2 k 2 )U 0 2 2 ( U G-G U ) dv ( U G-G U ) a ' n ds 0 v s s sε ' s (U G U -G ) n n (U sε G U (U -G )ds s n n sε G U -G )ds n n G U (U -G )ds n n as 16 - 4πU ( Po ) 0 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU n V’ ε n S P0 Fig. 3.2 Surface of integration 17 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • Choice of the adequate volume denoted as V ' and S ' satisfying the requirement of Green thm. a. The first requirement of Green thm. is that U, U, 2U, G, G and 2G exist in the volume of integration. v (U 2 G-G 2U )dv s (UG-GU )an ds b. The second requirement…(see Goodman P.42) U, U, 2U, G, G and 2G are continuous within the volume V ' 18 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 1 V U ( Po ) 4π e jkr01 U e jkr01 ( ) -U ( )ds s r01 n n r01 (3-4) This result is known as the integral theorem of Helmholtz and Kirchhoff, it plays an important role in the development of the scalar theory of diffraction, for it allows the field at any point P0 to be expressed in terms of the “boundary value” of the wave on any closed surface surrounding that point. 19 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 3.4 The Kirchhoff formula of diffraction by a planar screen S2 S1 Σ R r01 n P0 P1 Fig. 3.3 Kirchhoff formula of diffraction by a plane screen 20 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • 3.4.1 Application of the integral theorem 1 U ( Po ) 4 s [G 2 U G G -U ds s n n U U -U ( jkG)]ds G ( -jkU ) R 2 dw Ω n n e jkR uniformly bounded R e jkR U ( -jkU ) Rdw n As R→ U ( jkU ) R 0 lim R n 21 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • 3.4.2 The Kirchhoff boundary conditions Kirchhoff accordingly adopted the following assumptions [162]: U n 1. Across the surface , the field distribution U and its derivative 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 distribution U and its derivative U are identically n zero. These conditions are commonly knows as the Kirchhoff boundary conditions. 22 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • 3.4.3 The Fresnel-Kirchhoff diffraction formula P2 r01 r02 n P0 P1 Fig. 3.4 Point-source illumination a plane screen. 23 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU G(P1 ) 1 e jkr01 cos(n, r01 )( jk ) n r01 r01 jkr • Note: e jk cos(n, r01 ) 01 (3-5) r01 G G an n 1 jkr 1 jkr G ar ( e ) ar e ( jk ) e jkr (-1) (r 2 ) r r r 1 e jkr ar ( jk- ) 1 e jkr r r G a n (a r a n )( jk- ) r r 1 e jkr01 cos(a n a r )( jk) r01 r01 24 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 1 e jkr01 U U(P0 ) [ -jkU cos(n, r01 )]ds (Fowles) (3-6) 4π r01 n U ( P0 ) 1 j [ jk ( r21 r01 )] e r21r01 cos(n,r01 )- cos(n, r21 ) [ ]ds 2 Let the illumination source be a point source. e jkr01 U ( P1 ) U ' ( P1 ) ds r01 jkr21 1 Ae U' ( P1 ) [ jλ r21 cos(n, r01 )- cos(n, r21 ) ][ ] 2 25 Dr. Gao-Wei Chang (3-7) Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 3.5 The Rayleigh-Sommerfeld formulation diffraction • It is a well-known theorem of potential theory that if a twodimensional potential function and its normal derivative vanish together along any finite curve segment, then that potential function must vanish over the entire plane. Similarly, if a solution of the three-dimension wave equation vanishes on any finite surface element, it must vanish in all space. 26 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • 3.5.1 Choice of alternative Green’s function 1 G U U ( P0 ) )ds ( n G-U 4π S n 1 The conditions for validity of this equation are: 1. The scalar theory holds. 2. Both U and G satisfy the homogeneous scalar wave equation. Helmoltz equation) 3. The Sommerfeld radiation condition is satisfied. 27 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 1 U G U ( P0 ) -U )ds (G 4π S n n 1 1 4 G e jkr r G- G1 U U G( G -U )ds (G-U )ds 4π S ' 1 n n n n (Kirchhoff e jkr r (3-8) spherical wavelet) jk ~ r01 -e ~ r01 (Sommerfeld choose) 28 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU P0 r01 r02 n P0 P1 (Mirror) Fig. 3.5 Rayleigh-Sommerfeld formulation of diffraction by a plane screen 29 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU G-1 U ( P0 ) ds U 4π Σ n G - (P1 ) (P ) 2 1 n n (3-9) (refer to eq. (3-5)) jk ~r01 jkr e e Choose Green func. G ~ r r01 30 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU -1 G U Ι ( P0 ) ds U 2π Σ n -1 G U Π ( P0 ) ds U 2π Σ n jk ~ r01 -e G- ~ r r01 e jkr (3-10) jk ~ r01 e jkr e G ~ r r01 31 (3-11) Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • 3.5.2 The Rayleigh-Sommerfeld diffraction formula Rayleigh-Sommerfeld diffraction formula 1 U Ι ( P0 ) jλ 1 U Ι ( P0 ) jλ U ( P1 ) e r01 Σ U ( P1 ) jkr01 e jkr01 r01 S' cos(n, r01 )ds (3-12) cos(n , r01 )ds (3-13) Fresnel-Kirchhoff diffraction formula U ( P0 ) U ' ( P1 ) e Σ jkr01 r01 (3-14) ds jkr21 cos ( n , r ) cos ( n , r21 ) 1 Ae 01 U ' ( P1 ) [ ][ )]ds jλ r21 2 1 U ( P0 ) 4π Σ e jkr01 r01 [ U -jkU cos (n, r01 )]ds n 32 (3-15) Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • 3.6 Comparison of the Kirchhoff and Rayleigh-Sommerfeld theories P0 r01 r02 n P1 33 P0 Diffraction by a planar screen ∴ ~ r01 r01 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Kirchhoff (one singular point) G e jkr 01 r01 Sommerfeld (multiple singular point) G- G e jkr 01 r01 e jkr 01 r01 - e jk ~r01 ~r 01 jk ~r e 01 ~r 01 34 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • Scalar diffraction formula (general form) U ( P0 ) 1 U G 1 ( G -U ) ds [U Ι (P0 ) U Π (P0 )] 4π s n n 2 (Sommerfeld radiation condition) jkr G1 1 e 01 1 G U ( P0 ) ( -U ) ds U cos ( n r ) ds ( -U )ds 01 4π n jλ r01 2π n Σ 2 G n Σ Σ (Kirchhoff) 1 U 1 U U Π ( P0 ) (G )ds G ds 4π n 2π n Σ 2G (Kirchhoff) Σ 35 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • A comparison of the above equations leads us to an interesting and surprising conclusion: the Kirchhoff solution is the arithmetic average of the two Rayleigh-Sommerfeld solution. • In closing it is worth nothing that, in spite of its internal inconsistencies, there is one sense in which the Kirchhoff theory is more general than the Rayleigh-Sommerfeld theory. The latter requires that the diffracting screens be planar, while the former does not. 36 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 3.7 Further discussion of the HuygensFresnel principle • It expresses the observed field U( P0) as a superposition of diverging spherical Huygens-Fresnel wavelet exp ( jkr01 ) /r01 origination from secondary source located at each every point P1 within the aperture . 1 U ( P0 ) jλ Σ e jkr01 U ( P1 ) cos θds r01 37 (3-16) Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU U(P0 ) h( P0, P1 )U ( P1 )ds (3-17) Σ • where the impulse response h(P0 , P1 ) is given explicitly by 1 e jkr01 h( P0 , P1 ) cos θ jλ r01 r(t) LIT 38 y(t) Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 3.8 Generalization to nonmonochromatic wave (Nonomonochromatic time function) 1 U ( P0 ) jλ Σ e jkr01 U ( P1 ) cos θds r01 (3-22) For nonmonochromatic light U ( P0 , ) U ( P , ) jV e 1 Σ j2 πν ( r01 r0 1 ) V cos θds (3-23) where the phasor U ( P0 , ν) implies the Fourier transform (FT) of the disturbance u ( P0 , t ) with respect to (w, r, t) the temporal frequency . 39 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Note: Free space (nondispersive medium) R L L Vs C Note : The real, monochroma tic wave Monochromatic source u( Po ,t ) Re[U ( Po )e j 2 πνt ] Vs (t ) Vm cos Wot (or Vs (t ) Vme Vs (t ) Vne jW o t or the complex, monochroma tic wave ) u( Po ,t ) U ( Po )e j 2 πνt For nonmonochr omatic waves jnWo t u( Po ,t ) U ( Po , )e j 2 πνt d n - 40 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU u(P0 ,t) Σ or u(P0 , - ) cos θ [ 2πVr01 -j 2πv U ( P1, ν)e j 2πν(t r01 ) V ]ds (3-24) - r01 j 2πν(-( τ- )) cos θ (3-25) V [ -j 2πv U ( P1, ν)e ]ds 2πVr01 Σ - Since u(P1, - ) U ( P1, ν )e j 2 πνt dν d u(P0 , - ) -j 2πv U ( P1, ν )e j 2 πν(- τ )dν dτ d - dτ 41 - u(P1, - ) -j 2πv U ( P1, ν )e j 2 πν (- τ ) dν - Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Eq. (3-25) becomes u(P0 , - ) Σ or u ( P0 ,t ) r cos θ d u ( P1, -(-τ - 01 ))ds 2πVr01 dτ V Σ r01 cos θ d u ( P1, t- )ds 2πVr01 dt V (3-26) V is the velocity of propagation of the disturbance in a medium of refractive index n (v=c/n), and the relation νλ V or V has been used. 42 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 3.9 Diffraction at boundaries • A more physical point-of-view, first qualitatively expressed by Thomas Young in 1802, is to regard the observed field as consisting of a superposition of the incident wave transmitted through the aperture unperturbed, and a diffracted wave originating at the rim of the aperture. The possibility of a new wave origination in the material medium of the rim makes this interpretation a more physical one. • The applicability of a boundary diffraction approach in more general diffraction problems was investigated by Maggi [202] and Rubinowicz [249], who showed that the Kirchhoff diffraction formula can indeed be manipulated to yield a form that is equivalent to Young’s ideas. More recently, Miyamoto and Wolf [250] have extended the theory of boundary diffraction. 43 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 3.10 The angular spectrum of plane waves • • • Objective: To formulate scalar diffraction theory in a framework that closely resembles the theory of linear, invariant system. As we shall see, P1 P0 (1) if the complex field distribution of a monochromatic disturbance is Fourier-analyzed across any plane, the various spatial Fourier components can be identified as plane waves traveling in different directions always from that plane. (2) The field amplitude at any other point (or across any other parallel plane) can be calculated by adding the contributions of these plane waves, taking due account of the phase shifts they have undergone during propagation. 44 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Assume Z Equivalent to source Lens z z plane z0 Young’s experiment Assume Z Z=0 Source Screen 45 Dr. Gao-Wei Chang Screen Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 3.10.1 The angular spectrum and its physical interpretation Let the complex field, across z=0 plane, be represented by U(x, y, 0). Our ultimate objective is to calculate the resulting field U(x, y, z). u ( P0 , t ) U ( P0 )e j 2πνt Temporal freq. Phasor U ( x,y,0) A( f x ,f y ,0) Spatial freq. The Fourier transform of U(x, y, 0), i.e., its spectrum A( f x ,f y ,0) U ( x,y,0)e - j2π ( f x x f y y ) dxdy (3-27) - - And the inverse Fourier transform of its spectrum U ( x,y,0) - - A( f x , f y , 0)e j2π ( f x x f y y ) 46 df x df y Dr. Gao-Wei Chang (3-28) Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • To give physical meaning to the function in the integrand of the above integral, consider the plane wave j ( k r - 2πνt ) -j 2πνt p( x,y,z,t) 1 e P( x,y,z) e x k cos 1 • Where the phasor cos 1 2π 2π j (αx βy ) j rz P( x,y,z) ek r e λ e λ cos 1 y the position v ector r a x x a y y a z z and the wave vector 2 k (a x x a y y a z ) and α, β , γ are direction cosines which are interrelat ed thru. 47 Dr. Gao-Wei Chang z Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU γ 1-α 2 -β 2 Thus, across the plane z=0, β α j 2π ( x y ) λ λ P( x,y,0) e e j 2π ( f x x f yy ) We see that α β f x , f y , and f z λ λ 1-α 2 -β 2 λ The angular spectrum of the disturbance U (x,y,0) α β A( , ,0) λ λ U ( x,y,0)e β α - j 2π ( x y ) λ y - - 48 Z=0 Lens dxdy Dr. Gao-Wei Chang z z plane z0 Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • 3.10.2 Propagation of the angular spectrum Consider the angular spectrum of the disturbance A( α β , ,z ) λ λ U ( x,y,z)e β α x y) λ y dxdy (3-29) d d (3-30) - - U ( x,y,z) - j 2π ( U (x,y,z) - - α β A( , ,0)e λ λ β α j 2π ( x y ) λ y into the Helmholz eq. 49 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 2U k 2U 0 gives α β 2 2 α β 2 2 A( , ,z ) ( ) (1 - - ) A( , ,z ) 0 2 λ λ λ λ dz d2 (3-31) where 2 2 1 and 2 1 - 2 - 2 for all true direction cosines. An elementary solution of Eq. (3-31) can be written in the form α β α β j 2π( 1-α 2 -β 2 )z A( , ,z ) A( , ,0)e λ λ λ λ 50 Dr. Gao-Wei Chang (3-32) Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Finally, the substitution of Eq. (3-32) into Eq. (3-30) yields U ( x,y,z) - - α β j 2π( 1-α -β )z A( , ,0)e circ ( λ λ 2 2 α β j 2π ( x y ) λ λ d α 2 β 2 )e α β d λ λ (3-33) where the circ function limits the region of integration to the region within which 2 2 1 is satisfied. Note: When 2 2 1, and are no longer interpretable as direction cosines. Eq. (3-32) can be rewritten as α β α β A( , ,z ) A( , ,0)e - z λ λ λ λ Where 2 2 2 -1 51 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU Incident wave n1 sin θ1 n 2 sin θ2 Reflected n1 If θ1 θ2 and n1 n 2 n2 total internal refl. occurs. Transmitted (or refracted) (Including obsorbed wave ) 52 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 3.10.3 Effects of a diffracting aperture on the angular spectrum Define the amplitude transmittance function in the z=0 plane, t A ( x,y) U t ( x,y,0) U i ( x,y,0) Then U t ( x,y,0) U i ( x,y,0)t A ( x,y) And by the use of the convolution theorem. α β α β α β At ( , ) Ai ( , ) T ( , ) λ λ λ λ λ λ where 53 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU β α j 2 π ( x y) α β λ λ dxdy T( , ) t A ( x,y)e - - λ λ And is again the symbol for convolution. For example, if the incident wave illuminates the diffracting structure normally. α β α β Ai ( , ) ( , ) λ λ λ λ α β α β α β α β At ( , ) ( , ) T ( , ) T ( , ) λ λ λ λ λ λ λ λ 54 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • In this case, the transmitted angular spectrum is found directly by Fourier transforming the amplitude transmittance function of the aperture. • Note that, if the diffracting structure is an aperture that limits the extent of the field distribution, the is a broadening of the angular spectrum of the disturbance, from smaller the aperture, the broader the angular spectrum behind the aperture. 55 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU 3.10.4 The propagation phenomenon as a linear spatial filter • From Eq. (3-33) in Sec. 3.10.2, we have U ( x,y,z) - - 2 j ( 1- ( λf x ) 2 -( λf y ) 2 ) z α β j 2π ( λf x λf y ) 2 2 A( , ,0) circ ( ( λf x ) ( λf y ) ) e e df x df y λ λ α f where x λ and f y , and we have again explicitly introduced the bandwidth limitation associated with evanescent waves thru the use of a circ. function. z 1 2 2 2 2 1 - ( λf x ) - ( λf y ) ], f x f y exp[ j 2π H ( f x ,f y ) λ λ , otherwise 0 56 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU z or H ( f x ,f y ) exp[ j 2π 1 - ( λf x ) 2 - ( λf y ) 2 ] circ ( λf x ) 2 ( λf y ) 2 λ From Eq. (3-32) in Sec. 3.10.2 and Eq. (3-29) A( f x ,f y ,0) A( f x ,f y ,0) H( f x ,f y ) Thus the propagation phenomenon may be regarded as a linear, dispersive spatial filter with a finite bandwidth: 57 Dr. Gao-Wei Chang Optoelectronic Systems Lab., Dept. of Mechatronic Tech., NTNU • (1) Within the circular bandwidth, the modulus of the transfer function is unity but frequency-dependent phase shifts are introduced. • (2) The phase dispersion of the system is most significant at high spatial frequency and vanishes as both f x and f y approach zero. • • (3) For any fixed spatial frequency pair the phase dispersion increases as the distance of propagation z increases. In closing we mention the remarkable fact that despite the apparent different of their approaches, the angular spectrum approach and the first Rayleigh-Sommerfeld solution yield identical predictions of diffracted fields. 58 Dr. Gao-Wei Chang