Chapter 6 – Fourier Optics Gabriel Popescu University of Illinois at Urbana‐Champpgaign Beckman Institute Quantitative Light Imaging Laboratory http://light.ece.uiuc.edu Principles of Optical Imaging Electrical and Computer Engineering, UIUC ECE 460 –Optical Imaging 3.10 Lens as a phase transformer i . EEeo becomes important . How is the wavefront changed by a lens? b(x,y) (x,y) x22 y P α C1 R1 R2 b1 b2 b0 (,)x yknbxykbbxy (,) [o (,)] (3.32) glass air kbo k(1)(,) n b x y Chapter 3: Imaging 2 ECE 460 –Optical Imaging 3.10 Lens as a phase transformer . Let’s calculate b(x,y); assume small angles bR11() PC 1 RR22 () R R 11 2 11 1 1 x . Taylor expansion: 1| x xo 1 Small Angle Approx 2 (Gaussian) 22 (3.33) bR1111 R 1 22 xy22 . tan R1 22 . xy So: bx1(,y ) (3.34) 2R1 Chapter 3: Imaging 3 ECE 460 –Optical Imaging 3.10 Lens as a phase transformer bx(,y )bbxo 12 (,y ) bx (,y ) xy22 11 (3.35) bo 2 R12R . This is the thickness approximation . The phase φ becomes: ()(x, yk)(1)(o kb (1)()n b x, y) 22 xy 11 (3.36) o kn(1) 2 R12R 111 . But we know: (1)n f RR12 Chapter 3: Imaging 4 ECE 460 –Optical Imaging 3.10 Lens as a phase transformer k E(x, y) EE(x’(x, y) E '(xy , ) Exy ( , ) te ( xy , ) (3.37) . The lens transformation is: tei e k ixy()22 (3.38) iknb 2 f eeo . Chapter 3: Imaging 5 ECE 460 –Optical Imaging 3.10 Lens as a phase transformer . A lens transforms an incident plane wavefront into a parabolic shape . Note: f > 0 covergent lens f < 0 divergent f . So, if we know how to propagate through free space, then we can calculate field amplitude and phase through any imaging system Chapter 3: Imaging 6 ECE 460 –Optical Imaging 3.11 Huygens‐Fresnel principle y . Spherical waves: eikR . Wavelet: h R R 22 x z 222 xy . Rxyzz1 2 z z . We are interested close to OA, i.e. small angles 1 xy22 Rz 1 2 (3. 39) 2 z Chapter 3: Imaging 7 ECE 460 –Optical Imaging 3.11 Huygens‐Fresnel principle ikR . For amplitude11 is OK R z e R z 22 . For phase 1 xy kR kz 1 2 z z R The wavelet becomes: z ikz kx()22 y e i 22 h()(xy, ) e 2z (3. 40a) fxy()( , ) x y z 2 . Remember, for the lens we found: f ()xx k 22 ixy() (3.40b) io 2 f txye (, ) ee Negative Lens . Free space acts on the wavefront like a divergent lens (note “+” sign in phase) Chapter 3: Imaging 8 ECE 460 –Optical Imaging 3.11 Huygens‐Fresnel principle . At a given plane, a field is made of point sources y Ex( ,y )Ex ( ',y ') (xx ') (yy ')dx ' dy ' x . Eq 3.40 a‐b represent the impulse response of the system (free space or lens) . Recall linear systems (Chapter 2, page 12, Eq 2.16) . Final response ((p)output) is the convolution of the input with the impulse response (or Greeen’s function) . Nice! Space or time signals work the same! f (')(xxxdxfx 00 ')' ( ) Chapter 3: Imaging 9 ECE 460 –Optical Imaging 3.11 Huygens‐Fresnel principle y θ z U ()(,) U ()(xy, ) x Uxy(, ) U (, )( hx , y ) dd ik 22 ()()xy Ux(,y ) Ue (, )2z dd (3.41) Remember: V Chapter 3: Imaging 10 ECE 460 –Optical Imaging 3.11 Huygens‐Fresnel principle . Fresnel diffraction equation = convolution xy22 Rz1 . Fresnel diffraction equation is an approximation 2 2z of Huygens principle (17th century) 1 eikR(,) Ux(,)y Udd (,) cos(,) (3.42) iR (,) . ! Fresnel is good enough for our pourpose . Note: we don’ t care about constants A (no x‐y dependence) ik ()()xy22 Uxy()( , )( U (), ) e2z dd Chapter 3: Imaging 11 ECE 460 –Optical Imaging 3.12 Fraunhofer Approximation Z Z Z3 . One more approximation (far field) 1 2 x . The phase factor in Fresnel is: Near Field k 22 (,xy ) ( x ) ( y ) Fresnel 2z k 22 22 Fraunhofer ()()2()x yxy 2z (3.43) ≈ 0 . If zk() 22 , we obtain the Fraunhofer equation: i2 ()xy Ux(,y ) AU (, ) ez dd(3.44) . Thus eq. 3.44 defines a Fourier transform . Useful to calculate diffraction patterns ! Chapter 3: Imaging 12 ECE 460 –Optical Imaging 3.12 Fraunhofer Approximation x . Let’s define: f x z y f y z iff2( ) Uf(, f ) U (,). exy dd (3.45) xy . Example: diffraction on a slit x k z a Chapter 3: Imaging 13 ECE 460 –Optical Imaging 3.12 Fraunhofer Approximation x a, |x| < a/2 . One dimensional: Ux() a 0, rest . The far‐field is given by Fraunhofer eq: ifi2 xf x Uf()x Uxe () dx a a x 2 2 a . Similarity Theorem + [()]sin()x cfx : Uf()xx a sin() caf sin(afx ) a (3.46) afx x . fx z Chapter 3: Imaging 14 ECE 460 –Optical Imaging 3.12 Fraunhofer Approximation . Always measure intensity the diffraction pattern is: 2 2 2 sin(afx ) If()xx Uf () a (3.48) afx . Also Babinet’s Principle I(fx) fx() F [] 1()()[]f xF -2π -π π 2π afx Chapter 3: Imaging 15 ECE 460 –Optical Imaging 3.12 Fraunhofer Approximation f 1 . Note: sin(af ) sin x width of diffraction pattern x 1 a a . narrow slit: a1 . wide slit: a2 . Similarity Theorem ↔ uncertainty principle Chapter 3: Imaging 16 Quiz: What is the diffraction pattern from 2 slits of size a separated by d? a d ECE 460 –Optical Imaging 3.13 Fourier Properties of lenses U(x2,y2) U(x1,y1) U(x3,y3) U(x4,y4) OA F F’ z d1 d2 . Propagation: Fresnel U(x1,y1) U(x2,y2) Lens U(x2,y2)U(x3,y3) Fresnel U(x3,y3)U(x4,y4) Chapter 3: Imaging 18 ECE 460 –Optical Imaging 3.13 Fourier Properties of lenses ik 22 xx yy 21 2 1 2d1 . Ux(,22 y ) A 12 Ux (,) 11 ye dxdy 11 k ixy22 2 f 33 . Ux()( 33, y)( AUx 2333 (), y) e ik 22 xx yy 43 43 2d2 . U(, x44 y ) A 34 U (,) x 33 y e dx 33 dy (3.49) . Combining Eqs (3.49) is a little messy, but there is a special case when eqs simplify very useful Chapter 3: Imaging 19 ECE 460 –Optical Imaging 3.13 Fourier Properties of lenses U U . If d1 = d2 = f 1 4 FF’ff ixfyf2( 11xy ) U()( x44, y)( A 41 U 111 () x, y) e dx 11 dy (3. 50) xy44 ffxy; ff What is []U 2 []U 2 . Same eq as (3.45); now z f [*]UU . Lenses work as Fourier transformers UU * . Useful for spatial filtering Autocorrelation Chapter 3: Imaging 20 ECE 460 –Optical Imaging 3.13 Fourier Properties of lenses . Exercise: Use Matlab to FFT images (look up “fft2” in help) a) FFT iFFT Final Granting (amplitude) High‐pass FFT iFFT b) Final Band‐pass . Note the relationship between the frequencies passed and the details / contrast in the final image Chapter 3: Imaging 21 Fourier Optics Uinput Uoutput f f F F’ 2D FT pairs diffraction patterns (Using ImageJ) FT is in Log(Abs) scale!! Why? Time‐domain: sound • load your mp3 • plot time‐series • plot frequency amplitu de, phase, power spectrum, linear/ log • show frequency bands, i.e. “equalizer” • adjust and ppylay in real time Equalizer from mp3 player Space‐domain: image • load your image • display image • show 2D frequency amplitu de, phase, power spectrum, linear/ log • show rings of equal freq., “image equalizer” • adjust and disppylay in real time‐ example on next slide Fourier Filtering (ImageJ).