THE WAVE EQUATION 5.1. Solution to the wave equation in Cartesian coordinates Recall the Helmholtz equation for a scalar field U in rectangular coordinates U U r, ( r, ) r, 0, (5.1) Where is the wavenumber, defined as r, r, i r, n c r, i r, (5.) Assuming lossless medium ( 0 ) and decoupling the vacuum contribution ( 1 term, namely where k0 n ) from r, U r, k0 U r, k0 n r, 1 U r,, (5.3) c, we re-write Eq. to eplicitly show the driving. Note that Eq. 3 preserves the generality of the Helmholtz equation. The Green s function, h, (an impulse response) is obtained by setting the driving term (the right hand side) to a delta function, r, k0 hr, r y z h (3) (5.4) In order to solve this equation, we take the Fourier transform with respect to r, k k hk kh,, 1, (5.5) 0 1
where k kk. This equation breaks into three identical equations, for each spatial coordinate, as 1 k, h k k 0 1 1 1 k0 k k0 k k0 (5.6) To calculate the Fourier transform of Eq. 6, we invoke the shift theorem and the Fourier transform of function 1 k, ia f k a e f 1 isign k (5.7) Thus we obtain as the final solution 1 ik0 h, e, 0 (5.8) ik 0 The procedure applies to all three dimensions, such that the 3D solution reads h ik 0, e kr r, (5.9) where k is the unit vector, k k / k. Equation 9 describes the well known plane wave solution, which is characterized by the absence of amplitude modulation upon propagation. This is an infinitely broad wavefront that propagate along direction ˆk (Figure 1).
y r k Figure 5-1. Plane wave. 3
5.. Solution of the wave equation in spherical coordinates For light propagation with spherical symmetry, such as emission from a point source in free space, the problem becomes on dimensional, with the radial coordinate as the only variable, h h r, Ur, k, U k, (3) (3) r r 1 r 1 (1) r (5.10) Recall that the Fourier transform pair is defined in this case as h r h k kr k dk sinc 0 h k h r kr r dr sinc 0 (5.11) The Fourier properties of (3) r 1 k (3) r and etend naturally to the spherically symmetric case as (5.1) Thus, by Fourier transforming Eq. 4, we obtain the frequency domain solution, 1 h k, k k (5.13) 0 4
Not surprisingly, the frequency domain solutions for the Cartesian and spherical coordinates (Eqs. 6 and 13, respectively) look quite similar, ecept that the former depends on one component of the wave vector and the latter on the modulus of the wave vector. The solution in the spatial domain becomes 0 0 0 0 kr 1 sin hr, k dk k k kr ikr 1 1 1 e e r k k 0 0 kk0 i ikr 1 e 4ir k k dk. ikr dk (5.14) We recognize in Eq. 15 the Fourier transform of a shifted 1 k function, which we encountered earlier (Eqs. 7). Thus, evaluating this Fourier transform, we finally obtain Green s function for propagation from a point source, ikr e hr,, r 0 (5.15) r This solution defines a (outgoing) spherical wave. 5
z cos r 10 ;r y z r y Figure 5-. Spherical wave 6
Chapter 6 Fourier Optics Gabriel Popescu University of Illinois at Urbana Champaign Beckman Institute Quantitative Light Imaging Laboratory http://light.ece.uiuc.edu Principles of Optical Imaging Electrical and Computer Engineering, UIUC
3.10 Lens as a phase transformer i E E e becomes important o How is the wavefront changed by a lens? b(,y) P α y (,y) C 1 R 1 R b 1 b (, y) knb(, y) k[ b b(, y)] o b 0 (3.3) glass air kb k( n 1) b(, y) o
3.10 Lens as a phase transformer Let s calculate b(,y); assume small angles b R ( PC ) 1 1 1 R R R R 1 1 ( 1) 1 1 1 Taylor epansion: 1 o1 b1 R111 R1 tan So: b1 (, y) R 1 y y R 1 Small Angle Appro (Gaussian) (3.33) (3.34) 3
3.10 Lens as a phase transformer by (, ) b b(, y) b (, y) o 1 y 1 1 bo R1 R This is the thickness approimation The phase φ becomes: (, y ) o k ( n 1) b (, y ) y 1 1 o k ( n1) R R 1 1 1 But we know: ( n 1) f R1 R 1 (3.35) (3.36) 4
3.10 Lens as a phase transformer k E(,y) E ( E(,y) E '( y, ) Ey (, ) t( y, ) (3.37) The lens transformation is: i e te iknb o e. e e k i y f ( ) (3.38) 5
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 6
3.11 Huygens Fresnel principle Spherical waves: e Wavelet: h R ikr 1 R y z z y z y z R z We are interested close to OA, i.e. small angles 1 y R z 1 (3.39) 39) 1 z 7
For amplitude For phase 3.11 Huygens Fresnel principle kr 1 1 R z kz 1 The wavelet becomes: k i z is OK 1 z ( y ) e h ( y, ) e z z Remember, for the lens we found: ikz o t (, y) e e e z y ikr e R (3.40a) (, ) i k i y f ( ) R z z f y y f ( ) Freespace actsonthe the wavefront likea divergent lens (note + sign in phase) (3.40b) Negative Lens 8
3.11 Huygens Fresnel principle At a given plane, a field is made of point sources y Ey (, ) E ( ', y') ( ') ( y y') d' dy' Eq 3.40 a b represent the impulse response of the system (free space orlens) Recall linear systems (Chapter, page 1, Eq.16) Final response (output) is the convolution of the input with the impulse response (or Greeen s function) Nice! Space or time signals work the same! f ( ') ( ') d' f( ) 0 0 9
3.11 Huygens Fresnel principle y θ U (, ) U ( y, ) U(, y) U(, ) h(, y) dd ik ( ) ( y) z (3.41) U (, y ) U (, ) e d d z Remember: V 10
3.11 Huygens Fresnel principle Fresnel diffraction equation = convolution Fresnel diffraction equation is an approimation of Huygens principle (17 th century) R z1 y z ikr(, ) 1 e U(, y) U(, ) cos (, ) dd i R(, ) (3.4)! Fresnel is good enough for our pourpose Note: we don t care about constants A (no y dependence) ik ( ) ( y) z U (, y ) U (, ) e d d 11
3.1 Fraunhofer Approimation One more approimation (far field) The phase factor in Fresnel is: k ( y, ) ( ) ( y) zz k ( y ) ( ) ( y) z If z k( ) 0, we obtain the Fraunhofer equation: Z 1 Z Z 3 Near Field Fresnel (3.43) Fraunhofer i ( y ) z U(, y) A U(, ) e dd (3.44) Thus eq. 3.44 defines a Fourier transform Useful to calculate diffraction patterns! 1
3.1 Fraunhofer Approimation Let s define: f f y z z y z i ( f f ) y U( f, f ) U(, ). e dd y (3.45) Eample: diffraction on a slit k z a 13
3.1 Fraunhofer Approimation One dimensional: U( ) a, < a/ a 0, rest The far field is given by Fraunhofer eq: i f U( f ) U( ) e d a Similarity Theorem + [ ( )] sin c( f ) : U ( f ) a sin c ( af ) sin( af) a af f z (3.46) a a 14
3.1 Fraunhofer Approimation Always measure intensity the diffraction pattern is: sin( af) I( f) U( f) a af Also Babinet s Principle f( ) F[ ] 1 f ( ) ( ) F[ ] I(f ) (3.48) -π -π π π af 15
3.1 Fraunhofer Approimation f Note: 1 sin( af) sin width of diffraction pattern 1 a a narrow slit: a 1 wide slit: a Similarity Theorem uncertainty principle 16
Quiz: What is the diffraction pattern from slits of size a separated by d? a d
3.13 Fourier Properties oflenses U(, y ) U( 1, y 1 ) U( 3, y 3 ) U( 4, y 4 ) OA F F z d 1 d Propagation: Fresnel U( 1,y 1 ) U(,y ) Lens U(,y ) U( 3,y 3 ) Fresnel 3 3 4 4 U( 3,y 3 ) U( 4,y 4 ) 18
3.13 Fourier Properties oflenses y y 1 1 d 1 U(, y ) A U(, y ) e ddy ik 1 1 1 1 1 k i y 3 3 f 3 3 3 3 3 ik y y 4 3 4 3 d 4 4 34 3 3 3 3 U (, y ) A U (, y ) e U (, y ) A U (, y ) e d dy Combining Eqs (3.49) is a little messy, but there is a special case when eqs simplify very useful (3.49) 19
3.13 Fourier Properties oflenses If d 1 = d = f U 1 U 4 F f f F i ( f y f ) 4 4 41 1 1 1 1 1 (3.50) 1 1 y U (, y ) A U (, y ) e d dy 4 y4 f ; f y f f What is [ U ] [ U ] Same eq as (3.45); now z f [ U U *] Lenses work as Fourier transformers U U* Useful for spatial filtering Autocorrelation 0
3.13 Fourier Properties oflenses Eercise: Use Matlab to FFT images (look up fft in help) a) FFT ifft Final Granting (amplitude) High pass b) FFT ifft Final Band pass Note the relationship between the frequencies passed and the details / contrast in the final image 1
Fourier Optics U input U output F f f F
D FT pairs diffraction patterns (Using ImageJ) FT is in Log(Abs) scale!! Why?
Time domain: sound load your mp3 plot time series plot frequency amplitude, phase, power spectrum, linear/ log show frequency bands, i.e. equalizer adjust and play in real time Equalizer from mp3 player
Space domain: image load your image display image show D frequency amplitude, phase, power spectrum, linear/ log show rings of equal freq., image equalizer adjust and display in real time eample on j p y p net slide
Fourier Filtering (ImageJ)