MIT 2.71/2.710 Optics 10/31/05 wk9-a-1. The spatial frequency domain

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1 10/31/05 wk9-a-1 The spatial frequency domain

2 Recall: plane wave propagation x path delay increases linearly with x λ z=0 θ E 0 x exp i2π sinθ + λ z i2π cosθ λ z plane of observation 10/31/05 wk9-a-2

3 Spatial frequency angle of propagation? x λ electric field z=0 θ z plane of observation 10/31/05 wk9-a-3

4 Spatial frequency angle of propagation? x λ z=0 θ z plane of observation 10/31/05 wk9-a-4

5 Spatial frequency angle of propagation? The cross-section of the optical field with the optical axis is a sinusoid of the form sinθ E 0 exp i2π x λ φ + 0 i.e. it looks like ( i2π u φ ) E0 exp x + 0 where u sinθ λ is called the spatial frequency 10/31/05 wk9-a-5

6 2D sinusoids [ 2π ( ux vy) ] E cos + 0 [ i2π ( ux vy) ] E exp + 0 corresp. phasor y x 10/31/05 wk9-a-6

7 2D sinusoids [ 2π ( ux vy) ] E cos + 0 [ i2π ( ux vy) ] E exp + 0 corresp. phasor y x min max 10/31/05 wk9-a-7

8 2D sinusoids [ 2π ( ux vy) ] E cos + 0 [ i2π ( ux vy) ] E exp + 0 corresp. phasor y 1/u 1/v x min max 10/31/05 wk9-a-8

9 2D sinusoids [ 2π ( ux vy) ] E cos + 0 tanψ = Λ = u u v v 2 y Λ [ i2π ( ux vy) ] E exp + 0 corresp. phasor x ψ min max 10/31/05 wk9-a-9

10 10/31/05 wk9-a-10 Spatial (2D) Fourier Transforms

11 The 2D Fourier integral (aka inverse Fourier transform) g ( ) ( ) ( ) x, y G u, v e + i2 π ux+ vy = dudv 10/31/05 wk9-a-11 superposition complex weight, expresses relative amplitude (magnitude & phase) of superposed sinusoids sinusoids

12 The 2D Fourier transform The complex weight coefficients G(u,v), aka Fourier transform of g(x,y) are calculated from the integral ( ) ( ) ( ), v g x, y e i2 π ux+ vy G u = dxdy (1D so we can draw it easily... ) g(x) Re[e -i2πux ] Re[G(u)]= x dx 10/31/05 wk9-a-12

13 2D Fourier transform pairs (from Goodman, Introduction to Fourier Optics, page 14) 10/31/05 wk9-a-13

14 Space and spatial frequency representations SPACE DOMAIN g(x,y) i2 π ( ux+ vy) (, v) g( x, y) e G u = dxdy g + i2 π ( ux+ vy) ( x, y) G( u, v) e = dudv 2D Fourier transform SPATIAL FREQUENCY DOMAIN G(u,v) 2D Fourier integral aka inverse 2D Fourier transform 10/31/05 wk9-a-14

15 Periodic Grating /1: vertical y v x u Space domain y v Frequency (Fourier) domain x u 10/31/05 wk9-a-15

16 Periodic Grating /2: tilted y v x u Space domain y v Frequency (Fourier) domain x u 10/31/05 wk9-a-16

17 Superposition: multiple gratings + + Space domain Frequency (Fourier) domain 10/31/05 wk9-a-17

18 Superpositions: spatial frequency representation Space domain Frequency (Fourier) domain 10/31/05 wk9-a-18

19 Superpositions: spatial frequency representation 10/31/05 wk9-a-19 space domain spatial frequency domain g ( x, y) G ( u, v) = I{ g( x, y) }

20 Fourier transform properties /1 Fourier transforms and the delta function I I { δ ( x, y) } = 1 { exp[ i2π ( u x + v y) ]} = δ ( u u ) δ ( v v ) Linearity of Fourier transforms if I then 0 0 { g ( x, y) } = G ( u, v) and I{ g ( x, y) } = G ( u, v) 1 I { a g ( x, y) + a g ( x, y) } = a G ( u, v) + a G ( u, v) 1 1 for any pair of complex numbers 2 1 a 1, 1 0 a /31/05 wk9-a-20

21 Shift theorem (space frequency) I Shift theorem (frequency space) I Scaling theorem I 10/31/05 wk9-a-21 Fourier transform properties /2 { g ( x, y) } G( u, v) Let I = { g ( x x, y y )} = G( u, v) exp[ i π ( ux + vy )] { g( x, y) exp[ i2π ( ux + vy )]} = G( u u v v ) 1 ab u a v b { ( )} g ax, by = G, 0 0 0, 0 0 0

22 Modulation in the frequency domain: the shift theorem Space domain Frequency (Fourier) domain 10/31/05 wk9-a-22

23 Size of object vs frequency content: the scaling theorem Space domain Frequency (Fourier) domain 10/31/05 wk9-a-23

24 Convolution theorem (space frequency) I Fourier transform properties /3 { f ( x, y) } = F( u, v) and I{ h( x, y) } H ( u, v) Let I = Let g ( x, y) = f ( x, y ) h( x x, y y ) dx dy { g ( x, y) } = F( u, v) H ( u, v) Let (, v) = F( u, v ) H ( u u, v v ) du dv Q u Convolution theorem (frequency space) ( u, v) = I{ f ( x, y) h( x y) } Q, 10/31/05 wk9-a-24

25 Correlation theorem (space frequency) I Fourier transform properties /4 { f ( x, y) } = F( u, v) and I{ h( x, y) } H ( u, v) Let I = Let g ( x, y) = f ( x, y ) h( x x, y y) dx dy { ( )} ( ) * g x, y = F u, v H ( u, v) Let (, v) = F( u, v ) H ( u u, v v) du dv Q u Correlation theorem (frequency space) { } ( ) ( ) * u, v = I f x, y h ( x y) Q, 10/31/05 wk9-a-25

26 Spatial frequency representation 10/31/05 wk9-a-26 space domain 3 sinusoids Fourier domain (aka spatial frequency domain)

27 Spatial filtering 10/31/05 wk9-a-27 space domain 2 sinusoids (1 removed) Fourier domain (aka spatial frequency domain)

28 Spatial frequency representation 10/31/05 wk9-a-28 space domain ( x y) g, Fourier domain (aka spatial frequency domain) ( u, v) = I{ g( x y) } G,

29 Spatial filtering (low pass) removed high-frequency content 10/31/05 wk9-a-29 space domain Fourier domain (aka spatial frequency domain)

30 Spatial filtering (band pass) removed high- and low-frequency content 10/31/05 wk9-a-30 space domain Fourier domain (aka spatial frequency domain)

31 Example: optical lithography 10/31/05 wk9-a-31 original pattern ( nested L s ) mild low-pass filtering Notice: (i) blurring at the edges (ii) ringing

32 Example: optical lithography 10/31/05 wk9-a-32 original pattern ( nested L s ) severe low-pass filtering Notice: (i) blurring at the edges (ii) ringing

33 2D linear shift invariant systems Fourier input g i (x,y) transform G i (u,v) g o convolution with impulse response ( x y ) = g ( x, y) h( x x, y y) dxdy, i LSI multiplication with transfer function ( u v) G ( u, v) H ( u v) G, o, = i output g o (x, y ) transform inverse Fourier G o (u,v) 10/31/05 wk9-a-33

34 2D linear shift invariant systems Fourier input g i (x,y) transform G i (u,v) g o convolution with impulse response ( x y ) = g ( x, y) h( x x, y y) dxdy, i SPACE DOMAIN LSI multiplication with transfer function ( u v) G ( u, v) H ( u v) G, o, = i output g o (x, y ) transform inverse Fourier G o (u,v) 10/31/05 wk9-a-34 SPATIAL FREQUENCY DOMAIN

35 2D linear shift invariant systems Fourier input g i (x,y) transform G i (u,v) g o convolution with impulse response ( x y ) = g ( x, y) h( x x, y y) dxdy, i LSI multiplication with transfer function ( u v) G ( u, v) H ( u v) G, o, = i output g o (x, y ) transform h(x,y), H(u,v) are 2D F.T. pair inverse Fourier G o (u,v) 10/31/05 wk9-a-35

36 Sampling space and frequency δx :pixel size δu :frequency resolution u max space domain 10/31/05 wk9-a-36 x :field size Nyquist relationships: u max = 1 2δx δu = 1 2 x spatial frequency domain

37 The Space Bandwidth Product Nyquist relationships: from space spatial frequency domain: u max = 1 2δx from spatial frequency space domain: x 2 = 1 2δu x δx = 2u δu max N : 1D Space Bandwidth Product (SBP) aka number of pixels in the space domain 2D SBP ~ N 2 10/31/05 wk9-a-37

38 SBP: example 10/31/05 wk9-a-38 space domain δx = 1 x = 1024 u Fourier domain (aka spatial frequency domain) 4 = 0.5 δu = 1/1024 = max 10

The spatial frequency domain

The spatial frequency domain The spatial frequenc /3/5 wk9-a- Recall: plane wave propagation λ z= θ path dela increases linearl with E ep i2π sinθ + λ z i2π cosθ λ z plane of observation /3/5 wk9-a-2 Spatial frequenc angle of propagation?