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The Spatial Frequency Domain

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The Spatial Frequency Domain The spatial frequency domain MIT 2.71/2.710 Optics 10/31/05 wk9-a-1 Recall: plane wave propagation x path delay increases linearly with x λ ⎛ x E0 exp⎜i2π sinθ + ⎝ λ z ⎞ i2π cosθ ⎟ λ ⎠ θ z=0 z plane of observation MIT 2.71/2.710 Optics 10/31/05 wk9-a-2 Spatial frequency ⇔ angle of propagation? x λ electric field θ z=0 z plane of observation MIT 2.71/2.710 Optics 10/31/05 wk9-a-3 Spatial frequency ⇔ angle of propagation? x λ θ z=0 z plane of observation MIT 2.71/2.710 Optics 10/31/05 wk9-a-4 Spatial frequency ⇔ angle of propagation? The cross-section of the optical field with the optical axis is a sinusoid of the form ⎛ sinθ ⎞ E0 exp⎜i2π x +φ0 ⎟ ⎝ λ ⎠ i.e. it looks like sinθ E exp()i2π u x +φ where u ≡ 0 0 λ is called the spatial frequency MIT 2.71/2.710 Optics 10/31/05 wk9-a-5 2D sinusoids corresp. phasor E0 cos[]2π ()ux + vy E0 exp[i2π (ux + vy)] y x MIT 2.71/2.710 Optics 10/31/05 wk9-a-6 2D sinusoids corresp. phasor E0 cos[]2π ()ux + vy E0 exp[i2π (ux + vy)] y x min max MIT 2.71/2.710 Optics 10/31/05 wk9-a-7 2D sinusoids corresp. phasor E0 cos[]2π ()ux + vy E0 exp[i2π (ux + vy)] y 1/u 1/v x min max MIT 2.71/2.710 Optics 10/31/05 wk9-a-8 2D sinusoids corresp. phasor E0 cos[]2π ()ux + vy E0 exp[i2π (ux + vy)] u tanψ = y v 1 Λ = 2 2 u + v Λ x ψ min max MIT 2.71/2.710 Optics 10/31/05 wk9-a-9 Spatial (2D) Fourier Transforms MIT 2.71/2.710 Optics 10/31/05 wk9-a-10 The 2D2D Fourier integral (aka inverseinverse FourierFourier transformtransform) g()x, y = ∫ G(u,v) e+i2π (ux+vy) dudv superposition sinusoids complex weight, expresses relative amplitude (magnitude & phase) of superposed sinusoids MIT 2.71/2.710 Optics 10/31/05 wk9-a-11 The 2D2D Fourier transform The complex weight coefficients G(u,v), aka FourierFourier transformtransform of g(x,y) are calculated from the integral G()u,v = ∫ g(x, y) e−i2π (ux+vy) dxdy (1D so we can draw it easily ... ) g(x) Re[e-i2πux] Re[G(u)]=∫ x dx MIT 2.71/2.710 Optics 10/31/05 wk9-a-12 2D2D Fourier transform pairspairs (from Goodman, Introduction to Fourier Optics, page 14) MIT 2.71/2.710 Optics 10/31/05 wk9-a-13 Space and spatial frequency representations SPACE DOMAIN g(x,y) G()u,v = ∫ g (x, y ) e−i2π (ux+vy) dxdy g(x, y) = ∫ G(u,v) e+i2π (ux+vy) dudv 2D2D FourierFourier transformtransform 2D2D FourierFourier integralintegral aka inverseinverse 2D2D FourierFourier transformtransform SPATIAL FREQUENCY DOMAIN G(u,v) MIT 2.71/2.710 Optics 10/31/05 wk9-a-14 Periodic Grating /1: vertical y v x u Frequency Space (Fourier) domain y v domain x u MIT 2.71/2.710 Optics 10/31/05 wk9-a-15 Periodic Grating /2: tilted y v x u Frequency Space (Fourier) domain y v domain x u MIT 2.71/2.710 Optics 10/31/05 wk9-a-16 Superposition: multiple gratings ++ Frequency Space (Fourier) domain domain MIT 2.71/2.710 Optics 10/31/05 wk9-a-17 Superpositions: spatial frequency representation Frequency Space (Fourier) domain domain MIT 2.71/2.710 Optics 10/31/05 wk9-a-18 Superpositions: spatial frequency representation space domain spatial frequency domain g()x, y G(u,v) = ℑ{g(x, y)} MIT 2.71/2.710 Optics 10/31/05 wk9-a-19 Fourier transform properties /1 • Fourier transforms and the delta function ℑ{}δ ()x, y =1 ℑ{}exp[]i2π (u0 x + v0 y) = δ (u − u0 )δ (v − v0 ) • Linearity of Fourier transforms if ℑ{}g1()x, y = G1(u,v)() and ℑ{g2 x, y }= G2 (u,v) then ℑ{}()()a1g1()x, y + a2 g2 ()x, y = a1G1 u,v + a2G2 u,v for any pair of complex numbers a1, a2. MIT 2.71/2.710 Optics 10/31/05 wk9-a-20 Fourier transform properties /2 Let ℑ{}g()x, y = G(u,v) • Shift theorem (space → frequency) ℑ{}g()x − x0 , y − y0 = G(u,v)exp[− i2π (ux0 + vy0 )] • Shift theorem (frequency → space) ℑ{}g()x, y exp[i2π (ux0 + vy0 )] = G(u − u0 ,v − v0 ) • Scaling theorem 1 ⎛ u v ⎞ ℑ{}g()ax,by = G⎜ , ⎟ ab ⎝ a b ⎠ MIT 2.71/2.710 Optics 10/31/05 wk9-a-21 Modulation in the frequency domain: the shift theorem Frequency Space (Fourier) domain domain MIT 2.71/2.710 Optics 10/31/05 wk9-a-22 Size of object vs frequency content: the scaling theorem Frequency Space (Fourier) domain domain MIT 2.71/2.710 Optics 10/31/05 wk9-a-23 Fourier transform properties /3 Let ℑ{}f ()x, y = F(u,v) and ℑ{}h(x, y) = H (u,v) Let g()x, y = ∫ f (x′, y′)⋅h(x − x′, y − y′) dx′dy′ • Convolution theorem (space → frequency) ℑ{}g()x, y = F(u,v)⋅ H (u,v) Let Q()u,v = ∫ F(u′,v′)⋅ H (u − u′,v − v′) du′dv′ • Convolution theorem (frequency → space) Q()u,v = ℑ{f (x, y)⋅h(x, y)} MIT 2.71/2.710 Optics 10/31/05 wk9-a-24 Fourier transform properties /4 Let ℑ{}f ()x, y = F(u,v) and ℑ{}h(x, y) = H (u,v) Let g()x, y = ∫ f (x′, y′)⋅h(x′ − x, y′ − y) dx′dy′ • Correlation theorem (space → frequency) ℑ{}g()x, y = F(u,v)⋅ H * (u,v) Let Q()u,v = ∫ F(u′,v′)⋅ H (u′ − u,v′ − v) du′dv′ • Correlation theorem (frequency → space) Q()u,v = ℑ{f (x, y)⋅h* (x, y)} MIT 2.71/2.710 Optics 10/31/05 wk9-a-25 Spatial frequency representation space domain Fourier domain 3 sinusoids (aka spatial frequency domain) MIT 2.71/2.710 Optics 10/31/05 wk9-a-26 Spatial filtering space domain Fourier domain 2 sinusoids (1 removed) (aka spatial frequency domain) MIT 2.71/2.710 Optics 10/31/05 wk9-a-27 Spatial frequency representation space domain Fourier domain g()x, y (aka spatial frequency domain) MIT 2.71/2.710 Optics 10/31/05 wk9-a-28 G(u,v) = ℑ{g(x, y)} Spatial filtering (low–pass) removed high-frequency content space domain Fourier domain (aka spatial frequency domain) MIT 2.71/2.710 Optics 10/31/05 wk9-a-29 Spatial filtering (band–pass) removed high- and low-frequency content space domain Fourier domain (aka spatial frequency domain) MIT 2.71/2.710 Optics 10/31/05 wk9-a-30 Example: optical lithography original pattern mild (“nested L’s”) low-pass filtering Notice: (i) blurring at the edges (ii) ringing MIT 2.71/2.710 Optics 10/31/05 wk9-a-31 Example: optical lithography original pattern severe (“nested L’s”) low-pass filtering Notice: (i) blurring at the edges (ii) ringing MIT 2.71/2.710 Optics 10/31/05 wk9-a-32 2D2D linear shift invariant systems input output convolution with impulse response gi(x,y) go(x’, y’) inverse Fourier g x′, y′ = g x, y h x′ − x, y′ − y dxdy inverse Fourier o ()∫ i ( ) ( ) transform transform LSI Fourier Fourier transform transform multiplication with transfer function Gi(u,v) Go(u,v) Go (u,v) = Gi (u,v)( H u,v ) MIT 2.71/2.710 Optics 10/31/05 wk9-a-33 2D2D linear shift invariant systems SPACE DOMAIN input output convolution with impulse response gi(x,y) go(x’, y’) inverse Fourier g x′, y′ = g x, y h x′ − x, y′ − y dxdy inverse Fourier o ()∫ i ( ) ( ) transform transform LSI Fourier Fourier transform transform multiplication with transfer function Gi(u,v) Go(u,v) Go (u,v) = Gi (u,v)( H u,v ) SPATIAL FREQUENCY DOMAIN MIT 2.71/2.710 Optics 10/31/05 wk9-a-34 2D2D linear shift invariant systems input output convolution with impulse response gi(x,y) go(x’, y’) inverse Fourier g x′, y′ = g x, y h x′ − x, y′ − y dxdy inverse Fourier o ()∫ i ( ) ( ) transform transform LSI h(x,y), H(u,v) Fourier Fourier transform transform are 2D F.T. pair multiplication with transfer function Gi(u,v) Go(u,v) Go (u,v) = Gi (u,v)( H u,v ) MIT 2.71/2.710 Optics 10/31/05 wk9-a-35 Sampling space and frequency δx :pixel size δu :frequency resolution umax space ∆x :field size spatial domain frequency Nyquist 1 1 umax = δu = domain relationships: 2δx 2∆x MIT 2.71/2.710 Optics 10/31/05 wk9-a-36 The Space–Bandwidth Product Nyquist relationships: 1 from space → spatial frequency domain: u = max 2δx ∆x 1 from spatial frequency → space domain: = 2 2δu ∆x 2u = max ≡ N : 1D Space–Bandwidth Product (SBP) δx δu aka number of pixels in the space domain 2D SBP ~ N 2 MIT 2.71/2.710 Optics 10/31/05 wk9-a-37 SBP: example space domain Fourier domain δx =1 ∆x =1024 (aka spatial frequency domain) MIT 2.71/2.710 Optics −4 10/31/05 wk9-a-38 umax = 0.5 δu =1/1024 = 9.765625×10.
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