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 π λ ⎛ λ x θ E0 exp⎜i2 sin + ⎝ π z ⎞ i2 λcosθ ⎟ ⎠ 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 π θ ⎛ sin ⎞ E0 exp⎜i2 λ x +φ0 ⎟ ⎝ ⎠ i.e. it looks like π sinθ E exp()i2 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 1δ 1 umax = δu = domain relationships: 2 x 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 δ = max ≡ N : 1D Space–Bandwidth Product (SBP) x 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