[email protected] dept ece, ucb 425, ecee 232, x24661...

$: kelvin@colorado.edu Dept ECE, UCB 425, ECEE 232, x24661 ...ecee.colorado.edu/ecen5606/VUGRAPHS/FO-lab18-nup.pdf · Diffraction Theory Franhoffer and Fresnel Diffraction Coherent Optical$
ECEN 5696 Fourier Optics

Professor Kelvin WagnerDept ECE, UCB 425, ECEE 232, x24661

[email protected] you will learn in this lab

• Fourier transforms in 1-D time and 2-D space.

• Diffraction and imaging. Plane waves and k-space

– Propagation to the far field is given by a spatial Fourier transform

• A lens takes a Fourier transform

• 4F afocal imaging systems and spatial filtering in the Fourier plane

• Holographic spatial filtering for pattern recognition

– Dynamic polarization holography in doped dye-polymer

• Computer Generated HolographyKelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 1

Fourier Optics Learning Objectives

• Review Fourier transforms and develop deep intuitive understanding

• Generalize the Fourier transform to 2-D images and fields

• Construct arbitrary solutions to Maxwell’s Eqn as a superposition of plane waves

• Understand how waves propagate through space and are focused by lenses

• Develop a clear intuition for the propagation of plane wavesand Gaussian beams

• Understand the pattern matching capability of holographicFourier spatial filter

• Appreciate the capabilities of real-time dye-doped holographic material

• Extend the ideas of holography to computer generated and digital holography

Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 2

Suggested References for AdditionalReading

Texts and suggested references:

J. Goodman ,Introduction to Fourier Optics, 3rd Ed

J. Shamir,Optical Systems & Processes

J. Gaskill,Linear Systems, Fourier Transforms, and Optics

T. Cathey,Optical Information Processing and Holography

B. Saleh,Fundamentals of PhotonicsChapter 4

D. Brady,Optical Imaging and Spectroscopy, 2009

D. Voelz,Computational Fourier Optics: A MATLAB Tutorial, 2011

J. SchmidtNumerical Simulation of Optical Wave Propagation, 2011

N. GeorgeFourier Optics, 2012 on-line short manuscript

R.K. Tyson,Principles and Applications of Fourier Optics, 2014

Kedar Khare,Fourier Optics and Computational Imaging, 2016


Lecture Outline

Linear Systems and Fourier Transforms2-D Systems and Transforms, OperatorsWave Propagation, momentum spaceDiffraction Theory

Franhoffer and Fresnel DiffractionCoherent Optical Imaging, 4F afocal imaging, and Spatial FilteringHolographyOptical Information Processing and Optical Correlations

Aside into dynamic photoanisotropic holographic materialsComputer Generated Holography


2-D Linear Space Invariant Systems

General Shift Variant Linear Transformation

g(x, y) =

� ∞

−∞

� ∞

−∞f(x′, y′)h(x′, y′; x, y)dx′dy′

input impulses at differentx′, y′ yield different outputsh(x, y; x′, y′)

Space Invariant Linear System

g(x, y) =

� ∞

−∞

� ∞

−∞f(x′, y′)h(x− x′, y − y′)dx′dy′ g = h ∗ ∗f = F−1

xy {HF}

2-D impulse responseh(x, y)

Separable case

f(x)g(y) ∗ ∗q(x)r(y) = [f(x) ∗ q(x)] [g(y) ∗ r(y)]

Correlation in 2-D

Cfg(x, y)=

�f(x′, y′)g∗(x′−x, y′−y)dx′dy′=f ⋆⋆g=f(x, y)∗∗g∗(−x,−y) ↼⇁ FG∗

eg (f ∗∗g)⋆⋆(f ∗∗g) = (f ⋆⋆f)∗∗(g ⋆⋆g) ↼⇁ FG · (FG)∗ = |F |2|G|2


2D convolution with impulse response:Basis of bandlimited imaging

• Copy of Impulse Responseis “popped” at each shiftedaδ(x− x0, y − y0)

• Preserves amplitude andphase of each impulse

• Coherent summation of eachweighted positive and nega-tive sidelobe

• Observed image is modsquared of complex amplitude


The 1-D Temporal Fourier Transform:Definitions

Forward temporal Fourier transform (Hz)

G(f) =

�g(t)e−i2πftdt = F{g(t)}

Inverse transform

g(t) =

�G(f)ei2πftdf = F−1{G(f)}

Alternate definition using angular radian frequencyω = 2πf

Forward temporal Fourier transform (rad/sec)

G(ω) =

�g(t)e−iωtdt = F{g(t)} ω = 2πf

Inverse transform

g(t) =1

2π

�G(ω)eiωtdω = F−1{G(ω)} ≡ F−1

t {G(ω)} dω = 2πdf

Note that these FT functions are scaled versions of each other G(ω) = G(ω/2π)


Exactly Analogous 1-D Spatial FourierTransform: Definitions

Can similarly define FT in space using spatial frequency,u = fx [lines/mm], analogousto temporal frequencyf [Hz], or use wavevector,kx [rad/mm], analogous to angularfrequencyω [rad/sec].

Forward 1-D spatial Fourier transform

G(u) =

�g(x)e−i2πuxdx = Fx{g(x)}

Inverse 1-D spatial Fourier transform

g(x) =

�G(u)ei2πuxdu = F−1

x {G(u)}

or in terms of wavevectorkx

G(kx) =

�g(x)e−ikxxdx = F{g(x)} ≡ Fx{g(x)}

g(x) =1

2π

�G(kx)e

ikxxdkx = F−1{G(kx)} ≡ F−1x {G(kx)} ≡ F−1

kx{G(kx)}

Note that these FT functions are scaled versions of each other G(kx) = G(kx/2π)


2-dimensional Fourier transforms

rect(x, y) ↼⇁ sinc(u, v)rect(xa,

yb) ↼⇁ absinc(au, bv)

tri(x, y) ↼⇁ sinc2(u, v)

e−π(x2+y2) = e−πr2 ↼⇁ e−π(u2+v2) = e−πρ2

e−iπr2 ↼⇁ −ieiπρ2

δ(x, y) ↼⇁ 1(u, v)δ(x− x0, y − y0) ↼⇁ e−i2π(x0u+y0v)

δ(x)1(y) ↼⇁ 1(u)δ(v)ei2π(u0x+v0y) ↼⇁ δ(u− u0, v − v0)

cos[2π(u0x + v0y)] ↼⇁ 12 [δ(u−u0,v−v0)+δ(u+u0,v+v0)

comb(x, y) ↼⇁ comb(u, v)

circ(ra

)↼⇁ πa2J1(2πaρ)πaρ

Seperabilityf(x)g(y) ↼⇁ F (u)G(v)

RotationRθ{f(x, y)} ↼⇁ Rθ{F (u, v)}

Projection-Slice Theoremf(x, y) ∗ ∗Rθ{δ(x)1(y)} ↼⇁ F (u, v) · Rθ{1(u)δ(v)}


Separable functions

f(x, y) = p(x)q(y)

Fourier transform is separable because kernel is separable(This is the basis of the FFT)

F (u, v) =

�p(x)q(y)e−i2π(ux+vy)dxdy =

�p(x)q(y)e−i2πuxe−i2πvydxdy

=

�p(x)e−i2πuxdx

�q(y)e−i2πvydy = P (u)Q(v)

p(x)q(y) ↼⇁ P (u)Q(v)

Thus can use 1-D FT tables

eg

Π(x2

)e−π(y/a)2 ↼⇁ 2sinc2u · ae−π(av)2


Fourier Rotation Theorem

Rθ { } operator that rotates image byθ (CCW, RH) about origin

f ↼⇁ F

Rθ {f} ↼⇁ Rθ {F}Consider the rotation operating on a vector (image ofδ(x− x0, y− y0) )

Rθ

[x0y0

]⇒[x0 cos θ − y0 sin θx0 sin θ + y0 cos θ

]

Rθ

[10

]⇒[cos θsin θ

]

Rθ

[01

]⇒[− sin θcos θ

]

Rotated images in cartesian or polar component notation

f (x cosθ−y sinθ, x sinθ+y cosθ) ↼⇁ F (u cosθ−v sinθ, u sinθ+v cosθ)

g(r, θ − α) ↼⇁ G(ρ, φ− α)


Projection Slice Theorem

1DF

F2D

�f(x, y)dy = p0(x) 0 degree projection of realn domain

F (u, v) = F2D{f(x, y)} = F (r sin θ, r cos θ) = F (r, θ)

S0(u) = F1D{p0(x)} = F (r, 0) 0 degree slice of Fourier plane

Arbitrary angleθ0

pθ0(x′) =

�f(x′ cos θ0 − y′ sin θ0, x

′ sin θ0 + y′ cos θ0)dy′

Sθ0(u′)=F{pθ0(x′)}=F (u′, θ0)=F (u′ cosθ0, u

′ sinθ0)=R−θ0

{F{p0{Rθ0{f(x, y)}}}

∣∣∣∣∣v=0

}

f ∗ ∗Rθ0 {δ(x) · 1(y)}Fxy⇐⇒F · Rθ0 {1(u) · δ(v)}


Polar CoordinatesVery useful for circularly symetric functionsDepends on choice of origin

x = r cos θ u = ρ cosφ

y = r sin θ v = ρ sinφ

r =√x2 + y2 ρ =

√u2 + v2

θ = tan−1(yx

)φ = tan−1

(vu

) Fx

y

u

v

rθ

ρφ

dxdyrdrdθ

G(ρ, θ) =

� 2π

0

� ∞

0

g(r, θ)e−i2πrρ (cos θ cosφ + sin θ sinφ)︸︷︷︸rdrdθ

cos(θ − φ)

Separable caseg(r, θ) = gR(r)gΘ(θ)

F{g(r, θ)} =

� ∞

0

rgR(r)

� 2π

0

gΘ(θ)e−i2πrρ cos(θ−φ)dθdr

gΘ(θ) is periodic inθ and can be written as a Fourier series

gΘ(θ) =

∞∑

m=−∞cme

imθ cm =1

2π

� 2π

0

gΘ(θ)e−imθdθ


Rectangular to Polar


Fourier Bessel identify and HankelTransforms

use Bessel Fourier identity for FM modulation

eiα sin x =∞∑

m=−∞Jm(α)e

imx

to writee(−i2πrρ) sin(θ−φ+π/2) =

∑

m

Jm(2πrρ)eim(θ−φ)im

Thus

F{g(r, θ)} =

� ∞

0

rgR(r)

� 2π

0

gΘ(θ)∑

m′c′me

im′θ∑

m

Jm(2πrρ)eim(θ−φ)imdθdr

remember orthogonality of eigenfunctions� 2π

0

eim′θeimθdθ = δmm′ · 2π

=∞∑

m=−∞cm(i)

me−imφ · 2π� ∞

0

rgq(r)Jm(2πrρ)dr

︸︷︷︸Hankel transform GH(ρ) = Hm{gR(r)}


Circularly Symmetric

g(r, θ) = g0(r)1(θ) c0 = 1 ci = 0 ∀ i 6= 1

F{g(r, θ)} = 2π

� ∞

0

rg0(r)J0(2πρr)dr = G0(ρ) = H0{g0(r)}

circular symmetric in space↼⇁ circular symmetric in 2-D spatial frequency

g0(ar) ⇐⇒ 1

|a|2G0(ρ/a)

Circular Aperture

F{circ(ra

)}= |a|��2J1(2πaρ)

��aρ= |a|J1(2πaρ)

ρ


FT of unit circular disk

cyl(r) =

{1 r < 1

20 r > 1

2

area = π/4 0.5 1.0

circ(r)

cyl(r)

area=π

area=π/4 circ(r) =

{1 r < 10 r > 1

area = π

F{cyl(r)} = 2π

� .5

0

rJ0(2πrρ)dr r′ = 2πrρr = 0 → r′ = 0r = .5 → r′ = 2π.5ρ = πρ

Bessel identity� x

0 ηJ0(η)dη = xJ1(x)

= 2π

� πρ

0

r′

2πρJ0(r

′)dr′

2πρ=

1

2πρ2πρJ1(πρ) =

1

2ρJ1(πρ) =

π

4somb(ρ)

somb(ρ) = 2J1(πρ)

πρ

jinc(ρ) =J1(πρ)

2ρAperture of diameterD

F{cyl( r

D

)}=

D2π

4somb(Dρ) =

D��2ZZπ

��4

��4ZZπ

1

2��DρJ1(πDρ) = Djinc(Dρ)

First null of Airy patternDρ = 1.22Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 17

Jinc function

jinc x =J1(πx)

2xNulls occur at radii1.220, 2.233, 3.239, 4.241, 5.243 · · · ∼ n+ 1

4Peak isπ4 . jinc (.70576) = π

8 = 3dB width.Asymptotic expression forx > 3

jinc x ∼ cos[π(x− 3/4)]

πx√2x

Compare with slow assymptotic decay ofJ0(x) ∼√

2πx

cos(x− π/4) due to impulsiveHankel transform, while rapid decay ofjinc x is beacause transform has only a stepdiscontinuity. More rapid decay ofjinc 2x is due to even smoother form of its transform.Integral and volume� ∞

−∞jinc xdx = 1

� ∞

−∞jinc

√x2 + y2dxdy = 2π

� ∞

0

jinc r r dr = 1

1-D Fourier transform

Fx{jinc x} =√1− (2u)2rect u

2-D Fourier transform

Fxy{jinc√x2 + y2} = rect

√u2 + v2

0.5

cyl(ρ)

area=π/4

0.5-0.5

u

v

u


J0(πr) function in 2D and its FourierTransform


jinc (r) function in 2D and its FourierTransform


J20 (πr) and jinc 2(r) function in 2D and their2-D Fourier Transform

1st threesidelobes8dB, 10dB,and 12dBdown frompeak

1st threesidelobes18dB, 24dB,and 28dBdown frompeak


Matching a 2-D object with its Fouriertransform

Match the Object

a) b) c)

d) e) f)

g) h) i)

j) k) l)

With the 2-D Fourier transform

2 2 2

2 2 2

2 2 2

2 2 2


History of the Theory of Diffraction

Sommerfield defines diffraction as a deviation from rectilinear propagation that is notreflection or refractionHuygens in 1678 postulated secondary spherical sources

Newton in 1704 believed in corpuscular theory.Fought against acceptance of diffraction

Young in 1804 demonstrated 2-slit interference

Fresnel in 1818 developed an elegant theory of diffraction patternsPoisson objected since it predicted a bright spot in the shadow: “absurd”

Experimentally verified by Arago. One of the most profound predictions in sci-ence

Maxwell’s equations formulated in 1860

Kirchoff in 1882 showed that secondary sources were a consequence of the wave natureof light. Theory used 2 boundary values

Sommerfeld and Poincare showed Kirchoff’s theory had inconsistent boundary values

In 1896 Rayleigh-Sommerfeld formulated rigorous Green’s function diffraction theoryScalar theory with consistent boundary conditions


Linear Systems Viewpoint on Diffraction

Description of field propagating from input plane to output plane is linear

a1(x, y) =

�a0(x

′, y′)h(x′, y′;x, y)dx′dy′

Clearly system is space invariant. Shifting aδ(x, y) (or any other input) around in inputplane will shift output around identically. The propertiesof free space are identical atshifted locations

a1(x, y) =

�a0(x

′, y′)h(x− x′, y − y′)dx′dy′

whereh(x, y) is the impulse response of free spaceExpecth(x, y) to

1. Look like an expanding spherical wave2. Keep power normalized3. Conserve power flow

x’y’

xy

z

Convolution Theorem gives simpler Fourier domain description of diffraction

A1(u, v) = A0(u, v)H(u, v)

whereH(u, v) = Fxy{h(x, y)} is transfer function of free space


Scalar Diffraction Theory

Suppose we know the fieldEi(x, y) on some plane.What isE0(x

′, y′) on some other plane?

r

r’

R=r’-r

zx

y

n

Rayleigh-Sommerfeld

Eo(~r′) =

�S

Ei(~r)eikR

iλ|~R|cos(n, ~R)dS

Huygens waveletsnormalization90◦ phase shiftobliquity factor

Fresnel Approximation

R =√

(z′ − z)2 + (x′ − x)2 + (y′ − y)2 ≈ (z′ − z)

√1 +

(x′ − x)2 + (y′ − y)2

(z′ − z)2

≈ (z′ − z) +(x′ − x)2 + (y′ − y)2

2(z′ − z)

Using√1 + ǫ ≈ 1 + ǫ

2 − ǫ2

8


Fresnel Regimespherical Huygens wavelets≈ quadratic surfaceparaxialcos(n, ~R) ≈ 1 good to 5% accuracy forθ < 18◦

Fromǫ2/8 ≪ 1 we getz3 ≫ π4λ[(xo − xi)

2 + (yo − yi)2]2

For a 1cm object=⇒ z > 23cm

xi

yi

0

oy

ox

z

???

within phase factor useR ≈ z + (x′−x)2+(y′−y)2

2zin amplitude factor useR ≈ z

Eo(xo, yo) =−ieikz

λz

� �A

Ei(xi, yi)ei k2z (xo−xi)

2+(yo−yi)2dxidyi

=eikz

iλzei

k2z (x

2o+y2o)

� �A

Ei(xi, yi)ei k2z (x

2i+y2i )e−ikz (xoxi+yoyi)dxidyi


Rectangular aperture diffraction

dark=brightaperture

dark=dimobstruction

Solid:Exact RS

Dotted:Paraxial

Dashed:Assymptotic

z0 = w2/λ


Circular Aperture 3+1D Beam PropagationCrossection


Imaging with Fresnel Zone Plate

��z2o + h2 = (zo + δ)2 =

��z2o + 2zoδ +�

��0

δ2��z2i + h2 = (zi + δ′)2 =

��z2i + 2ziδ

′ + ��0

δ′2

δ =h2

2zoδ′ =

h2

2zi∆ = δ + δ′ =

h2

2zo+

h2

2ziis OPD

Successive zones with an aditional half wavelength OPD arelabeled as successive fresnel zones with radial boundarieshm

∆m =mλ

2=

h2m

2

(1

zo+

1

zi

)⇒ hm =

√mλ

1/f=√mλf

Area ofmth annulus bounded byhm−1 andhm

Am = πh2m − πh2

m−1 = π(mλf − (m− 1)λf) = πλf

circular apertures that consist ofN zones will sum on-axisfields out of phase with equal amplitude contributions

h1

h2

h3

h4

h5

h6h7

h8

zo zi

hm

δ δ’

ATOT = A1 − A2 + A3 − A4 + · · · ± AN =

{N odd ≈ A1 ⇒ ITOT = A2

1

N even ≈ 0 ⇒ ITOT = 0


2-D crosssections every 8λ from aD = 16λCircular Aperture BPM


Circular disk diffraction: Fresnel/Arago’sBright Spot


Fraunhofer Regime

z ≫ kx2imax

+ y2imax

2Approximate over the entire aperture bounded by±ximax and±yimax

eik2z (x

2imax

+y2imax) ≈ eiǫ ≈ 1

HeNeλ = .6328µm,

ximax ≈ 2.5cm =⇒ z > 1.6kmximax ≈ 100µm =⇒ z > 5cmximax ≈ 10µm =⇒ z > .05cm

Fraunhoffer Approximation

Eo(xo, yo) =−ieikz

λz

� �A

Ei(xi, yi)ei k2z [(xo−xi)

2+(yo−yi)2]dxidyi

=eikz

iλzei

k2z (x

2o+y2o)

� �A

Ei(xi, yi)e−i2πλz (xoxi+yoyi)dxidyi

Fourier Transform using scaled spatial frequenciesfx =xoλz andfy =

yoλz


Square Aperture

λ

RectangularAperture

Crosssectionsinc(x)=sinπx πx

sinc(x)sinc(y)far-field

Z =w /λ20

Illuminated by normal monochromatic plane wave

Ei(xi, yi) = Π(xiX

)(yiY

)

Far-field diffraction pattern

Eo(xo, yo) =eikz

iλzei

k2z (x

2o+y2o)Xsinc(Xfx)Y sinc(Y fy)

Far-field intensity pattern

Io(xo, yo) = |Eo(xo, yo)|2 =X2Y 2

λ2z2sinc2

(Xxoλz

)sinc2

(Y xoλz

)


Sinusoidal Phase grating

t(x, y) = eim2 sin(2πf0x)Π

(xw

)Π( yw

)

Use Bessel identity

eim2 sin(2πf0x) =

∞∑

n=−∞Jq

(m2

)ei2πqf0x

PhaseGrating

corrugatedwavefron

resolved intoplane wave k-space

Λ={

2π ΛK =---t θ

=⇒ T (u, v) = F {t(x, y)} =

∞∑

n=−∞Jq

(m2

)δ(u− qf0, y) ∗ ∗wsinc(wu)wsinc(wv)

=∞∑

n=−∞Jq

(m2

)w2sinc(w(u− qf0), wv)

Fraunhoffer diffraction U (x, y) = eikz

iλz ei k2z (x

2o+y2o)T

(xλz ,

yλz

)

I(x, y) =1

λ2z2

∞∑

n=−∞J2q

(m2

)w4sinc2(w(u− qf0), wv)


Square Wave Amplitude Grating

1/2+tm

1/2-tm

1.0

0

1/2

L

x

Square Wave Amplitude Transmission1/2

1/2-tm

1/2-tm2tm

2tm

-tm

tm

Amplitude square wave of periodL grating can be represented in various ways

t(x) = comb L(x) ∗[(12 − tm)Π

(xL

)+ 2tmΠ

(x

L/2

)]

Thus the FT is given by

T (u) =1

Lcomb 1/L(u)

[(12− tm)Lsinc(Lu) + ��2tmL/��2sinc

(Lu

2

)]

With Fourier orderscn = (12 − tm)sinc(n) + tmsinc

(n2

)

Fraction of light power diffracted into each first order is given by

|c1|2 =∣∣∣∣(12 − tm)

sin 1π

1π+ tm

sinπ/2

π/2

∣∣∣∣2

=

∣∣∣∣(12 − tm)0

1π+ tm

1

π/2

∣∣∣∣2

=

(2tmπ

)2

< 10.13%


Square Wave Amplitude Grating

T (u) =1

Lcomb 1/L(u)

[(12− tm)Lsinc(Lu) + tmLsinc

(Lu

2

)]

tm = 0.5 tm = 0.2


Monochromatic Wave Eqn: Helmholtz eqn

Each component of the vector wave eqn satisfies the scalar wave eqn

∇2u(~r, t)− 1

c2u(~r, t) = 0

For monochromatic waves

u(~r, t)=a(~r) cos[ω0t+Φ(x, y, z)]=ae−i[ω0t+Φ]+a∗e+i[ω0t+Φ]=Ae−iω0t+A∗e+iω0t= u+u∗

Since wave eqn is linear, we can just solve for one sideband, add the other later bytaking real part

˙u = (−iω)u = (−iω)Ae−iω0t ¨u = (−iω)2u = −ω2u = −ω2Ae−iω0t

Helmholtz eqn for monocromatic envelopeA(~r)

∇2A(~r)e−iω0t +ω2

c2A(~r)e−iω0t = 0 ⇒ (∇2 + k20)A(~r) = 0

Note if wave contains multiple temporal frequencies (say 2 to start, then arbitrary dis-tribution later). We can solve monochromatic isotropic Helmholtz for each temporalfrequency component seperately using an identical solution method and then find thetotal field amplitude by summing monochromatic components.


Postulate and propagate a plane wave soln

u(x, y, z, t) = a0e−i(ωt−~k·~r) = a0e

i~k·~re−iωt = A(~r)e−iωt

where~k = (kx, ky, kz) =2πλ(α, β, γ) = 2π

λk = 2π(u, v, w) = 2π(fx, fy, fz)

direction cosines

α = k · x = cos θx

β = k · y = cos θy α2 + β2 + γ2 = 1

γ = k · z = cos θz

Plug into Helmholtz eqn, use∇ · A = i~k · A and∇2A = (i~k) · (i~k)A = −k2A

−k2A +ω2

c2A = −

(k2 − ω2

c2

)A = 0

⇒ k2 =ω2

c2=

(2πν)2

c2=

(2π

λ

)2

|k| = 2π

λ

So we must choose the magnitude of the wavevector~k appropriately and with such achoice any monochromatic plane wave is a solution⇒ sphere of allowed~k


Propagation of a plane wave

Thus if we have a plane wave at any location (eg on a plane) we know how it propagatesboth forward and backwards eg between 2 planes

zθz

k

Plane wave produces equal 2-D linear phase factors across any parallel plane that sim-ply phase advance with propagation

At z = 0 linear phase factor due to a plane wave

A(x, y) = ei(kxx+kyy) u(x, y, t) = ei(kxx+kyy)e−iωt

At any otherz

A(x, y : z) = ei(kxx+kyy)eikzz u(x, y, z, t) = ei(kxx+kyy)eikzze−iωt

Wherekz =√

k20 − k2x − k2y


Transfer Function of Free Space

Plane wave u(x, y, z; t) = pa0e−i(ωt−~k·~r) + cc

2-D planar phase factorei~kt·r across a plane,z = 0 advances with a phase factoreikzz.

kk

tkkz

ktk

kz

An arbitrary wave can be decomposed a superposition of planewaves

u(x, y; 0) =

�U (kx, ky; 0)e

i(kxx+kyy)dkxdky

propagation between planes by phase advancing each plane wave component

kz =√

k20 − k2x − k2y ≈ k0 −k2x + k2y2k0

u(x, y; z) =1

(2π)2

�U (kx, ky; 0)e

i(kxx+kyy)eikzzdkxdky

U (kx, ky; z) = U (kx, ky; 0)eiz√

k20−k2x−k2y ≈ U (kx, ky; 0)eik0ze

−ik2x+k2y2k0

z


Angular Spectrum

A(fx, fy; 0) = Fxy {a(x, y; 0)}Consider a plane wave with wavevector|~k| = 2π/λ k = αx + βy + γz

p(x, y, z, t) = ei(~k·~r−ωt) = ei

2πλ (αx+βy)ei

2πλ γze−iωt

where~r = xx + yy + zz and~k = 2πλ (αx + βy + γz)

tip of the~k is constrained to lie on a sphere of radis2πλ

from Helmholtz eqn.

(∇2 + k2

)a = 0

Sinceα2 + β2 + γ2 = 1 =⇒ γ =√1− α2 − β2

When this 3-D plane wave strikes planez = 0 a 2-D linear phase factor will be ob-served.

E(x, y) = ei2πλ (αx+βy) = ei2π(fxx+fyy) = ei(kxx+kyy)

α = λfx β = λfy =⇒ γ =√1− (λfx)2 − (λfy)2


Completeness of the Fourier Integral

Any monochromatic physical 2-D distribution of field amplitude can be represented asa sum of complex sinusoids.

a(x, y) = F−1xy {A(fx, fy)} =

1

(2π)2

�A(kx.ky)e

i(kxx+kyy)dkxdky

=

�A(fx.fy)e

i2π(fxx+fyy)dfxdfy =

�A

(α

λ,β

λ

)ei

2πλ (αx+βy)dαdβ

Each of these components can be associated with a 3-D plane wave which solves theHelmholtz equation, so we know how it propagates. The solution throughout all ofthe following homogeneous half space can thus be found from thez component of thewavevector or equivalently from the direction cosineγ =

√1− α2 − β2.

kz(kx, ky) =2π

λγ =

√k20 − k2x − k2y

a(x, y : z) =1

(2π)2

� �A(kx, ky; 0)e

i(kxx+kyy)eikzzdkxdky

A(kx, ky; z) = A(kx, ky; 0)ei√

k20−k2x−k2yz = A(kx, ky; 0)Hz(kx, ky)


Propagation of the Angular Spectrum

A(fx, fy; z) = Fxy {a(x, y, z)} a(x, y, z) = F−1xy {A(fx, fy; z)}

a must satisfy Helmholtz eqn, and linearity of the differential eqn indicates

0 =(∇2 + k2

)a =

(∂2

∂x2+

∂2

∂y2+

∂2

∂z2+ k2

)a(x, y, z)

=

� �(∇2 + k2)A(fx, fy; z)e

i2π(fxx+fyy)dfxdfy

=

� � [∂2

∂z2A(fx, fy; z) +

[(i2πfx)

2 + (i2πfy)2 + k2

]A(fx, fy; z)

]ei2π(fxx+fyy)dfxdfy

=

� � [∂2

∂z2A(fx, fy; z) +

(2π

λ

)2 [1− α2 − β2

]A(fx, fy; z)

]ei

2πλ (αx+βy)d

α

λdβ

λ= 0

=⇒ ∂2

∂z2A

(α

λ,β

λ; z

)+

(2π

λ

)2 [1− α2 − β2

]A

(α

λ,β

λ; z

)= 0

Solution of this 2nd order DE is complex exponentials

A

(α

λ,β

λ; z

)= A

(α

λ,β

λ; 0

)ei

2πλ

√1−α2−β2z


Propagating versus Evanescent waves

Propagating waves:plane waves with phase advance ratekz

α2 + β2 < 1

Evanescent waves:real exponentials decay or grow

α2 + β2 > 1

A(α

λ,β

λ; z) = A(

α

λ,β

λ; 0)e−µz

µ = ±2πλ

√α2 + β2 − 1

choose the root that yields the physicallyreasonable decaying exponential solution.

k =k +ik =xk +izµ

t’ e

t’

kt’

kt

k Propagating Wave

EvanescentWave

kx

kz

e-µz

z

k2π/λ


FT and direction cosines

f(x, y) = cos(2πv0y) v0 = 1/Λy ⇐⇒ F (u, v) =1

2[δ(v− v0) + δ(v+ v0)]

x

y

u

v

}=Λy

y1/Λ =v0

y1/Λ =-v 0

Direction Cosine Spaceα = λuβ = λvγ =

√1− α2 − β2

sinφx =α1

k · x = cos(φx − π2)

sinφy =β1 = λ

Λy

(0,β )0

(0,−β )0

α

β

γφy

1

1

Unit Sphere


k-sphere

y

zφy

{φy

φy

λ

Λy

Λy

λ

sin φ = =βλΛy y 0

k

k

~k-spacekx =

2π

Λx

ky =2π

Λy

sinφx =kxk0

=2π/Λx

2π/λ=

λ

Λx

sinφy =kyk0

=2π/Λy

2π/λ=

λ

Λy

(0,-k )y

(0,k )y

zk

xkyk

yφ

2π ω λ c

=


Plane Waves

direction cosines of a plane wave

E(x, y, z) = E0pei2πλ (αx+βy+γz) = E0pe

i(kxx+kyy+kzz)

α2 + β2 + γ2 = 1 k2x + k2y + k2z = k20 =

(2πn

λ

)2

wherek0 = |~k| = 2πn/λ in medium of indexn.

k-space

k

2πn/λ

kz

kx

x

zθ

k

λ

kz

kx

In 2-dimensionsE(x, z; t) = A0pe

ik0(x sin θ+z cos θ)e−i2πνt + cc

whereα = sin θ/λ andγ = cos θ/λ.ν ≈ 5× 1014 Hz λ ≈ .63× 10−6m (HeNe)


Interference

k-space2πn/λ

Λ

ko

kr

Kg2E Eo r

E +Eo r2 2

I(x)

X

I(x) = |Eobj(x, z) + Eref(x, z)|2

= E2o + E2

r + EoE∗r po · p∗rei

~ko·~re−i~kr·~r + cc

= E2o + E2

r + 2EoEr cos[2πk0(αx + γz)]


Spherical Waves

x

z eikr

x

z e-ikr

Isophase surfaces are spherical,φ(r) = const, wherer2 = x2 + y2 + z2

Nonparaxial Spherical Wave

A(r, t) =Ao

reikre−iωt + cc =

Ao√x2 + y2 + z2

eik√

x2+y2+z2e−iωt + cc

Paraxial Regime z ≫ max(x, y) so that(x2 + y2)/z2 ≪ 1

r = z

√1 +

x2 + y2

z2≈ z +

x2 + y2

2z

using√1 + ǫ = 1 + ǫ/2− ǫ2/8 + ...

Paraxial Focusing Spherical Wave

A(r, t) =Ao

zei(kz−ωt)e−ikx

2+y2

2z + cc


Positive vs Negative phases

e−iωt describes a clockwise rotation of the phasor with time.Remember positive angles are CCW.

If we look at a snapshot of a wave, the portions emittedlater will have further advancedin the clockwise direction, and thus the phase will be morenegative

For a spherical wave diverging from a point, the movement away from the source movesto points on the wavefront emitted earlier, since they had topropagate farther to reachthat point. Thus the phase must increase in the positive sense as we move away fromthe origin.

eikr Expanding spherical waveei(kr−ωt) Spatiotemporal

eik2z (x

2+y2) Quadratic phase factor

e−ikr Focussing wave

e−i k2z (x2+y2) QPF

k

kx

x

zeikr

Wavefront emitted earlier

Wavefrontemitted later

x

ze

-ikr

x

ik xxe

k >0x

z

z

zQPF

QPF


Plane wave focussed by a lens

Plane WaveA(~r, t) = ei(kz−ωt)

F

Strike a lens

Ain(~r, t) = ei(kz−ωt)

Aout(~r, t) = ei(kz−ωt)e−ikx2+y2

2f = Ain(~r, t)t(x, y)

→ t(x, y) = e−ikx2+y2

2f

Symmetricbest form

collimatingbest formfocussingplano plano meniscusmeniscus Doublet

SymmetricTriplet

Thinlens


Imaging Lens

Field incident on lens,expanding spherical wave

A−(r) = eikx

2+y2

2d0 ei(kz−ωt)d1d0

P P’

Multiplies by lens phase factor and produces converging spherical wave

A+(r) = A−(r)t(r) = e−ikx

2+y2

2d1 ei(kz−ωt)

t(r) =A+(r)

A−(r)= e

−ik[x2+y2

2

(1d1+ 1

d0

)

Since we know the lens imaging law from classical optics

1

f=

(1

d1+

1

d0

)

We get as beforet(r) = e−ikx

2+y2

2f = e−iπx2+y2

λf


Lens phase factor

-R 2

R1

0∆

R2

R - R -x -y22 2 2

2

r

R -x -y22 2 2

R1

R -x -y12 2 2

{r= x +y2 2}

R - R -x -y12 2 2

1 ∆(x, y) = ∆0−R1

√1− x2+y2

R21

+R2

√1− x2+y2

R22

R

√1− x2 + y2

R2≈ R− x2 + y2

2R

Phase transmission function of a thin lens

t(x, y) = eikn∆0e−ik(n−1)x

2+y2

2

(1R1

− 1R2

)

1

f= (n− 1)

(1

R1− 1

R2

)

wheref= focal lengthDropping the phase factor

t(x, y) = e−i k2f (x2+y2)


Converging Spherical Wave Illuminating anAperture: Scaled FT at focus

Amplitude just after aperture of transmittancet0(x, y)

u(x′, y′;−d+) =A

de−ikde−i k2d(x

′2+y′2)t0(x′, y′)

Propagate through a distancez is given by a convolution

hz(x, y) =eikz

iλzei

k2z (x

2+y2)

t (x,y)0

d

x

z0

u(x, y; 0) = u(x, y;−d+) ∗ ∗hd(x, y)

=eikd

iλd

� �u(x′, y′;−d+)e

i k2d [(x−x′)2+(y−y′)2]dx′dy′

=eikd

iλdei

k2d(x

2+y2)

� �u(x′, y′;−d+)e

i k2d(x′2+y′2)e−i2πλd(xx

′+yy′)dx′dy′

Fx′y′{u(x′, y′;−d+)e

i k2d(x′2+y′2)

}∣∣∣ u=x/λd

v=y/λd

=A

iλd2ei

k2d(x

2+y2)T0

( x

λd,y

λd

)Since quadratic phase factors cancel

I(x, y; 0) =

(A

λd2

)2 ∣∣∣T0

( x

λd,y

λd

)∣∣∣2


Fourier Transforming with a Lens:Input placed against lens

y

x

yi

-x i

object placed against lens

A(x ,y )i i

FourierPlane

Field after lens when object is placed against lens

Ei(xi, yi) = A(xi, yi)tl(xi, yi) = A(xi, yi)e−i k2f (x

2i+y2i )

Propagating a distancef to the back focal plane

Eo(x, y) = eik2f (x

2+y2)

� �A(xi, yi)e

−i k2f (x2i+y2i )ei

k2f (x

2i+y2i )e−i2πλf (xxi+yyi)dxidyi

Proportional to scaled 2D FT ofA except for quadratic phase factor: eliminated bysquaring

Io(x, y) = ‖F{A(xi, yi)}u= xλf ,v=

yλf‖2


Fourier Transforming with a Lens:Input placed in front of lens

Spectrum of input

T (u, v) = F{At(x′, y′)}Propagates distanced to lens bymultiplying by paraxial TF

Ul(u, v) = T (u, v)e−iπλd(u2+v2)

Thus field at back focal plane is scaled FT d f

t(x’,y’)

y’

x’ x

yLens

f

uf(x, y) =ei

k2f (x

2+y2)

iλfUl

(x

λf,y

λf

)=

eik2f (x

2+y2)

iλfe−iπλd

[(xλf

)2+(

xλf

)2]

T( x

λd

y

λd

)

=ei πλf

(1−d

f

)(x2+y2)

iλf

� �t(x′, y′)e−i2πλf (xx

′+yy′)dx′dy′

Whend = f quadratic phase factor vanishes and we get exact scaled FT (phase flat)

uf(x, y) =A

λfF {t(x′, y′)}|u=x/λf,v=y/λf


Fourier Transforming with a point sourceimaging system

Expanding spherical wave strikes lens

u−(r) =a′

d0eikr

2/2d0ei(kz−ωt)

multiplied by lens to becomeconverging spherical wave

u+(r) = u−(r)tl(r) =a

d1e−ikr2/2d1ei(kz−ωt)

t(x’,y’)

y’

x’

Lensf

x

y

d01d

d

Where lens transmission istl(r) = e−ikr

22

(1d0+ 1

d1

)with 1

d0+ 1

d1= 1

f.

Propagate a distanced1 − d produces a sperical wave of radiusd that strikes mask

u(x′, y′;−d+) =a

d1

d1de−i k2d(x

′2+y′2)t(x′, y′)

Propagate through distanced, as before quadratic phase factor inside integral cancelsout yielding scaled FT with quadratic curvature

u(x, y; 0) =ei

k2d(x

2+y2)

iλd

a

dT( x

λd,y

λd

)


Fourier Diffraction Pattern Analysis

FF

g(x,y)

x

y y’

x’

Fourier Transform Lens

Analyze the Fourier power spectra and use to classify objectsKelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 58

4F imaging

Object FT Lens FT LensFT plane

1F

1F 2

F2

F

OutputImage

T1 =eik2f1

iλf1V[1/λf1]{F{}}

yxT =T1T2 =

eik2f1

iλf1V[1/λf2]

{F{eik2f2

iλf2V[1/λf1]{F{}}

}}=

eik2(f1+f2)

(−λ2f1f2)V[1/λf2]

{F{V[1/λf1]{F{}}

}}

(λf1)2V[λf1]{F{ · }}

= −f1f2eik2(f1+f2)V[f1/f2]{F{F{}}} = −f1

f2eik2(f1+f2)V[−f1/f2]{·}

scaled inverted imaging with no quadratic phase factor.

Afocal telescopic imaging system


Spatial Filtering

∆

δ

F2F1

Apertures in the Fourier Plane

t(x, y) = comb(x∆

)∗ Π(xw

)comb

( y∆

)∗ Π

( yw

)Π(xL

)Π(yL

)

E− (x′, y′) =1

iλF1F2D {At(x, y)}u=x′/λF1,v=y′/λF1

=1

iλF1∆comb (∆u)wsinc(wu) ∗∆comb (∆v)wsinc(wv) ∗ Lsinc(uL) ∗ Lsinc(vL)

E+(x′, y′) = E−(x

′, y′)Π(xδ

)δ′ =

1

∆=

δ

λF

=1

iλF1∆comb (∆v)wsinc(wv) ∗ Lsinc(uL) ∗ Lsinc(vL)

Output Imagem = F2F1

I(x”, y”) =

∣∣∣∣1

iλF2F{E+(x

′, y′)}∣∣∣∣2

=−1

λ2F1F2comb

( y

m∆

)∗ Π( y

mw

)Π( x

mL

)Π( y

mL

)


Fourier Filtering System

Collimator Object FT Lens FT LensFT planeSpatial Filter

1F

0F

1F 2

F2

F

OutputImage

Laser


Fourier Filtering : Selecting VerticalSpatial Frequency Components

W.Tom Cathey,Optical Information Processing and HolographyWiley , 1974


Fourier Filtering : separating a periodicobject from defects

E. Hecht and Zajac, Optics, Addison Welsley, 1974, 1997, 2016


Fourier Filtering : High Pass Filterwith DC block

W.Tom Cathey,Optical Information Processing and HolographyWiley , 1974

Microscopic metalic black dot carefully aligned over DC order

Size: few times diffraction limited DC beam widthλ/F# = λFD≈ 25µm

Also must be carefully aligned inz to unifomly blink off when translatedWhen behind Fourier plane moves in same direction as translationWhen in front of Fourier plane moves in opposite direction astranslation


Beamprop through Lens SystemsDouble slit diffraction and Fourier Transform Comparison of BPM with theory


BeamPropagation through 4F lens system


BeamPropagation through 4F lens system


4F lens system with Schlieren filterConverts Phase Modulation to Amplitude


4F lens system with Schlieren filterConverts Phase Modulation to Amplitude


4F lens system with Zernike Phase contrastdot

Converts Phase Modulation to Amplitude


4F lens system with Zernike Phase contrastdot

Converts Phase Modulation to Amplitude


Vander Lugt Complex Spatial FilterA. B. VanderLugt,Signal detection by complex spatial filtering, IEEE Trans. Inf. Theory IT-10, p139, 1964

Want to form a correlation integral

o(x′, y′) =

� �g(x, y)h∗(x− x′, y − y′)dx dy = F−1 {G(u, v)H∗(u, v)}

Need to perform a product of transforms. However, need to represent complex infor-mation in the transform plane.=⇒ use holography.

FF

g(x,y)

x

y y’

x’

Fourier Transform Lens

Plane wavereference beam

r(x′, y′) = rei2πλ (x′ sin θ+z′ cos θ)

∣∣∣z=0

a(x′, y′) =1

iλF

� �h(x, y)ei

2πλF (xx

′+yy′)dxdy =1

iλFH

(x′

λF,y′

λF

)


Vander Lugt Complex Spatial FilterExposure

The amplitude transmission of the mask is proportional to intensity and exposure time

t(x′, y′) = κT0 |a(x′, y′) + r(x′, y′)|2

t(x′, y′) = κT0

∣∣∣∣1

iλFH

(x′

λF,y′

λF

)+ rei

2πλ x′ sin θ

∣∣∣∣2

= κT0

[1

λ2F 2|H(u, v)|2 + |r|2

+1

iλFH

(x′

λF,y′

λF

)r∗e−i2παx′

+i

λFH∗(

x′

λF,y′

λF

)rei2παx

′]

= κT0

[1

λ2F 2|H(u, v)|2 + |r|2 + 2

λF|r| |H(u, v)| cos (2παuλF − ∠H(u, v))

]

Develop and reposition the filtermust be repositioned to a small fraction of the smallest resolvable feature of the transform∆x = λF/W . λ = .5µm,

W = 2.5cm,F = 25cm,∆x = 5µm .


Vander Lugt Complex Spatial FilterReadout

Convolution

D(x",y")

x

y y’

x’

Fourier Transform

Lens

Plane wavereference beam

FF

F

F

t(x’,y’)

g(x,y)

y"

x"

Correlation

DC

HologramFT Lens

b(x′, y′) =1

iλFG

(x′

λF,y′

λF

)b(u, v) =

1

iλFG(u, v)

Transmission of the field amplitude through the hologram

d(u, v) = b(u, v)t(u, v) =1

iλFG(u, v)

[kT0

(1

λ2F 2|H(u, v)|2 + |r|2

+1

iλFH(u, v)r∗e−i2παx′ +

i

λFH∗(u, v)rei2παx

′)]

=κT0

iλF

[G|r|2 + G|H|2

λ2F 2+GHr∗

iλFe−i2παx′ +

GH∗r

−iλFei2παx

′]


Vander Lugt Complex Spatial Filter OutputPlane

The final lens Fourier transforms this amplitude distribution. (note the coordinate in-version)

D(x”, y”) =κT0

−λ2F 2

[r2g(x”, y”) +

1

λ2F 2g(x”, y”) ∗ h(x”, y”) ∗ h∗(−x”,−y”)

+r∗

iλFg(x”, y”) ∗ h(x”, y”) ∗ δ(x− αλF )

+r

−iλFg(x”, y”) ∗ h(−x”,−y”) ∗ δ(x + αλF )

]

Remember, that this is an alternative representation of a correlation

g(x”, y”) ∗ h∗(−x”, y”) =

� �g(x, y)h∗ (−(x”− x),−(y”− y)) dx dy

=

� �g(x, y)h∗(x− x”, y − y”)dx dy = g(x”, y”) ⋆ h(x”, y”)

∗ represents convolution⋆ represents correlation


Optical Pattern recognition

FT Lens FT Lens

f(x,y)

Collimated Beam Filter

F*(fx,fy)

Output

A. B. VanderLugt, IEEE Trans. Inf. Theory IT-10, p139, 1964.


Rotated objects in Fourier Patternrecognition

FT Lens FT LensCollimated Beam

f(x,y)

Output

F*(fx,fy)

Filter

J. W. Goodman,Introduction to Fourier Optics, 1996


Discrimination and Invariance in OpticalCorrelation

• Rotation and Scale changes decrease correlation peak

• Build in Invariance by correlating aganst library of rotated and scaled prototypes

• Average filters use average across invariance class

• Too much averaging destroys recognition and discrimination


Correlation Peak results across invarianceclasses

• Poor discrimination

• Poor recognition

• Useless to average

=⇒ Edge enhance


Edge Enhanced Optical Correlation

• Edge enhance with DC block in Fourier plane

• Excellent recognition and discimination

• Edge enhanced prototypes can be averaged across invarianceclass

• Edge enhancing with a DC block is required for Fourier correlators to be useful


Edge Enhanced Optical Correlation Peaksacross invariance class

• Good discrimination

• Good recognition

• Average filters work

• 2N average filters vsN 2 prototype filters


Fourier Optic Filtering System in AOL


Holographic Vanderlugt Optical Correlator


Experimental Vanderlught Correlator inAOL


Dye Polymer holograms

is isomeris ground state

Reference wave

Object wave d

ks

kr

y

zo

(a)

y

zo

Diffracted beam

Transmitted beam

Probe beam

(b)


Azo-Dye Polymer holograms

S0

S1

1T

rapid ISC

( trans form )

( cis form )

hν1

τ−

(a)

N = N

NaO S3

H

NCH3 CH3

hν

τ-1

trans-form cis-form

(stable in darkness) (thermally unstable)

N = NH

NCH3

CH3

NaO S3

(b)


Polarization Volume Holograms

Vertical Horizontal

om = 1

(a)

I rx I sy

Vertical Horizontal

om = 4

I rx I sy

(c)

(b)

left circular

right circular

vert

ical

(re

fere

nce)

horiz

onta

l (si

gnal

)

left circular

right circular

vert

ical

(re

fere

nce)

horiz

onta

l (si

gnal

)

(d)

RightCircular

LeftCircular

om = 4

I r I s

m = 1o

RightCircular

LeftCircular

(a)

I r I s

(c)

right circular (reference)

left circular (signal)

vert

ical

horiz

onta

l

(b)

right circular (reference)

left circular (signal)

vert

ical

horiz

onta

l

(d)


Polarization Dependant TransitionProbability

X

Y

Z

θ

φO

a molecular axiselliptical polarizationa

b

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

YZ

X

YZ

X

YZ

X

YZ

X


Spatially Varying Anisotropic TransitionProbability and resulting orientational

population distribution

Transition probability

(b) (c)

Population distribution in the trans state

(a)

Vertical Vertical

Total amplitude field

RightCircular

LeftCircular

(a)

Total amplitude field

Transition probability

(b)

Population distribution in the trans state

(c)Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 89

Nonexponential Kinetics

t

Non

expo

nent

ial d

ecay

Free volume

p( )τ

τ

Gau

ssia

n di

strib

utio

n

Exponential vs Nonexponential Kinetics

Experimental Erasure & Writing


Computer Generated Holography (CGH)

Allows artificial synthesis of mathematical wavefront or Fourier filter not available asa real physical wave to be recorded by conventional holography.

1. Mathematically describe a wavefront or SDF

– Sample on a regular grid, obeying Nyquist criteria

2. Encode phase and amplitude as binary apertures

3. Plot (pen plotter, film recorder, laser printer, diffractive optic)

4. Copy and reduce onto optical medium

Review Articles

1. W.H. Lee, CGH, in Progress in Optics XVI, p. 291

2. W. Dallas, The computer in Optical Research, Springer, 1980, v41 Applied Physics

3. T. S. Huang, Digital Holography, Proc IEEE, v 59(9), p. 1335 1971.


Detour Phase CGH (Lohmann)

Lohman and Brown, AO, v. 5, p. 967, (1966) and AO, v.6 p. 1739, (1967)Binary transmission mask suitable for pen plotters and highcontrast photoreduction(or laser printer).No explicit reference added in mathematical formulation.Fourier Transform synthetic hologram for matched spatial filter. Uses FFT of discretepixelized object in computer.

G(j, k) =∑

m

∑

n

Ag(md, nd)ei2π(mj+nk)

• Divide intoN ×N subapertures .

• Plot filled rectangles in each with area or density∝|G(j, k)|

• Shift the position of the center of each rect from the cellcenter by an amount∝ ∠G(j, k)


Detour Phase Operation

Consider all apertures centered so the array of apertures acts just like a diffractiongrating

θ

λ

λ

d

sin θ = λ/d

θ

λ

d

∆λ/d

∆{2πλ/d

0

0

0

When one of the apertures is by a distance∆ then that portion of the wavefront picksup a phase2π∆/d


Lohmann hologram analysis

t(x, y) =∑

m

∑

n

Π

(x− ndx − φnmdx

2πm

w

)Π

(t−mdyAnmdy

)

dy

dx

cnm=φnmdx/2πm

hnm=Anmdy

w

Fourier transform of Lohmann hologram

T (u, v) =

� �t(x, y)e−i2π(ux+vy)dx dy

=∑

m

∑

n

wsinc(wu)e−i2πndxue−i2πcnmuhnmsinc(hnmv)e−i2πmdyv


Variants of Lohmann hologram

t(x, y) =∑

m

∑

n

Π

(x− ndx − cnm

wnm

)Π

(t−mdy

h

)

wnm =dx sin

−1Anm

πcnm =

dxφnm

2π

T (u, v) =

� �t(x, y)e−i2π(ux+vy)dx dy

=∑

m

∑

n

wnmsinc(wnmu)hsinc(hv)e−i2πcnmue−i2πndxue−i2πmdyv

=∑

m

∑

n

wnm

sinπ sin−1 dxAnmπ

πwnm

=∑

m

∑

n

(dxAnm

π

)hsinc(hv)e−i2πcnmue−i2πndxue−i2πmdyv

expand about first orderuc = 1/dx, vc = 0 to see desired term...


Interpolation

Lohmann technique plots aperture position corrsponding tophase at sampling point.Phase sampled at sampling aperture center

Aperture placed self consitently at position corresponding to phase function

Instead, plotting aperture at position where aperture phase = function phase gives moreaccurate representation with less reconstruction noise.Only possible for mathematically defined phase functions (instead of sampled phases)


Lohmann hologram represents samples on apolar grid


Example of Saturation in Lohmannholograms


Lee & Burkhardt delayed sampling

Decompose complex samples into real and imaginary parts.Now decompose into± real and± imaginary, only 2 of which are nonzero.At each sample location, plot a superpixel consisting of 4 stripes, +real, +imaginary,-real, -imaginary, with height of 2 nonzero apertures equalto corresponding amplitude

+r

+i

-r -i +-real, +-imaginary4-part decomposition

0,120,240 degree3-part decomposition

t(x, y) = Π

(x

dx/4

)∗[∑

n

∑

m

fr+(ndx, mdy)δ(x− ndx, y −mdy) + fr−(ndx +dx2, mdy)δ(x− ndx −

dx2, y −mdy)

+ fi+(ndx +dx4, mdy)δ(x− ndx −

dx4, y −mdy) + fi−(ndx +

3dx4

, mdy)δ(x− ndx −3dx4

, y −mdy)

]

Can be interpreted as a binarized version of a modulated carrier fringe


Example of phase flat Lee holograms


Example of phase random Lee holograms


Other types of CGH

2-level etched phase relief binary hologram

ideal

etchdeptherror

4-level etched binary optical element

0 1 2 3-1-2-3Ideal π

Etch deptherrors

Fourier Plane Diffracted Orders

Kinoforms∞ level phase onlyROACHReferenceless on-axis complex hologramClever use of color film as both amplitude and phase modulating hologram


Hologram Fidelity

MSE between objectfnm and recondtructiongnm overN ×M window

e = minλ

1

AB

A2−1∑

n=A2

B2−1∑

m=B2

|fnm − λgnm|2

λ is a complex scaling factor to removeDE = <g|f><g|g>

Efficiency of a CGH

NxM

AxB

cnm

Incident Power=∑N

n

∑Mm |1|2 = NM

trans.=∑N

n

∑Mm |cnm|2 = ηNM =

∑Nn

∑Mm |gnm|2

desired order=∑A

n

∑Bm |gnm|2

1’s in [0,1] object=∑A

n

∑Bm |gnmfnm|2

0’s in [0,1] object=∑A

n

∑Bm |gnm(1− fnm)|2

Histogram ofA× B region of interest

L U

u2σl

2σ

SNR =|U − L|2σ2u + σ2

l

CR =U

L


Projection onto Convex Sets

Constraints can be interpreted as subsets in Hilbert space

Convex subset contains all points on chords connecting two other points in the subsset

projection onto setA as operatorPA finds closest point in set. NotePAPAx = PAx.

Consider a family of constraintsC1, C2,... each forms a conves set in Hilbert space.

Co = ∩mi Ci is set intersection.

relaxed projectorTi = I + λi(Pi − I) λi ∈ {0, 2}Sequence of projectors,T = T1, T2, ...

T nx converges to a point satisfying constraints

FourierConstraints Real Space

Constraints


Fourier domain iterative optimization ofCGH

let gkl be CGH, Gmn be Fourier domain reconstruction

g Gkl mn

g’ G’kl

mn

2D FT

2D IFT

Apply CGHconstraints

apply constraintson reconstruction

Constraints on reconstruction

• over regionR of G• set zeroes to 0• binary high levels• symmetric for real CGH• autocorrelation at origin• let phase ofGmn freely vary• noise in other regions unimportant• maximize efficiency

CGH constraints

• Kinoform – phase only• Lohmann – polar complex samples• Lee – Cartesian complex samples• tranmsission< 1• multilevel phase –2n levels• binary amplitude (this one is hard)


[email protected] dept ece, ucb 425, ecee 232, x24661...

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