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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 3
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 4
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 5
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 6
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)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 7
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)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 8
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)}
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 9
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 10
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)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 11
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)}
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 12
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 13
Rectangular to Polar
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 14
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)}
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 15
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 16
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 18
J0(πr) function in 2D and its FourierTransform
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 19
jinc (r) function in 2D and its FourierTransform
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 20
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 21
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 22
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 23
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 24
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 25
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 26
Rectangular aperture diffraction
dark=brightaperture
dark=dimobstruction
Solid:Exact RS
Dotted:Paraxial
Dashed:Assymptotic
z0 = w2/λ
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 27
Circular Aperture 3+1D Beam PropagationCrossection
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 28
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 29
2-D crosssections every 8λ from aD = 16λCircular Aperture BPM
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 30
Circular disk diffraction: Fresnel/Arago’sBright Spot
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 31
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 32
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
)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 33
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)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 34
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%
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 35
Square Wave Amplitude Grating
T (u) =1
Lcomb 1/L(u)
[(12− tm)Lsinc(Lu) + tmLsinc
(Lu
2
)]
tm = 0.5 tm = 0.2
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 36
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.
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 37
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 38
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 39
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 40
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 41
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)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 42
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 43
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 44
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 45
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
=
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 46
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)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 47
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)]
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 48
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 49
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 50
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 51
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 52
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)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 53
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 54
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 55
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 56
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
)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 57
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 59
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
)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 60
Fourier Filtering System
Collimator Object FT Lens FT LensFT planeSpatial Filter
1F
0F
1F 2
F2
F
OutputImage
Laser
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 61
Fourier Filtering : Selecting VerticalSpatial Frequency Components
W.Tom Cathey,Optical Information Processing and HolographyWiley , 1974
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 62
Fourier Filtering : separating a periodicobject from defects
E. Hecht and Zajac, Optics, Addison Welsley, 1974, 1997, 2016
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 63
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 64
Beamprop through Lens SystemsDouble slit diffraction and Fourier Transform Comparison of BPM with theory
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 65
BeamPropagation through 4F lens system
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 66
BeamPropagation through 4F lens system
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 67
4F lens system with Schlieren filterConverts Phase Modulation to Amplitude
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 68
4F lens system with Schlieren filterConverts Phase Modulation to Amplitude
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 69
4F lens system with Zernike Phase contrastdot
Converts Phase Modulation to Amplitude
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 70
4F lens system with Zernike Phase contrastdot
Converts Phase Modulation to Amplitude
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 71
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
)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 72
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 .
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 73
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
′]
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 74
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 75
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.
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 76
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 77
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 78
Correlation Peak results across invarianceclasses
• Poor discrimination
• Poor recognition
• Useless to average
=⇒ Edge enhance
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 79
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 80
Edge Enhanced Optical Correlation Peaksacross invariance class
• Good discrimination
• Good recognition
• Average filters work
• 2N average filters vsN 2 prototype filters
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 81
Fourier Optic Filtering System in AOL
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 82
Holographic Vanderlugt Optical Correlator
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 83
Experimental Vanderlught Correlator inAOL
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 84
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)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 85
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)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 86
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)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 87
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 88
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 90
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.
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 91
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)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 92
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 93
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 94
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...
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 95
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)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 96
Lohmann hologram represents samples on apolar grid
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 97
Example of Saturation in Lohmannholograms
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 98
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 99
Example of phase flat Lee holograms
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 100
Example of phase random Lee holograms
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 101
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 102
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 103
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
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 104
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)
Kelvin Wagner, University of Colorado Graduate Optics/COSI Lab 2016 105
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