ECEN 5696 Fourier Optics
Professor Kelvin Wagner
Dept ECE, UCB 425, ECEE 232, x24661
[email protected]
What you will learn by completing this specialization
• Fourier transforms in time and space. 1D, 2D, 3D and 4D
• From Maxwell’s equations to diffraction and imaging
•Numerical techniques in wave optics
Aberations and Beam Propagation
•Holography and Optical Information Processing
• Spatio-Temporal Fourier Optics and multiple wavelengths
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 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 waves and Gaussian beams
• Compare, contrast, and analyze coherent and incoherent imaging systems
• Formulate a wave theory of aberations and visualize them
• Develop numerical techniques for optical beam propagation as one line of code
• Discover the use of optical correlations for pattern recognition
• Invent holography to record and transform optical fields
• Extend the ideas of holography to computer generated and digital holography
• Further generalize the Fourier approach to the case of broadband fields
• Utilize the Fourier decomposition to invent and evaluate novel optical systems
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 2
Suggested References for Additional
Reading
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 Photonics Chapter 4
D. Brady, Optical Imaging and Spectroscopy, 2009
D. Voelz, Computational Fourier Optics: A MATLAB Tutorial, 2011
J. Schmidt Numerical Simulation of Optical Wave Propagation, 2011
N. George Fourier 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 Fourier Optics Fall 2019 3
Course Outline
Linear Systems and Fourier Transforms
Sampling Theory and the Fast Fourier Transform (FFT)
2-D Systems and Transforms, Operators
Wave Propagation, momentum space
Diffraction Theory
Beam Propagation Method
Franhoffer and Fresnel Diffraction
Coherent Optical Imaging
Incoherent Imaging
Wave theory of aberrations
Holography
Computer Generated Holography
Digital Holography
Optical Information Processing
Synthetic Aperture Radar (SAR) and Tomography
Volume Holography: 3-D Fourier Transforms
Spatial and Temporal Fourier Optics: 4-D Fourier transforms
Vector Effects, Subwavelength Structures
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 4
Review of 1-D Fourier Transforms
Learning Objectives and Outcomes
•Remember integral definition of Fourier transform
– define operator representation
•Recognize equivalence of 1-D spatial FT and temporal FT
•Review some FT pairs of compact and singular functions
•Visually identify 1-D Fourier transform pair plots
• Summarize the properties of 1-D Fourier transforms
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 5
4 types of Fourier Transforms
R.A. Roberts, C.T. Mullis, Digital Signal processing, Addison Wesley 1987
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 6
The 1-D Temporal Fourier Transform:
Definitions Valid for finite support or
periodic signals
Forward temporal Fourier transform (Hz)
G(f) =
ˆ
g(t)e−i2πftdt = F{g(t)}
Inverse transform
g(t) =
ˆ
G(f)ei2πftdf = F−1{G(f)}
Alternate definition using angular radian frequency ω = 2πf
Forward temporal Fourier transform (rad/sec)
G(ω) =
ˆ
g(t)e−iωtdt = F{g(t)} ω = 2πf
Inverse transform
g(t) =
1
2π
ˆ
G(ω)eiωtdω = F−1{G(ω)} ≡ F−1{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 Fourier Optics Fall 2019 7
Exactly Analogous 1-D Spatial Fourier
Transform: Definitions
Can similarly define FT in space using spatial frequency, u = fx [lines/mm], analogous
to temporal frequency f [Hz], or use wavevector, kx [rad/mm], analogous to angular
frequency ω [rad/sec].
Forward 1-D spatial Fourier transform
G(u) =
ˆ
g(x)e−i2πuxdx = Fx{g(x)}
Inverse 1-D spatial Fourier transform
g(x) =
ˆ
G(u)ei2πuxdu = F−1x {G(u)}
or in terms of wavevector kx
G(kx) =
ˆ
g(x)e−ikxxdx = F{g(x)} ≡ Fx{g(x)}
g(x) =
1
2π
ˆ
G(kx)e
ikxxdkx = F−1{G(kx)} ≡ F−1x {G(kx)} ≡ F−1kx {G(kx)}
Note that these FT functions are scaled versions of each other G(kx) = G(kx/2π)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 8
1-dimensional Fourier transforms
Well defined continuous functions
rect(t) = Π(t) ↼⇁ sinc(f)
rect( tT ) = Π
(
t
T
)
↼⇁ T sinc(Tf)
sinc(t) ↼⇁ rect(f)
tri(t) = Λ(t) ↼⇁ sinc2(f)
sinc2(t) ↼⇁ tri(f)
e−πt
2
↼⇁ e−πf
2
e±iπt
2
↼⇁ e±iπ/4e∓iπf
2
e−|t| ↼⇁ 2
1+(2πf)2
e−|t|H(t) ↼⇁ 11+i2πf =
1−i2πf
1+(2πf)2
sechπt ↼⇁ sechπf
jinc(t) = J1(πt)2t ↼⇁
√
1− (2f)2Π(f)
1
|t|1/2 ↼⇁
1
|f |1/2
Singular functions in t or f
δ(t) ↼⇁ 1(f)
δ
(
t
a
)
= |a|δ(t) ↼⇁ |a|1(f)
δ(t− t0) ↼⇁ e−i2πt0f
1
iπt ↼⇁ sgn(−f)
u(t) = H(t) ↼⇁ 1
2
δ(f) + 1
i2πf
ei2πf0t ↼⇁ δ(f − f0)
cos(2πf0t) ↼⇁ 12 [δ(f−f0)+δ(f+f0)]
sin(2πf0t) ↼⇁ 12i [δ(f−f0)−δ(f+f0)]
comb(t) ↼⇁ comb(f)
comb
(
t
T
)
↼⇁ |T |comb(Tf)
tk ↼⇁
(−1
i2π
)k
δ(k)(f)
(
1
i2π
)k
δ(k)(t) ↼⇁ fk
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 9
Visual Fourier Transform Dictionary
Discontinuous Functions
rect(t)=P(t)
-1 0 1 2
time
sinc(t)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 10
Visual Fourier Transform Dictionary
Exponentially Decaying Functions
gausssian(t)
-2 0 2 4
time
gaussian(f)
-1 0 1 2
freq
exp(t)
-2 0 2 4
time
e-|t|
-2 0 2 4
time
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 11
Visual Fourier Transform Dictionary
Impulsive Functions
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 12
Visual Fourier Transform Dictionary
Singular Functions
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 13
The Temporal Fourier Transform:
Properties
Linearity
ag(t) + bh(t)↼⇁ aG(f) + bH(f)
Conjugation
g∗(±t)↼⇁ G∗(∓f)
Scale
g(αt)↼⇁
1
|α|G
(
f
α
)
Shift
g(t− t0)↼⇁ e−i2πft0G(f)
Modulation
ei2πf0tg(t)↼⇁ G(f − f0)
Derivative and Integration
dn
dxn
g(t)↼⇁ (i2πf)nG(f)
ˆ t
−∞
g(τ )dτ ↼⇁
1
i2πf
G(f) +
G(0)
2
δ(f)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 14
The 1-D Fourier Transform: Properties 2
Parseval’s Theorem
ˆ ∞
−∞
|g(t)|2dt =
ˆ ∞
−∞
|G(f)|2df
Convolution
ˆ
g(t′)h(t− t′)dt′ ↼⇁ G(f)H(f)
g(t)h(t) ↼⇁
ˆ
G(t′)H(t− t′)dt′
Correlation
ˆ
g(t′)h∗(t′ − t)dt′ ↼⇁ G(f)H∗(f)
ˆ
g(t′)g∗(t′ − t)dt′ ↼⇁ |G(f)|2
Fourier Integral
Ft{F−1t {g(t)}} = F−1t Ft{g(t)} = g(t)
Ft{Ft{g(t)}} = g(−t)
Kelvin Wagner, University of Colorado Fourier Optics Fall 2019 15
Operation x(t) = F−1f {X(f)} = F−1{X (ω)}
Ff⇐⇒ X(f) =
ˆ ∞
−∞
x(t)e−i2πftdt
F⇐⇒ X (ω) =
ˆ ∞
−∞
x(t)e−iωtdt
Linearity Ax(t) +By(t)
Ff⇐⇒ AX(f) +BY (f) F⇐⇒ AX (ω) +BY(ω)
Conjugation x∗(t)
Ff⇐⇒ X∗(−f) F⇐⇒ X ∗(−ω)
Scale x(at)
Ff⇐⇒ 1|a|X
(
f
a
) F⇐⇒ 1|a|X
(
ω
a
)
Mirror x(−t) Ff⇐⇒ X(−f) F⇐⇒ 2πX (−ω)
Duality X(t) or X (t) Ff⇐⇒ x(−f) F⇐⇒ x(−ω)
Time shift x(t− T ) Ff⇐⇒ e−i2πfTX(f) F⇐⇒ e−iωTX (ω)
Frequency shift e−i2πf0tx(t) = e−iω0tx(t)
Ff⇐⇒ X(f − f0) F⇐⇒ X (ω − ωs)
Time differentiation dx(t)
dt
Ff⇐⇒ i2πfX(f) F⇐⇒ iωX (ω)
d2x(t)
dt2
Ff⇐⇒ (i2πf)2X(f) F⇐⇒ iω)2X (ω)
dnx(t)
dtn
Ff⇐⇒ (i2πf)nX(f) F⇐⇒ (iω)nX (ω)
frequency differentiation (−i)ntnx(t) Ff⇐⇒ 1
(2π)n
dnX(f)
dfn
F⇐⇒ dnX (ω)
dωn
Time Integration
ˆ t
−∞
x(τ)dτ = u ∗ x Ff⇐⇒ X(f)
i2πf
+ 1
2
X(0)δ(f)
F⇐⇒ X (ω)
iω
+ πX (0)δ(ω)
frequency integration x(t)
t
Ff⇐
