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  • 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

    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

    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⇐

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