1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Optical Fourier Transform for Propagation

  1. Oct 9, 2015 #1
    1. The problem statement, all variables and given/known data
    The complex amplitudes of a monochromatic wave of wavelength ##\lambda## in the z=0 and z=d planes are f(x,y) and g(x,y), redprctively. Assume ##d=10^4 \lambda##, use harmonic analysis to determine g(x,y) in the following cases:
    (a) f(x,y)=1
    (d) ##f(x,y)=cos^2(\pi y / 2 \lambda)##

    2. Relevant equations
    Part (d):
    ##f(x,y)=cos^2(\pi y / 2 \lambda)=.5(1+cos(\pi y / \lambda)=.5(1+.5(exp(+i\pi y / \lambda)+exp(-i\pi y / \lambda)) ##
    Fourier Transform Equations:
    ##F(\nu_x,\nu_y)= \int_{-\infty}^\infty f(x,y)exp(-i2\pi (\nu_xx+\nu_yy))dxdy##
    ##f(x,y)= \int_{-\infty}^\infty F(\nu_x,\nu_y)=exp(+i2\pi (\nu_xx+\nu_yy))d\nu_xd\nu_y##
    Transfer Function of Free Space (Fraunhofer Approximation):
    ##g(x,y)=h_0exp(\frac{i\pi(x^2+y^2)}{\lambda d})F(\frac{x}{\lambda d},\frac{y}{\lambda d})##
    3. The attempt at a solution
    F(\nu_x,\nu_y)= \int_{-\infty}^\infty exp(-i2\pi (\nu_xx+\nu_yy)dxdy=\delta(\nu_x-0)\delta(\nu_y-0)
    g(x,y)=\int_{-\infty}^\infty F(\nu_x,\nu_y)H (\nu_x,\nu_y)exp(+i2\pi (\nu_xx+\nu_yy))d\nu_xd\nu_y\\
    Assume\;Fraunhofer\;Approx\; \lambda <<d\\
    g(x,y)=h_0exp(\frac{i\pi(x^2+y^2)}{\lambda d})F(\frac{x}{\lambda d},\frac{y}{\lambda d})\\
    g(x,y)=h_0exp(\frac{i\pi(x^2+y^2)}{\lambda d})\delta(\frac{x}{\lambda d})\delta(\frac{y}{\lambda d})\\
    g(x,y)=(i/\lambda d)exp(-ikd)exp(\frac{i\pi(x^2+y^2)}{\lambda d})\delta(\frac{x}{\lambda d})\delta(\frac{y}{\lambda d})\\
    g(x,y)=(i/\lambda d)exp(-ikd)[exp(\frac{i\pi x^2}{\lambda d})\delta(\frac{x}{\lambda d})][exp(\frac{i\pi y^2}{\lambda d})\delta(\frac{y}{\lambda d})]\\##

    I think because the function is f(x,y)=1, there should just be propogation through free space. This should go to g(x,y)= some phase shift. My guess is that having a delta function at the end is wrong and it should "disappear" somehow.

    I'm using the Fundamentals of Photonics, Bahaa E.A. Saleh (either edition).

    Thank you for your help!
  2. jcsd
  3. Oct 9, 2015 #2


    User Avatar
    Science Advisor
    Homework Helper

    Mathematically, your calculation does not show any fault. The FT of a constant is a delta function, however the physics involved should not allow your calculation to be true because Fraunhofer approximation is actually a further approximation within Fresnel approximation, in other words, by assuming Fraunhofer approximation, Fresnel approximation must be implied. And this Fresnel approximation does not allow the input field to be unbounded like it is in this problem.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Threads - Optical Fourier Transform Date
Fourier transform of the Helmholtz equation Jul 6, 2016
Fourier optics output May 14, 2015
4 Lens optical system/fourier transform Mar 2, 2014
Lab on fourier optics Jul 14, 2013
Fourier Optics Apr 1, 2010