ECEN 5696 – FOURIER OPTICS
Fall Semester 2009, University of Colorado at Boulder
Homework #5
Due Thursday, November 19
1. (Goodman 5-9) A unit-amplitude, normally incident, monochromatic plane wave
illuminates an object of maximum linear dimension D, situated immediately in front of a
larger positive lens of focal length f. Due to the positioning error, the intensity
distribution is measured across a plane at distance f   behind the lens (see figure 1).
How small must  be if the measured intensity distribution is to accurately represent the
Fraunhofer diffraction pattern of the object?

D
Figure 1
f
2. (Goodman 8-11) The VanderLugt method is used to synthesize a frequency-plane filter.
As shown in figure 2.a, a signal transparency with amplitude transmittance s(x,y) is
placed immediately against a positive lens and a photographic plate records the intensity
in the back focal plane. The amplitude transmittance of the developed plate is made
proportional to exposure, and the resulting transparency is placed in the system of figure
2.b. Assuming that the appropriate portions of the output plane are examined in each
case, what should the distance d between the object plane and the first lens of the
filtering system be in order to synthesize:
a. A filter with impulse response s(x,y)
b. A filter with impulse response s*(-x,-y)
film
Fig. 2 a.
S
f
f
s(x,y)
input
filter
output
Fig..2 b.
d
f
f
f
3. Wiener-Khinchin theorem
Prove the Wiener-Khinchin theorem, i.e. that the spectral density is the Fourier transform of the
autocorrelation function.
4. Michelson interferometer with partially coherent light
Consider a Michelson interferometer with a source possessing a spectrum as shown in the figure.
a. Calculate the maximum displacement  l that is possible to perform in one of the mirrors
(from the position of equal optical path length) before the interference fringes disappear.
b. Draw the average intensity at the center of the detector as a function of  l .
I ( )
0 .6 
0.61

5. Number of observable fringes in Young’s experiment
Determine the number of observable fringes in Young’s interferometer if each of the following
sources is used (assume full spatial coherence in all cases):
a. Filtered sunlight with   400  800nm
b. LED with   1μm,   50nm
c. He-Ne laser with   633nm ,   1MHz
6. Fourier-transform lens with incoherent light
Quasi monochromatic spatially incoherent light of uniform intensity illuminates a transparency
of intensity transmittance f(x,y) at the front focal plane of a lens. Determine an expression for the
intensity of the observed light at the back focal plane. Compare with the case of coherent light
illumination.
7. Coherence of light transmitted through a Fourier-Transform Optical System
Light from a quasi-monochromatic spatially incoherent source with uniform intensity is
transmitted through a thin slit of width 2a and travels between the front and back focal planes of
a lens. Determine an expression for the normalized mutual intensity in the back focal plane.