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Name:_________________________________
EE 8390: Fourier Optics
Final Exam
April 23, 2008
Take Home
Due Thursday May 8, in EE Office by 12:00 noon.
Exam Rules:
1. Scientific calculators and extra scratch paper may be used.
2. Exam is open book [class text], open notes. Do not discuss
this test with anyone.
3. Clearly designate results, e.g., by underlining them or
placing boxes around them.
4. Derived equations may be used, but some explanation of
their origin should be given
5. Numerical results (computer) may be used to validate and
plot results but full credit will only be earned for analytical
results unless otherwise directed.
6. State any assumptions you make.
7. Show your work.
8. Use extra paper as needed, present your solutions in an
organized fashion.
9. The Honor Code applies to this exam
Name:_________________________________
1)
Give 5 examples of “real world” effects, phenomena, or items that you can explain with
the techniques covered in this class. Give the “Reader’s Digest” version of the
explanation. For example if we had covered Rayleigh Scattering an excellent answer
might be: “The sky is blue because the shorter wavelengths (blue light) is scattered
preferentially as predicted by Rayleigh Scattering (1/4). This is also what makes sunsets
red as the sun passes through a thicker portion of the atmosphere scattering off most of
the blue light.” Be BRIEF and to the point. (15 points)
2)
Design a transparency which has a far field Fraunhofer pattern consisting of a single
square, inside the square is a sinusoidal carrier and outside the square is darkness. You
have at your disposal a grayscale printer with very fine 500,000 dots per inch and a
similar resolution “phase” printer which is capable of printing only 2 thicknesses of
material corresponding to 0 or Pi radians. Describe mathematically what pattern you
would record to achieve the desired Frauhofer pattern (you may pick a wavelength and
distance if you like, or simply leave it in variable form). Indicate what aspect of the
pattern results in the square, and what aspect of the pattern results in the sinusoidal
carrier (by sinusoidal carrier, I mean Sin if you look the amplitude, or Sin2 if you look at
the Intensity).(20 points)
3)
Problem 9-5 from the text (20 points)
4)
Design problem (45 points):
You’ve been hired by the Department of Defense to design an optical image processing
engine to look for tell-tale signs of mechanically planted land mines. You will need to
specify a design for a “Wedge-Ring” detector as described below, describe your light
source, pick an optical Fourier Transform configuration, and choose the choose the
cheapest (i.e., lowest performing) lens configuration which meets your performance
criteria. You will need to differentiate between the tell-tale signs of land mines, rows of
corn, and rows made by bailing hay. The information below will be useful:
Wedge Ring Detectors:
A typical wedge ring detector is shown in the Figure below. Its surface is divided into
two halves each containing a different detector geometry. Its use leverages the fact that
with an amplitude spatial light modulator the output of a Fourier Transform is Hermitian
Symmetric so the top half and the bottom half contain redundant information. The top
half of the detector contains rings. Each annulus integrates the light within two radii of
detector (defining the borders of one detector). Therefore, in a FT processor, the ring
with the highest signal indicates the range of spatial frequencies which contain the most
energy. The bottom half of the detector contains wedges. Each wedge integrates the
light within a given direction of spatial frequencies. Therefore, the combination of the
wedge signals and ring signals provides the frequency and direction of the spatial content
of the scene.
Name:_________________________________
Spatial Light Modulator:
You have at your disposal a grayscale amplitude LCD spatial light modulator which is
HDTV resolution (1080 x 608). It will function across the visible spectrum and
amplitude modulate pixels to 8 bit resolution. It has a diagonal measurement of 0.7
inches and unity fill factor of square pixels.
Tactical Camera:
The Image Sensor providing input to your processor provides 1080x608 resolution
images where the maximum resolution is 3 inches.
Fourier Transform Lenses:
You may select whatever focal length Fourier Transform Lens you desire. Also, you may
select whatever lens diameter you desire as long as it is at the focal length (F# at least 1).
The cost of the lenses goes up with decreasing F# so pick the largest F# lens you can
tolerate. Also, you must specify the quality of the surface. Quality will be specified in
terms of wavenumber of phase deviation. Assume (Note: this assumption is not universal
to systems but will allow you to work this problem) the form of the Wave error is:
x


W ( x)   1  cos  2    

 D   Q
Where D is the diameter of the lens aperture and Q is an integer representing the quality
of the lens. Q=1,2,4,8,20,40, or 100. The higher Q the better the lens and therefore the
more expensive. You can assume this wavefront error is present at the exit pupil of the
system. You should calculate the effects of this error numerically (NOT
ANALYTICALLY) to determine what the spectral resolution of your optical system.
Choose the cheapest lens (lowest performing) that will accomplish your goals. You may
elect to “stop down” an aperture on a larger lens of lower quality if you only need the
central portion of it. Note if you get stuck on the quality factor determination, the entire
problem can be worked in the absence of aberrations, this part should not hold you up
from the rest.
Targets and Clutter:
The target mechanical land mine placing machine leaves rows of disturbed earth which
are precisely 2.5 ft apart. In the same area are corn fields which are planted at 1ft row
spacings, and hay fields where the hay bailing machines leave rows space 4 feet apart.
Make sure your system has the spatial frequency resolution to differentiate corn, from
landmines, from hay fields, and can determine the direction of the land mine columns.
Name:_________________________________
You must specify:
 The configuration of the optical signal processing engine (sketch it out and
describe its functionality). What wavelength will you operate at?
 The parameters of the Fourier Transform Lens (F#, Diameter, focal length,
Quality factor and the determined spatial frequency resolution (size of aberration
PSF) (this is before you consider sampling on the detector).
 The dimensions of the wedge-ring detector (how many wedges, what size rings,
what size overall).
 Which detector wedge and ring elements will indicate which targets / clutter at
which orientations.
 Anything else which is critical to its operation…