FINAL EXAM

1. When an expanded beam of collimated light at an angle of 30 degrees to the optic axis passes through a lens of focal length 25 cm, how far from the lens is the image formed and how far from the optic axis is it located? [Assume the light is from a HeNe laser, l = 0.633 mm]
2. Use the frequency-shifting relation for a Fourier transform to demonstrate what must be done to a set of N data values (N is a power of 2) so that the transform display is centered with the zero frequency at the center of the display. [x(k)exp(i2pmk/N) <==> X(n-m)]
3. What is the Nyquist theorem and why is it important?
4. An expanded HeNe beam passes through a 2.5 cm by 2.5 cm image film negative. Just beyond the film is a lens of focal length 25 cm. A square grid of small calibrating grid marks 0.2 mm apart are in the film picture. Where would you put a small blocking aperture so that when an imaging lens is used to produce a final picture of the original scene, the calibrating grid marks are gone?
5. Link the following aperture types with the appropriate mathematical form of their Fourier transforms: rectangular slit, circular aperture, sine wave <===> Bessel function, sinc function, Kronecker delta functions.
6. What is the sensitivity vector g as used in holographic interferometry? How is it defined and if the directions involved are equally and oppositely oriented each at 20 degrees to the normal of the surface under study, what is its magnitude?
7. What is meant by phase unwrapping?
8. Given a set of four data values for a discrete function x(kT) which is periodic over a range of 4, compute the value of the discrete Fourier transform X(n/(NT)) for n = 2. The given data values are: 1, 2, 2, 0 and they span a complete period. (T may be taken as having a value of 1.)
9. A square cross-section rod of dimensions 1 cm x 1 cm x 5 cm is placed in a loading frame with the rod's long dimension vertical and one of its sides facing a camera. The rod surface is illuminated by a single He-Ne laser beam so that a double-exposure specklegram consisting of images taken before and after the rod is subjected to compression can be generated. The rod surface is 60 cm in front of a lens which images the surface onto a camera plate 30 cm behind the lens. To perform Fourier analysis of the specklegram, an expanded He-Ne beam is projected normally through the developed specklegram plate. A transform lens of focal length 25 cm is placed right after the specklegram. In the Fourier plane a small aperture is placed on the vertical axis 3.0 cm below the center of the Fourier transform, and when the specklegram is then imaged by another lens, the final image is seen to be divided into three equal parts by horizontal fringes. How far was the original rod compressed?
10. What is the difference between shadow and projection moire? What are the resolution requirements for the camera in each case? What are the requirements as to physical grid size in each case?

List of Equations

a sin q = ml

b sin q = ml

distances to satisfy far-field imaging

k = 2p/l

1/f = 1/s + 1/s'

M = -s'/s

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