Phys 198 April 30, 1997

# FINAL EXAM REVIEW QUESTIONS

## Diffraction Theory :

1. What is Huygen's theorem? What was its greatest failure?
2. What is meant by the obliquity factor? Who introduced it?
3. What is the usual distance equation that specifies the requirement for far-field diffraction?
4. What is the size of the Airy disk for a small aperture of diameter a illuminated by light of wavelength l ?
5. Explain the following statement: The far-field diffraction pattern of an aperture is the Fourier transform of its transmittance function.
6. When parallel light passes through a lens at an angle q to the optical axis, where is the image of the parallel light formed?
7. Do small objects in a transparency form data points closer to or farther from the optic axis than larger objects?

## Fourier Transforms :

1. When are Fourier frequency terms discrete harmonics and when are they continuous frequencies?
2. What is the Nyquist theorem?
3. For an object of 2 cm dimension, what is the minimum spatial resolution (in cm) when the object is digitized into an array of 512 samples?
4. How many harmonics will be present in the Fourier transform of the object described in the previous question?
5. What is the lowest (but not dc) spatial frequency involved in the transform of the object described above?
6. Assuming that in physical measurements all data is real, what is the smallest item in the object mentioned above that can be resolved ?
7. To display a Fourier transform so that the transform is centered in the display, what must be done to the input data?
8. What is a principal advantage of working in Fourier space?
9. When a lens of focal length f is used to bring in a far-field pattern to a convenient range, what is the relation between the radial distance r from the optic axis of a point in the Fourier transform, and the spatial frequency this location represents if the parallel coherent illuminating beam has wavelength l ?
10. 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 function.
11. If an image of dimension 2 cm contains an item of basic size 0.1 mm, what is the minimum sampling number that will satisfy the Nyquist condition?
12. A grid of small rectangular apertures is illuminated by a collimated and expanded laser beam. How would you obtain its Fourier transform optically? Where would you put a horizontal slit to filter the transform, and how would you get the transform of the filtered transform? What would it look like? Draw sketches.
13. Describe and sketch an optical low-pass filter. Do the same for a high-pass optical filter.

## Contouring with Moire :

1. What is the difference between shadow and projection moire?
2. Which requires the higher resolution imaging sensor?
3. What are the different requirements as regarding the necessary size of the master grid?
4. Why does one have a geometric term of tan q in its basic equation, while the other has sin q ?

## Holographic Interferometry :

1. Give at least five requirements of the experimental setup for obtaining holographic images.
2. What are some of the advantages and disadvantages of using holography for making measurements?
3. What are the three basic types of measurement in holographic interferometry?
4. What is the sensitivity vector and how is it defined? How is its magnitude obtained?
5. What is meant by the sign problem and how can it be solved?
6. Give a sketch of an experimental setup for holointerferometry that is suitable for in-plane measurement.
7. Give a sketch of an experimental setup for holointerferometry that is suitable for out-of-plane measurement.

## Speckle Metrology :

1. What is the difference between objective and subjective speckle?
2. What is the difference between speckle photography and speckle interferometry?
3. Which method has the greater resolution, photographic or interferometric?
4. Speckle photography is sensitive only to in-plane motion, while speckle interferometry can be made sensitive to either in-plane or out-of-plane motion. Draw sketches of appropriate setups for each case.
5. In analyzing specklegrams of speckle photography, what method of analysis gives point-by-point results? What analysis method gives full-field results? One method gives the magnitude and line-of-motion of the displacement recorded in the specklegram, and the other gives components of motion along a particular direction, but neither can give the sign of the motion. Discuss.
6. Draw a sketch of the setups appropriate for each method of analysis.

## Electronic Speckle Pattern Interferometry (ESPI) :

1. The basic advantages of ESPI are simplicity and economy compared to standard speckle pattern interferometry. What are some of the factors that bring simplicity and economy?
2. Draw sketches showing ESPI setups for in-plane and for out-of-plane measurements.
3. What is the common way of solving the sign problem in ESPI ?
4. What is meant by phase unwrapping? How is it linked to the Nyquist requirement and to the periodicity of trig functions?
5. If the maximum displacement limitation of ESPI is about half a speckle size, and the magnification of the optical system imaging the loaded surface onto the camera sensor is 1/3, what is the largest measurable motion of locations in the surface? (Assume speckle size is about 5 mm.)

## List of Equations

a sin q = ml

b sin q = ml

f = x/(lz)

 w = (Dn p)/tan q shadow moire w = (Dn p)/sin q projection moire

k = 2p/l

g = e2 - e1

g · d = Dn l

g = [0, 2]

 Sobj = 1.22 Ll/D objective speckle size Ssubj = 1.22 (M + !) l f/a subjective speckle size

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

m = -s'/s

 p = lD/d Young: speckle motion p = nlf/r Fourier: speckle motion Dy = p/M specimen motion

 w = nl/2 out-of-plane symmetric small angle beams u = nl/(2 sin q) in-plane motion symmetric beams