Phys 198 | April 30, 1997 |

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

- When are Fourier frequency terms discrete harmonics and when are they continuous frequencies?
- What is the Nyquist theorem?
- 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?
- How many harmonics will be present in the Fourier transform of the object described in the previous question?
- What is the lowest (but not dc) spatial frequency involved in the transform of the object described above?
- Assuming that in physical measurements all data is real, what is the smallest item in the object mentioned above that can be resolved ?
- To display a Fourier transform so that the transform is centered in the display, what must be done to the input data?
- What is a principal advantage of working in Fourier space?
- 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 ? - 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.
- 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?
- 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.
- Describe and sketch an optical low-pass filter. Do the same for a high-pass optical filter.

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

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

- What is the difference between
*objective*and*subjective*speckle? - What is the difference between speckle
*photography*and speckle*interferometry*? - Which method has the greater resolution,
*photographic*or*interferometric*? - 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.
- 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. - Draw a sketch of the setups appropriate for each method of analysis.

- 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? - Draw sketches showing ESPI setups for in-plane and for out-of-plane measurements.
- What is the common way of solving the
*sign problem*in ESPI ? - What is meant by phase unwrapping? How is it linked to the Nyquist requirement and to the periodicity of trig functions?
- 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.)

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** = **e**_{2} -**
e**_{1}

**g · d** = Dn l

g = [0, 2]

S_{obj} = 1.22 Ll/D |
objective speckle size |

S_{subj} = 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 |

**Send Mail or Comments:** matthysd@vms.csd.mu.edu

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