EE5621 Physical Optics -- Homework Assignment 5
Due date: 12 March 2009
Computer problem: Calculate the Fresnel diffraction patterns from circular and
square apertures.
1. Transfer function method: Use a computer to calculate and graph the
diffraction pattern from both a circular and square aperture illuminated by an onaxis plane wave by using the transfer function method of Fresnel propagation.
Assume the following:
Wavelength = 1 micrometer
Aperture radius = 1 mm (circle)
Aperture width = 1 mm (square)
Propagation distances = 1/50 meter, 1/3 meter, ½ meter, 1 meter, 10 meters, and 1
km
Note that some of the plots may have errors in them due to sampling issues. Note
which propagation distances are difficult to calculate.
2. Integral calculation method: Repeat the calculation in part 1 using the Fourier
transform form of the Fresnel diffraction integral.
Again, some of the plots corresponding to certain propagation distances may
contain sampling-induced errors. However, you should see that the distances that
produce errors in part 1 are different from the distances that produce errors in part
2.
3. Aliasing and sampling: From the results of the above calculations, answer the
following questions: 1) How fast do you need to sample the field in each of the
above methods, and at each of the propagation distances to avoid aliasing? 2) Is
there an advantage to using one technique over another (transfer function vs.
integral) with regard to sampling an aliasing? If so, at what propagation distances
is which method preferred?
Notes:
1)
Fourier transforms can be accomplished in programs like MatLab by
using the expression: b = fft(a) for a one-dimensional fast Fourier
transform or b = fft2(a) for a two-dimensional fast Fourier transform.
2)
The easiest way to calculate the diffraction pattern from a circular
aperture is probably to use a two-dimensional array containing a circle,
and utilizing two-dimensional FFT’s.
3)
Creating a circle can most easily be accomplished using the “meshgrid”
and “find” commands. Use Matlab “help” for definitions
4)
Zero padding of an FFT is often required to increase the resolution in the
frequency space. You will want to include significant zero padding in
this problem.
5)
You may find the command “fftshift” to be useful. This command shifts
the dc component of the Fourier transform into the center of the array
6)
The resulting diffraction pattern can be graphed in a variety of ways. A
one-dimensional slice through the pattern can be plotted in MatLab
using the “plot” command. A contour plot of the two-dimensional data
can be obtained using “contour”. The two-dimensional data can also be
plotted in a wire-frame plot using the command “mesh.” However, the
display that is closest to matching the observed optical pattern is
obtained by the command “image”, where the color table is set to blackand-white by using the command “colormap(gray)”.