1.    Fresnel diffraction on-axis

A plane wave of wavelength l = 500 nm is incident on an opaque  mask containing a 1-mm diameter hole.  Find all of the distances at which one can observe a diffraction pattern with a dark spot (zero intensity) at the center.  What size aperture would give dark spots at the same locations for microwaves (such as those used in a kitchen oven)?  For X-rays?   What about sound waves?

 Solution to problem 1.
 

2.    Zone plate

Design a Fresnel zone plate suitable for use as a lens with Cu Ka X-rays.  The (spherical) source of the X-rays is at 4 m from the zone plate, and the focal point should be at 1 m from the plate.  Make a scale drawing of the zone plate, and specify what material it should be made of.  How would you fabricate it?  Be specific (i.e. "by evaporation" would be an insufficient answer).

 Solution to problem 2.
 

3.    Apodization

We saw in class how modifying the transmission function of a slit can change its Fraunhofer diffraction pattern, specifically by suppressing the side lobes to improve the resolution.  We looked at the plain (square) slit, and the triangle function.  You are to pursue this further by looking at some other functions and finding out what characteristics of the slit funcion make for the “best” diffraction pattern (the one with which you could most easily resolve two closely-spaced sources).  Some functions you might try include (but are not limited to): a parabola, a cosine (i.e. one "bump" of a cosine pattern, between zeros), and a cosine squared.  The slit function does not need to be entirely smooth--you might try a Gaussian with a cut-off (since the Gaussian never goes quite to zero).

Consider at least three different slit transmision functions.  For each one, calculate the resulting diffraction pattern (intensity) and plot it.  Compare the plots (suitably normalized, i.e. with the same total transmitting area).  What can you conclude from the differences and similarities?  Which of your slit function produces the best pattern?  How might you go about fabricating a slit that has that transmission function?

 Solution to problem 3.
 

4.    Phase object

A square aperture of side a = 0.75 mm is half covered by a 0.4-mm-thick sheet of mica.  A plane wave of wavelength l = 500 nm is incident on it.  Calculate the resulting Fraunhofer diffraction pattern and make a graph of the intensity.  (The refractive index of muscovite mica is 1.552, for one orientation of the polarization.)

 Solution to problem 4.