Diffraction - Huygen - Obliquity factor

Hi,

I am an undergraduate physics student trying to explain some results I
observed in a microwave diffraction experiment. Instead of viewing the
diffraction pattern on some type of screen (as is often done when
visible light is used), I have a microwave receiver mounted on a
goniometer. The microwave emitter is linearly polarized, and is fairly
far from the aperture. Its center is lined up with the center of the
aperture, both vertically and horizontally.

The diffraction pattern I am observing is not explained by the standard
Fraunhofer analysis. To simplify the problem, I will consider just a
single slit aperture, whose width is slightly smaller than the
wavelength of the light.

The "standard" far-field approach would be to define the phase shift
as:

phase = (2*Pi/wavelength)*(d*Sin[theta])

where "d" is the distance along the slit. Then, to get the amplitude at
angle theta, I perform the following integral over d (from -slitWidth/2
to +slitWidth/2):

Integrate[ (A_0 / slitWidth) * Sin[w t + phase ]

The result is:

A = A_0 * Sin[w t] * Sin[Beta]/Beta

where Beta =

Pi * slitWidth * Sin[theta] / wavelength

This result predicts that the time-averaged amplitude will oscillate as
theta is increased, but that each subsequent peak will be smaller than
the previous peak. This result does *NOT* specify that the amplitude
has to go to zero at theta = Pi/2.

However, the observed intensity pattern does seem to always go to zero
at theta = Pi/2. I have been told that this is due to applying Huygen's
principle (which deals with propagation of scalar fields) to light (a
vector field). I was told that to correct my previous result (for
amplitude A), I can multiply it by an additional factor of Cos[theta].

I am trying to understand where this factor actually comes from. I
found the following source (among a few others):

http://fizz.phys.dal.ca/~hewitt/Web/PHYC3540/Lecture30.ppt

I read that Fresnel (and later Kirchoff) introduced a "obliquity
factor" to extend Huygen's principle. This factor made it so that each
wavelet was not a regular spherical emitter. Rather, each emitter sends
out waves whose intensity depends on the direction of propagation. The
mathematical expression for this "obliquity factor" is:

F(theta) = n^ (dot) r^ - n^ (dot) r'^

where n is a vector perpendicular to the plane of the aperture, r' is a
vector from the source to the aperture, r is a vector from the aperture
to the receiver, and theta. The "^" notation I used is supposed to be
read "hat", and go over the variable, but it's hard to represent these
things in ASCII. If you look at the Powerpoint file I referenced above,
I think the author may have meant to define r' from the aperture to the
source. Otherwise, when r and r' are parallel (theta = 0), F(theta)
would be equal to zero.

Anyway, when the emitter is far away and concentric with the aperture,
as in my case, the obliquity factor reduces to:

F(theta) = 1 + Cos[theta]

The normalized time-averaged intensity for a slit of infinitesimal
width would then be:

I = I_0 * (F/2 * Sin[Beta]/Beta)^2

which, for a slit of infinitesimal width, reduces to:

I = I_0 * (1/2)^2 * (1 + Cos[theta])^2

It still doesn't go to zero at theta = Pi/2. Rather, it only goes to
half of its original value. What am I missing?

For the record, my emitter puts out linearly polarized microwave
radiation at 2.86 cm, my slit width is 1.2 cm, my emitter is about 1m
from the aperture, and the radius of the goniometer (which the receiver
is mounted on) is about 1m. The relationship I was told to use ( I =
I_0 Cos[theta]^2 ) DOES fit the observed results, I just don't know how
to explain it. I think that I can neglect the curvature of the
wavefront arriving at the aperture.

.

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