Modeling Diffraction by a Circular Aperture Illuminated by a Diverging Light Beam
William Weiss, Marty Cohen and John Noé
Laser Teaching Center
Department of Physics & Astronomy
Stony Brook University
Introduction
Laser Teaching Center students are encouraged to be curious, and often just asking, "I wonder
what would happen if ..." can kickstart interesting research. Thus my project came about
through a chance observation while examining the diverging Gaussian wavefront emerging from a
singlemode optical fiber. While shining this light through a pinhole we observed an unexpected
and interesting diffraction pattern with an umistakeable dark spot in its center. As the distance
between the tip of the fiber and the pinhole was varied this pattern changed to others with a
bright or dark center surrounded by dark rings. Only at relatively large distances did we observe
the familiar Airy pattern that we initially expected.
Setup
The patterns were studied systematically by mounting a 100 micron diameter pinhole on a 3D
translation stage. This setup allowed the pinhole to be accurately centered in front of the fixed
fiber and the pinholefiber separation to be varied in steps as small as 25 microns from 0 to 20
mm. As before, the source feeding the fiber was a lowpower red HeNe laser (wavelength = 632.8
nm). The diffraction patterns were observed by eye and photgraphed on a small screen placed about
35 cm from the pinhole. The divergence of the light emerging from the fiber was measured in a
separate experiment by scanning a small photodetector across its profile. The profile was found
to have a Gaussian shape as expected, with a 1/e2 angular diameter of 2 x 0.08 radians. The
corresponding measured beam diameter at the tip of the fiber is 4.8 microns, in good agreement
with the manufacturers specification of 4.0 microns.
Schematic of setup. I used the 2 mirror coupling system in the lab with a lens to
focus the light into the fiber optic cable. The light emerges at the other end of the
fiber and is incident on the pinhole, which is attached to the translation stage.
Diverging Beam
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Fourier Optics
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The observed patterns can be understood by considering the Fresnel number N =
a^{2} / λL, where
a is the aperture radius, λ is the wavelength, and L is the fibertopinhole or
pinholetoscreen distance. Before the pinhole N is of order unity while after it N ~ 0.01. The
former value indicates that the phase of the light reaching the pinhole varies significantly with
radius, while N << 1 indicates that the path length from any point on the pinhole to a given
point on the screen is essentially the same (the Fraunhofer approximation). In the Fraunhofer
approximation each pattern is the Fourier transform (FT) of its (complex) aperture function,
T(r). The FT integrand reduces to the product of the Gaussian intensity function, the phase
function, and a zeroorder Bessel function. Mathematica was used to numerically evaluate the
integral across the screen and plot its square, the observable intensity distribution I(r), for a
number of appropriate fiberpinhole separations. Each plot took a few minutes. The calculated
distributions are in very good qualitative agreement with the sequence of observed and
photographed patterns. Work is in progress to extract numerical intensity profiles from the
images in order to make a more precise comparison.
