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What you see is what you get... except when the thing you are
looking at is either very small or very far away (all a matter
of perspective). Take, for instance, the light from a street
lamp projected through a hole the size of a fork tine in a
screen. If you place another screen just behind the hole what
you will see is an image of the hole. But as you move the
second screen further and further away, the image of the hole
starts to blur. Move the second screen far enough away, and
—
if your eyes are sensitive enough — you will see light and
dark rings appear. These rings are the diffraction pattern of
the hole in the first screen, known to mathematicians as the
Fourier transform of the hole, or at least the amplitude
thereof. Not many people can look at a diffraction pattern and
recognize what the original object is. Watson, Crick and
Franklin famously "decoded" xray diffraction patterns to
reveal the true structure of DNA. Thanks to the Fast Fourier
Transform — and fast microprocessors — Fourier transforms are
easy to invert, so that today, in principle, anyone could
recover the image of the original hole by simply inverting the
diffraction pattern, but for one problem: the observation is
of the amplitude of the Fourier transform, not the Fourier
transform itself. This is a central problem facing
crystallographers and anyone else that uses diffraction imaging
to infer the structure of the things through which light
passes.
Long-term IMA visitor Russell Luke (University of Delaware) has
been fascinated by mathematical algorithms for solving this
problem (known as the phase retrieval problem)
ever since his PhD thesis on the theory and practice of
wavefront reconstruction for NASA's James Webb Space
Telescope, Hubble's replacement. The phase retrieval problem
is a vexing instance of a nonconvex feasibility problem, for
which there is scant theory, though recent work by Luke and
collaborators is beginning to shed some light on this issue.
While at the IMA Luke attended as series of lectures by
organizer David Brady (Duke University) on optical imaging. At
one of the lectures Brady gave to the audience plastic glasses
which made the letters "DUKE" appear to the wearer whenever
Brady used
his laser pointer to emphasize important items in his talk on
the overhead screen. Brady's glasses were patented and
mass-produced by a manufacturer with a high-tech printer, but
Luke was confident that he could produce a less glamorous
version of the glasses with
nothing more than an ordinary laser printer, transparency film,
and mathematical software such as Matlab. With the help from
an instructional grant from the University of Delaware, and
space provided in the Mathematical Sciences' Modeling
Experiment and Computation Lab at UD, Luke built a diffraction
optical bench with which students can design their own glasses.
Following a very nicely-written recipe by Thad Walker
(University of Wisonsin), the student starts with the image she
want to appear through the glasses and then, with software
written by Luke, computes a binary mask that will produce the
desired image as the diffraction pattern of the mask. The
image is a very simple version of a hologram. Luke uses the
bench to teach mathematics students about the physical nature
of Fourier transforms, and to teach science/engineering
students about the mathemetical nature of diffraction. On a
deeper level, the lab provides a very tangible instance of some
of the frontiers of mathematical computation.
Figure 1: Grating (actual size) printed on transparency film
with ordinary laser printer.
Figure 2:Diffraction image (magnified) of grating
References
J.V. Burke and D.R. Luke, "Variational analysis applied to the
problem of optical phase retrieval’’, SIAM J. Contr. Optim., 42
(2003), pp. 576-595.
D.R. Luke, "Relaxed averaged alternating reflections for
diffraction Imaging", Inverse Problems, 21 (2005), pp. 37-50.
D.R. Luke, J.V. Burke, and R.J. Lyon, "Optical wavefront
reconstruction: Theory and numerical methods", SIAM Rev, 44
(2002), pp.169--224.
T. Walker, "Holography Without Photography," University of
Wisconsin, Department of Physics Technical Report (1998)
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