Posted August 2008
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" x-ray 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: Diffraction image (magnified) of grating
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).