Square root of Zero   Leave a comment

I earned my PhD by completing a research project that combined optics and fluid mechanics. Both topics have very interesting connections between phenomena we can directly experience and abstract mathematics. One example from optics is the fact that a diffraction pattern is the Fourier Transform of the object. Another is that diffraction can produce catastrophes and singularities.

It’s fairly straightforward to create ‘infinity in a coffee cup‘: the caustic is a surface in space on which the intensity of light is infinite. Caustics are created when multiple rays of light are all scattered into the same direction- for example, the pattern of light at the bottom of a pool. Mathematically, these patterns are called ‘catastrophes‘ or ‘singularities‘.

Here are some images of catastrophes- the dark lines crossing the concentric interference fringes are the singular surfaces (not the big annulus; that’s part of the lens):

If a caustic is a surface that is singular, then a wave dislocation is a surface that is zero. Wave dislocations can be created fairly easily in the lab- using an axicon or polarized light.

For whatever reason, I had some time recently to explore a very simple phenomenon: polarization. Polarization is an optical phenomena that has been exploited to enable the recent generation of 3D movies.

Controlling the polarization of light is important in many technologies besides entertainment- fiber optics, imaging, CD/DVDs; in addition, measuring the polarization has a lot of applications, for example in materials science, military target recognition, cryptography, thin film analysis, etc.

Most of optical stuff we have around the house (windows, glasses, etc) is completely oblivious to polarization. A few naturally-occuring minerals (calcite, cordierite) are sensitive to polarization and have very characteristic responses. Plastic displays a polarization effect, too- photoelasticity.

Overhead transparency sheets are becoming extinct, but they are clear plastic sheets about the size of an 8.5″ x 11″ sheet of paper, and in days gone by were often used in classrooms. The way they are made results in very uniform photoelasticity- in effect, the transparency sheets are a large biaxial crystal.

Those images above are of a transparency sheet sandwiched between two crossed polarizers. However, it’s not an image of the transparency (or the polarizers)- it’s an image of the back pupil plane of the lens I am using. This is a ‘conoscopic’ view of the transparency sheet. In a conoscopic image, what you see corresponds to different *directions*. That is, the very center is all the light that goes straight through the sample, and as you move off-center, you see whatever light got scattered into a particular direction regardless of *where* it passed through the sample. Mathematically, the conoscopic view is the Fourier transform of the orthoscopic (‘normal imaging’) view.

Sir Michael Berry has done a lot of interesting research. He, and his graduate advisor J.F. Nye, really developed the relationship between optics and catastrophes. They’ve also shown some simple ways to generate wave dislocations, and how the optical effects can be used to model lots of physical phenomena.

In this case, the optical phenomena correspond to a few interesting mathematical ideas: the square root of zero, geometric phase, matrix degeneracies, knots, and singular points. The mathematics allows a connection to be made between the images shown here and other phenomena such as rotations of spin-1/2 particles, quantum state preparation and measurement, and topological defects in condensed matter.

All that, from one piece of plastic in between two other pieces of plastic.


Posted November 13, 2010 by resnicklab in Physics, pic of the moment, Science

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