Fourier Optics, Hermite Functions, and Prolates
There are two ‘forms’ in which you can represent a function- in spatial form (where the variable is in metres, seconds, etc) and in frequency form (per metres, per seconds, etc.). The Fourier transform converts a function from its spatial representation into its frequency representation and the inverse Fourier transform converts a function from its frequency representation into its spatial representation. If you apply the Fourier transform to a function twice, you get the function flipped around the y-axis; three times and you get the inverse Fourier transform; four times and you get the original function back.
The fractional Fourier transform describes the ‘in-between’ states – for example, applying the Fourier transform ‘one-half times’, which, if you do twice, is the Fourier transform. The fractional Fourier transform takes an ‘angle’ parameter: the fractional Fourier transform of angle pi/2 is the Fourier transform, and angle 2pi is the same as four times the Fourier transform, which gives the original function.
The fractional Fourier transform has special applications in optics. A quadratic graded-index medium is a medium where the refractive index varies along its radius according to a certain function. If a function describes the distribution of light on some plane, then as this plane propagates along the medium the distribution of light will become the fractional Fourier transform of the original function with the angle depending on the distance.
My supervisor told me about a book on the topic which I borrowed from the library. Most of my time during the project was spent reading it and trying to understand all of the content. I found out about a lot of unexpected relations throughout the project, like the far-field diffraction pattern of an aperture being its Fourier transform, or the fractional Fourier transform’s relation to solutions of the Schrödinger equation in physics.
Our project focused on finding the behaviour of light as it propagated through a quadratic graded-index medium with regular apertures placed along it. We used some special functions called the Legendre polynomials and the Hermite-Gaussian functions to compute a matrix to model the transmission between two apertures. However, this matrix was infinitely large, so we had to cut it off at some point to actually calculate it. The larger we made it before cutting it off, the more accurate it was.
We checked the accuracy in two ways – we knew analytically that if we set the distance between the apertures to zero, we would get the identity matrix, which is 1s along the diagonal and 0s everywhere else. We did this, and the entries in the matrix approached their appropriate values as the dimension was increased (the entries in the matrix changed as the dimension did since they were calculated by a truncated infinite sum).
The other way was to find the eigenvectors of the matrix for the distance where the function would be exactly once Fourier transformed between the apertures. Analytically, we know the eigenfunctions of this to be the special prolate functions. When we reconstructed the eigenfunctions from the eigenvectors, we found that they were roughly similar to the prolates but not particularly precise
Most of our theoretical background was referenced from the following book:
Ozaktas, HM. et al. 2001, The Fractional Fourier Transform with Applications in Optics and Signal Processing, John WIley & Sons, Chichester
William Roland-Batty was a recipient of a 2018/19 AMSI Vacation Research Scholarship.