Many of the two-dimensional functions that we would be interested in in our optics class will have azimuthal symmetry. Examples are circular lenses and apertures. For the most part, we will consider only functions that do not vary with azimuth at all. However, the analysis tools that we develop here are generally applicable to other types of symmetry.

So far everything we have done has considered functions of only one independent variable, namely f(x). However, in much of optics, we have to deal with functions of two spatial variables, for example f(x, y). In this course we are only going to work with two coordinate systems. Primarily we are going to consider rectangular coordinates, i.e. x and y. However, occasionally we will also consider polar coordinates, g(ρ, φ). We will strive whenever possible to identify the coordinate system where...

It is very important to understand how to perform direct convolution, as well as to have a picture in your mind about graphical convolution and how it works. However, there is a vitally important theorem that relates the convolutional of two functions to their Fourier transforms. Consider the system that we’ve put together in Fig. 1. Our picture of linear systems tells us that we can compute the output in one of two ways. Either we can break up the input into a superposition of shifted and...

We have discussed the Fourier series and its relative, the Fourier integral. There are many specific forms that the Fourier integral can take, but the one that we are most interested in is known as the Fourier Transform.

1. Definition of Convolution; 2. Graphical Convolution

Operators are mathematical representations. For our purposes, these representations will model some real, physical process in mathematical notation. This operator will operate on functions in some vector space and produces outputs that lie in some other vector space. Often times (as will be the case most of the time this term), the output vector space and the input vector space are the same. The theoretical part of mathematical physics is the development of mathematical operators that capture...

Throughout this course (and in many areas of mathematical physics) it is extremely convenient to write one function of interest as a weighted superposition of another set of functions whose behavior we are familiar with. If our system is linear, we can then analyze the behavior of the system on our function of interest by breaking it up into its components parts and adding up the results. This is the basis of functional analysis which is closely related to linear algebra. For that reason,...

We will be working not just with functions, but with scaled and shifted versions of functions.