OPTI512R - Linear Systems and Fourier Optics

revised by Dr. Tyo Fall 2012

For many applications we find that it is much easier to represent our physical quantities in terms of complex numbers rather than just using real numbers alone. The concept of complex numbers comes from the continuation of functions such as square-root and logarithm that only apply to positive numbers in their traditional definitions. Figure 1 shows the function √ x. Other functions like ln x, sinx, etc., also have limits on their arguments, and complex numbers allow us to define these functions over the full range of real numbers.

revised Fall 2011

Consider again our systems view of the diffraction effects on an optical imaging system as depicted in Fig. 1. In our earlier discussion we had considered ideal, diffraction-limited optical systems that convert spherical waves incident on the entrance pupil originating at object position (xo, yo) into spherical waves leaving the exit pupil focused on image position (Mxo,Myo).

In the previous lecture we saw that the low-pass filter corresponding to coherent imaging was given by the pupil function, while the LPF for incoherent imaging with the same system was given by the autocorrelation function of the pupil. Some comparison is in order. First of all, the frequency where the OTF goes to zero is twice the frequency where the CTF goes to zero, meaning that higher spatial frequency fields can make it through the system. However, for binary pupils, the pass band of the CTF is generally flatter than the pass band of the OTF, meaning that the frequencies that do pass retain better contrast. These issues and more make comparison of the two types of imaging difficult.

We saw above that coherent systems are linear in field amplitude, which is complex. In our previous discussion, we constructed a LSI system defined by the convolution equation.

Up until this point, we have assumed that the fields we were dealing with were purely monochromatic. IN other words, they were ideal complex exponentials with time dependence ejωt, and they have zero bandwidth. This assumption is not realistic, as all physical optical fields will have some finite bandwidth that is related to randomness in the physical processes that generate the radiation. There are two classes of coherence that we are concerned with in general. The first class is temporal coherence and the second is spatial coherence. Temporal coherence describes the ability of light to interfere with a delayed version of itself. Spatial coherence describes the ability of a beam of light to interfere with a shifted version of itself. To test the former, we have to make two copies of the field, delay one, and recombine them. This is referred to as amplitude splitting. To test the latter, we sample the wavefront at two different locations, bring those fields into conjunction, and allow them to interfere. The second strategy is referred to wavefront splitting interferometry. These concepts are depicted schematically in Fig. 1.

Now that we know about the Fourier transforming properties of lenses at the back focal plane of the lens, we are prepared to take the next step and consider the performance of the lens more generally and at other surfaces that might be useful. Consider the system shown in Fig. 1. We know at this point that the diffraction problem is linear. Since it’s linear (but not necessarily shift invariant), we can break up the problem into a collection of point sources in the object plane (xo, yo), track the radiation from each of those point sources through the system, and add up the resulting fields at any particular observation position we choose. Let’s assume that a point source at position (xo, yo) produces an image h(xi, yi; xo, yo). This notation means that the image depends on both the location of the observation (xi, yi) and the location of the source (xo, yo). We can now write the fields in the image plane (xi, yi) in terms of the superposition integral.

We have considered thus far the diffraction of fields according to Huygen’s principle. We decompose the field at each plane into a superposition of spherical waves, and then we can propagate the fields by allowing those spherical waves to expand. The next step is to consider what happens when we introduce a lens into the optical path. Our treatment of the lens will follow the treatment of Goodman. At every point in the lens we consider the thickness of the lens, and treat the lens as something that introduces a phase delay in the fields that pass through that point.

When we look at Fresnel diffraction from circular apertures and objects we see some remarkable properties that stem from the rotational symmetry about the center of the circle. To maintain this symmetry we will assume for now that the source and observation points are located on the axis of symmetry as well. We begin with the Fresnel diffraction expression for the geometry depicted in Fig. 1.

In the last lecture we looked at Fraunhofer diffraction in the far-field of the aperture. Given that the aperture was limited by a circle of radius L1, we found that we are in the the Fraunhofer region when ....

In the previous lecture we examined plane wave solutions to the wave equation and discussed how an arbitrary field distribution in a given plane could be broken up into a superposition of plane waves. We could then propagate the fields by allowing each of those plane waves to propagate separately, then add everything up after some distance. Earlier this term we discussed the Heisenberg uncertainty principle that told us that it is not possible to know both the position and wavenumber of an electromagnetic wave infinitely precisely. In Lecture 15, we used a basis set that had zero uncertainty in wavenumber, i.e. we knew exactly the direction in which the wave was propagating. However, the plane wave has infinite extent, so the individual basis functions have no localization at all in position. For the rest of the course, we will use the dual representation of the fields. That is, we will choose a basis set that is precisely located in space, but provides no localization in wavenumber. That basis set is the spherical wave: we know exactly the location of the source, but the light propagates away in all directions. For the purposes of this discussion, we will assume that the fields are monochromatic and linearly polarized. This will allow us to use scalar diffraction theory, which ignores polarization in the computation.

In OPTI501 you have learned about Maxwell’s equations. In this course we are interested in reducing Maxwell’s equations to one particular form, the Helmholtz equation.

We learned in continuous linear systems how the convolution integral is defined.

We know that when the space function is continuous and aperiodic that it’s Fourier transform is also continuous and aperiodic. When the space function becomes periodic, we then have the Fourier series, which is discrete in the frequency domain. Likewise, we have just learned that when the function is discrete in space, its Fourier transform is continuous and periodic in the frequency variable. There is one block left - the block where both the space variable and the frequency variable are discrete. Based on symmetry, we would also expect the functions to be periodic in both space and in frequency. Is this the case too? We will see now that this last block is occupied by the discrete Fourier transform (or DFT).

Up until now, we have treated all functions as continuous functions. In other words, the function f(x) has a value for any x we choose. In practice, we often want to represent such continuous signals by a collection of samples or discrete values of the function. In this lecture we will discuss the circumstances under which we can ideally reconstruct the underlying continuous function only from a finite collection of samples.

This is an additive noise model, and we have assumed that the noise is independent from the signal. Our goal here is to develop a LSI system that can be used to filter out the noise, preserving the signal with the highest fidelity possible. The scenario is depicted in Fig. 1. There are many examples of applications where this is reasonable, such as additive detector noise in a camera focal plane array.

Consider the system flow diagram depicted in Fig. 1 that we have looked at before. When we consider the system in the frequency domain, we can think of the transfer function as modifying the frequency content of the input signal given by its Fourier transform F(ξ). Borrowing from the nomenclature of electronics, we often call these devices linear filters that filter the spectral properties of the input to create a desired output.

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 our function is separable.

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 weighted delta functions, pass each one through the system to get a superposition of shifted and weighted impulse responses h(x − x0), and then add them up through a convolution integral. Alternately, we can break up our input into a superposition of weighted complex sinusoids via the Fourier transform, pass each one through our system using the transfer function H(ξ), and then add them back up again through the inverse Fourier transform. We might ask ourselves, “what is the relationship between h(x) and H(ξ)? Given the notation we’ve chosen, we might guess that they are related by a Fourier transform.

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 the physical behavior that is observed in nature. Once the operators are defined, the rest is just math. In many cases the operators are accurate only up to some limit. For example, many of the systems that we experience in optics are not strictly linear, but can be modeled as such for inputs of low enough intensity.

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, many geometrical concepts can be leveraged in functional analysis to help understand what’s going on.

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