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...

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...

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...

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),...

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...

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...

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...

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.