Lecture 16.  Fraunhofer Diffraction

Lecture 22:  Frequency Domain Description of Imaging Systems

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.

Lecture 1 Notes: Complex Numbers Review and Mathematical Representation of Physical Quantities

Lecture 1: Complex Numbers

Lecture 12.  Sampling and Reconstruction

Lecture 13.  The Discrete Fourier Transform

Lecture 9.  The Hankel Transform and Radon Transform

Lecture 7.  The Convolution Theorem and Other Fourier Transform Properties

Lecture 6.  Fourier Transforms

Lecture 3: Harmonic Analysis and the Fourier Series

OPTI512R - Linear Systems and Fourier Optics

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).

Lecture 24 Notes: Effect of Aberrations on the Performance of Imaging Systems

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.

Lecture 23 Notes: Performance Comparison between Coherent and Incoherent Imaging Systems

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.

Lecture 22 Notes: Coherent and Incoherent Imaging Systems/CTF/OTF/MTF/PSF

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.

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.

Lecture 21 Notes: Coherence of Optical Fields

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.

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.

Lecture 20 Notes: Physical Optics Description of Imaging Systems

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.

Lecture 19 Notes: Fourier Transforming Properties of Lenses and Gaussian Cavities

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.

Lecture 18 Notes: Fresnel Diffraction from Circular Apertures

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 ....

Lecture 17 Notes: Fresnel Diffraction

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 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.

Lecture 16 Notes: Fraunhofer Diffraction

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.

Lecture 15 Notes: Plane wave spectrum and beams

My first Optics class was at the graduate level as a basis for future remote sensing endeavors. Needless to say, I wasn&#39;t prepared. While I used a textbook by Pedrotti for my Geometric optics, this lecture gave me a great understanding of wave optics. I would highly recommend this to anyone interested in Optics whose already taken a Fourier class. This professor&#39;s teaching methods and associated course notes are amazing.

GREAT Lectures!!!

This collection is great for optics students, but please make it a iTunes U Course~

Make it a course please

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