# Accuracy of Kirchhoff diffraction theory

1. Jul 13, 2010

### Sam_Goldberg

Hi guys,

My research project involves a paraboloidal telescope with a primary diameter of 25 meters, and a focal length of 10 meters. My goal is to come up with the mathematics and numerics to determine the surface profile error of the primary mirror. We will take a point source of 2mm wavelength light at a height of about 22 meters above the vertex of the paraboloid, shine it on the primary, and measure the reflected image (field amplitude and phase) using several detectors placed in the same vertical plane as the light source. We will construct a system of linear equations to relate the measured field to a piecewise constant function representing the deformations of the telescope primary.

The first step in doing something like this is to figure out the electric field pattern reflected by the primary. I have been treating this as a diffraction problem. Basically, I have been using the Fresnel-Kirchhoff diffraction formula and numerically integrating the formula over the primary. The Fresnel-Kirchhoff diffraction formula comes directly from the Kirchoff integral theorem, and assuming that the point source of light emits spherical waves. The Kirchoff integral theorem comes from the scalar Helmholtz equation.

Here is my question: I would like to have an idea just how accurate the Kirchhoff model of diffraction is, and whether it would be accurate enough to suit my needs. I would like to get the accuracy of my electric field calculations to 1 part in 1000, hopefully 1 part in 10000. But I have no idea just how accurate the Kirchhoff model would be, given the aforementioned parameters for the dimensions of the problem.

While the Kirchhoff model is the best I know how to use, it is an approximation. If it is not good enough, then what exactly is wrong with the Kirchhoff model, and what would be a better model to use? For example, would I need to go beyond scalar diffraction theory, using instead vector diffraction theory? What sources would you guys recommend, if any, to learn it?

Finally, where can I get a source that compares how accurate various diffraction models are? All of the articles I have been looking through present some pretty detailed mathematics without really explaining how much of an accuracy improvement they really offer. Is it possible to find any formulas that estimate the accuracy of Kirchoff, or more advanced diffraction theory? This would truly give me a better understanding of the situation.

Thanks,
-Sam.

2. Jul 13, 2010

### Sam_Goldberg

It appears as if I have gotten over 50 views without any responses. If anything (or perhaps everything) about my original post was unclear, please let me know and I will reexplain it. I've given the dimensions of the system, and would like to know how accurate Kirchoff diffraction theory would be, or where I could find a source to answer that question. I need to have a general picture of what the names of the different diffraction models are, how accurate they are, and some good places to learn about them. Also let me know if this is the correct place to post my question.

3. Jul 16, 2010

### Sam_Goldberg

anyone?

4. Jul 16, 2010

### Antiphon

Hi. I'll comment but I'm on an iPhone and it will take a while. Hold on.

5. Jul 16, 2010

### Antiphon

It will be highly accurate for the type of problem you describe; full illumination of a large surface in the near backscatter mode. You must take care to set up the proper vectorial relationships for the scattered field. It is not a scaler integral except in pure backscatter back to the far field.

The theory will not accurately compute edge diffraction or higher order diffraction that is not the result of an illuminated surface. The error will be small in your case, well under your 1000:1 criteria near thebfocal point, perhaps even a few orders of magnitude better.

You have not said anything about the surface of the reflector. You should include parallel and perpendicular reflection coefficients as a function of angle to handle the general case.

The vector Kirchoff physical optics formulation is very well presented in the Radar Cross Section handbook by Ruck. You need this reference if you are to undertake this calculation.

Reflector antennas are also designed using GTD/UTD. This theory extends geometrical optics to include diffraction effects. It is better at computing the diffraction terms but cannot be used on the primary reflector if you are in the GO caustic region (probably the case for you).

Your plan to form a linear matrix to represent surface figure errors is doomed to fail unless you are interested in deviations of less than 0.5mm or less. Even then it's dubious. You need to form the surfaces of constant phase in a plane to see the full phase error.

To get a feel for this, you need to refer to electrical engineering texts and journals, not physics or optics. You need to read for example the IEEE Journal "Transactions on Antennas and Propagation" to learn the state of the art. There are many good books too, go to the Artech House publishing site and search for anything connected to Antenna Theory, Scattering, and Radar Cross Section.

How's that for your first response?

6. Jul 17, 2010

### Sam_Goldberg

Thanks for the advice and the suggestions for sources to start with. I will definitely look at them. I have one more question, relating to the reliability of the general idea of the project.

You mentioned that you were doubtful that setting up a linear system would give intelligible results. Actually, I did not plan on making a complete map of the surface profile error. Maybe I should tell you about the application my professor had in mind.

Our telescope primary is composed of panels linked together. The entire paraboloid (with focal length 10 meters and diameter 25 meters) will be constructed from about 100 panels, and each of those panels is composed of subpanels (maybe 10 subpanels per panel, I actually don't have the number right now). We will have supports linked to the panels that will allow us to reposition them (both their piston and tilt). The goal of my project is to carry out the mathematics and numerics of measuring the error of each panel, so that we can reposition each panel periodically during the operation of the telescope. This way, our image will never be distorted by much.

My professor and I did not think that we would need to calculate a detailed map of the surface profile error, simply because all we can do is alter the piston and tilt of the panels. We thought it would only be necessary to solve for maybe 3 variables for each panel, because we only have limited control of the panels.

Is this argument valid? Or is there a problem in the above approach that we are overlooking?

7. Jul 20, 2010

### Antiphon

I see. I had assumed that you were making wavelength-scale corrections using coherent phase to count the "number of waves" of deviation as when you polish the surface of a lens.

The problem solution as you have explained it doesn't seem to me like it will have meaningful solutions but I could be wrong. I don't see a linear relationship between panel tilt and the error.

Pretend you want to solve this as a 1-d problem. You have a Fourier series with unknown phase offsets in some of the terms. How do solve for them using linear algebra?