# Reflector Analysis and Accuracy

New for August 2007! This page (and its companions) were contributed by Jackson Jones who is the proprietor of a new EM software tool company, Principles of Prediction, which is just getting started. Consider this page under construction, and come back soon to check on our progress!

## Reflector analysis & accuracy: what you need to know

Reflector analysis refers to the process of determining radiation patterns from antennae with reflectors. Often finite-element techniques are just too slow to be much help designing reflectors. Some expansion methods are used, but the bread and butter of reflector analysis consists of geometrical optics, the geometrical theory of diffraction (GO/GTD), physical optics, and the physical theory of diffraction (PO/PTD). We will discuss the important features of each of these methods, but let's address some common accuracy questions first.

1. What accuracy issues do I need to look for when using a reflector analysis program that uses GO/GTD, PO/PTD?

First, make sure all scatterers are at least five wavelengths in diameter. Generally, even following this guideline, it is important to check any design in a finite-element program for accuracy before manufacturing the reflector if it is close to five wavelengths. Some programs will not throw a warning if you try to model a system outside of the accuracy bounds of the underlying approximation, so you have to watch for this yourself.

Second, make sure your feed model is suitable for the desired accuracy. Many reflector analysis programs come with some canned feeds such as Gaussian beams, and other simple feed patterns. Some programs can model both feed and reflector with hybridized techniques, so you can use your exact feed model. However, this is not always possible. It is fairly common to use a Gaussian beam feed model.

2. I need a more accurate feed model than the Gaussian Beam. What do I do?

This can be a difficult question to answer. If you can afford it, buy software that lets you model both feed and reflector. If you can't, there are a host of issues to be dealt with. A common situation is to have one program which can model the feed, such as HFSS, and another to model the reflector system, such as GRASP. So the question becomes:

3. How do I import simulated or measured feed data into a reflector analysis program?

Some programs allow you to input a "tabulated feed." However, my experience has suggested this is a nearly worthless feature by itself. The reasons for this belief are somewhat technical, but important to understand. Generally, one inputs a tabulated pattern on a sphere of some size, possibly a far-field sphere. Then, one needs to recover the field in the proximity of the reflector to be modeled. How can this be done? With an expansion method.

Generally, one expands the tabulated data in specially normalized spherical harmonics. However, this leads to a host of accuracy problems. The idea is that one picks out coefficients in a series expansion of the fields given on the input sphere, and then assumes these coefficients will hold for all such spheres, thus allowing one to infer the entire feed radiation pattern. Let's look at this more closely. Since GRASP is a software package that markets the capability to input tabulated feed data using a spherical wave expansion, and their technical documentation is freely available, let's see what their documentation has to say about this technique. On page 102 of the technical description, they introduce the "minimum mode sphere." The spherical wave expansion is said to only be valid outside of this minimum mode sphere. The radius they give is r=N/k, where 'k' is the usual wavenumber and 'N' is the maximum polar mode used in the spherical wave expansion used to approximate the input tabulated feed pattern.

So what values of r are "typical"? Of course it depends on the wavenumber, but lets look more closely at the value of 'N.' 'N' is described as the "upper radial index of the SWE (Spherical Wave Expansion)." To my knowledge, the term "upper radial index"
is never defined anywhere in the documentation. But, since I am an egg-head, this does not stop me.

If we jump now to page 266 of the technical description, we are told that "the spherical modes have an azimuthal index m and a polar index n." What happened to this "upper radial index" that defines the accuracy of the spherical wave expansion technique? Now they seem to be saying that no such index exists! Indeed, the expansions given for the fields in terms of spherical waves involve summations only over the "azimuthal index" and the "polar index." If one looks closely, one sees the only radial dependence is the Hankel functions in the sum, thus this egghead infers that the index of the Hankel functions must be this "maximum radial index." Careful inspection shows the Hankel functions are indexed by the "polar index", 'n.' So we will take 'n' as our radial index.

Now, we are told that "in general, sufficient numerical accuracy is obtained when N=kr0." But what is r0? It is the radius of the minimum sphere which encloses the radiating feed structure. Our original minimum mode sphere had radius N/k but N=kr0 so this means the minimum mode sphere appears to come out to r0 This doesn't seem too bad at first and may be acceptable in some cases. However, my experience has shown that it can take significantly more modes than the minimum quoted in the technical description to accurately capture a real feed pattern for all but the simplest feeds. Indeed, the minimum is derived from the Nyquist sampling criterion (just as in Fourier expansions) to prevent aliasing, or inaccurate reproduction of the original pattern. But the theoretical number of modes to prevent aliasing has nothing to do with the number of modes needed to approximate any given pattern, and in general the number of modes needed to approximate the pattern is much higher, making the size of the minimum mode sphere so large as to make the entire procedure useless as the reflector will be inside the minimum mode sphere and accurate results will not be obtained. OK. So be careful about imported feed data.

4. I really need to import my feed data, is there anything I can do to fix this problem?

Well, first of all, you need to have a program to convert tabulated output from one program into tabulated input into another program. This is not so bad if you are a programmer, but can be a real pain if you are an engineer. It gets worse. The way to reduce the number of polar modes needed in the spherical wave expansion is to apply a "filter" to your tabulated data before inputting the data into a reflector analysis program. You need to filter out "noise." Generally, you need to decide on an angular region outside of which the simulated/measured feed pattern is not relevant to the performance characteristics of interest for the reflector system under consideration. Then, you need to spline damped spherical harmonics into the pattern replacing all data outside of the chosen region of interest. To really get the best accuracy requires experimentation in how many modes you need to capture the pattern in the region of interest, and then splining the pattern outside the region of interest into exactly the right sum of modes and coefficients obtained from experimenting with the regeneration of the pattern. You also may want to filter out 'noise' in the pattern inside the angular region of interest. Experimentation comparing results of finite element and other techniques can help determine the various parameters of the filter. OK. This is the kind of thing that you need an egghead to do, so hopefully at least the point is clear that spherical wave expansions are not always such a great idea.

5. The spherical wave expansion is not always accurate (rarely accurate in my experience), is there anything else I have to watch out for?

Yes. In general it is not a significant issue but for compact antennae one needs to take into account possible backscatter into the feed. In other words, if you have a feed very close to a reflector, the reflected field can interact with the field produced by the
feed itself. This effect is not taken into account in most reflector packages. The best way to check and see if this is relevant is to calculate the intensity of the field generated at the aperture of the feed and compare that to the intensity of the reflected field at the aperture of the feed. If these are close in magnitude, be careful - a standing wave may be set up that seriously degrades the system performance. Once again, there are no cut and dry answers other than to play it safe and check everything with finite element software.

#### 6. What about surface tolerances?

More to come!

Author : Jackson Jones