# On a Professional Note

Dear Visitor,

I am a electronic engineer specialising in antenna design and manufacture and have over the years published a couple of papers in technical journals and magazines that I would like to bring to the attention of my colleagues in this field. Specifically, I have published one paper in which I challenge the conventional wisdom on a principle applied in practically every antenna theory textbook, and also others that I think students may be unaware of and may contribute to their understanding of some concepts in antenna theory.

The papers I would like to discuss are listed below, with a complete but very modest list of my professional publications listed at the end.

- Aperture Theory and the Equivalence Principle
- An Illustrative Equivalence Theorem Example
- On Second-Order Asymptotic Expansions at a Caustic
- On the Equivalence of Geometrical-Optics Reflected Fields and the Stationary-Phase Solution of the Associated Radiation Integrals
- Circular Aperture Pattern Synthesis From Collapsed Equivalent Line-Source Distributions
- Analysis and Performance of a 12 Element Phase Steered Etched Dipole Array
- Complete list of publications

## 1. Aperture Theory and the Equivalence Principle

At a time when the powerful electromagnetic simulation packages of today were not yet readily available or very expensive, I developed an elementary Method of Moments program, based on the equivalence principle, for the design of basic antenna types restricted to two-dimensional profiles. When this was completed, I decided to add an aperture theory capability to the package, just to round it off. I had scarcely completed this when one of my former colleagues at the University of Pretoria, whom I had told about my software, sent one of his students to me to help him analyse a specific slot radiation problem. My software package allows one to set up the problem very quickly and I did so with the student looking over my shoulder. Much to my embarrassment, however, the result I got was utter nonsense, not even remotely correct. Convinced that I must have made a mistake in either the programming or my assumptions about the EM fields in the slot, I had to send the student home, mumbling some excuse. I could, of course, not leave the problem there and spent many evenings trying to figure out what the problem might be and studying various implementations of aperture field theory. Through trial and error I eventually came to the conclusion **that contrary to what is stated in practically all textbooks, one can indeed use the free space radiation integrals to solve aperture problems of arbitrary geometry, instead of having to use the geometry-specific Green’s function**. It took me a seemingly endless 6 years to get a paper with a proper formulation published.

Read more here

I have recently decided to take another look at the problem and have submitted a follow-on article that includes the following example of a reflector illuminated by a line source. The electromagnetic fields are calculated along a circular ‘aperture’ as indicated, and the original reflector is replaced with the new geometry and the magnetic current density radiating in its presence. The radiated patterns for the original and the artificial circular problem are almost indistinguishable. A significantly simpler proof of the concept is also presented. I am happy to report that the new paper has now been published.

Read more here

As a matter of interest, it took me about 19 years from the discovery of the problem (I initially thought that I had made a simple programming error somewhere), to finding the solution through trial and error, to at last reaching the publication of this final, much clearer proof-of-theory. The reasons why it took so long include that I first and foremost had a strenuous daytime job to do, along with the research for and writing and publication of my two books, as well as many other time-consuming interests (my Nadal-vs-Federer article, my Mathematical-Analysis-of-the-Star-of-Bethlehem article, not to mention the hundreds of hours of Matlab programming to extract continental outlines from, or converting to different projections, the numerous medieval maps I used in my Atlantis theory, along with seemingly endless minor additions to my website, and also the writing of several other professional articles as well). Just bringing my work to public attention through advertising and related discussion forums took an immense amount of time, whilst in-between all of this I still had to fit in a bit of a social life, mostly playing tennis, socialising with friends, watching movies and collecting and listening to music (that about sums up my social life - boring, isn't it?). The final proof-of-theory also did not just happen. I would every couple of years suddenly decide to attempt once more to derive a better formulation, only to fail again. However, in the end I managed to succeed (in my opinion), or as we say in Afrikaans, 'Aanhouer wen' ('He who perseveres, wins', more or less the same as 'Slow and steady wins the race'). But brother, was it slow ...

Some of the reviewer comments: "The approach proposed by the author is original and interesting", "Good finding and contribution", "The readability of the manuscript is excellent and the theoretical developments are clearly reported", "I think that this study is excellent and examples are enough and clear."

(a) Reflector illuminated by line source (b) Circular aperture and magnetic current density

Radiation patterns for original and equivalent geometries

## 2. An Illustrative Equivalence Theorem Example

As stated in the discussion of my paper on aperture theory, I devised a problem in which a plane wave propagates in free space, of which a circular part is replaced by a fictitious surface S with equivalent electric and magnetic surface currents superimposed on S. This example is ideal for evaluating the equivalence principle as the surface currents can be derived analytically and the radiation integrals can be evaluated by standard numeric methods. Regrettably some errors in the naming of the figure captions have crept in during the typesetting process of the original published paper and could not be corrected in time. For that reason I have reprinted the paper with the correct figure captions as can be viewed here. I am grateful to Professor Sembiam Rengarajan for his advice and contribution to the paper (text in grey).

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## 3. On Second-Order Asymptotic Expansions at a Caustic

When I began my PhD studies, I had long forgotten my undergraduate electromagnetic theory and had to catch up very quickly. On of the first problems I had to solve was how to calculate the high frequent electromagnetic fields at a caustic. I found an expression for the field at the caustic, but it did not yield the correct field away from the caustic, as it should. This little problem caused me a lot of self doubt in my capability to solve problems, and even whether I would ever be able to complete my PhD studies. Fortunately, and this is probably one of the aims of a postgraduate degree, I was able to derive a solution that was valid at and also away from the caustic. However, I was not quite happy that it was 100% correct, even though I had completed my studies (1990), and it remained in the back of my head for many years to come. After two half-hearted attempts at about four year intervals, another couple of years later I finally thought of a method that I could use to test my derivations - use the Method of Moments to calculate the fields and compare the results to the asymptotic expansion results. It took me an entire Christmas holiday break to derive the stationary phase and steepest descent solutions and present them in a paper. It then took another couple of years to get it published (2005).

This result is probably of very little if any practical importance, but it does present greater insight into the application of asymptotic techniques to scattering problems. Having said that, I am by no means an expert in this field and there still is a possibility that my derivations may not be 100% correct. Please do not hesitate to let me know if this is indeed the case. Regardless, it was sweet revenge on this nuisance little equation.

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## 4. On the Equivalence of Geometrical-Optics Reflected Fields and the Stationary-Phase Solution of the Associated Radiation Integrals

My PhD studies initially focussed on high frequency electromagnetic scattering problems, but I soon realised that I would have to find a means for verifying my calculations. The easiest way to do so was to analyse the scattering problem by means of the Method of Moments. As expected, the results were exactly the same. Nevertheless, I could not understand just why that would be the case and despite searching high and low, I could not find any discussion or proof of it anywhere in a textbook. I began playing around with the mathematics and was fortunate to be able to derive the proof I was looking for, for the geometrical optics reflection case. You can view it here (note that a typo had slipped in - Eq (2) should read Er = … ).

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## 5. Circular Aperture Pattern Synthesis From Collapsed Equivalent Line-Source Distributions

Anechoic chambers all have a limited far field range and in order to stretch that range in our chambers a bit, I wrote software for transforming a limited range measured pattern (amplitude and phase) to a far field pattern. In the process I noticed that, under certain conditions, one can collapse any aperture distribution to an equivalent line source distribution. I then realised that one could use the inverse process to synthesise circular aperture patterns (see paper here). I don't believe it has much practical value, but let me know if you can think of any.

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## 6. Analysis and Performance of a 12 Element Phase Steered Etched Dipole Array

At one stage of my career I conducted an investigation into phased array antennas and had to be able to predict the radiation patterns and input VSWR of an array of dipoles. Despite an extensive search, I could not find any papers or textbooks that presented exactly what I needed, and I had to make an attempt at deriving it myself. The result is shown in a paper I presented at a symposium (see here). I have no doubt that experts in the field does this type of derivation with their eyes closed, but it took me a while to get it done. Please let me know if you do not agree with the analysis and also if you know of any papers or textbooks in which it has been dealt with.

Read more here

## 7. Complete list

### Journals and Magazines

- A.J. Booysen, “On Second-Order Asymptotic Expansions at a Caustic,” Applied Computational Electromagnetics Society Journal, Vol. 20, No. 1, pp. 21-34, Mar. 2005.
- A.J. Booysen, “Circular Aperture Pattern Synthesis From Collapsed Equivalent Line-Source Distributions,” IEEE Trans. Antennas Propagat., Vol. 52, No. 11, pp. 2904-2911, Nov. 2004.
- A.J. Booysen, “Aperture Theory and the Equivalence Principle,” IEEE Antennas and Propagation Magazine, Vol. 45, No. 3, pp. 29-40, Jun. 2003.
- A.J. Booysen, “On the Equivalence of Geometrical-Optics Reflected Fields and the Stationary-Phase Solution of the Associated Radiation Integrals,” IEEE Antennas and Propagation Magazine, Vol. 44, No. 5, pp. 120-123, Oct. 2002.
- Riaan Booysen, “An Illustrative Equivalence Theorem Example,” IEEE Antennas and Propagation Magazine, Vol. 42, No. 6, pp. 132-135, Dec. 2000.
- A.J. Booysen, “A Physical Interpretation of the Equivalence Theorem,” IEEE Trans. Antennas Propagat., Vol. 48, No. 8, pp. 1260-1262, Aug. 2000.
- A.J. Booysen and C.W.I. Pistorius, “Electromagnetic Scattering by a Two-Dimensional Wedge Composed of Conductor and Lossless Dielectric,” IEEE Trans. Antennas Propagat., Vol. 40, No. 4, pp. 383-390, Apr. 1992.
- A.J. Booysen, C.W.I. Pistorius and J.A.G. Malherbe, “Reduction of the RCS of the Leading Edge of a Conducting Wing-Shaped Structure by Means of Lossless Dielectric Material,” Microwave and Optical Technology Letters, Vol. 4, No. 7, pp. 277-279, June 1991.
- A.J. Booysen and J.A.G. Malherbe, “Transmission Line Analysis of a Double Linear Taper in Rectangular Waveguide,” Microwave Journal, pp. 169-171, November 1989.

### Symposium papers

- A.J. Booysen, “Analysis and Performance of a 12 Element Phase Steered Etched Dipole Array,” IEEE/SAIEEE AP/MTTS-92 Proceedings, pp. 77-83, 1992.

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