70 IEEE Antennas and Propagation Magazine, Vol. 55, No. 4, August 2013

Postface

Joseph Bishop Keller

Professor EmeritusStanford University, Stanford, California, USA

In early 1944, I got a job in the Columbia University Divi sion of War Research, to work on sonar. My boss, Henry

Primakoff, asked me to help him analyze the scattering of sound by a dome. This is a thin, streamlined aluminum shell, which surrounds a projector. The projector emits a pulse of sound waves, and then acts as a hydrophone to detect any waves refl ected by a submarine. We obtained a Kirchhoff-like integral for the waves scattered by the dome, and evaluated it asymptotically for ka large. Here, 2k π λ= , where λ is the acoustic wavelength, and a is any lateral dimension of the dome.

I was pleased to fi nd that I could exactly derive the same result by Geometrical Optics. I used conservation of energy in a tube of refl ected or transmitted rays to get the amplitude. The phase at a point x was just ( )ks x , where ( )s x is the distance to x from the source, along an incident and refl ected or transmitted ray. This amplitude and phase yields the Geo-metrical Optics fi eld.

Later, as a graduate student of mathematics at NYU, I was disappointed to learn that my result was just a very special case of that of R. K. Luneberg. In his NYU lecture notes on the electromagnetic theory of optics (1948-49), he showed that the leading term in the short-wavelength asymptotic expansion of any time-periodic fi eld is just the Geometrical Optics fi eld. He derived this result, and higher-order terms, by Fourier transformation of a transient fi eld. Each discontinuity in the transient fi eld gave rise to a Geometrical Optics term (direct, refl ected, transmitted, etc.). Each discontinuous derivative gave rise to a higher-order term.

I then heard of the conformal-mapping method used by Busemann to analyze weak shock waves in aerodynamics. I realized that it could be used to solve the two-dimensional problem of diffraction of a step-function pulse by a wedge with Dirichlet or Neumann boundary conditions. My student, Albert Blank, and I solved this problem in 1950. Our solution contained the plane incident and refl ected discontinuities, as we expected. But in addition, it contained an expanding cir cular wavefront of radius ct , centered at the corner or edge of the wedge. Here, c is the propagation speed, and t is time after the incident discontinuity hits the edge. The fi eld was continuous across this circular wavefront, but it behaved like ( )1/2ct r− behind the wavefront, for r ct< , where r is dis tance from the edge. The fi eld thus had a singularity in its r derivative of order 1/2. Luneberg’s method shows that this singularity would yield a

term of order 1/2k− in the asymp totic expansion of the fi eld, but his expansion had no such term.

This made me realize that wavefronts diffracted from edges had to be included in the high-frequency or short-wavelength asymptotic expansion of fi elds. The rays normal to these wavefronts are edge-diffracted rays. When the incident wave hits the edge obliquely, these rays form cones, and the diffracted wavefronts are also cones. The rays satisfy the law of edge diffraction.

I developed these ideas into “The geometrical theory of diffraction,” which I presented at a Symposium on Microwave Optics at the Eaton Electronics Research Laboratory, McGill University, Montréal, Canada. It was published in the Pro-ceedings of the symposium in June, 1953. (The editor gratui-tously inserted the word “optics” after “geometrical” in the title.) This article described the edge-diffracted rays, the law of edge diffraction, the diffraction coeffi cients, etc.

Subsequently, I extended the theory to diffraction by curved smooth bodies, to diffraction in inhomogeneous media, to diffraction by vertices in three dimensions, etc. My stu dents, Robert Lewis, Bertram Levy, Bernard Seckler, Leo Levine, Robert Buchal, and many others, helped me in this work. We also used Braunbek’s idea to write the fi eld scat tered by a screen as a modifi ed Kirchhoff integral, with the exact current near the edge replacing the Kirchhoff value. Asymptotic evaluation of this integral gave the diffracted rays and fi eld with the correct diffraction coeffi cients. Then Lewis, my postdoc Daljit Ahluwalia, and our Dutch visitor, Johannes Boersma, constructed a uniform asymptotic theory of edge diffraction.

In 1962, in the Journal of the Optical Society of America, I published a review of some of this work entitled “Geometri cal Theory of Diffraction.” This was at the invitation of the Editor, Victor Twersky, who had been a student in one of my classes years before.

On the 50th anniversary of the publication of that paper, I am pleased to read in the papers in this volume how the theory has been developed and extended by so may others. The review by John L. Volakis shows how the theory was extended to apply to half-planes and wedges with impedance boundary conditions by T. Senior and by G. Malyuzhinets, as well as to structures with higher-order boundary conditions by many others. The work of Giuseppe Pelosi and Stefano Selleri describes how to solve wedge problems numerically as well as analytically. I.

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IEEE Antennas and Propagation Magazine, Vol. 55, No. 4, August 2013 71

Christiansen, N. C. Albertsen, and O. Breinbjerg describe the work in Denmark on GTD in the labo ratories of H. L. Knudsen and of E. B. Hansen, which led to new ways of understanding diffraction coeffi cients. Some of them visited me at NYU, and I often visited them in Lyngby. Y. Rahmat-Samii presents a very nice historical overview from his personal perspective. P. Ya. Ufi mtsev’s review of his extensive work on his PTD provides a valuable complement to the papers on the GTD.

Correction The following corrections should be made to the article by John W. Arthur, “The Evolution of Maxwell’s Equations from 1862 to the Present Day,” IEEE Antennas and Propagation Magazine, 55, 3, June 2013, pp. 61-81:

Page 61, column 1, the last line should read:

H is the magnetic field intensity.

Page 62, column 2, footnote 1: the reference given should be [4, p.460]

Page 64, Figure 2, at the end of the caption, the following credit should be included:

(Based on the digitized copy of the original article, available on the Royal Society of London’s Web site [4]).

Page 66, equation (G) should be:

( ) ( )0 1 equationfreex x y y z zG D D Dρ + ∂ + ∂ + ∂ =

Page 69, column 2, footnote 6: the font size of the word “different” should be the same as the rest of the text.

Page 78, column 1, the last line of the first paragraph should read:

...with the Lorentz force cast as f q Fυ=