User:Bci2versty/X-ray & Neutron/Uniform theory of diffraction

Fermat’s principle postulated in 1654 saying that no matter what kind of reflection or refraction to which a ray is subjected, the ray takes minimum time to travel from one point to another, then classical geometrical optics was established when Fermat’s principle was completed in mathematical theory. This is the fundamental theory describing the phenomenon of ray [1, p3]. However, classical geometrical optics is limited on solving high-frequency electromagnetics problems because the concepts of phase, wavelength, polarization and diffraction are not included.

Physicists introduced the electromagnetic wave theory with mathematical equations can explain what classical geometrical can not, but it has practical disadvantages and the concepts of geometrical optics were lost. Some time later, Luneberg and Kline extended the classical geometrical optics to modern geometrical optics (GO) by establishing connection between Maxwell’s electromagnetic field equations and geometrical concepts. However, GO is incapable of predicting the field in the shadow regions. For example, at shadow boundaries, the field intensity changes rapidly and GO cannot describe this behaviour correctly.

Electromagnetic fields have to be continuous everywhere, the discontinuities across the shadow boundaries does not exist in nature, and that is another reason why GO fails to describe the total electromagnetic field.

“Diffraction is the process whereby light propagation differs from the predictions of geometrical optics.” Joseph B. Keller explained the phenomenon of diffraction with this description. He introduced diffracted rays that behave like the ordinary rays of GO once they leave the edge and this laid the basis for what has become geometrical theory of diffraction (GTD).

In the lit region (y > 0) there is a ripple in the total field that is caused by the interference of the diffracted field with the incident field. The ripple decreases when the field is moving away from ISB because the strength of diffracted field is decreasing and the amplitude of the total field approaches the incident plane wave. Keller extended the GO to GTD by adding diffracted fields and he succeeded in correcting the deficiency in the GO that predicts zero fields in the shadow regions. However, the original form of the GTD suffers many problems. The most serious one is at shadow boundaries, where the GO fields fall abruptly to zero and the predicted diffracted fields become infinity.

Keller’s original GTD is not a uniform solution. It can predict the diffracted fields in regions away from the shadow boundaries, but become singular in the transition regions surrounding such boundaries.

The uniform theory of diffraction (UTD) developed by Kouyoumjian was divergent from Keller’s GTD theory but overcomes GTD’s defects that GTD is inapplicable in the vicinity of shadow boundaries. UTD performs an asymptotic analysis and by incorporating a transition function into the diffraction coefficient, the diffracted fields remain bounded across the shadow boundaries.

[1] D. A McNamara, C. W. I. Pistorius, and J. A. G. Malherbe, Introduction to The Uniform Geometrical Theory of Diffraction. Boston: Artech House, 1990 [2] M. Born, E. Wolf. Principle of Optics – Electromagnetic Theory of Propagation Interference and Diffraction of Light. London: Pergamon, 1959 [3] J. B. Keller, “One Hundred Years of Diffraction Theory” IEEE Transactions on Antennas and Propagation., vol. AP-33, No. 2, February 1985 [4] G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves. England: Peter Peregrinus Ltd. 1976 [5] M. J. Neve, High Frequency Asymptotic Techniques, 660.702 Applied Electro magnetics Course Notes. Department of Electrical and Electronic Engineering, The University of Auckland, New Zealand, 1997. [6] M.Abramowitz, I. Stegun, Handbook of Mathematical Functions. New York: Dover, fifth ed., May 1986 [7] J. Boersma, “Computational of Fresnel Integrals,” J. Math. Comp., vol. 14, p.380, 1960.