There are obvious needs for more efficient methods for the modeling of
diffraction and scattering problems—for instance, in the real-time resolution
of 2-D inverse problems in microelectronic process control, diffractive optics,
plasmonics, biophotonics and nanophotonics.
Perspectives for electromagnetic theory
The Rayleigh hypothesis has the potential to greatly simplify
the modeling of diffractive surface corrugations: After all, the
field solution is known; one just has to calculate the amplitudes
of the diffracted waves. This is usually done by matching the
fields at the interface, either directly or in the Fourier space.
(The method that Rayleigh himself used was an analytical
Computational methods based on the application of the
Rayleigh hypothesis have not been intensively developed during the past 40 years. The common belief was that it cannot
be the basis for the exact general modeling of actual gratings.
A remarkable exception is the C method—which is closely
related with the Rayleigh hypothesis. (In its embodiment, it is
a numerical Fourier-Rayleigh method.) It is successfully applicable to complex diffraction problems, and it is recognized as
a reference method for the analysis of metallic gratings with
smooth profiles. The smoothness of the profile is needed for an
effective application of the Fourier transform.
From the above theoretical derivation in the modal space, I
would expect that alternative methods based on the Rayleigh
hypothesis, but not using the Fourier transform, will significantly expand the class of structures that can be exactly solved.
These might include the modal expansion, the point-matching
technique and the T-matrix method.
The rehabilitation of Rayleigh’s hypothesis is, first of all, a
fundamental result in diffraction theory. The numerical demonstration of its validity and the given analytic proof are only
first steps in this direction. Further efforts are needed to exploit
its practical potential, both in the theoretical exploration of
the possible expansion of the class of admitted profiles and in
developing calculation methods that are capable of addressing ill-conditioned numerical problems that result from the
truncation of infinite series.
Very important in this regard is the possibility given by the
Rayleigh hypothesis of providing an analytical formulation
of the solutions up to close to the last step, where numerical
analysis is finally required to solve a large system, thus avoiding
the accumulation of errors typical of purely numerical methods and speeding up the calculation process dramatically.
What comes next
There are obvious needs for more efficient methods for the
modeling of diffraction and scattering problems, for instance,
in the real-time resolution of 2-D inverse problems in micro-
electronic process control, in diffractive optics as well as in basic
research and in plasmonics, biophotonics and nanophotonics.
The best known and most frequently used methods are the
finite-difference time domain and the Fourier-modal methods.
Alexander V. Tishchenko ( email@example.com) headed the research
team at the Institute of General Physics Moscow, where he developed analytical formulations of resonant grating problems on the basis of Rayleigh’s
hypothesis. He is now a professor at the Hubert Curien Laboratory, Université Jean Monnet of Saint-Etienne within the University of Lyon, France.
[ References and Resources ]
>> K Yasuura and Y. Okuno. J. Opt. Soc. Am. 72, 847-52 (1982).
>> G. Floquet. Ann. École Norm. Sup. 12, 47-88 (1883).
>> Lord Rayleigh (J. W. Strutt), Proc. R. Soc. London Ser. A 79, 399-
>> E. Loewen and E. Popov. Diffraction Gratings and Applications,
CRC; 1st edition (1997).
>> F. Bloch. Z. Physik 52, 555-600 (1928).
>> P.C. Waterman, Proc. Inst. Elect. Electron. Engrs. 53, 805-12
>> R. Petit and M. Cadilhac. C.R. Acad. Sci. Paris 262B, 468-71
>> B.H. T. Bates. IEEE Trans. Microwave Theory Tech. MT T- 15, 185-7
>> J. Chandezon et al. J. Opt. Paris 11, 235-41 (1980).
>> M.G. Moharam and T.K. Gaylord. J. Opt. Soc. Am. 72, 1385-92
>> A. Taflove. Computational Electrodynamics: The Finite-Difference
Time-Domain Method, Artech House (1995).
>> A.V. Tishchenko. Opt. Express 17, 17102-17117 (2009).
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