Energy-absorption calculus for multi-boundary diffraction gratings

 

A general energy-balance criterion for multi-boundary lossy periodical objects (general gratings) has been derived and verified numerically. In the general case, the difference A = 1 R T e 0 is called the absorption coefficient in the given diffraction problem with the sums of reflected and trans-mitted energies R and T, respectively. In addition to being physically meaningful, this expression is useful as one of the accuracy tests for computational codes. The energy criterion in the lossless case says A = 0. In the lossy case, one needs an independently calculated quantity to compare with A. For such a quantity, we use the absorption expression defined as the sum of volume or surface integrals. The equation for the absorption A of an electromagnetic field by a multilayer grating can be derived directly from Maxwell’s equations [1], or by the variational principle [2], or by applying the second Green’s identity to boundary functions for the contours in the upper and lower media [3]. By definition, the first part of integrals in the expression of A is 1 R, and the second, T, vanishes if the lower medium is absorbing or the lower boundary is perfectly conducting. The absorption expression in the explicit form which is based on scattering amplitude matrices has been added to the previous study to treat closed and separated boundaries, e.g. photonic crystals [4]. The sum A + R + T is actually the energy balance for an absorbing grating, and the extent to which it approaches unity is a measure of the accuracy of calculations. Maxwell equations being valid in the sense of distributions, the proposed general energy-balance criterion is valid in the same sense. The connection of the derived expression with the optical theorem and its application to non-periodical surfaces is discussed.

 

[1] Goray L.I., 2010, Application of the boundary integral equation method to very small wavelength-to-period diffraction problems, Waves Random Media, Vol. 20, pp. 569-586.

[2] Goray L.I. and Schmidt G., 2010, Solving conical diffraction grating problems with integral equations, J. Opt. Soc. Am. A, Vol. 27, pp. 585-597.

[3] Goray L.I., Kuznetsov I.G., Sadov S.Yu., and Content D.A., 2006, Multilayer resonant subwave-length gratings: effects of waveguide modes and real groove profiles, J. Opt. Soc. Am. A, Vol. 23, pp. 155-165.

[4] Goray L.I. and Schmidt G., 2012, Analysis of two-dimensional photonic band gaps of any rod shape and conductivity using a conical-integral-equation method, Phys. Rev. E, Vol. 85, pp. 036701-1-12.

© 2012 `Days on Diffraction‘, POMI.

 

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