Electromagnetic wave diffraction by semi-infinite discrete structures
The theory of wave diffraction by semi infinite set of obstacles is important for the electromagnetic field analysis in bounded photonic band-gap structures. Methods for solving the problems of wave diffraction by semi-infinite discrete structures have been developed much worse than the theory of wave diffraction by semi-infinite structures that have Wiener-Hopf geometry.
The spectral operator R of reflection by a semi-infinite periodic structure is most essential in the developed theory. This operator assigns to an incident field with some known space spectrum the reflected field generated by diffraction.
A semi-infinite system of partly transmitted layers is the one with the infinite number of identical obstacles. The neighbour obstacles are equally spaced at the range h along the Oz axis. This discrete semi-infinite structure isn’t a periodic one but can be transformed to self by space displacement h to the positive direction along the Oz axis and reducing one edge structure element. Such specific geometry can be used efficiently for solving the problem of wave diffraction by semi-infinite system of partly transmitted layers.
In the concept of this method it isn’t important what kind of obstacle would be a partially transmitted layer. For example it may be a plane-parallel dielectric slab, metal screen, or diffraction array. It is important that solution was known for the problem of wave diffraction by a single obstacle, i.e. operators were known for both the transmission and reflection by a single layer. Therefore further we will consider the semi-infinite system of the periodic strip gratings in the dielectric half-space. However all the below operator equations will be correct for any other kind of obstacles.
According to the Floquet theorem, the scattering electromagnetic field over and under a single grating can be represented by a series of partial waves. The vectors of space partial wave amplitudes in the region and satisfy to equations
where are the operators of transmission by the boundary :
are the operators of reflection by the boundary :
and are the propagation constants of the n-th space partial wave along the Oz axis in the free space and in the dielectric, accordingly; R is the operator of reflection by a semi-infinite system of strip gratings in the dielectric half-space with permittivity ; is the transform operator of space partial wave vector by the forward displacement along the Oz axis.
Let us analyse the field in the region to determine the operator in the above system of equations. In this region the equations for the vectors of space harmonic amplitudes are
Here and are the transmission and reflection operators of the single strip grating in the dielectric [L.N. Litvinenko and S.L. Prosvirnin, Spectral scattering operators in the problems of wave diffraction by plane screens. – Kiev: Naukova dumka, 1984. – 240 p. (in Russian)]; is the transform operator of amplitude vector of space partial waves by the field displacement h along the Oz axis. We can evaluate B, then C and b, and obtain the operator equation
to determine the unknown matrix operator R, I is the identity operator.
Equation (1) is the non-linear operator equation. The generalized Newton method can be used for solving this equation.
The electromagnetic field is the sum of incident wave and space partial waves in the free half-space over the semi-infinite structure. It can be written as the following expression
Complex amplitudes of reflection space partial waves could be found as
Thus, the solution of the problem is reduced to the solving of operator equation (1) relative to R by an iterate method.
Some numerical results respect to the electromagnetic wave diffraction by semi-infinite periodical structures of dielectric slabs and strip gratings may be found in the papers referenced below.
 Litvinenko L.N., Reznik I.I., Litvinenko D.L., Wave diffraction by semi-infinite periodical structure // Dokladi AN Ukraine, 1991, no. 6.
 Litvinenko L.N., Pogarsky S.A., Prosvirnin S.L., Wave diffraction by semi-infinite periodical structures, Journal of Infrared and Millimeter Waves, 1996, v.17, ïî.5, pp.897-903.
 Litvinenko D.L., Litvinenko L.N., Prosvirnin S.L., Reznik I.I., Wave diffraction by semi-infinite system of partially transmitted layers. - Proc. of the VI Inter. conf. on Mathem. Methods in Electromag. Theory, 1996, Lviv, Ukraine.- pp. 96 - 99.
 Ëèòâèíåíêî Ë.Í., Ïðîñâèðíèí Ñ.Ë., Àíàëèç äèôðàêöèè âîëí íà ïîñëåäîâàòåëüíîñòè èäåíòè÷íûõ ëåíòî÷íûõ ðåøåòîê. Ìíîãîâîëíîâûé ðåæèì, Ðàäèîôèçèêà è ðàäèîàñòðîíîìèÿ, 1999, ò.4, ¹3, 276-286 [PDF-1.14M].
 Lytvynenko Leonid, Prosvirnin Sergey, Schunemann Klaus, Wave scattering by semi-infinite structure Programme poster presentations, International Union of Radio Science XXVII-th General Assembly, Maastricht, Aug. 17-24, 2002, report B2P.12, paper 911.
 Lytvynenko L. M., Prosvirnin S. L., Schuenemann K., Wave diffraction by periodic multilayered structures, Radio Physics and Radio Astronomy, 2005, vol. 10, Special issue, pp. S186-S201 [PDF-246K] .