Multilayer model in optics. New analitic results.

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Multilayer model in optics. New analitic results. M.D.Kovalev BMSTU mdkovalev@mtu-net.ru

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Planar multilayer waveguide

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For ТЕ-waves propagating along Oz axis this is a boundary-value problem for the equation

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Reduced variables

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First example. 7 layers. The number of ТЕ-modes: K=6.

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Traditional dispersion equations –equations for the eigenvalues of the propagation constant Type 1 – equation, is obtained by equating to zero of the determinant of homogenius linear system due to boundary conditions. Type 2 ? equation, obtained by the known method of characteristic matrices This equations have too many terms if the number of layers is more then 4. Investigation of waveguides with many layers is now actual.

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The properties of the dispersion equations Th.1. Type 1 equation has roots, coinsiding with the refraction indexes of the inner layers of the waveguide. This roots may not be the eigenvalues of propagation constant. Th.2. The set of roots of type 2 equation is exactly the set of the eigenvalues of propagation constant. We propose a new one form of the dispersion equation. This equation in some known cases have no parasitic roots, and moreover it may be treated geometrically.

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Multilayer equation

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Homogenius variables,vectors

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Theorem 3. Vectors rotate counter-clockwise when is decreasing. Theorem 4. If then the directions of this vectors are converging to the direction of Ox axis.

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The multilayer equation in vector form

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The formulae for the number of TE-modes.

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The second example, K=7. The difference from the first example is only

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Thank you for the attention Ковалев М.Д., Число TE- и TM-мод в многослойном планарном волноводе со слоями двух типов. Электромагнитные волны и электронные системы, 2009, т. 14, № 2, С.4--17. Ковалев М.Д. Многослойное уравнение. Чебышевский сборник. Тула, т. 7, выпуск 2 (18), 2006, С. 99—106. Майер А.А., Ковалев М.Д. . Дисперсионное уравнение для собственных значений эффективного показателя преломления в многослойной волноводной структуре. ДАН, 2006, т. 407, №6, С. 766--769. Ковалев М.Д. , Многослойная модель в оптике и квантовой механике. ЖВМ и МФ, 2009, т. 49, №8, С. 1 -- 14. Ковалев М.Д., Об энергетических уровнях частицы в гребенчатой структуре. ДАН, 2008, Том 419, № 6, С. 1--5.