© M.D. Kovalev

Formulas for the number of optical TE and TM- modes in the planar dielectric waveguide, consisting of the arbitrary number of layers, are obtained. The refractive indices of the layers including the infinite outside ones may admit only two different values. The calculation of the number of modes is carried out by the method, previously developed by the author for calculating the number of energy levels of quantum particle in a comb-like potential field. It is based on the analysis of a new form of the dispersion equation, the so-called multilayer equation, and is realized firstly in the case of general position. The result is transferred to the general case of any thickness of layers and any two values of refraction indices. The multilayer equation in optical case is obtained by equating to zero of some function F depending on effective refractive index of the waveguide. A key property of function F for our calculations is its monotony on intervals of its continuity and a presence only of infinite discontinuities. The examples of applications of obtained formulas for waveguides, consisting of five layers and admitting up to ten different self modes are considered.