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The paper presents the formulation of the dispersion relations for regular planar waveguides and smoothly irregular integrated optical waveguides in forms of elementary continuous functions. The stable algorithm for computing the roots of the dispersion relations for regular and irregular waveguides, which is applicable in the case of real and complex coefficients of the phase retardation of waveguide modes, is developed. The theoretical and numerical method for studying the characteristics of inhomogeneous waveguide modes in the overcritical regimes is elaborated.

In contrast to the zero approximation of the adiabatic modes method, and moreover to the method of comparison waveguides, the fields of irregular waveguide modes of an integrated optical waveguide in the first approximation are described by a pair of equations of coupled oscillators allowing resonance solutions.