The paper considers a model of adiabatic waveguide modes in the zero approximation applied to the numerical solution of the problem of single-mode propagation of guided modes in a smoothly irregular integrated optical waveguide. Within the framework of the model, the solution to the Maxwell system of equations is reduced to a form that is expressed through the solution of a system of four ordinary differential equations and two algebraic equations for six components of the electromagnetic field. The multilayer structure of waveguides makes it possible to carry out one more stage of reducing the system of equations of the model to a homogeneous system of linear algebraic equations, the condition of nontrivial solvability of which specifies the dispersion equation. Auxiliary eigenvalue and eigenvector problems for describing adiabatic waveguide modes are solved. Example solutions for single-mode propagation of adiabatic waves are presented.
The paper considers the cross-sectional method and the small parameter method in application to the numerical solution of the problem of single-mode propagation of TE-modes in a smoothly irregular integrated optical waveguide invariant in the transverse direction. The cross-sectional method is formulated as a Helmholtz problem for a two-dimensional strip of smoothly irregular thickness. Using the Kantorovich method, the problem is reduced to a parametric problem for eigenvalues and eigenfunctions on a segment. The small parameter method is formulated as a Helmholtz problem on a two-dimensional plane, which in turn is also reduced to a parametric eigenvalue/eigenfunction problem on an axis. We implement various algorithms for the numerical solution of the resulting problems. Their coincidence high accuracy is shown with at the intersection of the definition domains.
In the case of guided and radiation modes of open waveguides, the Sturm-Liouville problem is formulated for self-adjoint second-order operators on the axis and the corresponding eigenvalues are real quantities for dielectric media. The search for eigenvalues and eigenfunctions corresponding to the leaky modes involves a number of difficulties: the boundary conditions for the leaky modes are not self-adjoint, so that the eigenvalues can turn out to be complex quantities. The problem of finding eigenvalues and eigenfunctions is associated with finding the complex roots of the nonlinear dispersion equation. Leaky modes, by analogy with radiation and guided modes, are considered as solutions of the Helmholtz equation. The presence of complex eigenvalues corresponding to the leaky modes leads to an infinite increase of eigen- functions corresponding to the leaky modes. In the present work, the leaky modes are considered as solutions of the problem for the wave equation and are analyzed as a wave process. Complex eigenvalues define the leaky modes as inhomogeneous waves. The proposed approach allows to obtain a mathematically sound representation of the leaky modes, within which one can see the region of existence of each particular leaky mode. The calculations of model structure, demonstrating the application of the described approach, are presented in the paper.
In this paper we investigate the solution of the problem of modeling the propagation of electromagnetic radiation in three-dimensional integrated optical structures, such as waveguide lenses. When propagating through three-dimensional waveguide structures the waveguide modes can be hybridized, so the mathematical model of their propagation must take into account the connection of TE- and TM-mode components. Therefore, an adequate consideration of hybridization of the waveguide modes is possible only in vector formulation of the problem. An example of three-dimensional structure that hybridizes waveguide modes is the Luneburg waveguide lens, which also has focusing properties.
If the waveguide lens has a radius of the order of several tens of wavelengths, its variable thickness at distances of the order of several wavelengths is almost constant. Assuming in this case that the electromagnetic field also varies slowly in the direction perpendicular to the direction of propagation, one can introduce a small parameter characterizing this slow varying and decompose the solution in powers of the small parameter. In this approach, in the zeroth approximation, scalar diffraction problems are obtained, the solution of which is less resource-consuming than the solution of vector problems. The calculated first-order corrections of smallness describe the connection of TE- and TM-modes, so the solutions obtained are weakly-hybridized modes.
The formulation of problems and methods for their numerical solution in this paper are based on the authors' research on waveguide diffraction on a lens in a scalar formulation.
In the paper we construct a method for approximate solution of the waveguide problem for guided modes of an open irregular waveguide transition. The method is based on straightening of the curved waveguide boundaries by introducing new variables and applying the Kantorovich method to the problem formulated in the new variables to get a system of ordinary second-order differential equations. In the method, the boundary conditions are formulated by analogy with the partial radiation conditions in the similar problem for closed waveguide transitions.
The method is implemented in the symbolic-numeric form using the Maple computer algebra system. The coefficient matrices of the system of differential equations and boundary conditions are calculated symbolically, and then the obtained boundary-value problem is solved numerically using the finite difference method. The chosen coordinate functions of Kantorovich expansions provide good conditionality of the coefficient matrices. The numerical experiment simulating the propagation of guided modes in the open waveguide transition confirms the validity of the method proposed to solve the problem.
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