The Derivation of the Dispersion Equations of Adiabatic Waveguide Modes in the Thin-Film Waveguide Luneburg Lens in the Form of Non-Linear Partial Differential Equation of the First Order

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Abstract


This paper presents a derivation of the dispersion equation for a three-layer integrated-optical Luneburg lens based on the method of adiabatic waveguide modes. From this equation there follows the relationship between the coefficient of phase deceleration and function, which determines the thickness of the irregular waveguide layer. The dispersion equation is represented in the form of non-linear partial differential equation of the first order with coefficients, depending on parameters. Among these parameters are regular waveguide layer thickness and optical parameters of the pending Luneburg lens. To represent the dispersion equation in the form of differential equations in partial derivatives, it is necessary to calculate a symbolic form the determinant of a matrix of 12th order, which determines the solubility of the system of linear algebraic equations, resulting from the boundary conditions. To calculate this determinant in analytical form a procedure of reduction of the system of linear algebraic equations with the use of the computer algebra system Maple is proposed.

M I Zuev

Joint Institute for Nuclear Research

Email: zuev.max@gmail.com
Laboratory of Information Technologies

E A Ayryan

Joint Institute for Nuclear Research

Email: ayrjan@jinr.ru
Laboratory of Information Technologies

J Buˇsa

Technical University in Ko.sice

Email: jan.busa@tuke.sk

V V Ivanov

Joint Institute for Nuclear Research

Email: ivanov@jinr.ru
Laboratory of Information Technologies

L A Sevastianov

Peoples’ Friendship University of Russia

Email: leonid.sevast@gmail.com
Telecommunication System Department

O I Streltsova

Joint Institute for Nuclear Research

Email: strel@jinr.ru
Laboratory of Information Technologies

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Copyright (c) 2013 Зуев М.И., Айрян Э.А., Буша Я., Иванов В.В., Севастьянов Л.А., Стрельцова О.И.

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