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An analysis of the B.P.M. approximation of the Helmholtz equation in an optical fiber. (English) Zbl 0626.65136
The electric field, due to a single frequency (\(\omega)\) light, inside an optical fiber is assumed to be of the form \(A(x,\omega)\exp.-ik_ 0x_ 3\), \((k_ 0=\omega n_ 0/c)\). The equation for the envelope function A(x,\(\omega)\), supposed to vary slowly over a single wave length is \[ (E)\quad -2ik_ 0(\partial A/\partial x^ 2_ 3)+(\partial^ 2A/\partial x^ 2_ 3)+\Delta_{\perp}A+(\omega^ 2/c^ 2)(n^ 2- n_ 0^ 2)A=0. \] Here \(\Delta_{\perp}\) is the transverse part of the Laplace operator. The refractive index n has small variation from the reference value \(n_ 0\). The ‘Beam Propagation Method (B.P.M.)’ algorithm is based on a discretization in \(x_ 3\) direction, in which, for the first and third intervals of length \(\Delta x_ 3/2\) each, the last term of eq. (E) is neglected and in the central interval \((\Delta x_ 3)\), the second term is droped, and further only the wave propagating in the \(x_ 3\) direction is retained. In this paper, the authors derive formally the continuous equation \((\Delta x_ 3\to 0)\), which is consistent with the B.P.M. algorithm. This equation is obtained by replacing the second and third terms of eq. (E) by a convolution term with A(x,\(\omega)\). Next, the authors give a mathematical framework that allows one to prove the existence and uniqueness of a solution of the B.P.M. equation. Finally, the long time behaviour of these solutions and the dependence of solutions on n are studied.
Reviewer: N.D.Sengupta
MSC:
65Z05 Applications to the sciences
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
78A40 Waves and radiation in optics and electromagnetic theory
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References:
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