An inverse problem in coupled mode theory

We study an inverse problem for the Zakharov-Shabat system which is motivated by an application to the design of co-directional couplers with prescribed response properties. An inverse problem in coupled mode theory Paul Sacks Department of Mathematics Iowa State University Ames, IA 50011 January 30, 2004 1 Statement of direct and inverse problem In this paper we are interested in an inverse problem for a system of ordinary differential equations which arises in the design of optical fiber devices. In the direct problem, we are given a coupling coefficient q(x), which is a bounded, complex valued function on [0, X]. For any k ∈ C and x ∈ [0, X] let A = A(x, k), B = B(x, k) denote the solution of ∂A ∂x = ikA+ qB ∂B ∂x = −ikB ± qA (1.1) with initial conditions A(0, k) = 1 B(0, k) = 0 (1.2) The corresponding inverse problem is to determine q(x) on [0, X] given the scattering data {B(X, k) : k ∈ R} (1.3) A closely related problem is the system (1.1) with boundary conditions A(0, k) = 1 B(X, k) = 0 (1.4) in which case the data for the inverse problem is {B(0, k) : k ∈ R} (1.5)