A dynamical formulation of one-dimensional scattering theory and its applications in optics |
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Authors: | Ali Mostafazadeh |
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Affiliation: | Department of Mathematics, Koç University, 34450 Sar?yer, Istanbul, Turkey |
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Abstract: | We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are given as solutions of a set of dynamical equations. By decoupling and partially integrating these equations, we reduce the scattering problem to a second order linear differential equation with universal initial conditions that is equivalent to an initial-value time-independent Schrödinger equation. We give explicit formulas for the reflection and transmission amplitudes in terms of the solution of either of these equations and employ them to outline an inverse-scattering method for constructing finite-range potentials with desirable scattering properties at any prescribed wavelength. In particular, we construct optical potentials displaying threshold lasing, antilasing, and unidirectional invisibility. |
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Keywords: | Scattering theory Transfer matrix Complex potential Spectral singularity Anti-laser Unidirectional invisibility |
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