Transfer Matrix Formulation of Scattering Theory in Two and Three Dimensions

Abstract

In one dimension one can dissect a scattering potential $v(x)$ into pieces $v_i(x)$ and use the notion of the transfer matrix to determine the scattering content of $v(x)$ from that of $v_i(x)$. This observation has numerous practical applications in different areas of physics. The problem of finding an analogous procedure in dimensions larger than one has been an important open problem for decades. We give a complete solution for this problem and discuss some of its applications. In particular we derive an exact expression for the scattering amplitude of the delta-function potential in two and three dimensions and a potential describing a slab laser with a surface line defect. We show that the presence of the defect makes the slab begin lasing for arbitrarily small gain coefficients.

Version to appear in Phys. Rev. A, 10 pages, 3 figures

References in corpus (6)

• Coherent Perfect Absorbers: Time-reversed Lasers
• The transfer matrix: a geometrical perspective
• Lasing Threshold Condition for Oblique TE and TM Modes, Spectral Singularities, and Coherent Perfect Absorption
• Perturbative Unidirectional Invisibility
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• Nonlinear Scattering and Its Transfer Matrix Formulation in One Dimension
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• Transmission of low-energy scalar waves through a traversable wormhole
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• Exact Solution of the Two-Dimensional Scattering Problem for a Class of $\delta$-Function Potentials Supported on Subsets of a Line
• Transfer matrix for long-range potentials
• Dynamical formulation of low-energy scattering in one dimension
• Dynamical Theory of Scattering, Exact Unidirectional Invisibility, and Truncated $\mathfrak{z}{e}^{-2i{k}_{0}x}$ potential
• Exactness of the first Born approximation in electromagnetic scattering
• Exceptional points and pseudo-Hermiticity in real potential scattering
• Can N-th Order Born Approximation Be Exact?
• Scattering Theory and $\mathcal{P}\mathcal{T}$-Symmetry
• Broadband directional invisibility
• Singularity-free treatment of delta-function point scatterers in two dimensions and its conceptual implications
• Propagating-wave approximation in two-dimensional potential scattering
• Potentials with Identical Scattering Properties Below a Critical Energy
• Modeling Nonlinear Optics with the Transfer Matrix Method
• Geometric scattering in the presence of line defects
• Scattering by a collection of $\delta$-function point and parallel line defects in two dimensions
• Renormalization of multi-delta-function point scatterers in two and three dimensions, the coincidence-limit problem, and its resolution
• Existence of the transfer matrix for a class of nonlocal potentials in two dimensions