Quasi-Exactly Solvable Scattering Problems, Exactness of the Born Approximation, and Broadband Unidirectional Invisibility in Two Dimensions

Abstract

Achieving exact unidirectional invisibility in a finite frequency band has been an outstanding problem for many years. We offer a simple solution to this problem in two dimensions that is based on our solution to another more basic open problem of scattering theory, namely finding scattering potentials $v(x,y)$ in two dimensions whose scattering problem is exactly solvable for energies not exceeding a critical value $E_c$. This extends the notion of quasi-exact solvability to scattering theory and yields a simple condition under which the first Born approximation gives the exact expression for the scattering amplitude whenever the wavenumber for the incident wave is not greater than $\alpha :=\sqrt{E_c}$. Because this condition only restricts the $y$-dependence of $v(x,y)$, we can use it to determine classes of such potentials that have certain desirable scattering features. This leads to a partial inverse scattering scheme that we employ to achieve perfect broadband unidirectional invisibility in two dimensions. We discuss an optical realization of the latter by identifying a class of two-dimensional isotropic active media that do not scatter incident TE waves with wavenumber in the range $(\alpha /\sqrt{2},\alpha ]$ and source located at $x=\infty$, while scattering the same waves if their source is relocated to $x=-\infty$.

Expanded published version, 7 pages, 1 figure

References in corpus (5)

• Transport in Bilayer Graphene: Calculations within a self-consistent Born approximation
• Invisibility and PT-symmetry
• Bidirectional invisibility in Kramers-Kronig optical media
• Unidirectionally Invisible Potentials as Local Building Blocks of all Scattering Potentials
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• Can N-th Order Born Approximation Be Exact?
• Dynamical formulation of low-frequency scattering in two and three dimensions
• Broadband directional invisibility
• Introducing a general method for solving electromagnetic radiation problem in an arbitrary linear medium