Addendum to “Unidirectionally invisible potentials as local building blocks of all scattering potentials”

Abstract

We give a complete solution of the problem of constructing a scattering potential v(x) that possesses scattering properties of one's choice at an arbitrary prescribed wavenumber. Our solution involves expressing v(x) as the sum of at most six unidirectionally invisible finite-range potentials for which we give explicit formulas. Our results can be employed for designing optical potentials. We discuss its application in modeling threshold lasers, coherent perfect absorbers, and bidirectionally and unidirectionally reflectionless absorbers, amplifiers, and phase shifters.Complex scattering potentials in one dimension provide a fertile ground for modeling various active optical systems. Among these are systems displaying spectral singularities [1][2][3][4] and unidirectional reflectionlessness and invisibility [5][6][7][8][9]. Spectral singularities correspond to scattering states that behave exactly like zerowidth resonances [1]. In optics they give rise to lasing at the threshold gain [10] while their time-reversal is responsible for coherent perfect absorption (CPA) or antilasing [11,12]. Unidirectional reflectionlessness and invisibility are also of great interest, because they offer means of realizing one-way linear optical devices [5,7]. The task of designing scattering potentials that support such desirable properties is clearly a problem of basic theoretical and practical importance. In this article we give a complete solution for this problem.Throughout this investigation, we use the transfer matrix of one-dimensional scattering theory [13] as our main tool. For a scattering potential v(x), the general solution of the Schrödiger equation, −ψ ′′ (x)+v(x)ψ(x) = k 2 ψ(x), or the Helmholtz equation, ψ ′′ (x)+k 2 n (x) 2 ψ(x) = 0, has the asymptotic form:where k is the wavenumber, n (x) is a refractive index that we can relate to v(x) via n (x) = 1 − v(x)/k 2 , and A ± and B ± are complex coefficients. The transfer matrix of v(x) (and n (x)) is the 2×2 matrix M satisfyingWe can express it in terms of the left and right reflection amplitudes, R l and R r , and the transmission amplitude T of v(x) according to [1]:Recall that the asymptotic scattering solutions of the above Schrödinger and Helmholtz equations have the formand the reflection and transmission coefficients are given by |R l/r | 2 and |T | 2 . We can use (1), (2) and (3) whereWe respectively refer to these conditions as "left-reflectionlessness" and "right-reflectionlessness." Clearly the condition for bidirectional reflectionlessness is R l = R r = 0. If a potential that is unidirectionally reflectionless at a wavenumber k 0 has a unit transmission amplitude, i.e., T = 1, at this wavenumber, we call it "unidirectionally invisible" [5]. Similarly, we use "left-invisible", "right-invisible", and "bidirectionally invisible" to mean that a potential is respectively leftreflectionless, right-reflectionless, and bidirectionally reflectionless and in addition satisfies T = 1 at k = k 0 .In this article we use the properties of the transfer matrix to addres...

Transfer matrix formulation of scattering theory in two and three dimensions

“…This is not only because of the composition property of the transfer matrix, which allows for the reduction of the scattering problem for a given potential to those of its truncations, but also because any discretization of this approach would yield matrix representations for the entries of the transfer matrix operator that could be substituted in (20) to determine T ± and consequently the scattering amplitude. This project was initiated during F. L.'s visit to Koç University Aug. [11][12][13][14][15][16][17][18][19][20][21][22][23][24]2015. We are indebted to the Turkish Academy of Sciences (TÜBA) for providing the financial support which made this visit possible and to Teoman Turgut for helpful discussions.…”

“…Next, we recall that ψ scat (r) = f (ϑ, ϕ) e ikr /r, where f is the scattering amplitude and (r, ϑ, ϕ) are the spherical coordinates of r with ϑ and ϕ respectively denoting the polar and azimuthal angles. Comparing this expression with the one given by (16) and (33) and using an argument given in Appendix F, we find…”

“…In order to see how M relates to the scattering amplitude of the potential v, we consider a left-incident wave and write the corresponding scattered wave in the form (16) where Θ is the Heaviside step function, i.e.,…”

“…These in turn give the scattered wave ψ scat according to (16) provided that we replace x by z. Next, we recall that ψ scat (r) = f (ϑ, ϕ) e ikr /r, where f is the scattering amplitude and (r, ϑ, ϕ) are the spherical coordinates of r with ϑ and ϕ respectively denoting the polar and azimuthal angles.…”

Perturbative unidirectional invisibility

“…The possibility of realizing it has attracted a lot of attention, because it provides a tool for constructing certain one-way optical devices [11]. Another remarkable property of unidirectionally invisible potentials is that they serve as the building blocks for constructing potentials with given scattering properties at a given wavenumber [12]. These observations provide ample motivation for a systematic study of the problem of characterizing scattering potentials displaying unidirectional invisibility.…”