Unidirectionally invisible potentials as local building blocks of all scattering potentials

We give a complete solution of the problem of constructing a scattering potential $v\left(x\right)$ that possesses scattering properties of one's choice at an arbitrary prescribed wave number. Our solution involves expressing $v\left(x\right)$ 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.

Figure 1

Graphs of real part of of $\mathfrak{n}\left(x\right)-1$ (solid blue curve) and the imaginary part of $\mathfrak{n}\left(x\right)$ (dashed purple curve) for $\mathfrak{n}\left(x\right)$ given by Eq. (23) with $\alpha =-{10}^{-4},\beta =1.759\times {10}^{-4}$, and $|k_{0}x|\le \pi$.

Figure 2

Graphs of the reflection and transmission coefficients $|R^l{|}^2$ (blue full curve), $|R^r{|}^2$ (black dotted curve) and ${\left|T\right|}^2$ (purple dashed curve) as functions of $k/k_0$ for the refractive index profile given by Eq. (23) with $\alpha =-{10}^{-4}$ and $\beta =1.759\times {10}^{-4}$. The sharp peak at $k=k_0$ is an obvious signature of a spectral singularity.

Figure 3

Graphs of $|R^l|$ (solid blue curve) and $|R^r|$ (dashed purple curve), and $|T-\sqrt{2}i|$ (dotted red curve) as functions of $k/k_0$ for the potential given by Eqs. (12) and (26), and $k_0=2\pi /\mu \mathrm{m}$. The fact that all of these quantities vanish for $k=k_0$ shows that this potential models a bidirectionally reflectionless amplifier that doubles the intensity of the incident wave upon transmission and induces a $\pi /2$ phase.