Non-Hermitian curved space via inverted wave equation

TL;DR

This work addresses the design of non-Hermitian, isotropic photonic media that actively shape wave amplitude, phase, and direction by inverting the wave equation to obtain a complex refractive index $n(x,y)$ from predefined fields. It introduces a curved-space analogue in which the real part of $n$ maps to geometric height and the imaginary part encodes spatial gain and loss, enabling a flexible inverse-design framework that includes sources and sinks. The authors demonstrate three NH-inspired capabilities: amplitude control, phase conversion (including planar-to-cylindrical and quadratic phase), and an isolator that achieves nonreciprocal wave shunting, all solved via explicit design equations and verified numerically. The approach promises practical implementations on silicon photonic platforms and broad applicability to advanced NH photonic devices such as CPA, invisibility, and lasing, expanding the inverse-design toolkit for nanophotonics. $n(x,y)$ is obtained from predefined waves using $n(x,y)=\sqrt{\frac{k_0^2 E_{\rm in}-\nabla^2 E_{\rm sc}}{k_0^2 E}}$, and phase-design employs $n=(k_0)^{-1}\sqrt{(S_x')^2+(S_y')^2-i(S_{xx}''+S_{yy}'')}$ with $E=\exp(iS)$.

Abstract

Inverting design method of solving passive graded materials from predefined amplitude and phase was developed along the line of transformation optics (TO), which however precludes the presence of source and sink in the pragmatic world. So in this Letter we extend such an inverse method to non-Hermitian media, offering more freedom to manipulate the wave flows. Our principle of a curved-space analogue picture powered with gain and loss, is exemplified by three types: amplitude controlling, phase conversion, and direction shunting. Our numeric examples showcase precise wave manipulation in a surprisingly simple manner, which goes beyond convectional paradigms of TO and is readily implementable in realistic photonic platform.

Figures (4)

• Figure 1: Amplitude modulation of a plane wave by two NH media: (a) gain medium vs. (d) loss medium in permittivity profiles; (b) electric field distribution in the gain medium with white arrows indicating the Poynting vector; (c) matched fields of computation and predefinition for both media, which also include comparison of varied $\beta$ values to tune the variance; (e) schematic of wave propagation in the gain medium (a); (f) imaginary parts of refractive index under different $\beta$ values for the gain medium (a), which aligns with a positive divergence of Poynting vector $\nabla\cdot\mathbf{S}>0$ representing a wave source negative in $\Im n(x, y)$.
• Figure 2: Phase conversion from $k_0x$ to $k_1x$ (with $k_1={2k_0}/{3}$) via a NH medium: (a) distribution of the permittivity; (b) electric field along $x$ axis for comparison between computed and predefined fields; (c) curved surface analogue with coloured imaginary. Phase conversion from $k_0x$ to $k_0^2x^2$ similarly: (d) - (f). In both panels (c) and (f), positive divergence of Poynting vector $\nabla\cdot\mathbf{S}>0$ represents a wave source with negative index $Im n(x, y)<0$, and a negative divergence $\nabla\cdot\mathbf{S}<0$ does a wave sink with positive imaginary index $\Im n(x, y)>0$.
• Figure 3: Conversion of plane-wave phase to cylindrical-wave phase in a NH medium. (a, b) Distribution of the complex permittivity and note the virtual point $(-b, 0)$ marked by circles; (c) electric field distribution; (d) comparison between predefined and simulated electric fields along the black line in (c); (e) schematic of wave propagation through the designed medium; (f) contour plots showing the real and imaginary parts of the refractive index, both with an arrow indicating the increasing directions. Parameter: $b=2$.
• Figure 4: Isolator medium: (a) material parameter uniform in vertical direction; (b) electric field distribution when the incident plane wave is deflected from $\theta\vert {n=0}$ to exit at $\theta\vert {n=10}$; (c) four power ratios of reflection and transmission (inset) in scattering matrix $\mathbf{S}$; (d) relative accuracy of the calculated field $E_{\rm c}$ compared to the predefined $E_{\rm p}$ at different exit angles, which is defined as $\xi(\theta) = 1 -{ [ \iint \vert E_{\rm c} - E_{\rm p}\vert^2 \, {\rm d}S} / { \iint \vert E_{\rm p}\vert^2\, {\rm d}S]^{1/2}}$.