Restricted Hilbert Transform for Non-Hermitian Management of Fields

Non-Hermitian systems exploiting the synergy between the properties of closed-conservative systems and open-dissipative (gain-loss) systems have recently become the playground to uncover unusual physical phenomena. Indeed, the spatial symmetry breaking in such systems allows tailoring of the wave propagation at will. Inspired by such a property, we propose a feasible approach based on local Hilbert transform to control the field flows in two- or higher-dimensional non-Hermitian systems. Moreover, we invent an iterative procedure to reduce the dimensionality of complex refractive-index parameter space to two, one, or zero dimensions, restricting the complex refractive index within practical limits. The proposal provides a flexible way to systematically design locally non-Hermitian systems realizable with a limited collection of realistic materials.

Figure 1

Locally modified non-Hermitian medium providing the Hilbert transform. (a) Scattering potential of the medium, $n_{\mathrm{Re}}(x,y)$, modified by local HT to mold the flow of light in the desired direction. Here, the optical medium is specifically tailored to create source and sink fields, which involves gain-loss regions, $n_{\mathrm{Im}}(x,y)$. The generated two-dimensional complex potential, $n(x,y)=n_{\mathrm{Re}}+i{n}_{\mathrm{Im}}$, provides spatially dependent directionality in the neighborhood of each spatial point to control field flows. (b) 2D complex refractive-index profiles of the modified HT medium, requiring a large number of materials corresponding to different spatial points. (c) Cross-section profile of the complex index.

Figure 2

Restrictions of the local HT for different restriction dimensionalities. (a) Mixture of two different materials with ${\tilde{n}}_1$, ${\tilde{n}}_2$; for instance, fixing the total density ${\tilde{n}}_1+{\tilde{n}}_2=d$ (solid or liquidlike) gives a line, and a mixture of two materials restricting the maximal density, ${\tilde{n}}_1+{\tilde{n}}_2<d$, gives an area in complex space. (b) Continuum of scalable metachips gives an available ring in n-complex space. (c) Limited collection of scaled-in-size meta-atoms gives a set of points in complex space (five points in this case).

Figure 3

Complex refractive-index distributions for sink directionality: (a) local HT, (b)–(d) Restricted local HT with hexagonal background pattern. In Restricted HT cases, the complex index profiles are obtained after 15 iterations. First row presents the real part of refractive index, second row depicts the imaginary part obtained from the local HT, and third and fourth rows depict the corresponding restricted index values and density distribution in complex space, respectively. Size of computation region is $32{\lambda }_0\times 32{\lambda }_0$ and generated distributions have spatial resolution of about $0.25{\lambda }_0$ with wavelength ${\lambda }_0$.

Figure 4

Convergence of Restricted local Hilbert transform. (a)–(c) First and second rows depict refractive indices in complex space after 1st and 12th iterations, respectively. Red and blue colors illustrate the desired and computed restricted complex refractive-index values obtained from local HT, respectively. (d) Correlation coefficients ($C_0$ corresponds to solid line; $C_1$ corresponds to dotted line) in terms of flow of the complex potential generated by HT during iterative procedures for different restriction dimensionalities: □, 2D → 2D; ○, 2D → 1D; *, 2D → 0D.

Figure 5

Field evolution in modified HT media for sink directionality. System is excited with a Gaussian beam placed at an arbitrary position within the structure as shown in (ai). Numerically calculated transient and final localized states are shown in (aii) and (aiii), respectively. Steady-state distributions for restricted cases are (b) a square area, (c) ring, and (d) a set of discrete points on the ring. Results indicate that a Gaussian source, initially located at an arbitrary position, localizes around the center in all cases after sufficient propagation time.