Coherent Control of Wave Scattering via Coincidences of Complex Spectra

TL;DR

The paper develops a general analytic framework for coherent control of multichannel wave scattering in complex geometries by introducing critically-constrained scattering modes (CCONs). Routing and demultiplexing are achieved by coordinating coincidences of CCON eigenfrequencies on the real axis, with topological protection ensuring robustness against parameter changes. A minimal-parameter design principle is derived, predicting how many tunable parameters are needed to realize overconstrained functionalities, and these predictions are validated through random-matrix theory, quantum-graph modeling, and full-wave simulations of a tunable chaotic cavity. The approach is general to all linear waves, enabling versatile devices for filtering, power division, and directional lasing with in situ reprogrammability and broad applicability to metamaterials and beyond.

Abstract

We introduce and validate a theoretical framework for coherent control of multichannel scattering of linear waves to route waves through complex geometries with multiple scattering. We show that steady-state perfect routing solutions are achievable at any frequency via tuning geometric param- eters so that multiple complex eigenfrequencies coincide on the real axis. The relevant complex spectra describe critically constrained scattering processes (CCONs), where a specific number of generically accessible outgoing channels are not excited due to destructive interference. Focusing on electromagnetic waves, we demonstrate in simulations high discrimination routing and demulti- plexing of signals in a multiport chaotic cavity with a number of tunable scatterers which can be predicted from theory. A similar approach can be used to implement other interesting functional- ities, such as filtering, power division and directional lasing. The method can be applied to other classical waves and also to quantum matter waves.

Figures (8)

• Figure 1: A schematic for different kinds of CCONs in a 3-port system and the corresponding submatrix of $S$ (in green), which must have zero determinant. (a) A schematic of a simple 1 port reflectionless mode $(\tilde{R}TT)$, (b) a 2 port Reflectionless mode $(\tilde{R}\tilde{R}T)$ and (c) A "dark" CCON $(ND\tilde{R}$): port 1 is "normal", and has both an input and reflected wave, port 2 is "dark" and has neither input or output, and port 3 is reflectionless. (d) A scatter plot of the zeros corresponding to the different CCON boundary conditions in (a)-(c), including CPA $(\tilde{R}\tilde{R}\tilde{R})$, and resonances $(TTT)$, obtained from a Random Matrix Theory model (see [ \ref{['title:main']}] for more details).
• Figure 2: R-zero tracking in a tunable chaotic cavity (inset) with two ellipsoidal scatterers of variable orientation, demonstrating tuning of an R-zero to a target real frequency (yellow dot) for an arbitrary initial configuration. (a) Tuning a single ellipsoid orientation (purple curves) will generate real frequency reflection dips, but not at the target frequency; optimizing with both scatterers (blue curve) gives a high quality solution. (b) Reflection coefficient of the cavity for three cases: (Solid blue curve) configuration optimized to show reflection dip at the target frequency, $0.975f_c$, where $f_c$ is the cutoff frequency of the waveguides. (Solid Purple curve) reflection spectrum corresponding to single parameter tuning at the configuration when an R-zero passes through the real axis randomly. (Gray dashed line) reflection spectrum at the starting point, showing no deep reflection dips.
• Figure 3: Example of a routing solution (see arrows) found by four parameter optimization, the minimum number of required parameters predicted by the theory, see Eq. (4). (a) Heat map of the optimized EM field, $H_z$, showing routing from port 1 to port 2 with discrimination $> 75 \; dB$. (b) Scattering matrix elements when exciting channel 1. Inset: Blow up of a region in the complex frequency plane near the real target frequency, showing the predicted coincidence of CCONs, $(\tilde{R}TT)$ and $(\tilde{R}T\tilde{R})$, to accuracy$<10^{-7}$.
• Figure 4: Demonstration of reprogrammable demultiplexing in chaotic cavity with 8 tunable scatterers (as predicted by theory). The target frequencies are $f_1 = 0.965 f_c$, at $f_2 = 0.975 f_c$. The data shown in fig. (a) correspond to the top routing schematic, with $f_1$ signal routed to port two and $f_2$ signal routed to port 3. The scattering matrix elements (left axis) show deep dips for $S_{11}$ at both $f_1$ and $f_2$, and for $S_{31}$ at $f_1$ and for $S_{21}$ at $f_2$. Symbols (right axis) are the CCON spectra (see legends) with a coincidence of $(\tilde{R}TT)$ and $(\tilde{R}T\tilde{R})$ at $f_1$ and of $(\tilde{R}TT)$ and $(\tilde{R}\tilde{R}T)$ at $f_2$, as predicted by theory. Panels (c-d) show heat maps of the EM field (Re{$H_z$}) with the desired routing patterns at $f_1,f_2$ for case (a). (b) Shows similar data to (a) for the bottom routing schematic, where the roles of ports 2 and 3 interchanged (we don't show field maps for this case).
• Figure 5: (a) Schematics for 4-port CCONs (top rows) and routing functions (second and third rows), which are coincidences of CCONs, as shown in the schematics. (b) Variation of Figure of Merit (FOM) for the various functions computed for a four-port chaotic cavity (reflection zero, single transmission zero, perfect reflection (T-symmetric case), partial routing, full routing) when optimized with a variable number of tuning parameters. In all cases excellent performance is found for the minimum number of parameters predicted by theory. Results are averaged over an appropriate ensemble of configurations of the tunable scatterers, (details are given in [ \ref{['title:main']}]).