How many channels can a photonic system support?

TL;DR

The work establishes a general method to bound the $n^{th}$ singular value of the electromagnetic Green function for arbitrarily structured photonic systems using the Courant-Fischer-Weyl min-max principle. By decomposing the transfer function as $\mathbf{G} = \mathbf{G}^{\circ} + \mathbf{G}^{\circ}\mathbf{T}\mathbf{G}^{\circ}$ and applying convex relaxations over the $\mathbf{T}$-operator or optimizing over subspaces, the authors produce computable, predictive limits on channel amplitudes that connect device structure to information-theoretic measures. Demonstrations on wavelength-scale 3D volumes illustrate bounds for waveguide-like mediators, metasurface-style configurations, and planewave-detection problems, with implications for Shannon capacity and Fisher information, and reveal how mediator size and geometry can substantially increase usable channels. Collectively, the results offer design-guiding insight into the tradeoffs of multi-channel photonic systems and highlight practical pathways for engineering multifunctional devices in communications and sensing.

Abstract

We develop a general method to bound the ordered singular values (channel amplitudes) of the electromagnetic Green function for arbitrarily structured linear photonic systems. The approach yields computable, quantitatively predictive, upper bounds on the $n^{th}$ singular value that capture the complexity of multi-channel tradeoffs from the device perspective. As an illustration of the practical value of the framework, indexed channel bounds are obtained for multi-wavelength scale three-dimensional volumes (up to $64\,λ^3$) and applied to common application classes related to waveguides, metasurfaces, and planewave detection. These results are immediately applicable to the calculation of information theoretic objectives such as Shannon capacity and Fisher information.

Figures (3)

• Figure 1: Channels and singular values. The figure illustrates the meaning of an electromagnetic channel in the sender-mediator-receiver convention used throughout the article. Each channel consists of a pair of connected input-output fields (one in the sender and one in the receiver) that are orthogonal to the fields of every other channel, along with a channel amplitude (the singular value $\sigma_{k}$) that dictates the strength of the mapping. Working in terms of channels leads to a completely decomposed description of communication (transmission). The lower part of the figure shows four physical phenomena, among many, that are described by functional sums over channel amplitudes: heat transfer molesky2020fundamental, Purcell enhancement chao2023maximum, lensing molesky2021comm, and Casimir torques strekha2022trace.
• Figure 2: Limit channel amplitudes for common three-dimensional design scenarios. (a) Upper bounds on the channel capacity of an arbitrarily structured "waveguide" (top: $\chi = 13.6 + 0.05i$, silicon-like mediator, bottom: $\chi = 0$, vacuum mediator) for different mediator volume lengths $l$ and cutoff amplitudes $c$. The mediator design volume (common schematic shown in the bottom pane) is always separated from the unstructured sender and receiver regions (on either side) by $\lambda/4$ . The purple $c = 3.0$ count line seen in the bottom panel is simply recopied from the top panel to aid comparison. (b) Upper bounds on achievable channel strengths for metasurface-like applications with varying component sizes and separation distances. The surface areas and thicknesses of the senders and receivers scale with that of the mediator for each scenario. Unannotated lines for the vacuum mediators match the annotated separations for the $\chi = 13.6 + 0.05i$ bounds. A common schematic showing the different dimensions of the sender, mediator and receiver for all four panels is shown in the top-left pane.
• Figure 3: Limit channel amplitudes for planewave detection. (a) Schematic for planewave angle detection metasurface design. The metasurface design region has dimensions $0.3\lambda \times C_y$. The detection region is a line segment of length $C_y$, separated from the metasurface by $\lambda/2$. The input space is spanned by a basis of $M$ planewaves with wavevectors at angles $\phi_l = \phi_{min} + (l-1) (\phi_{max}-\phi_{min})/(M-1)$. For the example shown, $\phi_{max,min} = \pm 15^\circ$. (b) Bounds and designs maximizing smallest singular value of $H$ for incident planewave detection, with $\phi_{min/max} = \pm 15^\circ$.

Theorems & Definitions (2)

• proof : Proof of $1|n$ inequality.
• proof : Proof of trace inequality