Hierarchical mean-field $\mathbb{T}$ operator bounds on electromagnetic scattering: Upper bounds on near-field radiative Purcell enhancement

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Hierarchical mean-field $T$ operator bounds on electromagnetic scattering: Upper bounds on near-field radiative Purcell enhancement

Sean Molesky, Pengning Chao, and Alejandro W. Rodriguez

Phys. Rev. Research 2, 043398 – Published 21 December 2020

Abstract

We present a general framework, based on Lagrange duality, for computing physical bounds on a wide array of electromagnetic scattering problems. Namely, we show that, via projections into increasingly localized spatial clusters, the central equality of scattering theory—the definition of the $T$ operator—can be used to generate a hierarchy of increasingly accurate mean-field approximations (enforcing local power conservation) that naturally complement the standard design problem of optimizing some objective with respect to structural degrees of freedom. Utilizing the systematic control over the spatial extent of local violations of physics offered by the approach, proof-of-concept application to maximizing radiative Purcell enhancement for a dipolar current source in the vicinity of a structured medium, an effect central to many sensing and quantum technologies, yields bounds that are often more than an order of magnitude tighter than past results, highlighting the need for a theory capable of accurately handling differing domain and field-localization length scales. Similar to related domain decomposition and multigrid notions, analogous constructions are possible in any branch of wave physics, providing a unified approach for investigating fundamental limits.

Received 15 September 2020
Accepted 24 November 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.043398

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Effective field theory Electromagnetism Nanophotonics Photonics Scattering theory

Abstract algebra Cluster methods Functional analytical methods Mean field theory Numerical approximation & analysis Self-consistent field theory

General PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Sean Molesky ^*, Pengning Chao ^*, and Alejandro W. Rodriguez^†

Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA

^*These authors contributed equally to this work.
^†arod@princeton.edu

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Vol. 2, Iss. 4 — December - December 2020

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Figure 1
Schematic hierarchical mean-field bounds and application to scattering cross sections. (a) Schematic of light scattering from a source near a structured (light blue) object enclosed within a cubic domain, which is partitioned into various subdomains (clusters). Like Maxwell's equations, the definition of the scattering $T$ operator, Eq. (1), imposes constraint relations that must be satisfied at every point in real space. Present approaches to electromagnetic limits [6, 22, 23, 24], however, only impose that this definition be respected on average, ignoring the possibility of unphysical local field variations—graphically depicted by the red (positive) and blue (negative) violations in the value of the imposed constraint equality within the smaller subdomains, otherwise satisfied on average when integrated over the entire volume. (b) By enforcing the validity of the definition of the scattering $T$ operator on successively smaller subdomains, dual bounds, $G$ given by Eq. (7), acquire a hierarchical order that mirrors the primal problem of optimizing some objective in terms of an increasing number of structural degrees of freedom, $L$ in Eq. (5). When strong duality holds, covering every case we have tested, the convex relaxation given by Eq. (7) determines an exact optimal mean-field solution respecting the associated set of constraints, and in the limit of point clusters and complete structural freedom, the two problem statements become equivalent. (c) Upper bounds on plane-wave scattering for any structure bounded (contained) by a sphere of radius $R$ . Lighter-colored lines result by asserting that real and reactive power is conserved globally, as in Refs. [6, 22]. Associated darker lines are produced by enforcing similar equalities over eight, evenly spaced, radial subdomains. A profile of one of the density-optimized structures is shown as an inset. Logarithmic color maps of the corresponding violation in the real part of Eq. (2), $Re [〈S| I_{x} |T〉] Z / λ$ versus $〈T| Sym [U I_{x}] |T〉 Z / λ$ , for one, two, and four (evenly spaced) shell clusters, in the plane perpendicular to both the incoming wave vector and direction of polarization, are shown below the panel.
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Figure 2
Upper bounds on radiative Purcell enhancement. Following the same conventions used in Fig. 1, but for a dipolar current source in the near field of an object, the imposition of relatively simple (radial) shell constraints is shown to substantially alter both the conditions under which resonant radiative Purcell enhancement (far-field emission compared to vacuum) can possibly occur [(a) and (b)] and to generally reduce the dependence of calculated limits on material properties [(a)–(c)]. In all panels, darker lines indicate achievable enhancement, for any structure fitting within a ball of radius $R$ , calculated with eight shell clusters of radii $r_{shell} / R = \{0.97, 0.95, 0.93, 0.91, 0.89, 0.87, 0.85, 0.6\}$ . Fainter lines, often more than an order of magnitude larger, describe similar results but supposing only a single cluster comprising the entire spherical domain. Dots mark enhancement values achieved by actual structures discovered through inverse design. Notable features are described in the main text. Radiative Purcell enhancements achievable with weak materials, $- 1 < Re [χ] < 1$ , are found to be unremarkable and hence are omitted from (c). Profiles and cross cuts of two representative density-optimized structures are displayed below (a) and (b). The color maps below (c) show spatial violations in the real part of the constraint equality of Eq. (2) for the parameter values marked in (b). Crucially, strong duality holds in all examples, indicating that use of more varied clusters (e.g., breaking spherical symmetry) should lead to even tighter bounds.
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Physical Review Research

Hierarchical mean-field T operator bounds on electromagnetic scattering: Upper bounds on near-field radiative Purcell enhancement

Sean Molesky, Pengning Chao, and Alejandro W. Rodriguez

Phys. Rev. Research 2, 043398 – Published 21 December 2020

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Hierarchical mean-field $T$ operator bounds on electromagnetic scattering: Upper bounds on near-field radiative Purcell enhancement