Abstract
We show how the central equality of scattering theory, the definition of the $\mathbb{T}$ operator, can be used to generate hierarchies of mean-field constraints that act as natural complements to the standard electromagnetic design problem of optimizing some objective with respect to structural degrees of freedom. Proof-of-concept application to the problem of 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 performance bounds that are frequently more than an order of magnitude tighter than all current frameworks, highlighting the irreality of these models in the presence of differing domain and field-localization length scales. Closely related to domain decomposition and multi-grid methods, similar constructions are possible in any branch of wave physics, paving the way for systematic evaluations of fundamental limits beyond electromagnetism.
Highlights
The study of structural design in photonics centers largely around three interconnected aims: explaining shared response features across many classes of geometries, discovering particular geometries with notable response characteristics, and characterizing the space of achievable responses and its dependence on constraints
Fields and structural variations set by Maxwell’s equations is nonconvex [21], the increasingly widespread use of numeric optimization has sharpened several open questions. It is rarely known how close the structures discovered by any particular algorithm approach the true optimum performance dictated by fundamental physical principles or to what extent response characteristics are determined by specific design choices
We have connected the necessity of imposing local field constraints to properly describe general scattering phenomena with the present shortcomings of recently developed programs for calculating fundamental performance limits on photonic devices via Lagrange duality [6,22,23,24], currently encompassing applications such as solar light trapping, enhancing radiative emission, near-field quenching, and a host of other engineering challenges related to electromagnetic power [6,22,23]
Summary
The study of structural design in photonics centers largely around three interconnected aims: explaining shared response features across many classes of geometries (e.g., effective medium theory [1] and topological photonics [2]), discovering particular geometries with notable response characteristics (e.g., high-efficiency antennas [3] and light-trapping films [4]), and characterizing the space of achievable responses and its dependence on constraints (e.g., the existence of fundamental limits [5] and scaling laws [6]). Accelerating over the last decade, the continued adoption of large-scale numerical methods has greatly simplified the collection of challenges related to discovery Techniques such as “density” (“topology”) or level-set optimization [7,8]—ideally matching structural degrees of freedom to the underlying computational discretization—have produced improved designs for applications varying from enhanced polarization control [9,10] and ultrathin optical elements [11,12,13] to wide band-gap photonic crystals [14,15,16] and topological materials [17,18,19]. When applied to situations where evanescent (near-field) wave effects dominate overall behavior (Fig. 2), these prior techniques produce bounds that are several orders of magnitude larger than, and exhibit markedly different trends from, what has been observed in any actual geometry, including those discovered by density optimization We remedy this issue, and elucidate fundamental connections between scattering theory, performance bounds, and structural optimization, by introducing the notion of multiscale T operator constraint hierarchies.
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