Abstract
A general framework for determining fundamental bounds in nanophotonics is introduced in this paper. The theory is based on convex optimization of dual problems constructed from operators generated by electromagnetic integral equations. The optimized variable is a contrast current defined within a prescribed region of a given material constitutive relations. Two power conservation constraints analogous to the optical theorem are utilized to tighten the bounds and to prescribe either losses or material properties. Thanks to the utilization of matrix rank-1 updates, modal decompositions, and model order reduction techniques, the optimization procedure is computationally efficient even for complicated scenarios. No dual gaps are observed. The method is well-suited to accommodate material anisotropy and inhomogeneity. To demonstrate the validity of the method, bounds on scattering, absorption, and extinction cross sections are derived first and evaluated for several canonical regions. The tightness of the bounds is verified by comparison to optimized spherical nanoparticles and shells. The next metric investigated is bi-directional scattering studied closely on a particular example of an electrically thin slab. Finally, the bounds are established for Purcell’s factor and local field enhancement where a dimer is used as a practical example.
Highlights
As the field of nanophotonics becomes more mature, interest is shifting away from the analysis of simple systems and toward the synthesis of structures with engineered electromagnetic behavior, e.g., maximal absorption [1], directional emission [2, 3], directed scattering [4, 5], fluorescence diplexing [6], waveguide power division [7], field confinement [8], and waveguide diplexing [9]
While such methods excel in the exploration of extremely broad design spaces and the discovery of nonintuitive solutions, there exists a strong need for analytic results which inform, direct, and truncate their computationally intensive calculations [10]
Physical bounds on performance objectives constitute a particular class of analytic results that aid inverse design in this way
Summary
As the field of nanophotonics becomes more mature, interest is shifting away from the analysis of simple systems (uniform waveguides, spheres, rods, etc.) and toward the synthesis of structures with engineered electromagnetic behavior, e.g., maximal absorption [1], directional emission [2, 3], directed scattering [4, 5], fluorescence diplexing [6], waveguide power division [7], field confinement [8], and waveguide diplexing (wavelength splitters) [9]. The purpose of this paper is to transfer this general approach to the field of optics, where the fundamental electromagnetic physics remain the same as in classical antenna theory but the metrics of interest shift from antenna parameters to quantities related to scattering, absorption, extinction, and local field enhancement To carry this out, we first develop a framework in which to discuss quantities of interest (e.g., scattered power, local fields) in terms of operators acting on current densities confined to a region of interest. We first develop a framework in which to discuss quantities of interest (e.g., scattered power, local fields) in terms of operators acting on current densities confined to a region of interest This general methodology is shared by previous work [7], though here we use notation and nomenclature based heavily on the numerical solution of integral equations in antenna theory [24] which has been used extensively used for developing bounds in that area [23]. Numerical examples are calculated for each derived bound and serve to demonstrate salient features
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