Global $\mathbb{T}$ operator bounds on electromagnetic scattering: Upper bounds on far-field cross sections

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Global $T$ operator bounds on electromagnetic scattering: Upper bounds on far-field cross sections

Sean Molesky, Pengning Chao, Weiliang Jin, and Alejandro W. Rodriguez

Phys. Rev. Research 2, 033172 – Published 30 July 2020

Abstract

We present a method based on the scattering $T$ operator, and conservation of net real and reactive power, to provide physical bounds on any electromagnetic design objective that can be framed as a net radiative emission, scattering or absorption process. Application of this approach to plane-wave scattering from an arbitrarily shaped, compact body of homogeneous electric susceptibility $χ$ is found to predictively quantify and differentiate the relative performance of dielectric and metallic materials across all optical length scales. When the size of a device is restricted to be much smaller than the wavelength (a subwavelength cavity, antenna, nanoparticle, etc.), the maximum cross-section enhancement that may be achieved via material structuring is found to be much weaker than prior predictions: the response of strong metals ( $Re [χ] ≪ 0$ ) exhibits a diluted (homogenized) effective medium scaling $\propto |χ| / Im [χ]$ ; below a threshold size inversely proportional to the index of refraction (consistent with the half-wavelength resonance condition), the maximum cross-section enhancement possible with dielectrics ( $Re [χ] > 0$ ) shows the same material dependence as Rayleigh scattering. In the limit of a bounding volume much larger than the wavelength in all dimensions, achievable scattering interactions asymptote to the geometric area, as predicted by ray optics. For representative metal and dielectric materials, geometries capable of scattering power from an incident plane wave within an order of magnitude (typically a factor of two) of the bound are discovered by inverse design. The basis of the method rests entirely on scattering theory and can thus likely be applied to acoustics, quantum mechanics, and other wave physics.

Received 27 March 2020
Revised 17 June 2020
Accepted 10 July 2020

DOI:https://doi.org/10.1103/PhysRevResearch.2.033172

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)

Electromagnetism Geometrical & wave optics Light propagation, transmission & absorption Metamaterials Mie scattering Photonics Plasmonics Scattering theory

Atomic, Molecular & OpticalGeneral Physics

Authors & Affiliations

Sean Molesky ^1,*, Pengning Chao ^1,*, Weiliang Jin ², and Alejandro W. Rodriguez^1,†

¹Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
²Department of Electrical Engineering, Stanford University, Stanford, California 94305, USA

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

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Vol. 2, Iss. 3 — July - September 2020

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Figure 1
Schematic of investigation. To what extent does the specification of an electric susceptibility ( $χ$ ) and a confining spatial domain for the design of a structured body (optical device), determine the maximum absorbed $P_{abs}$ , scattered $P_{sct}$ , and extinguished (total) $P_{ext} = P_{sct} + P_{abs}$ power that can be extracted from a known incident field?
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Figure 2
Scattering cross-section bounds for a plane wave incident on a compact object. The four panels show different aspects of scattering cross section bounds for any structure of electric susceptibility $χ$ that can be contained within a spherical boundary of $R / λ$ (e.g., cavities, nanoparticle, etc.). Dashed lines result from imposing the conservation of real power, as in Ref. [72]. Full lines result from additionally requiring the conservation of reactive power, as in Ref. [75]. As $R \to 0$ , limit values agree with (38) in all cases, and with (40) so long as $Im [χ] / |Re [χ]| ≳ 10^{- 4}$ . Dots in panels (a) and (b) mark scattering cross sections achieved in actual geometries discovered by inverse design, for $χ = - 10 + i 10^{- 1}$ and $χ = 16 + i 10^{- 6}$ , respectively. For the metal structure in (a), vertically aligned cross-hatched dots result from enforcing binarized permittivity profiles. (Binarization does not alter the dielectric results.) Two sample structures are shown as insets, with the plane wave incident from the more solid side of both designs, left side in (a), right side in (b), and aligned along the left-right symmetry axis. Unbounded cross sections are encountered only for fictitious metals with vanishing material loss, with the bounds exhibiting a weak, sublogarithmic, divergence as $Im [χ] \to 0$ . More practically, cross-section enhancements surpassing $\approx 200$ should not be expected. Descriptions of other major features are given in the accompanying text.
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Figure 3
Absorption cross-section bounds for a planewave incident on a compact object. The four panels provide analogous information as Fig. 2, but for absorbed rather than scattered power. Again, the dashed lines represent bounds determined by insisting only that real power is conserved, the solid lines result from additionally requiring the conservation of reactive power, and dots correspond to actual structures discovered by inverse design. For the example metal structure shown in (a), the planewave is incident from the solid side of the half shell (from the left in the first view and from the back left in the second). For the example dielectric structure, the planewave is incident from the left along the axis of symmetry of the mushroom cap in the full view, and from the back left in the cross cut. Small radii features are found to be in good agreement with the asymptotic predictions of (38) and (40) for $Im χ / |Re χ| ≳ 10^{- 4}$ . When the confining volume is small, $R / λ ≲ 1 / 20$ , the ability of a structured object to absorb radiation is found to be substantially weaker than past predictions [68, 71, 77]. As $R \to \infty$ , the geometric cross-section limit predicted by ray optics is recovered regardless of the material considered. Like scattering, these results indicate that absorption cross section enhancements larger than $\approx 200$ should not be expected.
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Figure 4
Absorption cross-section bounds for a plane wave incident on a periodic film. Absorption cross-section bounds for any periodic structure (e.g., grating, metasurface) of period $Λ$ and electric susceptibility (a) $χ = 13.6 + i 5 10^{- 2}$ (silicon at $0.8 μ m$ ) and (b) $χ = - 87 + i 10$ (gold at $1.5 μ m$ ) that can be contained within a planar boundary of thickness $t / λ$ . Dots correspond to actual absorption cross-section values discovered using inverse design following the algorithm described in Ref. [163]. A number of visualizations of the associated structures, along with a schematic of the absorption process investigated, are included as insets. All discovered designs with $t / λ < 10^{- 2}$ consist of a single layer, uniform in the normal direction. For larger thickness, optimization of three equally thick uniform layers is needed to achieve the reported absorption cross sections (eventually reaching absorption saturation). Like prior figures, solid lines are found by solving (17) under the constraint that total power is conserved, while dashed lines, coinciding with the solid line in (b), are found by only imposing the constraint of real power conservation. As shown in (a), total power conservation bounds for dielectric media depend on the square period supposed. Moving from left to right, the periods used in the inverse design optimizations are $Λ = \{5 λ, 5 λ, 5 λ, 5 λ, 4 λ, 2 λ, 3 λ / 2, 3 λ / 2, λ, λ\}$ . Results for metals ( $Re [χ] < 0$ ) do not depend on the periodicity of the system. Discussion of additional bound features relating solely to imposing the conservation of real can be found in Refs. [68, 72].
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Physical Review Research

Global T operator bounds on electromagnetic scattering: Upper bounds on far-field cross sections

Sean Molesky, Pengning Chao, Weiliang Jin, and Alejandro W. Rodriguez

Phys. Rev. Research 2, 033172 – Published 30 July 2020

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Global $T$ operator bounds on electromagnetic scattering: Upper bounds on far-field cross sections