Abstract
Scattering theory is the basis of all linear optical and photonic devices, whose spectral response underpins wideranging applications from sensing to energy conversion. Unlike the Shannon theory for communication channels, or the Fano theory for electric circuits, understanding the limits of spectral wave scattering remains a notoriously challenging open problem. We introduce a mathematical scattering representation that inherently embeds fundamental principles of causality and passivity into its elemental degrees of freedom. We use this representation to reveal strong constraints in the mathematical structure of scattered fields, and to develop a general theory of the maximum radiative heat transfer in the near field, resolving a longstanding open question. Our approach can be seamlessly applied to highinterest applications across nanophotonics, and appears extensible to general classical and quantum scattering theory.
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Introduction
Probing and harnessing the frequency dependence of electromagnetic scattering underlies atomic spectroscopy, molecular sensing, information and energy technologies, and more^{1,2,3,4}. A key pillar of electromagnetic scattering theory is the decomposition of scatterers into “resonators,” in which spectral response is determined by lifetimes and coupling coefficients (or suitable generalizations) of resonant modes^{5,6,7,8}. These “physical oscillators” enable complex scenarios to often be welldescribed by a small number of parameters, and they offer highaccuracy descriptive modeling. However, there is typically no limit on the possible number, lifetimes, or couplings of the modes, such that little can be said about their extreme limits. Mathematically, the difficulty in finding extreme limits arises because the set of all possible resonator designs is nonconvex. Hence physical oscillators provide little prescriptive guidance: what lineshapes are physically possible, and what are the ultimate limits of corralling broadband radiation?
In lieu of resonator decompositions, passivity, and causality have long been recognized as key constraints on broadband response in linear physical systems without gain^{9,10,11,12,13,14,15}. Causality is implied by passivity, so that one need not separately invoke it, and the foundations of linear system theory typically start with passivity^{12}. Passivitybased approaches to spectral response have yielded fundamental limits for matching networks in circuit theory^{16,17}, optical attenuation (e.g., in stellar grains^{18}), material susceptibilities^{19,20}, and more^{15}. Yet passivity itself is not a panacea, and electromagnetic scattering theory is a domain where its application has been met with limited success. Special linearamplitude, “opticaltheorem”like power quantities have bounds analogous to those for optical attenuation^{21,22}. But the general scattering properties of arbitrary systems are described by scattering matrices \({\mathbb{S}}\) that map input excitations at any number of “port” (powercarrying “channels” external to the scatterer) to their corresponding outputs, and scattering \({\mathbb{S}}\) matrices have few (if any) practical spectral limitations. Their analytic properties and representation theorems have been extensively studied, from dispersion relations^{13} and Blaschkeproduct representations^{13} to existence theorems for poles, zeros, and their generalizations^{23}, but known representations suffer from the same issue as their coupledmode counterparts: their degrees of freedom reside in nonconvex (and often unbounded) sets. This makes it difficult or impossible to identify optimal response, or upper limits thereof, across the physical design spaces of scientists and engineers.
The potential value of spectralresponse bounds is highlighted by a longstanding question in energy transport: what is the maximum rate at which two bodies can radiatively exchange heat in the near field? Going back many decades, it has been understood that radiative heat exchange in the near field can be substantially larger than its farfield counterpart^{24,25,26}, due to the enormous number of accessible evanescent channels in addition to propagating ones, yet the maximum extent of this enhancement—with ramifications for applications such as thermophotovoltaics^{27,28}, photonic refrigeration^{29}, and heatassisted magnetic recording^{30}—has been far less clear. Previous theoretical bounds^{22,31,32,33,34} have suggested strong materialelectrondensity dependencies, unbounded response for lowloss materials, and ordersofmagnitude gaps from known designs (>750X). The computational complexity of the problem has prohibited the application of largescale inverse design techniques, leaving unresolved whether current designs are suboptimal or the bounds are too loose.
In this article, we show that an alternative scattering matrix, the \({\mathbb{T}}\) matrix^{35}, can be represented by fictitious “mathematical oscillators” that are ideally suited for probing optimal spectral response. We show that passivity, in tandem with the specific interaction characteristics of materials with electromagnetic waves in low and highfrequency limits, leads to \({\mathbb{T}}\)matrix representations in terms of lossless Drude–Lorentz and Drude–Lorentzlike oscillators with matrixvalued (spatially nonlocal) coefficients. Crucially, the only degrees of freedom of these oscillators are their matrixvalued coefficients, which are constrained to a bounded, convex set. Such limitations must imply strong constraints on scattering response, which we use to identify a simple, general theoretical limit to nearfield radiative heat transfer. Our approach offers insights into why planar structures are better than sharptip patterns, why unconventional plasmonic materials should offer the largest enhancements, and yields materialindependent bounds within a small factor (5X) of stateoftheart designs.
Results
Passivity constraints and oscillator representation
In linear, timeinvariant electrodynamics, the \({\mathbb{T}}\) matrix is the linear operator that relates electromagnetic fields incident upon a scatterer to the polarization fields they induce^{35}. For simplicity of notation and exposition, we assume any standard spatial numerical discretization of sufficiently high accuracy; we collate the incident fields E_{inc}(x) into a vector e_{inc} and the polarization fields P(x) into a vector p, so that the frequencydomain (e^{−iωt} time convention) \({\mathbb{T}}\)matrix is defined by
the discrete analog of the convolution equation \({{{{{{{\bf{P}}}}}}}}({{{{{{{\bf{x}}}}}}}},\,\omega )=\int{\mathbb{T}}({{{{{{{\bf{x}}}}}}}},\,{{{{{{{{\bf{x}}}}}}}}}^{{\prime} },\,\omega ){{{{{{{{\bf{E}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}({{{{{{{{\bf{x}}}}}}}}}^{{\prime} },\,\omega )\,{{{{{{{\rm{d}}}}}}}}{{{{{{{{\bf{x}}}}}}}}}^{{\prime} }\). The \({\mathbb{T}}\) matrix can be derived from first principles via integral operators (cf. Supplementary Note 1 or ref. ^{35}), and its time derivative (or product with −iω) can be interpreted as an admittance matrix.
A passive scatterer in vacuum has a causal response function, such that it is analytic in the upperhalf plane and satisfied Kramers–Kronig (KK) relations^{15}. We write the KK relation in terms of \(\omega \mathbb{T}(\omega)\) to account for possible simple poles at zero: by Cauchy’s residue theorem, for ω in the UHP, \(\omega {\mathbb{T}}(\omega )=\frac{1}{i\pi }\int\nolimits_{\infty }^{\infty }\frac{{\omega }^{{\prime} }{\mathbb{T}}({\omega }^{{\prime} })}{{\omega }^{{\prime} }\omega }\,{{{{{{{\rm{d}}}}}}}}{\omega }^{{\prime} }\). (Physically, \({\mathbb{T}}(\omega )\) must decay as 1/ω^{2} at high frequencies, such that \(\omega {\mathbb{T}}(\omega )\) is square integrable.) Taking the Hermitian part of this equation yields \({{{{{{\mathrm{Re}}}}}}}\,\left[\omega {\mathbb{T}}(\omega )\right]=\frac{1}{\pi }\int\nolimits_{\infty }^{\infty }\frac{{\omega }^{{\prime} }{{{{{{\mathrm{Im}}}}}}}\,{\mathbb{T}}({\omega }^{{\prime} })}{{\omega }^{{\prime} }\omega }\,{{{{{{{\rm{d}}}}}}}}{\omega }^{{\prime} }\). Hence we can isolate the antiHermitian part of \({\mathbb{T}}(\omega )\) as its only degrees of freedom:
To further compress to positive frequencies only, we exploit symmetries of \({\mathbb{T}}(\omega )\). The Hermitian matrix \({\mathbb{Z}}={\omega }_{i}{{{{{{\mathrm{Im}}}}}}}\,{\mathbb{T}}({\omega }_{i})\) can be separated into its reciprocal part \({\mathbb{X}}=({\mathbb{Z}}+{{\mathbb{Z}}}^{T})/2\) and its nonreciprocal part \({\mathbb{Y}}=({\mathbb{Z}}{{\mathbb{Z}}}^{T})/2\). Realvalued timedomain fields require that \({\mathbb{T}}(\omega )={{\mathbb{T}}}^{*}(\omega )\), which implies that \({\mathbb{X}}(\omega )={\mathbb{X}}(\omega )\) and \({\mathbb{Y}}(\omega )={\mathbb{Y}}(\omega )\). Then algebraic manipulations of Eq. (2) give
We provide an alternative derivation of the same expression in the Methods section, by recognizing that \(i\omega {\mathbb{T}}\) is a passive admittance matrix, which implies a Herglotz–Nevanlinna representation^{36} that can be reduced to Eq. (3). For any scattering problem there are at least six matrices that satisfy an expression similar to Eq. (3): a scattering matrix, an impedance matrix, and an admittance matrix, each defined either in the volume or on a bounding surface. Yet only one of those six—the volume admittance matrix (essentially, \({\mathbb{T}}(\omega )\))—appears to be useful for wavescattering bounds. While Eq. (3) reduces the degrees of freedom to the antiHermitian part of \({\mathbb{T}}\), additional passivity considerations are needed to meaningfully constrain the possible scattering response.
The next constraints come directly from passivity. Passivity means that polarization fields do no net work. The work done by the incident fields on the polarization currents J is \(\frac{1}{2}{{{{{{\mathrm{Re}}}}}}}\,\int{{{{{{{{\bf{E}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}^{*}\cdot {{{{{{{\bf{J}}}}}}}}=\frac{1 }{2}{{{{{{\mathrm{Im}}}}}}}\,\int{{{{{{{{\bf{E}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}^{*}\cdot \omega {{{{{{{\bf{P}}}}}}}}\). Positivity of this expression implies that the antiHermitian part of \(\omega {\mathbb{T}}(\omega )\) is positive semidefinite, which we write \(\omega\,{{{{{{\mathrm{Im}}}}}}}\,{\mathbb{T}}(\omega )\ge 0\). (This is equivalent to the condition that admittance matrices have a positive semidefinite Hermitian part^{15}.) This means that \({\mathbb{X}}(\omega )+{\mathbb{Y}}(\omega )\ge 0\) for any realvalued ω. Using the symmetry relations for \({\mathbb{X}}(\omega )\) and \({\mathbb{Y}}(\omega )\) around ω = 0, we have the constraints \({\mathbb{X}}(\omega )+{\mathbb{Y}}(\omega )\ge 0\) and \({\mathbb{X}}(\omega ){\mathbb{Y}}(\omega )\ge 0\) at positive frequencies, which further imply \({\mathbb{X}}(\omega )\ge 0\). These constraints are convex (though still unbounded) in \({\mathbb{X}}(\omega )\) and \({\mathbb{Y}}(\omega )\).
The final key element is the identification of sum rules. Sum rules typically come from evaluation of KK relations in the limit ω → ∞ or ω = 0. At infinite frequency, the electrons of a material can be regarded as free, and material susceptibilities must scale as \(\chi (\omega )\to {\omega }_{p}^{2}/{\omega }^{2}\), where \({\omega }_{p}^{2}\) is proportional to the total electron density of the material^{13}. In this limit, the first Born approximation is asymptotically exact, and the polarization field is given by \({{{{{{{\bf{P}}}}}}}}\simeq \chi {{{{{{{{\bf{E}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}\simeq ({\omega }_{p}^{2}/{\omega }^{2}){{{{{{{{\bf{E}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}\) (in units where the freespace permittivity is 1), implying that the \({\mathbb{T}}\) matrix asymptotically approaches \(({\omega }_{p}^{2}/{\omega }^{2}){{\mathbb{I}}}_{V}\), where \({{\mathbb{I}}}_{V}\) is the identity matrix on the scatterer volume V. Inserting this limit into the KK relation derived before Eq. (2) yields the highfrequency sum rule,
This sum rule constrains the total contributions from \({{{{{{\mathrm{Im}}}}}}}\,{\mathbb{T}}(\omega )\) over all frequencies, a spatially resolved scattering generalization of the f sum rule for materialsusceptibility oscillator strengths^{37,38,39}. The nonreciprocal \({\mathbb{Y}}\) matrix makes no contribution to the integral due to its odd symmetry around ω = 0. Similarly, the lowfrequency asymptote is known: we can write \({\mathbb{T}}(\omega \to 0)={{\mathbb{T}}}_{0,V}\), where \({{\mathbb{T}}}_{0,V}\) is a Hermitian positive semidefinite matrix in the static limit. Inserting this expression into the KK relation derived before Eq. (2) yields a lowfrequency sum rule,
For design problems, one considers many possible scatterer domains V, each of which has different matrices on the righthand sides of the sum rules of (Eqs. (4), (5). How, then, can one accommodate many possible designs? Here we can again make the (critical) choice of the Hermitian/antiHermitian split in the KK relation, which, as we prove in Methods, endows the sum rules with a monotonicity property: enlarging V can only increase (in a positive semidefinite sense) the righthand sides of Eqs. (4), (5). Hence, for a designable domain D containing all possible scatterer subdomains, we can convert the equalities of Eqs. (4), (5) for specific volumes V into inequalities over the designable domain D.
We can unify the above properties to create a framework for fundamental limits. The \({\mathbb{T}}(\omega )\) matrix can always be written in the form of Eq. (3), while the realsymmetric matrix \({\mathbb{X}}(\omega )\) and the skewsymmetric matrix \({\mathbb{Y}}(\omega )\) are strongly constrained. We renormalize \({\mathbb{X}}\to (\pi /2){\mathbb{X}}\) and \({\mathbb{Y}}\to (\pi /2){\mathbb{Y}}\) to simplify the oscillator representation. Together, for the \({\mathbb{T}}\) matrix of any designed scatterer within a designable region D, we have:
The collective representation of Eq. (6) is the foundational result of our paper: the \({\mathbb{T}}\) matrix of any linear scattering body must be decomposable into a set of lossless oscillators, with matrixvalued coefficients satisfying definiteness conditions and constrained in total strength. The only degrees of freedom in the scattering process are the matrices \({\mathbb{X}}({\omega }_{i})\) and \({\mathbb{Y}}({\omega }_{i})\), both of which have strong constraints on the bandwidth over which they can be nonzero. The \({\mathbb{T}}(\omega )\) matrix is linear in these matrix degrees of freedom and the constraints are bounded convex sets. Hence this representation encodes the constraints of passivity and sum rules for electromagnetic scatterers in a mathematical structure that is ideally suited for optimization and fundamental limits.
For a first demonstration of the mathematical structure implied by this representation, we consider broadband scattering from an elliptical dielectric cylinder. To clarify the origin of the oscillators, we use a material with \(\chi={\omega }_{p}^{2}/({\omega }_{0}^{2}{\omega }^{2}ig\omega )\), with ω_{p} = 20, ω_{0} = 10, g = 0.01ω_{p}, which is nearly dispersionless with χ = 4 for ω between 0 and 1 (all frequencies in unit of 2πc/a) and consistent with the necessary highfrequency asymptotic response. The scattered electric field at various points within the scatterer, computed by fullwave simulations (cf. Supplementary Note 6), is shown in Fig. 1b, but is hard to interpret due to its seemingly random undulations. Advances in quasinormalmode (QNM) techniques suggest that one could accurately reproduce these fields with a modest number of QNMs^{8}, but that modeling capability does not imply an understanding of the extreme limits of what is possible. How many resonances can be excited? With what amplitudes, phases, and overlaps with powercarrying channels?
By contrast, consider the lineshapes of the Hermitian and antiHermitian parts of the \({\mathbb{T}}\) matrix (computed on a discretization of more than 37,000 spatial degrees of freedom), as depicted in Fig. 1c for the same three spatial locations and their cross terms. The lineshapes of the \({\mathbb{T}}\)matrix elements closely mimic the DrudeLorentzlike behavior of electronic transitions, but they arise not from real material oscillators, but from complex wavescattering behavior itself. The first three traces of Fig. 1c clearly show positive imaginary parts of varying widths, and real parts that transition from minima to maxima between the peaks of the imaginary parts, then transitioning back to minima where the imaginary parts peak. Hence the peaks tend to coincide (with the realpart peak slightly preceding the imaginarypart peak), and the characteristic lineshapes might be described as minimatomaximatotransition for the real parts and Lorentzianlike for the imaginary parts. The second set of three traces in Fig. 1c do not have exactly this pattern, because they have complexvalued residues that mix the real and imaginary parts. But their underlying “oscillatorlike" structure is still visible: one still sees peaks in one part nearly coinciding with (but slightly preceding) peaks in the counterpart, as well as Lorentzianlike lineshapes in one part being paired with minimatomaximatotransition lineshapes in the other. By contrast, no such structure arises in the scattered fields of Fig. 1b, because they simply do not have a representation resembling Eq. (6).
Collectively, the lineshape widths of the \({\mathbb{T}}\)matrix elements are nonzero thanks to the underlying resonant physics, but every frequency can and should (for our purposes) be interpreted as having its own, losslessoscillator amplitude, given by \(\omega \,{{{{{{\mathrm{Im}}}}}}}\,{\mathbb{T}}(\omega )\). The diagonal components have imaginary parts that must be positive. The offdiagonal components need not have positive imaginary parts, but they are constrained by the positivedefiniteness requirements of the entire matrix, as verified in Fig. 1d, which shows the positivity of the eigenvalues of \({{{{{{\mathrm{Im}}}}}}}\,{\mathbb{T}}(\omega )\). The final key component for meaningful constraints from such a representation is the sum rules, and their domain monotonicity property. Figure 1e shows the integrated response for three scattering bodies within the elliptical designable domain, showing both their convergence to the appropriate sumrule matrix constant as the integral is taken to infinity (the numerical integral converges to <1.7% error, as measured by the matrix Frobenius norm, using a 2000point GaussLegendre quadrature for frequencies from 0 to 40(2πc/a)), as well as the satisfaction of domain monotonicity between the sumrule matrices for the two subdomains of the elliptical domain. As a whole, these combined elements offer an ideal representation for identifying fundamental limits to spectral control.
Ultimate limits to NFRHT
Next, we apply our formulation to the question of maximal NFRHT. NFRHT, as depicted in Fig. 2a, poses prohibitive computational challenges—spatially and temporally incoherent, broadband thermal sources, exciting rapidly decaying near fields over large macroscopic areas—which have limited previous design efforts primarily to highsymmetry structures such as planar bodies^{40,41,42}. Numerous approaches have identified particular constraints with corresponding theoretical bounds^{22,31,32,33,34}, but as we show in Fig. 2b, there are ordersofmagnitude differences between the best structures and the best bounds^{22,34}. We label the bounds by their distinguishing attributes: in ref. ^{22} (“analyticity bound”), complexanalyticity played a central role, while in ref. ^{34} (“channel bound”), a decomposition into powercarrying channels was the starting point. Recently, it was discovered that a set of unconventional plasmonic materials offer significant (10X) improvements over the previous best planar structures^{43}, but otherwise, the field has been at an impasse, without a meaningful approach to either improve the best designs or tighten the bounds.
The \({\mathbb{T}}\) matrix formulation resolves this impasse. The heat transfer coefficient (HTC) between two bodies is the net flux rate (per area and per degree K) of electromagnetic energy passing between bodies at temperatures T and T + ΔT, as measured by the integral of power flux \((1/2){{{{\mathrm{Re}}}}}\,({{{{{\bf{E}}}}}}\times {{{{{{\bf{H}}}}}}}^{*}\cdot \hat{{{{{{\bf{n}}}}}}})\) through a separating plane with normal vector \(\hat{{{{{{{{\bf{n}}}}}}}}}\). The incoherent sources in body i with temperature T_{i} and susceptibility χ_{i}(ω), by the fluctuationdissipation theorem^{40}, are given by \(\langle {J}_{j}({{{{{\bf{x}}}}}},\omega )\,{J}_{k}^{*}({{{{{{\bf{x}}}}}}}^{{\prime} },\omega )\rangle=(4{\varepsilon }_{0}\omega /\pi ){{\Theta }}(\omega,{T}_{i})\,{{{{\mathrm{Im}}}}}\,[{\chi }_{i}(\omega )]{\delta }_{jk}\delta ({{{{{\bf{x}}}}}}{{{{{{\bf{x}}}}}}}^{{\prime} })\) at frequency ω, where \({{\Theta }}(\omega,{T}_{i})=\hslash \omega /\left({e}^{\hslash \omega /{k}_{{{{{{{{\rm{B}}}}}}}}}{T}_{i}}1\right)\) is the Planck spectrum, and k_{B} is the Boltzmann constant. There are a variety of mathematical transformations that we make to this problem to make it more amenable to optimization, detailed in Methods, such as using reciprocity to move the sources out of the hotter body and onto the dividing surface, exploiting spatial symmetries of the bounding domains (two halfspaces, allowing for any patterning within), as well as a nearfield generalization of the “optical theorem”^{44}. The key novelty, however, is our use of Eq. (6): once we have transformed the problem to an appropriate function of the twobody \({\mathbb{T}}\) matrix, we insert the representation theorem of \({\mathbb{T}}\) as a sum of positivesemidefinite matrix coefficients with Drude–Lorentz lineshapes. NFRHT at moderate or even high temperatures is dominated by lowfrequency response, so we only impose the lowfrequency sum rule. For a designable domain D of twohalf spaces, \({{\mathbb{T}}}_{0,D}=\alpha {{\mathbb{I}}}_{D}\), where α is a scalar function of the material susceptibility that is bounded above by 2. Once we insert the \({\mathbb{T}}\)matrix representation into the NFRHT expression, the resulting optimization problem over the infinite set of matrix oscillator coefficients has an analytical upper bound. Straightforward algebraic manipulations (cf. Methods) lead to an ultimate limit to nearfield radiative HTC given by
where d is the minimum separation between the bodies, T the temperature of the cooler body, and \(\beta=0.11(\alpha {k}_{B}^{2}/\hslash )=3.8\times 1{0}^{5}{{{{{{{{\rm{Wnm}}}}}}}}}^{2}/{{{{{{{{\rm{m}}}}}}}}}^{2}/{{{{{{{{\rm{K}}}}}}}}}^{2}\), a numerical constant. This limit cannot be surpassed by any geometric patterning, nor can exotic optical properties of any material alter its value.
Figure 2b compares our theoretical limit with the current stateoftheart, as well as the best known bounds. Whereas the gap between the optimal planar structures and the best previous bounds was at least 750X (and diverging to ∞ for some materials), the expression of Eq. (7) is only 5X larger than the best design. This bound has no material dependence, which resolves the problematic trend that if one orders the materials by their planar performance, as in Fig. 2b, the previous bounds tended to predict worse maximal performance from left to right. The resolution of this discrepancy is our use of the lowfrequency sum rule, which encodes a constraint on the local density of states seen by thermal emitters that depends only on their gap separation, independent of material. The \({\mathbb{T}}\)matrix approach predicts an optimal NFRHT frequency of \({\omega }_{\max }=2.57\frac{{k}_{{{{{{{{\rm{B}}}}}}}}}T}{\hslash }\), determined by the overlap of the Planck function with the Drude–Lorentz lineshape. The predictions are matched almost exactly by computationally optimized planar Drude metals or 2D heterostructures, as shown in Fig. 2c, d. For 300 K temperature, the spectra shown in Fig. 2c peak at almost exactly the optimal oscillator frequency, and the match persists across all relevant temperatures, as shown in Fig. 2d.
Although it seemed plausible (even likely) that nanostructuring may lead to enhanced NFRHT through fieldconcentration (lightningrod) effects, our sum rule explains why this is not the case: sharp tips can enhance the fields very close to a sharp tip, but not at the source location itself. The local density of states is proportional to the latter, and hence is not enhanced by lightningrod effects. To illustrate why sharptipbased (or related) structures are inferior, we design a numerical experiment. In NFRHT, after using reciprocity, the incident field arises from point sources along a separating plane between the bodies. For a single given dipole, the relevant lowfrequency sum rule constant is \({{{{{{{{\bf{e}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}^{T}{{\mathbb{T}}}_{0}{{{{{{{{\bf{e}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}={{{{{{{{\bf{e}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}^{T}{{{{{{{\bf{p}}}}}}}}\), i.e., the overlap between the (static) incident field and the (static) induced polarization. This is equivalent to the scattered field at the point source.
Figure 3a, b compares schematic depictions of sharptip versus planararea structures, while Fig. 3c shows finiteelement calculations for twodimensional analogs, with dipole sources of both possible polarizations between conducting wedges of arbitrary inner angles β, with the sources a distance d from either tip. The gray lines show the scattered fields near the tip (at distances 0.1d), which for the transverse polarization increase at smaller angles, i.e., sharper tips. This is the typical lightningrod effect. Yet these amplified fields play no role in determining the total level of broadband energy transfer; the static constant controlling the sum rule is proportional to the scattered field back at the source, shown in red. This quantity increases with the wedge inner angle, a result that must be true by our domain monotonicity theorem. Hence planar bodies (β = π) must have the largest possible frequencyintegrated response. The only remaining question is whether the frequency response can be tailored for maximum overlap with the Planck spectrum, but that question was answered above, affirmatively, by optimal material dispersion relations.
The closeness of the arbitrarystructure bound of Eq. (7) to the best planar structures arises despite quite different mathematical routes to these results. The translational symmetry of planar bodies implies conserved wavevectors and thus a set of evanescent planewave channels that are independent, with Landauerlike transmissivities^{42}. Such an approach cannot describe patterned structures. Instead, Eq. (7) culminates after using (generalized) reciprocity to move the sources from the hot body to the dividing surface, the sum rule to encapsulate the maximum densities of states seen by those sources, and the \({\mathbb{T}}\)matrix representation to constrain the possible scattering lineshapes. The striking similarity of the two results suggests that even when confronted by spatial and temporal incoherence, rapidly decaying fields, and large areas, the oscillator representation compactly captures the key physics of maximal response in the near field.
Discussion
In this article, we have introduced a framework for broadband electromagnetic scattering. The example of Fig. 1 showcases the joint consequences of passivity and sum rules on the structure of the electromagnetic \({\mathbb{T}}(\omega )\) matrix. We propose a recipe for identifying fundamental limits: rewrite any objective of interest in terms of the \({\mathbb{T}}(\omega )\) matrix, and then use the representation of Eq. (6) as the constraints. Our application of this framework to NFRHT offers clear guidance for the fundamental limits of radiative heat transfer and the physical mechanisms underlying them. The generality of our \({\mathbb{T}}\) matrix representation offers tantalizing prospects for wideranging applications across nanophotonics. Metasurfaces^{45,46}, for example, offer a compact form factor for optics. A central question is the extent to which metasurfaces can control incoming waves^{47,48,49}, across varying frequency and angular bandwidths, for applications from lenses to virtual and augmented reality. Similarly, techniques for imaging through opaque media have flourished with modern spatial light modulators^{50}, with a key open question being ultimate limits to spectral control. In photovoltaics and photodetection, the quest for everthinner devices must ultimately contend with fundamental limits, and similarly across almost every application of nanophotonics. There has long been a need to quantify ultimate limits to spectral control; our approach offers a theory to do so.
Our approach also dovetails seamlessly with a recent flurry of activity in understanding the limits controlling spatial degrees of freedom in nanophotonic systems^{51,52,53,54,55,56}. Transforming the typical Maxwell differential equations into a set of local conservation laws in space, for real and reactive power flows, leads to a mathematical form of the design problem that is amenable to systematic approaches to computational bounds. For a single frequency (or a small number of them^{57}), such conservation laws have shown powerful capabilities for identifying fundamental limits to spatial control. In these approaches, the degrees of freedom of the system are typically encoded not in the electric and magnetic fields, but rather in the electric and magnetic polarization currents that they induce. Those polarization fields are exactly those that are determined by the \({\mathbb{T}}\) matrix, which means that our spectral expansion of the \({\mathbb{T}}\) matrix should be seamlessly compatible with the spatial conservation laws proposed in ref. ^{52,53}. Together, the two approaches may enable a complete understanding of the spatiospectral limits of electromagnetic systems.
One might wonder why we have utilized the \({\mathbb{T}}(\omega )\) matrix, when the vast majority of photonics theory uses the scattering matrix \({\mathbb{S}}(\omega )\)? There are two reasons. First, in many scattering systems, incoming and outgoing waves are spatially distributed (e.g., spherical waves), requiring exquisite care with \({\mathbb{S}}\)matrix causality conditions, leading to (for example) phase shifts in the KK relations^{13}. It becomes unclear which degrees of freedom (if any) are necessary, sufficient, and have convex passivity constraints. The second issue is that there is not, as far as we know, a useful \({\mathbb{S}}\)matrix sum rule of a positive semidefinite quantity. Without such a sum rule, all response is unbounded. As discussed above, scatterervolume \({\mathbb{T}}\) matrices appear to be the unique scattering/impedance/admittance matrix where KK relations, passivity, and sum rules can all be combined into a bounded, convex set of constraints.
More broadly, the insight at the foundation of our framework, about the mathematical properties of scattering \({\mathbb{T}}\) matrices, can be directly applied to any classical wave equation. These techniques should be readily extensible to linear scattering problems in acoustics, elasticity, fluid dynamics, and beyond. The mathematical structure of the wave equation is similar in each case, and the resulting \({\mathbb{T}}\) matrices should therefore have similar representations. An interesting twist may arise in acoustic scattering theory, where materials with higherthanvacuum speeds of sound lead to “noncausal” scattering processes^{58} that have prevented the development of classical sum rules, and would appear to prohibit a corresponding \({\mathbb{T}}\) matrix representation. Yet the \({\mathbb{T}}\) matrix itself may offer a new route to complexanalytic response functions in exactly such scenarios. The reason higher sound speeds lead to “noncausal” response is that the scattered field appears at a location within the scatterer earlier than the incident wave itself. Hence, locally, the process appears noncausal. Yet the nonlocal nature of the \({\mathbb{T}}\) matrix may be precisely what is needed to resolve this paradox. A \({\mathbb{T}}\) matrix isolates the response at any point x to the contributions from the wave incident at each point \({{{{{{{{\bf{x}}}}}}}}}^{{\prime} }\) in the scatterer; each of which, individually, must be causal. Hence, not only should the \({\mathbb{T}}\) matrix be extensible to such scenarios; it may further resolve impediments that had previously stymied even simple sum rules in these fields. (Relatedly, wave scattering with any nontrivial/nonvacuum background has historically stymied sum rules, and this is another avenue of exploration with the \({\mathbb{T}}\) matrix.)
Finally, we speculate that the approach described here may even be extensible to quantum scattering. In the frequency domain, the key difference between quantum and classical scattering is the analytic structure of the governing equations. In classical wave equations, second derivatives in space are proportional to second derivatives in time, which lead to poles in the lower half of the complexfrequency plane and analyticity in the upper half. In quantum scattering, second derivatives in space are proportional to first derivatives in time, which leads to bounds states for negative real energies and branch cuts on the positive real axis. Our standard semicircular contours likely need to be replaced by “keyhole” contours^{13}, with the open question of whether there are meaningful sum rules that can be derived (perhaps dependent on boundstate properties, as in Levinson’s theorem^{59,60} for spherically symmetry potentials). If such sum rules could be derived, it is likely that an infiniteoscillator description could be used to identify fundamental limits for quantum scattering as well.
Methods
Domain monotonicity
In this section, we derive “domain montonicity” theorems for the matrices on the righthand sides of the sum rules of Eqs. (4), (5). Domain monotonicity is trivial for the highfrequency sum rule, as the righthand side is directly proportional to the identity matrix on V. Consider a domain D that contains V. How can we compare the two identity matrices? We can embed the identity matrix on V in a larger matrix on D, with zero elements for any spatial degrees of freedom in D and not in V. Hence, by direct comparison, we will have
proving that the highfrequency sum rule obeys domain monotonicity, implying that it can be converted to an inequality over any designable domain of interest.
Domain monotonicity for the lowfrequency (static) sum rule is less obvious. Here, we generalize the arguments of ref.^{22} to prove domain monotonicity. We need to prove that quantities of the form \({{{{{{{{\bf{x}}}}}}}}}^{{{{\dagger}}} }{{\mathbb{T}}}_{0,V}{{{{{{{\bf{x}}}}}}}}\) increase, for all x ≠ 0, when the domain V increases (i.e., contains all points of its original domain, and a nonzero volume of points outside of its original domain), for a positivesemidefinite static susceptibility. (Gyrotropic materials, with a nonreciprocal pole at zero, are materials that do not have such susceptibilities^{61}.) We can interpret the multiplication of \({\mathbb{T}}\) with x as the polarization field induced by an “incident field” x, and then multiplication on the left by x takes the overlap of that incident field with the polarization that it induces. Hence we will label our arbitrary vectors as e_{inc} instead of x, for clarity in the mathematical relations to follow, though we impose no constraints on the “incident field” and indeed allow it to be an arbitrary vector. In computing the response to such a vector, however, we can use a few important physical consequences of electromagnetism. In electrostatics, the fields (and \({\mathbb{T}}\) matrix) can be chosen to be realvalued, so that we can consider the objective as \({{{{{{{{\bf{x}}}}}}}}}^{T}{\mathbb{T}}{{{{{{{\bf{x}}}}}}}}\), without any conjugation.
We are interested in the quantity \(F={{{{{{{{\bf{e}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}^{T}{\mathbb{T}}{{{{{{{{\bf{e}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}={{{{{{{{\bf{e}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}^{T}{{{{{{{\bf{p}}}}}}}}\), and how it changes when the domain changes. We will consider only continuous, increasing changes in susceptibility: Δχ(x)≥0 everywhere. Hence a variation in F can be written
The polarization field is the solution of the volume (Lippmann–Schwinger) integral equation:
where \({{\mathbb{G}}}_{0}\) is the background (vacuum) Green’s function operator, ξ = − χ^{−1}, and e_{inc} is the incident field. The variation in p can be found by taking the variation of Eq. (10), which is: \(\left({{\mathbb{G}}}_{0}+\xi \right)\delta {{{{{{{\bf{p}}}}}}}}+(\delta \xi ){{{{{{{\bf{p}}}}}}}}=0\). Solving for δp:
Inserting this variation into the objective gives
Finally, from the equation ξ = − χ^{−1}, we have δξ = χ^{−1}(δχ)χ^{−1}, so that
which is nonnegative for any positive semidefinite δχ. Hence we have shown that
for any increases in the domain size or shape; since this is true for any vector e_{inc}, then variations in the electrostatic \({\mathbb{T}}\) matrix must themselves be monotonic. This means that given a scatterer Ω_{1} of any size and shape whose static \({\mathbb{T}}\) matrix is \({{\mathbb{T}}}^{(1)}(\omega=0)\), any other scatterer Ω_{2} whose volume encloses that of Ω_{1} must have a \({{\mathbb{T}}}^{(2)}(\omega=0)\) no smaller than \({{\mathbb{T}}}^{(1)}(\omega=0)\), i.e.:
when the scatterer domain Ω_{2} entirely encloses the scatter domain Ω_{1}.
Derivation of the NFRHT bound
To investigate radiative heat transfer from object 1 (bottom) to object 2 (top), we first break down the problem to power integrations at every frequency. The power flowing in the positive z direction across the middle separating plane (perpendicular to z) between the two objects is:
where the superscripts denote the current sources in the bottom object, whose amplitudes are dictated by the fluctuationdissipation theorem:
where \(Z(\omega,T)=\frac{4{\varepsilon }_{{{{{{{{\rm{0}}}}}}}}}\omega }{\pi }{{{{{{{\rm{Im}}}}}}}}{\chi }_{1}(\omega ){{\Theta }}(\omega,T)\), the susceptibility of the lower body is \({\chi }_{1}(\omega )=\frac{{\varepsilon }_{1}(\omega )}{{\varepsilon }_{{{{{{{{\rm{0}}}}}}}}}}1\), and Θ(ω, T) is the Planck distribution, \({{\Theta }}(\omega,T)=\hslash \omega /({e}^{\frac{\hslash \omega }{{k}_{{{{{{{{\rm{B}}}}}}}}}T}}1)\). The subscripts in r_{v} indicate the position vector lies in the volume of the emitter 1. Then the field correlations in Eq. (16) can be expressed in terms of the Green’s functions \({G}^{EJ}({{{{{{{\boldsymbol{r}}}}}}}},\,{{{{{{{{\boldsymbol{r}}}}}}}}}_{{{{{{{{\boldsymbol{v}}}}}}}}}^{{\prime} })\) and \({G}^{HJ}({{{{{{{\boldsymbol{r}}}}}}}},{{{{{{{{\boldsymbol{r}}}}}}}}}_{{{{{{{{\boldsymbol{v}}}}}}}}}^{{\prime} })\) applied to the thermal source correlations in Eq. (17).
Our bound will not distinguish between the x and y directions (which are symmetric in the bounding domain, even though of course they are not for many allowable patterns), in which case the upper bounds on either of the two terms in power integration in Eq. (16) are identical: \({{{{{{{\rm{Max}}}}}}}} [ {{{{{{{\rm{Re}}}}}}}}\int{{{{{{{\rm{d}}}}}}}}S{({E}_{x}^{{{{{{{{\boldsymbol{ J}}}}}}}}}({{{{{{{\boldsymbol{r}}}}}}}}))}{*}{H}_{y}^{{{{{{{{\boldsymbol{ J}}}}}}}}}({{{{{{{\boldsymbol{r}}}}}}}}) ]={{{{{{{\rm{Max}}}}}}}} [{{{{{{{\rm{Re}}}}}}}}\int{{{{{{{\rm{d}}}}}}}}S{({E}_{y}^{{{{{{{{\boldsymbol{J}}}}}}}}}({{{{{{{\boldsymbol{r}}}}}}}}))}{*}{H}_{x}^{{{{{{{{\boldsymbol{J}}}}}}}}}({{{{{{{\boldsymbol{r}}}}}}}}) ]\). Hence the maximum flux S(ω, T) equals the maximum of the function
We use reciprocity to transfer the flux evaluation of Eq. (18) on the surface S from sources in V to a field evaluation in V from sources on S. The background Green’s functions are reciprocal, i.e., \({G}_{ik}^{EJ}({{{{{\boldsymbol{r}}}}}},{{{{{\boldsymbol{r}}}}}}_{{{{{{\boldsymbol{v}}}}}}}^{\prime})={G}_{ki}^{EJ}({{{{{\boldsymbol{r}}}}}}^{\prime}_{{{{{{\boldsymbol{v}}}}}}},{{{{{\boldsymbol{r}}}}}})\), and \({G}_{ik}^{HJ}({{{{{\boldsymbol{r}}}}}},{{{{{\boldsymbol{r}}}}}}_{{{{{{\boldsymbol{v}}}}}}}^{\prime })={G}_{ki}^{EM}({{{{{\boldsymbol{r}}}}} }^{\prime} _{{{{{{\boldsymbol{v}}}}}}},{{{{{\boldsymbol{r}}}}}})\), so we can equate the fields at r produced by sources at r_{v} with fields at r_{v} produced by sources at r. In light of the correlations for currents sources inside the volume, Eq. (17), we can define the correlations for reciprocal current sources on the middle flux plane as
The amplitude ω is chosen so that \({\left({E}_{{{{{{{{\rm{inc}}}}}}}}}^{{J}_{x}}({{{{{{{{\boldsymbol{r}}}}}}}}}_{{{{{{{{\boldsymbol{v}}}}}}}}})\right)}{*}{E}_{{{{{{{{\rm{inc}}}}}}}}}^{{M}_{y}}({{{{{{{{\boldsymbol{r}}}}}}}}}_{{{{{{{{\boldsymbol{v}}}}}}}}})\) is independent of frequency, which will be important later. Simple insertion of the Green’s functions into Eq. (18) and the usage of reciprocity and Eq. (19) leads to a volumefield expression for F^{FHT}:
where V_{S} is exclusively the source volume. Equation (20) represents the total flux from an infinite plane of sources between the infinite bodies. An upper bound on this flux is given by the upper bound on the flux generated by a single set of point sources at a given position on the separating plane, multiplied by the (infinite) area of the plane. This allows us to easily switch to the quantity of interest in largearea NFRHT: the perarea radiative heat transfer, which is bounded above by the maximum flux from a single set of sources at a single position on the separating plane. This also resolves a second possible difficulty: how to represent the \({\mathbb{T}}\) matrix for infinite, extended structures? For point sources in the near field, there is no issue: the fields decay sufficiently quickly that the response is guaranteed to be wellbehaved. (Intuitively, one can imagine substituting large but finitesized structures at this stage, and later taking the limit as size goes to infinity. The rapid field decay ensures that the subsequent integrals converge, even in the infinitesize limit.)
We switch to vector notation now, using the notation of lowercase letters without the subscript v to represent field vectors on the domain of both objects. For example, the volume integral over the lower body in Eq. (20) becomes \({\left({{{{{{{{\boldsymbol{{e}}}}}}}_{v}}}}^{{J}_{x}}\right)}^{{{{\dagger}}} }{\mathbb{O}}{{{{{{{{\boldsymbol{{e}}}}}}}_{v}}}}^{{M}_{y}}\), where \({\mathbb{O}}\) has ones on its diagonal in the lower (source) volume and zeros everywhere else. We can write this integral out in terms of the \({\mathbb{T}}\) matrix:
Notice both \({\mathbb{T}}\) matrix and e_{inc} vectors are defined on the domain of both the top and bottom bodies. In Eq. (21) we defined the function \({\mathbb{E}}={{{{{{{\rm{Re}}}}}}}}\left({{{{{{{{\boldsymbol{e}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}^{{M}_{y}}{\left({{{{{{{{\boldsymbol{e}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}^{{J}_{x}}\right)}^{{{{\dagger}}} }\right)\) which is a rank2 matrix and can be decomposed into one positive eigenvalue term and one negative eigenvalue term:
with eigenvector q_{1,2} and eigenvalues λ_{1,2} given by
One can now see that our choice of source amplitudes in Eq. (19) leads to frequencyindependent eigenvalues of \({\mathbb{E}}\).
To bound the expression of Eq. (21), we will relax it in a few ways. (Interestingly, intensive numerical optimizations using manifoldoptimization techniques^{62,63} directly on Eq. (21) lead to the same upper limits that we derive below, suggesting that these “relaxations” are minimal and do not loosen the analysis given the constraints that we use, such as sum rules.) First, the \({\mathbb{E}}\) matrix defined by the two renormalized incident fields has one positive and one negative eigenvalue, per Eq. (22). Physically, we can interpret the negative sign of the second eigenvalue via the power expression of Eq. (21) containing \({\mathbb{E}}\), as the difference in powers absorbed for the two renormalized incident fields. This is of course bounded above by the absorption of only the first incident field, dropping the subtracted term, leaving only the contribution of the single positive eigenvalue of \({\mathbb{E}}\). Thus we have:
Next, we note that \({\mathbb{O}}\) indicates absorption only in the lower body; of course this quantity is bounded above by the total absorption in both bodies. This is represented mathematically as the constraint that \({\mathbb{O}}\le {\mathbb{I}}\), which implies:
Finally, the absorption in both bodies is less than the net extinction of the two bodies (their farfield scattered powers are positive, and essentially zero in the nearfield case, so that this relaxation is negligible). We can use a generalized “optical theorem” constraint to bound this quadratic absorptionlike quantity with a linear extinctionlike quantity. The idea is that absorption must be smaller than extinction: P_{abs}≤P_{ext}. Absorption is given in terms of \({\mathbb{T}}\)matrix by \({P}_{{{{{{{{\rm{abs}}}}}}}}}=\frac{\omega }{2}{{{{{{{\rm{Im}}}}}}}}({{{{{{{{\bf{e}}}}}}}}}^{{{{\dagger}}} }{{{{{{{\bf{p}}}}}}}})=\frac{\omega }{2{\varepsilon }_{{{{{{{{\rm{0}}}}}}}}}}\frac{{{{{{{{\rm{Im}}}}}}}}\chi }{ \chi \,{ }^{2}}{{{{{{{{\bf{e}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}^{{{{\dagger}}} }{{\mathbb{T}}}^{{{{\dagger}}} }{\mathbb{T}}{{{{{{{{\bf{e}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}\). Similarly extinction is given by \({P}_{{{{{{{{\rm{ext}}}}}}}}}=\frac{\omega }{2}{{{{{{{\rm{Im}}}}}}}}({{{{{{{{\bf{e}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}^{{{{\dagger}}} }{{{{{{{\bf{p}}}}}}}})=\frac{\omega }{2}{{{{{{{{\bf{e}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}^{{{{\dagger}}} }({{{{{{\mathrm{Im}}}}}}}\,{\mathbb{T}}){{{{{{{{\bf{e}}}}}}}}}_{{{{{{{{\rm{inc}}}}}}}}}\). Thus the “optical theorem” condition implies that for any \({\mathbb{T}}\) matrix,
Hence we can write
without introducing much relaxation. We can now rewrite Eq. (20) as
Surprisingly, the various transformations to this point have removed all explicit dependencies on material susceptibility χ_{1,2}(ω), with the only implicit dependence embedded in \({{{{{{{\rm{Im}}}}}}}}{\mathbb{T}}\). We will now focus on the upper bound for HTC, and the upper bound for RHT can be found by taking similar steps. To switch from the RHT to HTC bound computation, we just need to take the temperature derivative of the last expression to get
In our oscillator representation, we know that \(\omega {{{{{{\mathrm{Im}}}}}}}\,{\mathbb{T}}(\omega )\) is exactly the realsymmetric positivesemidefinite matrix \({\mathbb{X}}(\omega )\), which must satisfy the lowfrequency sum rule \(\int\nolimits_{0}^{\infty }{\mathbb{X}}({\omega }_{i})/{\omega }_{i}^{2}\le \alpha {{\mathbb{I}}}_{D}\). (The nonreciprocal part of \(\omega {{{{{{\mathrm{Im}}}}}}}\,{\mathbb{T}}(\omega )\) cannot contribute, as the NFRHT objective is symmetric around ω = 0, so that it can be written as the linear combination of positivefrequency contributions and their negativefrequency counterparts. The positive and negativefrequency contributions cancel for the nonreciprocal part due to its antisymmetry in frequency.) We renormalize \({\mathbb{X}}\) to simplify the sum rule: \({\mathbb{X}}({\omega }_{i})\to \alpha (\pi /2){\omega }_{i}^{2}{\mathbb{X}}({\omega }_{i})\), so that \(\int\nolimits_{0}^{\infty }{\mathbb{X}}({\omega }_{i})\le {{\mathbb{I}}}_{D}\). In terms of \({\mathbb{X}}({\omega }_{i})\), the total frequencyintegral HTC is
The optimization of Eq. (32), subject to the passivity constraint (\({\mathbb{X}}(\omega )\ge 0\)) and the sumrule constraint (\(\int\nolimits_{0}^{\infty }{\mathbb{X}}(\omega )\,{{{{{{{\rm{d}}}}}}}}\omega \le {{\mathbb{I}}}_{D}\)) is actually simple, thanks to the structure of the objective and representation. We form a basis \({\mathbb{Q}}\) whose first column is q_{1}, with all other columns orthogonal to q_{1}. If we write at every frequency \({\mathbb{X}}(\omega )={\mathbb{Q}}{{\mathbb{X}}}^{{\prime} }(\omega ){{\mathbb{Q}}}^{{{{\dagger}}} }\), then \({{{{{{\mathrm{Tr}}}}}}}\,\left[{q}_{1}{q}_{1}^{{{{\dagger}}} }{\mathbb{X}}(\omega )\right]={q}_{1}^{{{{\dagger}}} }{\mathbb{Q}}{{\mathbb{X}}}^{{\prime} }(\omega ){{\mathbb{Q}}}^{{{{\dagger}}} }{q}_{1}={\left({{\mathbb{X}}}^{{\prime} }(\omega )\right)}_{11}\). Hence only the (1, 1) element of \({{\mathbb{X}}}^{{\prime} }(\omega )\) contributes to the objective (due to the rankone nature of the excitation). The positive semidefinite property as well as the sum rule for \({\mathbb{X}}(\omega )\) are equivalent for \({{\mathbb{X}}}^{{\prime} }(\omega )\) (as the transformation was unitary). Hence we can rewrite the HTC bound as:
subject to the constraints \({{\mathbb{X}}}_{11}^{{\prime} }(\omega )\ge 0\) and \(\int\nolimits_{0}^{\infty }{{\mathbb{X}}}_{11}^{{\prime} }(\omega )\,{{{{{{{\rm{d}}}}}}}}\omega \le 1\). The maximization of an inner product subject to a “probability simplex” constraint^{64} has a simple solution: concentrate all of the response into the single degree of freedom where the objective vector is maximized. In particular, in this case, the optimal \({{\mathbb{X}}}_{11}^{{\prime} }\) is a delta function with unit amplitude at the frequency where \(\omega \frac{\partial {{\Theta }}(\omega,T)}{\partial T}\) is maximized. A simple calculation shows that this occurs for
which is exactly the nearfield Wien frequency that we found from HTC optimization for planar, unpatterned geometries^{43}. In terms of the dimensionless variable x_{opt} = 2.57, the HTC bound is:
Inserting the numerical prefactors, we arrive at the final bound:
where β = 3.8 × 10^{5}Wnm^{2}/m^{2}/K^{2}. For T = 300 K and d = 10 nm, HTC ≤ 1.1 × 10^{6} W/m^{2}/K, which is 5X the optimal planar performance. Hence this theoretical framework offers a close prediction to the best known designs, it predicts the optimal resonance frequency where the oscillatorstrength should be concentrated, and it explains why previous materialdependent predictions were incorrect.
Data availability
The datasets generated in this study are available at https://github.com/PhotonDesign/ScatteringOscillatorsResults.
Code availability
The simulation code used in this study is available at https://github.com/PhotonDesign/ScatteringOscillatorsResults.
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Acknowledgements
The authors thank Wenjin Xue and Hanwen Zhang for providing the fast integral solver for computing the \({\mathbb{T}}(\omega )\) matrix of the elliptical scatterer. This work was supported by: Air Force Office of Scientific Research Grant No. FA95502210393 (L.Z., O.D.M., general \({\mathbb{T}}\)matrix theory), Army Research Office Grant No. W911NF1910279 (L.Z., O.D.M., nearfield heat transfer analysis), Air Force Office of Scientific Research Grant No. FA95502210204 (F.M.), and Office of Naval Research Grant No. N000142212486 (F.M.).
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O.D.M. and F.M. conceived the initial idea. L.Z. and O.D.M. developed the formalism and example demonstrations. All authors discussed the technical aspects and features as well as possible extensions. All authors wrote the manuscript.
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Zhang, L., Monticone, F. & Miller, O.D. All electromagnetic scattering bodies are matrixvalued oscillators. Nat Commun 14, 7724 (2023). https://doi.org/10.1038/s41467023432212
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DOI: https://doi.org/10.1038/s41467023432212
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