## Abstract

We derive fundamental per-channel bounds on angle-integrated absorption and thermal radiation for arbitrarily structured bodies - for any given material susceptibility and bounding region - that simultaneously encode both the per-volume limit on polarization set by passivity and geometric constraints on radiative efficiencies set by finite object sizes through the scattering T operator. We then analyze these bounds in two practical settings, comparing against prior limits as well as near optimal structures discovered through topology optimization. Principally, we show that the bounds properly capture the physically observed transition from the volume scaling of absorptivity seen in deeply subwavelength objects (nanoparticle radius or thin film thickness) to the area scaling of absorptivity seen in ray optics (blackbody limits).

Original language | English (US) |
---|---|

Article number | 257401 |

Journal | Physical review letters |

Volume | 123 |

Issue number | 25 |

DOIs | |

State | Published - Dec 20 2019 |

## All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

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*Physical review letters*,

*123*(25), [257401]. https://doi.org/10.1103/PhysRevLett.123.257401

**T Operator Bounds on Angle-Integrated Absorption and Thermal Radiation for Arbitrary Objects**. In: Physical review letters. 2019 ; Vol. 123, No. 25.

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*Physical review letters*, vol. 123, no. 25, 257401. https://doi.org/10.1103/PhysRevLett.123.257401

**T Operator Bounds on Angle-Integrated Absorption and Thermal Radiation for Arbitrary Objects.** / Molesky, Sean; Jin, Weiliang; Venkataram, Prashanth S.; Rodriguez, Alejandro W.

Research output: Contribution to journal › Article › peer-review

TY - JOUR

T1 - T Operator Bounds on Angle-Integrated Absorption and Thermal Radiation for Arbitrary Objects

AU - Molesky, Sean

AU - Jin, Weiliang

AU - Venkataram, Prashanth S.

AU - Rodriguez, Alejandro W.

N1 - Funding Information: 0031-9007 1079-7114 American Physical Society 10.1103/PhysRevLett.123.257401 LETTERS Condensed Matter: Electronic Properties, etc. T Operator Bounds on Angle-Integrated Absorption and Thermal Radiation for Arbitrary Objects https://orcid.org/0000-0003-3575-5166 Molesky Sean 1 https://orcid.org/0000-0003-1946-1029 Jin Weiliang 2 Venkataram Prashanth S. 1 Rodriguez Alejandro W. 1 ,* Department of Electrical Engineering, 1 Princeton University , Princeton, New Jersey 08544, USA Department of Electrical Engineering, 2 Standford University , Stanford, California 94305, USA * Corresponding author. arod@princeton.edu 20 December 2019 20 December 2019 123 25 257401 11 July 2019 © 2019 American Physical Society 2019 American Physical Society We derive fundamental per-channel bounds on angle-integrated absorption and thermal radiation for arbitrarily structured bodies—for any given material susceptibility and bounding region—that simultaneously encode both the per-volume limit on polarization set by passivity and geometric constraints on radiative efficiencies set by finite object sizes through the scattering T operator. We then analyze these bounds in two practical settings, comparing against prior limits as well as near optimal structures discovered through topology optimization. Principally, we show that the bounds properly capture the physically observed transition from the volume scaling of absorptivity seen in deeply subwavelength objects (nanoparticle radius or thin film thickness) to the area scaling of absorptivity seen in ray optics (blackbody limits). National Science Foundation 10.13039/100000001 DMR-1454836 DMR 1420541 DGE 1148900 Cornell Center for Materials Research 10.13039/100008585 DMR1719875 Defense Advanced Research Projects Agency 10.13039/100000185 HR00111820046 Natural Sciences and Engineering Research Council of Canada 10.13039/501100000038 PDF-502958-2017 Motivated by the increasing control of light offered by micro- and nanoscale structuring [1,2] , impetus to find bounds analogous to the blackbody limit for geometries that violate the assumptions of ray optics (nanoparticles [3] , thin films [4] , photonic crystals [5,6] , etc.) has steadily grown over the past few decades. It is now well established that the absorption (radiative thermal emission) cross sections of a compact object can be much greater than its geometric area [7–12] (“super-Planckian” emission), and that deeply subwavelength films can achieve near unity absorptivity via surface texturing [13,14] . Limits applicable to all length scales and materials could both provide insight into these representative phenomena and guide efforts in related application areas such as integrated and metaoptics [15–17] , photovoltaics [18–21] , and photon sources [22–24] . Development of bounds for arbitrary objects has primarily followed two overarching strategies: modal decompositions based on quasinormal, Fourier, and/or multipole expansions [25–33] , relating absorption cross section to the number of excitable optical modes (channels), or material bounds, utilizing energy [34,35] and/or spectral sum rules [36–42] to constrain achievable polarization response. Separately, each of these approaches present challenges for photonic design. Modal decompositions incorporate the specific size and shape characteristics of a body through expansion coefficients, and hence, inherently, require some enumeration and characterization of the participating modes to determine the range of values these coefficients can take [26,39,40] . Although fundamental considerations (transparency, energy, size, etc.) can and have been used in this regard [28,33,41,43] , such cutoffs have yet to tightly bound potential coefficient values for arbitrary compact geometries, particularly when applied to metallic nanoparticles and antennas [27,35,44] . Conversely, material bounds set by intrinsic dissipation naturally reproduce the volumetric scaling of absorptivity characteristic of deeply subwavelength objects (and are highly accurate for the special case of weak polarizability in this regime [35] ). However, because such approaches intrinsically suppose an optimally large response field existing at all points within an arbitrary object for any incident field, the same volumetric scaling persists for all length scales. Consequently, material bounds rapidly become too loose beyond quasistatic settings, yielding unphysical divergences with both increasing object size and material response. In this Letter, we derive bounds on thermal radiation and absorption that combine these two approaches, linking the impact of material response with the influence of an object’s geometry through the scattering T operator. This leads to a per-channel limit on integrated absorption capturing both material and radiative losses through the singular values of the imaginary part of the vacuum Green function. The result is applicable to objects of any size, exhibiting a smooth transition in absorptivity from the volume scaling achievable in the quasistatic (deeply subwavelength) regime to the area scaling limit of macroscopic ray optics. Further, the bounds always asymptotically approach the ray optics limit when all characteristic lengths are large and diverge sublogarithmically (rather than linearly) with material quality for objects of finite extent, significantly reducing cross-section limits for typical optical media even when all characteristic lengths are small. Throughout, we compare the present results to prior bounds as well as structures discovered using topology optimization, realizing a variety of examples (metallic and dielectric) that nearly achieve the predicted limits. Derivation.— From the relations of scattering theory, both the power scattered from an incident field ( | E inc ⟩ ) and the thermal radiation emitted at a temperature T can be expressed in terms of the scattering T operator of an object and the vacuum Green function G vac [45] as P sct ( ω ) = k o 2 Z ⟨ E inc | T † Im [ G vac ] T | E inc ⟩ = k o 2 Z ⟨ E inc | Im [ T ] - T † Im [ V - 1 † ] T | E inc ⟩ (1) and H ( ω , T ) = Π ( ω , T ) Φ ( ω ) , Φ = 2 π Tr { Im [ G vac ] ( Im [ T ] - T † Im [ G vac ] T ) } . (2) Here, ω is the angular frequency, k o = 2 π / λ is the wave number, Z is the impedance of free space, Π ( ω ) = ℏ ω / [ exp ( β ℏ ω ) - 1 ] with ( β = 1 / k B T ) is the Planck energy of a harmonic oscillator, Tr [ ⋯ ] denotes the trace, Im [ T ] = ( T - T * ) / 2 i , and, by Kirchhoff’s law of thermal radiation, Φ is the object’s angle-integrated absorption [46] . (A synopsis of scattering formalism, along with a derivation of Eq. (2) , is provided in Supplemental Material [47] .) For a passive object, scattered power must be positive for any incident field. As such, Eq. (1) simultaneously dictates that all singular values of the T operator must be smaller than the material figure of merit ζ , ‖ T ‖ ≤ ζ = | χ ( ω ) | 2 Im [ χ ( ω ) ] , (3) which was similarly derived in Ref. [35] for polarization fields, and that Im [ T ] is positive definite. As Im [ G vac ] is real-symmetric positive definite, it can be expressed via a singular value decomposition as Im [ G vac ] = ∑ i ρ i | q i ⟩ ⟨ q i | , (4) where, as supported by our later analysis, each ρ i (eigenvalue) can be equated to the outgoing radiative flux of the i th mode—the i th radiative efficacy of the domain. The set { ρ i } plays an analogous role to the coupling coefficients { g i j } used by Miller in setting limits on far-field optical communication [41] . Consider Eq. (2) using this expansion, Φ = ( 2 / π ) ∑ i ρ i Im [ ⟨ q i | T | q i ⟩ ] - ρ i 2 | ⟨ q i | T | q i ⟩ | 2 - ( 2 / π ) ∑ { ( i , j ) | i ≠ j } ρ i ρ j | ⟨ q i | T | q j ⟩ | 2 . Now, take T opt to be a general operator described by the properties ( T opt ) T = T opt (reciprocity), ‖ T opt ‖ ≤ ζ (passivity), and Im [ T opt ] positive definite (passivity), ignoring all other physical constraints that any true T operator must satisfy. In this context, two characteristics of any maxima of Φ are clear. First, as ( ∀ i , j ) ρ i ρ j | ⟨ q i | T opt | q j ⟩ | 2 ≥ 0 , the appearance of any cross terms ( ⟨ q i | T opt | q j ⟩ ) will always decrease Φ . Therefore, to maximize Φ , a general operator T opt must be diagonalized in the basis of Im [ G vac ] , Eq. (4) . Second, the complex phase of ⟨ q i | T opt | q i ⟩ only influences the first (positive) piece of the sum, and so the value of Φ peaks when ( ∀ i ) atan ( Im [ ⟨ q i | T | q i ⟩ ] / Re [ ⟨ q i | T | q i ⟩ ] ) = π / 2 . Together, these two considerations show that achievable values of Φ are bounded by taking T opt to be diagonalized by Eq. (4) with purely imaginary eigenvalues: T opt = ∑ i i τ i | q i ⟩ ⟨ q i | with ( ∀ i ) τ i ∈ [ 0 , ζ ] . As such, Φ opt = 2 π ∑ i τ i ρ i - ( τ i ρ i ) 2 . (5) and maximizing the contribution of each τ i yields Φ opt = 2 π ∑ i { 1 / 4 ( ζ ρ i ≥ 1 / 2 ) ζ ρ i - ( ζ ρ i ) 2 else . (6) That is, based on the criterion ζ ρ i ≥ 1 / 2 , each channel in Eq. (6) produces either the Landauer limited contribution of 1 / 4 [48] or the material limited ζ ρ i - ( ζ ρ i ) 2 . Interpretation.— In terms of the T operator, the total power extracted from any incident field | E inc ⟩ by an object is P ext = k o ⟨ E inc | Im [ T ] | E inc ⟩ / ( 2 Z ) . Comparing with Eqs. (1) and (2) , Φ thus amounts to the difference of the extracted ( Im [ T ] ) and scattered ( T † Im [ G vac ] T ) power for free-space states. The separation of these two forms persists throughout the derivation of the bounds, representing the linear and quadratic terms of Eq. (5) . Φ opt results from their connected physics. In real space, Tr { Im [ G vac ] } = ∑ i ρ i is the integral of the local density of free-space states over the domain of the object. Following Eq. (6) , the total power that can be extracted by an object, the first term of Eq. (2) , is hence bounded by its ability to interact with radiative modes, Tr { Im [ T ] Im [ G vac ] } = ∑ i τ i ρ i , which is maximized (independently) under complete saturation of material response, ( ∀ i ) τ i = ζ . Relatedly, this form is also the result of applying the per-volume (shape independent) optical response limit of Ref. [35] to integrated absorption, and is similar to the light trapping bound of Ref. [34] . Because of these connections with prior work, Φ qs ( ω ) = ∑ i ζ ρ i = ζ ∫ V d r Im [ G vac ( r , r ) ] (7) serves as a useful comparison for Φ opt , and is subsequently referred to as the quasistatic bound . This name is chosen as Eq. (7) follows from the assumption that the interaction of the object with any incident field is identically material limited, which can occur in quasistatic settings. This does not mean that Φ qs is valid only under the quasistatic approximation. Like Φ opt , Φ qs is a mathematical bound derived from Maxwell’s equations, although Φ opt ≤ Φ qs for any selection of parameters. In Eqs. (5) and (6) this extracted power contribution is suppressed by scattering (radiative) losses, which are captured in the quadratic term in Eq. (2) as the coupling of the polarization currents generated within an object back to free-space modes: originating through the operator T † Im [ G vac ] T , each τ i represents the ability of the object to convert a given field into a current, and each radiative efficacy ρ i the conversion of a current into outgoing radiative flux. Equivalently, the presence of strong polarization currents, necessary for strong per-volume absorption, leads to radiative losses, and these losses limit possible absorption. If ζ ρ i > 1 / 2 , mirroring the observed dependencies of absorption ( ∝ V ) and scattering ( ∝ V 2 ) seen in highly subwavelength metallic antennas [26,30] , the growth of radiative losses with increasing τ i can potentially surpass the growth of the extracted power, inducing saturation. As both processes are rooted in the same conversion between radiative fields and polarization currents, this critical coupling occurs at the compelling value of τ i ρ i = 1 / 2 [61,62] , the probability of a maximally entropic Bernoulli process, resulting in the Landauer limit value of Φ opt = 1 / 4 . Analysis.— The practical usefulness of Eq. (6) stems from its favorable mathematical properties. Namely, Eq. (6) monotonically increases with ζ or any ρ i , and, as proved in Supplemental Material [47] , each ρ i increases if the object grows (domain monotonicity). This allows us to freely decouple any true object from an imagined encompassing region of space (bounding domain). A mismatch between the domain of the object and the domain of Im [ G vac ] must technically reduce ‖ T ‖ below ζ , but without any modification Eq. (6) remains an upper bound on Φ . That is, the result of Eq. (6) for any particular bounding domain is applicable to any object that can be enclosed (as well as any subdomain). The procedure for calculating Φ opt is straightforward for any bounding geometry (e.g., wires, disks, spheres, extended films, stars, disconnected patches, etc.). Precisely, the set of singular values { ρ i } of the domain can always be computed by forming a real-space matrix representation of Im [ G vac ] , Im [ G vac ] ( r ) = k o 3 4 π r [ ( sin ( r ) + cos ( r ) r - sin ( r ) r 2 ) I ¯ - ( sin ( r ) + 3 cos ( r ) r - 3 sin ( r ) r 2 ) r ^ ⊗ r ^ ] , (8) with every r multiplied by a hidden k o , and then performing a singular value decomposition of the result [49,63] . Here, to facilitate further investigation, we will focus on the high symmetry case of a ball where semianalytic evaluation is manageable (expressions for films, as well as minor additional details, are given in Supplemental Material [47] ). Nevertheless, we stress that determining Φ opt for domains lacking symmetry does not raise any meaningful computationally difficulties. For this geometry two types of singular values arise: ρ ℓ ( 1 ) = π R 2 4 ( ℓ + 1 2 ℓ + 1 [ J ℓ - 1 / 2 2 ( R ) - J ℓ + 1 / 2 ( R ) J ℓ - 3 / 2 ( R ) ] + ℓ 2 ℓ + 1 [ J ℓ + 3 / 2 2 ( R ) - J ℓ + 1 / 2 ( R ) J ℓ + 5 / 2 ( R ) ] ) , ρ ℓ ( 2 ) = π R 2 4 [ J ℓ + 1 / 2 2 ( R ) - J ℓ - 1 / 2 ( R ) J ℓ + 3 / 2 ( R ) ] , (9) where J ℓ ( - ) is the ℓ th Bessel function of the first kind with an additional factor of 2 π included in its argument, each ℓ (spherical harmonic) index has a multiplicity of ( 2 ℓ + 1 ) , and R is the radius of the ball normalized by the wavelength. Using standard properties of Bessel functions, it can be shown that for values of R ≫ ℓ , each of these singular values tends to the asymptote 2 π 2 R , and that for any combination of arguments ρ ℓ ( 1 ) < π ( ℓ + 1 ) ( π R ) 2 ℓ + 1 / [ 2 Γ 2 ( ℓ + 3 / 2 ) ] + 2 π ℓ ( π R ) 2 ℓ + 5 / [ ( 2 ℓ + 5 ) ( 2 ℓ + 3 ) Γ 2 ( ℓ + 5 / 2 ) ] and ρ ℓ ( 2 ) < 2 π ( π R ) 2 ℓ + 3 / [ ( 2 ℓ + 3 ) Γ 2 ( ℓ + 3 / 2 ) ] (asymptotically approached for small values of R ). These forms reveal two prescient general features. First, in the limit of small domains ( R ≪ 1 ), with “small” being determined by the value of ζ , only the first singular value of the first type contributes, and this triply degenerate (dipole) mode is responsible for the initial volume scaling necessitated by the physical meaning of the bounds. Second, the radial growth of the singular values shows that the saturation condition (impact of radiative losses) plays a major role in limiting radiative thermal emission and integrated absorption in wavelength scale volumes. [For ζ = 10 6 , Fig. 1(a) , radiative losses lead to order-of-magnitude deviations of Φ opt from Φ qs beyond R ≈ 0.003 λ .] As visually confirmed by Fig. 1(a) , as the domain grows, an increasing number of channels (multipoles) saturate, causing “steps” to appear in Φ opt , and these steps lead to successively larger deviation with Φ qs that ultimately regularize the initial volumetric scaling. Results for films, Fig. 1(b) , are qualitatively similar. However, since the domain is infinite, the steps associated with saturation are now blended into a continuum, and the large characteristic size limit is approached from below rather than above. From a practical perspective, the fact that Φ opt can achieve near ideal absorptivity for very small film thickness and moderate values of ζ is quite remarkable, a finding that is tacitly supported by a number of recent studies in 2D materials and metasurfaces [64–67] . Crucially, in either case, for any value of ζ , Φ opt asymptotes to a geometric perfect absorber (the blackbody limit). 1 10.1103/PhysRevLett.123.257401.f1 FIG. 1. Bounds on angle-integrated absorption and thermal radiation for arbitrarily structured bodies enclosed within compact and extended domains. Absorptivity ( Φ normalized by area A ) bounds Φ opt (orange lines) and Φ qs (purple lines), for a range of ζ = | χ | 2 / Im [ χ ] at a fixed wavelength λ . These quantities are shown as a function of the wavelength normalized radius R of an enclosing sphere (a) and thickness h of a semi-infinite film (b). Schematics of each setting are included as insets. Even for small characteristic lengths ( { R , h } ≤ 0.1 λ ), Φ opt is orders of magnitude smaller than Φ qs . The asymptotic behavior of the singular values also reveals general characteristics of the dependence of Φ opt on the material figure of merit ζ . Applying Stirling’s approximation to the bounding expressions given above, for ( ℓ ≫ e π R ) we have ρ ℓ ( 2 ) ≈ ( e π R / ℓ ) 2 ℓ + 1 / 4 and ρ ℓ ( 1 ) ≈ ( e π R / ℓ ) 2 ℓ + 3 / 2 , to arbitrary accuracy as ℓ becomes large. Fix R , and suppose that ζ = ρ k ( 2 ) ( ρ k ( 1 ) is analogous). Using the fact that e π R / ( k + ℓ ) < e π R / k , the remaining (unsaturated) linear contribution of Φ opt is then bounded by 9 ( e π R ) 3 / { 4 [ k 2 - ( e π R ) 2 ] } . Hence, as ζ saturates increasingly higher spherical harmonics, the contribution of the remaining unsaturated harmonics becomes increasingly small compared to the contribution of the newly saturated harmonic, ≈ ( 2 k + 1 ) / 4 . But, saturation of the ℓ th singular value (in the large ℓ limit) requires ln ( ζ 2 ) ≥ ( 2 ℓ + 1 ) ln ( ℓ e π R ) , (10) which has a sublogarithmic dependence between ℓ and ζ . Because of domain monotonicity, the above material scaling result for a ball is applicable to all compact (finite sized) objects. This bound on material quality scaling is well matched to the features of the Φ opt curves in Fig. 1(a) . Once the radius has surpassed ≈ λ , geometric increases in ζ ( × 10 2 ) produce relatively minute changes in the bounds. The same behavior also appears for smaller radii at larger values of ζ , but this range is not of great practical relevance since materials with ζ surpassing ≈ 10 8 are quite rare. For instance, in the optical to infrared, ω ∈ ( 0.5 – 15 ) μ m , ζ ( ω ) has a peak value of approximately 1.7 × 10 3 for gold, 2.4 × 10 3 for tungsten, 2.2 × 10 4 for silicon carbide, 6.8 × 10 9 for silicon, 3.3 × 10 7 for gallium arsenide, and 5.9 × 10 7 for gallium phosphide [68] . Optimizations.— Case evidence for the tightness of Eq. (6) is presented in Fig. 2 . Using a gradient topology optimization algorithm [2,50] , see Supplemental Material for details [47] , structures nearly achieving Φ opt have been discovered for two widely different domain sizes ( R = 0.05 λ and R = 0.5 λ ) and a variety of metallic and dielectric susceptibilities. In Fig. 2 , these media are grouped by imaginary susceptibility, corresponding to four different values of Im [ χ ] , { 0.5 , 1 , 2 , 4 } , with the remaining variation in ζ occurring due to Re [ χ ] . Explicit values of Re [ χ ] are given for circled points, providing a sense of the range considered. As was previously remarked by Miller et al. [35] , Φ qs is attained for a plane wave polarized along the axis of an ellipsoidal metallic nanoparticle, given a properly chosen aspect ratio. For small values of ζ this ratio is near unity and resonant metallic structures ( Re [ χ ] ≈ - 3 ) matching both bounds are easily discovered. As ζ moves to moderate values, the aspect ratio required for an ellipsoidal particle to match Φ qs becomes increasingly extreme. Because of our chosen spherical boundary, discovered structures begin to deviate considerably from Φ qs , but continue to come within a factor of 2 of Φ opt up to ζ = 10 3 . Past this point, numerical issues impede our present algorithms and it remains to be seen how much of the roughly order-of-magnitude headroom allowed by Φ opt is accessible. 2 10.1103/PhysRevLett.123.257401.f2 FIG. 2. Comparison of bounds with geometries discovered by inverse design. Absorptivity ( Φ over area A ) of structures discovered using gradient topology optimization for a variety of metallic (a) and dielectric (b) materials characterized by the material figure of merit ζ = | χ | 2 / Im [ χ ] . (See text for more information.) For comparison, the bounds Φ opt [Eq. (6) ] and Φ qs [Eq. (7) ] are also depicted. In (a), all structures are bound by a ball of radius R = 0.05 λ . For (b), the confining domain is a ball of R = 0.5 λ . The inset provides a visualization of the structure (exterior and planar cut) for the rightmost green square. The observation that optimized structures come within factors of unity of Φ opt provides case evidence of the tightness of Eq. (6) . Results for the larger domain, Fig. 2(b) , show similarly good agreement. An example structure is depicted in the right-hand inset (full view and planar cut), corresponding to the rightmost green square in the plot. Comparing with the assumptions made in deriving Eq. (6) , the T operator for this structure ( χ = 20 + 4 i , Φ = 0.60 Φ opt ) is indeed found to be nearly diagonal in the basis of Im [ G vac ] and has almost completely imaginary eigenvalues (for supporting data, see Supplemental Material [47] ). Remarks.— There are a few points that should be considered when using Eq. (6) , or comparing to prior literature. First, Φ opt is a bound on thermal emission and integrated absorption for a given domain and ζ factor. By choosing different geometries and material parameters, Eq. (6) can be applied to any desired context, but the confining volume is an essential feature. Second, there is no universal guarantee of tightness. Beyond the demonstrated agreement of the bounds with known quasistatic and ray optics asymptotics, the only a priori guarantee is domain monotonicity; there are likely volumes and material parameters where the value of Φ opt will be larger than the true Φ of any practical structure. Next, while we have only considered single wavelengths, there is no reason the bounds cannot be applied to finite frequency ranges. The derivation of Φ opt presented above does not incorporate any spectral sum rules (derived from causality), such as the fact that T opt should obey Kramers-Kronig dispersion relations, but for resonant absorption or thermal emission simply multiplying the bound by the width of the resonance should not produce a substantially looser bound than Φ opt at the peak wavelength. (As an expedient, taking Φ opt to be the peak value of a Lorentzian function of width Δ ω = ω Im χ / | χ | is likely a fair approximation.) Finally, as suggested in the introduction, Φ opt can be interpreted as the extension of prior multipole analysis [25–33] or communication limits [37,38] , to general domains with the crucial addition that an upper bound is set on the number modes which may contribute through the pseudo-rank of the imaginary part of the vacuum Green function ( Im [ G vac ] ) and the material figure of merit ( ζ ), Eq. (3) . We foresee this rank revealing capability potentially providing a number of benefits for future practical design and optimization. We also note that much of what has been developed in this Letter is applicable not only to generalized electromagnetic scattering (for incident plane waves or dipolar emitters with applications to solar cells, light-emitting diodes, and single-photon emitters) but also to quantum mechanics, acoustics, and other wave physics. This work was supported by the National Science Foundation under Grants No. DMR-1454836, No. DMR 1420541, and No. DGE 1148900, the Cornell Center for Materials Research MRSEC (Grant No. DMR1719875), the Defense Advanced Research Projects Agency (DARPA) under agreement HR00111820046, and the National Science and Engineering Research Council of Canada under PDF-502958-2017. The views, opinions, and findings expressed herein are those of the authors and should not be interpreted as representing the official views or policies of any institution. [1] 1 A. F. Koenderink , A. Alu , and A. Polman , Science 348 , 516 ( 2015 ). SCIEAS 0036-8075 10.1126/science.1261243 [2] 2 S. Molesky , Z. Lin , A. Y. Piggott , W. Jin , J. Vucković , and A. W. Rodriguez , Nat. Photonics 12 , 659 ( 2018 ). NPAHBY 1749-4885 10.1038/s41566-018-0246-9 [3] 3 P. Du and J. S. Yu , Chem. Eng. J. (Lausanne) 327 , 109 ( 2017 ). CMEJAJ 1385-8947 10.1016/j.cej.2017.06.069 [4] 4 J. K. Tong , W.-C. Hsu , Y. Huang , S. V. Boriskina , and G. Chen , Sci. Rep. 5 , 10661 ( 2015 ). SRCEC3 2045-2322 10.1038/srep10661 [5] 5 L. Zhu , A. P. Raman , and S. Fan , Proc. Natl. Acad. Sci. U.S.A. 112 , 12282 ( 2015 ). PNASA6 0027-8424 10.1073/pnas.1509453112 [6] 6 O. Ilic , P. Bermel , G. Chen , J. D. Joannopoulos , I. Celanovic , and M. Soljačić , Nat. Nanotechnol. 11 , 320 ( 2016 ). NNAABX 1748-3387 10.1038/nnano.2015.309 [7] 7 C. F. Bohren and D. R. Huffman , Absorption and Scattering of Light by Small Particles ( John Wiley & Sons , New York, 2008 ). [8] 8 M. I. Tribelsky , Europhys. Lett. 94 , 14004 ( 2011 ). EULEEJ 0295-5075 10.1209/0295-5075/94/14004 [9] 9 Z. Ruan and S. Fan , Appl. Phys. Lett. 98 , 043101 ( 2011 ). APPLAB 0003-6951 10.1063/1.3536475 [10] 10 S.-A. Biehs and P. Ben-Abdallah , Phys. Rev. B 93 , 165405 ( 2016 ). PRBMDO 2469-9950 10.1103/PhysRevB.93.165405 [11] 11 V. Fernández-Hurtado , A. I. Fernández-Domínguez , J. Feist , F. J. García-Vidal , and J. C. Cuevas , Phys. Rev. B 97 , 045408 ( 2018 ). PRBMDO 2469-9950 10.1103/PhysRevB.97.045408 [12] 12 D. Thompson , L. Zhu , R. Mittapally , S. Sadat , Z. Xing , P. McArdle , M. M. Qazilbash , P. Reddy , and E. Meyhofer , Nature (London) 561 , 216 ( 2018 ). NATUAS 0028-0836 10.1038/s41586-018-0480-9 [13] 13 M. A. Kats and F. Capasso , Laser Photonics Rev. 10 , 735 ( 2016 ). LPRAB8 1863-8899 10.1002/lpor.201600098 [14] 14 P. N. Dyachenko , S. Molesky , A. Y. Petrov , M. Störmer , T. Krekeler , S. Lang , M. Ritter , Z. Jacob , and M. Eich , Nat. Commun. 7 , 11809 ( 2016 ). NCAOBW 2041-1723 10.1038/ncomms11809 [15] 15 S. Kruk and Y. Kivshar , ACS Photonics 4 , 2638 ( 2017 ). APCHD5 2330-4022 10.1021/acsphotonics.7b01038 [16] 16 M. Khorasaninejad and F. Capasso , Science 358 , eaam 8100 ( 2017 ). SCIEAS 0036-8075 10.1126/science.aam8100 [17] 17 S. Hampson , W. Rowe , S. D. Christie , and M. Platt , Sens. Actuators B 256 , 1030 ( 2018 ). SABCEB 0925-4005 10.1016/j.snb.2017.10.041 [18] 18 E. Yablonovitch , J. Opt. Soc. Am. 72 , 899 ( 1982 ). JOSAAH 0030-3941 10.1364/JOSA.72.000899 [19] 19 H. A. Atwater and A. Polman , Nat. Mater. 9 , 205 ( 2010 ). NMAACR 1476-1122 10.1038/nmat2629 [20] 20 N. Zhang , C. Han , Y.-J. Xu , J. J. Foley IV , D. Zhang , J. Codrington , S. K. Gray , and Y. Sun , Nat. Photonics 10 , 473 ( 2016 ). NPAHBY 1749-4885 10.1038/nphoton.2016.76 [21] 21 D. Jariwala , A. R. Davoyan , J. Wong , and H. A. Atwater , ACS Photonics 4 , 2962 ( 2017 ). APCHD5 2330-4022 10.1021/acsphotonics.7b01103 [22] 22 J. D. Thompson , T. Tiecke , N. P. de Leon , J. Feist , A. Akimov , M. Gullans , A. S. Zibrov , V. Vuletić , and M. D. Lukin , Science 340 , 1202 ( 2013 ). SCIEAS 0036-8075 10.1126/science.1237125 [23] 23 T. Galfsky , H. Krishnamoorthy , W. Newman , E. Narimanov , Z. Jacob , and V. Menon , Optica 2 , 62 ( 2015 ). OPTIC8 2334-2536 10.1364/OPTICA.2.000062 [24] 24 N. Somaschi , V. Giesz , L. De Santis , J. Loredo , M. P. Almeida , G. Hornecker , S. L. Portalupi , T. Grange , C. Antón , J. Demory , C. Gómez , I. Sagnes , N. D. Lanzillotti-Kimura , A. Lemaítre , A. Auffeves , A. G. White , L. Lanco , and P. Senellart , Nat. Photonics 10 , 340 ( 2016 ). NPAHBY 1749-4885 10.1038/nphoton.2016.23 [25] 25 J. S. McLean , IEEE Trans. Antennas Propag. 44 , 672 ( 1996 ). IETPAK 0018-926X 10.1109/8.496253 [26] 26 R. E. Hamam , A. Karalis , J. D. Joannopoulos , and M. Soljačić , Phys. Rev. A 75 , 053801 ( 2007 ). PLRAAN 1050-2947 10.1103/PhysRevA.75.053801 [27] 27 D.-H. Kwon and D. M. Pozar , IEEE Trans. Antennas Propag. 57 , 3720 ( 2009 ). IETPAK 0018-926X 10.1109/TAP.2009.2025975 [28] 28 Z. Yu , A. Raman , and S. Fan , Proc. Natl. Acad. Sci. U.S.A. 107 , 17491 ( 2010 ). PNASA6 0027-8424 10.1073/pnas.1008296107 [29] 29 Z. Yu , A. Raman , and S. Fan , Opt. Express 18 , A366 ( 2010 ). OPEXFF 1094-4087 10.1364/OE.18.00A366 [30] 30 Z. Ruan and S. Fan , Phys. Rev. Lett. 105 , 013901 ( 2010 ). PRLTAO 0031-9007 10.1103/PhysRevLett.105.013901 [31] 31 J.-P. Hugonin , M. Besbes , and P. Ben-Abdallah , Phys. Rev. B 91 , 180202(R) ( 2015 ). PRBMDO 1098-0121 10.1103/PhysRevB.91.180202 [32] 32 Y. Jia , M. Qiu , H. Wu , Y. Cui , S. Fan , and Z. Ruan , Nano Lett. 15 , 5513 ( 2015 ). NALEFD 1530-6984 10.1021/acs.nanolett.5b02044 [33] 33 Y. Yang , O. D. Miller , T. Christensen , J. D. Joannopoulos , and M. Soljacic , Nano Lett. 17 , 3238 ( 2017 ). NALEFD 1530-6984 10.1021/acs.nanolett.7b00852 [34] 34 D. M. Callahan , J. N. Munday , and H. A. Atwater , Nano Lett. 12 , 214 ( 2012 ). NALEFD 1530-6984 10.1021/nl203351k [35] 35 O. D. Miller , A. G. Polimeridis , M. T. H. Reid , C. W. Hsu , B. G. DeLacy , J. D. Joannopoulos , M. Soljačić , and S. G. Johnson , Opt. Express 24 , 3329 ( 2016 ). OPEXFF 1094-4087 10.1364/OE.24.003329 [36] 36 R. Fuchs and S. Liu , Phys. Rev. B 14 , 5521 ( 1976 ). PLRBAQ 0556-2805 10.1103/PhysRevB.14.5521 [37] 37 O. D. Miller , C. W. Hsu , M. T. H. Reid , W. Qiu , B. G. DeLacy , J. D. Joannopoulos , M. Soljačić , and S. G. Johnson , Phys. Rev. Lett. 112 , 123903 ( 2014 ). PRLTAO 0031-9007 10.1103/PhysRevLett.112.123903 [38] 38 H. Shim , L. Fan , S. G. Johnson , and O. D. Miller , Phys. Rev. X 9 , 011043 ( 2019 ). PRXHAE 2160-3308 10.1103/PhysRevX.9.011043 [39] 39 Z. Ruan and S. Fan , Phys. Rev. A 85 , 043828 ( 2012 ). PLRAAN 1050-2947 10.1103/PhysRevA.85.043828 [40] 40 F. Alpeggiani , N. Parappurath , E. Verhagen , and L. Kuipers , Phys. Rev. X 7 , 021035 ( 2017 ). PRXHAE 2160-3308 10.1103/PhysRevX.7.021035 [41] 41 D. A. B. Miller , Appl. Opt. 39 , 1681 ( 2000 ). APOPAI 0003-6935 10.1364/AO.39.001681 [42] 42 D. A. B. Miller , J. Opt. Soc. Am. B 24 , A1 ( 2007 ). JOBPDE 0740-3224 10.1364/JOSAB.24.0000A1 [43] 43 C. Sohl , M. Gustafsson , and G. Kristensson , J. Phys. D 40 , 7146 ( 2007 ). JPAPBE 0022-3727 10.1088/0022-3727/40/22/042 [44] 44 M. Gustafsson , C. Sohl , and G. Kristensson , Proc. R. Soc. A 463 , 2589 ( 2007 ). PRLAAZ 1364-5021 10.1098/rspa.2007.1893 [45] 45 M. Krüger , G. Bimonte , T. Emig , and M. Kardar , Phys. Rev. B 86 , 115423 ( 2012 ). PRBMDO 1098-0121 10.1103/PhysRevB.86.115423 [46] 46 J.-J. Greffet and M. Nieto-Vesperinas , J. Opt. Soc. Am. A 15 , 2735 ( 1998 ). JOAOD6 0740-3232 10.1364/JOSAA.15.002735 [47] 47 See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.123.257401 , which includes Refs. [2,45,48–60], for a brief review of scattering theory, the inverse design method employed and mathematical support of the film results. [48] 48 S. Molesky , P. S. Venkataram , W. Jin , and A. W. Rodriguez , arXiv:1907.03000 [ Phys. Rev. B (to be published)]. [49] 49 A. G. Polimeridis , M. T. H. Reid , W. Jin , S. G. Johnson , J. K. White , and A. W. Rodriguez , Phys. Rev. B 92 , 134202 ( 2015 ). PRBMDO 1098-0121 10.1103/PhysRevB.92.134202 [50] 50 J. S. Jensen and O. Sigmund , Laser Photonics Rev. 5 , 308 ( 2011 ). LPRAB8 1863-8899 10.1002/lpor.201000014 [51] 51 B. A. Lippmann and J. Schwinger , Phys. Rev. 79 , 469 ( 1950 ). PHRVAO 0031-899X 10.1103/PhysRev.79.469 [52] 52 W. Jin , S. Molesky , Z. Lin , and A. W. Rodriguez , Phys. Rev. B 99 , 041403(R) ( 2019 ). PRBMDO 2469-9950 10.1103/PhysRevB.99.041403 [53] 53 W. Eckhardt , Physica (Amsterdam) 128A , 467 ( 1984 ). PHYADX 0378-4371 10.1016/0378-4371(84)90186-9 [54] 54 L. Tsang , J. A. Kong , and K.-H. Ding , Scattering of Electromagnetic Waves: Theories and Applications ( John Wiley & Sons , New York, 2004 ), Vol. 27 . [55] 55 K. Svanberg , SIAM J. Opt. 12 , 555 ( 2002 ). 10.1137/S1052623499362822 [56] 56 S. G. Johnson , https://nlopt.readthedocs.io/en/latest/ . [57] 57 W. Jin , A. G. Polimeridis , and A. W. Rodriguez , Phys. Rev. B 93 , 121403(R) ( 2016 ). PRBMDO 2469-9950 10.1103/PhysRevB.93.121403 [58] 58 A. Hochman , J. Fernandez Villena , A. G. Polimeridis , L. M. Silveira , J. K. White , and L. Daniel , IEEE Trans. Antennas Propag. 62 , 3150 ( 2014 ). IETPAK 0018-926X 10.1109/TAP.2014.2314734 [59] 59 N. Halko , P.-G. Martinsson , and J. A. Tropp , SIAM Rev. 53 , 217 ( 2011 ). SIREAD 0036-1445 10.1137/090771806 [60] 60 P.-G. Martinsson , V. Rokhlin , and M. Tygert , Appl. Comput. Harmon. Anal. 30 , 47 ( 2011 ). ACOHE9 1063-5203 10.1016/j.acha.2010.02.003 [61] 61 J. Pendry , J. Phys. A 16 , 2161 ( 1983 ). JPHAC5 0305-4470 10.1088/0305-4470/16/10/012 [62] 62 J. Pendry , J. Phys. Condens. Matter 11 , 6621 ( 1999 ). JCOMEL 0953-8984 10.1088/0953-8984/11/35/301 [63] 63 A. G. Polimeridis , F. Vipiana , J. R. Mosig , and D. R. Wilton , IEEE Trans. Antennas Propag. 61 , 3112 ( 2013 ). IETPAK 0018-926X 10.1109/TAP.2013.2246854 [64] 64 S. Thongrattanasiri , F. H. Koppens , and F. J. G. De Abajo , Phys. Rev. Lett. 108 , 047401 ( 2012 ). PRLTAO 0031-9007 10.1103/PhysRevLett.108.047401 [65] 65 G. M. Akselrod , J. Huang , T. B. Hoang , P. T. Bowen , L. Su , D. R. Smith , and M. H. Mikkelsen , Adv. Mater. 27 , 8028 ( 2015 ). ADVMEW 0935-9648 10.1002/adma.201503281 [66] 66 S. Kim , M. S. Jang , V. W. Brar , K. W. Mauser , L. Kim , and H. A. Atwater , Nano Lett. 18 , 971 ( 2018 ). NALEFD 1530-6984 10.1021/acs.nanolett.7b04393 [67] 67 J. Nong , H. Da , Q. Fang , Y. Yu , and X. Yan , J. Phys. D 51 , 375105 ( 2018 ). JPAPBE 0022-3727 10.1088/1361-6463/aad608 [68] 68 E. D. Palik , Handbook of Optical Constants of Solids ( Academic Press , New York, 1998 ), Vol. 3 . Publisher Copyright: © 2019 American Physical Society.

PY - 2019/12/20

Y1 - 2019/12/20

N2 - We derive fundamental per-channel bounds on angle-integrated absorption and thermal radiation for arbitrarily structured bodies - for any given material susceptibility and bounding region - that simultaneously encode both the per-volume limit on polarization set by passivity and geometric constraints on radiative efficiencies set by finite object sizes through the scattering T operator. We then analyze these bounds in two practical settings, comparing against prior limits as well as near optimal structures discovered through topology optimization. Principally, we show that the bounds properly capture the physically observed transition from the volume scaling of absorptivity seen in deeply subwavelength objects (nanoparticle radius or thin film thickness) to the area scaling of absorptivity seen in ray optics (blackbody limits).

AB - We derive fundamental per-channel bounds on angle-integrated absorption and thermal radiation for arbitrarily structured bodies - for any given material susceptibility and bounding region - that simultaneously encode both the per-volume limit on polarization set by passivity and geometric constraints on radiative efficiencies set by finite object sizes through the scattering T operator. We then analyze these bounds in two practical settings, comparing against prior limits as well as near optimal structures discovered through topology optimization. Principally, we show that the bounds properly capture the physically observed transition from the volume scaling of absorptivity seen in deeply subwavelength objects (nanoparticle radius or thin film thickness) to the area scaling of absorptivity seen in ray optics (blackbody limits).

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U2 - 10.1103/PhysRevLett.123.257401

DO - 10.1103/PhysRevLett.123.257401

M3 - Article

C2 - 31922767

AN - SCOPUS:85077239288

VL - 123

JO - Physical Review Letters

JF - Physical Review Letters

SN - 0031-9007

IS - 25

M1 - 257401

ER -