Owen Miller Yale Applied Physics Fundamental limits to optical response, via convex passivity constraints Owen Miller Yale Applied Physics Collaborators: Dr. Chia-Wei Hsu Prof. Steven Johnson Prof. John Joannopoulos Prof. Marin Soljacic Dr. Homer Reid Prof. Alejandro Rodriguez Brendan DeLacy Prof. Thanos Polimeridis Emma Anquillare Sponsor (12/2016+): Air Force Office of Scientific Research

How efficiently can light be absorbed? Principles Design Technology photonics Ni, Zhang et al. Science 249, 1310 (2015) How efficiently can light be absorbed? How quickly can you transfer energy? What are the strongest material configurations? mechanics X. Zheng et al. Science 344, 1373 (2014) quantum Kaminer et al. Nat. Phys. 11, 261 (2015) The design challenge: Lots of degrees of freedom in geometry, but what structure is best for given device & materials? And what performance & phenomena are possible?

Limits on what is possible: Examples Yablonovitch limit to solar-cell absorption enhancement (broadband, all-angle) Black-body limit on thermal radiation (ray optics, in far field for linear and/or equilibrium surface) Wheeler–Chu bounds on antenna quality factor Q (per V) Wiener / Hashin–Shtrikman / Bergman / Milton bounds on homogenized properties of composites … max 𝜔, shape 𝑓(𝜔) Scattering quantity, 𝑓 0 ∞ 𝑓 𝜔 d𝜔 0 ∞ 𝑓 res 𝜔 d𝜔 𝑄= 𝜔 Δ𝜔 Frequency, ω

Optical-Response Bounds via Convex Constraints Known scattered-power bounds Sum rules, using low- and high-frequency asymptotics Spherical-harmonic decomposition (𝜎∼ 𝜆 2 ) Optical theorem imposes convex constraints Material-dictated, shape-independent bounds Limits to optical response in absorptive systems High-radiative-efficiency plasmonics Radiative heat transfer bounds: near-field analogue of the blackbody limit Scattering-channel bounds, revisited Force/torque equivalents of 𝜎∼ 𝜆 2 And as I will discuss, always informed and driven by experimental considerations

Review: Maxwell & Materials polarization continuum, local, linear materials: 6x6 susceptibility χ(x,t) (breaks down for metals at < 10nm scales ⇒ nonlocal; or very strong fields ⇒ nonlinear) 𝜕 𝜕𝑡 →−𝑖𝜔 frequency domain: passive materials: 𝜔 Im 𝜒(𝑥,𝜔)>0 i.e., polarization currents can dissipate but not supply energy

Optical-Response Bounds via Convex Constraints Known scattered-power bounds Sum rules, using low- and high-frequency asymptotics Spherical-harmonic decomposition (𝜎∼ 𝜆 2 ) Optical theorem imposes convex constraints Material-dictated, shape-independent bounds Limits to optical response in absorptive systems High-radiative-efficiency plasmonics Radiative heat transfer bounds: near-field analogue of the blackbody limit Scattering-channel bounds, revisited Force/torque equivalents of 𝜎∼ 𝜆 2 And as I will discuss, always informed and driven by experimental considerations

EM-wave optical theorem [e.g. JD Jackson, Classical Electrodynamics] 1 2 Re 𝑬× 𝑯 ∗ Power/area = Poynting Einc Absorbed power: 𝑃 abs =− 1 2 Re 𝑆 𝑬× 𝑯 ∗ ⋅ 𝒏 n Scattered power: 𝑃 scat =+ 1 2 Re 𝑆 𝑬 scat × 𝑯 scat ∗ ⋅ 𝒏 Escat = E – Einc Extinction = absorption + scattering: 𝑃 ext = 1 2 Re 𝑆 𝑬 inc 𝑯 inc † 𝒏 ×𝑯 − 𝒏 ×𝑬 = 𝜔 2 Im 𝑠(𝜔) effective surface “currents” forward-scattering amplitude

Known bounds #1: Sum rules over 𝜔, 𝜆 𝑃 ext = 𝜔 2 Im 𝑠(𝜔) 𝑠(𝜔) analytic in upper half of complex-𝜔 plane Re 𝑠 𝜔 0 = 1 𝜋 𝒫 −∞ ∞ Im 𝑠 𝜔 𝜔− 𝜔 0 𝑑𝜔 Sohl, Gustafsson, et al., J. Phys. A 40, 11165 (2007) 𝜎 ext 𝜆 𝑉 d𝜆= 𝜋 2 𝛼 ′ 𝐸,𝜔=0 + 𝛼′ 𝑀,𝜔=0 E. M. Purcell, Astrophys. Jour. 158, 433 (1969) 𝜔 0 =0: 𝛼′ = polarizability / vol. 𝜔 𝑝 = “plasma frequency” 𝜎 ext 𝜆 𝑉 d𝜔= 𝜋 𝜔 𝑝 2 2𝑐 𝜔 0 →∞: (𝜒∼− 𝜔 𝑝 2 / 𝜔 2 as 𝜔→∞) R Gordon JCP 38, 1724 (1963) ZJ Yang et al. Nano Lett. 15, 7633 (2015)

Known bounds #2: spherical-harmonic-based limits 𝜎 ext ≤ 3 𝜆 2 2𝜋 scatterer (E dipole) incident planewave in vacuum Ω 𝜎 ext ≤ 5 𝜆 2 2𝜋 (E quadrupole) scattered power … 𝜎 ext ≤( 𝑁 2 +2𝑁) 𝜆 2 2𝜋 Arbitrary Shapes: (N multipoles) Kwon & Pozar IEEE TAP 57, 3720 (2009) Liberal et al. IEEE TAP 62, 4726 (2014) spherical-wave decomposition + assume no net outflow in a given mode (highly restrictive) = bound on 𝜎 abs spherical-wave decomposition + assume unitarity, i.e. no loss (highly restrictive) = bound on 𝜎 scat , 𝜎 ext Hamam, Soljacic et al. PRA 75, 053801 (2007) Spheres: Ruan and Fan APL 98, 43101 (2011)

Metals: Subwavelength, “plasmonic” resonances O. Benson, Nature 480, 193 (2011) Van Duyne, ARPC 58, 267 (2007) Raether, Surface Plasmons… (1988) Propagating surface modes Localized resonances Possible Applications: Key Tradeoff: Biosensors AN Grigorenko et. al. Nat. Mat. 12, 305 (2013) Re 𝜒<−1:  quasistatic modes (sub-l enhancements)  Im 𝜒 large, dissipative losses Hard Drives W. A. Challener et. al. Nat. Photon. 3, 220 (2009) Superlenses N. Fang et. al. Science 308, 534 (2005) …

Optical-Response Bounds via Convex Constraints Known scattered-power bounds Sum rules, using low- and high-frequency asymptotics Spherical-harmonic decomposition (𝜎∼ 𝜆 2 ) Optical theorem imposes convex constraints Material-dictated, shape-independent bounds Limits to optical response in absorptive systems High-radiative-efficiency plasmonics Radiative heat transfer bounds: near-field analogue of the blackbody limit Scattering-channel bounds, revisited Force/torque equivalents of 𝜎∼ 𝜆 2 And as I will discuss, always informed and driven by experimental considerations

Pabs and Pscat: Positive-Definite Quadratic Forms Maxwell for 𝜓: 𝑀𝜓+𝑖𝜔𝜒𝜓= 𝑱 𝑲 𝜓 inc 𝑉 𝑉 ext 𝑀= 𝑖𝜔 𝛻× −𝛻× −𝑖𝜔 Maxwell for 𝜓 scat : 𝜓 scat =𝜓− 𝜓 inc 𝑀 𝜓 scat +𝑖𝜔𝜒 𝜓 scat =−𝑖𝜔𝜒 𝜓 𝑖𝑛𝑐 𝑬 𝑯 𝑃 abs =− 1 2 Re 𝑆 𝑬× 𝑯 ∗ ⋅ 𝒏 = 1 2 𝑉 𝜓 ∗ 𝜔 Im 𝜒 𝜓 𝑃 scat =+ 1 2 Re 𝑆 𝑬 scat × 𝑯 scat ∗ ⋅ 𝒏 = 1 2 𝑉 ext 𝜓 scat ∗ 𝜔 Im 𝜒 𝜓 scat Passivity: 𝜔 Im 𝜒>0

Optical Theorem: 𝑃 abs + 𝑃 scat ∝ amplitude 𝑃 abs = 1 2 𝜓 ∗ 𝜔 Im 𝜒 𝜓 𝑉 𝑃 scat = 1 2 𝜓 scat ∗ 𝜔 Im 𝜒 𝜓 scat = 𝜔 2 Im 𝜓 inc † 𝜒𝜓− 𝜓 † 𝜒𝜓 𝑉 ext 𝑉 𝑃 abs 𝑃 ext = 𝑃 abs + 𝑃 scat = 𝜔 2 Im 𝜓 inc † 𝜒𝜓 power 𝑃 ext 𝑃 abs unphysical 𝑉 forward-scattering amplitude extinction (abs. + scat.) > absorption induced current 𝑃 ind

the rest is easy: Optimize desired objective subject to absorption ≤ extinction (typically a convex optimization problem, can be solved analytically)

General limits to optical response [Owen Miller et. al, Opt. Exp. 24, 3329 (2016)] By energy conservation, variational calculus and standard optimization theory (optimality conditions)… (∂Pabs/∂Pind = 0, etc.) 𝑃 abs , 𝑃 scat ≤𝛽𝜔(incident energy inside 𝑉) 𝜒 † Im 𝜒 −1 𝜒 For scalar, nonmagnetic 𝜒 + plane wave: 𝜎 abs 𝑉 , 𝜎 scat 𝑉 ≤𝛽 𝜔 𝑐 𝜒 2 Im 𝜒 𝛽= 1 1/4 absorption scattering Similar limit to power radiated by dipole at distance d, i.e. the local density of states (LDOS) 𝑑

How tight are these bounds? (by spectrum of Neumann-Poincare operator) One can prove that metallic (Re 𝜒<−1) structures can reach the absorption and scattering limits

Quasistatic resonance theory + sum rules + …algebra… “Warmup” Optimization: Silver ellipsoids Arbitrary-Shape Adjoint Optimization (≈1000 DOFs) + + 2000 𝜆 𝜎 tot /𝑉 (nm-1) 200 𝜆 2–20x improvement A miraculous coincidence? Quasistatic resonance theory + sum rules + …algebra… = analytical bound Expt. verification >6x 𝜎 ext (𝜔) 𝑉 ≲ 2 3 𝜔 𝑐 𝜒(𝜔) 2 Im 𝜒(𝜔) independent of shape! E. Anquillaire, Owen Miller et al., Opt. Exp. 24, 10806 (2016) O. D. Miller et al. PRL 112, 123903 (2014)

Metamaterials: Constrained by Bounds for Intrinsic Material [OD Miller et al. PRL 112, 157402 (2014)] Optimizations of Hyperbolic metamaterial (HMM) vs. thin films, in the near field showed: if design HMM structure with large near-field response, there is a thin film, of similar size, that has nearly equiv. response

“Best” materials vs. wavelength dashed lines: optimal ellipsoids require aspect ratios > 30:1

“Best” 2D materials vs. wavelength

Extension #1: High Radiative Efficiency Often want to optimize response such that radiative efficiency ( 𝑃 scat / 𝑃 ext ) > 𝜂 Example: solar cells, LEDs, … max 𝜓 𝑃 abs,scat,ext (𝜓) Nat. Mat. 9, 205 (2010) s.t. 𝑉 𝜓 † Im 𝜒 𝜓<(1−𝜂) Im 𝑉 𝜓 inc † 𝜒𝜓 [Pabs < (1-η)Pext] 𝑃 scat ≤𝜂 1−𝜂 𝜔 𝑈 inc 𝜒 2 Im 𝜒 𝑃 abs ≤ 1−𝜂 2 𝜔 𝑈 inc 𝜒 2 Im 𝜒

Extension #2: Inhomogeneous Background 𝜓 inc ~ 𝑒 𝑖𝑘𝑧 𝑧 𝜓 scat 𝜓 0 ~ 𝑒 𝑖𝑘𝑧 +𝑟 𝑒 −𝑖𝑘𝑧 = + 𝜒 1 𝜒 1 𝜒 2 𝜒 2 𝜒 2 𝑃 abs = 1 2 𝜓 ∗ 𝜔 Im 𝜒 𝜓 >0 𝑃 ext = 𝜔 2 Im 𝜓 inc † 𝜒𝜓 𝑉 1 𝑉 𝑃 scat = 1 2 𝜓 scat ∗ 𝜔 Im 𝜒 𝜓 scat >0 𝑉 2 + 𝑉 ext

Tantalizing Opportunity: Low-Loss “Dielectrics” 𝜆=600 nm 𝜒 Ag ≈−15+1𝑖 𝜒 Si ≈15+0.15𝑖

Hybrid dielectric–metal resonators with Yi Yang (MIT) Metal film: provides subwavelength (~quasistatic) resonance Dielectric nanoparticle: provides lateral confinement Previous hybrid approaches: Maksymov, Lalanne et al. PRL 105, 180502 (2010) Outlon, Zhang et al. Nat. Phot. 2, 496 (2008)

Hybrid plasmon–Bessel modes

Hybrid approach: high scattering efficiency Y. Yang, OD Miller,… In Preparation

Hybrid approach: scattering efficiency approaching upper bounds FO M sca = 𝜎 𝑠𝑐𝑎 𝑉 1 𝜂(1−𝜂)

Near-field enhancements w/ dielectric-on-metal Near-field spontaneous emission enhancement: Simultaneous large enhancement (>5k) and high radiative efficiency (90%, >75% photon) Previous all-metal approaches (theory): 10k & 20% Faggiani et al., ACS Phot. 2, 1739 (2015) 4k & ≈20% Mikkelson et al., Nat. Phot. 8, 836 (2014) 5k & <30% Yang et al. Nano Lett. 16, 4110 (2016)

Radiative Heat Transfer Far-field Kirchoff’s Law: absorptivity = emissivity = 1 “blackbody” definition (ray-optical) Stefan-Boltzmann 𝐻/𝐴≤ 1⋅Θ 𝜔,𝑇 / 𝜆 2 =𝜎 𝑇 4 image:Wikipedia Near field e.g. for future thermo-photovoltaic systems? In the near field, (evanescent) thermal transport can exceed “black-body” limit heat hot PV cell photons Difficulty: comp/expt progress is very recent Wang et. al. Nat Nano 9, 126 (2014)

Near-field radiative heat transfer: Milestones 𝑇 1 >0 𝑇 2 =0 Rytov / Polder / Van Hove (1950–1980’s) stochastic theory, plate–plate heat transfer, possibility of greater-than-blackbody transfer 𝐽 𝑖 𝑥,𝜔 𝐽 𝑗 𝑥 ′ ,𝜔 ∗ = 4 𝜀 0 𝜔 𝜋 Θ 𝜔, 𝑇 1 𝛿 𝑥− 𝑥 ′ 𝛿 𝑖𝑗 Im 𝜒(𝑥) J Pendry (1999): (unrealistic) theoretical bounds to plane–plane transfer PRB 78, 115303 G Chen et. al. (2008): first expt. measurements > blackbody transfer APL 92, 133106 Multiple groups (2008–2011): first rigorous sphere–sphere and sphere–plate theory S. Fan et. al. G. Chen et. al. PRB 84, 245431 (2011) PRB 77, 75125 (2008) SG Johnson et. al. (2011+): generic generic full-Maxwell solvers PRL 107, 114302 (2011) PRB 86, 220302 (2012)

Straightforward extension of limits to near-field heat transfer? dotted line = bounding surface Difficulty: sources are embedded within (arbitrary-shape) scattering body no conventional optical theorem

limits: two scattering problems + reciprocity redefine “incident” and “scattered” fields: Bound absorption in body 2, from unknown field Reciprocity: switch source + measurement points Bound the energy transmitted back to V1 𝐸 inc 1 (𝑥)= 𝑉 1 𝐺 1 𝑥, 𝑥 ′ 𝐽( 𝑥 ′ ) 𝐺 1 = Green’s function in the presence of body 1 𝐸 scat 1 (𝑥)= 𝑉 2 𝐺 1 𝑥, 𝑥 ′ 𝑃 ind ( 𝑥 ′ ) Note the similarities between this formulation and Kirchoff’s Law: we use reciprocity and

Upper limits to near-field heat transfer [ Owen Miller et al, Phys. Rev. Lett. 115, 204302 (2015) ] Heat flux at a given frequency 𝜔 is bounded by: 𝑮 𝟎 = vacuum Green’s function ~ 1 𝑟 3 in the near field for a minimum separation d: near-field enhancement ~ 1/d2 emission and absorption equally enhanced by |χ|2 / Im χ

Are the limits achievable in known structures? First consider simple structures and the generic limit: sphere-sphere reaches the limit sphere-plate off by 2x (pol. mismatch), correct scaling 𝑟≪𝑑≪𝜆 Drude metal, plasma frequency 𝜔 𝑝 and dissipation 𝛾=0.1 𝜔 𝑝

Are the limits achievable in known structures? What about extended (planar) structures? Φ 𝜔 𝐴 plate−plate ~ 1 𝑑 2 ln 𝜒 4 Im 𝜒 2

Are the limits achievable in known structures? What about extended (planar) structures? Promising avenue: periodic nanostructure interactions arrays of dipolar spheres interacting additively (overly idealized) to simultaneously achieve 𝜒 2 /Im 𝜒 (via particles) and 1/ 𝑑 2 (via array) enhancements

Reaching the limits: new possibilities in heat transfer Given optimal flux (and smallest bandwidth, Δ𝜔/ 𝜔 res , for a metal): 1010 limits radiative > conductive transport plates (in air) possible at: T=300K, d=30nm or T=1500K, d=0.5mm 107 1500K HT coeff, h (W/m2⋅K) 104 cond 300K air, 𝜅 𝑐𝑜𝑛𝑑 =0.026 W m⋅K 0.001 0.01 0.1 1 Separation d (mm) %$Note that in TPV conductive heat is waste. 𝑑 far-field heat transfer near-field heat transfer ≤𝜎 𝑇 4 𝐴 𝜆 𝑇 𝑑 2 𝜒 3 Im 𝜒 ≤𝜎 𝑇 4 𝐴 𝜎= Stefan–Boltzmann constant

Generalization of optical-communications limits What is the limit to information transmission between transparent media? bounded by the vacuum Green’s function D. A. B. Miller, “Communication with waves between volumes…” [Applied Optics 39, 1681 (1999)] generalization of the transparent-body communications limit, to include material effects: Radiative transfer = optical communication (by heat)

Optical-Response Bounds via Convex Constraints Known scattered-power bounds Sum rules, using low- and high-frequency asymptotics Spherical-harmonic decomposition (𝜎∼ 𝜆 2 ) Optical theorem imposes convex constraints Material-dictated, shape-independent bounds Limits to optical response in absorptive systems High-radiative-efficiency plasmonics Radiative heat transfer bounds: near-field analogue of the blackbody limit Scattering-channel bounds, revisited Force/torque equivalents of 𝜎∼ 𝜆 2 And as I will discuss, always informed and driven by experimental considerations

𝜎∼ 𝜆 2 bounds, revisited 𝑃 scat = 𝒄 𝑠 † 𝒄 𝑠 𝑃 ext =Re 𝒄 𝑖 † 𝒄 𝑠 𝑃 abs =− 1 2 Re 𝑆 𝑬× 𝑯 ∗ ⋅ 𝒏 = 1 4 𝑆 𝜓 † 𝑛 × − 𝑛 × 𝜓 Problem: Indefinite quadratic form (same form arises for 𝑃 scat ) 𝑆 𝜓 inc ( 𝑥 𝑆 )= 𝑐 𝑖 𝑸 𝒊 ( 𝑥 𝑆 ) 𝜓 scat ( 𝑥 𝑆 )= 𝑐 𝑠 𝑸 𝒊 ( 𝑥 𝑆 ) 𝑃 scat = 𝒄 𝑠 † 𝒄 𝑠 𝑃 ext =Re 𝒄 𝑖 † 𝒄 𝑠 𝑃 scat < 𝑃 ext 𝜎 abs , 𝜎 scat ≲ 𝑁 2 3 𝜆 2 2𝜋 JP Hugonin et al. PRB 91, 180202 (2015) OD Miller et al. Opt Exp 24, 3329 (2016) Vector SH 𝑸 𝒊 𝜓 inc 𝜓 scat

Progress, and New Questions Optical-response bounds by convex constraints Fundamental limits to optical response in absorptive systems; material FOM 𝜒 2 /Im 𝜒 Low-loss dielectric nanoparticles on metallic films for high-radiative-efficiency plasmonics Near-field analogue of the blackbody limit Opportunities Optical theorem underlies all-frequency sum rules & single-frequency bounds Systematic identification of convex constraints? Large-scale design to approach near-field bounds

Supporting Slides

Hybrid approach: robust to quantum quenching effects

Versatile Q-factors

Hybrid nanorod-on-metal

Design rules in the near field (1) In the absence of the absorber, 𝑉 2 , the fields emitted by 𝑉 1 into 𝑉 2 should be amplified by material enhancement ratio: Optimal-emitter condition 𝐸 inc ~ 𝜒 1 2 Im 𝜒 1 𝐸 dipole (2) With both bodies present, the currents induced in the absorber should be further enhanced by the second material enhancement ratio 𝑃 ind ~ 𝜒 2 2 Im 𝜒 2 𝐸 inc Optimal-absorber condition ~ 𝜒 1 2 Im 𝜒 1 𝜒 2 2 Im 𝜒 2 𝐸 dipole

∼ Why ? 1. Power balance in Maxwell’s equations 𝜒 2 Im 𝜒 Why ? 1. Power balance in Maxwell’s equations 𝑃 ext ∝Im 𝑉 𝐸 inc ∗ ⋅ 𝑃 ind 𝑃 abs = Im 𝜒 𝜒 2 𝑉 𝑃 ind ∗ ⋅ 𝑃 ind 2. Unitless measure of resistivity 𝜒 2 Im 𝜒 = 1 𝜀 0 𝜔 Re 𝜌 𝜔 𝐽= 1 𝜌 𝐸 ∼ + 𝑉 𝑅 – 𝑃= 𝑉 2 𝑅