1. 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

2. How efficiently can light be absorbed?
Principles
Design
Technology
photonics
Ni, Zhang et al. Science 249, (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, (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?

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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, (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)

9. 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, (2007)
Spheres:
Ruan and Fan APL 98, (2011)

10. 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) …

11. 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

12. 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
= 𝑉 ext 𝜓 scat ∗ 𝜔 Im 𝜒 𝜓 scat
Passivity: 𝜔 Im 𝜒>0

13. Optical Theorem: 𝑃 abs + 𝑃 scat ∝ amplitude
𝑃 abs = 𝜓 ∗ 𝜔 Im 𝜒 𝜓 𝑉
𝑃 scat = 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

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

15. 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

16. 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

17. 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, (2016)
O. D. Miller et al. PRL 112, (2014)

18. Metamaterials: Constrained by Bounds for Intrinsic Material
[OD Miller et al. PRL 112, (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

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

20. “Best” 2D materials vs. wavelength

21. 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

22. 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 = 𝜓 ∗ 𝜔 Im 𝜒 𝜓 >0
𝑃 ext = 𝜔 2 Im 𝜓 inc † 𝜒𝜓
𝑉 1 𝑉
𝑃 scat = 𝜓 scat ∗ 𝜔 Im 𝜒 𝜓 scat >0
𝑉 2 + 𝑉 ext

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

24. 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, (2010)
Outlon, Zhang et al. Nat. Phot. 2, 496 (2008)

25. Hybrid plasmon–Bessel modes

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

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

28. 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)

29. 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)

30. 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,
G Chen et. al. (2008): first expt. measurements > blackbody transfer
APL 92,
Multiple groups (2008–2011): first rigorous sphere–sphere and sphere–plate theory
S. Fan et. al.
G. Chen et. al.
PRB 84, (2011)
PRB 77, (2008)
SG Johnson et. al. (2011+): generic
generic full-Maxwell solvers
PRL 107, (2011)
PRB 86, (2012)

31. 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

32. 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

33. Upper limits to near-field heat transfer
[ Owen Miller et al, Phys. Rev. Lett. 115, (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

34. 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

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

36. 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

37. 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 𝜒 3 Im 𝜒
≤𝜎 𝑇 4 𝐴
𝜎= Stefan–Boltzmann constant

38. 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)

39. 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

40. 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, (2015)
OD Miller et al. Opt Exp 24, 3329 (2016)
Vector SH 𝑸 𝒊
𝜓 inc
𝜓 scat

41. 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

42. Supporting Slides

43. Hybrid approach: robust to quantum quenching effects

44. Versatile Q-factors

45. Hybrid nanorod-on-metal

46. 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 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 Im 𝜒 2 𝐸 inc
Optimal-absorber
condition
~ 𝜒 Im 𝜒 𝜒 Im 𝜒 2 𝐸 dipole

47. ∼ 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