Adjoint-Based Photonic Design: Optimization for Applications from Super-Scattering to Enhanced Light Extraction

Transcription

1 Adjoint-Based Photonic Design: Optimization for Applications from Super-Scattering to Enhanced Light Extraction Owen Miller Post-doc, MIT Applied Math PI: Steven Johnson Collaborators: Homer Reid (Math), Chia Wei Hsu (Harvard Physics), Wenjun Qiu (Physics), Brendan DeLacy (US Army ECBC), Marin Soljacic (Physics), John Joannopoulos (Physics)

2 Adjoint-Based Photonic Design: Optimization for Applications from Super-Scattering to Enhanced Light Extraction Owen Miller Post-doc, MIT Applied Math PI: Steven Johnson Collaborators: Homer Reid (Math), Chia Wei Hsu (Harvard Physics), Wenjun Qiu (Physics), Brendan DeLacy (US Army ECBC), Marin Soljacic (Physics), John Joannopoulos (Physics)

3 Zhang et al Nat. Comm. 3, 1180 (2012) Van Hulst et al Nat. Photon. 2, 234 (2008) Super-Scattering Particle Design Fundamental Question: How can we design nano-particles to maximally extinguish light, per unit of material volume/weight? Extinction = Absorption + Scattering Application: Obscurance (i.e. smokescreens) Troops concealed by smokescreens Nano-particle absorbers/scatterers have potential applications in Imaging Biomedicine Optical antennas Metamaterials

4 Previous work Primarily spherically-symmetric structures Fan et. al. APL 98, (2011) Soljacic et. al. Opt. Exp. 21, 1465 (2012) Some exploration of non-spherical particles: not systematic, or at different frequencies / different metrics Multipole Plasmon Resonances in Au Nanorods J. Phys. Chem. B, Vol. 110, No. 5, Nano-rings Nano-rods Nano-prisms Aizpurua et. al. PRL 90, (2003) Payne et. al. JCB 110, 2150 (2006) Jun et. al. Science 294, 1901 (2001) Kelly et. al. JPCB 107, 668 (2003)

5 The Computational Challenge Goal: maximize extinction / volume, σ V Need to explore large design space of non-spherical, three-dimensional structures For every structure, many frequencies (broadband performance) For every frequency, many incidence angles (random orientation) e

6 The Computational Challenge Goal: maximize extinction / volume, σ V Our Approach Need to explore large design space of non-spherical, three-dimensional structures Adjoint-based shape derivatives, within sophisticated optimizer ab-initio.mit.edu/nlopt For every structure, many frequencies (broadband performance) For every frequency, many incidence angles (random orientation) e

7 The Computational Challenge Goal: maximize extinction / volume, σ V Our Approach Need to explore large design space of non-spherical, three-dimensional structures Adjoint-based shape derivatives, within sophisticated optimizer ab-initio.mit.edu/nlopt For every structure, many frequencies (broadband performance) Complex-frequency transformation For every frequency, many incidence angles (random orientation) e

8 The Computational Challenge Goal: maximize extinction / volume, σ V Our Approach Need to explore large design space of non-spherical, three-dimensional structures Adjoint-based shape derivatives, within sophisticated optimizer ab-initio.mit.edu/nlopt For every structure, many frequencies (broadband performance) Complex-frequency transformation For every frequency, many incidence angles (random orientation) Boundary-element method Discretize surface only, not volume all angles essentially free homerreid.ath.cx/scuff-em/

9 Complex Frequency Transformation By optical theorem, extinction equals: σ = Im[f] f = forward scattering amplitude analytic in upper-half place (causality) Suppose we want to measure broadband performance: σ avg = Im f(ω) Δω/π ω ω Δω 2 dω Lorentzian window function analytic (no poles) one pole at ω 0 + iδω Contour integration: Im ω ω 0 + iδω σ avg = Im f(ω 0 + iδω) Re ω Many real w to One complex w!

10 Verification: many-to-one frequency transform Complex frequency = Complex materials ω 0 ω 0 + iδω ε, μ ε, μ ω 0 + iδω 1 + i Δω ω 0 Can be solved with existing FEM/BEM codes!

11 Beyond Spheres: Ellipsoids Optimizing σ over λ = 600,800 nm, random orientation V Among all multi-coated SiO 2 /Ag spheres, global optimum always SiO 2 Ag Very small (quasi-static) Silica sphere with single Silver coating σ avg V = σ ω V = 0.09nm 1 Δω/π ω ω Δω 2 dω Extending optimization to ellipsoids, how well can we do? R 1 R 2 Assume surface of revolution (i.e. spheroids, 2 degrees of freedom)

12 Equatorial Radius (nm) σ/v (nm -1 ) 60 Optimal Un-Coated, Ag Ellipsoids Optimization σ avg V [nm 1 ] Optimal Shapes oblate spheroid prolate spheroid Polar Radius (nm) (coated) sphere Wavelength (nm) Optimal Ellipsoid r 1 = 3nm, r 2 = r 3 = 45nm (r 1 at lower bound) σ avg = 0.53nm 1, 6x better than optimal sphere V Oblate (disk) > Prolate (needle) Actual sampling

13 Beyond Ellipsoids: Star-Shaped Structures Use spherical harmonics as basis functions for shapes r(θ, φ) = c lm Y lm (θ, φ) Adjoint shape derivatives: reciprocity in action l,m Example Structure Direct Simulation Adjoint Simulation Gradient = δf δx n = ε 2 ε 1 E E A + 1 ε 1 1 ε 2 D D A With only two simulations Derivative at every surface point!

14 Optimization #1: Ag near l=400nm Optimal Structure (<150 iterations) Top View Side View Dimensions 5nm How to think about structure? Inscribe tetrahedron in sphere Push in at centroids Performance equal for all 3 polarizations

15 Extinction / Volume (nm -1 ) Comparison of Optimal Structures Extinction / Volume (nm -1 ) Optimal General Shape Optimal Ellipsoid r 1 = r 2 = 2nm, r 3 = 10nm Wavelength (nm) Wavelength (nm) Roughly equal for all three polarizations y-, z-pol: very strong response x-pol: very weak

16 Optimal Structures Comparison: Total Response Extinction / Volume (nm -1 ) General Shape Ellipsoid Wavelength (nm) Almost exactly the same?! General shape optimum roughly 3% better than ellipsoidal optimum

17 More Optimizations General Ellipsoids (Discs)

18 Quasi-Static Resonances Solving Gauss Law: εe = ρ Resonant with respect to what? There is no frequency. Resonance with respect to permittivity. For a given structure, there are specific (negative & real-valued) permittivities for which a surface charge can exist, without an incident field Mathematical formulation: S ε x φ(x) = ρ ext (x) ε 0 ε 1 2π ε 1 + ε 0 ε 1 ε 0 σ x = n φ x + F x, y σ y dy S for φ = 0: F x, y = n x x y x y 3 ε n + 1 ε n 1 σ n x = 1 2π F x, y σ n y dy S

19 Quasi-Static Resonances Solving Gauss Law: εe = ρ Resonant with respect to what? There is no frequency. Resonance with respect to permittivity. For a given structure, there are specific (negative & real-valued) permittivities for which a surface charge can exist, without an incident field Spheres Three modes at ε = 2 Each mode contributes to σ/v through dipole strength p n and coupling c n = 1 1 ε n 1 1 ε 1 Ellipsoids Two modes at ε π r max 4 r min One mode at ε 8 π r min r max Images are surface charge densities of respective modes

20 Quasi-static particle design Suppose we want to design particle for maximum extinction at ε(ω 0 ). How? (1) Want at least one resonance ε n Re[ε ω 0 ] (2) Want polarizability concentrated at ε n (i.e. should not be wasted at other permittivities) How does an oblate spheroid look, from this perspective? Two dipole modes at ε 1 = ε 2 = 6.9 One dipole mode at ε 3 = 0.3 So we have two strong, degenerate resonances. But we re losing a significant part of polarizability at -0.3, right? Not quite.

21 Sum rules Fuchs ( ), for collections of homogeneous particles: #1: sum rule on polarizabilities modes n #2: weighted sum of resonant permittivities modes n p n,α = 1 1 p 1 ε n = 1 n 3 α = x, y, z p n = avg. p n,α for all x,y,z This is the crucial sum rule (barely noticed in literature). Very roughly speaking, weighted average of ε k has to = 2 The primary upper bound discussed in the literature has been the integrated extinction per volume: σ ext (λ) V = π 2 γ = static Tr[γ] But this is shape-dependent, polarizability does not provide a general limit

22 Quasi-static upper bound Constrained optimization: Maximize σ(ω 0 ) V = 1 3 ω 0 c Subject to (1-3) p n,α = 1 (4) modes n modes n n 1 1 ε n 1 p 1 ε n = 1 n ε ω 0 α [x, y, z] 1 p n Sum Rule #1 Sum Rule #2 (5) ε n < 0 Uniqueness Theorem

23 Quasi-static upper bound Constrained optimization: Maximize σ(ω 0 ) V Subject to (1-3) p n,α = 1 (4) = 1 3 modes n modes n ω 0 c n 1 1 ε n 1 p 1 ε n = 1 n ε ω 0 α [x, y, z] 1 p n Sum Rule #1 Sum Rule #2 (5) ε n < 0 Uniqueness Theorem For ε ω 0 < 2: σ ext (ω 0 ) For a material permittivity ε = ε r + iε i and susceptibility χ = ε 1: V ω 0 c ω 0 c χ 4 χ 2 + χ r ε r 2 ε i χ i ε r ε i, 1 Given material and frequency, this bounds extinction for any possible shape!

24 Mode Polarizability What do the optimal structures have in common? Permittivity Eigenvalue Permittivity Eigenvalue Permittivity Eigenvalue They all have roughly degenerate eigenmodes at the desired permittivity; all other modes have zero dipole moment, except as required by sum rules All mode calculations courtesy of excellent MNPBEM Toolbox:

25 whereas e.g. a tetrahedron has resonances:

26 The left-hand side can roughly be interpreted as: # of full-strength resonances (ideal max. 3, for 3 polarizations) About the general upper bound: Ellipsoids are nearly optimal; they even hit the upper bound in three limits (ε, ε = 2, ε 0) There are structures that do better than ellipsoids; can possibly take manufacturability into account From the optimizations, perhaps the true upper bound is even closer to the ellipsoid than the one derived here

27 Mode Polarizability What do the optimal structures have in common? Permittivity Eigenvalue Permittivity Eigenvalue Permittivity Eigenvalue They all have roughly degenerate eigenmodes at the desired permittivity; all other modes have zero dipole moment, except as required by sum rules All mode calculations courtesy of excellent MNPBEM Toolbox:

28 Ideal Materials Upper bound allows shape-independent calculation of ideal materials Ag Al Au Cu The lines that are not labeled: Cr, Co, Ir, Li, Mo, Ni, Os, Pd, Pt, Rh, Ta, Ti, W, V material data: refractiveindex.info (Palik, Rakic, Sopra-SA)

29 Key Points Moving from spheres to ellipsoids: 6x improvement Moving from ellipsoids to arbitrary shapes: <1.34x improvement even theoretically possible There are shapes superior to ellipsoids; can be found through computational shape optimization Ideal Materials for the Visible New, Quasi-Static Extinction Limit σ(ω 0 ) V 2 3 ω 0 c χ 4 χ 2 + χ r χ i