1 / 33

From electrons to photons: Quantum-inspired modeling in nanophotonics

From electrons to photons: Quantum-inspired modeling in nanophotonics. Steven G. Johnson , MIT Applied Mathematics. Nano-photonic media ( l -scale). strange waveguides. & microcavities. [B. Norris, UMN]. [Assefa & Kolodziejski, MIT]. 3d structures . [Mangan, Corning].

Patman
Download Presentation

From electrons to photons: Quantum-inspired modeling in nanophotonics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. From electrons to photons: Quantum-inspired modeling in nanophotonics Steven G. Johnson, MIT Applied Mathematics

  2. Nano-photonicmedia (l-scale) strange waveguides & microcavities [B. Norris, UMN] [Assefa & Kolodziejski, MIT] 3d structures [Mangan, Corning] synthetic materials optical phenomena hollow-core fibers

  3. 1887 1987 Photonic Crystals periodic electromagnetic media can have a band gap: optical “insulators”

  4. dielectric spheres, diamond lattice photon frequency wavevector Electronic and Photonic Crystals atoms in diamond structure Periodic Medium Bloch waves: Band Diagram electron energy wavevector interacting: hard problem non-interacting: easy problem

  5. Electronic & Photonic Modelling Electronic Photonic • strongly interacting —tricky approximations • non-interacting (or weakly), —simple approximations (finite resolution) —any desired accuracy • lengthscale dependent (from Planck’s h) • scale-invariant —e.g. size 10   10 Option 1: Numerical “experiments” — discretize time & space … go Option 2: Map possible states & interactions using symmetries and conservation laws: band diagram

  6. + constraint eigen-state eigen-operator eigen-value Fun with Math First task: get rid of this mess 0 dielectric function e(x) = n2(x)

  7. Electronic & Photonic Eigenproblems Electronic Photonic simple linear eigenproblem (for linear materials) nonlinear eigenproblem (V depends on e density ||2) —many well-known computational techniques Hermitian = real E & w, … Periodicity = Bloch’s theorem…

  8. A 2d Model System dielectric “atom” e=12 (e.g. Si) square lattice, period a a a E TM H

  9. Periodic Eigenproblems if eigen-operator is periodic, then Bloch-Floquet theorem applies: can choose: planewave periodic “envelope” Corollary 1: k is conserved, i.e.no scattering of Bloch wave Corollary 2: given by finite unit cell, so w are discrete wn(k)

  10. Solving the Maxwell Eigenproblem Finite celldiscrete eigenvalues wn Want to solve for wn(k), & plot vs. “all” k for “all” n, constraint: where: H(x,y) ei(kx – wt) Limit range ofk: irreducibleBrillouin zone 1 Limit degrees of freedom: expand H in finitebasis 2 Efficiently solve eigenproblem: iterative methods 3

  11. ky kx Solving the Maxwell Eigenproblem: 1 Limit range ofk: irreducible Brillouin zone 1 —Bloch’s theorem: solutions are periodic in k M first Brillouin zone = minimum |k| “primitive cell” X G irreducible Brillouin zone: reduced by symmetry Limit degrees of freedom: expand H in finite basis 2 Efficiently solve eigenproblem: iterative methods 3

  12. Solving the Maxwell Eigenproblem: 2a Limit range ofk: irreducible Brillouin zone 1 Limit degrees of freedom: expand H in finite basis (N) 2 solve: finite matrix problem: Efficiently solve eigenproblem: iterative methods 3

  13. Solving the Maxwell Eigenproblem: 2b Limit range ofk: irreducible Brillouin zone 1 Limit degrees of freedom: expand H in finite basis 2 — must satisfy constraint: Planewave (FFT) basis Finite-element basis constraint, boundary conditions: Nédélec elements [ Nédélec, Numerische Math. 35, 315 (1980) ] constraint: nonuniform mesh, more arbitrary boundaries, complex code & mesh, O(N) uniform “grid,” periodic boundaries, simple code, O(N log N) [ figure: Peyrilloux et al., J. Lightwave Tech. 21, 536 (2003) ] Efficiently solve eigenproblem: iterative methods 3

  14. Solving the Maxwell Eigenproblem: 3a Limit range ofk: irreducible Brillouin zone 1 Limit degrees of freedom: expand H in finite basis 2 Efficiently solve eigenproblem: iterative methods 3 Slow way: compute A & B, ask LAPACK for eigenvalues — requires O(N2) storage, O(N3) time Faster way: — start with initial guess eigenvector h0 — iteratively improve — O(Np) storage, ~O(Np2) time for p eigenvectors (psmallest eigenvalues)

  15. Solving the Maxwell Eigenproblem: 3b Limit range ofk: irreducible Brillouin zone 1 Limit degrees of freedom: expand H in finite basis 2 Efficiently solve eigenproblem: iterative methods 3 Many iterative methods: — Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …, Rayleigh-quotient minimization

  16. Solving the Maxwell Eigenproblem: 3c Limit range ofk: irreducible Brillouin zone 1 Limit degrees of freedom: expand H in finite basis 2 Efficiently solve eigenproblem: iterative methods 3 Many iterative methods: — Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …, Rayleigh-quotient minimization for Hermitian matrices, smallest eigenvalue w0minimizes: minimize by preconditioned conjugate-gradient(or…) “variational theorem”

  17. a Band Diagram of 2d Model System(radius 0.2a rods, e=12) frequencyw(2πc/a) = a / l irreducible Brillouin zone G G X M M E gap for n > ~1.75:1 TM X G H

  18. Preconditioned conjugate-gradient: minimize (h + a d) — d is (approximate A-1) [f + (stuff)] The Iteration Scheme is Important (minimizing function of 104–108+ variables!) Steepest-descent: minimize (h + a f) over a … repeat Conjugate-gradient: minimize (h + a d) — d is f+ (stuff): conjugate to previous search dirs Preconditioned steepest descent: minimize (h + a d) — d = (approximate A-1) f ~ Newton’s method

  19. The Iteration Scheme is Important (minimizing function of ~40,000 variables) no preconditioning % error preconditioned conjugate-gradient no conjugate-gradient # iterations

  20. E|| is continuous E is discontinuous (D = eE is continuous) Any single scalare fails: (mean D) ≠ (anye) (mean E) Use a tensore: E|| E The Boundary Conditions are Tricky e?

  21. The e-averaging is Important backwards averaging correct averaging changes order of convergence from ∆x to ∆x2 no averaging % error tensor averaging (similar effects in other E&M numerics & analyses) resolution (pixels/period)

  22. Gap, Schmap? a frequencyw G G X M But, what can we do with the gap?

  23. Intentional “defects” are good microcavities waveguides (“wires”)

  24. (Same computation, with supercell = many primitive cells) Intentional “defects” in 2d

  25. Microcavity Blues For cavities (point defects) frequency-domain has its drawbacks: • Best methods compute lowest-w bands, but Nd supercells have Nd modes below the cavity mode — expensive • Best methods are for Hermitian operators, but losses requires non-Hermitian

  26. Time-Domain Eigensolvers(finite-difference time-domain = FDTD) Simulate Maxwell’s equations on a discrete grid, + absorbing boundaries (leakage loss) • Excite with broad-spectrum dipole ( ) source Dw Response is many sharp peaks, one peak per mode signal processing complexwn [ Mandelshtam, J. Chem. Phys.107, 6756 (1997) ] decay rate in time gives loss

  27. Signal Processing is Tricky signal processing complexwn ? a common approach: least-squares fit of spectrum fit to: FFT Decaying signal (t) Lorentzian peak (w)

  28. There is a better way, which gets complex w to > 10 digits Fits and Uncertainty problem: have to run long enough to completely decay actual signal portion Portion of decaying signal (t) Unresolved Lorentzian peak (w)

  29. There is a better way, which gets complex w for both peaks to > 10 digits Unreliability of Fitting Process Resolving two overlapping peaks is near-impossible 6-parameter nonlinear fit (too many local minima to converge reliably) sum of two peaks w = 1+0.033i w = 1.03+0.025i Sum of two Lorentzian peaks (w)

  30. Idea: pretend y(t) is autocorrelation of a quantum system: time-∆t evolution-operator: say: Quantum-inspired signal processing (NMR spectroscopy):Filter-Diagonalization Method (FDM) [ Mandelshtam, J. Chem. Phys.107, 6756 (1997) ] Given time series yn, write: …find complex amplitudes ak & frequencies wk by a simple linear-algebra problem!

  31. …expand U in basis of |(n∆t)>: Umn given by yn’s — just diagonalize known matrix! Filter-Diagonalization Method (FDM) [ Mandelshtam, J. Chem. Phys.107, 6756 (1997) ] We want to diagonalize U: eigenvalues of U are eiw∆t

  32. Filter-Diagonalization Summary [ Mandelshtam, J. Chem. Phys.107, 6756 (1997) ] Umn given by yn’s — just diagonalize known matrix! A few omitted steps: —Generalized eigenvalue problem (basis not orthogonal) —Filter yn’s (Fourier transform): small bandwidth = smaller matrix (less singular) • resolves many peaks at once • # peaks not knowna priori • resolve overlapping peaks • resolution >> Fourier uncertainty

  33. Do try this at home Bloch-mode eigensolver: http://ab-initio.mit.edu/mpb/ Filter-diagonalization: http://ab-initio.mit.edu/harminv/ Photonic-crystal tutorials(+ THIS TALK): http://ab-initio.mit.edu/ /photons/tutorial/

More Related