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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].

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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

Nano-photonicmedia (l-scale)

strange waveguides

& microcavities

[B. Norris, UMN]

[Assefa & Kolodziejski,

MIT]

3d

structures

[Mangan,

Corning]

synthetic materials

optical phenomena

hollow-core fibers

photonic crystals

1887

1987

Photonic Crystals

periodic electromagnetic media

can have a band gap: optical “insulators”

electronic and photonic crystals

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

electronic photonic modelling
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

fun with math

+ constraint

eigen-state

eigen-operator

eigen-value

Fun with Math

First task:

get rid of this mess

0

dielectric function e(x) = n2(x)

electronic photonic eigenproblems
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…

a 2d model system
A 2d Model System

dielectric “atom”

e=12 (e.g. Si)

square lattice,

period a

a

a

E

TM

H

periodic eigenproblems
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)

solving the maxwell eigenproblem
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

solving the maxwell eigenproblem 1

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

solving the maxwell eigenproblem 2a
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

solving the maxwell eigenproblem 2b
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

solving the maxwell eigenproblem 3a
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)

solving the maxwell eigenproblem 3b
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

solving the maxwell eigenproblem 3c
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”

band diagram of 2d model system radius 0 2 a rods e 12

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

the iteration scheme is important

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

the iteration scheme is important19
The Iteration Scheme is Important

(minimizing function of ~40,000 variables)

no preconditioning

% error

preconditioned

conjugate-gradient

no conjugate-gradient

# iterations

the boundary conditions are tricky

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?

the e averaging is important
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)

gap schmap
Gap, Schmap?

a

frequencyw

G

G

X

M

But, what can we do with the gap?

intentional defects are good
Intentional “defects” are good

microcavities

waveguides (“wires”)

microcavity blues
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

time domain eigensolvers finite difference time domain fdtd
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

signal processing is tricky
Signal Processing is Tricky

signal processing

complexwn

?

a common approach: least-squares fit of spectrum

fit to:

FFT

Decaying signal (t)

Lorentzian peak (w)

fits and uncertainty

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)

unreliability of fitting process

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)

quantum inspired signal processing nmr spectroscopy filter diagonalization method fdm

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!

filter diagonalization method fdm

…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

filter diagonalization summary
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

do try this at home
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/