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

From electrons to photons: Quantum-inspired modeling in nanophotonics

Steven G. Johnson, MIT Applied Mathematics

Nano-photonic media (-scale)

synthetic materials

strange waveguides

3d structures

hollow-core fibersoptical phenomena

& microcavities

[B. Norris, UMN] [Assefa & Kolodziejski, MIT]

[Mangan, Corning]

1887 1987

Photonic Crystalsperiodic electromagnetic media

2-D

periodic intwo directions

3-D

periodic inthree directions

1-D

periodic inone direction

can have a band gap: optical “insulators”

Electronic and Photonic Crystalsatoms in diamond structure

wavevector

elec

tron

ene

rgy

Per

iod

ic M

ediu

mB

loch

wav

es:

Ban

d D

iagr

amdielectric spheres, diamond lattice

wavevector

phot

on f

requ

ency

interacting: hard problem non-interacting: easy problem

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

r ∇ ×

r E =−

1c

∂∂t

r H =i

ωc

r H

r ∇ ×

r H =ε

1c

∂∂t

r E +

r J =i

ωc

εr E

0

dielectric function (x) = n2(x)

First task:get rid of this mess

∇ ×

1ε

∇ ×r H =

ωc

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2 r H

eigen-operator eigen-value eigen-state

∇ ⋅r H =0

+ constraint

Electronic & Photonic Eigenproblems

∇ ×

1ε

∇ ×r H =

ωc

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2 r H

Electronic Photonic

€

−h2

2m∇ 2 + V

⎛

⎝ ⎜

⎞

⎠ ⎟ψ = Eψ

simple linear eigenproblem(for linear materials)

nonlinear eigenproblem(V depends on e density ||2)

—many well-known computational techniques

Hermitian = real E & , … Periodicity = Bloch’s theorem…

A 2d Model System

square lattice,period a

dielectric “atom”=12 (e.g. Si)

a

a

E

HTM

Periodic Eigenproblemsif eigen-operator is periodic, then Bloch-Floquet theorem applies:

r H (

r x ,t)=e

ir k ⋅r x −ωt( ) r

H r k (r x )can choose:

periodic “envelope”planewave

Corollary 1: k is conserved, i.e. no scattering of Bloch wave

Corollary 2: given by finite unit cell,so are discrete n(k) r H r k

Solving the Maxwell Eigenproblem

H(x,y) ei(kx – t)€

∇+ ik( ) ×1

ε∇ + ik( ) × Hn =

ωn2

c 2Hn

€

∇+ ik( ) ⋅H = 0

where:

constraint:

1

Want to solve for n(k),& plot vs. “all” k for “all” n,

Finite cell discrete eigenvalues n

Limit range of k: irreducible Brillouin zone

2 Limit degrees of freedom: expand H in finite basis

3 Efficiently solve eigenproblem: iterative methods

QuickTime™ and aGraphics decompressorare needed to see this picture.

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Photonic Band Gap

TM bands

Solving the Maxwell Eigenproblem: 11 Limit range of k: irreducible Brillouin zone



—Bloch’s theorem: solutions are periodic in k

kx

ky

first Brillouin zone= minimum |k| “primitive cell”

€

2π

a

M

X

irreducible Brillouin zone: reduced by symmetry

Solving the Maxwell Eigenproblem: 2a1 Limit range of k: irreducible Brillouin zone

2 Limit degrees of freedom: expand H in finite basis (N)


H =H(xt) = hmbm(xt)m=1

N

∑ solve: ˆ A H =ω2 H

Ah=ω2Bh

Aml = bmˆ A bl Bml = bm blf g = f* ⋅g∫

finite matrix problem:

Solving the Maxwell Eigenproblem: 2b1 Limit range of k: irreducible Brillouin zone



€

(∇ + ik) ⋅H = 0— must satisfy constraint:

Planewave (FFT) basis

H(xt) = HGeiG⋅xt

G∑

€

HG ⋅ G + k( ) = 0constraint:

uniform “grid,” periodic boundaries,simple code, O(N log N)

Finite-element basisconstraint, boundary conditions:

Nédélec elements[ Nédélec, Numerische Math.

35, 315 (1980) ]

nonuniform mesh,more arbitrary boundaries,

complex code & mesh, O(N)[ figure: Peyrilloux et al.,

J. Lightwave Tech.21, 536 (2003) ]

Solving the Maxwell Eigenproblem: 3a1 Limit range of k: irreducible Brillouin zone



Ah=ω2Bh

Faster way:— start with initial guess eigenvector h0

— iteratively improve— O(Np) storage, ~ O(Np2) time for p eigenvectors

Slow way: compute A & B, ask LAPACK for eigenvalues— requires O(N2) storage, O(N3) time

(p smallest eigenvalues)

Solving the Maxwell Eigenproblem: 3b1 Limit range of k: irreducible Brillouin zone



Ah=ω2BhMany iterative methods:

— Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …, Rayleigh-quotient minimization

Solving the Maxwell Eigenproblem: 3c1 Limit range of k: irreducible Brillouin zone



Ah=ω2BhMany iterative methods:

— Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …, Rayleigh-quotient minimization

for Hermitian matrices, smallest eigenvalue 0 minimizes:

ω02 =min

h

h' Ahh'Bh

minimize by preconditioned conjugate-gradient (or…)

“variationaltheorem”

Band Diagram of 2d Model System(radius 0.2a rods, =12)

E

HTM

a

freq

uenc

y

(2π

c/a)

= a

/

X

M

X M irreducible Brillouin zone

r k


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Photonic Band Gap

TM bands

gap forn > ~1.75:1

The Iteration Scheme is Important(minimizing function of 104–108+ variables!)

Steepest-descent: minimize (h + f) over … repeat

€

02 = min

h

h' Ah

h'Bh= f (h)

Conjugate-gradient: minimize (h + d)— d is f + (stuff): conjugate to previous search dirs

Preconditioned steepest descent: minimize (h + d) — d = (approximate A-1) f ~ Newton’s method

Preconditioned conjugate-gradient: minimize (h + d)— d is (approximate A-1) [f + (stuff)]

The Iteration Scheme is Important(minimizing function of ~40,000 variables)


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ÑÑ Ñ Ñ Ñ ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ

J

JJ

JJ

JJ

J J J J J J J JJ JJJJJJJJJJJJ

0.000001

0.00001

0.0001

0.001

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0.1

1

10

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1000

10000

100000

1000000

1 10 100 1000

# iterations

% e

rror

preconditionedconjugate-gradient no conjugate-gradient

no preconditioning

The Boundary Conditions are Tricky

E|| is continuous

E is discontinuous

(D = E is continuous)

Any single scalar fails: (mean D) ≠ (any ) (mean E)

Use a tensor

€

−1 −1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

E||

E

The -averaging is ImportantQuickTime™ and aGraphics decompressorare needed to see this picture.

B B

B

B

BB

B

B

B B

BB

B

J

JJ

J

J J

JJ J

J J

J J

HH

H H H H H H HH H H H

0.01

0.1

1

10

100

10 100resolution (pixels/period)

% e

rror

backwards averaging

tensor averaging

no averaging

correct averagingchanges order of convergencefrom ∆x to ∆x2

(similar effectsin other E&M

numerics & analyses)

Gap, Schmap?

a

freq

uenc

y

X M


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Photonic Band Gap

TM bands

But, what can we do with the gap?

Intentional “defects” are good

3D Photonic Crystal with Defects

microcavities waveguides (“wires”)

Intentional “defects” in 2dQuickTime™ and aGraphics decompressorare needed to see this picture.

a




(Same computation, with supercell = many primitive cells)

Microcavity Blues

For cavities (point defects)frequency-domain has its drawbacks:

• Best methods compute lowest- 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)

Simulate Maxwell’s equations on a discrete grid,+ absorbing boundaries (leakage loss)

• Excite with broad-spectrum dipole ( ) source

Response is manysharp peaks,

one peak per modecomplex n [ Mandelshtam,

J. Chem. Phys. 107, 6756 (1997) ]

signal processing

decay rate in time gives loss


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E

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E

E

EE

EEEEEEEEEEEEEEEEEEEEEEEEE0

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Signal Processing is Tricky

complex n

?

signal processing


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Decaying signal (t) Lorentzian peak ()

FFT

a common approach: least-squares fit of spectrum

fit to:

€

A

(ω −ω0)2 + Γ 2


E E E E E

E

E E E E E0

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Fits and UncertaintyQuickTime™ and aGraphics decompressorare needed to see this picture.

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Portion of decaying signal (t) Unresolved Lorentzian peak ()

actual

signalportion

problem: have to run long enough to completely decay

There is a better way, which gets complex to > 10 digits

Unreliability of Fitting Process

= 1+0.033i

= 1.03+0.025i

sum of two peaks

Resolving two overlapping peaks isnear-impossible 6-parameter nonlinear fit

(too many local minima to converge reliably)

Sum of two Lorentzian peaks ()

There is a better way, which gets

complex for both peaksto > 10 digits

Quantum-inspired signal processing (NMR spectroscopy):

Filter-Diagonalization Method (FDM)[ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]

Given time series yn, write:

€

yn = y(nΔt) = ake−iωk nΔt

k

∑

…find complex amplitudes ak & frequencies k

by a simple linear-algebra problem!

Idea: pretend y(t) is autocorrelation of a quantum system:

€

ˆ H ψ = ih∂

∂tψ

say:

€

yn = ψ (0) ψ (nΔt) = ψ (0) ˆ U n ψ (0)

time-∆t evolution-operator:

€

ˆ U = e−i ˆ H Δt / h

Filter-Diagonalization Method (FDM)[ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]

€

yn = ψ (0) ψ (nΔt) = ψ (0) ˆ U n ψ (0)

€

ˆ U = e−i ˆ H Δt / h

We want to diagonalize U: eigenvalues of U are ei∆t

…expand U in basis of |(n∆t)>:

€

Um,n = ψ (mΔt) ˆ U ψ (nΔt) = ψ (0) ˆ U m ˆ U ˆ U n ψ (0) = ym +n +1

Umn given by yn’s — just diagonalize known matrix!

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 known a priori • resolve overlapping peaks • resolution >> Fourier uncertainty

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/

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

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