From electrons to photons: Quantum-inspired modeling in...

From electrons to photons: Quantum-inspired modeling in nanophotonics

Steven G. Johnson, MIT Applied Mathematics

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Nano-photonic media (λ-scale)

synthetic materials

strange waveguides

3d structures

hollow-core fibersoptical phenomena

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

MIT]

[Mangan, Corning]

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Photonic Crystalsperiodic electromagnetic media

1887 19872-D�

periodic in�two directions�

3-D�

periodic in�three directions�

1-D�

periodic in�one direction�

can have a band gap: optical “insulators”

Electronic and Photonic Crystalsatoms in diamond structure

wavevector

dielectric spheres, diamond lattice

wavevector

phot

on fr

eque

ncy

elec

tron

ener

gy

Peri

odic

Med

ium

Ban

d D

iagr

amB

loch

wav

es:

interacting: hard problem non-interacting: easy problem

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Electronic & Photonic ModellingElectronic 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 ⇒ λ ×10

Option 1: Numerical “experiments” — discretize time & space … go

Option 2: Map possible states & interactionsusing symmetries and conservation laws: band diagram

Fun with Math

r ∇ ×

r E = − 1

c∂∂t

r H = i ω

c

r H

r ∇ ×

r H = ε

1c

∂∂t

r E +

r J = i

ωc

εr E

0

dielectric function ε(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

PhotonicElectronic

−

h2

2m∇2 + V

⎛

⎝ ⎜

⎞

⎠ ⎟ ψ = Eψ

∇ ×

1ε

∇ ×r

H =ωc

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2 r H

simple linear eigenproblem(for linear materials)

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

—many well-knowncomputational techniques

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

A 2d Model System

dielectric “atom”ε=12 (e.g. Si)

square lattice,period a

a

a

ETMH

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:

planewaveperiodic “envelope”

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

Corollary 2: given by finite unit cell,so ω are discrete ωn(k)

r H r k

Solving the Maxwell Eigenproblem

∇ + ik( )×1ε

∇ + ik( )× Hn =ωn

2

c 2 Hn

∇ + ik( )⋅ H = 0constraint:

Finite cell discrete eigenvalues ωn

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

00.10.20.30.40.50.60.70.80.91

Photonic Band Gap

TM bands

where: H(x,y) ei(k⋅x – ωt)

Limit range of k: irreducible Brillouin zone1

2 Limit degrees of freedom: expand H in finite basis

3 Efficiently solve eigenproblem: iterative methods

Solving the Maxwell Eigenproblem: 1Limit range of k: irreducible Brillouin zone1

—Bloch’s theorem: solutions are periodic in k

kx

ky2

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

πaΓ

M

X

irreducible Brillouin zone: reduced by symmetry

2 Limit degrees of freedom: expand H in finite basis

3 Efficiently solve eigenproblem: iterative methods

Solving the Maxwell Eigenproblem: 2aLimit range of k: irreducible Brillouin zone1

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

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

N

∑ ˆ A H = ω 2 Hsolve:

Ah = ω 2Bhfinite matrix problem:

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

3 Efficiently solve eigenproblem: iterative methods

Solving the Maxwell Eigenproblem: 2bLimit range of k: irreducible Brillouin zone1

2 Limit degrees of freedom: expand H in finite basis

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

Finite-element basisPlanewave (FFT) basisconstraint, boundary conditions:

H(x t ) = HGeiG⋅xt

G∑ Nédélec elements

[ Nédélec, Numerische Math.35, 315 (1980) ]HG ⋅ G + k( )= 0constraint:

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) ]

3 Efficiently solve eigenproblem: iterative methods

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Solving the Maxwell Eigenproblem: 3aLimit range of k: irreducible Brillouin zone1

2 Limit degrees of freedom: expand H in finite basis

3 Efficiently solve eigenproblem: iterative methods

Ah = ω 2BhSlow 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

(p smallest eigenvalues)

Solving the Maxwell Eigenproblem: 3bLimit range of k: irreducible Brillouin zone1

2 Limit degrees of freedom: expand H in finite basis

3 Efficiently solve eigenproblem: iterative methods

Ah = ω 2BhMany iterative methods:

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

Solving the Maxwell Eigenproblem: 3cLimit range of k: irreducible Brillouin zone1

2 Limit degrees of freedom: expand H in finite basis

3 Efficiently solve eigenproblem: iterative methods

Ah = ω 2BhMany iterative methods:

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

for Hermitian matrices, smallest eigenvalue ω0 minimizes:

ω02 = min

h

h' Ahh' Bh

minimize by preconditionedconjugate-gradient (or…)

“variationaltheorem”

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

freq

uenc

yω

(2πc

/a)

= a

/ λ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Photonic Band Gap

TM bands

MΓ X M Γirreducible Brillouin zone

Γ r k

E gap forn > ~1.75:1

TMXH

Origin of the Band GapHermitian eigenproblems:

solutions are orthogonal and satisfy a variational theorem

Electronic Photonic

field oscillationsfield in high ε

• minimize kinetic + potential energy

(e.g. “bonding” state)

• minimize:

ω 2 = minr E

∇ ×r E

2

∫ε

r E

2

∫c 2

• higher bands orthogonal to lower —must oscillate (high kinetic) or be in low ε (high potential)

(e.g. “anti-bonding” state)

Origin of Gap in 2d Model System

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Photonic Band Gap

TM bands

E

HTM

Γ X M Γ

Ez

– +

Ez

lives in high ε

orthogonal: node in high ε

gap forn > ~1.75:1

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

ω02 = min

h

h' Ahh'Bh

= f (h)

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

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

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

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

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

E E E E E E E E E EE EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE

Ñ

ÑÑ

Ñ Ñ Ñ Ñ Ñ Ñ ÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑÑ

J

JJ

JJ

JJ

J J J J J J J JJ JJJJJ

J

JJJJJJ

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

10

100

1000

10000

100000

1000000

1 10 100 1000

% e

rror

preconditionedconjugate-gradient no conjugate-gradient

no preconditioning

# iterations

The Boundary Conditions are TrickyE|| is continuous

E⊥ is discontinuous(D⊥ = εE⊥ is continuous)

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

Use a tensor ε:ε

ε

ε−1 −1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

E||

E⊥

ε?

The ε-averaging is Important

B B

B

B

B

B

B

B

BB

B

B

B

J

JJ

J

J J

JJ J

J J

J J

HH

H H HH

H H HH

HH H

0.01

0.1

1

10

100

10 100

resolution (pixels/period)

% e

rror

backwards averaging

tensor averaging

no averaging

correct averagingchanges orderof convergencefrom ∆x to ∆x2

(similar effectsin other E&M

numerics & analyses)

Gap, Schmap?

freq

uenc

yω

Γ X M Γ0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Photonic Band Gap

TM bands

a

But, what can we do with the gap?

Intentional “defects” are good

microcavities waveguides (“wires”)

Intentional “defects” in 2d

a

(Same computation, with supercell = many primitive cells)

Microcavity Blues

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

• Best methods compute lowest-ω bands,but Nd supercells have Nd modesbelow 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

∆ω

Response is manysharp peaks,

one peak per mode

signal processingcomplex ωn [ Mandelshtam,

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

decay rate in time gives loss

Signal Processing is Trickysignal processing

complex ωn

?

EEEEEE

E

E

E

E

E

E

E

EEEEEEEEEEEEEEEEEEEEEEEEEEE0

50

100

150

200

250

300

350

400

450

0 0.5 1 1.5 2 2.5 3 3.5 4

EE

E

E

E

E

E

E

EEEEE

E

E

E

EEEEEE

EEEEEEEEEEEE

EEEEEEEEE

EEEEEEEEEEEEEEEEEEEEEEE

EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 1 2 3 4 5 6 7 8 9 10

FFT

a common approach: least-squares fit of spectrum

fit to:A

(ω −ω0)2 + Γ2

Decaying signal (t) Lorentzian peak (ω)

Fits and Uncertaintyproblem: have to run long enough to completely decay

E E E E E

E

E E E E E0

5000

10000

15000

20000

25000

30000

35000

40000

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

EE

E

E

E

E

E

E

E

EEE

E

E

E

E

E

E

E

EEE

E

E

E

E

E

E

E

EEE

E

E

E

E

E

E

EEEEE

E

E

E

E

E

EEEEE

E

E

E

E

E

EEEE

E

E

E

E

E

E

EEEEE

E

E

E

E

E

EEEE

E

E

E

E

E

E

EEEEE

E

E

E

E

E

EEEE

E

E

E

E

E

E

EEEE

E

E

E

E

E

E

EEEEE

E

E

E

E

EEEEE

E

E

E

E

E

EEEEEEE

E

E

E

EEEEE

EE

E

E

E

EEEEE

EE

E

E

E

EEEEE

EE

E

E

EEEEEE

EEE

E

EEEEEEEEEE

EEEE

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10

actual

signalportion

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

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

Unreliability of Fitting ProcessResolving two overlapping peaks is

near-impossible 6-parameter nonlinear fit(too many local minima to converge reliably)

E E E E E E E E E E E E E E E E E E E EE

E

E

E

E

E E

E

E

E

EE

E E E E E E E E E E E E E E E E E E E0

200

400

600

800

1000

1200

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

ω = 1+0.033i

ω = 1.03+0.025i

sum of two peaks

Sum of two Lorentzian peaks (ω)

There is a better way, which gets

complex ωfor both peaksto > 10 digits

Quantum-inspired signal processing (NMR spectroscopy):

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

yn = y(n∆t) = ake−iω k n∆t

k∑Given time series yn, write:

…find complex amplitudes ak & frequencies ωkby 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(n∆t) = ψ(0) ˆ U n ψ(0)

time-∆t evolution-operator: ̂ U = e−i ˆ H ∆t / h

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

yn = ψ(0) ψ(n∆t) = ψ(0) ˆ U n ψ(0) ̂ U = e−i ˆ H ∆t / h

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

…expand U in basis of |ψ(n∆t)>:

Um,n = ψ(m∆t) ˆ U ψ(n∆t) = 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/