From electrons to photons: Quantum-inspired modeling in...
Transcript of From electrons to photons: Quantum-inspired modeling in...
From electrons to photons: Quantum-inspired modeling in nanophotonics
Steven G. Johnson, MIT Applied Mathematics
Image removed due to copyright reasons.
Nano-photonic media (λ-scale)
synthetic materials
strange waveguides
3d structures
hollow-core fibersoptical phenomena
& microcavities[B. Norris, UMN] [Assefa & Kolodziejski,
MIT]
[Mangan, Corning]
Image removed due to copyright reasons.
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
Image removed due to copyright reasons. Image removed due to copyright reasons.
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
Image removed due to copyright reasons.
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/