From electrons to photons: Quantum-inspired modeling in nanophotonics

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From electrons to photons Quantum-inspired
modeling in nanophotonics

• Steven G. Johnson, MIT Applied Mathematics

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Nano-photonic media (l-scale)
strange waveguides
microcavities
B. Norris, UMN
Assefa Kolodziejski, MIT
3d structures
Mangan, Corning
synthetic materials
optical phenomena
hollow-core fibers
3
Photonic Crystals
periodic electromagnetic media
can have a band gap optical insulators
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Electronic and Photonic Crystals
atoms in diamond structure
Periodic Medium
Bloch waves Band Diagram
electron energy
wavevector
interacting hard problem
non-interacting easy problem
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Electronic Photonic Modelling
Electronic
Photonic
strongly interacting tricky approximations
non-interacting (or weakly), simple
approximations (finite resolution) any
desired accuracy
lengthscale dependent (from Plancks 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
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Fun with Math
First task get rid of this mess
0
dielectric function e(x) n2(x)
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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 Blochs
theorem
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A 2d Model System
dielectric atom e12 (e.g. Si)
square lattice, period a
a
a
E
TM
H
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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)
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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 of k irreducible Brillouin zone
1
Limit degrees of freedom expand H in finite basis
2
Efficiently solve eigenproblem iterative methods
3
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Solving the Maxwell Eigenproblem 1
Limit range of k irreducible Brillouin zone
1
Blochs 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
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Solving the Maxwell Eigenproblem 2a
Limit range of k 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
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Solving the Maxwell Eigenproblem 2b
Limit range of k 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
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Solving the Maxwell Eigenproblem 3a
Limit range of k irreducible Brillouin zone
1
Limit degrees of freedom expand H in finite basis
2
Efficiently solve eigenproblem iterative methods
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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
(p smallest eigenvalues)
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Solving the Maxwell Eigenproblem 3b
Limit range of k 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
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Solving the Maxwell Eigenproblem 3c
Limit range of k 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 w0
minimizes
minimize by preconditioned conjugate-gradient
(or)
variational theorem
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Band Diagram of 2d Model System(radius 0.2a
rods, e12)
a
frequency w (2pc/a) a / l
G
X
M
G
irreducible Brillouin zone
M
E
gap for n gt 1.751
TM
X
G
H
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Origin of the Band Gap
Hermitian eigenproblems solutions are
orthogonal and satisfy a variational theorem
Electronic
Photonic
field oscillations
minimize kinetic potential energy (e.g.
bonding state)
minimize
field in high e
higher bands orthogonal to lower must
oscillate (high kinetic) or be in low e (high
potential) (e.g. anti-bonding state)
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Origin of Gap in 2d Model System
orthogonal node in high e
Ez
lives in high e
G
X
M
G
Ez
E
gap for n gt 1.751
TM
H
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The Iteration Scheme is Important
(minimizing function of 104108 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 Newtons
method
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The Iteration Scheme is Important
(minimizing function of 40,000 variables)
no preconditioning
error
preconditioned conjugate-gradient
no conjugate-gradient
iterations
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The Boundary Conditions are Tricky
e?
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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 EM numerics
analyses)
resolution (pixels/period)
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Gap, Schmap?
a
frequency w
G
X
M
G
But, what can we do with the gap?
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Intentional defects are good
microcavities
waveguides (wires)
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Intentional defects in 2d
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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
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Time-Domain Eigensolvers(finite-difference
time-domain FDTD)
Simulate Maxwells 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
complex wn
Mandelshtam, J. Chem. Phys. 107, 6756 (1997)
decay rate in time gives loss
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Signal Processing is Tricky
signal processing
complex wn
?
a common approach least-squares fit of spectrum
fit to
FFT
Decaying signal (t)
Lorentzian peak (w)
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Fits and Uncertainty
problem have to run long enough to completely
decay
actual
signal portion
Portion of decaying signal (t)
Unresolved Lorentzian peak (w)
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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 10.033i
w 1.030.025i
Sum of two Lorentzian peaks (w)
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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!
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Filter-Diagonalization Method (FDM)
Mandelshtam, J. Chem. Phys. 107, 6756 (1997)
We want to diagonalize U eigenvalues of U are
eiw?t
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Filter-Diagonalization Summary
Mandelshtam, J. Chem. Phys. 107, 6756 (1997)
Umn given by yns just diagonalize known matrix!
A few omitted steps Generalized eigenvalue
problem (basis not orthogonal) Filter yns
(Fourier transform) small bandwidth smaller
matrix (less singular)
resolves many peaks at once peaks not
known a priori resolve overlapping peaks
resolution gtgt Fourier uncertainty
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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/