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

2. 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. can have a band gap: optical “insulators”
Photonic Crystals
periodic electromagnetic media
1887
1987
can have a band gap: optical “insulators”

4. Electronic and Photonic Crystals
atoms in diamond structure
dielectric spheres, diamond lattice
wavevector
photon frequency
Periodic
Medium
Band Diagram
Bloch waves:
electron energy
wavevector
interacting: hard problem
non-interacting: easy problem

5. Electronic & Photonic Modelling
• 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

6. Fun with Math get rid of this mess First task:
dielectric function e(x) = n2(x)
eigen-operator
eigen-value
eigen-state
+ constraint

7. Electronic & Photonic Eigenproblems
nonlinear eigenproblem
(V depends on e density ||2)
simple linear eigenproblem
(for linear materials)
—many well-known
computational techniques
Hermitian = real E & w, … Periodicity = Bloch’s theorem…

8. A 2d Model System dielectric “atom” square lattice, a a TM
e=12 (e.g. Si)
square lattice,
period a a a E TM H

9. 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)

10. 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 2
Limit degrees of freedom: expand H in finite basis 3
Efficiently solve eigenproblem: iterative methods

11. Solving the Maxwell Eigenproblem: 1
Limit range of k: irreducible Brillouin zone 1
—Bloch’s theorem: solutions are periodic in k M kx ky
first Brillouin zone
= minimum |k| “primitive cell” X G
irreducible Brillouin zone: reduced by symmetry 2
Limit degrees of freedom: expand H in finite basis 3
Efficiently solve eigenproblem: iterative methods

12. Solving the Maxwell Eigenproblem: 2a
Limit range of k: irreducible Brillouin zone 1 2
Limit degrees of freedom: expand H in finite basis (N)
solve:
finite matrix problem: 3
Efficiently solve eigenproblem: iterative methods

13. Solving the Maxwell Eigenproblem: 2b
Limit range of k: irreducible Brillouin zone 1 2
Limit degrees of freedom: expand H in finite basis
— 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) ] 3
Efficiently solve eigenproblem: iterative methods

14. Solving the Maxwell Eigenproblem: 3a
Limit range of k: irreducible Brillouin zone 1 2
Limit degrees of freedom: expand H in finite basis 3
Efficiently solve eigenproblem: iterative methods
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)

15. Solving the Maxwell Eigenproblem: 3b
Limit range of k: irreducible Brillouin zone 1 2
Limit degrees of freedom: expand H in finite basis 3
Efficiently solve eigenproblem: iterative methods
Many iterative methods:
— Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …,
Rayleigh-quotient minimization

16. Solving the Maxwell Eigenproblem: 3c
Limit range of k: irreducible Brillouin zone 1 2
Limit degrees of freedom: expand H in finite basis 3
Efficiently solve eigenproblem: iterative methods
Many iterative methods:
— Arnoldi, Lanczos, Davidson, Jacobi-Davidson, …,
Rayleigh-quotient minimization
for Hermitian matrices, smallest eigenvalue w0 minimizes:
“variational
theorem”
minimize by preconditioned
conjugate-gradient (or…)

17. Band Diagram of 2d Model System (radius 0.2a rods, e=12)
frequency w (2πc/a) = a / l
irreducible Brillouin zone G X M G M E
gap for
n > ~1.75:1 TM X G H

18. 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
Preconditioned conjugate-gradient: minimize (h + a d)
— d is (approximate A-1) [f + (stuff)]

19. The Iteration Scheme is Important
(minimizing function of ~40,000 variables)
no preconditioning
% error
preconditioned
conjugate-gradient
no conjugate-gradient
# iterations

20. The Boundary Conditions are Tricky
E|| is continuous
E is discontinuous
(D = eE is continuous)
Any single scalar e fails:
(mean D) ≠ (any e) (mean E)
Use a tensor e:
E|| E e?

21. The e-averaging is Important
backwards averaging
correct averaging
changes order
of convergence
from ∆x to ∆x2
% error
no averaging
tensor averaging
(similar effects
in other E&M
numerics & analyses)
resolution (pixels/period)

22. Gap, Schmap? a
frequency w G X M G
But, what can we do with the gap?

23. Intentional “defects” are good
microcavities
waveguides (“wires”)

24. Intentional “defects” in 2d
(Same computation, with supercell = many primitive cells)

25. 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

26. 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
signal processing
Response is many
sharp peaks,
one peak per mode
complex wn
[ Mandelshtam,
J. Chem. Phys. 107, 6756 (1997) ]
decay rate in time gives loss

27. Signal Processing is Tricky
complex wn ?
a common approach: least-squares fit of spectrum
fit to:
FFT
Decaying signal (t)
Lorentzian peak (w)

28. Fits and Uncertainty
problem: have to run long enough to completely decay
actual
signal
portion
Portion of decaying signal (t)
Unresolved Lorentzian peak (w)
There is a better way, which gets complex w to > 10 digits

29. 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
There is a better way, which gets
complex w
for both peaks
to > 10 digits
w = i
w = i
Sum of two Lorentzian peaks (w)

30. Given time series yn, write:
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!
Idea: pretend y(t) is autocorrelation of a quantum system:
say:
time-∆t evolution-operator:

31. Filter-Diagonalization Method (FDM)
[ Mandelshtam, J. Chem. Phys. 107, 6756 (1997) ]
We want to diagonalize U: eigenvalues of U are eiw∆t
…expand U in basis of |(n∆t)>:
Umn given by yn’s — just diagonalize known matrix!

32. 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

33. Do try this at home Bloch-mode eigensolver:
Filter-diagonalization:
Photonic-crystal tutorials (+ THIS TALK):
/photons/tutorial/