| 1 |
Maxwell's Equations and Linear Algebra |
Converted Maxwell's equations to Hermitian eigenproblem (for linear, lossless media). Introduced Dirac |H> notation and adjoints. Derived real eigenvalues and orthogonal eigenvectors for Hermitian operators. Showed Maxwell operator to be Hermitian, positive semi-definite, and thus real ω and orthogonal |H>. Compared with quantum mechanics, noted scale invariance. |
 Joannopoulos, John D., Robert D. Meade, and Joshua N. Winn. Photonic Crystals: Molding the Flow of Light. Princeton, NJ: Princeton University Press, July 3, 1995, chapter 2. ISBN: 0691037442. |
| 2 |
Modes of a Metal Box and Mirror Symmetry |
Solved toy example of 1d Hz(x) field between 2 metal plates. Discussed symmetry, and showed that a mirror plane σ implies eigenfields that are even or odd under the mirror plane, corresponding to a mirror-reflection operator that commutes with the eigen-operator. Defined how vector fields rotate, and showed that H is not a vector but a pseudo-vector - it picks up an extra factor of -1 under improper rotations (rotations of a mirror image, which have determinant -1). For this reason the H field "looks" odd when the E field is even, and vice versa, but the fields "really" have the same symmetry.
Began a more complicated example of a field in a 2d metal box - here, there are many symmetry operations and to relate them we must use representation theory. |
Representation Theory Summary (PDF) |
| 3 |
Symmetry Groups, Representation Theory, and Eigenstates |
Representation theory. Proved that eigenstates transform as representations, and showed how this applies to 2d metal box (for which the states fall into three of the five possible representations). Discussed representations, equivalent and irreducible representations, conjugacy classes, and character tables. Showed how to get character table using the orthogonality rules without knowing the representations; found the character table for C4v (symmetry group of square). |
For more information and background on these concepts, and more proofs, see:
Inui, Tetsuro, Yukito Tanabe, and Y. Onodera. Group Theory and Its Applications in Physics (Springer Series in Solid-State Sciences). 2nd corrected ed. New York, NY: Springer-Verlag, February 1996. ISBN: 3540604456.
Or any similarly-titled book, e.g.:
Falicov, L. M. Group Theory and Its Physical Applications. Chicago, IL: University of Chicago Press, June 1966. ISBN: 0226235408. |
| 15 |
Line and Surface Defects in 2d, Numerical Methods Introduction |
Line-defect and surface states in 2d crystals, and projected band diagram. Showed why there is always some terminations that give rise to surface states.
Introduced numerical methods for solving continuous eigenproblems. Discussed choice of basis (finite-difference, planewave, finite element) and reduction to finite matrix via Galerkin method. Talked about scaling of dense matrix methods and sparse/iterative methods for N×N matrices. Showed how iterative methods apply to planewave problem via FFTs.
Began discussing preconditioned conjugate-gradient minimization of the Rayleigh quotient to find the lowest eigenvalues and corresponding eigenvectors. Started with steepest descent, then Newton's method, then preconditioned steepest descent. Didn't quite get to conjugate gradients. |
Suggested Reading: Bai, Z., J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, eds. Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. Philadelphia, PA: SIAM, 2000. ISBN: 0898714710. The full text of this book can be found online.
Numerical Eigensolver Methods for Maxwell's Equation (PDF)
A Longer Talk on Numerical Eigensolver Methods for Maxwell's Equation (PDF) |
| 18 |
Perfectly Matched Layers (PML), Filter Diagonalization |
Continued discussion of PML: talked about numerical reflections and implementation of frequency-dependent ∈ and µ via introduction of auxiliary fields.
Discussed filter-diagonalization methods (FDM) in a Fourier basis, for computing resonant frequencies and loss rates from time-domain simulation.
Began discussion of three-dimensional photonic crystals. Introduced cubic/fcc/diamond lattices and reciprocal lattice/Brillouin zone. Contrasted fcc and diamond lattices of spheres. |
Filter Diagonalization (PDF) |
| 22 |
Bistability in a Nonlinear Filter, Periodic Waveguides |
Analyzed optical bistability in nonlinear filter/cavity.
Introduced hybrid systems combining band gaps and index guiding. 1d-periodic waveguides in 2d and 3d, projected band diagrams, guided modes, and "gaps". |
Solijacic, M., et al. "Optimal bistable switching in nonlinear photonic crystals." Phys Rev E 66, no. 5 (2002): 055601(R) |