|1||Maxwell's equations and linear algebra||
Textbook: Chapter 2
For a more sophisticated treatment of Hilbert spaces, adjoints, and other topics in functional analysis, a good text is: Goldberg, Israel, Seymour Goldberg, and Marinus Kaashoek. Basic Classes of Linear Operators. Boston, MA: Birkhauser Verlag, 2004. ISBN: 9783764369309.
Notes on the Algebraic Structure of Wave Equations (PDF)
|2||Modes of a metal box and mirror symmetry||Textbook: Chapter 2, chapter 3 (first section)|
|3||Symmetry groups, representation theory, and eigenstates||
Textbook: Chapter 3 (for a basic overview of the consequences of symmetry)
Refer to Innui or Tinkham for a more in-depth discussion.
|4||Translational symmetry, waves, and conservation laws||Textbook: Chapter 3 (section on translational symmetry)|
|5||Total internal reflection and the variational theorem||Textbook: Chapter 3 (sections on index guiding and variational theorem)|
Discrete translations and Bloch's theorem
Textbook: Chapter 3 (section on discrete translation symmetry)
For a similar theorem in 3d, see Bamberger, A., and A. S. Bonnet. "Mathematical Analysis of the Guided Modes of an Optical Fiber." SIAM J Math Anal 21 (1990): 1487-1510.
|7||Bloch's theorem, time reversal, and diffraction||Textbook: Chapter 3 (sections on mirror symmetry/polarization and time-reversal symmetry)|
|8||Photonic band gaps in 1d, perturbation theory||
Textbook: Chapter 2 (section on perturbations), chapter 4 (introduction, sections on origin of the gap, and final section on omnidirectional reflection), chapter 10 (last section, discusses reflection, refraction, and diffraction)
For the same derivation of perturbation theory, see "time-independent perturbation theory" in any quantum-mechanics textbook.
See any book on optics or advanced electromagnetism for Brewster's angle.
|9||1d band gaps, evanescent modes, and defects|
|10||Waveguides and surface states, omni-directional reflection|
|11||Group velocity and dispersion||Textbook: Chapter 3 (section on phase and group velocity, see footnotes in that section for a derivation of group velocity from this perspective)|
|12||2d periodicity, Brillouin zones, and band diagrams||Textbook: Chapter 5 (2d photonic crystals), appendix B (reciprocal lattice and Brillouin zone)|
|13||Band diagrams of 2d lattices, symmetries, and gaps||
Textbook: Chapter 5
Notes on Coordinate Transforms in Electromagnetism (PDF)
|14||Triangular lattice, complete gaps, and point defects|
|15||Line and surface defects in 2d, numerical methods introduction|
|16||Conjugate-gradient, finite-difference time-domain (FDTD) method|
|17||More FDTD: Yee lattices, accuracy, Von-Neumann stability||Taflove, A., and S. C. Hagness. Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, Inc., 2005. ISBN: 9781580538329.|
|18||Perfectly matched layers (PML), filter diagonalization||Notes on Perfectly Matched Layers (PDF)|
|19||3d photonic crystals and lattices||
Textbook: Chapter 8
Watts, M. R., S. G. Johnson, H. A. Haus, and J. D. Joannopoulos. "Electromagnetic Cavity with Arbitrary Q and Small Modal Volume without a Complete Photonic Bandgap." Optics Letters 27 (2002): 1785-1787.
|20||Haus coupled-mode theory, resonance, and Q||Textbook: Chapter 10|
|21||Coupled-mode theory with losses, splitter / bend / crossing / filter devices|
|22||Bistability in a nonlinear filter, periodic waveguides|
|23||Photonic-crystal slabs: gaps, guided modes, waveguides||Textbook: Chapter 8|
Cavities in photonic-crystal slabs
|25||Hollow-core and solid-core photonic-bandgap fibers||Textbook: Chapter 9 (sections on index-guiding holey fibers, hollow-core holey fibers, and Bragg fibers)|