Readings listed below from 'Textbook' are from the required course text: Amazon logo Joannopoulos, John D., Steven G. Johnson, Robert D. Meade, and Joshua N. Winn. Photonic Crystals: Molding the Flow of Light. Princeton, NJ: Princeton University Press, 2008. ISBN: 9780691124568.

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: Amazon logo 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

MPB demo

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 Amazon logo 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

Photonic-crystal fibers

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)