Mathematical Methods in Nanophotonics

Readings

Readings listed below from ‘Textbook’ are from the required course text: 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.

LEC #	TOPICS	READINGS
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)
6	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	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
24	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)