Mathematical Methods in Nanophotonics
High-level approaches to understanding complex optical media, structured on the scale of the wavelength, that are not generally analytically soluable. The basis for understanding optical phenomena such as photonic crystals and band gaps, anomalous diffraction, mechanisms for optical confinement, optical fibers (new and old), nonlinearities, and integrated optical devices. Methods covered include linear algebra and eigensystems for Maxwell's equations, symmetry groups and representation theory, Bloch's theorem, numerical eigensolver methods, time and frequency-domain computation, perturbation theory, and coupled-mode theories.
Spring 2008
Find out what solid-state physics has brought to Electromagnetism in the last 20 years. This course surveys the physics and mathematics of nanophotonics—electromagnetic waves in media structured on the scale of the wavelength. Topics include computational methods combined with high-level algebraic techniques borrowed from solid-state quantum mechanics: linear algebra and eigensystems, group theory, Bloch's theorem and conservation laws, perturbation methods, and coupled-mode theories, to understand surprising optical phenomena from band gaps to slow light to nonlinear filters. Note: An earlier version of this course was published on OCW as 18.325 Topics in Applied Mathematics: Mathematical Methods in Nanophotonics, Fall 2005.
Readings
- 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: 9783540604457.
- Tinkham, Michael. Group Theory and Quantum Mechanics . Mineola, NY: Dover Publications, 2003. ISBN: 9780486432472.
- 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.
- 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.
- LEC #: 2 · TOPICS: Modes of a metal box and mirror symmetry · READINGS: Textbook: Chapter 2, chapter 3 (first section)
- Textbook: Chapter 3 (for a basic overview of the consequences of symmetry)
- LEC #: 4 · TOPICS: Translational symmetry, waves, and conservation laws · READINGS: Textbook: Chapter 3 (section on translational symmetry)
- LEC #: 5 · TOPICS: Total internal reflection and the variational theorem · READINGS: Textbook: Chapter 3 (sections on index guiding and variational 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.
- LEC #: 7 · TOPICS: Bloch’s theorem, time reversal, and diffraction · READINGS: Textbook: Chapter 3 (sections on mirror symmetry/polarization and time-reversal symmetry)
- 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.
- LEC #: 11 · TOPICS: Group velocity and dispersion · READINGS: Textbook: Chapter 3 (section on phase and group velocity, see footnotes in that section for a derivation of group velocity from this perspective)
- LEC #: 12 · TOPICS: 2d periodicity, Brillouin zones, and band diagrams · READINGS: Textbook: Chapter 5 (2d photonic crystals), appendix B (reciprocal lattice and Brillouin zone)
- Textbook: Chapter 5
- LEC #: 17 · TOPICS: More FDTD: Yee lattices, accuracy, Von-Neumann stability · READINGS: Taflove, A., and S. C. Hagness. Computational Electrodynamics: The Finite-Difference Time-Domain Method . Norwood, MA: Artech House, Inc., 2005. ISBN: 9781580538329.
- 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.
- LEC #: 20 · TOPICS: Haus coupled-mode theory, resonance, and Q · READINGS: Textbook: Chapter 10
- LEC #: 23 · TOPICS: Photonic-crystal slabs: gaps, guided modes, waveguides · READINGS: Textbook: Chapter 8
- LEC #: 25 · TOPICS: Hollow-core and solid-core photonic-bandgap fibers · READINGS: Textbook: Chapter 9 (sections on index-guiding holey fibers, hollow-core holey fibers, and Bragg fibers)
Prerequisites & course information
Same subject as 8.315[J] Prereq: 8.07 , 18.303 , or permission of instructor Acad Year 2027-2028: G (Spring) Acad Year 2028-2029: Not offered 3-0-9 units
Sources
- Course catalogue ↗2026–2027
- MIT OpenCourseWare ↗Spring 2008
- MIT OCW syllabus ↗
- MIT OCW readings ↗
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.