Mathematical methods in nanophotonics

Lectures: 3 sessions / week, 1 hour / session

Tired of doing electromagnetism like it's 1865?

Find out what solid-state physics has brought to Electromagnetism II (8.02 or 8.022) in the last 20 years, in this new course surveying the physics and mathematics of nanophotonics - electromagnetic waves in media structured on the scale of the wavelength.

In this regime, which is the basis for everything from iridescent butterfly wings to distributed-feedback lasers and integrated optical devices to the next generation of optical fibers, the 140-year-old analytical techniques you learned in Electromagnetism II aren't very useful. Instead, we will cover 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.

Advanced Analytic Methods in Science and Engineering (18.305) or permission of instructor. (Basically, some experience with partial differential equations and linear algebra.) This is a graduate-level course aimed at beginning graduate students and suitably advanced undergraduates.

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.
This book is at an undergraduate level, and 18.369 is somewhat more advanced, but the book should provide a useful foundation. You can read the book online.

Useful (but not required) books for this course:

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.

Problem sets are assigned on a roughly weekly basis and are worth one third of the total grade. The collaboration policy on homework is as follows:

First, talk to anyone you wish, and read anything you wish (with the exception of homework solutions from previous terms, which are strictly verboten). In fact, you are encouraged to discuss the course material and the homework problems with your classmates - often the best way to learn something is to force yourself to explain it to someone else. However, before you discuss a homework problem with a classmate or look for related information in some reference, you are expected to first make a solid effort at it on your own.

Second, after you discuss a homework problem with a classmate or read related information in some other reference, you must write up the solution on your own, in your own words, starting from something close to a blank sheet of paper.

There is one midterm exam that covers the first half of the course. There is no final exam, but there is a final project due at the end of the course.

Students must choose a published paper on an interesting photonic-crystal phenomenon, replicate the results of that paper using the MIT numerical software, and then extend the results in some interesting way. Students must write one 5-10 page paper based on their project.

Discrete translations and Bloch's theorem

MPB demo

Problem set 2 due

Problem set 3 out

Problem set 3 due

Problem set 4 out

Problem set 4 due

Problem set 5 out

Problem set 5 due

Problem set 6 out

Cavities in photonic-crystal slabs

Photonic-crystal fibers

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