18.369, Spring 2009

Mathematical Methods in Nanophotonics

Prof. Steven G. Johnson, Dept. of Mathematics


This is the home page for the 18.369 course at MIT in Spring 2009, where the syllabus, lecture materials, problem sets, and other miscellanea are posted.

You can also download the course announcement flyer, and visit this photonic-crystal tutorial page to find materials for past lectures by SGJ on related subjects. This course was previously offered as 18.325 in Fall 2005 (also on OpenCourseWare) and as 18.369 in Spring 2007 and also in Spring 2008.

Tired of doing electromagnetism like it's 1865?

Find out what solid-state physics has brought to 8.02 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 8.02 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.

For beginning graduate students and advanced undergraduates.


Lectures: MWF 2–3pm (2-102). Office Hours: TR 4:30–5:30 (2-388).

Probable topics: Methods: linear algebra & eigensystems for Maxwell's equations, symmetry groups and representation theory, Bloch's theorem, numerical eigensolver methods, time and frequency-domain computation, perturbation theory, coupled-mode theories, waveguide theory, adiabatic transitions. Optical phenomena: photonic crystals & band gaps, anomalous diffraction, mechanisms for optical confinement, optical fibers (new & old), nonlinearities, integrated optical devices.

Grading: 33% problem sets (weekly/biweekly). 33% mid-term exam (April 6). 34% final project (proposal due April 13, project due May 13).

Books: Photonic Crystals: Molding the Flow of Light (Second Edition). (This book is at an undergraduate level, and 18.369 is somewhat more advanced, but the book should provide a useful foundation.)

Useful (but not required) books in reserve book room: Photonic Crystals: Molding the Flow of Light by Joannopoulos et al. (only the first edition, however). Group Theory and Its Applications in Physics by Inui et al., and Group Theory and Quantum Mechanics by Michael Tinkham.

Final projects: A typical project will be to find some interesting nanophotonic structure/phenomenon in the literature (chapter 10 of the book may be a helpful guide to some possibilities), reproduce it (usually in 2d only, so that the simulations are quick), using the numerical software (Meep and/or MPB) introduced in the course/psets, and extend/analyze it in some further way (try some other variation on the geometry, etc.). Then write up the results in a 5 to 10 page report (in the format of a journal article, with references, figures, a review of related work, etcetera)—reports should be written for a target audience of your classmates in 18.369, and should explain what you are doing at that level. Projects should not be a rehash of work you've already done for your research, (but may be some extension/digression thereof).

Prerequisites: 18.305 or permission of instructor. (Basically, some experience with partial differential equations and linear algebra. e.g. 8.05, 8.07, 6.013, 3.21, 2.062.) This is a graduate-level course aimed at beginning graduate students and suitably advanced undergraduates.

Supplementary lecture notes: Notes on the algebraic structure of wave equations and Notes on Perfectly Matched Layers (PMLs).

Lecture Summaries and Handouts

Lecture 1: 4 Feb 2009

Handouts: syllabus (see also this web page), problem set 1 (due 15 Feb.), collaboration policy

Motivation and introduction: this class is about electromagnetism where the wavelength is neither very large (quasi-static) nor very small (ray optics), and the analytical and computational methods we can use to understand phenomena in materials that are structured on the wavelength scale.

We start by setting up the source-free Maxwell equations as a linear eigenproblem, which will allow us to bring all of the machinery of linear algebra and (eventually) group theory to bear on this problem without having to solve the PDE explicitly (which is usually impossible to do analytically).

Notational introductions: Hilbert spaces (vector space + inner product), notation for inner products and states (magnetic fields etc.). Defined the adjoint (denoted †) of linear operators.

Further reading: See chapter 2 of the textbook. For a more sophisticated treatment of Hilbert spaces, adjoints, and other topics in functional analysis, a good text is Basic Classes of Linear Operators by Gohberg et al.

Lecture 2: 6 Feb 2009

Defined Hermitian operators, and showed that the Maxwell eigen-operator ∇×ε-1∇× is Hermitian for real ε (by showing that ∇× is Hermitian). Proved that Hermitian operators have real eigenvalues and that the eigenvectors are orthogonal (or can be chosen orthogonal, for degeneracies).

Comparison to quantum mechanics; talked about scale invariance, etc.

Further reading: See chapter 2 of the textbook.

Lecture 3: 9 Feb 2009

Simple one-dimensional example of fields in metallic cavity, showed that consequences match predictions from linear algebra. Discussed consequences of symmetry, and in particular showed that mirror symmetry implies even/odd solutions. Discussed subtleties of mirror symmetries for electromagnetism: although the E and H fields seem to have opposite symmetry, they don't, because H is a pseudovector. Defined general rotation operators for vector and pseudovector fields.

Further reading: Chapter 3 of the text.

Lecture 4: 11 Feb 2009

Handouts: representation theory summary

Gave a simple 2d example of fields in a 2d metal box, and showed that the symmetries are more complicated, and may include degeneracies. In order to understand this, we need to understand the relationship of different symmetry operations to one another — this relationship is expressed more precisely by the group of symmetry operators. Defined groups, and group representations, and proved that all eigenfunctions can be chosen to transform as partner functions of an irreducible representation of the symmetry group. As an example, even and odd functions is a mirror-symmetric system correspond to the two irreducible representations of that group.

Defined conjugacy classes, which break the group operations down into subsets that are related by symmetry. Introduced the character table of a group, the table of the traces ("characters") of the irreducible representations (which are constant with a given conjugacy class and representation).

Lecture 5: 13 Feb 2009

Handouts: pset 1 solutions and pset 2 (due in 2 weeks)

Using the rules from the representation theory handout, built up the character table for the symmetry group of the square (called C4v). Then, looked at the eigenfunction solutions that we previously had for this case, and showed how we could classify them into the various irreducible representations. Conversely, showed how, using the character table, we can "guess" what the corresponding eigenmodes must look like (or at least the sign pattern). Predicted a couple of field patterns for modes we hadn't seen yet. Then, showed that some of the apparent double degeneracies are actually accidental, and that we could decompose them into one-dimensional representations, and in fact obtained some of the predicted field patterns.

Lecture 6: 17 Feb 2009

Looked at the projection operator in more detail and gave some graphical examples of how we can use it to decompose a function into partner functions.

Proved in general that the irreducible representation is conserved over time in a linear system, by showing that the projection operator commutes with the time-evolution operator.

Lecture 7: 18 Feb 2009

Defined the time-evolution operator explicitly via an exponentiated operator on the 6-component vector-field (E, H). Showed that the time-evolution operator is unitary in an appropriate inner product, and that this leads to conservation of energy.

Showed that for continuous translational symmetry, the representations are exponential functions exp(ikx) for some number k (possibly complex, but real for unitary represenatations). Concluded that the solutions of Maxwell's equations in empty space are planewaves, and discussed the corresponding dispersion relation.

Further reading: See chapter 3 of the book (section on translational symmetry).

Lecture 8: 20 Feb 2009

Explained how conservation of the exp(-ikx) representation, which gives conservation of k, leads immediately to Snell's law at a flat interface.

Introduced dielectric waveguides, via the simple 2d example of a high-ε region surrounded by a low-ε region, invariant in the x direction. Showed that the solutions far from the waveguide lead to a continuous region, the light cone, and argued (proof to come later) that the higher-ε region pulls down localized guided modes below the light cone. Since they are localized, they form discrete bands as discussed in a previous lecture.

Introduced the variational theorem (or minimax theorem), which arises for any Hermitian eigenproblem. Noted that this result leads to iterative computational methods to find the lowest (and subsequent) eigenvalue.

Further reading: chapter 3 of the book, sections on index guiding and variational theorem.

Lecture 9: 23 Feb 2009

Proved the variational theorem (at least for finite-dimensional spaces), and more generally showed that extrema of the Rayleigh quotient are eigenvalues.

Discussed the variational theorem as it appears for the Maxwell eigenproblem, and its relation to the corresponding theorem in quantum mechanics where it has a physical interpretation as minimizing the sum of kinetic and potential energy.

Used the variational theorem to prove the existence of index-guided modes (in two dimensions, for the TE polarization), for any translation-invariant structure where ε is increased "on average" in a localized region, for an appropriate definition of "on average."

Considered related theorems in quantum mechanics: an arbitrary attractive potential will always localize a bound state in 1d or 2d, but not in 3d, and sketched a simple dimensional argument in 1d and 3d (but not 2d, which is a difficult borderline case). Discussed the related theorem for 3d waveguides (2d localization), and the case of substrates where the theorem does not apply and the fundamental modes has a low-ω cutoff. p>Started by considering a periodic "waveguide" in two dimensions: a sequence of dielectric rods in air. By analogy with ray-optics and total-internal reflection, it seems that this could not support guided modes. However, it does, and to understand this we need to understand discrete translational symmetry.

Further reading: chapter 2 of the textbook, section on variational theorem, and chapter 3 on discrete translation symetry. For a similar theorem in 3d, see Bamberget and Bonnet [J. Math. Anal, 21, 1487 (1990)], also see K. K. Y. Lee, Y. Avniel, and S. G. Johnson, "Rigorous sufficient conditions for index-guided modes in microstructured dielectric waveguides" [Opt. Express 16, p. 9261, 2008] for a further generalization.

Lecture 10: 25 Feb 2009

Showed that the representations of the discrete translation group are again exponentials, and thereby proved Bloch's theorem: the eigenfunctions can be chosen in the form of a planewave multipled by a periodic function.

As a corollary, the Bloch wavevector k is conserved, and explained how this relates to a famous mystery from the 19th century: why electrons in a pure conductor act almost like a dilute gas. Talked a bit about the history of periodic structures in solid-state physics and electromagnetism, from Lord Rayleigh (1887) to Eli Yablonovitch (1987).

Showed that k is periodic, and that k is only preserved up to addition of reciprocal lattice vectors. Sketched out the consequences for the band structure and how a photonic band gap depends on this periodicity.

Discussed reflection (specular and diffractive) from a periodic surface, and minimum-frequency/maximum-wavelength cutoffs for various diffracted orders.

Further reading: Chapter 3 of the book, section on discrete translational symmetry. Chapter 10 in the book, last section (the one on reflection, refraction, and diffraction).

Lecture 11: 27 Feb 2009

Handouts: MPB demo (see also the MPB home page) and example files: 2dwaveguide.ctl and 2dwaveguide-periodic.ctl; problem set 2 solutions.

Gave demo of MPB eigensolver software for 2d dielectric waveguide (add meep on Athena, currently only available on Athena/Linux, e.g. in the clusters or via ssh to linux.mit.edu).

Further reading: The MPB web page, and Appendix D of the course notes (on numerical methods).

Lecture 12: 2 March 2009

Handouts: pset 3 (due Friday, 13 March).

Lecture 17: 13 March 2009

Handouts: pset 3 solutions

Lecture 18: 16 March 2009

Handouts: pset 4 (due Monday, 30 March); see also files bandgap1d.ctl and defect1d.ctl

30 March 2009

Handouts: pset 4 solutions

30 March 2009

Handouts: pset 5 (due Monday, 20 April); see also files sq-rods.ctl, sq-rods.ctl, rod-transmission.ctl