18.369, Spring 2010

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 2010, 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, Spring 2008, and Spring 2009.

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 (2-136). Office Hours: Thurs. 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 7, see below). 34% final project (proposal due April 14, project due May 12).

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 in previous terms 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), and several other PDF files that will be made available as the term progresses.

Previous mid-terms: fall 2005 and solutions, spring 2007 and solutions, spring 2008, spring 2009 and solutions.

Lecture Summaries and Handouts

Lecture 1: 3 Feb 2010

Handouts: syllabus (this web page), introductory slides, problem set 1 (due 12 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. In that situation, there are very few cases that can be solved analytically, but lots of interesting phenomena that we can derive from the structure of the equations.

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: 5 Feb 2010

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). The Maxwell operator is also positive semidefinite, and it follows that the eigenfrequencies are real.

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

Further reading: See chapter 2 of the textbook.

Lecture 3: 8 Feb 2010

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: 10 Feb 2010

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 in a mirror-symmetric system correspond to the two irreducible representations of that group.

Lecture 5: 11 Feb 2010

Handouts: pset 1 solutions, pset 2 (due Friday 26 February).

Define conjugacy classes, which break the group operations down into subsets that are related by symmetry. Introduce 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).

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

Lecture 6: 16 Feb 2010

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.

Showed that for continuous translational symmetry, the representations are exponential functions exp(ikx) for some number k (real for unitary representations; in weird cases, k may be a nondiagonalizable matrix with imaginary eigenvalues, but these solutions are not needed in periodic or translationally invariant systems). Concluded that the solutions of Maxwell's equations in empty space are planewaves, and discussed the corresponding dispersion relation.

Further reading: The Inui and Tinkham texts have more information on projection operators (both on reserve at the library). To get exponential functions from representations, I relied on the fact that any nonzero (anywhere continuous) function f(x) with f(x+y)=f(x)f(y) must be an exponential; proofs of this are summarized on Wikipedia. See also chapter 3 of the textbook for a more basic discussion of translational symmetry. References on cases where you cannot ignore the possibility of polynomially growing generalized eigenfunctions can be found in this paper by Klein et al.

Lecture 7: 17 Feb 2010

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.

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

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.

Derived Poynting's theorem in order to define electromagnetic energy and flux in general, and showed that we got the same quantity as we did from unitarity. For time-harmonic fields, showed that |E|2/2 and |H|2/2 and Re[E*×H]/2 are time averages of the corresponding real oscillating fields Re(E) and Re(H). Showed that the time-average energies in the E and H fields are the same.

Further reading: See my Notes on the algebraic structure of wave equations for a general discussion of many wave equations, showing that they share the common form dψ/dt D ψ where D is anti-Hermitian. For Poynting's theorem, see any graduate-level book on electromagnetism, e.g. Jackson's Classical Electrodynamics. The result is summarized in chapter 2 of the textbook. Beware that matters are more complicated for dispersive media (ones in which ε and μ depend on ω), as discussed in Jackson.

Lecture 8: 19 Feb 2010

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 min–max theorem), which arises for any Hermitian eigenproblem. Proved the variational theorem (at least for finite-dimensional spaces), and more generally discussed the derivation (in chapter 2 of the book) that all 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."

Further reading: chapter 2-3 of the book, sections on index guiding and variational theorem.

Lecture 9: 22 Feb 2010

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.

Discrete translational symmetry:

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 (chapter 7 of the book), and to understand this we need to understand discrete translational symmetry.

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: electrons in a pure conductor act almost like a dilute gas, because they scatter only from impurities/imperfections that break the periodicity.

Further reading: chapter 2 of the textbook, section on variational theorem, and chapter 3 on discrete translation symmetry. For a similar theorem regarding 2d localization in 3d waveguides, 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: 24 Feb 2010

Derived the periodicity of the Bloch wavevector k in one dimension. Adding 2π/a does not change the irrep, and is only a relabeling of the eigensolution. The conservation of the irrep now corresponds to conservation of k up to multiples of 2π/a. As an application, discussed reflection (specular and diffractive) from a periodic surface, and minimum-frequency/maximum-wavelength cutoffs for various diffracted orders.

Briefly discussed band diagram and guided modes for periodic waveguide in 2d (a sequence of dielectric rods in air). We will return to this in more detail in chapter 7 of the book. Discussed the concept of the (first) Brillouin zone, and the irreducible Brillouin zone, although a more general definition will have to wait until we get to 2d periodicity (chapter 5).

Considered interaction of rotational symmetries with k: showed that rotations R transform a solution at k into solution at Rk, and hence ω(k)=ω(Rk).

By conjugating the eigenequation, for real ε, showed that ω(k)=ω(−k) in general, even for structures without mirror symmetry. Connected this to time-reversal symmetry: the conjugated mode corresponds to running time backwards, which still solves the same Maxwell's equation. One way to break time-reversal symmetry is by introducing absorption loss (complex ε, which time-reverses into gain). Alternatively, briefly mentioned magneto-optic materials (complex-Hermitian ε, neglecting absorption) and why a static magnetic field can (locally) break time-reversal symmetry, and of use for Faraday isolators.

Further reading: Chapter 3 of the textbook. Diffractive reflections are discussed at the end of chatper 10 in the textbook. For an interesting application of magneto-optic materials, which break time-reversal symmetry, see this page on one-way waveguides.

Lecture 11: 26 Feb 2010

Handouts: problem set 3 (due March 12).

Discussed the consequence of mirror symmetry in 2d: modes separate into two polarizations, TM (Hx, Hy, Ez) and TE (Ex, Ey, Hz).

Began new topic: photonic band gaps in one dimension.

First, gave overview of history (starting with Lord Rayleigh, 1887) and applications. Then, sketched band structure and identified gaps.

Origin of the photonic band gap in 1d: starting with uniform medium, considered qualitatively what happens when a periodic variation in the dielectric constant is included. First, the bands "fold" onto the Brillouin zone, which is just a relabelling in the uniform medium. Second, the degeneracy at the edge of the Brillouin zone is broken because one linear combination (cosine) is more concentrated in the high-ε material than another linear combination (sine). Thus, any infinesimal periodicity opens a (possibly small) gap.

Further reading: Chapter 3 in the book, sections on mirror symmetry/polarization and time-reversal symmetry. Chapter 4 in the book, introduction and sections on origin of the gap.

Lecture 12: 1 Mar 2010

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-x86, e.g. in the clusters or via ssh to linux.mit.edu).

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

Lecture 13: 3 Mar 2010

A quantitative estimate of the size of the band gap in 1d, via perturbation theory. In particular, derived first-order perturbation theory for the eigenvalue of any Hermitian operator with some small change, by expanding the eigenvalue and eigenfunction as power series in the change and solving order-by-order. We then write down this perturbative expression for the Maxwell operator, and see that the fractional change in frequency is just the fractional change in index multiplied by the fraction of electric-field energy in the changed material.

Using first-order perturbation theory, computed the size of the band gap for a 1d periodic structure to first order in Δε. Defined the "size" of the gap in a dimensionless way as a fraction of mid-gap.

Discussed optimum parameters at low-index-contrast, and generalized to "quarter-wave condition" to maximize gap for arbitrary index contrast.

Further reading: For the same derivation of perturbation theory, see "time-independent perturbation theory" in any quantum-mechanics text, e.g. Cohen-Tannoudji. See also the section on small perturbations in chapter 2 of the book. See chapter 4 of the book on the origin of the 1d gap, and on the special formulas for quarter-wave stacks in 1d (discussed in more detail in Yeh's Optical Waves in Layered Media).

Lecture 14: 5 Mar 2010

Handout: Revised representation theory handout, including item at end about product representations.

Degenenerate perturbation theory: noted that I actually "cheated" in the previous calculation because in deriving first-order perturbation theory I had assumed a unique expression for the unperturbed mode (up to constant factors), i.e. a non-degenerate eigenfunction. For the k-fold degenerate case, we actually have to solve a small k×k eigenproblem first to diagonalize the perturbation, although we can often do this by symmetry.

More generally, discussed product representation theory and the origin of selection rules in perturbative expressions of this sort (for integrals of products of three partner functions of various irreps).

More discussion of localization of modes by defects in 1d crystals, discussing how a positive Δε "pulls down" a mode from the upper edge of the gap, and a negative Δε "pushes up" a mode from the lower edge. A bit of discussion of the general case and the importance of dimensionality.

Began discussing out-of-plane propagation; immediate consequence is that TE and TM modes separate, and now we have both kx and ky.

Further reading: Degenerate perturbation theory is derived in most quantum texts (e.g. Cohen-Tannoudji). See chapter 4 of the text on defect modes, and section on out-of-plane propagation. For a variational proof of localization by defects in gaps for Schrodinger's equations, see our recent paper.

Lecture 15: 8 Mar 2010

Handout: projected TM band diagram from multilayer film (corrected from figure 10 of chapter 4 in the book).

Off-axis propagation, projected band diagrams for multilayer films, Fabry-Perot defect modes, and surface states.

Further reading: See chapter 4 of the book.

Lecture 16: 10 Mar 2010

Handout: TE/TM projected band diagram and omnidirectional reflection (from book chapter 4, figure 15)

Omnidirectional reflection: sketched TM/TE projected band diagram for multilayer film and identified the possibility of a range of omnidirectional reflection from air (i.e. a range of 100% reflection for all incident angles and polarizations of incident propagating waves, as long as translational symmetry is not broken). Identified the two key criteria that the index contrast be large enough and that the lower of the two mirror indices be larger than that of the ambient medium (air). Explained how the latter condition, and the odd shape of the TE projected band diagram, arise from Brewster's angle.

Wave propagation velocity: defined phase velocity (along homogeneous directions) and group velocity. Explained why phase velocity is not uniquely defined in a periodic medium (and even in a uniform waveguide it can easily be infinite). Showed that group velocity is the velocity of propagation of wave packets, by considering a narrow-bandwidth packet and Taylor-expanding the dispersion ω(k) to first order. Another viewpoint is that group velocity is the energy-propagation velocity (in a lossless medium), and explained the general principle that the velocity of any "stuff" can be expressed as the ratio of the flux rate of the stuff to the density of the stuff...our task in the next lecture will be to derive this ratio for the group velocity. Began the derivation by relating the derivative of ω to the derivative of the operator Θk, which is equivalent to the Hellman–Feynman theorem of quantum mechanics or to first-order perturbation theory (exact for infinitesimal perturbations) or to "k-dot-p theory" of solid-state physics.

Further reading: See chapter 4 of the book, final section on omnidirectional reflection; see any book on optics or advanced electromagnetism for Brewster's angle (e.g. Jackson or Hecht). See chapter 3 of the book, section on phase and group velocity. See the footnotes in that section, e.g. Jackson, Classical Electrodynamics, for a derivation of group velocity from this perspective and other information.

Lecture 17: 12 Mar 2010

Handouts: problem set 3 solutions and problem set 4 (due March 29); MPB example files bandgap1d.ctl and defect1d.ctl.

Applied the Hellmann-Feynman theorem to our Θk eigenproblem to show that the group velocity dω/dk is precisely the energy velocity (ratio of energy flux to energy density, averaged over time and the unit cell). Proved that this group velocity is always ≤c for ε≥1.

Discussed cases in which the group velocity can be greater than c: lossy media (complex ε) and evanescent waves (complex k). In these cases, however, the "group velocity" does not correspond to a velocity of energy/information transport. Showed how group velocity relates to phase delay dφ/dω and gave a complex-k example in which this gives a deceptive "superluminal" result of "zero" time delay.

New topic: 2d photonic crystals. Defined the primitive lattice vectors, the Bravais lattice, and the reciprocal lattice. Discussed relationship of the reciprocal lattice to a generalized Fourier series, and reinterpreted the periodicity of solutions in reciprocal space via the Fourier-series expansion of the Bloch envelope.

Further reading: chapter 3 of the book (section on velocity). Appendix B of the book on the reciprocal lattice.

Lecture 18: 15 Mar 2010

Handouts: triangular-lattice Brillouin zone (from appendix B of the book)

Discussed group-velocity dispersion, qualitatively, and defined the dispersion parameter D that characterizes the rate of pulse spreading; you will investigate this more quantitatively in homework. Brief discussion of dispersion compensation.

New topic: the Brillouin zone(s).

In 1d, we already saw the simplest example of a Brillouin zone, the interval [-π/a,+π/a]. Showed that in the square lattice, things are similarly simple: the natural Brillouin zone is just a square "unit cell" centered on the origin, with diameter 2π/a. Showed how the symmetries of the structure can reduce this to an "irreducible Brillouin zone" (I.B.Z) that is just a triangle, and gave the canonical Γ/X/M names for the corners of this triangle. Pointed out that there are four equivalent M points and two equivalent X points, by periodicity in k space; there is also a Y point that is the 90-degree rotation of the X point, whose solutions are related (in a symmetric structure) but are not the same as at X.

Began more careful discussion of Brillouin zones, by looking at the triangular lattice. Defined lattice vectors, found reciprocal lattice vectors, and showed that the reciprocial lattice is also triangular but rotated 30°. Noted that the "unit cell" of the lattice, however it is chosen, does not have the full symmetry, motivating us to seek a better definition of the first Brillouin zone.

Showed how to construct the first Brillouin zone (and the second Brillouin zone, etc.) via perpendicular bisectors between reciprocal lattice points. (The generalization of this to non-periodic structues is called a Voronoi cell, and in the real lattice it is called a Wigner–Seitz cell.) Showed that B.Z. contains no equivalent k points (not including the B.Z. boundaries), and all inequivalent k points (if you include the B.Z. boundaries). Showed that the B.Z. has the full symmetry of the point group. We can therefore construct the irreducible Brillouin zone (I.B.Z.), which is the B.Z. reduced by all of the symmetries in the point group (+ time reversal), and are the only k we need to consider.

Gave the examples of the square-lattice B.Z. and the triangular-lattice B.Z., constructed in this way, and reduced the latter to the I.B.Z. for a 6-fold symmetrical (C6v) structure.

Further reading: beginning of chapter 5 of the book (2d photonic crystals), and appendix B on the reciprocal lattice and Brillouin zone. For a discussion of dispersion in telecommunications systems, a good references is Ramaswami and Sivarajan, Optical Networks: A Practical Perspective (Academic Press, 1998).

Lecture 19: 17 Mar 2010

Handout: 2d-crystal figures (from chapter 5 of the textbook).

Considered the TM band diagram of the square lattice of rods (figure 1 of the handout). Discussed the origin of the gap from the variational theorem (explaining the band-edge field patterns in figure 2), and the reason for a minimum index contrast to get a gap (the differing periodicities and hence differing gaps in different directions).

Considered the space group at various k points in the I.B.Z., where k breaks some of the symmetry. Showed that Γ and M have the full symmetry of the lattice, whereas X has a reduced symmetry group. Furthermore, from the symmetry of the points between Γ and M or Γ and X, explained why we have zero group velocity at the X and M points, and why the local maxima (usually) lie along the I.B.Z. boundaries.

Point-defect states in the square lattice of rods. Either decreasing the radius of a rod to push up a "monopole" state, or increasing the radius of a rod to pull down a "dipole" state. Showed how we can easily predict the qualitative field patterns and symmetries from the corresponding bands of the bulk crystal.

Further reading: textbook, chapter 5.

Lecture 20: 19 Mar 2010

Further discussion of point defects: related the defect modes to the 5 irreps of the C4v symmetry group, and showed how we can easily guess the field patterns and degeneracies that we will get (at least for low-order modes in defects that are not too big).

Discussed losses of point-defect states surrounded by finite crystals. Showed that the solutions must be (approximately) exponentially decaying in time, and that the exponential decay rate (the loss rate) in turn decreases exponentially with the number of crystal periods. In this sense, it behaves almost like a solution with a complex eigenfrequency...but of course, there are no such solutions: it is really a (Lorentzian) superposition of real-ω non-localized modes. Briefly discussed the concept of "leaky modes" as a pole in the complex plane.

Line-defect states and waveguides in 2d photonic crystals. Projected band diagrams for the line defect, and the guided mode. Emphasize differences from index-guiding (can guide in air) and Fabry-Perot waveguides (even if we break translational symmetry, light can only scatter forwards or back—the waveguide effectively forms a one-dimensional system).

Surface states in 2d crystals. Began discussing diffraction/reflection/refraction at interfaces, and introduced the need for an isofrequency diagram (contour plot of ω in reciprocal space).

Further reading: textbook, chapter 5. Diffraction/refraction at interfaces is discussed at the end of chapter 10.

Lecture 21: 29 Mar 2010

Handouts: figure 14 and figure 15 from book chapter 10; pset 4 solutions

Discussed reflection/diffraction/refraction at 2d crystal interfaces, following closely the treatment at the end of chapter 10 in the book. Relationship of isofrequency diagrams, group velocity, and conservation of k||. Briefly discussed negative refraction, flat-lens imaging, supercollimation. Talked a little about metamaterials, in the limit λ>>a, where the crystal can be replaced by a homogenized effective medium; for the most part this course deals with the regime where λ is comparable to a.

Going back to band gaps, pointed out that there is no TE gap for this structure covering all wavevectors. The reason has to do with the boundary conditions on the electric field: showed that at an interface, the parallel component of E is continuous while the perpendicular component of D is continuous. Using this fact, for the TE polarization, where the field lines cross a dielectric interface, the field energy is "pushed out" of the dielectric, which lowers variational denominator. This makes it more difficult to get a large contrast (gap) between bands than for the TM polarization.

Began talking a little about dual structures of holes where the TE bands do have a gap.

Further reading: chapter 10 ("Reflection, Refraction, and Diffraction" section), and chapter 5 (on TM vs. TE gaps). See also the atlas of band gaps in appendix C.

Lecture 22: 31 Mar 2010

Briefly discussed structures that have a TE gap, and the triangular lattice-of-holes structure that has an overlapping TE+TM gap.

Briefly discussed three-dimensional photonic crystals: similar mathematical concepts, but no TE/TM distinction, so gaps are harder to get and require more intricate structures. At this point, the design of 3d crystal structures is mostly dominated by fabrication questions; each fabrication technology gives rise to different structures. Conceptually and mathematically, however, not much else changes, so I'm going to skip further discussion of 3d crystals for the purposes of 18.369.

New topic: Computational photonics. Began by categorizing computational methods along three axes: what problem is solved, what basis/discretization is used to reduce the problem to finitely many unknowns, and how are the resulting finitely many equations solved? Discussed three categories of problems: full time-dependent Maxwell solvers, responses to time-harmonic currents J(x) e-iωt, and eigenproblems (finding ω from k or vice-versa). Discussed three categories of basis/discretization: finite differences, finite elements, and spectral methods. Also discussed (briefly) the analogous bases in integral-equation methods (e.g. boundary elements). Emphasized that there is no "best" method; each method has its strengths and weaknesses, and there are often strong tradeoffs (e.g. between generality/simplicity and efficiency).

Further reading: Chapter 5 (TE gap structures) and chapter 6 (3d gaps). See appendix D on computational photonics.

Lecture 23: 2 Apr 2010

Explained the Galerkin method to turn linear differential/integral equation, plus a finite-basis approximation, into a finite set of N equations in N unknowns. Showed that Galerkin methods preserve nice properties like positive-definiteness and Hermitian-ness, but generally turn ordinary eigenproblems into generalized ones (unless you happen to have an orthonormal basis).

Talked about solving the frequency-domain eigenproblem in a planewave (spectral) basis, ala MPB. One big motivation for using a planewave basis is that it makes it trivial to enforce the transversality constraint (∇ċH=0), which is diagonal in Fourier space. Derived the Fourier-series representation of the Θk operator on the unit cell.

In order to solve this equation, we could simply throw it directly at Matlab or LAPACK (LAPACK is the standard free linear-algebra library that everyone uses). With N degrees of freedom, however, this requires O(N2) storage and O(N3) time, and showed that this quickly gets out of hand. Instead, since we only want a few low-frequency eigenvalues (not N!), we use iterative methods, which start with a guess for the solution (e.g. random numbers) and then iteratively improve it to converge to any desired accuracy. Most iterative solvers require only a black-box routine that computes matrix times vector, so later on when we derive a fast way to operate Θk we can exploit that.

For Hermitian eigenproblems, one class of iterative techniques is based on minimizing the Rayleigh quotient: given any starting guess, if we "go downhill" in the Rayleigh quotient then we will end up at the lowest eigenvalue and corresponding eigenvector. We can find subsequent eigenvalues/eigenvectors by deflation: repeating the process in the subspace orthogonal to the previous eigenvectors. A very simple optimization technique is steepest-descent: repeated line searches in the downhill direction given by the gradient of the Rayleigh quotient. Showed explicitly how we can evaluate this gradient using only matrix times vector operations. In practice, there are better optimization methods for this problem than steepest descent, such as the nonlinear conjugate-gradient method, but they have a similar flavor.

The key to applying iterative methods efficiently for this problem is to use fast Fourier transforms to perform the Θk matrix-vector product in O(N log N) time and O(N) storage, as explained in the next lecture.

Further reading Textbook, appendix D. Spectral methods, Galerkin, etcetera: J. P. Boyd, Chebyshev and Fourier Spectral Methods. Iterative eigensolver methods: Bai et al, Templates for the Solution of Algebraic Eigenvalue Problems; also Numerical Linear Algebra by Trefethen and Bau (readable online with MIT certificates).

Lecture 24: 5 Apr 2010

Explained how Θk operator can be computed in O(N log N) time and O(N) storage, for N degrees of freedom, using fast Fourier transform (FFT) algorithms. In particular, explained how the Fourier transforms over the unit cell, evaluated on a uniform grid, can be expressed in terms of a discrete Fourier transform and hence evaluated via FFTs. (There are a lot of other important details as well, such as preconditioning, which I elected to skip.)

Switched problems to time-domain solvers: find the time-dependent fields in response to an arbitrary time-dependent current, for some initial conditions. This is the most general solution technique, and can handle things like nonlinearities and time-dependent media in which frequency is not conserved (a problem for frequency-domain methods). On the other hand, when a more specialized method (e.g. a frequency-domain eigensolver) is available, often it is easier and more bulletproof than using the most general tool.

In particular, talked about finite-difference time-domain (FDTD) methods, in which space and time are broken up into uniform grids. Started with 1+1 dimensions (1 space + 1 time). Derived the second-order accuracy of center-difference approximations, and in order to utilize this in FDTD concluded that we need to store H and E on grids staggered in time and space. Wrote down the general "leapfrog" scheme for time-stepping the fields.

Further reading: Appendix D of the book. FFTs on Wikipedia gives a decent survey. For more details on MPB in particular, see this paper on MPB. For FDTD in general, see e.g. Allen Taflove and Susan C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2005). See also our free FDTD software: Meep.

Mid-Term Exam: 7 Apr 2010

The mid-term exam will be held in room 3-343. You will have two hours, and can come either from 1-3pm or from 2-4pm, whichever fits your schedule.

You can bring one 8×11 sheet of notes containing whatever you want, and also the representation theory handout.

Previous mid-terms: fall 2005 and solutions, spring 2007 and solutions, spring 2008, spring 2009 and solutions.

Spring 2010 midterm and solutions. Mean: 36/50±7.6/50.

Lecture 25: 9 Apr 2010

Handouts: Notes on PML, Notes on coordinate transforms in electromagnetism

Finished discussion of FDTD discretization: covered Yee lattices in 2d and 3d, Courant/Von-Neumann stability analysis, numerical dispersion and anisotropy. (In practice, numerical errors are usually dominated by discontinuities and singularities at material interfaces.)

Lecture 26: 12 Apr 2010

Started discussing boundary conditions and perfectly matched layers (PML). Introduced PML as an analytic continuation of the solution and equations into complex coordinates in the direction perpendicular to the boundary. Showed how this transforms oscillating solutions into decaying ones without introducing reflections (in theory). Showed how we transform back to real coordinates, and the entire PML implementation can be summarized by a single equation: ∂/∂x → (1+iσ/ω)−1∂/∂x, where σ(x) is some function that is positive in the PML and zero elsewhere, characterizing the strength of the decay.

Discussed implementation of PML in the time domain, showing how (in general) auxiliary differential equations may be required to deal with terms i/ω (which correspond to integrals after Fourier transformation).

Discussed how any coordinate transformation (including the complex one for PML) can be represented as merely a change in ε and μ, while keeping Maxwell's equations in Cartesian form. (See handout for proof.) This can be used to derive the "UPML" formulation of PML as anisotropic absorbing materials, and for neat theoretical results such as "invisibility cloaks."

Discussed fact that UPML isn't purely an anisotropic absorber: there is absorption for transverse fields, but gain for longitudinal fields. Luckily, most propagating modes are mostly transverse, so the absorption wins. Briefly mentioned an interesting identity which proves that mostly longitudinal fields, where PML blows up, coincide with unusual "backward-wave" modes where group and phase velocity are opposite (see this paper by Loh et al.).

Further reading: handouts from lecture 25.

Lecture 27: 14 Apr 2010

Limitations of PML. Discussed fact that PML is no longer reflectionless in discretized equations, but this is compensated for by turning on the absorption (e.g.) quadratically over a wavelength or so. Noted that PML requires Maxwell's equations to be invariant in the direction ⊥ to the PML, which excludes PML. Note also the backward-wave case from last lecture where PML blows up (although this never occurs for index-guided waveguides). In these cases, the only thing left is to turn on an absorption sufficiently slowly.

New topic: temporal coupled-mode theory. Started with a canonical device, a waveguide-cavity-waveguide filter, and showed how the universal behavior of device in this class can be derived from very general principles such as conservation of energy, parameterized only by the (geometry-dependent) frequency and lifetime of the cavity mode.

Further reading: chapter 10 of the book.

Lecture 28: 16 Apr 2010

Demo of Meep FDTD code (installed on Athena/Linux machines: add meep). In particular, went through the tutorial in the Meep manual, and covered the basic techniques to find transmission/reflection spectra and resonant modes (see also the introduction section of the manual).

Lecture 29: 21 Apr 2010

Continued with temporal coupled-mode theory. Finished deriving the coupled-mode equations for the waveguide-cavity-waveguide system. Showed that the transmission is always a Lorentzian curve peaked at 100% (for symmetric decay) with a width inversely proportional to the lifetime, and showed that this happens because of a resonant cancellation in the reflected wave. Defined the dimensionless lifetime Q (the quality factor).

More examples of coupled-mode theory: External loss and resonant absorption, application to photovoltaic absorption.

Further reading: chapter 10 of the book.

Lecture 30: 23 Apr 2010

Coupled-mode theory with a nonlinear Kerr cavity, and derivation of optical bistability.

Briefly discussed a variety of other possibilities that can be analyzed with coupled-mode theory: side-coupled resonance and Fano resonances, channel-drop filters, ring resonators.

Further reading: section of chapter 10 on nonlinear cavity. See also e.g. the Bloembergen (1965) and Agrawal (2001) textbooks listed in the bibliography. See the "some other possibilities" section of chapter 10 for references on the other possibilities discussed in lecture.

Lecture 31: 26 Apr 2010

New topic periodic dielectric waveguides (chapter 7).

Reviewed periodic dielectric waveguides, which we've seen once or twice before: periodic replication of the light cone and bands below that which flatten out at the edge of the Brillouin zone. Incomplete gaps: ranges of frequencies where there are no guided modes (but still light-cone modes).

Partial confinement of light by defects, and intrinsic radiation losses due to coupling to light-line mode. Tradeoff between localization and loss (due to Fourier components inside the light cone). Discussed two mechanisms for large radiation Q despite the incomplete gap: delocalization and cancellation.

Further reading: Chapter 7 of the book on periodic waveguides, and chapter 8 on delocalization/cancellation mechanisms.

Lecture 32: 28 Apr 2010

Photonic-crystal slabs: band gaps, symmetry/polarization, and line-defect waveguides. Microcavities (very similar to analysis in periodic dielectric waveguides).

Further reading: chapter 8 of the book.

Lecture 33: 30 Apr 2010

Application of photonic-crystals without defects to LEDs: using the periodicity to couple to vertical radiation. Also, enhancement of spontaneous emission due to slow light at band edges.

Defined density of states and local density of states, at first directly via the number of frequency eigenvalues in a given frequency interval. Then derived connection to the diagonals of the imaginary part of the Green's function, and hence to the power radiated by a dipole current source.

Discussed peaks in density of states for resonant modes. Derived Van Hove singularities at band edges.

Further reading: For LED applications of photonic-crystal slabs, see this paper and this company. For Van Hove singularities, see e.g. Solid State Physics by Ashcroft and Mermin. The connection between the trace of the imaginary part of the Green's function and the local density of states, in the case of quantum mechanics, is derived in e.g. Green's Functions in Quantum Physics by Economou (Springer, 2006).

Lecture 34: 3 May 2010

Handouts: figures from chapter 9.

New topic: photonic-crystal fibers. Discussed the various types, and focused on the case of index-guiding holey fibers and their properties.

The short-wavelength scalar approximation and its consequences for holey fibers.

Further reading: chapter 9 (section on index-guiding holey fibers and the scalar limit). For a rigorous derivation of the scalar limit, see this 1994 paper by Bonnet-Bendhia and Djellouli.

Lecture 34: 5 May 2010

Discussed consequences of the scalar limit. First for a dielectric waveguide with a square cross-section (which maps to the square TM metallic cavity of pset 2), and then for a holey fiber with a solid core (which maps to a 2d metallic photonic crystal). In both cases, applied product representation theory to the relationship between the scalar LP modes and the vector modes.

Discussed the origin of band gaps in the holey-fiber light cone, from the scalar limit, and band-gap guidance in hollow-core fibers. Emphasized the importance of the band gap lying above the light line of air.

Lecture 35: 7 May 2010

Origin of band gap in Bragg fibers, and role of cylindrical symmetry. Guided modes and polarizations, and analogy with hollow-core fibers. The TE01 and HE11 modes.

Losses in hollow-core fibers: cladding-related losses (which decrease with core radius) vs. intermodal-coupling losses (which increase with core radius). For the specific example of absorption loss, derived relationship between losses per unit distance and group velocity and the fraction of the field energy in the cladding. Sketched simple argument that the fraction of the field energy in the cladding, and hence cladding-related losses, scales inversely with the cube of the core radius.

Further reading: chapter 9.

Lecture 36: 10 May 2010

Handout: first few pages of our 2002 paper

Going full-circle back to the beginning of the course, we again derive an algebraic (linear operator / eigenproblem) formulation of Maxwell's equations. This time, however, we do so for constant-ω separating out the z derivative and the corresponding k component (for z-periodic structures) kz (denoted β). That is, we write Maxwell's equations in the form:

A ψ = -i B ∂ψ/∂z

where ψ is a four-component vector field consisting of (Ex, Ey, Hx, Hy), and A and B are linear operators. This is the most convenient formulation for considering problems of propagation in the z direction along a waveguide, where perturbations may break translational symmetry but frequency is still conserved. Showed that A andB are Hermitian (but not positive-definite) for real ε and μ.

Considered the properties of the eigenproblem for translation-invariant A: Aψ=βBψ. Because B is indefinite, this only has real eigenvalues β when <ψ,Bψ> is nonzero. Physically, showed that <ψ,Bψ> corresponds to the time-average power flowing in the z direction, which is positive for +z propagating modes, negative for -z propagating modes, and zero for evanescent (complex β) modes. Derived an orthogonality relationship for the modes, which for real ε and μ ends up being an unconjugated inner product.

Briefly considered the periodic-A case, in which the eigenoperator is replaced by A+iB∂/∂z, which is still Hermitian.

Further reading: For a traditional treatment of this subject, which does not stress the algebraic aspects (or even write down an explicit eigenproblem for that matter), see Marcuse, Theory of Dielectric Optical Waveguides (1978). Alternatively, for a textbook treatment of the formulation given here and in the paper linked above, see also Skorobogatiy and Yang, Fundamentals of Photonic Crystal Guiding (2008).

Lecture 37: 10 May 2010

Derived coupled-wave theory for perturbed waveguides: expand ψ in the eigenmodes of a given cross-section, then solve for a set of ODEs relating the coupling coefficients. For simplicity, considered only the case where the unperturbed waveguide is translation-invariant. (This treatment is analogous to time-dependent perturbation theory in quantum mechanics.)

First considered small perturbations ΔA, such as fabrication disorder. In this case, as an expansion basis we use the unperturbed waveguide, and derive a set of ODEs for the expansion coefficients. Showed that, for example, a periodic perturbation of period a will (to lowest order) only couple modes with Δβ=2π/a: this is known as quasi phase matching, and is equivalent to the band-folding picture derived earlier in the course.

Second, considered slowly varying perturbations in A, such as a waveguide taper. Derived the coupled-mode equations, related to the rate of change ∂A/∂z. To lowest order, related the scattered power to a Fourier transform of the rate of change of the waveguide, and hence proved the adiabatic theorem: the scattered power into other modes goes to zero as any change becomes more and more gradual (assuming a nonzero Δβ between the modes). In the same way, explained that the rate of approach of the adiabatic limit is determined by the smoothness of the rate of change, particularly at the endpoints.

Further reading: For the adiabatic theorem, generalized to the more difficult case of periodic waveguides, see our 2002 paper. For small ΔA in periodic waveguides, see our 2004 paper.