18.369, Spring 2012

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 2012, 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, Spring 2009, and Spring 2010.

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. 4–5pm (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, see below). 34% final project (proposal due April 11, project due May 16).

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 (available in the library): 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 (e.g.) 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, ideally Phys. Rev. A style), with a literature, 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, spring 2010 and solutions.

Lecture Summaries and Handouts

Lecture 1: 8 Feb 2012

Handouts: syllabus (this web page), introductory slides, 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. The basics of electromagnetism in macroscopic media (ε and μ) are covered in any non-freshman electromagnetism textbook, e.g. Classical Electrodynamics by Jackson or Introduction to Electromagnetism by Griffiths.

Lecture 2: 10 Feb 2012

Handouts: pset 1 (due next Friday)

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.

Simple one-dimensional example of fields in metallic cavity, showed that consequences match predictions from linear algebra.

Further reading: See chapter 2 of the textbook.

Lecture 3: 13 Feb 2012

Discussed scale invariance of Maxwell's equations, and the fact that if we scale up the whole system by a factor of s then the solutions are the same, just with wavelengths scaled up by s (frequencies scaled by 1/s). This comes from the multiplicative nature of the Maxwell operator, and is very different for additive operators like the Schrodinger operator in quantum mechanics (see end of chapter 2 in the text).

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 ÔR for vector and pseudovector fields.

Further reading: Chapter 3 of the text.

Lecture 4: 15 Feb 2012

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, irreducibility, and partner functions, conjugacy classes and most of the other things on the handout, with some examples (the square symmetry group and the mirror symmetry group). (Covered everything on the handout except Great Orthogonality Theorem, character tables, projection operators, and product representations.)

We will show next that eigenfunctions are partner functions of representations of the symmetry group. For example, even and odd functions in a mirror-symmetric system correspond precisely to the two irreducible representations of the {E,σ} group.

Further reading: Chapter 3 of the text, but this doesn't get into representation theory. See e.g. Group Theory and Its Applications in Physics by Inui et al. or Group Theory and Quantum Mechanics by Michael Tinkham, or any book with a similar title.

Lecture 5: 17 Feb 2012

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

Proved that all eigenfunctions can be chosen to transform as partner functions of an irreducible representation of the symmetry group, with the dimension of the representation given by the degree of degeneracy of the eigenvalue. Proved that all representations derived from a given eigenvalue are equivalent. Noted that orthonormal eigenfunctions give a unitary representation (outlined proof but did not work it through).

If the representation is irreducible, then the degeneracy comes from the symmetry of the system. If the representation is reducible, then we call it an accidental degeneracy (not coming from symmetry). Accidental degeneracies rarely happen by accident—usually the degeneracy has somehow been forced—so generically we only expect degeneracies if there are >1 dimensional irreps.

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).

Build the simple character table for the {E,σ} mirror-symmetry group, reprising the previous result that in mirror-symmetric systems we expect even/odd eigenfunctions, and we don't expect (non-accidental) degeneracies (unless there are additional symmetries).

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.

Further reading: See e.g. Group Theory and Its Applications in Physics by Inui et al. or Group Theory and Quantum Mechanics by Michael Tinkham, or any book with a similar title. Note that any book like this typically has an appendix full of character tables for different common symmetry groups (whose nomenclature can take some getting used to).

Lecture 6: 21 Feb 2012

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

Used the projection operator to classify the modes of the square cavity, and in particular found that some of the modes are accidental degeneracies. In this way, we are able to find representatives of all five irreps. Conversely, by looking at the irreps, we can guess some of the types of eigenfunctions that should appear, inferring the sign pattern from the character table.

Showed how we can apply the projection operator to "random" functions to find partners of different irreps, even without an eigenproblem. And, once we have partner functions, we can obtain representation matrices for each irrep (useful for 2+ dimensional irreps). As an example, looked at ψ(x,y)=1, x, and x2; found in particular that the 2d irrep transforms like {x,y}, i.e. the ordinary 2d rotation matrices.

Began talking about how projection operators give us conservation laws.

Further reading: The Inui and Tinkham texts have more information on projection operators (both on reserve at the library).

Lecture 7: 22 Feb 2012

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.

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: 24 Feb 2012

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.

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

Lecture 9: 27 Feb 2012

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.

Lecture 10: 29 Feb 2012

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.

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.

Lecture 11: 2 Mar 2012

Handouts: pset 2 solutions, pset 3 (due Friday 16 March).

Reviewed result from last lecture 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. Defined the primitive lattice vectors. 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.

Qualitative description of the resulting band diagrams in 1d-periodic systems.

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. This means that we only need to look at the "unit cell" in k-space to get the band diagram. Discussed the concept of the (first) Brillouin zone in 1d, although a more general definition will have to wait until we get to 2d periodicity (chapter 5). Defined the reciprocal lattice vectors, and gave examples for 1d periodicity and for a 2d square lattice.

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.

Explained how these symmetries lead to mirror symmetries about the center and edge of the 1d Brillouin zone, and corresponding extrema of the bands at thes places. We therefore only need to look at the smallest nonredundant part of the Brillouin zone: the irreducible Brillouin zone.

Further reading: Chapter 3 of the textbook. See appendix B for more on the reciprocal lattice, a topic we will return to later.

Lecture 12: 5 Mar 2012

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

As an application of conservation of k (up to addition of reciprocal lattice vectors) in periodic systems, discussed reflection (specular and diffractive) from a periodic surface, and minimum-frequency/maximum-wavelength cutoffs for various diffracted orders.

Began new topic: photonic band gaps in one dimension.

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 13: 7 Mar 2012

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

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 14: 9 Mar 2012

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 15: 12 Mar 2012

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 d-fold degenerate case, we actually have to solve a small d×d 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).

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 2010 paper.

Lecture 16: 14 Mar 2012

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.

Computationally, e.g. in MPB, we often compute localized cavity modes by imposing periodic boundary conditions in a supercell consisting of many unit cells, plus a defect. Explained how, in the absence of a defect, such a supercell leads to the original band structure "folded" into the new Brillouin zone. A defect then pulls one of these foldings into the gap, with a cosine-like dispersion relation that becomes flat exponentially fast as the supercell size is increased.

Further consideration of defects in supercells. Although before we considered this to be a computational artifact, it can also be introduced intentionally: a periodic sequence of cavities, forming a "coupled-cavity waveguide" (CCW) or "coupled-resonator optical waveguide" (CROW). (Strictly speaking, we don't have a "waveguide" here because we are in one dimension and there is no lateral confinement, but the same ideas apply with some caveats in higher dimensions.) Showed how the cosine dispersion curve can be derived very generally from a tight-binding analysis for an abstract sequence of coupled cavities. For exponentially localized modes, one then obtains a bandwidth and group velocity (slope) that decrease exponentially with the cavity separation, with an inflection point at the center of the bandwidth where there is zero group velocity dispersion.

Defects surrounded by a finite crystal are more tricky. Intuitively, they will "leak out" slowly from the defect via their evanescent tails. Mathematically, the topic of such "leaky modes" is quite tricky. To deal with such situations, it is useful to have a new tool, the local density of states (LDOS), essentially a measure of the "local" eigenfrequency spectrum that is very useful in aperiodic systems.

Started by defining the density of states DOS(ω): just a delta function for each eigenfrequency (assuming a finite domain, i.e. "electromagnetism in a box", i.e. a point spectrum). Obtained the per-period DOS of a periodic system by starting with a supercell of P periods and taking P to ∞, and in this case we find that the DOS is a mostly continuous function of ω. At the edges of band gaps, however, solved for the DOS by locally expanding the bands as quadratics, to obtain a 1/√Δω divergence: a Van Hove singularity. The next step will be to obtain a DOS per-point not just per period: the LDOS.

Further reading: For coupled-cavity waveguides see also Yariv et al., "Coupled-resonator optical waveguide: a proposal and analysis," Optics Letters 24, 711–713 (1999). For off-axis propagation in multilayer films, see chapter 4 of the course notes. For Van Hove singularities, see e.g. Solid State Physics by Ashcroft and Mermin.

Lecture 17: 16 Mar 2012

Handouts: notes on DOS and LDOS

Defined a local density of states (LDOS) by weighting the DOS by ε|E|2 of the corresponding eigenfields, normalized so that the LDOS integrates to the DOS. This weights each eigenvalue in the spectrum by its coupling to the electric-field energy at a given point, and intuitively gives a local measure of the DOS. This particular measure is especially useful, however, because it turns out to correspond to a simple physical quantity: the power radiated by a dipole current at that point. This in turn helps us to understand why the LDOS is important for things like spontaneous emission, which can be modelled semiclassically as random dipole current sources—the varying LDOS means that spontaneous emission (and related phenomena like lasing) can be either enhanced or suppressed depending on the surrounding structure.

Derived the connection between the power radiated by a dipole and the LDOS. Defined the (dyadic) Green's function via the inverse of ∇×∇×−ω2ε, expanded it in terms of the eigenfunctions, and applied the standard trick of defining a lossless system as the limit of a lossy system as the losses go to zero (which has the side effect of automatically imposing the boundary condition of zero incoming fields at infinity), and using this obtained the LDOS expression (up to a constant coefficient).

Further reading: 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 18: 19 Mar 2012

Handouts: pset 3 solutions, pset 4 (due Mon April 2) [see files bandgap1d.ctl and defect1d.ctl]

Off-axis propagation, projected band diagrams for multilayer films, Fabry-Perot defect modes, and surface states. Omnidirectional reflection for the TM polarization.

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

Lecture 19: 21 Mar 2012

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 after spring break will be to derive this ratio for the group velocity.

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 20: 23 Mar 2012

Handout: BEM lecture notes

Guest lecture by Dr. Homer Reid: surface-integral equations (SIEs) and boundary-element methods (BEMs) in electromagnetism.

Further reading: SCUFF-EM, Homer's free BEM code for electromagnetic scattering problems.

Lecture 21: 2 April 2012

Handout: pset 4 solutions

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

Further reading: chapter 3 of the book (section on velocity). For a discussion of dispersion (and dispersion compensation) as it applies in optical fibers, see e.g. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective.

Lecture 22: 4 April 2012

Handout: 2d square/triangular-lattice Brillouin zones (from appendix B of the book)

New topic: 2d periodicity

Reviewed Bloch's theorem, the primitive lattice vectors, the Bravais lattice, the primitive reciprocal lattice vectors, and the reciprocal lattice, for 2d periodicity. Reviewed the periodicity in k-space (reciprocal space),

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.

Midterm: 6 April 2012

Midterm exam and solutions. (Closed-book, only the representation-theory handout was allowed.)

Lecture 23: 9 April 2012

Handout: figures 2 and 3 from book, chapter 5

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.

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: textbook, chapter 5.

Lecture 24: 11 April 2012

Handout: figures 16—20 from chapter 5 of the book

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.

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 25: 13 April 2012

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).

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.

Lecture 26: 18 April 2012

Handouts: figure 14 and figure 15 from book chapter 10

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.

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). 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 10 (refraction etc), and appendix D.

Lecture 27: 20 April 2012

Handouts: slides on band-structure calculations

Frequency-domain eigensolvers, e.g. MPB.

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.

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 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.

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. In practice, there are better optimization methods for this problem than steepest descent, such as the nonlinear conjugate-gradient method, and preconditioning, but they have a similar flavor. Showed the effect of the different iteration schemes on convergence rate (see handout).

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

A planewave basis actually converges exponentially fast if everything is a smooth (analytic) periodic function, but this is not true if ε is discontinuous (as it usually) is: the Fourier series of a discontinuous function converges only at a linear rate (error ~ 1/#terms in 1d). The planewave basis is dual to a uniform grid under a discrete Fourier transform (DFT), so we can equivalently think of "staircasing" of interface, and in general the question is what ε to assign to pixels straddling the boundaries. Intuitively, boundary pixels should be assigned some intermediate ε value, which is equivalent to discretizing a smoothed structure—but then we face the problem that the act of smoothing changed the structure, and itself introduces a 1st-order error in general. Argued (see handout) that the right thing to do is to assign an anisotropic ε to interfaces: one can show that the proper anisotropic ε corresponds to a smoothing that introduces zero 1st-order error, and hence leads to 2nd-order convergence as shown in the handout.

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). See e.g. this paper on subpixel-smoothing and perturbation theory.

Lecture 28: 23 April 2012

Handouts: slides on FDTD and resonant-mode calculations

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: a Yee grid. Wrote down the general "leapfrog" scheme for time-stepping the fields.

Derived the CFL stability condition (in 1d) relating Δt to Δx. This arises from the simple fact that, if the scheme is to converge as Δx and Δt go to zero, it must support solutions that propagate at speed c, but the maximum speed at which information propagates in the grid is Δx/Δt, and hence Δx/Δt>c (in 1d). As a consequence, if we make the spatial discretization finer, we must also make the time discretization finer. e.g. in 3d this means that doubling the spatial resolution increases the total simulation time by (at least) a factor of 16=24.

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.

Further reading: See e.g. these notes on finite-difference approximations for the basic ideas of center differences etc. For FDTD in general, see e.g. Allen Taflove and Susan C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, 2005). For the CFL condition in general, see e.g. this book chapter by Trefethen. See also our free FDTD software: Meep, and in particular the introduction and tutorial sections of the Meep manual.

Lecture 29: 25 April 2012

Handouts: slides on FDTD and resonant-mode calculations (from previous lecture). slides reviewing PML and limitations thereof

Motivated the interest in resonant modes, and defined the dimensionless lifetime Q (for "quality factor"); see handout.

Went through the resonant-mode calculation part of the Meep tutorial. The basic idea is to hit the cavity with a short (broad-bandwidth) pulse that excites all the modes, and then to extract the resonant frequencies and lifetimes from analyzing the response (a sum of decaying sinusoids). In principle, this could be done by running for a long time and then plotting the Fourier transform: each resonant mode would be a Lorentzian peak, whose width and location give the lifetime and frequency. In practice, this would require a very long simulation to resolve very narrow peaks, so instead we use a fancier type of signal processing called "filter diagonalization", implemented by the free Harminv package and integrated with Meep.

Discussed absorbing boundary conditions and perfectly matched layers (PML); see notes below and slide handout above. 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 fact (to be proved in next lecture) that 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. This means that PML can be interpreted simply as an anisotropic absorbing material ("UPML").

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. (Any absorption, turned on slowly enough, has negligible reflections; this idea is used e.g. in anechoic chambers.) Note that PML requires Maxwell's equations to be invariant in the direction ⊥ to the PML, which excludes photonic crystals from having any true PML. Briefly discussed (see slides for more detail) on how this fact has sometimes been confused in the literature, since the lack of a true PML can be disguised if you turn on the PML gradually enough (over many periods).

Further reading: Notes on PML; see also e.g. the discussion of PML in Taflove's book.

Lecture 30: 27 April 2012

Handouts: Notes on coordinate transforms in electromagnetism, slides on transformation optics

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."

New topic: temporal coupled-mode theory (TCMT). Started with a canonical device, a waveguide-cavity-waveguide filter, and began to derive 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.

Began by parameterizing the unknowns: the amplitude A in the cavity and the incoming/outgoing wave amplitudes s in each channel k, normalized so that |A|2 is energy in the cavity and |s|2 is power. Then wrote down the most general linear time-invariant equation relating A and the incoming wave from a single input port: A(ω)=g(ω) s1+(ω), where g(ω) is some function of frequency in the frequency domain (a type of Green's function or generalized susceptibility). The key assumption of TCMT is resonance: we assume that there is a resonant mode, corresponding to a pole in g(ω) (or the LDOS) at a complex frequency ω0−i/τ, and that 1/τ<<ω0 so that g(ω)≈α/(iω-iω0−1/τ), where α/i is the residue of the pole (i.e. g is dominated near ω0 by the contribution of the pole, and far from ω0 the amplitude A is so small that we will neglect it). In time domain, this corresponds to a simple ODE dA/dt = -i ω0A − A/τ + αs1+, where α is a constant to be determined.

Further reading: chapter 10 of the book.

Lecture 31: 30 April 2012

More discussion of the ODE dA/dt = -i ω0A − A/τ + αs1+ from the last lecture. This describes a resonant mode that, in the s1+=0 of no sources, is simply exponentially decaying: A(t)=A(0) e−iω0t−t/τ. Discussed why this is reasonable, on physical grounds, if 1/τ<<ω0 so that the field decays very little in one optical period. In this case, it is nearly an eigenfunction, in which case the fields everywhere in the cavity are nearly proportional to A. This implies that both the energy U and the outgoing Poynting flux P are nearly proportional to |A|2, but in that case −P=dU/dt is proportional to U and hence U is exponentially decaying.

Furthermore defined the quality factor Q of the cavity, which is simply a dimensionless lifetime Q=ω0τ/2.

Also wrote down the most general linear time-invariant relation for the outgoing amplitude s1−=βs1++γA for some constants β and γ in the freuqency domain. If we are only interested in the response of the system near resonance, then we can approximate β and γ by their values at ω0, in which case they are constants and the s1−=βs1++γA is valid in the time domain as well.

What remains is to eliminate the unknowns α, β, and γ. Do this by the method in chapter 10 of the book: apply energy conservation to find γ=√(2/τ) (up to an arbitrary phase choice), and then time-reversal symmetry (or reciprocity) to find β=−1 and γ=√(2/τ). Hence, the only geometry- and physics- dependent parameters in the problem are ω0 and τ.

Showed that the transmission in a waveguide-cavity-waveguide system 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.

Further reading: chapter 10 of the book.

Lecture 32: 2 May 2012

More examples of coupled-mode theory: A waveguide splitter, external loss (and the need for and absorption Q much larger than the total Q for small loss) and resonant absorption (maximizing absorption, with application to photovoltaics).

Further reading: chapter 10 of the book.

Lecture 33: 4 May 2012

Handouts: slides on TCMT resonant-mode calculations (also in ppt format)

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.

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).

Further reading: chapter 10 of the book, and chapter 7 of the book on periodic waveguides.

Lecture 34: 7 May 2012

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, chapter 8 on loss mechanisms.

Lecture 35: 9 May 2012

Handouts: pages 144–195 of my photonic-crystal tutorial slides

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 36: 11 May 2012

Handouts: pages 197–236 of my photonic-crystal tutorial slides

New topic: photonic-crystal fibers. Discussed the various types from the handouts: photonic-bandgap vs. index guiding, and 2d-periodic vs. Bragg fibers (concentric "1d" crystals). Emphasized the importance of the band gap lying above the light line of air, the role of rotational symmetry in Bragg fibers, and the meaning of the light cone.

Further reading: Chapter 9 of the book.

Lecture 37: 14 May 2012

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

Discussed consequences of the scalar limit. First for a dielectric waveguide with a square or rectangular 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. Noted that holey fibers will support only a finite number of guided modes (and can even be "endlessly single mode" for the right parameters).

Discussed the origin of band gaps in the holey-fiber light cone, from the scalar limit, and band-gap guidance in hollow-core fibers.

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 (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 38: 16 May 2012

Handouts: notes, first few pages of our 2002 adiabatic-theorem 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.

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.)

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: See especially the first few pages of our 2002 paper. 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). For small ΔA in periodic waveguides, see our 2004 paper.