This is the home page for the 18.369 course at MIT in Spring 2014, 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, Spring 2010, and Spring 2012.

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 inmedia 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 (E17-128). **Office Hours:**
Thurs. 4–5pm (E17-416).

**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 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, spring 2012 and solutions.

**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. Proved that self-adjoint/Hermitian operators (Ô=Ô^{†}) have real eigenvalues.

**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.

**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.

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 *Ô _{R}*
for vector and pseudovector fields.

**Further reading:** Chapter 3 of the text.

**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.

**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 *C*_{4v}). 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).

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 x^{2};
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).

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.

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, means that planewaves are produced by a line-current source J_{z}(x,y)=δ(x) e^{-i(ky-ωt)} in 2d, assuming outgoing (radiation) boundary conditions.

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

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

Introduced the **variational theorem** (or 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."

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.

**Handouts:** pset 2 solutions, pset 3 (due Friday 14 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 R**k**, 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.

Discussed the consequence of mirror symmetry in 2d: modes separate into two polarizations, TM (*H _{x}*,

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.

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 how perturbation theory can be used to derive the effect of absorption losses (to lowest order): adding a small imaginary part to ε yields a corresponding imaginary part in ω, giving loss or gain depending on the sign.

**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*).

**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).

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.

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.) (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. Explained why this cannot be an eigenfunction in
any usual sense: both any leaky (complex-ω) mode must *grow
exponentially* far from the defect. Physically, this is because
an observer far away is seeing radiation from the leaky mode as it
was in the past, when the mode was exponentially larger. Really,
we will see that such "modes" are local approximations that take into
account only a single pole (= resonance) in the Green's function.

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.

**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
∇×∇×−ω^{2}ε,
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).

**Handouts:** TE/TM projected band diagram and omnidirectional reflection (from book chapter 4, figure 15), pset 3 solutions, pset 4 (due Friday April 4) [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.

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.

**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). 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).

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.

Applied the Hellmann-Feynman theorem to our
Θ_{k} 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).

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:** 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. 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*.

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

From the energy velocity expression derived in the last lecture, proved that this group velocity is always ≤c for ε≥1. (At a deeper level, it turns out that you can prove this for any passive media.) Also gave a simple proof that the "front velocity" (the rate at which the wave "front" of nonzero fields can move) is bounded by the upper bound of the energy velocity.

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.

**Further reading:** A classic reference on wave velocity is
L. Brillouin, *Wave Propagation and Group Velocity* (Academic
Press, New York, 1960), but the equality of energy and group velocity
in electromagnetism goes back to M. Abraham in 1911 [Nuovo Cimento I,
443]. The case of periodic nondispersive media can be found in the
textbook, and goes back at least as far as P. Yeh,
J. Opt. Soc. Am. 69, 742 (1979). Generalizations to passive and lossy
media can be found
in Glasgow
(2001), including the proof of the front velocity which I gave in
class (which goes back in various forms to other authors); passive
lossless dispersive media are discussed in e.g. A. D. Yaghjian,
IEEE Trans. Antennas and Propagation 55, 1495 (2007).
Recent work by Welters et al. generalizes this to passive lossy
periodic media and other extensions. For 2d periodicity, see the beginning of chapter 5 of the book (2d
photonic crystals), and appendix B on the reciprocal lattice and
Brillouin zone.

**Handout:** figures 2 and 3 from book, chapter 5

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 (C_{6v}) structure.

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.

**Further reading:** beginning of chapter 5 of the book (2d
photonic crystals), and appendix B on the reciprocal lattice and
Brillouin zone.
See this paper for
some counterexamples and further discussion regarding the occurrence
of band extrema at the edges of the I.B.Z.

**Handout:** figures from chapter 5 and 6 of the book

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.

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.
Related the defect modes to
the 5 irreps of the C_{4v} symmetry group, and showed how we
can easily guess the field patterns and degeneracies that we will get.

**Further reading:** textbook, chapter 5 and 6

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

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.

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*.

**Further reading:** Chapter 5, chapter 10 (section on reflection, refraction, and diffraction).

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).

**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(N^{2}) storage and O(N^{3}) 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 Θ_{k}
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. See our paper on MPB for more detail on planewave-based eigensolvers for electromagnetism. 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.

**Handout:** see slides from Lecture 23

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). Mentioned 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=2^{4}.

**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.

**Handout:** see slides from Lecture 23

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.

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.

**Handouts:** Notes on coordinate transforms in electromagnetism; see also 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."

Went through *Sources and Integral Equations* slides from Lecture 23, including the Principle of Equivalence, surface integral equations (SIEs) and boundary element methods (BEMs).

**Further reading:** For the principle of equivalence and its application to wave sources in FDTD, see our review article
Electromagnetic Wave Source Conditions (excerpted from *Advances in FDTD Computational Electrodynamics: Photonics and Nanotechnology*, 2013). For a free/open-source BEM code, see SCUFF-EM by Homer Reid.

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_{k±} in
each channel *k*, normalized so that |A|^{2} is energy in
the cavity and |s_{k±}|^{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) s_{1+}(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 ω_{0}−i/τ, and that
1/τ<<ω_{0} so that
g(ω)≈α/(iω-iω_{0}−1/τ),
where α/i is the residue of the pole (i.e. g is dominated near
ω_{0} by the contribution of the pole, and far from
ω_{0} the amplitude A is so small that we will neglect
it). In time domain, this corresponds to a simple ODE
dA/dt = -i ω_{0}A − A/τ + αs_{1+},
where α is a constant to be determined.

Also wrote down the most general linear time-invariant relation for
the outgoing amplitude
s_{1−}=βs_{1+}+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 ω_{0}, in which
case they are constants and the s_{1−}=βs_{1+}+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 ω_{0} 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.

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

More examples of coupled-mode theory: A waveguide splitter, a waveguide crossing, 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).

Analyzed the time-delay in a resonant filter and showed that: it is given by dφ/dω (the derivative of the phase) if we have a narrow-band pulse near the transmission maximum (so that amplitude is independent of frequency to first order), and that this time delay is precisely τ (the lifetime of the cavity mode).

Furthermore defined the **quality factor** Q of the cavity, which is simply a dimensionless lifetime Q=ω_{0}τ/2.

**Further reading:** chapter 10 of the book and references therein. For waveguide splitters, see also Fan et al. (2001) and Manolatou et al. (1999); for crossings, see also Johnson et al. (1998).

Brief overview of nonlinear optics. Discussed nonlinear susceptibilities and polarization, along with various phenomena in a Kerr mediumL self-phase modulation (SPM) and cross-phase modulation (XPM), third-harmonic generation (THG), and four-wave mixing (FWM). There is also second-harmonic generation (SHG) and difference-frequency generation (DFG) in a second-order nonlinear medium.

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

**Further reading:** TCMT slides from above, chapter 10 of the book, and Soljacic (2002) on bistability. We also have several papers on TCMT for frequency conversion in nonlinear cavities: Rodriguez (2007), Hashemi (2009), Burgess (2009), Ramirez (2011), and Bi (2012).

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). Symmetry and polarization.

**Further reading:** chapter 7 of
the book.

**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).

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:** Chapters 7 and 8 of the book.

**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.

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.