This is the home page for the 18.369 course at MIT in Spring 2010, where the syllabus, lecture materials, problem sets, and other miscellanea are posted.

You can also download the course announcement flyer, and visit this photonic-crystal tutorial page to find materials for past lectures by SGJ on related subjects. This course was previously offered as 18.325 in Fall 2005 (also on OpenCourseWare) and as 18.369 in Spring 2007, Spring 2008, and Spring 2009.

Tired of doing electromagnetism like it's 1865?

Find out what solid-state physics has brought to 8.02 in the last 20 years, in this new course surveying the physics and mathematics of nanophotonics—electromagnetic waves 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 (2-136). **Office Hours:**
Thurs. 4:30–5:30 (2-388).

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

**Grading**: 33% problem sets (weekly/biweekly). 33% mid-term
exam (April 7, see below). 34% final project
(proposal due April 14, project due May 12).

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

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

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

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

Supplementary lecture notes: Notes on the algebraic structure of wave equations and Notes on Perfectly Matched Layers (PMLs), and several other PDF files that will be made available as the term progresses.

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

**Handouts:** syllabus (this web page), introductory slides, problem set 1 (due 12 Feb.), collaboration policy

Motivation and introduction: this class is about electromagnetism
where the wavelength is neither very large (quasi-static) nor very
small (ray optics), and the analytical and computational methods we
can use to understand phenomena in materials that are structured on
the wavelength scale. In that situation, there are very few cases
that can be solved analytically, but lots of interesting phenomena
that we can derive from the *structure* of the equations.

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

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

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

Defined Hermitian operators, and showed that the Maxwell
eigen-operator
∇×ε^{-1}∇× is
Hermitian for real ε (by showing that ∇× is
Hermitian). Proved that Hermitian operators have real eigenvalues and
that the eigenvectors are orthogonal (or can be chosen orthogonal, for
degeneracies). The Maxwell operator is also positive semidefinite,
and it follows that the eigenfrequencies are real.

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

**Further reading:** See chapter 2 of the textbook.

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

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

**Handouts:** representation theory summary

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

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

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

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

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

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

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

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

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

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

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

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

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

Introduced the **variational theorem** (or min–max
theorem), which arises for any Hermitian eigenproblem. Proved the
variational theorem (at least for finite-dimensional spaces), and more
generally discussed the derivation (in chapter 2 of the book) that
all extrema of the Rayleigh quotient are eigenvalues.

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

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

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

Considered related theorems in quantum mechanics: an arbitrary attractive potential will always localize a bound state in 1d or 2d, but not in 3d, and sketched a simple dimensional argument in 1d and 3d (but not 2d, which is a difficult borderline case). Discussed the related theorem for 3d waveguides (2d localization), and the case of substrates where the theorem does not apply and the fundamental modes has a low-ω cutoff.

**Discrete translational symmetry:**

Started by considering a periodic "waveguide" in two dimensions: a sequence of dielectric rods in air. By analogy with ray-optics and total-internal reflection, it seems that this could not support guided modes. However, it does (chapter 7 of the book), and to understand this we need to understand discrete translational symmetry.

Showed that the representations of the discrete translation group
are again exponentials, and thereby proved **Bloch's theorem**: the
eigenfunctions can be chosen in the form of a planewave multipled by a
periodic function. As a corollary, the Bloch wavevector **k** is
conserved, and explained how this relates to a famous mystery from the
19th century: electrons in a pure conductor act almost like a dilute
gas, because they scatter only from impurities/imperfections that
break the periodicity.

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

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

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

Considered interaction of rotational symmetries with **k**:
showed that rotations R transform a solution at **k** into solution
at 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.

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

**Handouts:** problem set 3
(due March 12).

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

Began new topic: **photonic band gaps in one dimension**.

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

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

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

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

Gave demo of MPB eigensolver software for 2d dielectric waveguide
(`add meep`

on Athena, currently only available on
Athena/Linux-x86, e.g. in the clusters or via ssh to
`linux.mit.edu`

).

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

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*).

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

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

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

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

Began discussing out-of-plane propagation; immediate consequence is
that TE and TM modes separate, and now we have both
*k _{x}* and

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

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

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

**Further reading:** See chapter 4 of the book.

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

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

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

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

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

Applied the Hellmann-Feynman theorem to our
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). Proved
that this group velocity is always ≤c for ε≥1.

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

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

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

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

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

New topic: the Brillouin zone(s).

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

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

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

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

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

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

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

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

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

**Further reading:** textbook, chapter 5.

Further discussion of point defects: 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
(at least for low-order modes in defects that are not too big).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

In order to solve this equation, we could simply throw it directly
at Matlab or LAPACK (LAPACK is the standard free linear-algebra
library that everyone uses). With N degrees of freedom, however, this
requires O(N^{2}) storage and O(N^{3}) time, and
showed that this quickly gets out of hand. Instead, since we only
want a few low-frequency eigenvalues (not N!), we use **iterative
methods**, which start with a guess for the solution (e.g. random
numbers) and then iteratively improve it to converge to any desired
accuracy. Most iterative solvers require only a black-box routine
that computes matrix times vector, so later on when we derive a fast
way to operate Θ_{k} we can exploit that.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

**Further reading:** handouts from lecture 25.

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

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

**Further reading:** chapter 10 of the book.

Demo of Meep FDTD code
(installed on Athena/Linux machines: `add meep`

). In
particular, went through the tutorial in the Meep manual, and covered
the basic techniques to find transmission/reflection spectra and
resonant modes (see also the introduction section of the manual).

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

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

**Further reading:** chapter 10 of the book.

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

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

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

New topic **periodic dielectric waveguides** (chapter 7).

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

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

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

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.

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

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

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

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

**Handouts:** figures from chapter 9.

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

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

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

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

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

Origin of band gap in Bragg fibers, and role of cylindrical
symmetry. Guided modes and polarizations, and analogy with
hollow-core fibers. The TE_{01} and HE_{11} modes.

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

**Further reading:** chapter 9.

**Handout:** first few pages of our 2002
paper

Going full-circle back to the beginning of the course, we again
derive an algebraic (linear operator / eigenproblem) formulation of
Maxwell's equations. This time, however, we do so for
constant-ω separating out the *z* derivative and the
corresponding **k** component (for *z*-periodic structures)
*k _{z}* (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
(*E _{x}*,

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

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

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

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

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

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

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