This is the home page for the 18.369 course at MIT in Spring 2008, 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.

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 (2-146). **Office Hours:**
TR 4:30–5:30 (2-388).

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

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

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

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

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

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

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

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

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

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

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

**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.

**Handouts:** photonics collage
(illustrating many interesting devices/geometries), used to point out
(below) that one thing that many such devices have in common is that
symmetries are everywhere.

Defined positive definite/semidefinite operators, and showed that the Maxwell operator is positive semidefinite and hence has nonnegative eigenvalues, and hence that the eigenfrequencies ω are real (for real ε>0). Compared to Schrodinger equation of quantum mechanics; showed that Maxwell's equations are scalable (multiplying dimensions by 10 just divides ω by 10, for same ε).

Illustrated with simple 1d example: electromagnetic modes trapped
between two perfect-metallic mirrors. This leads to cosine solutions
*H _{z}*, whose orthogonality and
completeness correspond to that of the familiar Fourier cosine series.
Brief discussion of why the eigenvalues are discrete (and cases in
which they are not discrete).

Noted that all eigensolutions are even or odd functions for this problem, as a consequence of the mirror symmetry. Argued that understanding the impact of symmetry is crucial for a wide variety of devices in photonics (and other areas), and is one of the key analytical tools that we have in complex geometries that cannot be solved analytically (see handout).

Began to explore the consequences of symmetry more rigorously, first by proving that a symmetry corresponds to an operator that commutes with the eigen-operator. Then, showed that for mirror symmetries this results in eigenfunctions that must be even or odd (for nondegenerate eigenvalues).

Started discussion of how vector fields transform, by noting that we have an apparent problem already in this case: when the magnetic field looks even, the electric field looks odd, and vice versa. We will deal with this next time by showing that the magnetic field is not a vector, it is a pseudovector.

**Further reading:** chapter 2 of the book (properties of
harmonic modes; scaling properties; discrete vs. continuous
frequencies; comparison to quantum mechanics) and chapter 3 (first
section).

**Handouts:** representation-theory
quick-reference

Reviewed simple argument for why (well-behaved) Hermitian operators on finite domains have discrete spectra (see also chapter 2 of the book), and constructed an (artificial) counter-example of a crazy operator with a finite domain and a dense point spectrum (for which the eigenfunctions at a finite frequency oscillate arbitrarily fast).

Defined rotations (proper and improper) more generally as 3×3 real orthogonal matrices, and defined the corresponding linear operations on the Hilbert spaces of scalar functions and vector functions (vector fields). Showed that if the electric field is a vector, that the magnetic field is a pseudovector (multipled by -1 under improper rotations), and why this is important in looking at the symmetry of eigenmodes.

Looked at more complicated example of modes in a 2d square metallic cavity, and showed that looking at mirror planes individually does not reveal the complete structure of the symmetry of the solutions. That structure will be revealed with the help of group representation theory.

Introduced the concept of the space group (rotations + translations that leave the system invariant).

**Further reading:** See chapter 3 of the book for a
basic overview of the consequences of symmetry, and the books by Inui
et al. or Tinkham for a more in-depth discussion.

Reviewed space group and representations, and defined the point group and symmorphic space groups. Proved that all eigenstates can be chosen to transform as an irreducible reprensentation of the space group. Argued that degeneracy typically comes from 2×2 (or larger) irreducible representations, and that the other possibility, an "accidental" degeneracy unrelated to symmetry, is very unlikely (and typically must be forced deliberately).

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

**Handouts:** problem set 1 solutions, problem set 2 (due 29 Feb.)

Using the rules from the representation theory handout, built up the character table for the symmetry group of the square (called
*C*_{4v}). Then, looked at the
eigenfunction solutions that we previously had for this case, and
showed how we could classify them into the various irreducible
representations. Conversely, showed how, using the character table,
we can "guess" what the corresponding eigenmodes must look like (or at
least the sign pattern). Predicted a couple of field patterns for
modes we hadn't seen yet. Then, showed that some of the apparent
double degeneracies are actually accidental, and that we could
decompose them into one-dimensional representations, and in fact
obtained some of the predicted field patterns.

Proved that any function (not just eigenstates) can be decomposed into a sum of partner functions of the irreducible rotation. Example: any function can be decomposed as a sum of an even function and an odd function (the irreducible representations of the {E,σ} group). Defined the projection operator P that gives the component of a given function that transforms as a given representation (or a given partner function of a given representation); see also problem set 2.

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

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

Guest lecturer: Yidong Chong.

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

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

Briefly reviewed previous lecture, and noted that the fact that the
representation functions are exponential functions follows from the
property D(x+y)=D(x)D(y) that must hold for any representation D: the
only functions with that property are exponentials (as long as we
restrict ourselves to rpresentations that are *anywhere*
continuous, or even measurable).

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

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

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

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

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.

Described how the variational theorem leads directly to a computational method of Rayleigh-quotient minimization, which we will discuss further later in the course.

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

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

**Further reading:** chapter 2 of the textbook, section on
variational theorem, and chapter 3 on discrete translation symetry.
For a similar theorem in 3d, see Bamberget and Bonnet
[*J. Math. Anal*, **21**, 1487 (1990)].

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

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

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

Showed how index-guiding arises in periodic waveguides. (We will consider these waveguides again in much more detail later. See also chapter 7 in the course notes.)

Considered interaction of space group with **k**: showed that
rotations R transform solution at **k** into solution at R**k**,
and that only the point group matters for this analysis. Showed how
this reduces the first Brillouin zone (loosely, the "unit cell" of
**k** around the origin) to the irreducible Brillouin zone of
inequivalent **k** under the point group.

**Further reading:** Chapter 3 of the book, section on
discrete translational symmetry, and section on rotational symmetry.

**Handouts:** MPB demo (see also the MPB home page) and example files: 2dwaveguide.ctl and 2dwaveguide-periodic.ctl; problem set 2 solutions and problem set 3 (due March 14).

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

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

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

Review rotational symmetry and its consequences on the band diagram
and on the Bloch modes/eigenproblem at a particular **k**.

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

Noted that ω(**k**)=ω(−**k**) in general,
even for structures without mirror symmetry. Derived this from
time-reversal symmetry (conjugating the eigenequation, for real
ε). Brief mention of magneto-optic materials
(complex-Hermitian ε) and why a static magnetic field can
(locally) break time-reversal symmetry, and of use for Faraday
isolators.

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.

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

Began preparation for 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. Discussed connection to the variational theorem, and to Hellmann-Feynman theorem for the derivative of the eigenvalue. 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.

**Further reading:** Chapter 10 in the book, last section (the
one on reflection, refraction, and diffraction). 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.

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.

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.

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

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

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 medium 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.

**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 31); MPB example files bandgap1d.ctl and defect1d.ctl.

Derived Poynting's theorem, defining the electromagnetic energy density and flux (Poynting vector) so that energy (electomagnetic + mechanical) is conserved.

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 result of
"zero" time delay.

**Further reading:** chapter 3 of the book (section on
velocity). For a similar derivation of Poynting's theorem, see any
advanced text on electromagnetism, e.g. Jackson or Griffiths.

Discussed group-velocity dispersion, qualitatively, and defined the dispersion parameter D; you will investigate this more quantitatively in homework.

New topic: 2d photonic crystals. Defined the square lattice, derived its reciprocal lattice (also square) and described its first Brillouin zone and irreducible Brillouin zone (for a square lattice of e.g. circular rods that do not break any of the symmetry). Labelled the Γ, M, and K high-symmetry points in the I.B.Z. Sketched the band diagram for TM modes, noted degeneracies at Γ and M, and discussed the origin of the band gap. Because the periodicity in different directions is not the same, there is a minimum index contrast to get a gap. The variational theorem lets us easily predict (qualitatively) what the first two bands look like, predict the degeneracy at M, and understand the origin of the gap.

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:** 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).

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 diagram, and in the real lattice it is called a Wigner-Seitz cell.) Gave examples of square lattice and triangular lattice.

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.

Showed the I.B.Z. for the square lattice (with the canonical labels Γ, X, and M for the corners), and sketched the corresponding band diagram. Argued that if we are only interested in band extrema (e.g. band for band gaps) then it is usually sufficient to only plot around the boundaries of the I.B.Z.

Showed I.B.Z. for triangular lattice (with the canonical labels Γ, M, and K for the corners)

**Further reading:** Appendix B of the book.

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.

Reviewed another way to think about the gap, using the variational theorem. Showed how we can easily guess what the lowest two bands look like at the X and M points, and how the big contrast between them leads to a gap.

In contrast, pointed out that there is no TE gap for this structure (at least, it never overlaps the TM gap). The reason has to do with the boundary conditions that we derived earlier when were were discussion out-of-plane propagation in 1d: 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.

To get a large TE gap, we need the dual structure: air holes in
high dielectric, to give contiguous "veins" in which the field lines
can run without crossing an interface. For example, gave example of
square lattice of air holes, which has a TE gap...but no (overlapping)
TM gap. The problem now is that we have *too much* high
dielectric, giving the TM modes too much room to be concentrated in
the dielectric while remaining orthogonal. To get a gap for both
polarizations, it turns out that one possibility is a triangular
lattice of air holes in dielectric.

Sketched the field/sign patterns for the M and K points in the triangular lattice, to illustrate what those points mean.

**Further reading:** Chapter 5 of the book.

**Handouts:** pset 4 solutions

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).

**Further reading:** Chapter 5 of the book.

**Handouts** Spring 2007 midterm and solutions;
Fall 2005 midterm and solutions

Surface states in 2d crystals, the influence of crystal termination, and derived (in a handwavy way) the fact that there must always be some termination that has a surfaces state.

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, and from the different representations (giving rise to monopole, dipole, quadrupole, etc. patterns).

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.

**Further reading:** Chapter 5 of the book.

Finite-difference time-domain methods (FDTD). Overview of Yee-lattice staggered-grid discretization, center-difference approximations and accuracy, and the Courant stability criterion (via Von Neumann analysis).

**Further reading:** Allen Taflove and Susan C. Hagness,
*Computational Electrodynamics: The Finite-Difference Time-Domain
Method* (Artech, 2005).

**Handouts:** midterm exam, problem set
5 (due Monday, 21 April) and rod-transmission.ctl example file for
Meep; (sq-rods.ctl and tri-rods.ctl example files for MPB).

Two hours in room 2-146, from 2–4pm.

**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. In the next
lecture, we will show how this coordinate transform can be represented
simply by an anisotropic ε and μ (the so-called "UPML").

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 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.

Showed 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. This can be used to derive the "UPML" formulation, and for neat
theoretical results such as "invisibility cloaks."

**Further reading:** handouts from lecture 22.

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).

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. Showed that the transmission is always a Lorentzian curve
peaked at 100% (for symmetric decay) with a width inversely
proportional to the lifetime. Defined the dimensionless lifetime
*Q* (the quality factor).

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

More examples of coupled-mode theory. External loss and resonant absorption. Phase shifts and time delays.

Reviewed some basics of nonlinear optics: the nonlinear
susceptibilities χ^{(2)} (Pockels effect) and
χ^{(3)} (Kerr effect), self/cross-phase modulation,
harmonic generation, four-wave mixing, etc.

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

**Further reading:** section of chapter 10 on nonlinear cavity.
See also e.g. the Bloembergen (1965) and Agrawal (2001) textbooks
listed in the bibliography.

New topic: **Periodic dielectric waveguides and incomplete
gaps**, corresponding to chapter 7 in the book.

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. Impact of symmetry: TE-like and TM-like modes. 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). Coupled-mode theory for a lossy filter, and condition that
radiation *Q* must be much greater than total *Q* (inverse
of fractional bandwidth) to get high transmission at resonance.

**Further reading:** chapter 7 of the book, and section of
chapter 10 on filters with lossy cavities.

Photonic-crystal slabs: band gaps, symmetry/polarization, and line-defect waveguides.

**Further reading:** chapter 8 of the book.

Microcavities on photonic-crystal slabs. Delocalization and cancellation. Examples of cavities.

Discussed why 3d exponential confinement of zero-loss modes in slabs is impossible, from a simple Fourier-space argument.

Also briefly discussed the possibility of using specialized materials and mode-matching to construct cavities with infinite Q and finite modal volume without using a complete gap.

**Further reading:** chapter 8 of the book. For the last topic
of mode-matching, see this paper by
Watts et al.

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

**Further reading:** chapter 9 (section on index-guiding holey fibers).

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

**Further reading:** chapter 9 (sections on scalar approximation).

Gaps and guided modes in hollow-core holey fibers. The air light line, surface states, and termination.

Gaps (light cone) of Bragg fibers, and the importance of rotational symmetry.

**Further reading:** chapter 9 of the book (sections on
hollow-core holey fibers and Bragg fibers).

Guided modes and polarizations in Bragg fibers. Discusses losses from cladding (especially material absorption) and scaling with core radius in hollow-core fibers.

**Further reading:** chapter 9 of the book (sections on Bragg fibers and cladding losses).