This is the home page for the 18.369/8.315 course at MIT in Spring 2018, 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, Spring 2012, Spring 2014. and Spring 2016.

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 3–4pm (2-135). **Office Hours:**
Thurs. 4–5pm (2-345). TA/grader: Ethan Jaffe.

**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 18**: 2-4pm *or* 3-5pm in 13-1143). 34% final project
(proposal due April 13, project due May 16).

**Books**: Photonic Crystals:
Molding the Flow of Light (Second Edition) (**readable online**). (This book is at an
undergraduate level and 18.369/8.315 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. (readable online via MIT), 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 (other useful free/open-source software: SCUFF-EM and S4), and extend/analyze it in some further way (try some
other variation on the geometry, etc.). Then write up the results in a 7-15 page report (in the format of a journal article, ideally Phys. Rev. A style, including figures, a comprehensive review of related work, etcetera)—reports
should be written for a target audience of your classmates in 18.369/8.315,
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**: 18.303, 8.07, 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 are all good background.) 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; spring 2014 and solutions.

**Handouts:** syllabus (this web page), introductory slides, collaboration policy, pset 1 (due Wed Feb 14).

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 and Hermitian operators (Â=Â^{†}).

**Further reading:** See chapter 2 of the *Photonic Crystals* textbook for Maxwell's equations as an eigenproblem etc. 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.

Showed that the Maxwell
eigen-operator
∇×ε^{-1}∇× is
Hermitian for real ε (by showing that ∇× is
Hermitian). The Maxwell operator is also positive semidefinite for ε>0,
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.

**Handouts:** representation theory summary

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 *R̂*
for vector and pseudovector fields.

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, which we will get to next time.

**Further reading:** Chapter 3 of the photonic-crystals text.
See the Inui textbook, or many similar sources, on group theory; the
most helpful in this context are the many "group theory in physics"
books.

**Handouts:** pset 2 (due Mon, Feb 26), pset 1 solutions

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). (Briefly 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.

Claimed that all eigenfunctions can be chosen to transform as partner functions of an irreducible representation of the symmetry group (also called a "basis" of the representation), with the dimension of the representation given by the degree of degeneracy of the eigenvalue. Proof next lecture.

**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. (especially sections 4.1, 6.1, and 6.2) or *Group Theory and Quantum Mechanics* by Michael Tinkham (especially sections 3-1 and 3-6),
or any book with a similar title.

Proved that all eigenfunctions can be chosen to transform as partner functions of an irreducible representation of the symmetry group (also called a "basis" of the representation), 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.

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:** Character tables for all of the common symmetry groups are tabulated in both textbooks and online, e.g. see this page on the C4v group. See Inui section 6.6 on projection operators.

**Handout:** notes on decomposition of functions into partner functions

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 to "sketch" the qualitative features that we expect
to find in the eigenfunctions. 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.

Next time, we will show the irreducible representation is conserved over time in a linear system, by showing that the projection operator commutes with the time-evolution operator.

**Handout:** notes on time evolution and conservation laws

The irreducible representation is "conserved" over time in a linear system, because 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, for lossless materials, and that this
leads to conservation of energy.

**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 briefly in Jackson. A much more
complete review of passive dispersive media, including the
consequences of passivity for causality etcetera, can be found in our
2014
paper Speed-of-light
limitations in passive linear media: see in particular the
discussion of passivity in section II.B and of "dynamical" energy density
in section V.A.

More from last lecture's handout.

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.

Talked a little about a more general formulation of the time-dependent problem, including arbitrary dispersion, and derived that the susceptibility χ needs to be a "passive convolution operator" for energy to be non-increasing. It turns out that this has lot of interesting consequences, which we mostly won't have time to get into.

**Translational symmetry:** 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).

**Further reading:** Textbook, chapter 3 on continuous translational symmetry.

**Handouts:** pset 2 solutions,
pset 3 (due March 9)

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.

Introduced the "reduced" eigenproblem Θ_{k}=e^{-ikx}Θe^{ikx} for the modes of a particular wavevector **k**. This is also Hermitian, its solutions ω(k) yield the dispersion relation (or *band structure*) of the problem.

Explained how mirror symmetry in *z* means that *z*-invariant solutions in "2d" structures ε(x,y) can be segregated into two polarizations: even H_{z}-polarized (what the book calls "TE") and odd E_{z}-polarized (what the book calls "TM"). (Note that the literature is split on the terminology here: many authors call the former TM and the latter TE.)

**Further reading**: See the book, chapter 3, on index guiding and the variational principle.
(See e.g. Jackson's *Classical Electrodynamics* for a more
traditional viewpoint on dielectric waveguides, focused on the two
cases that can be solved analytically, and Marcuse's *Theory of
Dielectric Optical Waveguides* for an expanded version of this.
See e.g. Ramaswami and Sivarajan, *Optical Networks* for a nice
practical overview of dielectric waveguiding in modern
telecommunications.)

Continued discussing **dielectric waveguides**, via the simple 2d example
of a high-ε region surrounded by a low-ε region,
invariant in the x direction. Explained 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 in order to stay orthogonal.

Introduced the **min–max or "variational" theorem**, which arises for any Hermitian eigenproblem. Proved the
variational theorem (with the simplifying assumption of a basis of
eigenfunctions), and mentioned the derivation (in
chapter 2 of the book) that all extrema of the Rayleigh quotient are
eigenvalues. We will use this theorem to derive general conditions
under which guided modes are guaranteed to arise in dielectric
waveguides.

Used the min–max theorem to prove the existence of index-guided
modes (in two dimensions, for the TE/H_{z} 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**: See the book, chapter 2, on the variational principle.
See these
notes on localization in a different scalar-wave equation via the
same variational method. Bamberget and Bonnet (1990) is a classic paper on the theory of dielectric waveguiding. Lee (2008) is an extension of the variational proof to Maxwell's equations in much more complex periodic waveguides and photonic-crystal fibers. A point source (delta-function current) in a dielectric waveguide will generally excite *both* guided (below the light line) and radiating (above the light line) solutions, and you might be interested in this animation of a point source in a dielectric waveguide (ε=4 surrounded by air) (see also the larger version of the same animation).

Considered related localization 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); see also these notes. Discussed the related theorem for 3d waveguides (2d localization); see the Bamberget and Bonnet and Lee papers above.

**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 mentioned 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.

**Further reading**: Chapter 3 of the textbook on 1d periodicity, and chapter 7 of the textbook on periodic waveguides.

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. Talked about a
few different ways to think about this: 2π/a gives the same irrep,
it gives the same PDE for the Bloch modes, and the Bloch modes can be
expanded as a Fourier series with components at k+2πm/a for all
integers m.

Discussed the band diagram, light cone, and guided modes of a periodic dielectric waveguide.

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.

**Further reading:** Chapter 3 and appendix B of the textbook on
1d periodicity. Chapter 7 on periodic dielectric waveguides. Chapter
10 has a section on reflection/diffraction from a periodic surface.

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. Defined the reciprocal lattice
vectors, and gave examples for 1d periodicity and for a 2d square
lattice, and a 2d hexagonal 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).

Discussed the "little group of **k**:" the subset of the
rotations that preserve **k** (up to addition of a reciprocal
lattice vector), hence the symmetry group of Θ̂ₖ. This group is
important to understand the solutions that are possible at
each **k**, and will be especially important in higher dimensions
to understand at which **k** non-accidental degeneracies are
possible.

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

**Handouts:** pset 3 solutions, pset 4 (due Fri March 23; the files bandgap1d.ctl and defect1d.ctl are used)

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.

Began new topic: **photonic crystals in one dimension**. Sketched
the form of the dispersion relation (band structure) and explained several
qualitative features we can predict without solving:

- From symmetry and periodicity, ω(k)=ω(-k)=ω(k+2π/a), and consequently ω(π/a-k)=ω(π/a+k) (i.e., ω is symmetric around both k=0 and k=π/a)
- Because ω(k) is smooth except at crossings, we expect the bands to have zero slope at k=0 and k=&pi/a (around which they are even-symmetric)
- ωa<<1 corresponds to the long-wavelength limit, at which the waves don't "see" the periodic structure and instead see some "average" homogeneous ε (an "effective medium" or "metamaterial"; also called "homogenization theory"). Hence around (ω,k)=(0,0) the bands should approach straight lines (the solutions in a homogeneous medium).
- Because the 1d Maxwell equations are a 2nd-order ODE, and the eigenproblem at a given ω is a 2-point boundary-value problem, there can be only two solutions at each ω. It follows that the bands ω(k) must be
*monotonic*in (0,π/a). - Because the symmetry group at each k has at most a mirror symmetry, which has no 2d irreps, we do not generically expect degeneracies. (By playing with the parameters, it turns out that you can force accidental degeneracies at k=0.)
- The combination of the last two points mean that we generically expect
*band gaps*to arise: ranges of ω in which there are no solutions (at least, not at real values of k). - Although I haven't shown it yet, it turns out the bands have slopes that alternate in sign: the first band has positive slope in (0,π/a), the second band negative, the third positive, and so on.

Next time, we will use perturbation theory to derive the magnitude of the gap and other features by starting with a homogeneous medium and then adding a little bit of periodic ε contrast.

**Further reading:** Chapter 4 of the textbook.

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.

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*).

To apply perturbation to the opening of the gap, there is a slight
complication because the cos(πx/a) and sin(πx/a) modes of the
unperturbed (homogeneous) system are degenerate. This creates an
ambiguity: to which linear combination of these eigenfunctions (i.e.,
what 0-th order solution) do we apply our perturbation analysis to?
For a general perturbation with L-fold degeneracy, there is a
technique called *degenerate perturbation theory* that solves
this problem by reducing it to an L×L eigenproblem. Here,
however, we can solve the problem by symmetry: since the perturbed
problem still has mirror symmetry around x=0, we know that the
perturbed eigenfunctions must still be even/odd, so we must start with the
cos/sin solutions.

Discussed reflection of light from a semi-infinite 1d crystal, at a frequency in the gap. We have no propagating solutions in the crystal, so by conservation of energy we must have 100% reflection. However, showed by analytical continuation of the band edge that we expect exponentially decaying "evanescent" solutions in the crystal, with a complex wavevector k ≈ sqrt(Δω/α) + π/a, where Δω is how far we are into the gap and α is the band-edge curvature.

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. In general, we expect a discrete set of frequencies for such "bound states", and the number of bound states in a given gap should grow asymptotically proportional to the volume of the defect (the diameter in 1d), but for an arbitrarily weak defect we still generically expect at least one bound state.

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 reading:** For evanescent waves and defect cavities,
see chapter 4 of the textbook. For a proof of localization in gaps by arbitrarily weak defects in 1d and 2d, for the Schrodinger equation, see Parzygnat et al. (2010).

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

Gave demo of MPB eigensolver software for 2d dielectric waveguide (available on Athena, e.g. in the clusters or via ssh to
`athena.dialup.mit.edu`

).

In the handout, I gave the plotting code in Matlab. Because the output of MPB is either comma-delimited text (ω vs. k) or HDF5 (field data), essentially any modern plotting software should be able to handle it. In class, I used Julia, a Matlab-like language with C-like performance in an IJulia notebook with the PyPlot plotting library (which calls the Python matplotlib under the hood). You can use whatever you want.

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

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...we will show that
in the electromagnetic case, this ratio gives exactly dω/dk. In
particular, we will apply 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).

**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 the Fourier perspective; see also
my notes
on wave velocity and Fourier transforms
from 18.303. A
much more complete review of velocity in lossy and dispersive media
can be found in our 2014
paper Speed-of-light limitations in passive linear media.

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).

From the energy velocity expression, 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.

Discussed group-velocity dispersion, qualitatively.

**Further reading:** See chapter 3 of the book for a similar Hellman-Ferynman approach, and the Welters (2014) paper for much more on group and front velocity bounds.
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*.

**Handouts:** TE/TM projected band diagram and omnidirectional reflection (from book chapter 4, figure 15), pset 4 solutions, pset 5 coming soon.

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

**Further reading:** For off-axis propagation in multilayer
films, see chapter 4 of the textbook. 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).

**Handout:** pset 4 solutions, pset 5 (due next Monday),
figures 2 and 3 from book, chapter 5, 2d square/triangular-lattice
Brillouin zones (from appendix B of the book)

Finished discussing surface states and Fabry-Perot waveguides from last lecture.

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), the Brillouin zone, and the irreducible Brillouin zone.

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). 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:** 2d photonic-crystal slides

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). Showed that Γ and M have the full symmetry of the lattice, whereas X has a reduced symmetry group, which is why we have non-accidental degeneracies at those points.

Influence of boundary conditions on TE vs TM gaps, and why TE gaps prefer "hole" (connected) structures to "rod" (disconnected) structures. Mentioned TE+TM gap in hexagonal lattice of rodes. Discussed why higher symmetry (Brillouin zones closer to circles/spheres) typically makes it easier to get a gap. (In 3d, the closest B.Z. to a sphere is from an fcc lattice, and most 3d gaps are in fcc-like structures.)

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.

**Handout:** computational photonics slides, pset 5 solutions

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.

The finite-difference frequency-domain method (FDFD), guest lecture by Wonseok Shin.

**Further reading:** Free MaxwellFDFD code and FD3D code by W. Shin. See e.g. these notes on finite-difference approximations for the basic ideas of center differences etc.

**Handout:** See slides from lecture 22.

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.

Mentioned, but did not derive, the CFL stability condition (in 1d) relating Δt to Δx. 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}.

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 later, see notes below) 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:** 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.
Notes on PML; see also e.g. the discussion of PML in Taflove's book. Notes on coordinate transforms in electromagnetism; see also slides on transformation optics

Went over the Principle and Equivalence, mode sources, and integral-equation methods; see slides from lecture 22.

**Further reading:** SCUFF-EM, Homer Reid's free BEM code for electromagnetic scattering problems. See also this SCUFF video tutorial (June 2018).

**Handouts:** Section 4.4 (LDOS) of Electromagnetic Wave Source Conditions.

Alternatively, an approach that lets us talk about the "local" spectrum of finite periodic structures, open resonators, and other lossy cases, is the **local density of states**. Began discussing section 4.4 of the handout (DOS and LDOS), and showed that the LDOS is also proportional to the power radiated by a dipole source at a given position and frequency: this latter definition has the advantage of being much easier to generalize, and easier to connect to other physical processes like spontaneous emission or antennas.

Discussed the "principle of limiting absorption:" the "right" way to define a "lossless" system is to add a little bit of loss everywhere and take the limit as this loss goes to zero from above (Im ωε = 0^{+}). This allows us to rigorously deal with poles on the real-ω axis, and also automatically gives us outgoing ("radiation" or "Sommerfield") boundary conditions. It also allows us to correctly derived the LDOS connection to the power.

**Further reading:** See the Snyder and Love textbook for leaky modes via saddle-point ("steepest-descent")
integration. For the limiting absorption principle, see
e.g. Schulenberger
and Wilcox (1971). See section 4.4 of the handout and references
therein for more information on DOS and LDOS.

**Handout:** TCMT slides

Continued discussing LDOS (from the notes): connected to the famous Q/V formula for Purcell enhancment, and discussed the per-period LDOS and DOS in periodic systems with the connection to van Hove singularities.

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 with a high-level overview (see slides), with derivations to come next time.

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

Analyzed the time-delay in a waveguide-cavity system described by TCMT 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 2τ (the lifetime of the cavity mode).

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

TCMT derivation.

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.

Analyzed effect of absorption in resonant filter, and why narrow-band filters require very low loss. Discussed resonant absorption by forcing the decay rates to match ("impedance matching").

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

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

More applications of TCMT (see slides). Nonlinear effects.

New topic **periodic dielectric waveguides** and **photonic-crystal slabs** (chapters 7-8).

**Further reading:** chapter 7-8 of the book, slides on photonic-crystal slabs.

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.

Finished discussing delocalization and cancellation mechanisms for high Q slab cavities.

**Further reading:** chapter 8 of the book, tutorial slides on photonic-crystal slabs.

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.

**Further reading:** chapter 9 of the book, tutorial slides on photonic-crystal fibers.

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.

**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.

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

**New topic (see notes)**: 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}*,

Discussed orthogonality of modes and unconjugated "inner products," propagating vs. evanescent modes (showing that the latter carry zero power)

β eigenproblem for z-periodic problems.

Coupled-wave equations for *nearly* uniform cross sections: small perturbations, slowly varying perturbations, and the adiabatic theorem.

Connected the adiabatic limit to the rate of convergence of the Fourier transform of the rate of change. See section 2.1 of Oskooi et al. (2012).

**Slides:** my slides from some research seminars

Discussed a basic picture of lasers: the Maxwell-Bloch equations, the SALT nonlinear eigenproblem for the steady-state modes, recent SALT computational methods, and laser linewidth limits due to quantum/thermal noise.

**Further reading:** See e.g. Haken's *Laser Theory* for the Maxwell-Bloch equations, Ge, Chong, & Stone (2010) on SALT, Esterhazy et al. (2014) on computational SALT, Pick et al. (2014) and references therein on laser linewidth.