18.369, Spring 2007

Mathematical Methods in Nanophotonics

Prof. Steven G. Johnson, Dept. of Mathematics


This is the home page for the 18.369 course at MIT in Spring 2007, 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)

Tired of doing electromagnetism like it's 1865?

Find out what solid-state physics has brought to 8.02 in the last 20 years, in this new course surveying the physics and mathematics of nanophotonics—electromagnetic waves in media structured on the scale of the wavelength.

In this regime, which is the basis for everything from iridescent butterfly wings to distributed-feedback lasers and integrated optical devices to the next generation of optical fibers, the 140–year-old analytical techniques you learned in 8.02 aren't very useful. Instead, we will cover computational methods combined with high-level algebraic techniques borrowed from solid-state quantum mechanics: linear algebra and eigensystems, group theory, Bloch's theorem and conservation laws, perturbation methods, and coupled-mode theories, to understand surprising optical phenomena from band gaps to slow light to nonlinear filters.

For beginning graduate students and advanced undergraduates.


Lectures: MWF 2–3pm (2-139). 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 13). 34% final project (proposal due April 18, project due May 16).

Books: Useful (but not required) books in reserve book room: Photonic Crystals: Molding the Flow of Light by Joannopoulos et al. (course notes are the second edition of this!), Group Theory and Its Applications in Physics by Inui et al., and Group Theory and Quantum Mechanics by Michael Tinkham. Course notes (available from MIT CopyTech): Photonic Crystals: Molding the Flow of Light, Second Edition (preprint).

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).

Lecture Summaries and Handouts

Lecture 1: 7 Feb 2007

Handouts: syllabus, problem set 1 (due 16 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 started 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), Dirac bra-ket notation |H> for the field ("state") and <A|B> for the inner product. Defined the adjoint (denoted †) of states (kets) and linear operators. Defined Hermitian operators, and showed that the Maxwell eigen-operator ∇×ε-1∇× is Hermitian for real ε (by showing that ∇× is Hermitian).

Further reading: See chapter 2 of the course notes. For more information on bra-ket notation, see any quantum-mechanics textbook (the classic text on this notation is The Principles of Quantum Mechanics by Dirac). 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.

Lecture 2: 9 Feb 2007

Proved properties of Hermitian eigenproblems: real eigenvalues and orthogonal eigenvectors. Showed that the Maxwell eigen-operator is, in addition, positive semi-definite (for ε>0), leading to real frequencies ω. 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 Hz, whose orthogonality and completeness correspond to that of the familiar Fourier cosine series. Brief discussion of generalized functions (distributions) and why they are the relevant definition of "function" in this context. Noted that all eigensolutions are even or odd functions for this problem, as a consequence of the mirror symmetry.

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.

Lecture 3: 12 Feb 2007

Handouts: representation-theory quick-reference

Explored the consequences of symmetry: starting with the 1d example, showed that a mirror plane implies eigenstates that are either even or odd (at least for non-degenerate modes, and degenerate modes can be chosen to be even or odd).

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) and defined the concept of a representation of the group, with a simple example of the 1d mirror-symmetry group {E,σ}.

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

Lecture 4: 14 Feb 2007

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).

Based on the eigenfunctions we derived for the square, started building up the character table for the group (called C4v). (This is actually the opposite of what normally does: one normally starts with the character table to predict what the solutions will look like.)

Lecture 5: 16 Feb 2007

Handouts: problem set 1 solutions, problem set 2 (due 28 Feb.)

Discussed going from the character table to predict things about the solutions, which is the usual thing that one does. Explained how we can construct the character table (and the representation matrices, if desired) a priori using the rules from the 2nd page of the representation-theory handout from lecture 3.

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). Gave an example for C4v using a cartoon example function.

Proved that partner functions of different representations, or different partner functions of the same representation, are orthogonal (regardless of whether they are eigenfunctions).

Began examining the case of translational symmetry. Showed that a representation D(x) of a (1d) translation by x must satisfy D(x+y) = D(x) D(y)...we will then prove that this results in exponential functions.

Lecture 6: 20 Feb 2007

Prove that the irreducible representations of the translation group are exponential functions, assuming that they are anywhere continuous. [In fact, one only need assume that they are Lebesgue measurable, as proved in e.g. Edwin Hewitt and Karl Stromberg, Real and Abstract Analysis (Springer: New York, 1965), exercise 18.46.]

Plugging this into Maxwell's equations, we see that we must get plane-wave solutions in any homogeneous medium: field ~ exp(ikx - iωt). The divergence constraint tells us that the plane-wave must be transverse (amplitude ⊥ k), and the eigenequation giving the disprsion relation &omega=c|k|.

The projection operator for the exp(-ikx) representation of the translation group is just the Fourier transform.

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.

Explained how conservation of the exp(-ikx) representation, which gives conservation of k, leads immediately to Snell's law at a flat interface. Similarly, conservation of ω follows from translational invariance in time (time invariance) in a linear system.

Further reading: chapter 3 of the course notes, section on continuous translational symmetry.

Lecture 7: 21 Feb 2007

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.

Argued that, in general, a Hermitian operator with localized eigenmodes must lead to discrete eigenvalues, in order to preserve orthogonality. This leads to the discrete guided bands in a waveguide, the discrete harmonics in a piano string, and the discrete energy levels of an electron bound to a Hydrogen atom, among many other things.

Introduced the variational theorem, which arises for any Hermitian eigenproblem. Quick and dirty proof, using completeness. (You will give a more general proof for pset 3.) "Dirty", because completeness is sometimes proved using the variational theorem—sketched proof of completeness [similar to one in Courant and Hilbert, Methods of Mathematical Physics, ch. 6 (Wiley: New York, 1953)].

A more formal and general discussion of completeness can be found if you look up "spectral theorem" in any functional analysis book, such as the one by Gohberg et al. (see above). Gohberg et al. do not use the variational theorem, but on the other hand they only prove the spectral theorem for "compact" Hermitian operators; the Θ operator in the Maxwell eigenproblem is not compact (but its inverse is, more or less).

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.

Further reading: chapter 2 of the course notes, section on variational theorem.

Lecture 8: 23 Feb 2007

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

Discussed a famous theorem of quantum mechanics: an arbitrarily weak potential well localized bound states in 1d and 2d, but not 3d; "proved" this in 1d and 3d by a dimensional argument. Explained how an analogous theorem arises in electromagnetism, by which an arbitrarily weak rise in ε can localize waveguide (index-guided) modes in 2d (1d cross-section) and 3d (2d cross-section). Proved sufficient conditions for 2d waveguiding by the variational theorem.

Discussed (qualitatively) effects of substrates/asymmetry (introduces a lower-frequency cutoff) and 3d (two "polarizations", degenerate if symmetry group supports it).

Lecture 9: 26 Feb 2007

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.

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.)

Began looking at consequence of periodicity on diffraction.

Further reading: Chapter 3 of the course notes, section on discrete translational symmetry. Chapter 10 of the course notes, section on refraction and diffraction.

Lecture 10: 28 Feb 2007

Handouts: problem set 2 solutions, problem set 3 (due Monday, March 12).

Continued analysis of diffractive reflections from periodic surfaces, and showed the existence of a minimum-frequency cutoff for each diffracted order, and connection to Fourier series of 1/ε.

Considered interaction of space group with k: showed that rotations R transform solution at k into solution at Rk, 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.

Considered how k breaks some of the symmetry, and therefore reduces the space group of Θk. Gave example of the waveguide, where k breaks some one of the mirror symmetries.

Further reading: Chapter 10 of the course notes, section on refraction and diffraction. Chapter 3 of the course notes.

Lecture 11: 2 March 2007

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

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

Gave demo of MPB eigensolver software for 2d dielectric waveguide (add meep on Athena, currently only available on Athena/Linux).

Lecture 12: 5 March 2007

Finished MPB demo, by looking at 2dwaveguide-periodic.ctl example (above): a periodic sequence of dielectric cylinders in air, which supports guided modes exponentially confined to the rods. We'll discuss this kind of structure in more detail later in the course (see also chapters 7–8 in the course notes).

Began discussing numerical methods for electromagnetism, broken into five steps:

  1. Pick the problem you want to solve: here, the eigenproblem Θk. (Alternatives include simulating Maxwell's equations in time, or finding the field in response to some current.)
  2. Reduce to a finite domain: here, the periodic unit cell. (Alternatives include some sort of absorbing boundary or expressing the unknowns in terms of surfaces instead of volumes.)
  3. Expand the unknowns Hk in terms of a finite-basis approximation. (e.g. a finite grid or finite elements, or a truncated Fourier series.)
  4. Express the problem in terms of these finite unknowns (e.g. via a Petrov-Galerkin, weighted-residual method.)
  5. Solve for the unknowns. (e.g. by iterative methods.)

Today, we covered steps 1–3. (This is only a brief overview...one could easily spend an entire course on these kinds of methods.)

Further reading: appendix D of the course notes.

Lecture 13: 7 March 2007

Handout: Numerical eigensolver methods for Maxwell's equation (slides taken from a longer talk on the same subject).

Discussed weighted residual methods, in which we obtain an N×N set of (linear) equations for the unknown basis coefficients by requiring the inner product with a set of N weight functions |wm> to be zero. If the weight functions are delta functions, this gives a collocation method; if the weight functions are the same as the basis functions, this gives a Galerkin method. Showed that the finite-dimensional equations in a Galerkin method inherit any nice properties, such as being Hermitian and/or positive-definite, of the original operator. Showed the form of the operator that we obtain for the planewave expansion.

Discussed various methods for solving the linear equations. e.g. dense-matrix methods (ala LAPACK) are O(N2) in storage and O(N3) in time, making them impractical for large 3d problems. Mentioned sparse direct solvers, but mostly focused on iterative linear solvers. In particular, showed how both linear equations (Ax=b) and eigenvalue problems (Ax=λx), for Hermitian A, can be expressed as a minimization problem.

Began discussion of a few numerical minimization techniques. Started with steepest-descent, which I don't recommend per se but which is the basis of many other gradient-based methods. Then considered Newton's method, which requires you to compute the inverse Hessian (second-derivative matrix) and is thus impractical for large problems. By analogy, however, an approximate inverse Hessian is useful, and this leads to preconditioned steepest-descent methods. To be continued...

Further reading: appendix D of the course notes. J. P. Boyd, Chebyshev and Fourier Spectral Methods (Dover, 2000) online. Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods by Barrett et al.. Templates for the Solution of Algebraic Eigenvlue Problems by Bai et al..

Lecture 14: 9 March 2007

Continued discussion of iterative minimization techniques for solving linear equations. Began with a few brief remarks about preconditioners (a hard problem, still the subject of active research), and pointed out that diagonal (Jacobi) preconditioners turn out to work well for many problems using a spectral basis. Then explained why steepest-descent methods are suboptimal—two subsequent search directions only search two lines, rather than a whole plane. Summarized the idea behind conjugate-gradient methods (which many other courses at MIT cover in detail; see also the introduction by Shewchuk, below). Showed some numerical results (see handout from lecture 13) illustrating the dramatic difference made by preconditioning and conjugate-gradient.

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. Finally, 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).

Further reading: J. R. Shewchuck, "An Introduction to the Conjugate Gradient Method Without the Agonizing Pain" (1994). See also the Templates books linked above. For time-reversal symmetry, see chapter 3 of the course notes. For one-dimensional periodic structures, see chapter 4 of the course notes.

Lecture 15: 12 March 2007

Handouts: problem set 3 solutions; problem set 4 (due Friday, 23 March); example files bandgap1d.ctl and defect1d.ctl for problem set 4.

Brief mention of magneto-optic materials and why a static magnetic field can (locally) break time-reversal symmetry.

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).

To make this quantitative, we derive first-order perturbation theory for the eigenvalue of any Hermitian operator with some small change. 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 4 of the course notes. For the same derivation of perturbation theory, see "time-independent perturbation theory" in any quantum-mechanics text, e.g. Cohen-Tannoudji.

Lecture 16: 14 March 2007

Given first-order perturbation theory, we apply it to derive the band gap in our slightly-perturbed periodic medium. We define the concept of fractional gap, etc.

More generally, we must employ degenerate perturbation theory to analyze broken degeneracies like this: in the case of a d-fold degeneracy, we show that first-order perturbation theory leads to a small d×d eigenproblem for the new (first-order) eigenvalues and (zeroth-order) eigenstates.

Discussion of the general concept of avoided eigenvalue crossings, or anti-crossings, and gave a simple mechanical example of two coupled pendula.

Re-considered the source of the band gap from another perspective, that of the variational theorem: the lowest band "wants" to concentrate in the high-ε material, and the next band wants to do the same but is forced out due to the orthogonality constraint. This line of reasoning, while less precise, lends itself better to two- and three-dimensional periodicity.

Lecture 17: 16 March 2007

There are many analytical results for one-dimensional (univariate) ε structures, but we do not go through them in detail because they do not generalize well to higher dimensions. (The basic analytical technique is that of "transfer-matrix methods", whose closest generalizations in higher dimensions are purely numerical methods such as boundary-element methods.) We will, however, summarize a few of the most important results, which are derived in detail by, e.g., P. Yeh, Optical Waves in Layered Media (Wiley: New York, 1988).

Discussed the "optimal" two-layer periodic structure: the quarter-wave stack, which leads to a maximal gap (for two periodic layers) and maximum confinement strength. In this case, there are analytical formulas for the mid-gap frequency and the gap size. At mid-gap, each layer is a quarter-wavelength thick (for the wavelength λ/n in the material), hence the name. Gave a simple intuition for why this choice is good, in terms of all reflected waves adding up exactly in phase at mid-gap (although of course the band gap extends over a range of frequencies).

Introduced the topic of evanescent waves inside the band gap, which you can see, for example, in a semi-infinite multilayer film. Derived their exponential decay rate by analytic continuation of the band edge, which shows that the attenuation rate generally increases as one goes deeper into the gap.

We can see that a homogeneous region sandwiched between two semi-infinite multilayer films—a Fabry-Perot cavity—will support localized cavity modes. More generally, we argue that creating any defect that breaks the periodicity can introduce cavity modes, and for small defects these cavity modes are closely associated with the band-edge states by analytic continuation.

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: chapter 4 of the course notes.

Lecture 18: 19 March 2007

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

Considered the meaning of "negative" group velocity (negative slope) in the dispersion relation. There are still waves going in both direction if we look at the entire Brillouin zone; this is the first instance (of many) where the direction of k does not determine the direction of propagation. However, for light incident on an interface perpendicular to the periodicity, this can lead to "backward" refracted waves in the crystal.

More generally, this refraction leads to the question of off-axis propagation. We now look at the multilayer film as a two-dimensional structure periodic in x, and begin by considering propagation in the y direction (kx=0). The analysis is very similar to that of the waveguide modes. We argue that the lowest TM band will always lie beneath the lowest TE band, by showing that the boundary conditions on E at dielectric interfaces affect the confinement of the electric-field energy (variational theorem denomenator) ε|E|2. In particular, when E is parallel to the interface then ε|E|2 is more confined in the high-ε material, and conversely when E is perpendicular to the interface. (The same thing happened in the 2d dielectric waveguide from pset 3.)

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

Lecture 19: 21 March 2007

Opened with review of defect-mode cavity in crystal—the fact that it has no (conserved) wavevector, and is just a delta function in the density of states between two continua (the propagating modes far away). Also reviewed the relation to a supercell, or coupled-cavity sequence: in that case there is a wavevector, but the Brillouin zone is getting smaller and smaller and so it approaches the continua+delta case for isolated cavities.

Projected band structure for off-axis propagation (TM polarization only): showed how the two irreducible Brillouin-zone edges project, and then interpolated in between to get continuum limit. Explained why bandwidths approach zero and continua approach straight lines: the light lines of the high-index material (corresponding to index-guided modes) and of the low-index material (corresponding to Fresnel-reflected resonances).

Next, considered projected band diagram of defect, and showed that we must get a defect band(s) in the gap, corresponding to a waveguide mode. If the ε of the defect differs from εhi, then the defect band must intersect the continuum. Such waveguide modes differ substantially from their index-guided counterparts, in at least three ways: they can be guided in a lower-ε medium (e.g. air, decreasing material interactions), they always have a low-frequency cutoff, and the group velocity approaches zero at this cutoff (increasing material interactions).

Argued that at an interface of the crystal, we can get surface states: guided on one side by the gap, and on the other side by the light line. In fact, we always get surface states for some termination (which we will justify in more detail later), whereas there are also some terminations where there are no surface states.

Considered TM reflection from a semi-infinite crystal, and showed that for the TM polarization there is always a range of ω giving omnidirectional reflection (100% reflection from every angle). The same is only sometimes true of the TE polarization, which we will consider in the next lecture.

Further reading: chapter 4 of the course notes.

Lecture 20: 23 March 2007

Handouts: problem set 4 solutions

Omnidirectional reflection for both TE and TM polarizations: argued that TE is the limiting factor, based on the existence of Brewster's angle. At this angle, the TE reflection from every interface is zero, leading to a straight line in the projected band diagram along which there are no gaps. If this line is above the light line, we are sunk—an omnidirectional reflector will be impossible. On the other hand, if the low-index material is > the ambient index, then this does not occur, and if we have a large enough index contrast then we will have an omnidirectional mirror (as was first proposed in 1998 by Winn, Fink, et al). Discussed limitations of omnidirectional mirrors—they don't work if we break translation symmetry, e.g. if the source is too close (unless we substitute something equivalent like rotation symmetry).

Began discussion of group velocity. First, introduced phase velocity ω/k in a homogeneous medium, which is the most obvious (but not the most useful) definition of wave "velocity". Then, gave classic derivation of group velocity by considering the propagation of a broad wave packet (narrowly peaked in Fourier space) and Taylor-expanding ω(k) to first order.

Looking for another definition of velocity, considered the case of a flowing fluid. There, we can define the velocity at any point as the ratio of mass flux (mass/area/time) to density (mass/volume), which corresponds to our usual concept of velocity. Argued that we should do something similar for electromagnetism, and that this will give an alternate (equivalent) derivation of group velocity that is sometimes easier to work with...

Derived Hellman-Feynman theorem, which gives an exact expression for the derivative of the eigenvalue of a Hermitian operator via first-order perturbation theory. We will use this to compute dω/dk.

Further reading: end of chapter 4 of the course notes (on omnidirectional reflection), and the section of chapter 3 of the course notes on velocity.

Lecture 21: 2 April 2007

Handouts: problem set 5 (due Monday, 9 April).

Continued group velocity discussion. Applied Hellman-Feynman theorem to derive expression for dω/dk by differentiating the eigen-operator, and show that we get a ratio of energy flux (the Poynting vector) to energy density, time-averaged and averaged over the unit cell. This is only true for real ε and real k (we also assumed frequency-independent ε, but this can be included if desired); in other cases, the definition of velocity itself is more complicated. (Gave some examples of cases where one can get a "group velocity" greater than the speed of light by violating these constraints, but in those cases group velocity is not energy velocity.)

Sketched the derivation of electromagnetic energy density and flux; see any textbook (e.g. Jackson, Classical Electrodynamics) for a complete proof and discussion of dispersive and non-dispersive cases.

Proved that the group velocity is bounded above by the speed of light, assuming ε is at least 1 in addition to the constraints above. (Proof involved the Cauchy-Schwarz inequality, which we also proved.)

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

Further reading: chapter 3 of the course notes (section on velocity).

Lecture 22: 4 April 2007

Some concluding remarks about group-velocity dispersion. Defined "normal" (D>0) and "anomalous" (D<0) dispersion, and showed how we can easily get both in waveguides. Pointed out that the dispersion parameter diverges at a band edge (e.g. in a Fabry-Perot waveguide) and mentioned how this can be used for dispersion compensation.

Contrasted group and phase velocity, where the latter is not uniquely defined in a periodic system.

New topic: Multidimensional periodicity, especially two-dimensional periodicity (corresponding to chapter 5 in the notes).

Define the lattice and reciprocal lattice for general d-dimensional lattice, and showed how to get primitive reciprocal lattice vectors from the primitive latitice vectors; gave example of square lattice in two dimensions (where reciprocal lattice is also a square lattice).

To minimize our computational (or analytical) efforts, we only want to consider k vectors in some unit cell of the reciprocal lattice. However, we also want a region that preserves the point group, so that we can reduce it by the rotational symmetries. This leads us to define the first Brillouin zone: the region in k space that is closer to the origin than to any other reciprocal lattice point.

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.) Gave example of square lattice.

Further reading: chapter 5 of the course notes, and appendix B (on the Brillouin zone etc.)

Lecture 23: 6 April 2007

Continued discussion of (first) Brillouin zone (B.Z.). 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.

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.

Discussed the origin of the band gap via a similar perturbative, "band folding" argument to what we did in 1d. Considerd in particular the Γ–X line: showed that now we need to consider "folding" of additional bands from ky=n2π/a for integers n, and that this leads to additional anti-crossings and local minima. Showed that the gap along Γ–X first opens at a frequency ωa/2πc=0.5, versus ωa/2πc=1/√2 for Γ–M, and this means that the gaps are not overlapping to start with. We need some minimum index contrast to get an overlapping gap for all directions in 2d, unlike 1d.

Further reading: chapter 5 of the course notes

Lecture 24: 9 April 2007

Handouts: problem set 5 solutions

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. This leads us to the subject of triangular lattices in general. We show that the reciprocal lattice is also a triangular lattice, but rotated by 30 degrees, and the Brillouin zone is a hexagon—this is the first example of a case where the B.Z. is not a simple unit cell of the reciprocal lattice.

Further reading: chapter 5 of the course notes, and also appendix B on the reciprocal lattice.

Lecture 25: 11 April 2007

Further consideration of the triangular lattice, its Brillouin zone, and its irreducible Brillouin zone. Explained why the threshold index contrast for a gap is generally smaller for a triangular lattice than a square lattice: its Brillouin zone is more nearly circular. Sketched out what the fields look like at the M and K points, showing what periodicities/sign patterns these k points imply.

Gave a somewhat bad sketch of the fields to explain why the first TM band is degenerate at K; nevertheless, there is a complete gap (all directions and all polarizations).

Started considering 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).

Further reading: chapter 5 of the course notes.

Mid-term Exam: 13 April 2007

The mid-term exam is in room 2-102, from 1pm to 4pm. You have two hours, so you can either come from 1–3pm or 2–4pm, whichever fits your schedule.

Lecture 26: 18 April 2007

Handouts: midterm solutions, problem set 6 (due Monday, 30 April). Files for pset6: sq-rods.ctl, tri-rods.ctl, line-defect.ctl, rod-transmission.ctl.

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).

Outlined a simple device that one can make by combining waveguides and cavities: a filter, formed by a waveguide coupled to a cavity coupled to another waveguide. Described the basic behavior of this system—a Lorentzian peak at the cavity frequency—which we will derive analytically soon.

Returned to the topic of numerical simulations. So far, we have been using frequency-domain eigensolvers, which numerically correspond to solving a linear eigenproblem. One could also do a frequency-domain response to some current J at a given ω, which corresponds to solving a linear equation. The most general technique, however, is a time-domain simulation, in which one solves the full Maxwell equations as a function of time, given some time-dependent currents etc. Outlined how time-domain methods can be used to efficiently compute transmission spectra by Fourier-transforming the response to a short pulse.

Further reading: chapter 5 of the course notes. We will continue studying devices in much greater detail next week, corresponding to chapter 10. See Appendix D of the course notes for a summary of frequency- and time-domain simulation methods.

Lecture 27: 20 April 2007

Gave demo of Meep, our (finite-difference) time-domain simulation package, which is installed on Athena in the meep locker. Went through tutorials discussion computation of CW (constant-ω, "continuous-wave") field patterns, transmission spectra from response to short pulses, reflection spectra, and resonant modes.

Further reading: Appendix D of the course notes, and the Meep Home Page (especially the manual's introduction and tutorial).

Lecture 28: 23 April 2007

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).

Lecture 29: 25 April 2007

Handouts: Notes on coordinate transformation & invariance in electromagnetism

Perfectly Matched Layer (PML) absorbing boundary regions. We derive this by first deriving how Maxwell's equations transform under general coordinate transformations (a quite useful result in its own right), and then by considering complex coordinate transformations.

Lecture 30: 27 April 2007

We introduce temporal coupled-mode theory, which allows us predict universal behaviors of a whole class of devices formed by coupling a set of cavities and waveguides, parameterized by only a few geometry-dependent unknowns (typically, the frequencies and decay rates of the cavities).

Began by studying a canonical problem: two waveguides coupled at their ends to a resonant cavity between them. Using the key approximation of weak coupling (the cavity decays slowly), plus conservation of energy and time-reversal symmetry, we derive the coupled-mode equations. We show that the transmission spectrum is completely characterized by the cavity frequency ω0 and its decay rate(s) 1/τ.

Further reading: Course notes, chapter 10.

Lecture 31: 30 April 2007

Handouts: problem set 6 solutions (see also skew-rods.ctl)

Continued examination of the waveguide-cavity-waveguide problem. Relate the 100% transmission on resonance to a destructive interference between two reflection mechanisms (direct reflection, and energy leaking backwards from the cavity). Defined the quality factor Q, and related it to the fractional bandwidth at half-maximum, the energy decay rate (or lifetime), and the ratio of outgoing power to energy in the cavity. Argued that in general we can express 1/Q (a dimensionless decay rate) as a sum of 1/Qk for lifetimes Qk associated with individual loss channels.

Consider how we get <100% transmission when we have asymmetric decay rates, or when we include an external loss mechanism (e.g. a radiation loss, or an absorption loss). Show that this drop in transmission is proportional to the ratio of the associated Q's. This means that for a narrow-bandwidth device the loss Q must be very large (much larger than ω/bandwidth).

As another example, considered how we can obtain 100% resonant absorption, e.g. for a detector or a solar cell, at least within a narrow bandwidth, by matching the loss Q to the reflection Q.

Further reading: Course notes, chapter 10.

Lecture 32: 2 May 2007

Briefly sketched some further applications of coupled-mode theory. Considered waveguide splitters, bends, Fano resonances, channel-drop filters.

Began considering a nonlinear filter, where there is a nonlinear material in the cavity (or everywhere, but the strongest nonlinear effects will occur in the cavity where the field is maximum). Introduced nonlinearities in terms of a power-series expansion of the susceptibility χ, defined Pockels and Kerr effects, and explained how harmonic generation arises.

Further reading: Course notes, chapter 10—see section on "Some other possibilities."

Lecture 33: 4 May 2007

Derived optical bistability for a Kerr-nonlinear cavity coupled to two waveguides (or other input/output channels). Show that we get a cubic equation for the output power, which results in a highly nonlinear hysteresis effect.

Gave an overview of hybrid structures: periodic waveguides and photonic-crystal slabs, which combine index-guiding with photonic band gaps. This results in an imperfect confinement mechanism, at least for cavities in three dimensions, but leads to structures that are much easier to fabricate than three-dimensionally periodic photonic crystals with complete band gaps.

Started by looking at a 2d model system, a periodic sequence of dielectric squares: periodic in one direction (x), and index-guided in the other direction (y). We actually already studied this sort of structure in lecture 9 and problem set 3, so this is really review.

Further reading: Course notes, chapter 10—see section on bistable switching. Chapter 7.

Lecture 34: 7 May 2007

Periodic dielectric waveguides. Reviewed their projected band diagrams, the light cone, and guided modes. Discussed the relation of mirror symmetry planes to TE-like and TM-like mostly-polarized fields.

Point-defect resonant modes in periodic waveguides. Discussed intrinsic radiation loss, and tradeoff between localization and loss (and its origin in the localization of the Fourier transform under the light cone). Discussed effect of substrate on cavity losses (due to the substrate pulling down the light cone). Cavity Q vs. number of periods.

Introduced photonic-crystal slabs, which have 2d periodicity and index-guiding in the z direction. Started sketching the projected band diagram of a slab with a triangular lattice of holes (a "hole slab").

Further reading: Chapters 7 and 8 in the course notes.

Lecture 35: 9 May 2007

Further discussion of photonic crystal slabs. Discussed hole slabs and rod slabs, the effect of the finite thickness of the slab, the existence of an optimal thickness (from the perspective of gap size), and the importance of "polarization" (TE-like vs. TM-like, really even and odd symmetry with respect to z=0).

Line-defect waveguides, and the similarity and differences with respect to two-dimensional photonic crystals. The effect of the substrate and symmetry breaking.

Point defects: resonant cavities and intrinsic radiation loss. Just as before, we have a tradeoff between localization and loss. Briefly discussed the general (difficult) optimization problem for Q (at a fixed modal volume V, or vice versa). Discussed the mechanism of multipole cancellation which becomes apparent when you optimize Q over various parameters.

Further reading: Chapter 8 in the course notes.

Lecture 36: 11 May 2007

Introduction to photonic-crystal fibers: optical fibers with a periodic "cladding" surrounding their core. The cladding can be either 2d-periodic (e.g. holey fibers, with a lattice of air holes) or 1d-periodic (concentric rings, a.k.a. Bragg fibers). The guiding mechanism can be either a band gap (a.k.a. photonic-band-gap fibers), which can even guide light within an air core, or ordinary index-guiding.

Discussed index-guided photonic-crystal (holey) fibers, in particular a holey fiber with a missing hole at the center. Sketched the band diagram, and pointed out a couple of mysterious features: there are at most 4 modes at all frequencies (including 2 doubly degenerate modes), and 3 of those modes lie nearly on top of one another. Explained these features by deriving the rigorous scalar limit: at kz goes to ∞, the modes are described by a scalar wave equation with zero boundary conditions in the holes, exactly like a 2d TM system with perfect-metal rods. Explained how this gives rise to "endlessly single mode" waveguides (as first derived by Birks et al.), and how it also lets us predict the "LP modes" (linearly polarized modes) that we get for kz. Discuss applications, e.g. nonlinear fibers.

Further reading: Chapter 9 in the course notes.

Lecture 37: 14 May 2007

Band-gap guidance in photonic-crystal fibers. Discussed how the band gap arises from the scalar limit, and how portions of the band gap above the air light line allow us to guide light primarily in air. Discuss the importance of crystal termination and surface states.

Discuss Bragg fibers, and how the band gap arises from the large-radius limit (where the layers are essentially flat multilayer films). Emphasize the importance of rotational symmetry, which means that the modes go as exp(i mφ) where m is a conserved integer quantity (the "angular momentum quantum number"). This means that we need not consider light propagating in the azimuthal (φ) direction far away. Discuss the analogy with hollow metallic tube waveguides, and the various polarization and hybrid-polarization classifications.

Discussed losses of hollow-core fibers, in two categories. First, losses that come from the field penetration into the cladding (e.g. absorption, roughness, etc.), which decrease with core radius. Second, losses associated with coupling between modes (e.g. bending loss) which increase with core radius. Argued that the scalar limit implies that the cladding-related losses decrease, asymptotically, inversely with the cube of the core radius.

Further reading: Chapter 9 in the course notes.

Lecture 38: 16 May 2007

Coupled-wave theory in waveguides, to describe coupling between modes due to bends, tapers, roughness, etc.

Derived the constant-ω Schrodinger-like eigenequation from Maxwell's equations, in which kz, often called β in this context, is the eigenvalue. This is a generalized Hermitian eigenproblem, but is not positive definite. Show that this leads to complex-β eigenmodes: the evanescent modes. Show that the orthogonality relation and normalization is closely related to the power flowing through the waveguide.

Derived coupled-mode theory for a waveguide with a continuously-changing cross section, and proved the adiabatic theorem: if the waveguide cross section changes very slowly, the eigenmode coefficients are constants (the modes are "adiabatically" transformed).

Further reading: S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Physical Review E, vol. 66, p. 066608, 2002.