Mathematical Methods in Nanophotonics

Overview

This is the home page for the 18.325 course at MIT in Fall 2005, 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.

Syllabus

Lectures: TR 12:30–2pm (26-168). 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 and/or biweekly). 33% mid-term exam (November 1). 34% final project (proposal due Nov. 10, project due Dec. 13).

Books: Useful (but not required) books in reserve book room: Photonic Crystals: Molding the Flow of Light by Joannopoulos et al., Group Theory and Its Applications in Physics by Inui et al.

Lecture Summaries and Handouts

Lecture 1: 8 Sep 2005

Converted Maxwell's equations to Hermitian eigenproblem (for linear, lossless media). Introduced Dirac |H> notation and adjoints. Derived real eigenvalues and orthogonal eigenvectors for Hermitian operators. Showed Maxwell operator to be Hermitian, positive semi-definite, and thus real ω and orthogonal |H>. Compared with quantum mechanics, noted scale invariance.

Erratum: I forgot to note another approximation that I made: we assume that ε is independent of ω. The dependence ε(ω) is called material dispersion; in many cases, we can neglect this variation for a transparent material if we are looking at a small enough range of ω. (Effects of dispersion will be discussed later in the term.) Assuming ε(ω) constant is equivalent to assuming that the material responds instantaneously to an applied field.

Further reading: Photonic Crystals: Molding the Flow of Light by Joannopoulos et al., chapter 2.

Lecture 2: 13 Sep 2005

Handouts: Problem Set 1 (due Thursday, 22 September — DATE CHANGED) and Representation Theory Summary

Solved toy example of 1d Hz(x) field between 2 metal plates. Discussed symmetry, and showed that a mirror plane σ implies eigenfields that are even or odd under the mirror plane, corresponding to a mirror-reflection operator that commutes with the eigen-operator. Defined how vector fields rotate, and showed that H is not a vector but a pseudo-vector—it picks up an extra factor of -1 under improper rotations (rotations of a mirror image, which have determinant -1). For this reason the H field "looks" odd when the E field is even, and vice versa, but the fields "really" have the same symmetry.

Began a more complicated example of a field in a 2d metal box—here, there are many symmetry operations and to relate them we must use representation theory. Erratum: The representation-theory handout did not define classes as precisely as it should have. The class should be a "maximal" set of conjugate elements, in that it should contain every element of the original group that is conjugate (as defined in the handout) to any member of the class. (The version linked to above has been revised to correct this ambiguity.)

Lecture 3: 15 Sep 2005

Representation theory (see handout from previous lecture). Proved that eigenstates transform as representations, and showed how this applies to 2d metal box (for which the states fall into three of the five possible representations). Discussed concepts in handout — representations, equivalent and irreducible representations, conjugacy classes, and character tables. Showed how to get character table using the orthogonality rules without knowing the representations; found the character table for C4v (symmetry group of square).

For more information and background on these concepts, and more proofs, see Group Theory and Its Applications in Physics by Inui et al. (in the reserve book room). (Or any similarly-titled book, e.g. Group Theory and Its Physical Applications by L. Falicov.)

Lecture 4: 20 Sep 2005

Derived exp(-ikx) representations for translational symmetry group, and used this to find planewave solutions of Maxwell's equations in homogeneous space. Derived conservation laws: showed that the representation of the field is invariant as a function of time.

Using this, derived Snell's law (conservation of interface-parallel k). The result is actually much more general than the usual Snell's law, however, because we can use it in cases where the ray-optics picture is invalid. (It will become even more interesting when we look at discrete periodicity).

Lecture 5: 22 Sep 2005

Handout: Problem Set 1 Solutions

Considered a two-dimensional dielectric waveguide, and derived the "total-internal-reflected" modes (except this is not ray optics). Introduced the far-reaching concept of the light cone and the fact that higher-dielectric regions introduce discrete guided bands below the light cone that correspond to exponentially localized states.

Derived the variational theorem in Hermitian eigenproblems. Our proof relied on completeness (you'll derive an alternate proof for homework); conversely, we showed how (under certain conditions on the operator), completeness can be derived from the variational theorem. Showed the form(s) taken by the variational theorem for electromagnetism, and (crudely) outlined a simple numerical method based on minimizing the variational Rayleigh quotient.

Proved the existence of guided modes in any 2d waveguide (localization in one direction), under very weak conditions on the dielectric function, using the variational theorem.

Discussed the generalization to 3d waveguides (localization in two directions). Argued that the presence of a substrate on one side of the waveguide creates a low-frequency cutoff for guiding.

Lecture 6: 27 Sep 2005

Handouts: Problem Set 2 (due Thursday 6 October — note change), MPB demo (individual files: 2dwaveguide.ctl, 2dwaveguide-periodic.ctl)

Derived the representations exp(-ika) for discrete translational symmetry with period a, and thereby found Bloch's theorem: in a periodic system, the eigensolutions can be chosen of the form exp(ikx) multiplied by a periodic "Bloch envelope" function. Discussed relation to quantum mechanics and the mystery of the "free electron gas" model from the 19th century. Derived that the Bloch wavevector k, in turn, is periodic with periodicity 2π/a and defined the first Brillouin zone (at least in 1d...the more complete definition will wait until we get to 2d-periodic systems).

Demo'ed the MPB eigensolver software, which is installed on Athena (see above handouts). Computed the band diagram and eigenmodes of the simple 2d waveguide structure we already considered in class. Also showed a more complicated system, a 1d-periodic sequence of 2d dielectric cylinders, and showed that this also has guided modes (quite different from a naive total-internal-reflection picture).

For more information on MPB, see the MPB home page and the MPB manual. Hopefully, you shouldn't need to know much Scheme (just think of it as a file format with lots of parentheses), but see these Guile and Scheme links (GNU Guile is the Scheme implementation used in MPB).

Lecture 7: 29 Sep 2005

Reviewed Bloch's theorem, and stated it in its more-general 3d-periodic form, and showed how e.g. continuous translational symmetry is simply a special case. Described how the other space-group symmetries interact with k (i.e. the translational symmetries), and two effects in particular. First, the point group (space group ignoring translations) relates the states at one k to states at other k rotated by the symmetries, so that we only need to compute the eigenstates in a small region of k to get the eigenstates and eigenvalues everywhere. Second, the Bloch-envelope eigenstates at a particular k transform according to the symmetries of Θk, which are a subset of the space group of the whole system (k breaks some of the symmetry in general).

Also discussed time-reversal symmetry. Reversing time is equivalent to conjugating the eigenproblem, and from this we saw that the k and -k eigenstates have the same eigenvalue. Time-reversal symmetry is broken, however, if ε is complex—in particular, we still have a Hermitian problem if ε is complex-Hermitian, but we don't have time-reversal symmetry. This can happen when you have an external magnetic field, and the resulting magneto-optic effect can be used to make optical isolators.

Considered diffraction from a 1d-periodic surface, and showed that for sufficiently high frequencies one gets additional reflected beams for additional diffracted orders, corresponding to Fourier components of the dielectric function. This comes from the fact that, in a periodic system, k is only conserved up to multiples of 2π/a.

Introduced (but did not derive) the concept of photonic band gaps in one-dimensional systems, resulting in DBR mirrors (hence iridescent colors in nature), Fabry-Perot cavities, DFB lasers, and so on. Sketched the band structure of an archetypical 1d-periodic system.

Lecture 8: 4 October 2005

Handouts: figures for 1d-periodic structures, from book

Derived the origin and general characteristics of the photonic band gap and band diagram in one dimension, starting with a uniform system and consdidering the effect of a small perturbation.

Derived 1st-order perturbation theory for the eigenvalues of a perturbed Hermitian operator, both the non-degenerate and the degenerate case. (See also "time-independent" or "stationary" perturbation theory in any quantum-mechanics text.) Applied to Maxwell's equations, and found the first-order correction in the frequency from a small change Δε. Used this to compute the size of the first band gap in a weakly periodic system.

Lecture 9: 6 October 2005

Handouts: Problem set 2 solutions, Problem set 3 (due Tuesday 18 October).

Sample MPB files for problem set 3: bandgap1d.ctl, defect1d.ctl.

Began by reviewing 1d gap structures, and how lowest "dielectric band" is concentrated in high-ε layer while next "air band" is forced out by orthogonality, hence the gap. Discussed numerical results (from 4 Oct handout).

Discussed the traditional quarter-wave stack, which leads to maximum field attenuation in the band gap, and gave some useful analytical formulas for the layer thicknesses, mid-gap frequency, and gap size.

By analytic continuation near the band edge, showed that states in the band gap are exponentially decaying, plus a oscillating-sign phase. Furthermore, discussed why larger gaps generally lead to stronger decay. However, gave a couple of counter examples of systems with very large gaps which may have only weak decay: chirped gratings and random gratings (Anderson localization).

Showed how, by introducing a defect in the periodicity, one can trap localized defect modes (with a finite number of discrete frequencies), which are either "pushed up" or "pulled down" from either end of the gap, depending on the type of defect.

Discussed how in actual computation with periodic boundary conditions (ala MPB), defects are computed via supercells, and that this leads to "folded" band structures where many bands must be computed before you get to the localized state.

In the supercell, the defect band is nearly flat, with slope decreasing exponentially with the supercell size. Discuss how this can actually be used in order to form "coupled-cavity waveguides" [ Yariv et al., Opt. Lett 24, 711 (1999) ] that have low velocity (slope) and a zero-dispersion point. Derived the cosine-like dispersion relation using a tight-binding approximation, based on very general considerations (Hermitian, mirror symmetry, linearity, weak coupling).

Lecture 10: 13 October 2005

Examined off-axis propagating in 1d-periodic structures: projected band diagrams, Fabry-Perot waveguides, surface states, and omnidirectional mirrors. Much reference were made to the book-figure handout from lecture 8.

Began with the off-axis band diagram, which causes the TM and TE bands to split—explained why TM bands lie lower than TE bands, due to the discontinuous boundary conditions of ε|E|2.

Next considered the projected band diagram in which the ω for all values of kz (z = periodic direction) are plotted as a function of ky (y = parallel to layers), resulting in continuum regions. Related asymptotic behavior at large ky to ray-optics limit, explaining asymptotic slope and narrowing bandwidths.

Next considered a Fabry-Perot waveguide, formed by a defect in the periodic structure, giving rise to a guided band in the gap as a function of ky. Discussed criteria for whether this guided band intersects the continuum regions. Noted that we can now guide light in a lower index region (even air), quite different from index guiding.

Considered the surface states that arise at an interface—they are confined by index-guiding with respect to the homogeneous medium on one side of the interface and by the band gap with respect to the periodic structure on the other side of the interface. Claimed (without proof, yet), that there is always some crystal termination that gives rise to surface state(s).

Considered the criteria for omni-directional reflection from a multilayer film. Showed that for TM polarization, one always has a range of omni-directional reflection, but for TE polarization one may not, depending on the materials. Discussed the importance of Brewster's angle in the TE projected band diagram.

Lecture 11: 18 October 2005

Handouts: Problem set 3 solutions (see also defect1dsol.ctl), Problem set 4 (due Thursday 27 October), figures from chapter 5 of Photonic Crystals: Molding the Flow of Light

Discussed group velocity dω/dk and dispersion.

Gave classic derivation of group velocity for a medium uniform in one direction, by considering the propagation of a narrow-bandwidth pulse.

In a general periodic medium (for real, nondispersive ε and real wavevector k), showed that dω/dk = flux/energy-density = energy velocity. Did this by first deriving the Hellman-Feynman theorem for the derivative of the eigenvalue of a Hermitian operator, and applied this to compute dω/dk (which was then related to average flux/energy by various vector manipulations).

Then showed that |dω/dk| is at most c, for ε at least 1 (for real, nondispersive ε and real wavevector k) in a general periodic medium. This was a straightforward application of a few inequalities, including the triangle inequality and the Cauchy-Schwarz inequality for inner products.

Discussed "superluminal" situations with non-Hermitian systems (gain/loss or evanescent fields) and for strongly dispersive ε. Weakly dispersive ε will be handled in homework.

Discussed impact of group-velocity ("chromatic") dispersion (frequency-dependent group velocity), and defined the dispersion parameter D. Discussed divergence of D at zero-group-velocity point in Fabry-Perot waveguide, or at any band edge.

Closed by preparing for 2d and 3d periodicity: derived/defined the primitive reciprocal lattice vectors Gi.

Lecture 12: 20 October 2005

Reviewed group velocity, noted that phase velocity is not uniquely defined in a periodic system due to non-uniqueness of k vector.

Considered the 2d-periodic square lattice (in 2d). Defined its lattice vectors and found that the reciprocal lattice is also square lattice.

(Noted relation between reciprocal lattice and Fourier series: the Fourier transform of a 2d periodic function is a series of delta functions at the reciprocal lattice vectors. Periodicity of the Bloch wavevector k arises naturally in this description.)

Defined the first Brillouin zone (B.Z.) (and 2nd, 3rd, ...) = pts closer to k=0 than to any other reciprocal lattice vector, and showed that this gives us all the inequivalent points and preserves the full rotational symmetry of the lattice. Showed how to construct it by using perpendicular bisectors between k=0 and other reciprocal lattice points.

Constructed the first Brillouin zone in 1d, and also for the 2d square lattice. In these simple cases, the B.Z. is simply the unit cell of the recip. lattice, centered on k=0, but this is not always the case! Defined irreducible B.Z. (I.B.Z.), and constructed I.B.Z. for square lattice of circular rods (C4v symmetry).

Defined special points Γ, X, and M for sq. lattice, and derived the space group in each of the different k regions. Discussed band diagram in 2d, and explained why we normally plot ω vs. k around the I.B.Z. boundary.

Showed TM band diagram of sq. lattice of dielectric rods in air (from handout), and pointed out that non-accidental degeneracies only occur at Γ and M points (which have C4v symmetry and thus have a 2-dimensional irreducible representation).

Lecture 13: 25 October 2005

Continued with TM band diagram of sq. lattice of dielectric rods in air (from handout). Explained flatness of band edges at Γ, X, and M, and derived sufficient symmetry conditions for zero slope at those poitns. Discussed exception of non-zero slope at ω=0 Γ point, and relation to effective-medium theory.

Origin of 2d TM gap. Explained why infinitesimal periodicity does not give a gap (unlike 1d), and discussed how band diagrams "fold" in 2d. In this case, there is a minimum index contrast of about 1.73:1 to get a TM gap. Explained gap in terms of variation theorem and orthogonality of modes, and compared with computed D field plots (see handout). Explained why boundary conditions prevent TE gap in this structure. Showed how TE gap arises in a square lattice of dielectric veins in air (which has no TM gap for these parameters).

Introduced idea that triangular lattice of holes can give simultaneous TE/TM gap. Introduced triangular lattice, gave its lattice vectors, and derived its reciprocal lattice (a triangular lattice rotated 30°).

Lecture 14: 27 October 2005

Handouts: Problem set 4 solutions

Continued studying triangular lattice. Found the B.Z. and I.B.Z. (with corners Γ, M, and K), and showed that it has C6v symmetry (which has 2 two-dim. irreduc. representations). Showed that this B.Z. is more circular than B.Z. of sq. lattice, which makes it easier to have a gap (for rods, index contrast must be only 1.3:1 to get a TM gap).

Showed that, in both sq. and tri. lattices, M point is a standing wave pattern along the next-nearest neighbor direction(s). Showed that K point has 2nd-nearest-neighbor periodicity and C3v symmetry.

Showed TM/TE band diagram for tri. lattice of holes (from handout), which has complete TM+TE gap. Explained why first TM bands are doubly degenerate at K.

Discussed point defect modes in a square lattice, formed by changing the radius of a single rod. Plotted localized mode frequencies vs. rod radius. Explained why reducing rod radius pushes up monopole mode from lower band at M, and increasing rod radius pulls down doubly-degenerate dipoles modes from upper band at X, and then pulls down more modes. Related the defect modes to the irreducible representations of the C4v symmetry group.

"Mid-term" Exam: 1 November 2005

Closed-book/notes, 1.5hrs. You may bring one standard 8.5"×11" sheet of paper with anything written on it that you find useful. Covers all material up to and including lecture 14 (but no point defects in 2d crystals).

The exam — 1 problem plus any 3 of last 4, 25 points each. (Still a bit too long, sorry.)

Problem Set 5, due Tuesday, 8 Nov. 2005. This is the same as the mid-term, but now you have a chance to think a bit more deeply about the problems. It will be graded as an ordinary pset, separate from the mid-term. Do all 5 problems.

Lecture 15: 3 November 2005

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

Line-defect and and surface states in 2d crystals, and projected band diagram. Showed why there is always some terminations that give rise to surface states.

Introduced numerical methods for solving continuous eigenproblems. Discussed choice of basis (finite-difference, planewave, finite element) and reduction to finite matrix via Galerkin method. Talked about scaling of dense matrix methods and sparse/iterative methods for N×N matrices. Showed how iterative methods apply to planewave problem via FFTs.

Began discussing preconditioned conjugate-gradient minimization of the Rayleigh quotient to find the lowest eigenvalues and corresponding eigenvectors. Started with steepest descent, then Newton's method, then preconditioned steepest descent. Didn't quite get to conjugate gradients.

Suggested reading: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide; Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, eds. (SIAM: Philadelphia, 2000).

Lecture 16: 8 November 2005

Handouts: Problem set 6 (due Thurs 17 Nov.) (see also rod-transmission.ctl sample file for problem 4), and Pset 5/Exam solutions.

Introduced the finite-difference time-domain (FDTD) method, and demo'ed our "Meep" time-domain code. See: the Meep homepage for a manual, tutorial, etcetera. (On Athena, `add mpb`.) In particular, we go over concepts that are also described in the introduction and tutorial of the Meep manual: computation of Green's functions, transmission spectra, and resonant modes.

Lecture 17: 11 November 2005

More in-depth discussion of FDTD methods. Introduced the Yee lattice in 1d/2d/3d, analyzed accuracy of center-difference approximations vs. forward/backward differences, and explained how boundary discontinuities degrade the nominal quadratic accuracy. Von-Neumann stability analysis and the Courant factor relating temporal and spatial discretizations. Introduced perfectly-matched layer (PML) absorbing boundaries.

Lecture 18: 15 November 2005

Handouts: Filter diagonalization, figures from chapter 6 of (revised) Photonic Crystals book, fcc Brillouin zone

Continued discussion of PML: talked about numerical reflections and implementation of frequency-dependent ε and μ via introduction of auxiliary fields.

Discussed filter-diagonalization methods (FDM) in a Fourier basis (see handout), for computing resonant frequencies and loss rates from time-domain simulation. See also harminv, our free FDM software.

Began discussion of three-dimensional photonic crystals. Introduced cubic/fcc/diamond lattices and reciprocal lattice/Brillouin zone (see handouts). Contrasted fcc and diamond lattices of spheres.

Lecture 19: 17 November 2005

Handout: Problem set 6 solutions

Continued discussion of 3d crystals. Introduced "Yablonovite" and analyzed its symmetry and irreducible Brillouin zone. Also discussed Woodpile, natural opal, and inverse-opal crystals, and concluded with this MIT structure.

In MIT structure (which is simpler to visualize since layers resemble 2d crystals), discussed line and point defects and surface states.

Lecture 20: 22 November 2005

Handout: cavity/waveguide devices

Concluded 3d-crystal discussion with description of multi-photon lithography and interference-lithography schemes.

Began introduction of Haus coupled-mode theory (see also H. Haus, Waves and Fields in Optoelectronics, chapter 7), to analyze devices combining waveguides and resonant cavities like those in handout.

Started with simple system of waveguide coupled to cavity, and showed that while the reflection is always 100%, there is a time delay proportional to the cavity lifetime for frequencies near resonance. Then analyzed waveguide-cavity-waveguide system, and showed 100% Lorentzian transmission peak at resonance. Defined quality factor, Q, and gave various interpretations.

Lecture 21: 29 November 2005

Continued coupled-mode theory discussion. Considered imperfect waveguide-cavity-waveguide filter systems, with asymmetry and/or losses. Analyzed splitter, bend, and crossing from handout. Analyzed resonant-absorption system where one wants to absorb 100% of incident light. Analyzed channel-drop filter (from handout).

Lecture 22

Analyzed optical bistability in nonlinear filter/cavity [see also Soljacic et al., Phys. Rev. E 66 055601(R) (2002) ].

Introduced hybrid systems combining band gaps and index guiding. 1d-periodic waveguides in 2d and 3d, projected band diagrams, guided modes, and "gaps".

Lecture 23

Localized defect modes (resonant modes) in 1d-periodic waveguides. Quality factor (Q) and losses, relation to Fourier decomposition, effect of substrates.

2d-periodic photonic-crystal slabs with vertical index guiding: gaps, symmetries, effects of slab thickness. Line-defect waveguides in rod and hole slabs. Losses from asymmetry, disorder.

Lecture 24

Handout: photonic-crystal fibers

Localized point-defect modes in photonic-crystal slabs. Mechanisms for increasing Q: delocalization and cancellation.

Photonic-crystal fibers. Introduce conventional silica fiber and its limitations. Introduce hollow-core fibers: 2d-periodic (gaps, guided mdoes, surface states, effect of termination); 1d-periodic "Bragg fibers" (gaps, conservation of angular momentum, comparison with metal waveguides, TE/TM/hybrid modes, suppression of cladding absorption and perturbation theory, fabrication by Fink et al.).

Lecture 25

Continue discussion of hollow-core fibers from above.

Solid-core fibers: band diagrams and modes, endlessly single-mode, enhanced nonlinearities. Effective area (with vectorial corrections) to determine strength of nonlinearities. The scalar limit and LP modes (asymptotic field patterns, symmetry and degeneracies, finite vs. infinite # modes, and origin of band gaps).