1 
Maxwell's Equations and Linear Algebra 
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 semidefinite, and thus real ω and orthogonal H>. Compared with quantum mechanics, noted scale invariance. 

2 
Modes of a Metal Box and Mirror Symmetry 
Solved toy example of 1d H_{z}(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 mirrorreflection operator that commutes with the eigenoperator. Defined how vector fields rotate, and showed that H is not a vector but a pseudovector  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. 
Problem set 1 out 
3 
Symmetry Groups, Representation Theory, and Eigenstates 
Representation theory. 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 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 C_{4v} (symmetry group of square). 

4 
Translational Symmetry, Waves, and Conservation Laws 
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 interfaceparallel k). The result is actually much more general than the usual Snell's law, however, because we can use it in cases where the rayoptics picture is invalid. (It will become even more interesting when we look at discrete periodicity). 

5 
Total Internal Reflection and the Variational Theorem 
Considered a twodimensional dielectric waveguide, and derived the "totalinternalreflected" modes (except this is not ray optics). Introduced the farreaching concept of the light cone and the fact that higherdielectric 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 lowfrequency cutoff for guiding. 
Problem set 1 due 
6 
Discrete Translations and Bloch's Theorem
MPB Demo 
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 2dperiodic systems).
Demo'ed the MPB eigensolver software. Computed the band diagram and eigenmodes of the simple 2d waveguide structure we already considered in class. Also showed a more complicated system, a 1dperiodic sequence of 2d dielectric cylinders, and showed that this also has guided modes (quite different from a naive totalinternalreflection picture). 
Problem set 2 out 
7 
Bloch's Theorem, Time Reversal, and Diffraction 
Reviewed Bloch's theorem, and stated it in its moregeneral 3dperiodic form, and showed how e.g. continuous translational symmetry is simply a special case. Described how the other spacegroup 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 Blochenvelope eigenstates at a particular k transform according to the symmetries of Θ_{k}, which are a subset of the space group of the whole system (k breaks some of the symmetry in general).
Also discussed timereversal symmetry. Reversing time is equivalent to conjugating the eigenproblem, and from this we saw that the k and k eigenstates have the same eigenvalue. Timereversal symmetry is broken, however, if Ε is complex  in particular, we still have a Hermitian problem if e is complexHermitian, but we don't have timereversal symmetry. This can happen when you have an external magnetic field, and the resulting magnetooptic effect can be used to make optical isolators.
Considered diffraction from a 1dperiodic 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 onedimensional systems, resulting in DBR mirrors (hence iridescent colors in nature), FabryPerot cavities, DFB lasers, and so on. Sketched the band structure of an archetypical 1dperiodic system. 

8 
Photonic Band Gaps in 1d, Perturbation Theory 
Derived the origin and general characteristics of the photonic band gap and band diagram in one dimension, starting with a uniform system and considering the effect of a small perturbation.
Derived 1storder perturbation theory for the eigenvalues of a perturbed Hermitian operator, both the nondegenerate and the degenerate case. (See also "timeindependent" or "stationary" perturbation theory in any quantummechanics text). Applied to Maxwell's equations, and found the firstorder correction in the frequency from a small change Δ∈. Used this to compute the size of the first band gap in a weakly periodic system. 

9 
1d Band Gaps, Evanescent Modes, and Defects 
Began by reviewing 1d gap structures, and how lowest "dielectric band" is concentrated in high∈ layer while next "air band" is forced out by orthogonality, hence the gap. Discussed numerical results.
Discussed the traditional quarterwave stack, which leads to maximum field attenuation in the band gap, and gave some useful analytical formulas for the layer thicknesses, midgap frequency, and gap size.
By analytic continuation near the band edge, showed that states in the band gap are exponentially decaying, plus a oscillatingsign 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 "coupledcavity waveguides" (Yariv, et al. Opt Lett 24, 711 (1999).) that have low velocity (slope) and a zerodispersion point. Derived the cosinelike dispersion relation using a tightbinding approximation, based on very general considerations (Hermitian, mirror symmetry, linearity, weak coupling). 
Problem set 2 due
Problem set 3 out 
10 
Waveguides and Surface States, Omnidirectional Reflection 
Examined offaxis propagating in 1dperiodic structures: projected band diagrams, FabryPerot waveguides, surface states, and omnidirectional mirrors. Much reference were made to the bookfigure from lecture 8.
Began with the offaxis 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 ∈E^{2}.
Next considered the projected band diagram in which the ω for all values of k_{z} (z = periodic direction) are plotted as a function of k_{y} (y = parallel to layers), resulting in continuum regions. Related asymptotic behavior at large k_{y} to rayoptics limit, explaining asymptotic slope and narrowing bandwidths.
Next considered a FabryPerot waveguide, formed by a defect in the periodic structure, giving rise to a guided band in the gap as a function of k_{y}. 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 indexguiding 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 omnidirectional reflection from a multilayer film. Showed that for TM polarization, one always has a range of omnidirectional 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. 

11 
Group Velocity and Dispersion 
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 narrowbandwidth pulse.
In a general periodic medium (for real, nondispersive ∈ and real wavevector k), showed that dω/dk = flux/energydensity = energy velocity. Did this by first deriving the HellmanFeynman 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 ∈ at least 1 (for real, nondispersive e and real wavevector k) in a general periodic medium. This was a straightforward application of a few inequalities, including the triangle inequality and the CauchySchwarz inequality for inner products.
Discussed "superluminal" situations with nonHermitian systems (gain/loss or evanescent fields) and for strongly dispersive ∈. Weakly dispersive∈ will be handled in homework.
Discussed impact of groupvelocity ("chromatic") dispersion (frequencydependent group velocity), and defined the dispersion parameter D. Discussed divergence of D at zerogroupvelocity point in FabryPerot waveguide, or at any band edge.
Closed by preparing for 2d and 3d periodicity: derived/defined the primitive reciprocal lattice vectors G_{i}. 
Problem set 3 due
Problem set 4 out 
12 
2d Periodicity, Brillouin Zones, and Band Diagrams 
Reviewed group velocity, noted that phase velocity is not uniquely defined in a periodic system due to nonuniqueness of k vector.
Considered the 2dperiodic 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 (C_{4v} symmetry).
Defined special points Γ, X, and Μ 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, and pointed out that nonaccidental degeneracies only occur at Γ and Μ points (which have C_{4v} symmetry and thus have a 2dimensional irreducible representation). 

13 
Band Diagrams of 2d Lattices, Symmetries, and Gaps 
Continued with TM band diagram of sq. lattice of dielectric rods in air. Explained flatness of band edges at Γ, X, and Μ, and derived sufficient symmetry conditions for zero slope at those points. Discussed exception of nonzero slope at ω=0 Γ point, and relation to effectivemedium 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. 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°). 

14 
Triangular Lattice, Complete Gaps, and Point Defects 
Continued studying triangular lattice. Found the B.Z. and I.B.Z. (with corners Γ, Μ, and Κ), and showed that it has C_{6v} symmetry (which has 2 twodim. 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 nextnearest neighbor direction(s). Showed that K point has 2ndnearestneighbor periodicity and C_{3v} symmetry.
Showed TM/TE band diagram for tri. lattice of holes, which has complete TM+TE gap. Explained why first TM bands are doubly degenerate at Κ.
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 doublydegenerate dipoles modes from upper band at X, and then pulls down more modes. Related the defect modes to the irreducible representations of the C_{4v} symmetry group. 
Problem set 4 due 


Midterm Exam covering all material up to and including lecture 14 (but no point defects in 2d crystals). 
Problem set 5 out 
15 
Line and Surface Defects in 2d, Numerical Methods Introduction 
Linedefect 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 (finitedifference, 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 conjugategradient 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. 

16 
Conjugategradient, Finitedifference Timedomain (FDTD) Method 
Finish conjugategradient discussion. Also talk about subpixel averaging and convergence.
Introduced the finitedifference timedomain (FDTD) method, and demo'ed our "Meep" timedomain code. 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. 
Problem set 5 due
Problem set 6 out 
17 
More FDTD: Yee Lattices, Accuracy, VonNeumann Stability 
More indepth discussion of FDTD methods. Introduced the Yee lattice in 1d/2d/3d, analyzed accuracy of centerdifference approximations vs. forward/backward differences, and explained how boundary discontinuities degrade the nominal quadratic accuracy. VonNeumann stability analysis and the Courant factor relating temporal and spatial discretizations. Introduced perfectlymatched layer (PML) absorbing boundaries. 

18 
Perfectly Matched Layers (PML), Filter Diagonalization 
Continued discussion of PML: talked about numerical reflections and implementation of frequencydependent ∈ and µ via introduction of auxiliary fields.
Discussed filterdiagonalization methods (FDM) in a Fourier basis, for computing resonant frequencies and loss rates from timedomain simulation.
Began discussion of threedimensional photonic crystals. Introduced cubic/fcc/diamond lattices and reciprocal lattice/Brillouin zone. Contrasted fcc and diamond lattices of spheres. 

19 
3d Photonic Crystals and Lattices 
Continued discussion of 3d crystals. Introduced "Yablonovite" and analyzed its symmetry and irreducible Brillouin zone. Also discussed Woodpile, natural opal, and inverseopal crystals.
In MIT structure (which is simpler to visualize since layers resemble 2d crystals), discussed line and point defects and surface states. 
Problem set 6 due 
20 
Haus Coupledmode Theory, Resonance, and Q 
Concluded 3dcrystal discussion with description of multiphoton lithography and interferencelithography schemes.
Began introduction of Haus coupledmode theory (see also Haus, H. Waves and Fields in Optoelectronics. Chapter 7.), to analyze devices combining waveguides and resonant cavities.
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 waveguidecavitywaveguide system, and showed 100% Lorentzian transmission peak at resonance. Defined quality factor, Q, and gave various interpretations. 

21 
Coupledmode Theory with Losses, Splitter / Bend / Crossing / Filter Devices 
Continued coupledmode theory discussion. Considered imperfect waveguidecavitywaveguide filter systems, with asymmetry and/or losses. Analyzed splitter, bend, and crossing. Analyzed resonantabsorption system where one wants to absorb 100% of incident light. Analyzed channeldrop filter. 

22 
Bistability in a Nonlinear Filter, Periodic Waveguides 
Analyzed optical bistability in nonlinear filter/cavity.
Introduced hybrid systems combining band gaps and index guiding. 1dperiodic waveguides in 2d and 3d, projected band diagrams, guided modes, and "gaps". 

23 
Photoniccrystal Slabs: Gaps, Guided Modes, Waveguides 
Localized defect modes (resonant modes) in 1dperiodic waveguides. Quality factor (Q) and losses, relation to Fourier decomposition, effect of substrates.
2dperiodic photoniccrystal slabs with vertical index guiding: gaps, symmetries, effects of slab thickness. Linedefect waveguides in rod and hole slabs. Losses from asymmetry, disorder. 

24 
Cavities in Photoniccrystal Slabs
Photoniccrystal Fibers 
Localized pointdefect modes in photoniccrystal slabs. Mechanisms for increasing Q: delocalization and cancellation.
Photoniccrystal fibers. Introduce conventional silica fiber and its limitations. Introduce hollowcore fibers: 2dperiodic (gaps, guided mdoes, surface states, effect of termination); 1dperiodic "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.). 

25 
Hollowcore and Solidcore Photonicbandgap Fibers 
Continue discussion of hollowcore fibers.
Solidcore fibers: band diagrams and modes, endlessly singlemode, 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). 
Project due 