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Introduction to Photonic Crystals: Bloch’s Theorem, Band Diagrams, and Gaps (But No Defects) Steven G. Johnson and J. D. Joannopoulos, MIT 3rd February 2003 1 Introduction Photonic crystals are periodically structured electromagnetic media, generally possessing photonic band gaps: ranges of frequency in which light cannot prop- agate through the structure. This periodicity, whose lengthscale is proportional to the wavelength of light in the band gap, is the electromagnetic analogue of a crystalline atomic lattice, where the latter acts on the electron wavefunction to produce the familiar band gaps, semiconductors, and so on, of solid-state physics. The study of photonic crystals is likewise governed by the Bloch- Floquet theorem, and intentionally introduced defects in the crystal (analo- gous to electronic dopants) give rise to localized electromagnetic states: linear waveguides and point-like cavities. The crystal can thus form a kind of per- fect optical “insulator,” which can confine light losslessly around sharp bends, in lower-index media, and within wavelength-scale cavities, among other novel possibilities for control of electromagnetic phenomena. Below, we introduce the basic theoretical background of photonic crystals in one, two, and three dimensions (schematically depicted in Fig. 1), as well as hybrid structures that combine photonic-crystal effects in some directions with more-conventional in- dex guiding in other directions. (Line and point defects in photonic crystals are discussed elsewhere.) Electromagnetic wave propagation in periodic media was first studied by Lord Rayleigh in 1887, in connection with the peculiar reflective properties of a crystalline mineral with periodic “twinning” planes (across which the dielectric tensor undergoes a mirror flip). These correspond to one-dimensional photonic crystals, and he identified the fact that they have a narrow band gap prohibiting light propagation through the planes. This band gap is angle-dependent, due to the differing periodicities experienced by light propagating at non-normal incidences, producing a reflected color that varies sharply with angle. (A similar effect is responsible for many other iridescent colors in nature, such as butterfly wings and abalone shells.) Although multilayer films received intensive study 1 2-D periodic in two directions 3-D periodic in three directions 1-D periodic in one direction Figure 1: Schematic depiction of photonic crystals periodic in one, two, and three directions, where the periodicity is in the material (typically dielectric) structure of the crystal. Only a 3d periodicity, with a more complex topology than is shown here, can support an omnidirectional photonic bandgap over the following century, it was not until 100 years later, when Yablonovitch and John in 1987 joined the tools of classical electromagnetism and solid-state physics, that the concepts of omnidirectional photonic band gaps in two and three dimensions was introduced. This generalization, which inspired the name “photonic crystal,” led to many subsequent developments in their fabrication, theory, and application, from integrated optics to negative refraction to optical fibers that guide light in air. 2 Maxwell’s Equations in Periodic Media The study of wave propagation in three-dimensionally periodic media was pi- oneered by Felix Bloch in 1928, unknowingly extending an 1883 theorem in one dimension by G. Floquet. Bloch proved that waves in such a medium can propagate without scattering, their behavior governed by a periodic envelope function multiplied by a planewave. Although Bloch studied quantum mechan- ics, leading to the surprising result that electrons in a conductor scatter only from imperfections and not from the periodic ions, the same techniques can be applied to electromagnetism by casting Maxwell’s equations as an eigenproblem in analogue with Schro¨dinger’s equation. By combining the source-free Fara- day’s and Ampere’s laws at a fixed frequency ω, i.e. time dependence e−iωt, one can obtain an equation in only the magnetic field ~H: ~∇× 1 ε ~∇× ~H = (ω c )2 ~H, (1) where ε is the dielectric function ε(x, y, z) and c is the speed of light. This is an eigenvalue equation, with eigenvalue (ω/c)2 and an eigen-operator ~∇ × 2 1 ε ~∇× that is Hermitian (acts the same to the left and right) under the inner product ∫ ~H∗ · ~H ′ between two fields ~H and ~H ′. (The two curls correspond roughly to the “kinetic energy” and 1/ε to the “potential” compared to the Schro¨dinger Hamiltonian ∇2 + V .) It is sometimes more convenient to instead write a generalized Hermitian eigenproblem in the electric field ~E, ~∇× ~∇× ~E = (ω/c)2ε ~E, which separates the kinetic and potential terms. Electric fields that lie in higher ε, i.e. lower potential, will have lower ω; this is discussed further in the context of the variational theorem of Eq. (3). Thus, the same linear-algebraic theorems as those in quantum mechanics can be applied to the electromagnetic wave solutions. The fact that the eigen- operator is Hermitian and positive-definite (for real ε > 0) implies that the eigenfrequencies ω are real, for example, and also leads to orthogonality, vari- ational formulations, and perturbation-theory relations that we discuss further below. An important difference compared to quantum mechanics is that there is a transversality constraint : one typically excludes ~∇ · ~H 6= 0 (or ~∇ · ε ~E 6= 0) eigensolutions, which lie at ω = 0; i.e. static-field solutions with free magnetic (or electric) charge are forbidden. 2.1 Bloch waves and Brillouin zones A photonic crystal corresponds to a periodic dielectric function ε(~x) = ε(~x + ~Ri) for some primitive lattice vectors ~Ri (i = 1, 2, 3 for a crystal periodic in all three dimensions). In this case, the Bloch-Floquet theorem for periodic eigenproblems states that the solutions to Eq. (1) can be chosen of the form ~H(~x) = ei~k·~x ~Hn,~k(~x) with eigenvalues ωn(~k), where ~Hn,~k is a periodic envelope function satisfying: (~∇+ i~k)× 1 ε (~∇+ i~k)× ~Hn,~k = ( ωn(~k) c )2 ~Hn,~k, (2) yielding a different Hermitian eigenproblem over the primitive cell of the lat- tice at each Bloch wavevector ~k. This primitive cell is a finite domain if the structure is periodic in all directions, leading to discrete eigenvalues labelled by n = 1, 2, · · ·. These eigenvalues ωn(~k) are continuous functions of ~k, forming discrete “bands” when plotted versus the latter, in a “band structure” or dis- persion diagram—both ω and ~k are conserved quantities, meaning that a band diagram maps out all possible interactions in the system. (Note also that ~k is not required to be real; complex ~k gives evanescent modes that can exponen- tially decay from the boundaries of a finite crystal, but which cannot exist in the bulk.) Moreover, the eigensolutions are periodic functions of ~k as well: the solution at ~k is the same as the solution at ~k + ~Gj , where ~Gj is a primitive reciprocal lattice vector defined by ~Ri · ~Gj = 2piδi,j . Thanks to this periodicity, one need only compute the eigensolutions for ~k within the primitive cell of this reciprocal 3 lattice—or, more conventionally, one considers the set of inequivalent wavevec- tors closest to the ~k = 0 origin, a region called the first Brillouin zone. For example, in a one-dimensional system, where R1 = a for some periodicity a and G1 = 2pi/a, the first Brillouin zone is the region k = −pia · · · pia ; all other wavevectors are equivalent to some point in this zone under translation by a multiple of G1. Furthermore, the first Brillouin zone may itself be redundant if the crystal possesses additional symmetries such as mirror planes; by eliminat- ing these redundant regions, one obtains the irreducible Brillouin zone, a convex polyhedron that can be found tabulated for most crystalline structures. In the preceding one-dimensional example, since most systems will have time-reversal symmetry (k → −k), the irreducible Brillouin zone would be k = 0 · · · pia . The familar dispersion relations of uniform waveguides arise as a special case of the Bloch formalism: such translational symmetry corresponds to a period a → 0. In this case, the Brillouin zone of the wavevector k (also called β) is unbounded, and the envelope function ~Hn,k is a function only of the transverse coordinates. 2.2 The origin of the photonic band gap A complete photonic band gap is a range of ω in which there are no propa- gating (real ~k) solutions of Maxwell’s equations (2) for any ~k, surrounded by propagating states above and below the gap. There are also incomplete gaps, which only exist over a subset of all possible wavevectors, polarizations, and/or symmetries. We discuss both sorts of gaps in the subsequent sections, but in either case their origins are the same, and can be understood by examining the consequences of periodicity for a simple one-dimensional system. Consider a one-dimensional system with uniform ε = 1, which has planewave eigensolutions ω(k) = ck, as depicted in Fig. 1(left). This ε has trivial periodic- ity a for any a ≥ 0, with a = 0 giving the usual unbounded dispersion relation. We are free, however, to label the states in terms of Bloch envelope functions and wavevectors for some a 6= 0, in which case the bands for |k| > pi/a are translated (“folded”) into the first Brillouin zone, as shown by the dashed lines in Fig. 2(left). In particular, the k = −pi/a mode in this description now lies at an equivalent wavevector to the k = pi/a mode, and at the same frequency; this accidental degeneracy is an artifact of the “artificial” period we have chosen. In- stead of writing these wave solutions with electric fields ~E(x) ∼ e±ipix/a, we can equivalently write linear combinations e(x) = cos(pix/a) and o(x) = sin(pix/a) as shown in Fig. 3, both at ω = cpi/a. Now, however, suppose that we per- turb ε so that it is nontrivially periodic with period a; for example, a sinusoid ε(x) = 1+∆ ·cos(2pix/a), or a square wave as in the inset of Fig. 2. In the pres- ence of such an oscillating “potential,” the accidental degeneracy between e(x) and o(x) is broken: supposing ∆ > 0, then the field e(x) is more concentrated in the higher-ε regions than o(x), and so lies at a lower frequency. This oppo- site shifting of the bands creates a band gap, as depicted in Fig. 2(right). (In fact, from the perturbation theory described subsequently, one can show that for ∆ � 1 the band gap, as a fraction of mid-gap frequency, is ∆ω/ω ∼= ∆/2.) 4 k p/a p/a-p/a -p/a w w k gap k a Figure 2: Left: Dispersion relation (band diagram), frequency ω versus wavenumber k, of a uniform one-dimensional medium, where the dashed lines show the “folding” effect of applying Bloch’s theorem with an artificial period- icity a. Right: Schematic effect on the bands of a physical periodic dielectric variation (inset), where a gap has been opened by splitting the degeneracy at the k = ±pi/a Brillouin-zone boundaries (as well as a higher-order gap at k = 0). 5 sin (px/a) cos (px/a) a n hi gh n lo w Figure 3: Schematic origin of the band gap in one dimension. The degener- ate k = ±pi/a planewaves of a uniform medium are split into cos(pix/a) and sin(pix/a) standing waves by a dielectric periodicity, forming the lower and up- per edges of the band gap, respectively—the former has electric-field peaks in the high dielectric (nhigh) and so will lie at a lower frequency than the latter (which peaks in the low dielectric). By the same arguments, it follows that any periodic dielectric variation in one dimension will lead to a band gap, albeit a small gap for a small variation; a similar result was identified by Lord Rayleigh in 1887. More generally, it follows immediately from the properties of Hermitian eigensystems that the eigenvalues minimize a variational problem: ω2 n,~k = min ~E n,~k ∫ ∣∣∣(~∇+ i~k)× ~En,~k∣∣∣2∫ ε ∣∣∣ ~En,~k∣∣∣2 c 2, (3) in terms of the periodic electric field envelope ~En,~k, where the numerator mini- mizes the“kinetic energy”and the denominator minimizes the“potential energy.” Here, the n > 1 bands are additionally constrained to be orthogonal to the lower bands: ∫ ~H∗ m,~k · ~Hn,~k = ∫ ε ~E∗ m,~k · ~En,~k = 0 (4) for m < n. Thus, at each ~k, there will be a gap between the lower “dielectric” 6 bands concentrated in the high dielectric (low potential) and the upper “air” bands that are less concentrated in the high dielectric: the air bands are forced out by the orthogonality condition, or otherwise must have fast oscillations that increase their kinetic energy. (The dielectric/air bands are analogous to the valence/conduction bands in a semiconductor.) In order for a complete band gap to arise in two or three dimensions, two ad- ditional hurdles must be overcome. First, although in each symmetry direction of the crystal (and each ~k point) there will be a band gap by the one-dimensional argument, these band gaps will not necessarily overlap in frequency (or even lie between the same bands). In order that they overlap, the gaps must be suffi- ciently large, which implies a minimum ε contrast (typically at least 4/1 in 3d). Since the 1d mid-gap frequency ∼ cpi/a√ε¯ varies inversely with the period a, it is also helpful if the periodicity is nearly the same in different directions—thus, the largest gaps typically arise for hexagonal lattices in 2d and fcc lattices in 3d, which have the most nearly circular/spherical Brillouin zones. Second, one must take into account the vectorial boundary conditions on the electric field: moving across a dielectric boundary from ε to some ε′ < ε, the inverse “potential” ε| ~E|2 will decrease discontinuously if ~E is parallel to the interface ( ~E‖ is continuous) and will increase discontinuously if ~E is perpendicular to the interface (ε ~E⊥ is continuous). This means that, whenever the electric field lines cross a dielectric boundary, it is much harder to strongly contain the field energy within the high dielectric, and the converse is true when the field lines are parallel to a bound- ary. Thus, in order to obtain a large band gap, a dielectric structure should consist of thin, continuous veins/membranes along which the electric field lines can run—this way, the lowest band(s) can be strongly confined, while the up- per bands are forced to a much higher frequency because the thin veins cannot support multiple modes (except for two orthogonal polarizations). The veins must also run in all directions, so that this confinement can occur for all ~k and polarizations, necessitating a complex topology in the crystal. Ultimately, however, in two or three dimensions we can only suggest rules of thumb for the existence of a band gap in a periodic structure, since no rigorous criteria have yet been determined. This made the design of 3d photonic crystals a trial and error process, with the first example by Ho et al. of a complete 3d gap coming three years after the initial 1987 concept. As is discussed by the final section below, a small number of families of 3d photonic crystals have since been identified, with many variations thereof explored for fabrication. 2.3 Computational techniques Because photonic crystals are generally complex, high index-contrast, two- and three-dimensional vectorial systems, numerical computations are a crucial part of most theoretical analyses. Such computations typically fall into three cate- gories: time-domain “numerical experiments” that model the time-evolution of the fields with arbitrary starting conditions in a discretized system (e.g. finite- difference); definite-frequency transfer matrices wherein the scattering matri- 7 ces are computed in some basis to extract transmission/reflection through the structure; and frequency-domain methods to directly extract the Bloch fields and frequencies by diagonalizing the eigenoperator. The first two categories intuitively correspond to directly measurable quantities such as transmission (although they can also be used to compute e.g. eigenvalues), whereas the third is more abstract, yielding the band diagrams that provide a guide to in- terpretation of measurements as well as a starting-point for device design and semi-analytical methods. Moreover, several band diagrams are included in the following sections, and so we briefly outline the frequency-domain method used to compute them. Any frequency-domain method begins by expanding the fields in some com- plete basis, ~H~k(~x) = ∑ n hn ~bn(~x), transforming the partial differential equation (2) into a discrete matrix eigenvalue problem for the coefficients hn. Truncating the basis to N elements leads to N × N matrices, which could be diagonal- ized in O(N3) time by standard methods. This is impractical for large 3d systems, however, and is also unnecessary—typically, one only wants the few lowest eigenfrequencies, in which case one can use iterative eigensolver meth- ods requiring only ∼ O(N) time. Perhaps the simplest such method is based directly on the variational theorem (3): given some starting coefficients hn, one iteratively minimizes the variational (“Rayleigh”) quotient using e.g. precondi- tioned conjugate-gradient descent. This yields the lowest band’s eigenvalue and field, and upper bands are found by the same minimization while orthogonal- izing against the lower bands (“deflation”). There is one additional difficulty, however, and that is that one must at the same time enforce the (~∇+i~k)· ~H~k = 0 transversality constraint, which is nontrivial in three dimensions. The simplest way to maintain this constraint is to employ a basis that is already transverse, for example planewaves ~h~Ge i ~G·~x with transverse amplitudes ~h~G · (~G + ~k) = 0. (In such a planewave basis, the action of the eigen-operator can be computed via a fast Fourier transform in O(N logN) time.) 2.4 Semi-analytical methods: perturbation theory As in quantum mechanics, the eigenstates can be the starting point for many analytical and semi-analytical studies. One common technique is perturbation theory, applied to small deviations from an ideal system—closely related to the variational expression (3), perturbation theory can be exploited to consider effects such as nonlinearities, material absorption, fabrication disorder, and ex- ternal tunability. Not only is perturbation theory useful in its own right, but it also illustrates both old and peculiarly new features that arise in such analyses of electromagnetism compared to scalar problems such as quantum mechanics. Given an unperturbed eigenfield ~En,~k for a structure ε, the lowest-order correction ∆ω(1)n to the eigenfrequency from a small perturbation ∆ε is given 8 by: ∆ω(1)n = − ωn(~k) 2 ∫ ∆ε ∣∣∣ ~En,~k∣∣∣2∫ ε ∣∣∣ ~En,~k∣∣∣2 , (5) where the integral is over the primitive cell of the lattice. A Kerr nonlinearity would give ∆ε ∼ | ~E|2, material absorption would produce an imaginary fre- quency correction (decay coefficient) from a small imaginary ∆ε, and so on. Similarly, one can compute the shift in frequency from a small ∆~k in order to determine the group velocity dω/dk; this variation of perturbation theory is also called k · p theory. All such first-order perturbation corrections are well known from quantum mechanics, and in the limit of infinitesimal perturbations give the exact Hellman-Feynman expression for the derivative of the eigenvalue. However, in the limit where ∆ε is a small shift ∆h of a dielectric boundary between some ε1 and ε2, an important class of geometric perturbation, Eq. (5) gives a surface integral of | ~E|2 on the interface, but this is ill-defined because the field there is discontinous. The proper derivation of perturbation theory in the face of such discontinuity requires a more careful limiting process from an anisotropically smoothed system, yielding the surface integral: ∆ω(