Prop. QUMI..&err. 1994,Vol. 18, pp. 153-200 Copyright 0 I!394ElsevicrScience Ltd Printed in Great Britain. All rights reserved. 0019-6727194 $26.00
BANDGAPS IN PERIODIC STRUCTURES
PIERRE R. VILLENEUVE* and MICHBL Prcti Centre d’optique, Photonique et Laser, Universitk Laval, Qu&ec GlK 7P4, Canada
Abstract-Photonic bandgaps are defined as frequency intervals for which propagation of electromagnetic waves is forbidden in all 471steradians within a dielectric structure with a periodic index of refraction. Such structures consist, for example, of dielectric spheres in suspension or air holes in a dielectric material, with a spatial period comparable to the electromagnetic wavelength. The principal feature of periodic structures is their ability to perturb the density of electromagnetic states within the structures. Since photonic bandgap materials can essentially suppress all states, the radiative dynamics within the materials can be strongly modified. By changing the atom-field radiative coupling, photonic bandgap materials could lead to the inhibition of spontaneous emission; if a local defect is introduced within the structure, it will behave like a high-Q microcavity. The existence of bandgaps can be predicted from a classical treatment of the vector wave equation. The use of the plane-wave expansion method can lead to accurate results but introduces two problems related to the dielectric discontinuities and the plane-wave cutoff. Experimental investigations at microwave frequencies have demonstrated many of the properties of photonic bandgap structures.
CONTENTS 1. Introduction 1.1. Basic concept 1.2. History 2. Theoretical Analysis of Photonic Bandgaps 2.1. Three-dimensional structures 2.1.1. Face-centered-cubic structure 2.1.2. Diamond structure 2.1.3. Simple-cubic structure 2.2. Two-dimensional structures 2.2.1. Triangular structure 2.2.2. Square structure 3. Accuracy of the Numerical Calculations 3.1. Numerical representation of the dielectric structure 3.2. Supergaussian dielectric function 3.2.1. One-dimensional structures 3.2.2. Two-dimensional structures 3.2.3. Three-dimensional structures 4. Applications of Photonic Bandgap Materials 4.1. Suppression of spontaneous emission 4.2. High-Q single-mode microcavities 4.3. Observation of Anderson localization 4.4. Signal propagation and control 5. Experimental Investigations 5.1. Samples 5.2. Experimental setups 5.2.1. Anechoic chamber method 5.2.2. Coherent microwave transient spectrosc :opy method 5.2.3. Waveguide method 6. Conclusion References
154 154 155 158 160 160 164 166 168 170 173 174 174 177 179 180 182 183 183 184 186 187 188 188 189 189 192 196 197 198
*Present address: Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. 153
P. R. VILLENEUVE and M. Pm& 1.
1.1. Basic Concept
Waves in a periodic diffusing medium undergo multiple scattering when their wavelengths are comparable to that of the period medium. For example, electrons-or electronic waves-undergo multiple scattering in a crystal from atoms distributed periodically in space. While constructure interference gives rise to allowed electronic states such as valence and conduction bands, destructive interference gives rise to forbidden states. The allowed electronic energies form bands separated by gaps of forbidden energy states or electronic bandgaps. By analogy to electrons in crystals, photons-or electromagnetic waves-undergo multiple scattering in a dielectric structure whose index of refraction (or dielectric constant) is spatially modulated with a spatial period comparable to the electromagnetic wavelength. In some dielectric structures with a three-dimensional periodicity, there are no propagation modes in any direction for a range of frequencies, giving rise to a complete ‘photonic bandgap’. These structures consist, for example, of high-index dielectric spheres suspended in a low-index dielectric background or air holes in a high-index dielectric material, with lattice constants comparable to the electromagnetic wavelength. At optical wavelengths, the dielectric structures are about a thousand times larger than the crystal structures of atoms. The physical basis of electronic and photonic bandgaps is the same, namely the coherent interference of waves scattered from periodically distributed scatterers in space. While electronic bandgaps relate to electrons in a crystal, photonic bandgaps relate to photons in a periodic dielectric structure. We can expect that many features of electrons will have a photonic counterpart, such as Brillouin zones, dispersion relations, complete energy gaps, impurity bands, etc. However, there are important differences. First, the wave nature of the electron required for the interference effects depends on a quantum-mechanical approach; the electron is treated as a scalar wave with a DeBroglie wavelength. The electronic wave function must satisfy Schrodinger’s scalar equation while the electromagnetic wave function must satisfy Maxwell’s vector equations. Furthermore, electrons are fermions while photons are bosons and electronelectron interactions are strong while there are no photon-photon interactions at intensities of practical interest. Although the word ‘photon’ is used, the appearance of bandgaps arises from a strictly classical treatment of the problem. The propagation (or absence of) can be computed from the classical vector wave equation in a material with a periodic index. The principal feature of periodic structures is their ability to perturb the density of electromagnetic states within the structure. Since photonic bandgap (PBG) materials can essentially suppress all states and since the atom-field radiative coupling depends on the electromagnetic environment, then PBG materials could greatly affect the radiative dynamics within the structures and lead to significant changes in the properties of optical devices. The propagation of electromagnetic waves in periodic dielectric structures is not a new topic. Devices such as optical gratings, volume holograms and distributed feedback lasers are well known and have been studied for some time. However, the propagation of light in these devices is forbidden only for a small range of wavevectors and the index contrasts are typically of the order of 0.1% of the average index. The creation of a complete photonic bandgap in all 4n steradians requires a three-dimensional periodicity in the microstructure and a large index contrast typically of the order of 2: 1 or greater. It is noted here that standard simplifications such as the slowly varying envelope approximation commonly used in the analysis of devices with small index contrasts cannot be used for the analysis of photonic bandgaps since the fields in a bandgap can be strongly attenuated within as little as one or two periods of the microstructure.
1.2. History Although the existence of electronic bandgaps has been known for some time, the first suggestion of a complete gap for electromagnetic waves has only recently been made by Yablonovitch”’ at Bellcore and John”’ at Princeton in 1987. The suggestion that periodic structures could give rise to complete electromagnetic bandgaps prompted research to find a geometry that would yield the largest gap for the smallest index contrast. At the outset it was suggested that the first Brillouin zone in reciprocal space should be as spherical as possible.* The face-centered-cubic (fee) structure was presented as being the best candidate since its first Brillouin zone is closer to a sphere than all other common crystal structures. Simple analytical conditions for the generation of photonic bandgaps predicted the existence of a gap for an index contrast as low as 1.21 to l(‘) and 1.46 to l(*) but these analyses did not take into consideration certain aspects of the problem such as field polarization. The index contrasts of 1.21 and 1.46 would then be more appropriately labeled ‘lower limits’. Although Yablonovitch and John had chosen to work with the fee structure, the spatial distribution of the dielectric material in real space still needed to be determined. The first suggestion”’ called for a three-dimensional checkerboard pattern with cubic etch pits one-quarter wavelength in length. A laborious experimental cut-and-try method was undertaken at Bellcore by Yablonovitch and Gmittefi3T4’where many fee structures were fabricated from low-loss dielectric materials. The structures were made of either dielectric spheres in suspension or spherical air holes in high-index dielectric materials. The experiments were carried out at microwave frequencies since the fabrication of ordered dielectric arrays at optical wavelength scales was (and still remains) a challenge. The propagation of electromagnetic waves was tested through the periodic structures. In 1989, a full gap was thought to be observed in a structure with spherical air holes in a material of index 3.5 but the finite size of the sample did not allow sufficient resolution to detect symmetry-induced degeneracies in the dispersion relation. Instead of generating a complete gap, the periodic structure gave rise to a pseudogap with a significant reduction of the density of states. Meanwhile, theoretical investigations were being carried out. Initially, a scalar approximation was used to compute the dispersion relations of periodic structures.(5-9) The vector wave equation was reduced to a scalar equation analogous to Schrodinger’s equation used in electronic band theory. Although the comparison between photons and electrons was attractive, it rapidly became apparent (‘O)that the scalar theory used for the computation of electronic bands in crystals was inadequate for the computation of photonic bandgaps. By neglecting the vector nature of electromagnetic waves, theoretical results did not agree with the experiments. For example, the existence of large photonic gaps was predicted in fee structures of dielectric spheres in suspension while none were observed in the experiments. A full vector analysis was then undertaken to solve Maxwell’s equations by taking into account the vector nature of the electromagnetic field. In 1990, Leung and Lit&“) and Zhang and Satpathy”” computed the dispersion relations for the fee structure with spherical air-filled atoms in a dielectric background and found the symmetry-induced degeneracies. To solve this degeneracy problem, Ho et Cal. used a diamond structure-an fee structure with two atoms per Wigner-Seitz unit cell-to lift the degeneracies in the dispersion relation and to open a full gap instead of a pseudogap. The use of two atoms in the fee Wigner-Seitz cell broke the perfect spherical symmetry and lifted the degeneracies. It then became apparent(14’ that the use of any nonspherical atom in the fee structure would lift the degeneracies. Photonic bandgaps were predicted to be as large as 29% of the central frequency for an index contrast of 3.6 to 1 in the diamond structure. (13)It was also predicted that the gap should remain open for an index contrast as low as 2.1 to 1. Many other structures have been studied such as
for using the most sphere-like
zone will be presented
P. R. VILLENEIJVE and M. Pm&
square and circular rods in a simple-cubic structure(‘5’ and have been found to yield photonic bandgaps. Theoretical calculations have shown that the structure requiring the lowest index contrast to yield a gap (known to date) is a type of diamond structure requiring an index contrast of 1.87 to l.(‘@ Researchers were also concerned with the feasibility of fabricating these periodic dielectric structures. Instead of growing a structure one period at a time and etching individual holes, Yablonovitch et ~1.(‘~*“-‘~’ presented a method that only required drilling long slanted holes into the top surface of a solid dielectric slab. The intersection of the holes below the surface produced an fee structure of nonspherical air-filled atoms. Such a structure was fabricated at microwave frequencies and the gap width was close to 20% of its center frequency in a material with a refractive index of 3.6. This structure solved two important problems that experimentalists were facing at the time: (i) the fee structure was made of non-spherical atoms which lifted the degeneracies and opened a complete gap, and (ii) the structure could be fabricated on the scale of optical wavelengths by reactive ion etching. Preliminary attempts at microfabricating this structure were carried out in 1992 at Bellcore.(20’ A search for photonic bandgaps has also been carried out in two-dimensional structures (or more accurately structures with a two-dimensional periodicity). The search was motivated by two observations: (i) two-dimensional structures should be easier to fabricate than their three-dimensional counterparts, and (ii) experiments performed in one-dimensional microcavities and waveguides had shown that lower-than-three-dimensional structures could exhibit a strong reduction of the density of allowed states in a structure.(2” The two-dimensional structures were made of parallel dielectric rods in air or cylindrical air holes in a dielectric material. Propagation was studied in the plane perpendicular to the rods. As in three-dimensional structures, the objective was to find the geometry that would yield the largest gap for the smallest index contrast. In 1991, Plihal et &(22.23’published the dispersion relations for arrays of long parallel rods with circular cross-sections whose intersection with the perpendicular plane formed a square or triangular lattice. Meade et ~1.c~~’ and Villeneuve and Piche(25’ showed that the triangular lattice (also known as the two-dimensional hexagonal-closepacked lattice) could give rise to a full gap in the perpendicular plane. The use of nonspherical atoms in three-dimensional structures prompted Villeneuve and PichC to use rods with asymmetric cross-sections to lift some of the degeneracies in the two-dimensional dispersion relations. Instead, this has resulted in a reduction of the forbidden gaps and an increase of the minimum index contrast required to open a gap. Other structures, such as arrays of air rods with square cross-section in a dielectric material, have been found to yield two-dimensional bandgaps. (25v26) The effects of the finite length of the dielectric rods on the photonic bandgaps have been considered by Maradudin and McGurn.(27’ Experiments in two-dimensional structures were carried out also at microwave frequencies. The structures were fabricated with standard machine tools in materials with a high microwave refractive index. Robertson et ul.(28-30’ investigated the photonic band structures and surface modes with the coherent microwave transient spectroscopy (COMITS) technique which uses optoelectronically pulsed antennas to generate picosecond electromagnetic transients. This method allowed them to measure the dispersion relations for a wide range of frequencies. McCall et u~.P’-~~’used a waveguide scattering chamber to investigate bandgaps, defect modes and localization in arrays of dielectric rods. In 1992, theorists began to question the accuracy of their numerical results. The computation of photonic dispersion relations had been carried out with the plane-wave method which expanded the field and the dielectric function in infinite series of plane waves. The electromagnetic problem was reduced to an infinite-dimensional eigenvalue problem which was solved with standard matrix-diagonalization methods. This required the storage and manipulation of large matrices. While an infinite series expansion of the electric and magnetic fields should have led to the same solutions-since electromagnetic fields are
in periodic dielectric structures
complementary-the truncation of the matrix equation affected the solutions. Sijziier et ~1.“~’ and Villeneuve and PichC’35’computed photonic bandgaps using both the electric and magnetic fields and compared the convergence of each method. The number of plane waves used in the expansion was dictated by the available computer time and memory. Another problem associated with the plane-wave method was the poor representation of discontinuous functions near the discontinuities. Sbziier et a1.(34)used gaussian functions which provided good convergence of the eigenvalues. To approximate a discontinuous function, Villeneuve and Picht(35’ used continuous high-order supergaussian functions and showed that the plane-wave expansion method could lead to reliable and accurate results. Other numerical methods which do not require the storage and handling of large matrices have been presented for computing photonic bandgaps. For example, Meade et ~1.‘~~’ used special features of certain matrix operations in order to cut down on the amount of computer memory required to solve the problem. Pendry et al. and Stefanou et al. presented a finite-element method which allowed them to determine the transmission coefficient through finite-sized slabs of periodic(37”S’ or disordered(39) material without storing large matrices. Ever since the concept of photonic bandgaps was suggested in 1987, researchers have been looking to understand the conditions necessary to open and maximize bandgaps. Simple analytic criteria were suggested for the minimum index contrast required to open a gap(‘,*) However, the phenomenon and for the volume fraction required to maximize the gaps. (34,4o) of photonic bandgaps has proven to be too complicated to be explained by a simple formula. Although it has been accepted at the outset that the fee structure would be the best candidate to yield photonic bandgaps (since its Brillouin zone is closest to a sphere), some authors”5*4” have suggested that the fee lattice would not necessarily be special and that the emergence of gaps would be related to the strength of the dielectric contrast and to the connectivity of the dielectric components of the structure. The role of the connectivity has also been addressed by Meade et al. (42)from a field analysis in the periodic structure. The principal feature of PBG structures is their ability to affect the radiative dynamics within the structure. At the outset, Yablonovitch”) had suggested that photonic crystals could change the properties of the radiation field in such a way that there would be no electromagnetic modes available in the dielectric structure. If the radiative transition frequency of an excited atom was to lie within the bandgap then the atom would be incapable of coupling its energy to the surrounding medium due to the absence of available modes. The atom would stay in an excited state leading to the inhibition of spontaneous radiative decay. Since spontaneous emission is a major source of energy loss, speed limitation and noise, its suppression could strongly enhance the performance of optical devices such as semiconductor lasers. However, it is not desirable to eliminate spontaneous emission entirely in devices requiring at least one electromagnetic mode to operate. The introduction of a single defect in a perfect periodic structure would generate such a mode. The field would be localized around the defect and its frequency would lie within the gap. The single defect could be made either by adding or removing some dielectric material in one unit cell. Yablonovitch et uI.(~~.~) and Meade et ~1.‘~~’have compared a defect mode to a Fabry-Perot resonator made of two quarter-wave multilayer dielectric mirrors separated by one-half wavelength; the half-wavelength layer plays the role of a defect and gives rise to a standing-wave mode localized around the defect. Theoretical calculations of localized defect modes in three-dimensional structures have been carried out by Yablonovitch et u1.(43-44) and Meade et uI.(~‘)using a supercell method and by Leungc4@using a Green’s function method and vector Wannier functions. In 1991, these modes were observed experimentally in two- and three-dimensional structures: McCall et Al. observed a localized mode within the forbidden gap of a perfect two-dimensional array of dielectric rods in which a single rod had been removed; Yablonovitch et uZ.(~~.~) observed a localized mode in an fee structure of nonspherical air holes in which a small slice of the dielectric material had been removed in one unit cell. Localized modes have also been
P. R. VILLENEWVE and M. Pm&
predicted by Meade et a1.(47)and observed by Robertson et al@’ at the surface of some photonic crystals. PBG structures with a single defect could be used for the fabrication of new optical devices such as single-mode light-emitting diodes with high quantum efficiency and zero threshold.(4v~50) PBG structures also offer a new environment for the analysis of atomic physics. In 1988, Kurizki and Genack”‘) suggested that atomic interactions could be profoundly modified in PBG structures. In a series of publications, John and Wang(40’52-54) investigated the radiative properties of a single atom whose decay frequency lies within the forbidden gap. John and Wang showed that the photon that an atom emits could tunnel through many lattice cells before being reflected and reabsorbed by the atom leading to strong self-dressing of the atom by its own localized radiation. They also indicated that the photon would be localized around the atom and would form a stable bound state with the atom analogous to electron-impurity-level bound states which occur in the gap of a semiconductor. If several atoms were placed within the periodic structure then neighboring atoms could interact via the exponentially attenuated field localized around each atom. The localized electromagnetic modes would then broaden into a photonic impurity band which would exhibit novel nonlinear optical properties.o4) The problem of radiative coupling between atoms in a PBG material has also been addressed by Kurizki.(55) Mossberg and Lewenstein@) indicated that, by suppressing certain radiative transitions, PBG structures could dramatically modify the radiative properties of strongly driven atoms. Dowling and Bowden (57)studied the beat radiation from dipoles whose radiation frequencies are near the edge of a photonic bandgap and Kurizki et al.‘S8’ suggested that atomic properties could also be modified when atoms interact with a local defect mode. Disordered periodic structures could also modify the atomic radiative dynamics. Experimental observations of a change in the fluorescence lifetime of molecules in a disordered aqueous suspension of microspheres have been reported by Martorell and Lawandy@@‘) and Tong et aZ.@‘)Although the colloidal system did not generate a complete photonic bandgap, the density of allowed states was significantly reduced. In addition, John has indicated(2.40) that a small disorder in the periodic structure could also induce localization of electromagnetic modes in the forbidden gap. Other applications call for the use of PBG structures in semiconductor laser mode control@) and highly efficient planar antennas.(63*61) The concept of photonic bandgaps in periodic dielectric structures has been extended by Dowling and Barut to other periodic systems such as sonic waves in fluids with periodic density variations’65’ and neutral magnetic dipoles in a periodic magnetic field.@@ Finally, it must be noted that by 1990, PBG structures were beginning to attract much attention from the general scientific community. Several articles have been published in popular scientific magazines such as Nature,@7’ Physics Today,(40-68’ Optics and Photonics News,(‘8~63,69-72) Physics World (73)and New Scientist.(74J In this review, we begin by presenting a theoretical analysis of photonic bandgaps in Section 2 followed by an analysis of the precision of the theoretical calculations in Section 3. Sections 4 and 5 are devoted to some of the important applications and experimental investigations of PBG materials. We end the review with a presentation of the future directions and prospects for PBG material research in Section 6. 2. THEORETICAL
We start by formulating the electromagnetic problem in an infinite nonabsorbing periodic dielectric structure. This can be done either with the magnetic or electric field. Although both
Photonic bandgaps in periodic dielectric structures
formulations should lead to the same solutions, it will be shown that numerical results depend on the actual formulation of the problem. The wave equations are
1 -VxX(r,t) =w2%+,2) [ c (r) Vx[Vxd(r,t)]=~~~~(r)&(r,f)
where the dielectric function t: (r) is position-dependent that
and periodic with lattice vector R such
L(r - R) = 6 (r).
Since the eigenfunctions of Eqns. (1) and (2) must satisfy Floquet’s theorem, they can be expanded in a series of plane waves of the form X’(r,
t ) = exp( - iot ) c H, exp[i (k + G) - r]
where w is the frequency, k the wave vector and G the reciprocal-lattice vector. By substituting Eqns. (4) and (5) into Eqns. (1) and (2) respectively, we find the following infinite-dimensional matrix equations for the Fourier coefficients Hc and E, : (k+G)
where ccc = t (G - G’) is the Fourier transform of c (r) and tfcc = q (G - G’) is the Fourier transform of the inverse of c (r); the coefficients qcc can be found either by Fourier transforming directly the function c -l(r))li 1,12~lS,22,23,2M7.34.35,41) (inverse expansion method) or by Fourier transforming the function .Z(r) and then inverting the resulting matrix ~cG’(‘o~‘s~‘6~24~34~35~4’~42~45~ (Ho’s method). Both formulations of rlcc are identical in infinite-dimensional systems. Equations (6) and (7) are the dispersion relations for electromagnetic waves of frequency o and wavevector k. Equation (6) is a standard eigenvalue problem while Eqn. (7) is of the form Ax = 23x which is a generalized Hermitian eigenvalue problem (since A and B are Hermitian matrices and B is positive definite). By choosing the origin such that L (- r) = t: (r), cGbecomes real and the matrix B becomes real symmetric. Hence, Eqn. (7) can be solved with standard numerical methods.* Equation (7) could have been written in the standard form B -‘Ax = Lx but since B -‘A is not Hermitian, this would have introduced additional computational complexity. To solve Eqns. (6) and (7), the infinite-dimensional eigenvalue problems must be truncated. In a d-dimensional structure (d = 1,2 or 3), the size of the matrix system scales as 2N d where N is the number of plane waves of the expansion along each coordinate axis. The factor of two appears due to the two possible polarizations of light in the structure. If we use the inverse expansion method to solve Eqn. (6), then Eqns. (6) and (7) will yield identical results only in the asymptotic limit of an infinite number of plane waves. However, if Ho’s method is used to solve Eqn. (6), Eqns. (6) and (7) will yield identical eigenvalues and eigenvectors when the same finite number of plane waves is used in the expansion, (76)The results presented in Sections 2.1 and 2.2 are computed from Eqn. (6) using Ho’s method. It will be shown in Section 3 that Ho’s method leads to a faster convergence of the eigenvalues than the inverse expansion method. *We have used the routine DGVLSP from the commercial IMSL library. JFQE1*,*--E
P. R. VILLENEWEand M. Pm&
Most of the vocabulary and notation used in this section is borrowed from solid state physics. Readers not familiar with the concepts of crystal structures and reciprocal lattices should refer to introductory solid state textbooks. 2.1. Three-Dimensional
Many three-dimensional structures have been studied in the hope of finding the geometry that would yield the largest gap for the smallest index contrast. The actual geometry of the dielectric crystal appears only in the position-dependent function c(r) of Eqn. (6). It is evaluated on a fine grid in the real-space Wigner-Seitz (WS) unit cell and is then Fourier transformed into reciprocal space. We can write the periodic dielectric function as ~(r)=x~,exp(iG*r)
where &c= -
and V,,, is the volume of the WS cell. By using Eqn. (9), we can study any periodic arrangement of ‘atoms’ with any shape and filling fraction. For some structures, the integral in Eqn. (9) can be evaluated analytically; however, in the case of odd-shaped geometries (including structures where atoms overlap) the integral must be computed numerically. The filling fraction of each material in the structure is obtained from a numerical integration over the real-space unit cell. The dielectric structures can be made of either high-dielectric atoms in a low-dielectric background or low dielectric holes in a high-dielectric background. A convenient choice for the low-dielectric material is air (C = 1); although the calculations presented in this section are valid for materials with any dielectric constant, we will often refer to dielectric/air structures for simplicity. 2.1.1. Face-centered-cubic structure. First we consider a face-centered-cubic (fee) array of spheres with radius R, and dielectric constant E, embedded in a background material of dielectric constant Q,. The spheres occupy a fractionf, of the total volume. In the fee structure, the maximum proportion of the available volume which may be filled by the spheres before overlapping is 74% which corresponds to the close-packed condition. When L, > E*, the dielectric spheres are in suspension in a low-dielectric material. If the size of the spheres is smaller than close-packed then the spheres must be supported in position by the background material. When the filling fraction of the dielectric spheres is larger than 74%, the spheres overlap and the structure becomes self-supporting. On the other hand, when cn < Q,, the structure is self-supporting when the low-dielectric spheres are not touching each other but the structure remains self-supporting beyond the close-packed condition until the spheres occupy 97% of the available volume, at which point the background material becomes disconnected and forms little islands of high-dielectric material. If we use the notation Sy for the filling fraction of the spheres in the close-packed condition and f $" when the background material becomes disconnected, then both the spheres and background material are connected when fP
+I& + 3fak- %)
where x = IGIR,. The above equation is valid for any lattice with one spherical atom per WS real-space cell when f,
Photonic bandgaps in periodic dielectric structures
as shown in Fig. la. Repeated translation of the WS cell in real space generates an fee structure. However, there are no restrictions as to the spatial distribution of the dielectric material in the cell. In this section, the fee structure contains one spherical atom centered in each WS cell. The choice of the parametersf,, L, and cb has been motivated by experiments.“V4) Several fee structures have been fabricated with various filling fractions and index contrasts. In fee arrays of dielectric spheres in suspension (c, > cb), both experimental and theoretical investigations were unable to generate a complete gap. On the other hand, experiments indicated the presence of a gap in an fee structure of overlapping spherical air holes occupying 86% of the total volume (f, >fP) in a dielectric material of refraction index 3.5 (6. < Lo). The structure is shown in Fig. lb. Theoretical investigations for this structure agreed with most of the experimental results except for wave vectors close to two symmetry points; Fig. lc shows the theoretical dispersion relations for the fee array of air spheres along the symmetry lines of the first Brillouin zone. The graph is called the band diagram-in analogy with electronic band theory-and gives the allowed electromagnetic states in the dielectric structure as a function of the wave vector k. The first Brillouin zone and its symmetry points are shown in Fig. Id. The band diagram is shown in the standard reduced-zone scheme of the first Brillouin zone; all the wavevectors k can be shifted into the first Brillouin zone by adding or subtracting a suitable reciprocal lattice vector G such that k’ = k + G lies within the first Brillouin zone. This can easily be seen from Eqns. (4) and (5); if we let &(r) = exp[i (k + G) *r] then eL’(r) = exp[i (k’ + G) . r] = exp[i (k + G’) * r] = &(r) for any vector G. The band diagram is computed from Eqn. (6) using Ho’s method with N = 9 expansion terms along each coordinate axis. The frequency is normalized with respect to 27rc/L where L is the unit cell length. The photonic band diagram can be scaled to any frequency of the spectrum simply by scaling the unit cell length of the dielectric crystal to the wavelength of the electromagnetic field. The dispersion curves in Fig. lc correspond to travelling waves in the dielectric structure. The velocity of energy propagation or group velocity is given by am/a k which corresponds to the slope of the dispersion curves. In the long wavelength limit, the dispersion is linear (w = u lkl) as in a continuous medium where the velocity u of the energy propagation is
FIG. la. Wigner-Seitz real space unit cell of the fee lattice.
P. R. VILLENEUVEand M. Prcti
FIG. lb. fee crystal structure consisting of overlapping spherical air holes in a dielectric background material. The spheres are located at the corners and center of each face of a cube and they occupy 86% of the available volume.
5 g &
0.4 0.2 0.0
xu FIG. lc. Band diagram
of the fee structure
FIG. Id. First
shown in Fig. lb. The dielectric index of 3.5.
zone and symmetry
of the fee lattice.
has a refractive
independent of frequency. In a gap, there are no purely real values of k that satisfy the dispersion relations. However, there exists a complex value of k whose imaginary part will scale inversely with the attenuation length within the crystal. In a gap, the wave is exponentially attenuated through successive periods. The attenuation can be significant within as few as one or two lattice periods. At most symmetry points of the Brillouin zone, the group velocity is zero and the solutions are standing waves, analogous to Bragg reflection of X-rays; we recall that when the Bragg condition is satisfied, a standing wave is set up through successive reflections back and forth between parallel planes of atoms in the crystal and there is no net transmission of energy. We say that a gap exists if, for a given frequency, there are no allowed modes for any wavevector k or in other words, there is a gap between the nth and (n + 1)th bands when w, + , (k) > co,,(k’) for all k and k’. The experimental search was focused on finding a gap between the second and third bands. The photonic band structure was probed by measuring the transmission of electromagnetic waves incident on the structure. These waves were either TE-polarized (electric field parallel to the surface) or TM-polarized (electric field parallel to the plane of incidence). The lowest bands along the T-L and T-X lines are double degenerate since TE- and TM-polarized waves are degenerate along those lines. Reflection symmetry of the crystal breaks the degeneracy along the other symmetry lines. However, the second and third bands remain degenerate at the W point (independent of the dielectric contrast and filling fraction of the spheres) and they cross each other along the U-X line near the symmetry point U preventing a gap from opening between the second and third bands in the fee structure.* To overcome this problem, Ho et ~1.“~’ suggested changing the symmetry of the lattice in order to break the degeneracy at the W point induced by the crystal symmetry. They chose to keep the fee lattice (since it is the most sphere-like) but to fill the WS cell not with one spherical atom but two atoms, as in the diamond structure. Before presenting the results for the diamond structure, we note that in the band diagram shown in Fig. lc, a gap is almost completely open between the eighth and ninth bands except at the X and W symmetry points for which the condition C+(X) > w,(W) is not satisfied. However, had the filling fraction of the air spheres been smaller than 86%, the experimenters might have had a better chance of finding a complete gap-not between the second and third bands but at higher frequencies between the eighth and ninth. For example, when the filling fraction of the air spheres is 74% (close-packed condition), a complete gap opens up between the eighth and ninth bands. The structure is shown in Fig. 2a. It is very similar to the one used by Yablonovitch and Gmitter(3*4’except that the filling fraction of the air spheres is 12% smaller. The size of the gap is relatively small (around 4.7%) and would have been difficult to detect. The size of the gap is defined as the ratio of the gap width (between the highest point of the eighth band and the lowest point of the ninth band) to the midgap frequency. It increases significantly as the dielectric constant of the dielectric material increases, as shown in Fig. 2b. It is not very likely that complete gaps could be found at higher frequencies since the band density increases as a function of frequency. We now present the rationale for using the most sphere-like Brillouin zone. The Brillouin zone of the fee structure shown in Fig. Id is closer to a sphere than all other common crystal structures since it has the smallest ratio of the distances between the origin and the closest and farthest points on the surface of the Brillouin zone. In the fee structure, the W symmetry point (farthest) is about 29% farther from the origin than the L point (closest). Hence a gap at the W point is more likely to occur at a higher frequency than a gap at the L point. This can be seen in Fig. lc in spite of the degeneracy at the W point. In order to favor a gap to occur at a common frequency for all values of k, every point on the Brillouin zone should be equally distant from r. *The finite size of the crystal
used in the experiment
did not allow adequate
to observe these degeneracies.
P. R. VILLENEUVE and M. Pm&
FIG. 2a. fee crystal structure consisting of spherical air holes in a dielectric background material. The spheres are in the close-packed condition and occupy 74% of the total volume.
2.1.2. Diamond structure. The diamond structure is an fee lattice with two atoms per WS unit cell positioned along the ( 1,1,1) direction giving to the diamond structure a preferred axis. The maximum proportion of the available volume which may be filled by the spheres before overlapping is only 0.34 which is 46% of the filling fraction for the close-packed fee structure with one atom. The diamond structure of air spheres is self-supporting when the filling fraction of the spheres is greater than close-packed; both the spheres and background material are connected and the structure remains self-supporting up to a filling fraction of 99% at which point the background material becomes disconnected. When the filling fraction is smaller than close-packed, the spheres are disconnected and the integral in Eqn. (9) can be computed analytically. The expansion coefficients become EG =
$,6~ + 3f,(c, - Lb)
sm x--:cos xcos +x+G,+G) (f,GfZp) 1
where the parameters have the same definition as before and L is the unit cell length. Figure 3 shows the band diagram for a diamond structure of overlapping air spheres occupying 8 1%
&b FIG.2b. Size of the bandgap in an fee structure of spherical air atoms as a function of the dielectric constant st, of the background material. The size of the bandgap is defined as the ratio of the gap width to the midgap frequency. The gap occurs between the eighth and ninth bands. (Fig. 2b from Ref. 34.)
Photonic bandgaps in periodic dielectric structures
0.8 9 g
FIG. 3. Photonic band diagram of the diamond structure consisting of overlapping air spheres in a dielectric material with a refractive index of 3.6. The air spheres occupy 81% of the total volume. The first Brillouin zone is shown in Fig. Id.
of the total volume. The high-dielectric material has a refractive index of 3.6. Most of the bands along the I-L, I-X and X-W symmetry lines are doubly degenerate, which favors the opening of full gaps. A complete gap appears between the second and third bands. The size of the gap with respect to its midgap frequency is 29%. In contrast with the fee structure, the diamond structure can also give rise to a complete bandgap in an array of dielectric spheres in air (as opposed to air spheres in a highdielectric material). Figure 4a shows the size of the gap with respect to its midgap frequency as a function of the filling fraction of the spheres in a diamond structure for both cases of air spheres in a dielectric material and dielectric spheres in air. The maximum gap to midgap ratio is 15.7% for slightly overlapping dielectric spheres occupying 37% of the total volume whereas the maximum gap can reach 28.8% for air spheres occupying 81% of the available volume. In both structures, the refractive index of the high-dielectric material was 3.6. In the structure of air spheres, the gap exists when both the air spheres and the background material are connected, or fP
P. R. VILLENEUVE and M. PICI& 0.30
._ 5 L
FIG. 4. (a) Size of the photonic bandgap in a diamond structure as a function of the filling fraction of the spheres for arrays of dielectric spheres and air spheres. In both cases, the refractive index of the high-dielectric material was 3.6. (b) Size of the gap as a function of the index contrast between the spheres and the background material. The broken line corresponds to air spheres occupying 8 1% of the total volume while the solid line corresponds to dielectric spheres with a filling fraction of 34%. The size of the gap is defined as the ratio of the gap width to the midgap frequency. The gap appears between the second and third bands for both air and dielectric spheres. (From Ref. 13.)
2.1.3. Simple-cubic structure. The simple-cubic (SC) structure has a cubic WS cell which we choose to fill with either a cubic or spherical atom. When the WS cell is filled with a cubic atom, the SC array of dielectric cubes remains disconnected for any filling fraction (unless f, = 1 in which case the structure is a continuum of dielectric constant 6,). On the other hand, the SC array of air cubes is self-supporting for every filling fraction since the cubes cannot overlap. The expansion coefficients in eqn. (9) are
Cc = +,&lJ +&:,
We have carried out a search with a wide range of parameters but complete bandgaps were not found in simple-cubic arrays of either dielectric cubes or air cubes. In light of the results
for the fee and diamond structures, we could have anticipated the absence of bandgaps since the a- and b-type materials cannot both be connected. In the case of spherical atoms, the close-packed condition is reached when the filling fraction of the spheres is 52%. Both the spheres and background material remain connected up to a filling fraction of 97% at which point the background material becomes disconnected and forms star-shaped islands with six points., When the filling fraction is lower than S2%, the expansion coefficients cG in Eqn. (9) are given by Eqn. (10) since Eqn. (10) is valid for any periodic arrangement with one spherical atom per unit cell below closepacking. Again the existence of full gaps has not been reported for arrays of dielectric atoms in air. On the other hand, a complete gap can be generated in an SCarray of air spheres. Figure 5a shows a simple-cubic structure of overlapping air spheres occupying 81% of the total volume. The corresponding band diagram is shown in Fig. 5b for the case of a dielectric material with a refractive index of 3.6. The first Brillouin zone and its symmetry points are shown in Fig. 5c. A complete gap appears between the fifth and sixth bands. The parameters were chosen to be identical to those used in the previous section for the diamond structure; while the size of the gap in the diamond structure was 29%, the SCstructure gave rise to a 6.4% gap. Figure 6 shows the size of the gap as a function of the radius of the air spheres in the SC structure. The a- and b-type materials are connected when n < R, < &n which corresponds to 0.52
FIG. 5a. Simple-cubic crystal structure consisting of overlapping spherical air holes in a dielectric material. The spheres are located at the corners of a cube and they occupy 8 1% of the total volume.
P. R. VILLENEUVE and M. PICI&
0.2 0.1 0.0
FIG. 5b. Photonic band diagram of the sc structure shown in Fig. 5a. The dielectric material has a refractive index of 3.6.
connected and Q should be larger than q-actually, around twice as large or more. This is not a sufficient condition but can serve as a guideline. In terms of the filling fraction of the u-type material, the condition translates to f z
FIG. 5c. First Brillouin zone and symmetry points of the sc lattice.
TABLE1. Fraction of the available volume occupied by the u-type material in the close-packed condition (fP) and when the b-type material becomes disconnected (ft%). The filling fractions are given for three different crystal structures
fee diamond SC (spherical
0.7405 0.3401 0.5236
0.965 + 0. I % 0.994 + 0.1% 0.965 k 0.1%
FIG. 6. Size of the photonic
bandgap in a simple-cubic structure as a function of the radius of the air spheres in the dielectric material of index 3.6 (in units of L/2n). The size of the gap is defined as the ratio of the gap width to the midgap frequency. The a- and b-type materials are connected when I[ < R, &c. The gap appears between the fifth and sixth bands. (From Ref. 15.)
TE- and TM-polarized waves are coupled in three-dimensional structures-and even degenerate along certain symmetry lines-they are decoupled in two-dimensional structures lowering the probabilities for the generation of a gap common to both polarizations. By decoupling the fields into two orthogonal polarizations, Eqns. (6) and (7) can be rewritten such that the corresponding matrix systems scale as N2 for each polarization, reducing considerably the computational time and memory required to reach convergence. If the magnetic field is transverse to the rods (TM-polarization), Eqns. (6) and (7) reduce to: (13) (14) where $G = (k + G)x Hs - (k + G), Hx; . However, if the electric field is transverse to the rods (TE-polarization), Eqns. (6) and (7) become ;q&k+G).(k+G’)H,,.-dH,=O
(k + G)fEG - (k + G),(k + G),E,G - o ’ C cG~E,G,= 0. G
We note that in this last case, Eqns. (16a) and (16b) are coupled which gives rise to a
matrix system of dimensions 2N*. The above equations can be solved with either Ho’s method or with the inverse expansion method. We have elected again to work with Ho’s method.
P. R. VILLENEUVE and M. PIC&
In analogy to three-dimensional structures, the two-dimensional periodic function can be expanded in a series identical to the one given in Eqns. (8) and (9). These equations will allow us to consider any periodic array of rods with any shape and filling fraction. 2.2.1. Triungdur structure. First we consider an array of rods with circular cross-section of radius R, and dielectric constant E, located at the corners of a regular triangle (or equivalently at the corners and center of a regular hexagon) in a background material of dielectric constant cb. The cylinders occupy a fractionf, of the total volume. The maximum proportion of the available volume which may be filled by the cylinders before overlapping is 0.91; when the filling fraction is smaller than close-packed, the rods are isolated and the background material is connected throughout the structure; when the filling fraction is greater than 0.91, the cylinders overlap and the background material becomes disconnected. It is not possible for both a- and b-type materials to be connected in a given structure. This is radically different from the behaviour of the three-dimensional structures presented in Section 2.1 and will play an important role in the presence or absence of bandgaps in two-dimensional structures. When the filling fraction of the rods is smaller than close-packed, the expansion terms in Equation (9) can be computed analytically and are given by (17) where J, (x) is the first-order Bessel function of x. Equation (17) is valid for any two-dimensional lattice with one circular rod per WS cell when f,
FIG. 7a. Crystal structure consisting of cylindrical air holes with circular cross-section on a two-dimensional triangular lattice. The air rods occupy 83% of the total volume. The thickness of the ribs between the cylinders is only 4% of the lattice constant.
Photonic bandgaps in periodic dielectric structures
1.0 0.6 ?? $ 0.6 :g 0.4 0.2 0.0
FIG. 7b. Photonic band diagram of the triangular structure shown in Fig. 7a. The solid and broken curves correspond to TM- and TE-polarizations respectively. The dielectric material has a refractive index of 3.5.
FIG. 7c. First Brillouin zone and symmetry points of the triangular lattice.
0.65 0.75 0.85 filling fraction FIG. 8. Size of the two-dimensional photonic bandgap in a triangular lattice of cylindrical air holes as a function of the filling fraction of the air rods. The background material has a refractive index of 3.5. The size of the gap is defined as the ratio of the gap width to the midgap frequency and the asymmetry coefficient a is defined as the ratio of the two semiaxes of the cross-section of the rods. When a = 1, the cross-section is circular.
P. R. VILLENEUVEand M. PICA
air rods. The gap disappears when the rods reach the close-packing condition, which also corresponds to the point where the background material becomes disconnected. The largest gap is 17.4% occurring when], = 0.83 or equivalently when R, = 0.48L where L is the lattice constant or unit cell length. Seeing that the rods are close-packed when R, = OSL then the maximum gap occurs when the ribs between the cylinders are very thin; in fact, the thickness of the ribs is only 4% of the lattice constant. We have found also that the gap remained open for an index contrast as low as 2.6. We note that there seems to be another gap in Fig. 7b at higher frequencies. The size of this gap is only 1.2% which is not much larger than our numerical error. In addition to investigating bandgaps in arrays of rods with perfectly circular cross-section, we have studied the effects of asymmetry on the size of the gaps. We have defined an asymmetry coefficient a as the ratio of the two semiaxes of the elliptical cross-section of the rods. The ellipses were elongated along the (1 ,O) direction* and still located at the corners of a regular triangle. Figure 8 shows the size of the gap in an array of rods with an asymmetry coefficient of 0.80 and 0.70. When tl = 0.80, the dielectric structure becomes partly disconnected whenf, = 0.73 and forms long parallel corrugated dielectric planes separated by air. The structure becomes completely disconnected when f, = 0.97. When 01= 0.70, the partial and complete break-ups occur when f0 = 0.63 and 0.98, respectively. By increasing the asymmetry of the rods, the structure becomes partially disconnected for a larger range of filling fractions. In general, the introduction of asymmetry has the effect of reducing the size of the gap. Moreover, we have found that asymmetric structures require a larger index contrast to open a gap than symmetric structures. When c( = 0.70 or 0.80, the maximum gap occurs when the structure is partially disconnected, whereas the maximum gap in a symmetric structure occurs when the dielectric material is completely connected. In all the structures we have investigated, whether connected or partly disconnected, the gaps were maximum when the thickness of the dielectric ribs between the air columns was very small (in the order of a few percent of the lattice constant). Although gaps can exist in partially disconnected structures, the gaps disappear when the structures become completely disconnected. Prompted by results for three-dimensional structures, we have also used asymmetric rods in an attempt to lift the degeneracy of the first and second TM-bands at the K symmetry point in Fig. 7b. We have indeed been able to lift the degeneracy but instead of opening a gap, the first and second bands crossed each other near the K symmetry point. The band crossing could have been avoided by aligning the ellipses along the (0,l) direction instead of the (1 ,O) direction. In this case, a gap for TM-polarization could have been opened across the entire Brillouin zone but it would not have overlapped with a gap for TE-polarization. Similar band degeneracies were also lifted in arrays of dielectric rods by using rods with an elliptical cross-section but again we have not been able to generate a full gap common to both TEand TM-polarizations. As we have indicated above, the index contrast must be greater than 2.6 in order to open a two-dimensional gap in the triangular lattice.? At first glance, it is somewhat surprising that two-dimensional structures require a larger index contrast to generate a gap in a two-dimensional plane than three-dimensional structures in all 47r steradians. We would have expected that the minimum index contrast in two-dimensional structures be at most equal to that of three-dimensional structures. In order to elucidate this problem, we consider the
“We assumed initially that $L/Z) where L is the tThe triangular lattice with that requires the lowest
the corners of the regular triangle were located at the coordinates (0,O) (L,O) and (L/2, lattice constant. one circular rod per WS cell will actually turn out to be the two-dimensional structure index contrast to open a gap.
in periodic dielectric structures
propagation in an fee structure along the plane P normal to the (l,l,l ) direction. If propagation is forbidden in all 472steradians then it must also be forbidden in plane P. Yet the intersection of plane P and the atoms of the fee structure forms a triangular lattice identical to that presented in this section. So why is it that propagation is forbidden in the plane P of an fee lattice with an index contrast as low as 2 while a two-dimensional triangular lattice requires an index contrast larger than 2.6? Part of the answer lies in the “connectivity” of the different materials in the dielectric structure. In three-dimensional structures, dielectric channels and air channels exist throughout the structure whenf? c f,
- 6) sin(&kJGAsin(&h%J :,h%x k.hiLG,
As in the triangular lattice, bandgaps were not found in square arrays of dielectric rods in air with either square or circular cross-section. Nevertheless, bandgaps can be generated in arrays of cylindrical air holes in a dielectric material for both square and circular cross-sections. The gaps occur between the third and fourth TM-bands in both structures and between the second and third TE-bands when the cross-section of the rods is square and second and third TE-bands when the cross-section is circular. When the cross-section of the rods is circular, the close-packing condition is reached when the rods occupy 79% of the volume. When the filling fraction of the rods is smaller than 79%, the expansion coefficients cc in Eqn. (9) are given by Eqn. (17). Figure 9 shows the size of the two-dimensional gap in a square array of cylindrical air rods in a dielectric material with a refractive index of 4. The gap disappears when the background material becomes disconnected cf, = 0.79). The largest computed value of the gap is 7.1% which is less than half of
FIG. 9. Size of the two-dimensional photonic bandgap in a square lattice of cylindrical air holes as a function of the filling fraction of the air rods. The background material has a refractive index of 4. The size of the gap is defined as the ratio of the gap width to the midgap frequency and the asymmetry coefficient a is defined as the ratio of the two semiaxes of the cross-section of the rods.
P. R. VILLENEUVE and M. PICHB
the largest gap generated in a triangular array with a refractive index of 3.5. The difference would have been even greater had the gap in the triangular array been computed with a refractive index of 4. As in triangular lattices, we have investigated the effects of elliptical rods on the size of the gaps. The ellipses were aligned along the (1,O) direction or equivalently along the (0,l) direction. Figure 9 shows the size of the gap in a square array of elliptical rods with an asymmetry coefficient of 0.98 and 0.97. Since an asymmetry coefficient of 1 corresponds to a perfectly circular cross-section, we can see from Fig. 9 that a small degree of asymmetry gives rise to a significant reduction of the size of the gap; the largest gap decreases by over 40% simply by introducing an asymmetry of 3% in the cross-section of the rods. The gaps disappear when the dielectric structures become completely disconnected. When tl = 0.98, a partial break-up occurs when f,= 0.77 and the structure becomes completely disconnected when f, = 0.80; when o! = 0.97, the partial and complete break-ups occur when f,= 0.76 and 0.81, respectively. Finally, we have noted that when tl < 0.94, the square array can no longer generate a gap for both polarizations. This is very different from the case of the triangular lattice for which the maximum gap was reduced by only 26% with an asymmetry coefficient of 0.80. 3. ACCURACY
OF THE NUMERICAL
The results presented in the previous section were computed numerically using the plane-wave expansion method. This method introduced two problems which affected the accuracy of the solutions: (i) the infinite eigenvalue equation had to be truncated to be solved, and (ii) the dielectric function was poorly estimated near the discontinuities. By truncating the infinite expansion, the high frequency components were removed. Seeing that the Fourier transform of a discontinuous function has wide tails, then a large number of expansion terms are needed to accurately reconstruct the dielectric function. Furthermore, the partial Fourier sum has large amplitude fluctuations close to the discontinuities; it tends to overshoot and undershoot the dielectric constants at the interface. The use of more expansion terms does not remove these overshoots and undershoots but merely moves them closer to the point of discontinuity. This is known as the Gibbs phenomenon. The Gibbs phenomenon becomes especially important when the volume fraction of the dielectric material is small. When f, is close to f fsc,the dielectric ribs between the air atoms are very thin and the reconstruction of the dielectric function can be significantly different from the actual structure. In order to examine the accuracy of the numerical results, we will compare the rate of convergence of the gaps as a function of the number of expansion terms using both Eqns. (6) and (7). We will show that Ho’s method for solving Eqn. (6) yields the fastest convergence. Furthermore, we will present an alternative to the discontinuous step-function representation of the dielectric structure by using supergaussian functions. These functions will allow us to overcome the Gibbs phenomenon and also eliminate spatial fluctuations of the dielectric function. 3.1. Numerical Representation of the Dielectric Structure In a d-dimensional structure, the size of the matrix system scales as 2Nd where N is the number of plane waves of the expansion along each coordinate axis and the factor of two is a result of the two possible polarizations of light in the structure. Some authors have worked with matrices on the order of 750 to 3200, where in some cases an unequal number of plane waves (from 7 to 13) were used along each coordinate axis. By neglecting high-frequency terms, the truncated dielectric function was altered and led to large spatial fluctuations of the dielectric constant-in some cases large enough for the dielectric constant
to be locally negative. We emphasize here that the size of the matrix system increases as the number of terms used in the expansion to the power of the dimensionality of the structure. Therefore, a complete analysis of convergence in three-dimensional structures would require more working memory than is currently available in today’s supercomputers; we would need, for example, close to 100 GBytes simply to manage a matrix generated from using 35 plane waves along each axis. Although we have chosen ‘typical’ photonic structures for the analysis of the convergence, we must note that the rate of convergence depends on the actual structure and is affected by the volume fraction of the dielectric components especially near the close-packing condition. Furthermore, the accuracy decreases as the value of the frequency increases. In order to minimize the numerical error, we have chosen to use the same number of expansion terms along each axis and an equal number of even and odd terms. The numerical calculations presented in this review have been carried out on a CONVEX C-220 vector computer. With an allowed working space of 100 MBytes, the maximum number of expansion terms that we could use was 1787,41 and 11 for one-, two- and three-dimensional structures, respectively. The number of expansion terms has proven to be sufficient to reach convergence in one- and two-dimensional structures but we have not been able to draw definite conclusions for three-dimensional structures. Our objective is to study the rate of convergence of the two numerical methods used for solving Eqn. (6) as a function of the number of expansion terms. We recall that one method called for the expansion of the dielectric function (Ho’s method) while the other called for the expansion of its inverse (inverse expansion method). Each method yields a different representation of the dielectric function. Figure 10a shows the function c -l(r) reconstructed from the truncated series CGqGG exp(iG’ . r) using the inverse expansion method. We have chosen to show a one-dimensional structure for visual simplicity. We have also chosen to use the ‘standard’ step-function representation for the dielectric structure with a small filling fraction of the high-dielectric material, since photonic bandgaps generally occur in structures with a small solid fraction. The dielectric structure in Fig. lob is reconstructed from the matrix elements qGc for IGI = 0. It is reconstructed over one lattice period from N = 7, 13 and 21 plane waves. The values of the dielectric constants before the expansion were 1 and 9. Although the amplitude of the oscillations at the base of the dielectric function I is reduced with respect to L -l(r), the value of the high dielectric constant can be significantly overestimated as shown in Fig. lob. Furthermore, the discontinuous nature of the dielectric function is poor when we use a small number of plane waves. This method is impractical when
1.25 1.00 c 0.75 V ‘; w 0.50 0.25 0.00 -7T
FIG. 10a. Inverse ofa one-dimensional dielectric function reconstructed with N = 7, 13 and 21 plane waves as a function of the position over one lattice period. JPQE,*,2--F
P. R. VILLENEUVEand M. PICHB
FIG. lob. Dielectric function obtained by inverting the function in Fig. 10a. The values of the dielectric constants before the expansion are 1 and 9 and the high-dielectric material occupies 16% of the total volume (or a thickness of approximately n/3).
the dielectric contrast is high since the function 6 -l(r) is very small and its expansion can become negative (crossing the zero) giving rise to singularities in the dielectric function, as shown in Fig. lob when N = 13. Although the full width at half-maximum of the dielectric function should be equal to that of its inverse (since it is a square wave), we notice from Fig. 10 that the plane wave representation of the dielectric structure introduces a large difference between the two widths. Since the size of the bandgaps is highly sensitive to the volume fraction of each dielectric component, the accuracy of the results will be greatly affected. A large number of plane waves will be required for the dielectric function to converge to a square wave. Figure 11a shows the reconstruction of the same dielectric structure but using Ho’s method with 13 plane waves. Ho’s method has the effect of breaking the symmetry of the matrix cGG by taking its inverse; for example u (G, = G;) is no longer equal to q (G, = G;) if G, # G,. Hence in the summation of Eqn. (6), there is a different representation of L -l(r) for each G. Each curve of Fig. 1la corresponds to a different value of G. When IGI is large, the truncated series can be significantly different (e.g. smaller dielectric contrast) from the actual dielectric structure. Figure 11b shows the reconstruction of the dielectric function L (r) from the initial coefficients cGG before inversion for IGI = 0. Ho’s method leads to a more accurate reconstruction of the dielectric structure than the inverse expansion method and it does not yield infinite values of the dielectric function nor negative values of its inverse. 1.25 ,
FIG. 1la. Inverse of the same dielectric function as in Fig. 10 using Ho’s method. corresponding to N = 13 is shown, where each curve corresponds to a different
7T Only the case value of (Cl.
Photonic bandgaps in periodic dielectric structures
reconstructed the initial coefficients kc. before the matrix inversion with N = 7, 13 and 21.
3.2. Supergaussian Dielectric Function
The discontinuous dielectric step-function is reconstructed from a Fourier series of continuous sine and cosine functions. Hence the Fourier expansion of the step-function cannot be uniformly convergent. The series overshoots the actual values at the simple discontinuities and as the number of expansion terms is increased, the overshoots and undershoots increase while moving closer to the discontinuity. The dielectric function can even become negative. On the other hand, when the number of expansion terms is small, there are large spatial fluctuations and again the dielectric function can be locally negative. In order to overcome these problems, we have chosen to write the dielectric function with a supergaussian function of high order. High-order supergaussians are well-behaved and can be made to go from maximum to minimum in a small fraction of the electromagnetic wavelength. Their smooth and almost-discontinuous behaviour make them ideal for numerical analyses. They more closely resemble the real dielectric function than do ‘noisy’ step-functions. Although most of the physics happens at the discontinuities, we will show that by increasing the order of the supergaussians (hence approaching a discontinuous function), the use of supergaussian functions can lead to very accurate results. We write the dielectric function in one unit cell as (19) where w is the waist of the supergaussian, n a positive even integer and 6, and $, the dielectric constants of the atoms and background, respectively. A supergaussian of order n = 2 is simply a gaussian function. The Fourier coefficients of Eqn. (9) are given by drexp[-(]r]/w)“]exp(-iG c,=c,&@+&,-a,) cell fcell
They were computed with a gaussian quadrature over the finite domain of one lattice cell with over 20 integration points per harmonic oscillation. As the number of expansion terms was increased, the number of harmonic oscillations of the dielectric function increased, hence requiring a larger number of sampling points to accurately estimate the integral. Although this was a time-consuming process, we only needed to compute the expansion coefficients once for any given structure, regardless of frequency. Alternatively, the expansion coefficients could have been evaluated with Fast Fourier Transforms (FFTs) which would have cut
P. R. VILLENEUVEand M. PIC&
down on the computation time but would have introduced aliasing problems typical to FFTs. Figure 12 shows the truncated step- and supergaussian functions using N = 27 and 501 expansion terms along each axis. In analogy to step-functions, high-order supergaussians require high-frequency terms for an accurate reconstruction of the function; we have found that the number of expansion terms must be greater than roughly twice the order (N 2 2~2). We emphasize that if this condition is not met, the truncated dielectric function will have large fluctuations. By comparing the dotted curves corresponding to the condition N <.2n in Figs 12a and b, we can see that the expansion of step- and supergaussian functions yield the same results. To improve the series expansion of c(r), we can either increase N or reduce n. By increasing N, the large spatial variations can be suppressed but unlike supergaussians, the series expansion of step-functions does not converge uniformly and the expansion overshoots the dielectric constants at the discontinuities as shown in Fig. 12b. If instead we choose to reduce n, then the sharpness of the discontinuity is reduced. Since the discontinuities play a vital role in the electromagnetic properties of periodic structures, we must choose a high supergaussian order. In order to determine the accuracy of the photonic band structure with supergaussian atoms, we will first consider lower-dimensional structures (d < 3) which will allow us to use a large number of expansion terms, since the matrix dimension scales as 2Nd. These lower-dimensional structures will give rise to the same qualitative behaviour at the discontinuities as do three-dimensional structures. We choose to consider these geometries simply to
P!./\ _,,__ ,_,,,, ho
240 501 ----- 30 501
4 2 _...
i i $7 ,lj ,.. “/ ! .”.._s \/ I
i B c:y\, Y,.., I..___.“_‘-‘ii:._,.z I
FIG. 12. Dielectric function reconstructed with N = 27 and 501 expansion terms along each axis as a function of the position over one lattice period. The dielectric function is written in terms of (a) supergaussians of order n = 30 and 240 and (b) a step-function. The dielectric constants and filling fraction are the same as in Fig. 10.
Photonic bandgaps in periodic dielectric structures
compare the convergence of the different methods. We will then present results for three-dimensional structures although the number of expansion terms will be insufficient to reach full convergence. 3.2.1. One-dimensional structures. First, we consider one-dimensional structures such as thin film multilayer dielectric structures consisting of infinite parallel dielectric planes. We investigate the propagation only along the axis perpendicular to the dielectric planes (or periodic axis). In these structures, the two orthogonal polarizations are degenerate along the periodic axis in analogy to the polarization degeneracy of light incident on a dielectric mirror at normal incidence. As expected from the theory of one-dimensional dielectric layered structures, multiple gaps can be opened even for a small dielectric contrast. The size of the gaps will converge as the order of the supergaussian increases, i.e. as the dielectric function approaches a discontinuous function. We show in Fig. 13a the size of the first gap as a function of the supergaussian order, for an order of up to 240 in a structure with a dielectric contrast of 9. We have used N = 501 plane waves in the expansion which guarantees that the condition N 2 2n introduced above is met everywhere. We present the results for Eqn. (6) using both numerical methods, namely Ho’s method and the inverse expansion method. The corresponding dielectric function is shown in Fig. 12. An increase of the order of the supergaussian corresponds to an increase in the sharpness of the discontinuity. Figure 13a shows that as the dielectric function approaches a discontinuous function, the size of the gaps converges to an asymptotic value. It is therefore reasonable to assume that supergaussians can yield accurate results for photonic bandgaps. The size of the first gap is over 60% of its center frequency and converges within a relative accuracy of lo-’ for n > 18 using Ho’s method and for n > 120 using the inverse expansion method. An analysis of the second and third bands (not shown) has indicated that the order of the supergaussian had to be greater than 240 in order for the gaps to reach convergence. If we were to use a supergaussian of order n = 120 and N 2 2n in three-dimensional structures, we would need over lo6 GBytes of memory to store the corresponding matrix which is not accessible in today’s computers-not by a long shot. We are therefore left with the following question: although we need N 2 2n expansion terms to accurately reconstruct the supergaussian, do we need as many plane waves for the gap to reach convergence? Fortunately, we will show that the answer is no; actually the number of plane waves is significantly smaller than 2n. From the results of Fig. 13a, we choose n = 240 to guarantee convergence. Figure 13b shows the size of the first gap as a function of the number of
2 45 c3 % 30 .g 15 Cn 0 0
Order of Supergaussian FIG. 13a. Size of the first gap in a one-dimensional periodic structure as a function of the order of the supergaussian. The number of expansion terms was N = 501 which satisfied the condition N 2 2n for all n.
P. R. VILLENE~Mand M. Ptcr-tk 62
N FIG. 13b. Size of the first gap as a function of the number of expansion terms for both the step and supergaussian functions. The order of the supergaussian was n = 240. In 13a and 13b, the solid circles are computed with the inverse expansion method while the hollow circles are computed with Ho’s method. The dielectric structure is shown in Fig. 12.
expansion terms for a supergaussian of order 240 and for a step-function using again both methods. In examining these figures, we must pay attention to the rate of convergence and the respective extrapolated values, for each method. First let us consider the rate of convergence. We can see that the size of the first gap converges within a relative accuracy of lo-’ for N < 46 using Ho’s method and 6 x lo-’ using the inverse expansion method.
We note that there is a minimum order to open a gap; here the minimum order is n = 6 with Ho’s method and n = 8 with the inverse expansion method. This is consistent with observations made in Ref. 34 where bandgaps were not seen in structures with a gaussian dielectric function n = 2. From the results obtained in Section 3.2.1, we choose an order of 240 and compute the size of the two-dimensional gap as a function of the number of expansion terms. The size of the gap is shown in Fig. 14b. The gap converges within a relative accuracy of lo-’ for N = 11 with Ho’s method and N > 37 with the inverse expansion method, which indicates again that the use of Ho’s method with a small number of expansion terms is sufficient to yield accurate results, consistent with the results for one-dimensional structures. We have also shown in Fig. 14b the size of the gap using the step-function expansion of the dielectric function. The relative difference between the extrapolated values for the step-function and supergaussian function is less than one percent, using both methods. Figures 14a and 14b also show that the size of the bandgaps computed with the inverse expansion method can be close to the extrapolated value when the plane wave reconstruction of the dielectric structure is poor (small n or small N). As the representation of the dielectric function improves, the inverse expansion method actually leads to worse results before
Order of Supergaussian 20 g
2 16 Cl 2
1 “4 ‘4
i I i
b--f%J-9-e)-~-*~-e-9-~ ... ._. _. i
101 L/a 3
’ ’ ’ ’
.S~~= 8F__. Q ..~i_= 0 ’ ’ ’ ’ s ’ ’ 1 ’
N FIG. 14. Size of the first bandgap across all symmetry points in a two-dimensional triangular lattice as a function of (a) the order of the supergaussian when N = 39 and (b) the number of expansion terms along each axis for a supergaussian of order n = 240, and for a step-function. In (a) and (b), the hollow symbols are computed with Ho’s method while the solid symbols are computed with the inverse expansion method. The air rods of circular cross-section are embedded in a material with an index constant of 3.5. The air rods occupy 75% of the total volume.
P. R. VILLENEUVEand M. PICHB
converging to the asymptotic value. These results can be interpreted qualitatively from the representation of the dielectric structure shown in Fig. 10 where, as the number of plane waves increases for the inverse expansion method, the inverse dielectric function becomes negative before converging to an adequate representation. 3.2.3. Three-dimensional structures. Before presenting the results for three-dimensional structures, we emphasize again that supergaussian functions are useful only in the case where a sufficiently large number of terms is used in the expansion, otherwise the use of supergaussians will lead to similar spatial fluctuations and overshoots as those generated by step-functions. In order to improve the representation, it would then be necessary to lower the order of the supergaussian although at the expense of the sharpness of the discontinuity. We have shown that in one- and two-dimensional structures, the improvement of the spatial representation of the dielectric function with supergaussian functions did not lead to significant changes of the results; neither the small fluctuations of the reconstructed dielectric function nor the Gibbs phenomenon played a vital role in the solutions regardless of the numerical method used. In three-dimensional structures, our limited available computer memory did not justify the use of supergaussian functions. However, it is interesting to compare the performance of both numerical methods for the computation of the bandgaps in a three-dimensional geometry such as dielectric spheres at closepacking in a diamond structure. This structure was presented in Section 2.1.2. Figure 15 shows the size of the three-dimensional bandgap as a function of the number of expansion terms. Again, the two methods lead to different results but as we have indicated above, the number of expansion terms is not sufficient to draw a definite conclusion. We must re-emphasize that Ho’s method has consistently led to a faster convergence for all of the lower-dimensional structures that we have studied, since the numerical representation of the dielectric structure converged faster with Ho’s method than with the inverse expansion method. Therefore, it appears reasonable to conclude that the extrapolated values with Ho’s method are closer to the ‘true’ value than those with the inverse expansion method. In order to overcome the problems generated from the truncation of the infinitedimensional system (due to limited computer working memory), other computational methods will have to be used where the storage and manipulation of large matrix systems will not be 20
FIG. IS. Size of the bandgap across all symmetry points in a three-dimensional diamond structure as a function of the number of expansion terms along each axis for a step-function. The hollow squares are computed with Ho’s method while the solid squares are computed with the inverse expansion method. The dielectric spheres are embedded in air and occupy 34% of the total volume (close-packing) and have a refractive index of 3.6.
in periodic dielectric structures
required. One such method has been suggested by Meade et al.Q@Another method could be an iterative process based on the propagation of a set of eigenfunctions in a periodic structure, reshaping the mode after each period, in analogy to the beam propagation method used for computing the eigenmodes of optical resonators and for studying the propagation in waveguides and optical fibres. This method would be well adapted to Fast Fourier Transforms and to the Prony or Krylov algorithms which would significantly reduce the computer time and memory required to find the eigenvalues. Yet another approach for solving the eigenvalue problem would be to use a finite-element method which matches the field on either side of the dielectric boundaries. Such a method has been presented by Pendry et a1.(37*39) and Stefanou et a1.c3@
The principal feature of periodic dielectric structures is their ability to perturb the density of electromagnetic states within the structure. Since PBG materials can essentially suppress all states, the radiative dynamics within the structures can be strongly modified. 4.1. Suppression of spontaneous emission Since spontaneous emission is an important process at optical frequencies, it can significantly affect the performance of semiconductor optical devices. Spontaneous emission is a major source of energy loss, speed limitation and noise. Hence, much attention has been devoted to its reduction and suppression. The rate of spontaneous radiative decay of an atom (or molecule) depends on the frequency of the radiative transition-the lifetime of an excited state is proportional to the square of the radiative transition wavelength-and on the density of electromagnetic modes available to the photon at that frequency. In the case of a single two-level atom coupled to the electromagnetic field, the rate of transition y between an initial state Ii) and final state If) is given by Fermi’s golden rule:
(21) where Hint is the interaction part of the Hamiltonian that couples the atom to the field and p(o) is the density of modes at frequency o. In free space, there is an infinite number of available modes and the probability of finding an atom in an excited state will decay exponentially with time. However, if the density of available modes is altered from the free-space density, then the atomic radiative dynamics will be affected.* For instance, in the case of two closely-spaced parallel mirrors separated by a distance d, there are no allowed radiation-field modes polarized parallel to the mirrors if d -c A/2. In the ideal case, the mirrors are infinite and perfectly reflecting such that there are no cavity losses. Hence, if an atom with a radiative transition wavelength II is placed between the two mirrors and if the atom’s transition dipole moment is parallel to the mirrors, it will be unable to decay by a radiative process. The atom will stay in an excited state until it loses its energy through nonradiative processes such as phonon relaxation and Auger recombinations, or higher-order radiative processes such as two-photon transitions. It has been known for some time (“I that boundary conditions can alter the emission rate of excited atoms, but experimental evidence of this effect has only been demonstrated 20 years ago by Drexhage.(78) He observed changes of the spontaneous emission rate of organic dye
*The analysis of atomic radiative dynamics in an environment conditions is known as Cavity Quantum Electrodynamics.
whose density of states is altered by boundary
P. R. VILLENEUVP and M. RcI-&
molecules deposited on dielectric films over a metallic mirror. The open geometry of his setup left the dye molecules partly coupled to free-space modes which prevented him from observing a significant reduction of spontaneous emission. Since then, many experimental investigations of the reduction of spontaneous decay rates have been carried out at microwave and optical frequencies in simple systems. Successful demonstrations of the reduction of spontaneous emission in a variety of different systems have been reported-see for example Refs 21, 79-8 I and references therein. One system in particular has been the object of much attention, namely the parallel-plate metallic optical microresonator. Although these resonators are conceptually simple and easy to build at microwave frequencies, the observation of a significant reduction of spontaneous emission in the optical regime has been hampered by the difficulties of controlling and observing the emission of atoms near surfaces. An atom between the plates of an open-sided microresonator is partly coupled to the free-space modes. In order to isolate an atom completely, the atom can be placed in a closed three-dimensional metallic cavity. If the dimensions of the cavity are comparable to the atomic transition wavelength then the available electromagnetic modes will consist of a spectrally discrete set of modes. If there are no modes available at the atomic transition frequency then atomic spontaneous radiative decay can essentially be suppressed. At microwave frequencies, three-dimensional metallic cavities can be fabricated such that all states are suppressed. However, at optical frequencies, metals have very high losses. In analogy to metallic microcavities, PBG materials can suppress all states within the structure with the additional advantage of being usable at optical frequencies since they are made of low-loss dielectrical materials. If the radiative transition frequency of an excited atom lies within the bandgap then spontaneous radiative decay is inhibited. The suppression of spontaneous emission in dielectric materials could have a considerable effect on semiconductor lasers and solid state electronics since the lifetime of the carriers is limited by electron-hole radiative recombination processes. An important application of the suppression of spontaneous emission has been suggested by Yablonovitch.(4q,50) Since the quantum efficiency of laser diodes is limited by energy loss through random spontaneous emission in all 47~steradians, Yablonovitch suggested using PBG materials to suppress all unnecessary spontaneous emission such that the quantum efficiency into the laser mode would be very large. By driving a laser diode with a correlated flow of electrons, the number of output photons per unit time intervals would be constant. This is known as photon-number-state squeezing. 4.2. High-Q Single-Mode Microcavities Although the density of electromagnetic states can be suppressed in perfect PBG crystals, optical devices such as lasers require at least one (and hopefully only one) mode to operate. The introduction of a single defect in the otherwise perfect periodic structure will generate such modes. The frequencies of these modes-alled defect modes or localized modes-lie within the gap and the corresponding fields are localized around the defect. The amplitude of the fields decays exponentially away from the defect; for defect modes whose frequencies are near the center of the gap, the localization can actually be very strong and the fields can decay significantly within as little as one lattice constant. A single defect can be introduced in the crystal either by adding or removing some dielectric material in one unit cell. In either case, there is a minimum defect volume required to generate a localized mode. Although the threshold volume is almost 10 times larger if dielectric material is added instead of removed, the defects only need to be a fraction of the optical wavelength in size. As we have shown in Section 2, the connectivity of the dielectric materials plays a vital role in the existence of photonic bandgaps. Hence, it is not surprising that a very
Photonic bandgaps in periodic dielectric structures
defect will affect the bandgap; if a dielectric material is locally disconnected by the removal of a small segment of a dielectric rib, a local mode will be able to exist. that the removal of dielectric It has been predictedC4’) and experimentally verified (43*44) material in a PBG structure will generate a single mode in the gap while the addition of extra material will give rise to several modes. The experimental results will be presented in Section 5. The frequency of the localized modes can be tuned by changing the size of the defect. The localized modes should have essentially a zero bandwidth in an infinite dielectric crystal but in a finite-sized crystal, the localized modes will be able to tunnel out to the surface; the exponentially attenuated field amplitude will be nonzero at the surface of the crystal giving rise to a finite width of the frequency spectrum.* The local defect can be interpreted as a microcavity of dimensions comparable to the optical wavelength surrounded by periodic dielectric ‘mirrors’ reflecting radiation in all 47~steradians back into the microcavity. Yablonovitch et uZ.(~~@‘) and Meade et aLC4’)have compared the defect modes in a PBG structure to the modes of a one-dimensional Fabry-Perot resonator made of two quarter-wave multilayer dielectric mirrors separated by one-half wavelength or equally to a one-dimensional quarter-wave periodic structure with a quarter wavelength of phase slip (or defect) at the center. Such a Fabry-Perot resonator gives rise to a standing-wave mode with a narrow linewidth localized at the phase slip. Since PBG crystals can reflect all radiation back into the microcavity, the only losses occur from the absorption of the dielectric material itself. A measure of the internal losses and external coupling of a mode in a cavity is given by the Q-factor (or quality factor). It is determined by the number of optical cycles required for the radiation field energy in the cavity to decay by a factor of e- ’ . By using low-loss dielectric materials, the Q-factor of the localized modes can be very large. Hence, PBG materials can give rise to large-Q single-mode microcavities covering a range of frequencies limited only by our ability to find low-loss dielectric materials with a refractive index sufficiently large to generate a photonic bandgap at the desired frequency. The spectral bandwidth of the mode is determined by the microcavity Q-factor which, in a PBG structure, depends only on the radiative and non-radiative losses of the dielectric material. Yablonovitch has suggested that high-Q single-mode PBG microcavities could be used for the fabrication of reliable and thresholdless light-emitting diodes. In analogy to light-emitting diodes, a voltage can be applied to the PBG crystal. The fraction p of the total spontaneous emission coupled into a given mode defines the quantum efficiency of the microcavity. In a typical cavity, B is smaller than unity and spontaneous emission is distributed into several modes. In the case of high-Q PBG microcavities, there exists only one mode, hence /? is essentially unity. In this limit, the output into the single mode increases linearly with pump power. The efficiency-defined as the ratio of the output to input powers-can be larger than 50%‘49*M) in spite of the absence of gain in the PBG microcavity; the efficiency is comparable to that of ‘typical’ laser diodes commercially available. In comparison, typical commercial laser diodes currently in use are approximately 100 to 400 microns in length with a cross-section between 1 and 20 ,um’, resulting in 50 to 200 longitudinal modes within the spectral bandwidth of the gain medium, and a /? value between 10e4 and 10m5. In these laser diodes, single-mode output results from gain saturation and mode competition. In Fabry-Perot microresonators used for vertical cavity surface emitting lasers, there is only one high-Q longitudinal mode within the gain bandwidth but the open-sided cavity allows the atoms to be coupled to free-space modes. small
*The presence of several defects
also have the effect of increasing defects.
through tunneling between localized modes on neighboring
of the localized
P. R. VILLENEUW and M. Pm&
The thickness of the active region is less than one micron and the p value is between 10-l and 10p3. Although high-Q microcavities can generate a steady output of photons, the absence of gain-hence of stimulated emission-in the PBG microcavity leads to a spectrally incoherent output signal dominated by random spontaneous emission. Instead of removing a dielectric rib, a gain medium could have been introduced in the PBG structure. However, the introduction of extra dielectric material would generate several modes, the lowest of which being doubly degenerate if the defect is centered in the WS cell.~43-45)Furthermore, the presence of several modes in the microcavity would have the effect of reducing the fraction of spontaneous emission coupled into each mode (p < 1). As a consequence, there would be a threshold pump power below which the rate of spontaneous emission into any one mode would be larger than that of stimulated emission. As the pump power increased, the output power of the resonator would display a sharp increase when the rate of stimulated emission exceeded the rate of spontaneous emission. More details on the properties of PBG single-mode light-emitting diodes and on the fabrication techniques suggested for the fabrication of a defect in a periodic microstructure can be found in Refs 49 and 50. High-Q PBG microcavities can also be compared to metallic microwave cavities such as those presented in Section 4.1. As we have noted, closed metallic cavities can be fabricated to support only one mode; however, they cannot be used at optical frequencies since metals become very lossy reducing significantly the Q-factor. PBG microcavities have the advantage of being usable at optical frequencies. atom Finally, we note that John and Wang have suggested (52-54)that an individual would also be capable of binding a local mode similar to those generated by a defect in a PBG crystal. The mode would display the same exponentially localized behaviour as defect modes. 4.3. Observation of Anderson Localization As mentioned in Section 1, electrons (or electronic waves) undergo multiple scattering in a crystalline solid. Constructive interference of these waves will allow electrons to propagate resulting in electrical conductivity. If some disorder is added to the crystal structure, then electrical conductivity is affected such that some electrons will undergo destructive interference and will be localized in the disordered crystal. This phenomenon was first mentioned in 1958 by Anderson (**) but experimental observation of such electronic localization has been difficult due to electronelectron interactions and electron-photon scattering. John has suggested(*) that PBG materials would be ideal for the observation of the optical counterpart of electronic Anderson localization. Since there are no photon-photon interactions at intensities of practical interest and since disordered dielectric microstructures could probably be fabricated more easily than ordered structures, the localization of light could be achieved. In the strong localization limit occurring in a perfectly periodic structure, a local mode is bound to an atom-as we have seen in Section 4.2. As the disorder of the structure is increased, photonic bandgaps will disappear and will be replaced by pseudogaps in which the density of states is small (but nonzero). The states in the pseudogaps are strongly localized but very sensitive to the disorder of the structure. Experimental investigations of localized states have been carried out at optical frequencies in three-dimensional random arrangements of scatterers.@3) Localized states have been observed at microwave frequencies in two-dimensional disordered arrays of rods.(33) to observe a reduction of the rate Disordered periodic structures have also been used (59m6’) of spontaneous emission at optical frequencies. Although the disordered periodic structures
did not suppress all the electromagnetic states in the structure, the density of states was significantly smaller than that of free space. 4.4. Signal Propagation and Control In microwave and millimeter wave integrated circuits, signals are radiated off a chip into free space by a planar antenna. However, the power radiated into the dielectric substrate can be significantly larger than the power radiated into the air. Brown et al.‘@’ have suggested using a PBG material as the antenna substrate such that, if the driving frequency of the antenna lay within the gap, there would be no power radiated into the substrate. This concept was verified experimentally CM)for a bow-tie antenna on an fee substrate at microwave frequencies. It was shown that the power was predominantly radiated into the air rather than into the substrate; However, the driving power of the antenna was not completely radiated into the air; power was lost through evanescent modes at the air/substrate interface.c4’) PBG materials could also be used in two-dimensional waveguide structures for wide-angle branching. Since Y-junctions are key components in integrated optical circuits, many investigations have been carried out to achieve a large coupling efficiency into the branches of a waveguide. However, efficient coupling has been limited to small-angle branching since the radiation tends to leak out of the waveguide at the junction. By placing the Y-junction within a two-dimensional PBG material as shown in Fig. 16, the coupling efficiency could be significantly enhanced. If the frequency of the propagating mode in the waveguide would lie within the bandgap, then the radiation could not leak out of the waveguide. Other examples of signal propagation and signal control such as delay lines and nonreciprocal devices have also been suggested. @4)The ability of PBG materials to suppress the density of states in the bandgap can be used for several other applications. We will present some of these potential applications in Section 6.
FIG. 16. Y-Junction surrounded by a PBG material. If the frequency of the radiation propagating in the waveguide (black) lies within the gap, then the radiation cannot leak out into the dielectric material (prey).
P. R. VILLENEUVE and M. PICH~
Before theorists had even begun developing a rigorous analysis of photonic bandgaps the concept had already been tested experimentally. in periodic dielectric structures, Initially, the lack of reliable theoretical results had left experimenters to tackle the problem with a tedious cut-and-try approach. The experiments were carried out in the microwave regime; this allowed the PBG structures to be fabricated with conventional machine tools since the lattice constant of PBG structures scales as the electromagnetic wavelength. Furthermore, dielectric materials with large dielectric constants could readily be found at microwave frequencies in order to satisfy the requirement for high-dielectric contrasts. 5.1. Samples The fabrication of PBG crystals at optical wavelengths remains a great challenge that will need to be solved if PBG crystals are ever to be used to their fullest potential. Many state-of-the-art reactive ion etching techniques are currently being used(85-87)in an attempt at fabricating two- and three-dimensional periodic microstructures. One problem lies in the difficulty of producing long holes with a constant diameter. In a preliminary attempt,“‘) the hole diameter was diverging as a function of etching depth and the microstructure was only a few periods deep. It has been noted, (49) however, that the fields can be significantly attenuated within as few as one or two periods and that photonic crystals need not be many layers thick to observe photonic bandgaps. In the meantime, experimental analyses must be carried out at longer wavelengths since the dielectric structures can be fabricated with conventional machine tools. The samples used in the experiments were typically several centimeters in length and had over 10 periods along each direction. Yablonovitch and Gmitter”’ built several fee structures from low-loss dielectric materials. They fabricated arrays of up to 8000 dielectric spheres made of A&O, (n N 3.06) each with a radius close to 6 mm supported in position by thermal-compression-molded dielectric foam with a microwave refractive index of 1.Ol . Several structures were fabricated with different filling fractions (from low filling fractions up to close-packed) by changing the spacing between the dielectric spheres. These structures failed to produce a complete bandgap which overlapped over all wavevectors in the Brillouin zone. Yablonovitch and Gmitter also fabricated fee structures made of spherical air holes in a dielectric material. They drilled hemispheres onto opposite faces of dielectric slabs with a microwave refractive index of 3.5 using spherical drill bits. The slabs were then stacked such that the hemispheres faced one another forming an fee array of spherical boles.(3) By increasing the size of the drill bits, the filling fraction of the air holes was extended beyond the close-packing limit. In some of their samples, the overlapping holes extended through the walls of the WS cells allowing one to see right through the crystal along certain directionsthis was also the case in Figs 1b and 5a. As it was pointed out in Section 2.1.1, band crossing in the dispersion relation near the W and U symmetry points prevented a complete gap from opening in these fee structures. Following the theoretical results of Ho et al., (I31Yablonovitch et a1.(‘4-‘7-‘9)found that the symmetry-induced degeneracies in the fee structure could be lifted by using nonspherical atoms. They fabricated an fee structure by placing a mask with a triangular array of holes on the top surface of a solid dielectric slab and by drilling three sets of holes at 35.26” from the vertical and 120” about the azimuth. The long cylindrical holes intersected each other below the surface at the center of the WS cell producing an fee structure of non-spherical elongated air-filled atoms. Yablonovitch er al. fabricated three structures using this process with three different hole diameters in a dielectric material with a microwave refractive index
of 3.6. They observed a gap close to 20% of its center frequency in the structure made of 22% dielectric material. An important feature of this fabrication process is its practicality since it does not involve drilling thousands of holes individually. It could even lend itself to microfabrication by using reactive ion etching to produce the holes. By covering the dielectric material with a mask containing a triangular array of holes, the periodic structure could be fabricated by tilting the material by 35.26” and making three etching operations at 120” about the azimuth. The projection of the circular holes in the mask would result in cylindrical holes with oval cross-section. For a very thin mask, the eccentricity of the oval holes would be l/d. Qian and Leungc7’) predicted that the size of the gap would actually be increased with respect to the gap generated by cylindrical holes with circular cross-section. In the case of periodic structures with a two-dimensional lattice, the samples were made of arrays of independent dielectric rods with circular cross-section which were arranged in either a square or triangular array. The filling fraction was changed by using different sized rods or by moving them closer or farther apart. Other two-dimensional structures were fabricated by drilling long parallel cylindrical holes perpendicular to the top surface of a dielectric block with a large index constant. The holes were arranged in a square or triangular lattice and the filling fraction was changed by increasing or reducing the size of the drill bits. 5.2. Experimental Setups The different setups used in the analysis of photonic bandgaps are essentially based on the same approach, namely the measurement of the field amplitude (and phase) transmitted through the PBG crystals along specific directions. All experiments to date have been carried out at microwave frequencies. 5.2.1. Anechoic chamber method. The setup used by Yablonovitch and Gmitter is shown in Fig. 17. It consists of a long anechoic chamber made of microwave absorbing material. The sample is placed at one end of the chamber and a spherical wave is emitted at the other by a monopole antenna. The signal propagates down the chamber and by the time it reaches the crystal its wavefront is essentially planar. This plane wave then passes through the sample and onto the receiving monopole. The setup is operated in a pump-probe configuration. Part of the signal of a sweep oscillator is modulated before being radiated into the chamber by the monopole antenna. The
X-YRECORDER FIG. 17. Experimental setup used by Yablonovitch and Gmitter. A sweep oscillator feeds a IO-dB splitter. Part of the signal is modulated (MOD) before being radiated into a long anechoic chamber by a monopole antenna. The other part of the signal is used to measure the amplitude and phase differences through the crystal. (From Ref. 3.)
P. R. VILLENEUVE and M. Pm4
other part of the signal is used to measure the amplitude and phase differences between the input and output signals through the crystal. The sweep oscillator can generate a continuous fixed-frequency signal over a frequency range of 1 to 20 GHz. One of the photonic crystals studied by Yablonovitch and Gmitter is shown in Fig. lb along with its dispersion relation. In the experiment, the actual sample had a lattice constant of 1.27 cm.(3) In order to measure the dispersion relation experimentally, propagation had to be investigated along all the symmetry lines of the Brillouin zone. The sample was positioned in the chamber such that the radiation was incident on the (l,O,O) face of the crystal at a constant angle. The electric field was polarized either parallel to the surface or parallel to the plane of incidence. The frequency of the incident wave was then swept. As the frequency was increased, the experimenters identified the frequency at which the transmission through the crystal was cut off and at which it reappeared. These frequencies corresponded to the lower and upper limits of the gap, respectively. In order to map out the dispersion relation, Yablonovitch and Gmitter also needed to determine the value of the wavevector within the structure. Since the frequencies at the edge of a gap correspond to wavevectors lying on the surface of the Brillouin zone, and since the component of k parallel to the surface was conserved upon transmission through the crystal surface, then k was uniquely defined (up to a reciprocal lattice constant). In order to map out the band diagram for all values of k, the crystal was then rotated slightly and the procedure was repeated. The entire procedure was performed for both polarizations of the incident wave yielding two lower bands and two upper bands (one for each polarization). With this method, Yablonovitch and Gmitter were able to measure only the bands at the edge of the gap since they were relying on the absence of transmission through the crystal to identify the bands. Furthermore, bands along the T-X and T-L symmetry lines could not be measured since they were not lying on the surface of the Brillouin zone. Finally, we note that Yablonovitch and Gmitter studied several photonic crystals with this setup, including the fee structure with nonspherical air holes. A similar setup was used by Meade et al. (24) to investigate photonic bandgaps in two-dimensional structures. Plane wave radiation was sent on a triangular array of cylindrical air holes drilled into a dielectric material with a microwave refractive index close to 3.6. The air rods had a diameter of 0.99 cm and the lattice constant was 1.04 cm. Propagation through the sample was investigated in the plane perpendicular to the air rods. The electric field was polarized either parallel or perpendicular to the rods. A gap common for both orthogonal polarizations was found between 13 and 15.5 GHz. In addition to investigating photonic bandgaps in perfect dielectric crystals, Yablonovitch et al. studied the generation of localized modes in three-dimensional structures with a single defect.(43+44) In order to excite every possible mode in the crystal, a point source illuminationclose to the sample-was used instead of a single-k plane wave. Since the wavefront of the point source was spherical, every point on the wavefront had a different component of the wavevector parallel to the direction of propagation in the crystal. This allowed the generation of essentially all wavevectors for each frequency. In these experiments, an fee structure was chosen with nonspherical air holes in a dielectric material with a refractive index close to 3.6. The structure and its fabrication process were described above in Section 5.1. The lattice constant of the crystal used in the experiment was 1.1 cm and the hole diameter was 0.52 cm. The defects were made either by breaking one of the interconnecting dielectric ribs or by adding a single dielectric sphere in a hole of the fee lattice. The samples were placed in a microwave absorbing chamber and monopole antennas were placed within the chamber on either side of the dielectric crystal. Figure 18a shows the transmitted amplitude through the crystal in the absence of a defect as a function of the microwave frequency. The sample was aligned along the (l,l, 1) direction between the antennas. A gap was found from 13 to 16 GHz.
Figure 18b shows the transmitted amplitude through the crystal with a missing slice in one of the dielectric ribs. The defect was located at the center of the WS cell. The transmission spectrum was essentially unaffected by the defect except for the appearance of a single mode in the center of the gap. At the defect-mode frequency, propagation through the crystal was allowed and radiation hopped from one antenna to the other.(43*44) The mode energy was localized around the defect, as we have indicated in Section 4.2. The defect mode in Fig. 18b is labelled ‘deep acceptor’ in analogy to acceptor atoms in doped semiconductors. The word ‘deep’ refers to the distance of the mode frequency from either side of the bandgap. Figure 18c shows the transmitted amplitude through the crystal with an extra dielectric sphere centered in an air hole. Yablonovitch et al. observed that the addition of a single
15 Frequency in GHz
FIG. 18. (a) Transmitted amplitude through an fee crystal in the (l,l,l) direction as a function of frequency. The sample is made of asymmetric air holes in a dielectric material with a refractive index of 3.6. The air holes occupy 78% of the total volume. (b) Transmitted amplitude through the fee structure with a missing slice in one of the dielectric ribs centered in the WS cell. (c) Transmitted amplitude through the fee structure with an extra dielectric sphere centered in an air hole. (From Ref. 43.) JpQE,*,2--G
P. R. VILLENEWE and M. PI&
SANPLE FIG. 19. Experimental setup used by Robertson er al. based on the COMITS technique. Part of an optical pulse is converted into a short current transient which propagates down a stripline antenna and is radiated into the air. The emitted radiation is collimated by a hemispherical lens then passes through the sample. The transmitted signal is then focused onto the receiving antenna by another hemispherical lens. The other part of the optical pulse is used to measure the amplitude and phase changes through the sample. (From Ref. 28.)
dielectric sphere led to the generation of multiple modes in the gap.(43,44) Since the volume of the defect in Fig. 18~ was only slightly larger than the threshold volume, the defect modes were close to the upper gap-edge. 5.2.2. Coherent microwave transient spectroscopy method. A different experimental setup was used by Robertson et al. (*a-3’)to investigate the photonic band structure and surface modes of periodic dielectric structures. The method is based on the coherent microwave transient spectroscopy (COMITS) technique@*) which uses optoelectronically pulsed antennas to generate picosecond electromagnetic transients.* An important feature of this experimental method lies in its very broad frequency range (15-140 GHz) and good polarization sensitivity (60: 1).(*‘) Furthermore, the phase of the transmitted signal can easily be found which allows the dispersion relations to be measured for the entire frequency range. The setup is shown in Fig. 19. The transmitter and receiver consist of identical coplanar stripline antennas with exponentially-tapered ends. The experiment is set up in a pump-probe configuration. The transmitter is triggered by a short optical pulse ( - 1.5 ps) focused between the coplanar striplines and generates a short current transient. This pulse propagates down the stripline and is radiated into the air at the exponentially-tapered end of the antenna. A hemispherical fused-silica lens is used to collimate the emitted radiation which propagates in free-space and then passes through the sample. Another hemispherical lens is used to focus the transmitted signal onto the receiver. The time-dependent voltage induced in the receiver is measured as a function of the time delay between the pump and probe pulses. The transmitted amplitude can be found as a function of frequency by taking the Fourier transform of the time-dependent signal. The phase is given by the ratio of the Fourier transformed signals with and without the sample. Although the setup can be used to measure dispersion relations in three-dimensional structures, Robertson et al. chose to investigate structures with a two-dimensional lattice since they were easier to fabricate. The samples were made of (i) long low-loss ceramic rods (n - 2.98) with a radius of 0.37 mm(28.29), and (ii) long cylindrical holes of radius 0.75 mm in a low-loss Emerson and Cummings Stycastmaterial (n - 3.60).“‘) The dielectric rods and cylindrical holes were 10 cm long and were arranged in a square or triangular lattice. In order to avoid undesirable end effects, the height and width of the samples were chosen to be larger than the collimated beam size (- 3 cm). The samples were positioned between the antennas *Similar
are used in microwave
Photonic bandgaps in periodic dielectric structures
such that the radiation propagated in the plane perpendicular to the cylinders. Furthermore, the samples could be rotated such that the electric field of the emitted radiation was aligned either parallel (TM-polarization) or perpendicular (TE-polarization) to the axis of the cylinders. An example of the transmitted amplitude spectra is shown in Fig. 20 for TM- and TE-polarizations. It is obtained from the numerical Fourier transform of the time-dependent signal. In the case shown in Fig. 20, the sample was made of dielectric rods arranged in a square lattice with a lattice constant of 1.87 mm; the array was 7 rows deep along the direction of propagation and 25 rows wide. Propagation was studied in the (1,O) direction of the lattice. A large gap for TM-polarization can be seen in Fig. 20a from 45 to 70 GHz and two dips of the transmitted amplitude around 100 and 125 GHz indicate the presence of smaller gaps. Figure 20b shows a large attenuation around 70 GHz and a complete gap between 100 and 110 GHz. Since the receiver had a time-window of only 200 ps, the spectral resolution of the signal was close to 5 GHz which accounts for the poor resolution of the smaller gaps. This time-window limitation was also responsible for setting an upper limit on the length of the samples; had the dimension of the sample along the direction of propagation been too large, the transmitted signal reaching the receiver would have extended beyond the 200 ps detection interval, resulting in a loss of information. In order to measure the dispersion relations for any frequency, the information acquired from the transmitted amplitude alone is not sufficient; it is also necessary to know the phase difference between the input and output signals. Since the phase information is preserved with the COMITS method, it is possible to plot the band structure as shown in Fig. 21. The sample, propagation direction and polarizations are identical to those used in Fig. 20. By comparing Figs 20 and 21, large amplitude attenuations can be associated to the gaps in the dispersion relations. Figure 21 also shows the theoretical dispersion relations calculated with the methods presented in Section 2. Although the agreement between theory and experiment is excellent for most bands, theoretical calculations predict the existence of certain bands which the experiments have failed to detect. The experimenters found (28*2g) that the excitation of these bands by the planar wavefront of the incident radiation on the sample is forbidden by symmetry. They believe that the undetected modes do exist in the crystal but that the modes could not be excited in their experiment. The COMITS method was also used to observe surface modes on periodic dielectric structures. Meade et al. suggested that nonradiative modes with exponentially decaying fields
FREQUENCY (GM) FIG. 20. Transmitted amplitude (solid circles) through a square array of dielectric rods as a function of frequency. The broken lines correspond to the reference amplitude spectra in the absence of a sample. The transmitted amplitude is measured through 7 rows of rods along the (1,O) direction. The electric field is polarized (a) parallel and (b) perpendicular to the axis of the rods. The rods have a diameter of 0.37 mm and a refractive index of 2.98. The lattice constant is 1.87 mm. (From Ref. 28.)
P. R. VILLENEUVEand M. PICHI?
BAND 4 _____ ___-----
100 .---•---•__ 75 -
BAND IL_/B_ #a-=
FIG. 21. Dispersion relations of solid circles are determined from The broken lines are computed perpendicular to the rods.
-e_n_N_o_=__.___ _,&AiD 2 C--m_
4.2 0.4 12.5 WAVEVECTOR@I-1)
a square array of dielectric rods along the (1,O) direction. The the amplitude transmission and phase change through the crystal. theoretically. The electric field is polarized (a) parallel and (b) The sample is identical to that of Fig. 20. (From Ref. 28.)
on both sides of the air/crystal boundary could exist on the surface of properly terminated photonic crystals.(47) Robertson et al. observed such modes in two-dimensional crystals with certain terminations.(48’ In their experiment, microwave radiation was coupled to the surface mode with a phase-matching prism. The surface mode could not radiate into the air since it could not satisfy wavevector and frequency conservation, nor could it radiate into the crystal since its frequency was in the bandgap. (47,48) Surface modes were found in a square array of dielectric rods terminated by hemicylindrical rods while surface modes were not found on arrays terminated by full cylindrical rods. 5.2.3. Waveguide method. A third experimental setup was used by McCall et a1.(3’.32)to investigate the photonic bandgaps in two-dimensional structures. The experiments were carried out in a microwave scattering chamber which behaved essentially like a parallel-plate metallic waveguide. The plates were separated by a distance of 1 cm and the experiment was carried out for TM-polarizations over a frequency range of 6 to 20 GHz. In this range of frequencies, the chamber operated in a single TEM mode and the chamber behaved like a two-dimensional system since the electric field had essentially no variation between the plates along the transverse direction.* The samples were made of short low-loss dielectric rods (length N 1 cm) arranged in a square”‘) or triangular (32)lattice and were placed between the parallel plates. A continuous fixed-frequency signal was sent into the chamber and propagated down through the arrangement of rods. The transmitted power was detected at the other end of the chamber. The rods were positioned such that the radiation propagated in the plane perpendicular to the rods and the electric field was polarized parallel to the rods (TM-polarization). *For frequencies higher than 20 GHz, there would be large variations of the field amplitude across the metal plates and the constant-field approximation would no longer be valid. Furthermore, in order to avoid the appearance of higher-order modes, the spacing between the plates would need to be reduced.
-30 -35 -la 45 -50 -55 8
(a) -15 -20 -25 -u) -35 -40 45 -xl -55
(b) -15 -20 -25 -30 -35 A0 -05 -!a -55 8
FIG. 22. Transmitted power through a triangular array of dielectric rods as a function of frequency. The transmitted power is measured along the (a) (1,O) and (b) (0,l) directions. The electric field is polarized parallel to the axis of the rods. The rods have a diameter of 0.96 cm and a refractive in,-lpY nf? n ThP ,ntt;,.,x rr\nrtsnt:a 1 77nm I,., T..,,.,:tt,A_~.,Pr tl...,...A. +I.,*,:"""..I,., "__"..
P. R. VILLENEWE and M. PIC&
An example of the experimental results is shown in Fig. 22 for a triangular array of alumina composite dielectric rods (n - 3.0) with a 0.48 cm radius and a 1.27 cm lattice constant. The transmitted power is given in Figs 22a and 22b as a function of the frequency injected into the chamber for two different lattice orientations. The strong attenuation through the sample corresponds to the second photonic bandgap; a smaller gap should appear centered at 6.3 GHz. There is good agreement with theoretical calculations except along the (0,l) direction in which case a gap was predicted to appear up to 12.8 GHz. However, the upper limit of the (0,l) gap in Fig. 22b is not well defined; this could be caused by excess noise due to a depolarization of the field in the chamber and by coupling into higher frequency modes. McCall et al. also used the waveguide method to investigate localized states caused by a structure defect. A single rod was removed from the center of the triangular array. The
Dismncc (ii unitsof d)
(4 FIG. 23. Spatial
energy distribution in the triangular array of dielectric rods used in Fig. 22~. The with respect to the missing rod is given in units of lattice constants along the directions indicated in the insets. (From Ref. 32.)
Photonic bandgaps in periodic dielectric structures
transmitted power is shown in Fig. 22c for radiation incident along the same direction as in Fig. 22a. A sharp peak appears within the gap at 11.2 GHz which corresponds to a localized-or defect-state. In order to investigate the spatial energy distribution of the defect state, several small holes were drilled into the top plate of the chamber and the energy density was measured throughout the array by inserting a tuned probe in the holes. The energy density is shown in Fig. 23 for two different angles of incidence on the structure. The rapid fall-off of the energy density around the defect indicates a strong localization of the defect state; the decay length is close to one lattice constant. The experimenters have observed”‘) that defect modes decay more rapidly (i.e. localization is stronger) in a triangular lattice than in a square lattice. 6. CONCLUSION We have presented a review of the theoretical and experimental investigations carried out on photonic bandgaps in periodic dielectric structures. These periodic structures have the ability of perturbing (and even suppressing) the density of electromagnetic states within the structures. Ideally, the density of states could be designed to satisfy any requirement simply by fabricating the appropriate PBG structure. In addition to the applications presented in Section 4, PBG materials could be used to enhance nonlinear effects in dielectric materials. Since radiative decay of excited atoms is dominated by first-order processes such as one-photon transitions, PBG materials could reduce one-photon transitions and increase the probability of observing higherorder decay. Since it is possible to fabricate a structure for which the transition at frequency w would lie within the gap while the transition at frequency to would lie outside of the gap, PBG materials could be used for the fabrication of devices such as two-photon lasers operating at frequency 50. Previous attempts at fabricating such lasers have been hampered by one-photon transitions which quenched most of the gain. PBG materials could sufficiently reduce the probability of first-order radiative transitions and allow the observation of two-photon lasing. These lasers would have remarkable properties such as short-pulse generation, emission of squeezed states and frequency tuning. PBG materials could also be used for the generation of highly-directional radiation. For example, propagation in an array of rods could be suppressed in the plane perpendicular to the rods and in a wide angle on either side of the plane. Propagation would then be restricted to a small cone along the axis of the rods. This continuous incoherent directional radiation would be well suited for imaging. Finally, PBG materials could be used in integrated optics as filters and polarizers in two-dimensional waveguide structures or as dispersive components in lasers. The numerous and remarkable applications of PBG materials justify the considerable effort currently being devoted to the fabrication of PBG structures at optical wavelengths and to the understanding of the underlying physical processes. Although there are no simple analytical criteria for the generation of photonic bandgaps, there seems to be three necessary conditions which must be satisfied: (i) both high- and low-dielectric materials must be connected throughout the structure, (ii) the index contrast between the two materials must be greater than 2, and (iii) the photonic crystal must be made mostly of low-dielectric material. The diamond structure is the only known exception to the first condition in three-dimensional structures since a gap has been predicted in an array of disconnected dielectric spheres. However, we noted in Section 2.1.2 that the disconnected spheres were still closely packed when the gap disappeared. This indicates that the first condition remains an excellent guideline. Furthermore, we recall that the accuracy of the threedimensional calculations was poor, which could affect the accuracy of our conclusions and
P. R. VILLENEUVE and M. PIcHf.
explain in part the discrepancy between the first condition and the results for the diamond structure. In two-dimensional structures, the first condition can never be satisfied. In these structures, it seems that only the high-dielectric material need be connected. We have seen that the gaps are largest when the dielectric ribs between the air columns are thin. In general, most two- and three-dimensional structures consisting of disconnected dielectric atoms failed to generate bandgaps. This is not surprising since we have found that bandgaps were also absent in arrays of air holes when the volume fraction of air was sufficiently large to disconnect the dielectric material. The remaining islands of dielectric material behaved like starshaped dielectric atoms distributed periodically in space similar to arrays of dielectric spheres, except that the atoms had odd shapes. Further investigations will need to be carried out in order to find structures generating a complete gap in all 47~steradians which would be easy to fabricate. One such structure could be made of a dielectric slab with a two-dimensional array of holes, operating like a waveguide. While propagation would be forbidden in the plane of the dielectric slab if the frequency was to lie within the gap, radiation would also be confined (or guided) within the slab since it would not be able to satisfy the condition for frequency and wavevector conservation at the surface of the slab. More investigations will also be needed in order to develop active devices such as lasers. The lifetime of excited atoms in PBG materials will need to be determined in order to find such parameters as energy level population, gain, pumping threshold, etc. The effective polarizability associated to the dipole transition of the excited atoms will need to be determined and will provide access to the line shape, frequency shifts and transition rates.(90,9” It will also be important to determine the lifetime of excited atoms in structures which give rise to an incomplete three-dimensional gap such as twodimensional PBG materials or three-dimensional structures with an insufficient index contrast.
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