Abstracts of Talks

Periodically Time Dependent Photonic Crystals. B. ALEXANDROV, I. Z. Kostadinov, Faculty of Physics, University of Sofia, 5 J. Bourchier Blvd., 1126 Sofia, Bulgaria.
- - - We consider, external two parametrical fast periodic variation with time of the dielectric constant in the first place, of homogeneous media, and secondly of one-dimensional photonic crystal. The fast periodical time dependence, leads to parametric resonance-like phenomena which together with space periodicity changes completely the nature of the photonic band structure. In the case of simultaneously space and time periodicity we have an electromagnetic waves spectrum essentially described in an equivalent rectangle defining the Brillouin zone of the quasimomentum and quasienergy. Effects of incommensurability of the two internal lengths - space Lx and time Lt lattice constants, are present.

Theory of Light Scattering Through a Periodic Array of Nanorods Embedded in a Slab. D. ANDRE, A. Dereux, J. -P. Vigneron, Institute for Studies in Interface Sciences, Facultes Universitaires Notre-Dame de la Paix, Rue de Bruxelles 61, B-5000, Namure, Belgium, C. Girard, Laboratoire de Physique Moleculaire, Universite de Franche-Comte, F-25030 Besancon cedex, France.
- - - In the last few years, the interest in the study of light scattering by mesoscopic and microscopic scatterers was renewed in different contexts. The development of scanning near-field optical microscopy has clearly demonstrated the importance of theoretical simulations to support a better understanding of the experimental results [1]. In the context of the discovery of photonic band structures (similar to electronic band structures in solids), it was demonstrated that theoretical methods were useful to optimize geometries exhibiting a full bandgap [2].
These experimental frameworks evidenced the limits of the classical description of optical phenomena. Theoretical methods were then developed to meet the specific needs of the mesoscopic regime. Among them, the Green's dyadic method associated to perturbation theory proved to be a very powerful and versatile technique to deal with a broad range of near-field optical phenomena [3]. This method consists in a simultaneous resolution of discrete Lippman-Schwinger and Dyson's equations in order to determine the near-field. The far-field can then be deduced in order to calculate reflection and transmission coefficients.
In this work, we computed the scattering by a two-dimensional lattice of cylinders embedded in a film. The axis of the cylinders is perpendicular to the surface of the film, and the periodicity is parallel to the surface. In order to solve this problem, we divided the resolution into two steps. In the first step, the electromagnetic field and the dyadic Green's function of an homogeneous slab are computed. The second step applies a two-dimensional Bloch's theorem to compute the near-field in the array of nanorods from which we calculate the reflection and the transmission in the far-field. Our results show some finite-size effects of the system.

This work was performed in the framework of the Human Capital and Mobility research network Near-Field Optics for Nanoscale Science and Technology initiated by the European Community. The Institute for Studies in Interface Sciences acknowledges support from the Walloon Ministry for Technology. The Laboratoire de Physique Moleculaire is Unite Associee au CNRS 772.

[1] For a recent review of scanning near-field optical microscopy, see D. Courjon and C. Bainier, Rep. Progr. Phys. 57, 989 (1994).
[2] Photonic Band Gaps and Localization, ed. by C. M. Soukoulis, NATO ASI Series B 308, Plenum Press (New York, 1993).
[3] O. J. F. Martin, C. Girard and A. Dereux, Phys. Rev. Lett. 74, 526 (1995).
[4] F. Forati, A. Dereux, J. P. Vigneron, C. Girard and F. Scheurer, submitted to NFO-3.

2D Photonic Crystal with Multiple Refractive-Index Steps. T. BABA, T. Matsuzaki, Yokohama National University.
- - - The 2D photonic crystal is attracting attention because the expected spontaneous emission control is much more effective that in planar microcavities and the experiment in the optical frequency range is much easier than for 3D. 2D structures so far studied have an abrupt refractive index step between optical atoms and the background medium. In this report, we propose the structure having multiple index steps. We numerically show that the conventional 2D structure constructed by arranged circular rods comes to exhibit new photonic gaps inside the 2D plane (2D photonic gaps) by assuming an intermediate index region around the rods. This type of structure will extend the possibility of the existence of a photonic gap.
It has been shown that circular holes exhibit 2D photonic gaps. =46or these structures, however, the fabrication technique of very thin (<0.05$/mu$m for semiconductor structures), smooth and abrupt walls with the aspect ratio over 20 are required to obtain the photonic gap for the aimed optical frequency. Some other structures exhibit gaps for either TE or TM polarization but not the absolute gap that means the overlap of TE and TM gaps. What we found in this study is the band diagram changes and absolute gaps appear by assuming the intermediate index region. As the intermediate index region, we can suppose some dielectric film such as Si$_3$N$_4$ formed around the rods. The diameter of rods designed for the optical frequency are larger than 0.1$\mu$m. This seems easy to fabricate in comparison with the conventional structures. Electron beam lithography and ECR dry etching technique are now being developed to realize this type of structure. Some experimental results will be presented at the meeting.

[1] R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Appl. Phys. Lett. 61, 495 (1992).
[2] T. Baba and M. Koma, Jpn. J. Appl. Phys. 34, 1405 (1995).
[3] T. Baba and T. Matsuzaki, MPE '95, Tokyo J2 (1995).
[4] T. Krauss, Y. P. Song, S. Thomas, C. D. W. Wilkinson, and R. M. DelaRue, Electron. Lett. 30, 1444 (1994).

The Interaction of EM Radiation with Metallic Systems. P. M. BELL, J. B. Pendry, Condensed Matter Theory Group, Blackett Laboratory, Imperial College, London SW7 2BZ, L. Martin-Moreno, Instituto de Ciencia de Materiales (CSIC), Universidad Autonoma de Madrid, 28049 Madrid, Spain.
- - - The interaction of EM waves with metals can produce very interesting behavior, particularly when the metallic system has a rich structure. The most obvious example of this being the absorption of light by the silver colloid in a photographic film. This talk will discuss how to use the OPAL suite of codes to describe the optical response of complex metallic surfaces through such quantities as the reflection and transmission coefficients and the photonic band structure. Furthermore, the ability of the OPAL codes to visualize the behavior of the EM field provides additional insights into the behavior of light interacting with metals.

Energy Transport Properties Random Media. K. BUSCH (1,2), C. M. Soukoulis (2), 1. Institut fur Theorie der Kondensierten Materie, Universitat Karlsruhe, 76128, Karlsruhe, Germany, 2. Ames Laboratory*, and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011.
- - - We present a new method for efficient, accurate calculations of the transport properties of random media. It is based on the principle that the wave energy density should be uniform when averaged over length scales larger than the size of the scatterers. This new scheme captures the effects of the resonant scattering of the individual scatterer exactly, as well as the interparticle interactions in a mean-field sense. It has been successfully applied to both "scalar" and "vector" classical wave calculations. Results for the energy transport velocity give pronounced dips for low concentration of scatterers, while as the concentration increases the dips are smeared out in excellent agreement with experiment. This new approach is of general use and can be easily extended to treat different types of wave propagation in random media.

*Operated for the U.S. Department of Energy by Iowa State University under contract no. W-7405-eng-82. This work was supported by the Director for Energy Research, Office of Basic Energy Sciences.

Defect States in Photonic Band Gap Materials. K. BUSCH (1,2), C. T. Chan (2), C. M. Soukoulis (2), 1. Institut fur Theorie der Kondensierten Materie, Universitat Karlsruhe, 76128, Karlsruhe, Germany, 2. Ames Laboratory*, and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011.
- - - The various analytical and numerical techniques used to obtain the band structure and the defect modes in photonic band gap materials will be presented. First, we will restrict ourselves to one-dimensional systems, since there are analytical results and one easily can check the different numerical techniques. Extensions to two and three dimensions will be discussed. The role of nonlinear defects will also be discussed.

*Operated for the U.S. Department of Energy by Iowa State University under contract no. W-7405-eng-82. This work was supported by the Director for Energy Research, Office of Basic Energy Sciences.

Two-Dimensional Photonic Band Gaps: New Hexagonal Structures. D. CASSAGNE, C. Jouanin, D. Bertho, Groupe d'Etude des Semiconducteurs CC.074, Universite Montpellier II, France.
- - - Periodic dielectric structures have been recently proposed to inhibit spontaneous emission in semiconductors. From this suggestion, the new concepts of photonic band gaps and photonic crystals have been developed. Zero-threshold lasers, wave guides, and polarizers are promising applications. A new class of two-dimensional periodic dielectric structures with hexagonal symmetry is investigated in order to obtain photonic band gap materials. This set has the hexagonal symmetry and contains, in particular, several structures previously discussed. The photonic band gap structure is related to the basic properties of the materials and some features of the opening of the gaps are explained. By varying the crystal pattern, we show how band gaps common to E and H polarizations appear for a new design of two-dimensional periodic dielectric structures. The dependence of these gap widths with the filling patterns is studied and potential application for the creation of photonic crystals in the optical domain is discussed.

Order-N Method for Photonic Band Gap Systems. C. T. CHAN, Ames Laboratory* and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011.
- - - The eigen-modes for electromagnetic waves in an inhomogeneous dielectric medium can be obtained with a method that scales linearly ("order-N") with the size of the system. The method employs discretization of the Maxwell equations in both the spatial and the time domain and the integration of the Maxwell equations in the time domain. The spectral intensity can then be obtained by a Laplace transform. The method can obtain dispersive relationships as well as photonic densities of states. We will apply the method to a few problems of current interest, including the photonic band structure of a periodic dielectric structure, the effective dielectric constants of some 3-dimensional and 2-dimensional systems, and the defect states of a periodic dielectric structure with structural defects.

Work done in collaboration with Q. L. Yu and K. M. Ho.

*Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under contract no. W-7405-eng-82. This work was supported by the Director for Energy Research, Office of Basic Energy Sciences.

Calculation of Band Structure in Photonic Media Using Modal Expansion and R-Matrix Propagation Techniques. J. MERLE ELSON, P. Tran, Research and Technology Division, Naval Air Warfare Center Weapons Division, China Lake, CA 93555-6001.
- - - We describe a method to calculate dispersion associated with complex structures described by regions of different permittivity. This includes, for example, arrays of cylinders or spheres. We use a coupled mode method with an R-matrix, or recursive, propagation scheme. This method is quite simple to implement by dividing the complex structure into sublayers.
The R-matrix propagation algorithm is comparatively free of numerical instability that exists for certain T-matrix propagation schemes. Unlike the T-matrix approach which seeks a matrix that relates the electric and magnetic field on one side of a layer to the electric and magnetic field on the other side of the layer, the R-matrix method relates the electric field of both sides of a layer to the magnetic field on both sides. A recursive formula is derived to allow stacking up successive sublayers to form a global R-matrix for any structure of interest. The field inside each sublayer is expanded in terms of its modes and this is very simple for arbitrary profiles, requiring only the diagonalization of a matrix to obtain the eigenvalues.
Using this basic approach, we can calculate the dispersion of photonic media two different ways. First, for a given frequency, we calculate transmission through a photonic structure of finite thickness and this yields the phase of the transmitted field. We compare this phase to the comparable phase in the absence of the photonic structure. The difference in phase is related to the frequency-dependent complex refractive index. By looking at this phase difference versus frequency, we can deduce dispersion curves.
A second approach applies to a photonic structure of infinite thickness. We apply the Floquet theorem to calculate the band structure. This approach is quite simple, requiring only the eigenvalues of a matrix. We then look for eigenvalues of unit magnitude, equate to the Floquet exponential term, and calculate the wave vector for a given frequency. To obtain the band structure, we change frequency and repeat the process.
Numerically, we have compared our results favorably with other published results.

Fabrication of Three-Dimensional Photonic Band Gap Material by Deep X-Ray Lithography. G. FEIERTAG, W. Ehrfeld, H. Freimuth, R. Weiel, IMM Institut fur Mikrotechnik GmbH, Carl-Zeiss-Strasse 18-20, D-55129 Mainz, Germany, G. Kiriakidis, C. M. Soukoulis, Foundation for Research and Technology Hellas (FORTH), P.O. Box 1527, Heraklion, Crete, Greece.
- - -Yablonovitch et al. [1], by implementing the diamond structure proposed by Ho et al. [2], constructed the "three cylinder" dielectric structure which exhibits a photonic band gap in the whole Brillouin zone. This structure can be fabricated by mechanically drilling three sets of holes 35 degrees off vertical into the top of a solid dielectric. However, extremely small dimensions required for high frequencies cannot be achieved. This problem can be solved by using deep x-ray lithography which allows the fabrication of structures usable up to the infrared range. Three irradiations were performed, whereas the tilted arrangement of mask and resist was rotated each time by 120 degrees. Resist layers with a thickness of 500 microns were irradiated at the DCI storage ring in Orsay, France. The lattice constants of our structures were 227 and 114 microns corresponding to midgap frequencies of 0.75 and 1.5 Thz, respectively.
Since the dielectric constant of the PMMA, was not high enough for the formation of a photonic band gap, a molding step was applied. The holes in the resist structure were filled with a solution of polysilazane in tetrahydrofuran.
After the evaporation of the solvent, the samples were pyrolyzed at 1100 C N$_2$ atmosphere. The resist decomposes into Co$_2$ and H$_2$O, whereas polysilazane is transformed into a SiCN ceramic. A lattice of ceramic rods corresponding to the holes in the resist structure remained.

[1] E. Yablonovitch, T. J. Gmitter, K. M. Leung, Phys. Rev. Lett 67, 2295 (1991).
[2] K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990); C. T. Chan, K. M. Ho, and C. M. Soukoulis, Europhys. Lett, 16, 563 (1991).

Fabrication of 2-D Photonic Crystals with Infrared Band Gaps in Porous Silicon. U. GRUENING, V. Lehmann, Siemens AG, ZFE T ME 1, Otto-Hahn-Ring 6, 81730, Munchen, Germany.
- - -The fabrication of photonic band gap materials at micron and submicron length scales still poses some problems. Macroporous silicon is a way to meet the requirements of regularity and high index contrast for a 2-D infrared photonic material. The controlled formation of pores in an n-type silicon by light-assisted electrochemical etching in hydrofluoric acid can lead to a regular pattern of uniform holes with minimal changes of the pore diameter both between neighboring pores and with depth. To show the feasibility of the approach, we etched a 340 $\mu$m deep 2-D square lattice of circular air rods with a lattice constant of 8 $\mu$m in an n-type silicon substrate. This structure possesses individual photonic gaps for both polarizations in the infrared region between 250 and 500 cm$^{-1}$ (20 - 40 $\mu$m). The transmission spectra between 50 and 650 cm$^{-1}$ were in good agreement with the theoretical calculated structure. The pore formation technique should allow the fabrication of photonic lattices with a complete 2-D band gap in the middle and near infrared.

Effective Waveguides in Periodic Structures with Slowly Varying Parameters. S. A. Bulgakov, Instituto de Ciencia de Materiales, Facultad de Ciencias, C-3, Cantoblanco, E-28049, Madrid, Spain and V. V. KONOTOP, Departamento de Fisica, Universidade da Madeira, Praca do Municipio, 9000, Funchal, Portugal.
- - - Using the formalism of the Wannier functions we develop the theory of the waveguide in one-dimensional periodic structures under smooth modulations. The properties of such waveguides and possibilities of their practical realizations are investigated numerically. Some models allowing analytical treatment are also considered.

Fabrication of 2-D PBG Structures in GaAs/AlGaAs. T. F. KRAUSS, R. M. DeLaRue, Department of Electronics and Electrical Engineering, The University of Glasgow, Glasgow G12 8LT, Scotland. - - - We discuss several issues regarding the fabrication of two-dimensional photonic bandgap (PBG) waveguide structures in the GaAs/AlGaAs semiconductor system, in particular relating to pattern generation and etching technology. Most 2-D PBG structures reported so far, e.g., [1] consist of circular features placed in a hexagonal grid. According to theoretical predictions [1,2], the most promising geometry to exhibit a full (i.e., for both polarizations) PBG is a `honeycomb' lattice with high (approximately 80\%) air-filling ratio. Considering the periodicity of around 400nm, which is required for operation of such a structure at 850nm, the walls between adjacent circular voids become very thin (below 10nm), which can not practically be controlled in a lithographic process. We, therefore, employ hexagonal features and have demonstrated structures with high (65-70\%) air-filling ratios down to 160nm periodicity and 20nm wall width [3]. We use a silica-mask and a special low-damage RIE process with SiCl$_4$ to transfer the pattern into the semiconductor. One of the main problems of a 2-D waveguide structure, however, is the fact that the third dimension is finite, sub-micronic in most cases. Scattering out of the plane of the photonic lattice, which is not an issue in theoretical calculations that assume the third dimension to be of infinite extent, is an important loss-mechanism which can easily screen any PBG effects. We, therefore, propose a structure with very narrow (10's of manometers) air-gaps that ensures both the required high refractive index contrast and wave guiding with minimal scattering losses.

[1] J. M. Gerard, A.Izrael, J. Y. Marzin and R. Padjen: `Photonic bandgap of two-dimensional dielectric crystals', Solid State Electronics, 37, 341-1344, 1994.
[2] R. D. Meade, K. D. Brommer, A. M. Rappe and J. D. Joannopoulos: 'Existence of a photonic band gap in two dimensions', Appl.Phys.Lett., 61, 495-497, 1992.
[3] T. Krauss, Y.P.Song, S.Thoms, C.D.W.Wilkinson and R.M.DelaRue, 'Fabrication of 2-D photonic bandgap structures in GaAs/AlGaAs', El. Lett., 30, pp.1444-1446, 1994.

Photonic Band Structures of 1D and 2D Periodic Metallic Systems in the Presence of Dissipation. V. KUZMIAK, Institute of Radio Engineering and Electronics, Czech Academy of Sciences, Chaberska 57, 182 51 Praha 8, Czech Republic, A. A. Maradudin, Department of Physics, University of California, Irvine, CA 92717.
- - - In this talk we describe a plane wave method for calculating photonic bandstructures of systems which consist of metallic components in which dissipation is introduced through the use of a complex dielectric function of the form $\epsilon (\omega ) = 1 - \omega_p^2 / [\omega (\omega + iy)]$, where $\omega_p$ is the plasma frequency of the conduction electrons and $\gamma = 1/\tau$ is an inverse electron relaxation time. In this case the reduction of the band structure calculation to the solution of a single standard eigenvalue problem is not possible, but it can be transformed into a generalized eigenvalue problem. For structures with filling fractions f $le$ 1%, the generalized eigenvalue problem is reduced to a problem requiring the solution of sets of nonlinear simultaneous equations which correspond to the diagonal terms of the matrix in the plane wave representation with the non-diagonal elements taken into account perturbatively. This approach yields complex values of the frequency w for each value of the two-dimensional wave vector $\vec{k}_{\|}$. In addition to the real part $\omega_R$ of the calculated complex frequency $\omega$, we can also determine the attenuation $k(\vec{k}_{\|}$ of each mode as it propagates through the system. We first consider a model system represented by a one-dimensional, periodic array of alternating layers of vacuum and a metal characterized by the complex dielectric function described above. Then we apply this method to obtain the photonic band structures of an infinite array of parallel, infinitely long metallic rods whose intersections with a perpendicular plane form a square or triangular lattice. The real part of the resulting band structure displays an absolute band gap below the lowest frequency band, and nearly dispersionless bands in the frequency range in which $\epsilon (\omega )$ is negative, in the absence of damping. In the case of E-polarization, the imaginary part of the resulting bandstructure, the life time of each mode, as a function of the wave vector $\vec{k}_{\|}$ exhibits a non-zero minimum for the lowest band at the $\Gamma$-point($\vec{k}_{\|}$= 0), which is identical to the inverse conduction electron relaxation time used in the dielectric function $\epsilon (\omega )$. In contrast, in the case of H-polarization the solution for the lowest band at the $\Gamma$-point gives the travelling free space plane wave with zero attenuation. The imaginary part of the photonic band structure also yields nearly dispersionless lifetimes as counterparts of the flat bands found in the real part of the photonic band structure.

Impurity Modes from Frequency Dependent Dielectric Impurities in Photonic Band Structures. M. G. Khazhinsky, A. R. McGURN, Department of Physics, Western Michigan University, Kalamazoo, MI 49008.
- - - The impurity modes of photonic band structures are calculated using Green's function methods. Both 1-d (layered slabs) and 2-d (square lattice of rods) periodic systems formed of vacuum and frequency-independent dielectric materials are considered. Impurities are added to these systems by the introduction of frequency-dependent dielectric material. In the 1-d system, a slab is replaced by an impurity slab and in the 2-d system, a rod is replaced by an impurity rod to form impurity modes in a photonic band gap. Particular attention is paid to computing impurity modes associated with the dielectric resonances of the frequency dependent dielectric material of the impurities. Results are presented for periodic arrays of frequency independent dielectric materials with GaAs impurities.

Photonic Band Structures and Resonant Modes. P. J. ROBERTS, P. R. Tapster, T. J. Shepherd, Defense Research Agency, St. Andrews Road, Great Malvern, Worcs. WR14 3PS, United Kingdom.
- - - This paper reports on calculations of dispersion relations and scattering characteristics of 3-dimensionally periodic dielectric structures. Particular interest is paid to dielectric distributions which give rise to complete photonic band-gaps; frequency regions in which propagation of electromagnetic radiation is strongly inhibited. Also reported is the electromagnetic properties of systems which comprise stacked photonic crystals with overlapping photonic stop-bands. These systems have been studied experimentally recently, and can display scattering characteristics attributable to the presence of resonant modes at frequencies within the union of the stop-band frequency ranges of the individual crystals.
The numerical methods employed are the plane-wave expansion method (for dispersion relations only), and real-space transfer-matrix/scattering-matrix approach (for dispersions relations and scattering information). Thus far, systems with diamond crystal symmetry have been concentrated on, with the numerical methods optimized to treat this crystal class. A study of systems with tetragonal symmetry, which can be naturally fabricated in layer-by-layer growth techniques (forming "woodpile"-type geometries), and which have been shown to possess particularly wide band-gaps, is currently underway.

Photonic Band Structures of Atomic Lattices. R. SPRIK (1), A. Lagendijk (1,2), 1. van der Waals-Zeeman Laboratorium, Universiteit van Amsterdam, Valckenierstraat 65-67, 1018 XE Amsterdam, The Netherlands, 2. FOM Institute for Atomic and Molecular Physics, 1098 SJ Amsterdam, The Netherlands.
- - - The first steps towards three-dimensional lattices of laser trapped atoms have been successfully taken with the use of laser cooling techniques [1]. The propagation of light with wavelengths near the optical resonances in the atoms is dominated by multiple scattering from occupied unit cells and will lead to the formation of well defined optical band structures when all cells are filled.
The band formation for the propagating light is similar to photonic band structures in periodic three-dimensional dielectrics ('photonic crystals') [2]. The main difference is the strong resonant character of the scatterers near an optical resonance. Furthermore, in the limit of weak light fields (Rayleigh limit) and if recoil effects are ignored, the propagation of the light should conserve energy and is coherent. Some of these aspects are also encountered in photonic crystals with dielectric and metallic components [3], although dissipation is present in these systems.
The proper description of the photonic band structure of the filled optical lattice can be based on the t-matrix of the energy conserving resonant dipole scatterer both in the scalar approximation to the Maxwell equations and the full vector form [4]. This t-matrix is the exact classical representation for a two level dipolar atom. It explicitly depends on the energy but not on the wave vectors of the incoming and outgoing waves. This reflects the isotropic character of the scattering from the point-like atom. With the use of formalisms developed in solid state band calculations [5] to exploit the symmetry of the lattice, the photonic band structure of the optical atomic lattice can be calculated by diagonalizing the secular matrix for plane waves in the crystal [6]. The coupling terms in the matrix are essentially determined by the phase shift given by the t-matrix and are equal for all the plane waves.
The resultant band structure has two important features. The dispersive effects of the scatterers cause a distortion of the band structure and the formation of gaps near the Brillouin zone. Furthermore, near the resonance frequency $\omega_0$ of the two level system, a full band gap develops. The width depends on the coupling strength, but even for small coupling, the gap is present. This band is essentially a polariton-like propagation in the crystal. When $\omega_0$ is tuned close to the Brillouin zone, combinations of distortion and polariton gap occur. Also, the density-of-states and the real space eigenfunctions are calculated to evaluate the influence of the atomic lattice on the radiation lifetime.
Although atomic transitions associated with other multipole moments than the dipole transition are usually very weak under far field conditions, these transitions can play a role in the photonic band structures when the unit cell dimensions are comparable with the range of the higher multipole radiation. These transitions are easily incorporated in the calculations. In particular, the quadrupole transitions associated with the $\ell$=2 angular momentum of the photon, give rise to bands which have similar character as the d-band states of electrons in transition metals.

[1] P. Verkerk et al., "Dynamics and Spatial Order of Cold Cesium Atoms in a Periodic Optical Potential," Phys. Rev. Lett 68, 3861 (1992).
[2] see e.g., NATO Workshop, "Photonic Band Gaps and Localization," ed. by C. M. Soukoulis, Plenum Press, NY (1993).
[3] J. B. Pendry, "Photonic Band Structure," J. Mod. Opt 41, 209 (1994).
[4] Th. M. Nieuwenhuizen, A. Lagendijk, B. A. van Tiggelen, "Resonant Point Scatterers in Multipole Scattering of Classical Waves," Phys. Rev. Lett. A 169, 191 (1992); B. van Tiggelen, "Multiple Scattering and Localization of Light," Ph.D. Thesis, University of Amsterdam, (1992).
[5] A. Gonis, "Green Functions for Ordered and Disordered Systems," Studies in Mathematical Physics vol. 4, ed. by E. van Groesen, E. M. de Jager, North Holland (1992).
[6] R. Sprik, A. Lagendijk, "Coupled Dipole Approximation of Photonic Band Structures in Colloidal Crystals," contribution QWA6, European Quantum Electronics Conference (1994).

Inhibition of Spontaneous Emission in the Photonic Band Gap. V. SRIVASTAVA, School of Physics, University of Hyderabad, Hyderabad - 500046, India.
- - - We have studied the interaction of two dipoles, separated by distance R, through a radiation of resonance frequency wo which happens to fall in the photonic band gap. The known concepts of condensed matter physics responsible for opening of an electronic band gap are adapted to the photonic case and the origin and nature of evanescent photonic modes having frequencies in the photonic band gap are discussed. It is suggested that the resonant dipole-dipole interaction (RDDI) shall be suppressed if mediated by a frequency in the photonic band gap, in case the separation between the dipoles is larger than the localization length of the photon involved. Further, it is suggested, the RDDI may be enhanced at the band edge where the photons, are piled up at the atomic sites due to Bragg reflection.

Photonic Band Structure Calculation of Systems Possessing Kerr Nonlinearity. P. TRAN, Code C02313 (474400D), Research and Technology Division, Naval Air Warfare Center Weapons Division, China Lake, CA 93555.
- - - An FFT version of the Finite Difference Time Domain (FDTD) technique is used to calculate the band structure of photonic crystals possessing Kerr non-linearity, i.e., a system where the displacement field is related to the electric field by the constitutive equation D(r) = [e(r) + x(r) $\vertE(r)\vert^2 ]$ E(r). The FDTD technique integrates the two coupled time-dependent Maxwell's equations on a grid by using finite difference approximation to the curl and time derivative. Using FFT to evaluate the curl is more accurate (although requires more computing time). It also reduces the need for a staggered grid (where the magnetic and electric field are evaluated at different grid points), hence, reducing the complexity of the programming. Furthermore, one does not need to worry about edge effect, since the FFT automatically enforces the periodicity on the solution so one can integrate as long in time as necessary to get the frequency resolution required. Details of the technique as well as results of the calculation for a two-dimensional system consisting of infinitely long circular rods arranged in a square lattice will be presented.

Nonlinear Phenomena in Layered Superlattices. G. P. TSIRONIS, University of Crete and Institute of Electronic Structure and Laser, P.O. Box 1527, 711 10 Heraklion, Crete, Greece.
- - - We will analyze the stationary properties of a nonlinear Kronig-Penny model which constitutes a series of delta functions with strength that is modulated according to an amplitude square law. This model describes the propagation of EM waves in a one-dimensional array of thin dielectric slabs where the Kerr coefficient is non negligible and are placed periodically (or aperiodically, in general) in a linear dielectric material. We utilize the point-like nature of the nonlinearity in the model to cast the problem into the form of an area preserving nonlinear map. We study the periodic, quasiperiodic, and disordered cases and calculate the transmission coefficient through a finite segment. We compare with the linear (unmodulated) case and find the transparent and opaque regimes as a function of the nonlinearity coefficient and the geometry. We show that a device can be constructed with optical limiting properties.

Photonic Gap in Complex One- and Two-Dimensional (2D) Systems, Periodically Modulated Waveguides, and Photonic Stub Tuners. R. Akis, P. VASILOPOULOS, F. Sezikeye, Concordia University, Montreal, Canada.
- - - The behavior of the even gaps, as a function of the thickness of a dielectric layer inserted in the cell of a 1D system, is similar to that of a conventional two-layered system but that of the odd gaps is not as most of them are quite wider than those of quarter-wave (QW) stack. In 2D systems the maximum first gap is obtained when P$_0$ = area* dielectric constant is the same in both materials. Adding interstitial structure to the center of sides of a unit cell gives approximately the results of P$_0$. In waveguides of width L$_y$=a, with a periodic refractive index profile, the gap becomes much larger than the 1D QW gap for sufficiently short a. A mode-matching technique [1] is used for perfectly conducting boundaries along y and a finite-element technique [2] for a dielectric confinement. These waveguide gaps can be further enhanced by adding stubs that results in an interference between waves propagating through the main waveguide and those reflected from the stubs. Transmission results are also presented for waveguides. Depending on the stub lengths, very narrow bands can appear in the gaps in a controlled manner.

[1] H. Wu et al., Phys. Rev. B 44, 6351 (1991).
[2] M. Sigalas et al., Phys. Rev. B 48, 14121 (1993).

Strong Dynamic Effects on the Diffraction of Photonic Colloidal Crystals. W. L. VOS (1), R. Spirk (1), A. van Blaaderen (2), A. Imhof (3), A. Lagendijk (1,4), G. H. Wegdam(1), 1. van der Waals-Zeeman Laboratory, Universiteit van Amsterdam, 1018 XE Amsterdam, The Netherlands, 2. AT&T Bell Laboratories, Murry Hill, NJ 07974, 3. Van 't Hoff Laboratorium, Universiteit Utrecht, 3584 CH Utrecht, The Netherlands, 4. FOM Institute for Atomic and Molecular Physics, 1098 SJ Amsterdam, The Netherlands.
- - - Lately, there has been a growing interest in photonic band structures and the possibility of fabricating optical bandgaps, which hold exciting prospects such as inhibition of spontaneous emission or localization of light [1,2]. Photonic band structures occur when light waves travel through a dielectric structure with a refractive index that is periodically modulated on length scales comparable to the wavelength and are analogous to band structures of electrons in atomic crystal. A promising route to fabricating optical photonic crystals on a large scale is to use self-organizing colloidal crystals [3]. Optical diffraction experiments of photonic colloidal crystals are essential in deriving structural parameters and moreover, yields information about the photonic strength of the crystal. However, this technique is as yet not on an equal level of sophistication compared to x-ray diffraction of atomic crystals. Therefore, optical diffraction experiments have been performed as a function of wavelength on colloidal crystals with various refractive index modulation depths, leading to the photonic limit. With increasing photonic strength, apparent Bragg spacings derived from the measured scattering angles are found to depend on wavelength, which is unphysical. Because the index modulations are much stronger than typically encountered in x-ray diffraction (10$^{-4}$), even the dynamical diffraction theory [4] does not remedy these phenomena. Much better results are obtained when the photonic crystals are described with an average refractive index (Maxwell-Garnett) with point scatterers, or by describing a set of lattice planes with a stratified dielectric model.

[1] Development and Applications of Materials Exhibiting Photonic Band Gaps, Ed. by C. M. Bowden, J. P. Dowling, and H. O. Everitt, J. Opt. Soc. Am. B 10, 280 (1993).
[2] Photonic Band Gaps and Localization, Ed. by C. M. Soukoulis, Plenum, New York (1993).
[3] See e.g., P. N. Pusey, in "Liquids, Freezing, and the Glass Transition," Ed. by D. Levesque, J.-P. Hansen, and J. Zinn-Justin (Elsevier, Amsterdam, 1990).
[4] W. H. Zachariasen, "Theory of X-ray Diffraction in Crystals" (Wiley, New York, 1945).

Optical Measurements of Photonic Band Structure in fcc Colloidal Crystals. I. I. TARHAN, G. H. Watson, Department of Physics and Astronomy, University of Delaware, Newark, DE 19716.
- - - Polystyrene colloidal crystals form three-dimensional periodic dielectric structures which can be used for photonic band structure measurements in the visible regime. Kossel lines obtained from these crystals reveal the underlying photonic band structure of the lattice in a qualitative way. Also, Kossel lines can be used for locating the symmetry points of the lattice for exact positioning of the samples for transmission measurements. From these measurements, the photonic band structure of an fcc crystal has been obtained along the directions between the L-point and the W-point. Also, a modified Mach-Zehnder interferometer has been developed for accurately measuring relative phase shifts of light propagating in photonic crystals to determine the dispersions resulting from photonic band structure near the band edges.

Transition from Ballistic to Diffusive Behavior for Multiply Scattered Waves. P. Sheng (1,2), Z. Q. ZHANG (1), 1. Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, 2. Exxon Research and Engineering Co., Annandale, NJ.
- - - We consider acoustic pulse propagation through a slab of scattering medium with thickness L. The results of direct solution of the Bethe-Salpeter integral equation with the ladder-diagram vertex are presented. The numerical solution presents, for the first time, a clear first-principle picture of how acoustic waves make the transition from ballistic to diffusive behavior. It is found that the diffusive limit, for the transmitted wave, is not well established even for L = 10 $\ell_s$, where $\ell_s$ denotes the scattering mean free path. The frequency correlation function of the transmitted acoustic field has been calculated. It is found that in the diffusive limit, the correlation function is well described by a single dimensionless scaling variable, $\Delta \omega L^2 /D$, where D is the wave diffusion constant and $\Delta \omega$ is the frequency separation, in good agreement with prior analytical results. However, when L = 1, few $\ell_s$ the scaling behavior breaks down. In this formalism absorption effect can also be incorporated, and is shown to have a significant impact on the consequent calculated correlation function.

Last updated: Fri Jun 9 10:34:27 CDT 1995
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