**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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.

**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.