Photonic Band Gap Materials

During the past five years, there has emerged a new class of materials called photonic band gap materials or, more simply, photonic crystals. The underlying concept behind these materials stems from early notions by Yablonovitch [1] and John [2]. In a nutshell, the basic idea is to design materials so they can effect the properties of photons in much the same way ordinary solids or crystals effect the properties of electrons. Now, the properties of electrons are governed by Schroedinger’s equation (1)$$ \left\{ { - \frac{{{{\nabla }^{2}}}}{2} + V(r)} \right\}\psi (r) = E\psi (r) $$ and properties of photons by Maxwell’s equations, which can be cast in a form very reminiscent of the Schroedinger equation, (2)$$ \left\{ {\nabla \times \frac{1}{{\varepsilon (r)}}\nabla \times } \right\}H(r) = {{\omega }^{2}}H(r) $$.

A new class of layer-by-layer structures that have full three-dimensional photonic band gaps is described. Each layer has a particularly simple arrangement of one-dimensional rods. These new structures have gaps of appreciable width. The dependence of the gap on the dielectric contrast, filling ratio, cross sectional geometry of the rods, and the structure of the unit cell is systematically studied. The gap is relatively insensitive to the exact geometrical properties of the rods in each layer. Such layer-by-layer structures have already been fabricated over a wide range of microwave and millimeter wave frequencies. Other structures with three-dimensional photonic bandgaps are surveyed.

In May 1992, I accepted a position at Iowa State University (ISU) and started working on photonic band gap materials with Gary Tuttle, assistant professor of electrical engineering. By that time, the classical papers in the field had already been published [1, 2]. Concepts like conduction and valence bands, acceptor and donor defects [3], had been introduced and widely used. There were series of articles in Physical Review Letters, about physical mechanisms around these materials [4,5]. People were excited about possible “photonic” applications and experiments, where the spontaneous emission would be reduced to nil at optical frequencies. To my surprise, all of the experimental work was limited by the original work done by Eli Yablonovitch [2] which was only performed at 15 GHz!! There had been some work by the IBM group to test two-dimensional alumina rods which showed a band gap around 70 GHz, but only for one of the polarizations[6]. I could not find any other experimental work which showed a full band gap in all directions. Besides, there were no published efforts to push the frequency performance to a frequency higher than the original 15 GHz.

Yablonovitch et al. [1] constructed the “three cylinder” dielectric structure which exhibits a photonic band gap in the whole Brillouin zone. They implemented the diamond structure proposed by Ho et al. [2,3]. This structure can be fabricated mechanically by drilling three sets of holes 35° off vertical into the top of a solid dielectric. However, extremely small dimensions, which are required for high frequencies, cannot be achieved. This problem can be solved by using deep X-ray lithography, which allows the fabrication of structures that can be used up to the infrared range. Three irradiations were performed in which the tilted arrangement of mask and resist was rotated each time by 120°. PMMA resist layers with a thickness of 500 microns were irradiated at the DCI storage ring in Orsay, France. The lattice constants of the 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 is not high enough for the formation of a photonic band gap, a moulding step must be applied. The holes in the resist structure were filled with a solution of polyvinylsilazane in tetrahydrofuran. After the evaporation of the solvent, the samples were pyrolyzed at 1100°C under N₂ atmosphere. The resist decomposes into CO₂, CH₄, CO, and H₂O, whereas polyvinylsilazane is transformed into a SiCN ceramic. A lattice of ceramic rods corresponding to the holes in the resist structure remained.

We aim in this chapter to provide an introduction to the rich tapestry of physical phenomena involving wave propagation in wavelength-scale periodic structures, and to explore briefly their significance in present day research and technology.

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 are useful for locating symmetry points of the lattice for exact orientation of the crystals. Prom transmission measurements the photonic band structure of an fcc crystal has been obtained along the directions between the L-point and the W-point. A modified Mach-Zehnder interferometer has also been developed for accurately measuring relative phase shifts of light propagating in photonic crystals to determine the dispersion resulting from photonic band structure near the band edges.

We have performed optical diffraction studies on colloidal crystals with large refractive index mismatches up to 1.45 and polarizibilities per volume as large as 0.6. These conditions push colloidal crystals into the regime where strong coupling between photonic crystals and the light field occurs. It is found that the photonic band structures result in apparent Bragg spacings that strongly depend on the wavelength of light. The dynamical diffraction theory that correctly describes weak photonic effects encountered in X-ray diffraction, also breaks down. Two simple models are presented that give a much better description of the diffraction of photonic crystals.

The simplest and most fundamental system for studying radiation-matter coupling is a single two-level atom interacting with a single mode of an electromagnetic field in a cavity. It received a great deal of attention shortly after the maser was invented, but at that time, the problem was of purely academic interest since the matrix elements describing the radiation-atom interaction are so small. The field of a single photon is not sufficient to lead to an atom field evolution time shorter than the other characteristic times of the system, such as the excited state lifetime, the time of flight of the atom through the cavity, and the cavity mode damping time. It was therefore not possible to test experimentally the fundamental theories of radiation-matter interaction, which predict, among other effects, (a)a modification of the spontaneous emission rate of a single atom in a resonant cavity,(b)oscillatory energy exchange between a single atom and the cavity mode, and(c)the disappearance and quantum revival of Rabi nutation induced in a single atom by a resonant field.The situation has drastically changed in the last few years with the introduction of frequency-tunable lasers, which can excite large populations of highly excited atomic states characterized by a high principal quantum number n of the valence electron. These states are generally called Rydberg states, since their energy levels can be described by the simple Rydberg formula.

There is a growing interest in recent years for the propagation of acoustic (AC) and elastic (EL) waves in random and periodic composite materials [1–15]. The interest among solid state physicists is mainly connected to the question of existence or not of spectral gaps in periodic systems or localized states in disordered systems in analogy with what happens to the electron wave propagation.

We have investigated the electromagnetic properties of a 3-dimensional wire mesh in a geometry resembling covalently bonded diamond. The frequency and wave vector dispersion show forbidden bands at those frequencies v₀, corresponding to the lattice spacing, just as dielectric photonic crystals do. But, they have a new forbidden band which commences at zero frequency and extends, in our geometry, to approximately one-half v₀, acting as a type of plasma cut-off frequency. Wire mesh photonic crystals appear to support a longitudinal plane wave, as well as 2 transverse plane waves. We identify an important new regime for microwave photonic crystals, an effective medium limit, in which electromagnetic waves penetrate deeply into the wire mesh through the aid of an impurity band.

Recently, there has been growing interest in the development of Photonic Band Gap (PBG) materials [1–21]. These are periodic dielectric materials exhibiting frequency regions where electromagnetic (EM) waves cannot propagate. The reason for the interest on PBG materials arises from the possible applications of these materials in several scientific and technical areas such as filters, optical switches, cavities, design of more efficient lasers, etc. [1, 2]. Most of the research effort has been concentrated in the development of two-dimensional (2D) and three-dimensional (3D) PBG materials consisting of positive and frequency independent dielectrics [1–18] because, in this case, one can neglect the possible problems related to the absorption [15, 19]. However, there is more recent work on PBG materials constructed from metals [20, 21] which suggests that these metallic structures may be very useful in the low frequency regions. In these regions, the metals become almost perfect reflectors.

The concept of a transfer matrix is extremely simple: if we know electric and magnetic fields in the x-y plane at z=0, then we can use Maxwell’s equations at a fixed frequency to integrate the wavefield and find electric and magnetic fields in the x-y plane at z=c. In fact, if we assume that both B and D have zero divergence, we need only know two components of each field: let us say, (1.1)$$ F\left( {z = 0} \right) = \left[ {{{E}_{x}}\left( {z = 0} \right),{{E}_{y}}\left( {z = 0} \right),{{H}_{x}}\left( {z = 0} \right),{{H}_{y}}\left( {z = 0} \right)} \right] $$ Then, (1.2)$$ F\left( {z = c} \right) = T\left( {c,0} \right)F\left( {z = 0} \right) $$ defines the transfer matrix, T.

The optical properties of a two-dimensional array of metallic particles on a dielectric slab have been investigated by a number of authors, mainly because of the possible applications of such systems in coating technology. In most cases the metallic particles are randomly distributed on the substrate surface and vary both in volume and shape [1], although in some instances a periodic arrangement in space of nearly identical particles has been achieved [2]. Usually, the average diameter of the metallic particles is of the order of 100Å and together the non-overlapping particles cover 30–70% of the substrate surface. The traditional analysis of such experiments is based on the Maxwell-Garnett theory and various extensions of it [1]. Essentially, the particles are replaced by interacting dipoles; an effective (local) field is evaluated by the use of the Clausius-Mossotti equation, and an effective dielectric function for the composite medium is obtained. This analysis (we refer to as the dipolar or electrostatic approximation) breaks down when the wavelength of the incident radiation is relatively small, of the same order of magnitude as the size of the particles and/or the interparticle distance, or when the volume occupied by the particles is about half or more of the total volume. We shall deal with this problem in section 2. We shall represent the particles by spheres characterised by a certain dielectric function and we shall assume they are arranged periodically on a plane.

Recently there has been renewed interest in the electromagnetic (EM) properties of complex dielectric materials. One outcome of this interest has been the application of novel methods in solving Maxwell’s equations. One of these methods, the transfer matrix technique, is particularly suitable for the study of the propagation of EM waves. In this work we show how it is possible to apply it to the study of several properties, such as energy loss of charged particles passing close to corrugated surfaces, absorption of light in metallic systems and enhanced Raman scattering in rough surfaces. As an example we study these properties in an array of metallic cylinders and explore how they vary as a function of the cylinders packing fraction.

This paper reports on schemes for the accurate determination of the electromagnetic properties of photonic crystals with arbitrary underlying lattice symmetry, using a real space transfer matrix method. The schemes are applied to hexagonal crystals and diamond-symmetry crystals, and results are compared with those obtained using a plane wave expansion method. The transfer matrix method is then applied to systems which comprise stacked finite-thickness photonic crystals with different but overlapping photonic stop bands, between which there can exist a planar cavity. Such ultra wide band gap structures can display scattering characteristics attributable to the presence of resonant modes at frequencies within the intersection of the stop band frequency ranges of the individual crystals. An initial study is presented of two stacked hexagonal crystals whose invariant axes are parallel. Results for stacked diamond-symmetry photonic crystals axe imminent.

The great majority of the existing calculations of photonic band structures have been limited to periodic structures, either discrete or continuous, fabricated from a dielectric characterized by a positive, real, frequency-independent dielectric constant ∊ _A , embedded in a dielectric matrix characterized by a dielectric constant ∊ _B that is also, positive, real, and frequency-independent [1]. The restriction to components characterized by such dielectric constants seems overly restrictive, since one can envision periodic two-and three-dimensional structures fabricated from metal wires or metal particles, or from rods and particles of polar semiconductors or ionic crystals, all of which are characterized by frequency-dependent dielectric functions, that can be negative in certain frequency ranges. It might be expected that the dispersion curves for electromagnetic waves propagating through such systems (their photonic band structures) may display interesting features, especially in the frequency ranges in which the dielectric functions of the embedded components are negative.

We present a new approach for calculating the dispersion curves of electromagnetic waves in periodic media which contain metallic components characterized by a complex, frequency-dependent dielectric function.The formalism is based on the use of a position-dependent dielectric function and the plane wave technique. Because of the complex form of the dielectric function the reduction of the band structure calculation to the solution of a single standard eigenvalue problem is not possible. Instead, a generalized eigenvalue problem has to be solved. At low filling fractions of the metallic components (f ≤ 1%) the generalized eigenvalue problem is reduced to the problem of solving sets of nonlinear simultaneous equations which correspond to the diagonal terms of the matrix equation in the plane wave representation, with the non-diagonal elements taken into account perturbatively. The resulting complex band structure yields besides the dispersion curves also the attenuation of each mode as it propagates through the system. The method has been applied to the calculation of the photonic band structures of electromagnetic waves propagating through both 1D and 2D periodic systems. The real part of the photonic band structure for small filling fractions is not significantly affected by the presence of the dissipation.Both parts of the complex photonic band structure exhibit different behaviour depending on the polarization of the electromagnetic waves.

In a recent paper, Elson and Tran [1] applied a multilayer modal method with an R-matrix propagator to calculate diffraction intensity from gratings and dispersion in photonic media. This calculation was done in Fourier space where the fields and photonic media are manifest as Fourier transforms of spatially variable quantities. With regard to photonic media, the results of this method [1] agree with experimental data for transmission through a seven-row deep square arrangement of round cylinders. In the present work, a similar calculation is done in real space, where modal expansion solutions are not required. Like the previous work, [1] the method presented here divides the photonic medium into sublayers and the R-matrix algorithm is applied to propagate the solutions through the inhomogeneous media. Unlike the previous work, we do not need to compute eigenvalues and eigenvectors to find modal solutions. This can be an important advantage since such computations can be time consuming.

This paper reviews three applications we have investigated using conventional (i.e., all dielectric) photonic crystals at frequencies up to about 30 GHz: (1) microwave mirrors, (2) substrates for planar antennas, and (3) photonic-crystal heterostructures. In each case, an important characteristic of the photonic crystal is that the reflection at frequencies in the stop band is distributed over at least one lattice constant in depth. Thus, the heat generated by residual dielectric absorption is distributed over a much larger volume than the heat generated by surface losses in a metal mirror, enabling a lower operating temperature. An additional characteristic of the photonic crystal, essential to the antenna application, is that its stop band is three-dimensional and thus rejects the majority of power radiated by an antenna mounted on its surface. This makes the planar antenna much more efficient than the same antenna placed on a homogeneous substrate made from the same dielectric material as the photonic crystal. A key factor in the ultimate practicality of these applications is the development of new types of photonic crystals that are superior structurally to conventional crystals or that display enhanced stop-band characteristics. To widen the stop band, we have studied a photonic-crystal heterostructure consisting of a stack of monoperiodic sections having different lattice constants. The resulting structure is shown to have a stop band of nearly one octave.

The radiation patterns of a dipole antenna on a photonic crystal substrate are measured and compared with dipole antennas on dielectric substrates. The layer-by-layer photonic crystal with a gap between 12 and 15 GHz is utilized. The angular distribution of radiated power is optimized by varying the position, orientation and driving frequency of the antenna. Virtually no radiated power is lost to the photonic crystal resulting in gains and radiation efficiencies larger than on other conventional dielectric substrates.

We discuss recent progress in our effort to develop a high gradient accelerator cavity based on Photonic Band Gap (PBG) concepts. Our proposed cavity consists of a two-dimensional (2-D) photonic lattice, composed of either dielectric or metal scatterers, bounded in the third dimension by flat conducting (or superconducting) plates. A defect introduced to the lattice, usually a removed scatterer, produces a defect mode with fields concentrated at the defect site and decaying exponentially in all directions away from the defect site. The defect mode is designed to resonate at frequencies in the 2–20 GHz range, where metals can still be used to confine the energy with minimal loss. We present in this paper some of the technical considerations which have arisen relevent to this application, and to PBG structures in general. In particular, we focus on measurements and calculations carried out for a 2-D metal PBG cavity.

We introduce and analyse a new type of resonant microcavity consisting of a channel waveguide and a one-dimensional photonic crystal. A band gap for the guided modes is opened and a state is created within the gap by adding a single defect in the periodic system. An analysis of the eigenstates in the system shows that a strong field confinement of the defect state can be achieved with a modal volume less than half of a cubic half-wavelength. The coupling efficiency to this mode will be shown to exceed 80%. As a proof of concept, we present a feasibility study for the fabrication of these microcavities in an air-bridge configuration with micron-sized features using semiconductor materials.

We discuss the fabrication of two-dimensional photonic bandgap (PBG) lattices in a GaAs/AlGaAs semiconductor and present a waveguide structure that allows their optical characterisation. The one-dimensional case has been used as a vehicle for exploring, theoretically and experimentally, the buildup of a PBG in a semiconductor-air photonic lattice. By designing a lattice with a high semiconductor fill-fraction, the scattering loss has been reduced to a tolerable level and true PBG effects have been observed.

A structure with a refractive index that varies periodically in three dimensions can fail to transmit light of certain “forbidden” frequencies, despite being non-absorbing. This is because Bragg diffraction, constructive multiple-beam interference from a periodic array of scatterers, takes place for all directions; only an exponentially-decaying evanescent field is supported. These frequencies ω are the full photonic band-gaps (PBGs) of the structure. Light that attempts to enter the structure is expelled and returns to the medium from which it came. If that medium is free space, the structure behaves as a totally-reflecting mirror; if the medium is a buried excited atom wanting to emit a photon at a forbidden frequency, the spontaneous emission is suppressed[1].

Electronic surface states, whose energies lie within the semiconductor band-gap, are normally regarded as a nuisance to be avoided in semiconductor lasers. This is particularly true of micro-pillar laser arrays where the surface-area-to-volume ratio is large. Similarly, intra-photonic-band-gap interface or surface states will reduce the effectiveness of a photonic bandgap material by introducing intra-band-gap states into which unwanted spontaneous emission and lasing can occur [1,2]. These photonic defect states are not, however, always undesirable. Examples include: i) the DFB laser mode supported by a structural defect at the centre of a uniform Bragg mirror — the resonant frequency of this mode lies within the photonic bandgap of the Bragg mirror [3]; ii) the surface-guided Bloch modes (SGBM) confined at the surface of multilayer stacks [4]; and iii) Bragg waveguide modes (BWGM) in which total internal reflection is replaced by Bragg reflection between two multilayer stacks [4]. A general feature of defect modes is a phase velocity that is highly sensitive both to optical frequency and to the “strength” of the local aperiodicity that defines the defect. Small compositional and structural changes can radically alter the position of the mode within the stop-band, providing an effective tuning mechanism. Owing to these and other unique properties, defect modes may provide the basis for the development of a versatile new family of optoelectronic devices.

The fabrication of photonic band gap materials at micron and submicron lengthscales 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 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 µm deep 2-D square lattice of circular air rods with a lattice constant of 8 µm 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 µm). 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.

In recent years experimental and theoretical studies of artificially manufactured, lossless periodic dielectric media called Photonic Band Gap (PBG) Materials or Photonic Crystals, have experienced a dramatically increased amount of interest[1, 2]. This can be attributed to their highly unusual, yet rather easily controllable properties which, firstly, offer exciting and challenging new problems for basic research and, secondly, give rise to numerous device applications.

A recent problem of interest in the study of photonic band structures has been that of a single dielectric impurity in an otherwise periodic photonic band structure. This problem has been studied using supercell methods [1], transfer matrix methods [2,3], and Green’s function methods [4–6]. In this paper we will look at the exact solution of the single impurity problem using Green’s function methods. Results will be presented for one-dimensional (layers) and two-dimensional (periodic array of rods) systems for impurities with frequency dependent dielectric constants substituted into periodic dielectric media composed of frequency independent dielectric materials. The impurities will be substitutional impurities (replacement of a single slab or rod by that of another dielectric composition). Particular attention will be given to obtaining theoretical results using dielectric constants appropriate to materials which have been used in recent experimental studies of photonic systems.

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, waveguides, 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 the widths of these gaps with the filling patterns is studied and potential application for the creation of photonic crystals in the optical domain is discussed.

The existence of a photonic band gap (PBG), in materials where the refractive index varies periodically, gives rise to many interesting and potentially useful properties, including the localization of light[1], the inhibition of radiation [2], etc., see Ref. 3 and references cited therein. These properties become more pronounced when the photonic gap is made large. Accordingly, the search for crystals with large PBGs has been extensive[1–4]. However, to our knowledge this search has had a limited success as it has identified structures with significant PBGs described by a gap-to-midgap frequency ratio of only about 20%.

Within the framework of the classical electromagnetic scattering theory, this paper develops a practical evaluation scheme to accurately compute the light scattered by a two-dimensional array of particles, with finite extension in the direction perpendicular to the periodicity.

Guided modes in slab-like structures consisting of dielectric material surrounded by 2D photonic band gap material drilled in the same dielectric are studied by the supercell method. Focusing on E-polarized modes ($$ \vec{H} $$ transverse to the rods), we outline (i) the existence of a guide with mid-gap mode much narrower than the classical quarter-wave layer of the 1D case, (ii) the role of the relative “phases” of the boundary corrugations for contradirectional coupling, and (iii) the symmetries of modes at k = 0 and how it appears in the succession of modes.

Experimental evidence of second harmonic generation in a macroscopically centrosymmetric lattice formed by spherical particles of optical dimensions is presented. Second harmonic light is scattered from the surface of these spherical particles. A simple theoretical model based on scattering in the Rayleigh-Gans approximation indicates constructive interference of light scattered at the second harmonic frequency leads to a plane wave front propagating in the direction of the incident beam. The implications of the addition of defects in a controlled manner will be discussed in the framework of the photonic band gap theory. Along these lines we will discuss in detail the simple case of a truncated periodic lattice in one dimension. We will present additional experimental results showing enhancement and suppression of the radiation of a dipole sheet oscillating at the second harmonic frequency embedded in a 1-dimensional periodic structure with a defect.

Photonic crystals that possess a Photonic Band Gap (PBG) are systems with very intersting physical properties and, in addition, are very important in applications. As a result of numerous efforts [1, 2, 3, 4, 5, 6] both in the experimental and theoretical direction it is now well established that PBG systems can be realized in practice. Possible applications include resonance cavities, perfect refractive mirrors, antennas and electromagnetic wave filters with broad frequency range from ultraviolet to microwave. Most PBG studies so far have addressed almost exclusively PGB structures where nonlinearity in the dielectric constant of the constituting matertial is negligible.[7] One reason for this is that in the presence of nonlinearity the superposition principle does not hold any more, rendering the calculations more difficult and also turning familar concepts such as that of bands and gaps essentially meaningless. In the present paper we present recen-t results obtained through a simple periodic model concerning nonlinear effects in one dimensional PBG crystals and other nonlinear periodic lattice systems.[8, 9] The nonlinear Kronig-Penney (KP) model that we use shows that in the electromagentic case of wave propagation in one dimension occurrence of shifts in the linear stop gaps as a result of nonlinearity, the onset of bistability as well as switching properties.[10]

It is shown that in a periodic medium with smoothly modulated parameters, electromagnetic waves can propagate along effective waveguides. Such waveguides are interpreted in terms of defect modes. Respective theory based on the Wannier function expansion is developed. Some examples and relative effects are considered.

The Finite Difference Time Domain (FDTD) technique dates back to 1966 when it was first developed by Yee [1]. Since then it has been widely used to calculate the radar cross section of objects as well as normal modes of wave guides. Chan et al. [2] recently applied this technique to calculate the band structure of photonic crystal with excellent results. The motivation for their use of this technique is the fact this technique scales linearly with system size. Systems having random defects, which destroy any periodicity, are often studied using the “super cell” method where the system is assumed to be periodic with a very large period. In such a case, it is important the technique for band structure calculation scales favorably with system size. The plane wave expansion technique, for example, is impractical since it scales as N3 where N is the system size. In this paper I show how the FDTD technique can be extended to calculate the band structure of nonlinear photonic crystals, in particular, those which possess Kerr nonlinearity.

There are at least two technological revolutions in the twentieth century which owe their existence to atomic and condensed matter physics. The first was the semiconductor revolution which resulted in the emergence of the electronics industry. The second was a direct consequence of the invention and applications of the laser. A third revolution, may yet emerge from the discovery of “high temperature” superconductivity. In each of these developments, the fundamental elementary particles or “actors” are electrons and photons. The properties of semiconductors, lasers, and superconductors are fundamentally a consequence of the dynamical and cooperative behaviour of electrons and photons in carefully controlled environments. Electrons are strongly interacting fermions which are readily localized by the positive charge of an atomic nucleus. The semiconductor crystal facilitates a highly controlled delocalization of the electronic wavefunction. This selective flow of charge enables a semiconductor device to perform its function. Photons are weakly interacting bosons which propagate readily. The laser facilitates the emergence of a cooperative or coherent state of a large collection of photons. In a superconductor, electrons in a metal form bosonic, Cooper pairs which condense into a cooperative state analogous to the coherent state of photons in a laser field.

We present a new method for efficient, accurate calculations of the transport properties of random media 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 multiple scattering 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.

Light scattering from resonant two-level atoms on three-dimensional lattices can be described by a classical energy conserving t-matrix. The optical band structure of lattices filled with such atoms are calculated exactly in scalar approximation and displays, e.g., the formation of polariton gaps. The Einstein coefficient for spontaneous emission in a system with an inhomogeneous dielectric constant is shown to be proportional to part of the total density of states. This part is calculated for the dipolar lattice model.

A 1D random photonic lattice has been studied both by the plane-wave method and by the transfer matrix approach for arbitrary contrast in its dielectric permittivity. The role and competition of different scattering mechanisms such as Bragg diffraction, one scatterer resonances, and Bragg remnants have been investigated. It is shown that strong localization results not only from Bragg diffraction and disorder in the scatterers distribution, but also Bragg remnants produce localization for a medium with small contrast. Based on the properties of the random lattice in the light frequency band, highly sensitive sensor could be devised.

One of the most important properties of photonic band-gap (PBG) materials is the confinement of light whose frequency lies in the bandgap [1]. Many of the envisioned applications of PBG materials rely on the degree to which this confinement characteristic is realized. It follows, therefore, that while it is important to examine the special properties of PBG materials in its perfect form, it may also be important to study the wave behavior in imperfect PBG materials, so as to gain the knowledge about PBG material requirements in practical applications. In particular, it is of interest to see what happens to waves when there is a density of imperfections, or impurities, in PBG materials.

Wave propagation in random media has been the focus of considerable attention in the past decade [1,2]. Various interesting wave phenomena have been uncovered, and studied, e.g., the coherent backscattering effect[3], short-range and long-range spatial intensity correlations [4,5], and dynamical intensity correlations [6], etc. In case when the sample size is much greater than transport mean free path, great simplification in theoretical treatments has been found in the diffusion approximation. The diffusion approach has been successful in describing most of the phenomena observed, including the coherent backscattering cone and its sample size dependence [7], and the spatial intensity correlations [8,9], etc. It works well even in the case when there are internal reflections in the random media [10,11]. The application to the coherent backscattering of the scalar acoustic waves is also excellent [12]. It should be noted that the diffusive property of light has recently been used to advantage as a new optical imaging technique [13].