Session V35: Photonic Crystals

A Dynamical Effective Medium Theory for Meta-materials

We present a new dynamical effective medium theory in the framework of two-parameter coherent potential approximation. While conventional homogenization theories will fail near resonances, our theory gives simultaneously accurate effective permittivity and permeability near resonant frequencies. The theory can also describe the absorption. Our assertions are supported by the band structure and transmission calculations of periodic composites in both two and three dimensions.   

The gap structures and wave functions of two-dimensional quasicrystals for classical waves.

By using the sparse-matrix canonical-grid (SMCG) method, we have performed large-scale multiple scattering calculations to study the gap structures and wave functions of two-dimensional quasicrystals for classical waves. As sample size is increased, we find self-similar-like evolution in the gap structures. We also find the self-similar state and large-size localized states. All these findings arise from the quasiperiodic long-range order. A self-similar state at band edge is identified in a large sample containing 33919 scatterers. A coherent picture is presented to describe the relationship between the self-similar-like evolution of gap structures and the formation of self-similar states.   

Electromagnetic wave propagation in media whose permittivity varies periodically in time

We have developed a general theory for propagation of plane electromagnetic waves in a medium with permittivity that is varying periodically in time. The Bloch-Floquet theorem dictates that these are a superposition of harmonic modes whose frequencies differ by $2\pi /T$, where $T$ is the period of $\varepsilon (t)$. For arbitrary periodicity, the dispersion relation $\omega (t)$ for the ``Bloch frequency'' is given in terms of the roots of an infinite determinant whose elements depend on the Fourier coefficients of $\varepsilon (t)$. For small variation of $\varepsilon (t)$ around an average $\varepsilon _0 $,$\omega (t)$ is characterized by regions of the wave vector $k$ that are forbidden for propagation. These are centered at \textit{$\omega $} and $k$ values that are, respectively, integer multiples of $\pi /T$ and of $\pi \varepsilon _0^{1/2} /cT$. The widths of the gaps are proportional to the corresponding Fourier coefficients of $\varepsilon (t)$. In the special case of square-periodic variation of $\varepsilon (t)$, there is no need to recur to a perturbational calculation, because the dispersion relation can be derived analytically, with no approximations. Again, we find wave vectors gaps whose edges are located at the frequencies $\omega =0,\pi /T,2\pi /T,...$ .   

Achieving a Photonic Band-Edge Near Visible Wavelengths by Copper Coating on 3D Tungsten Photonic-lattices.

The fabrication of all-metal three-dimensional (3D) photonic lattices (PLs) is a subject that has raised great interest due to their potential in energy applications [1]. However, there are very few metals are suitable for the architecture of 3D PLs with a photonic band-edge near visible wavelengths. 3D tungsten (W) photonic lattice in ``wood-pile'' architecture has been successfully fabricated [1]. But the intrinsic limitation of W prevents the photonic band-edge to go beyond 2$\mu $m. We show that by material engineering this limitation can be overcome. We coated a very thin layer of copper (Cu) conformally onto the W PLs and modified their optical properties. The Cu thin films were coated on the PLs by electroless deposition. This thin Cu film ($\sim $70 nm) has dramatic effects on the photonic band-gap behavior of the W PLs. After the coating of Cu, the photonic lattice has a much larger photonic band-gap. The band-edge of the photonic lattice is shifted from $\lambda \sim $2$\mu $m to $\lambda \sim $750nm. Our experimental data agree with the predictions by finite difference time domain calculations. This method provides a new route for tailoring photonic properties and it should work for 3D PLs constructed from other materials. \newline [1] J. G. Fleming, S.Y. Lin, I. El-Kady, R. Biswas and K. M. Ho, \textbf{\textit{Nature}} \textbf{417}, 52-55 (2002).   

From Photonic Crystals to Photonic Metamaterials. I

We present a very general mean-field theory for a photonic crystal (dielectric or metallo-dielectric) with arbitrary (3D) Bravais lattice and arbitrary inclusions within the unit cell. The material properties are characterized by a generalized conductivity at every point in the unit cell. Averaging over many unit cells for very small Bloch wave vectors, we derive the macroscopic response for the metamaterial. This, in general, turns out to be the bianisotropic response, specified in terms of permittivity and permeability tensors and, also, ``crossed'' D/H and B/E tensors. These four tensors are derived in terms of the content of the unit cell. However, in case of inversion symmetry these ``crossed'' tensors vanish. Moreover, for cubic symmetry the system becomes isotropic, described by scalar, frequency-dependent, permittivity and permeability.The response is essentially local, although, spatial dispersion can also be accounted for. Our mean-field theory is valid for optical photonic bands, as well as acoustic bands.   

From Photonic Crystals to Photonic Metamaterials. II

We have applied the mean-field theory (I) described in the preceding abstract to metallo-dielectric photonic crystals with cubic symmetry and for different forms of the metallic inclusions. In particular, we have calculated analytically the effective permittivity and permeability for periodic composites with ``cages'' of thin metallic wires or ``3D crosses'' and for arrays of small metallic or dielectric spheres in vacuum. We compare our results for the permittivity of such systems with former theories and find agreement as far as the real part is concerned. Our results for the imaginary part of the permittivity differ from other works and those for the permeability are new. In fact, the extinction coefficient that we derive for small metallic spheres is three orders of magnitude greater than the prediction of the Maxwell-Garnett theory. We are in the process of applying our theory to other interesting metallo-dielectric systems.   

Quantum Hall effect analogs in photonic crystals: semi-classical treatment of ``chiral'' (unidirectional) edge modes

Previously, we have shown that ``photonic crystals'' (periodic metamaterials that transmit electromagnetic waves) made of non- reciprocal media (Faraday effect) can theoretically possess channels that allow light to propagate in one direction only. These channels are the direct photonic analog of the ``chiral edge states'' of electronic systems exhibiting a quantum Hall effect. Here, we construct exactly soluble models of photonic systems having these properties. The models considered here correspond to photonic systems with smoothly varying domain walls across which the Faraday coupling changes. The spectrum of bound states of this model contains bi-directionally propagating modes localized to the interface, and ``zero modes,'' in which light remains localized to the interface and propagates only in a forward direction perpendicular to it. In the semiclassical treatment of this model, we show that the quantization condition for the bound states has an additional contribution from a Z2 Berry phase factor picked up by the modes as they encircle points of degeneracy.   

Influence of disorder on the band structure of a photonic crystal

We investigate both theoretically and experimentally the influence of disorder on the optical mode spectra of chains of coupled micropillar cavities (photonic dot resonators - PDR). These quasi one-dimensional structures were fabricated by lateral patterning of planar Bragg microcavities. Maxwell equations were solved by finite elements methods. The disorder is introduced by systematic variation of the cavity sizes. For periodic chains without disorder (equal PDRs) angle-resolved photoluminescence shows well-pronounced photonic band gaps. We show that even small disorder drastically modifies the properties of the optical modes and their spectra. Instead of set of propagating energy bands with Bloch states, impurity bands with Anderson-like localized states appear. The dependences of the energy bands and their intensities on cavities sizes are investigated and explained.   

Photonic band-gaps in two dimensional semiconductor-dielctric composite structures

This paper reports the multiple band-gaps in the two-dimensional semiconductor-dielectric photonic crystals of several compositions: semiconductor (dielectric) cylinders in the dielectric (semiconductor) background. We consider both square lattice and triangular lattice arrangements and compute extensive band structures using a plane-wave method within the framework of an efficient standard eigenvalue problem for both E- and H-polarizations. The whole range of filling fractions has been explored to claim the existence of the lowest (the so-called acoustic band gap) and multiple higher-frequency band gaps within the first thirty to forty bands for various compositions. The completeness of the existing band gaps is substantiated by computing the band structures via detailed scanning of the principal symmetry directions covering periphery as well as the interior of the irreducible part of the first Brillouin zone and through the computation of the density of states. In general, the composition made up of doped semiconducting cylinders in the insulating background is found to be the optimum case for both geometries. Such semiconductor-dielectric photonic crystals which are shown to possess huge lowest band gaps below a threshold frequency (the plasma frequency) have an advantage over the dielectric photonic crystals in the emerging technology based on the photonic crystals.   

Fabrication of Highly Reflecting Si/SiO$_{2}$ Bragg Mirrors Using Transferred Nanomembranes

A long term goal of electronics and photonics is the integration of both on the same materials platform and substrate. Silicon and silicon- dioxide are an excellent photonic materials pair, due to the superior properties of the oxide-silicon interface, and due to the large index-of- refraction contrast in this materials pair. We present a novel method to fabricate Si/SiO$_{2}$ mirrors based on transfer of freely released silicon nanomembranes and subsequent thermal oxidation. The surface roughness of the transferred membranes is examined using atomic force microscopy. RMS roughnesses as small as 0.25nm are achieved. The reflectivity is also measured for our transferred membranes, and an increase in reflection is easily observed as the number of stacked membranes is increased. The experimental reflectivity data match theoretical calculations within 10{\%}. We discuss the possibility of aligning many membranes to make complicated photonic structures.   

Tuning of the Optical Response of a Si/SiO$_{2}$ Superlattice at the Wavelength of $1.54\mu$m

We have studied a one-dimensional photonic crystal of alternating layers of silicon and silicon oxide (and, also, air), fixing the wavelength at $1.54\mu $m, of great importance for communications in the near infrared. We assume ~strong doping of the Si layers, the density of free electrons reaching values up to 3X10$^{20}$/cm$^{3}$. Using a realistic plasma model that takes into account electrons, holes, and the corresponding attenuation, our calculations lead to the conclusion that the reflectance and transmittance can depend very strongly on the carrier density. With a careful choice of parameters, a change in the reflectance from 0 to 90{\%} can be attained in the vicinity of a photonic band edge, with only a moderate change in the impurity concentration.   

Improved Photonic Bandgap Structures via Symmetry Breaking

We have previously reported on our efforts to develop a semiconductor based photonic bandgap nano-membrane device with 3D MEMS features which is designed to reconfigure the photonic crystal structure into different types of waveguide devices such as switches, modulators, delay lines, etc.$^{1}$ As design and fabrication interact there is tension between what one would like to make and what one can actually make. We have found that in many cases, by breaking some of the symmetry, it is possible to shift gaps around and to introduce new gaps. We discuss various examples of this. As an example with especially good performance, we discuss the case of rotated square holes in a square lattice. 1. ``Design and Fabrication of a Reconfigurable Photonic Bandgap Waveguide Device with MEMS Features,'' Weimin Zhou, Monica Taysing-Lara, Gerard Dang, Lorna Harrison, David Mackie, Matthew Ervin, and Peter Newman, presented at CLEO 2004.   

Wannier Function Expansion of Localized States in Photonic Crystals

We present a theoretical treatment of localized electromagnetic modes in infinite photonic crystals. Our basis states are local vector Wannier functions calculated from the unperturbed crystal eigenstates. Analogous to the calculation of localized electronic states by expansion in terms of electronic orbitals, this Wannier basis is used to expand photonic crystal defect states. The localized nature of the basis states is critical so that the basis can be truncated after a small number of neighbor lattice sites in the vicinity of the defect. We present results that verify the eigenmodes of the crystal and examine defect modes. An analysis is done to determine the convergence of the mode as a function of the number of basis states included in the computation. This formalism equally treats localized states that are bound in a photonic band gap and states resonant with the propagating photon states of the infinite crystal. The present method has certain advantages with regard to computational complexity, spatial resolution, and the efficient accommodation of non-spherical geometric features.   

Tangent formulation for the band structure of one-dimensional N-period layered photonic crystals

At symmetry points of the Brillouin zone, the two-layer Kronig-Penney (KP) problem has even and odd parity solutions that are expressible with tangents and cotangents. Similar solutions are derived here for an arbitrary number of layers. Namely, the eigenvalue-eigenvector problem for the energy spectra and wave functions of arbitrary, one-dimensional, N-period layered systems is formulated in terms of tangents only. The resulting equations are compact, algorithmically simple, numerically stable, and physically appealing. The derived secular equation is Hermitian and only of order N$\times $N, i.e., half the size of the KP secular equation. The eigenfrequency condition has physically attractive geometric representation in terms of N-sided figures such as triangles and tetrahedrons for N=3. The analytic power of the formalism is demonstrated by diagonalizing the secular equation for N=3, finding a factored form of the KP equation, and deriving analytic eigenfrequency conditions and analytic wave functions for the three layer problem. The analyticity of the formalism should aid the band gap engineering of the band structure and wave functions of multilayer structures. The numerical ease of implementation is demonstrated by calculating the eigenfrequencies and wave functions for a three-layer photonic crystal.   

Temperature tuning of two-dimensional photonic crystals in the presence of phonons and a plasma of electrons and holes

We have theoretically studied the electromagnetic transmittance in finite samples of InSb-based two-dimensional photonic crystals. Due to the temperature dependence of the intrinsic carrier concentration in the semiconductor, our square arrays of parallel InSb cylinders in air give rise to tunable transmission spectra. As the temperature increases from 200 to 290 K, we find that the midgap frequencies move up in frequency while the widths of the gaps diminish, in agreement with the bulk band structure. We find that absorption affects considerably the transmittance intensity. In order to achieve significant tuning of the transmission, appropriate structural parameters and spectral regions are proposed. We have also studied transmission of light through two-dimensional photonic crystals of finite width, with the radii of the cylinders in the middle row altered in comparison to the host cylinders.