Homogenization of 3D finite photonic crystals with heterogeneous permittivity and permeability

We consider a heterogeneous magneto-dielectric photonic crystal and derive the so-called ‘homogenized Maxwell system’ via the multi-scale method and provide ad hoc proofs for the convergence of the electromagnetic field towards the homogeneous one using the notion of two-scale convergence. The homogenized medium is described by anisotropic matrices of permittivity and permeability, deduced from the resolution of two annex problems of electrostatic type on a periodic cell. Noteworthily, this asymptotic analysis also covers the case of photonic crystals with non-cuboidal periodic cells. We solve numerically the associated system of partial differential equations with a method of fictitious charges and a finite element method (FEM) in order to exhibit the matrices of effective permittivity and permeability for given magneto-dielectric periodic composites. We then compare our results in the 2D case against some Fourier expansion approach and provide duality relations in the case of magneto-dielectric checkerboards. We further compute some low-frequency eigenmodes of a photonic crystal fiber with metallic outer boundary and compare them with the eigenmodes of a corresponding effective anisotropic waveguide, thanks to the FEM. Finally, we derive the effective properties of a 3D photonic crystal both through classical homogenization (solving numerically two decoupled annex problems) and Bloch wave homogenization. In the case of spherical inclusions, the latter approach amounts to evaluating the slope of the first band around the origin on a Bloch diagram which we compute using finite edge elements.     

Natural photonic crystals

Photonic structures appeared in nature several hundred millions years ago. In the living world, color is used for communication and this important function strongly impacts the individual chances of survival as well as the chances to reproduce. This has a statistical influence on species populations. Therefore, because they are involved in evolution, natural color-generating structures are – from some point of view – highly optimized. In this short review, a survey is presented of the development of natural photonic crystal-type structures occurring in insects, spiders, birds, fishes and other marine animals, in plants and more, from the standpoint of light-waves propagation. One-, two-, and three-dimensional structures will be reviewed with selected examples.     

Graded photonic crystals

We present a concept of graded photonic crystals used to enhance the control of light propagation. Gradual modifications of the lattice periodicity make it possible to bend the light at the micrometer scale. This effect is tailored by parametric studies of the isofrequency curves. As a demonstration, we propose a two-dimensional graded photonic crystal that could provide frequency-selective tunable bending.     

Metallic photonic crystals

Metallic photonic crystals (PC), also originally called artificial dielectrics, have properties different from those of their dielectric homologues. They are of strategic interest for the microwave domain where they exhibit sufficiently weak loss in addition to their robustness, conformability and low-cost fabrication. In this paper, we review some recent results on metallic photonic crystals and their potential applications to microwave circuits and antennas. After recalling spectral properties of metallic PC, we successively address ultra-compact structures such as high-impedance surfaces, electrically controllable photonic band gaps and left-handed materials. Finally, we discuss new opportunities offered by metallic PCs in the optical domain. To cite this article: J.-M. Lourtioz, A. de Lustrac, C. R. Physique 3 (2002) 79–88     

Design of photonic bands for opal-based photonic crystals

To make a device from an opal—or otherwise—the photonic bands and the optical properties derived from them are needed. Knowing the effects of different parameters defining the opal geometry and different possible modifications of its structure are needed, too. An accurate definition of the device will be required to obtain a good performance. With this aim, the optics of light with a wavevector in the vicinity of the L point in the Brillouin zone and its coupling to bare opals band structure are presented. An important aspect is the transition from finite to infinite crystal and the study of size effects on the bands. It is possible to substantially alter the photonic band structure of an opal-based system, while maintaining the lattice structure, simply by growing layers of other materials with an appropriate refractive index. Here, it is shown how, by the growth of accurately controlled thin layers of silicon and germanium, and further processing, one can induce the opening of two complete photonic band gaps (PBGs) in an opal structure. Finally, the possibility to fabricate a simple device consisting in a planar waveguide will be shown. By means of a very simple and inexpensive procedure, engineered planar defects acting as microcavities have been realized. These can be viewed as a particular case of a much more general class of heterostructures that can be grown by combining opal vertical deposition and chemical vapour deposition of oxides. A further step is made by applying electron beam lithography to provide lateral definition and facilitate three-dimensional structuring.     

Two-dimensional function photonic crystals

In this paper, we have studied two-dimensional function photonic crystals, in which the dielectric constants of medium columns are the functions of space coordinates , that can become true easily by electro-optical effect and optical kerr effect. We calculated the band gap structures of TE and TM waves, and found the TE (TM) wave band gaps of function photonic crystals are wider (narrower) than the conventional photonic crystals. For the two-dimensional function photonic crystals, when the dielectric constant functions change, the band gaps numbers, width and position should be changed, and the band gap structures of two-dimensional function photonic crystals can be adjusted flexibly, the needed band gap structures can be designed by the two-dimensional function photonic crystals, and it can be of help to design optical devices.     

One-dimensional opal photonic crystals

One-dimensional opals are 1D self-assembled close packed colloidal crystals consisting of monodisperse colloidal globules. Polystyrene globules with sizes in the 1.9–10 μm range sit on a flat substrate and touch two neighbors in diametrally opposite contact points. These opals are quasi-1D photonic crystals. Optical modes, including whispering gallery modes of individual globules, coupled collective modes, and nanojet-induced modes, are visualized in 1D opals.     

蝴蝶与光子晶体

近年来 ,科学家们发现有着各种各样颜色的动物 ,如有花斑的甲壳虫、美丽的小鸟和五彩斑斓的蝴蝶等 ,它们身上的彩虹都来自于一种称为光子晶体 (photoniccrystals)的微结构 .通常的颜料是由其本身的化学成分在吸收或反射一定频率的光线后呈现出各种颜色的 ,但光子晶体与此不同 ,它对光的反射是来自于本身的物理结构的功能 ,也就是说 ,光子晶体具有周期性的空穴或凸隆阵列 ,这种有一定尺寸的阵列 ,在晶体本身完全是无色的情况下 ,也能反射兰色光线 ,并吸收其他颜色的光线 .这是由于光子晶体的阵列从不同角度去看时会有一些微小的差别 ,而从不…     

Chalcogenide glass photonic crystals

All-optical switching devices are based on a material possessing a nonlinear optical response, enabling light to control light, and are enjoying renewed interest. Photonic crystals are a promising platform for realizing compact all-optical switches operating at very low power and integrated on an optical integrated circuit. In this review, we show that by making photonic crystals from a highly nonlinear chalcogenide glass, we have the potential to integrate a variety of active devices into a photonic chip. We describe the fabrication and testing of two-dimensional Ge₃₃As₁₂ Se₅₅ chalcogenide glass photonic crystal membrane devices (waveguides and microcavities). We then demonstrate the ability to post-tune the devices using the material photosensitivity. In one proposal we hope to introduce a double-heterostructure microcavity using the photosensitivity alone.     

Resonant three-dimensional photonic crystals

The theory of the exciton-polariton band structure of a resonant three-dimensional photonic crystal is developed for an arbitrary dielectric contrast and an arbitrary effective mass of an exciton excited in a composite material. The calculation is performed for a periodic array of semiconductor balls embedded in a dielectric matrix. The position of the lower polariton dispersion branches is shown to depend monotonically on the exciton effective mass and to be governed by the interaction of light with the first several states of a mechanical exciton quantum-confined within each ball. The effect of excitonic states on the band gap of a photonic crystal in the [001] direction is considered analytically in terms of a two-wave approximation.     

Near-field probing of photonic crystals

Photonic crystals form an exciting new class of optical materials that can greatly affect optical propagation and light emission. As the relevant length scale is smaller than the wavelength of light, sub-wavelength detection forms an important ingredient to obtain full insight in the physical properties of photonic crystal structures. Spatially resolved near-field measurements allow the observation of phenomena that remain hidden to diffraction-limited far-field investigations. Here, we present near-field investigations in both collection and illumination modes that highlight the power of local studies. We show how propagation losses are unambiguously determined and that light detected in far-field transmission can actually contain contributions from different, sometimes unexpected, local scattering phenomena. Simulations are used to support our findings. Furthermore, it is shown that local coupling of light to a thick three-dimensional photonic crystal is position-dependent and that the spatial distribution of the coupling efficiency itself is frequency-dependent.     

Physical optics of photonic crystals

A one-dimensional harmonic model of a photonic crystal is considered using the methods of physical optics. The theoretical formalism is based on the notion of the Fresnel volume reflection and the system of two first-order differential equations, which are equivalent to the wave equation. Using the Rayleigh layer as an example, it is shown that the volume reflection plays a role of the friction, similar to the friction in oscillations of a pendulum, and, in a strongly inhomogeneous medium, can suppress field oscillations and turn the group velocity to zero. In the approximation of small modulation factor, the models of two, four, and six waves are considered. In the two-wave model, the dispersion relation contains a zone of inhomogeneous waves, whose width is determined by the Fresnel reflection coefficient from one period. The refinement in terms of the four-wave or six-wave model yields only a small correction to the position of the zone, retaining its width unchanged. The wavenumber as a function of frequency is described by a circle inside the zone of inhomogeneous waves and by a hyperbola outside this zone. Mathematically, the method used is significantly simpler than those based on the application of the Floquet theorem to the wave equation. It is shown that the notion of forbidden zones is inconsistent with respect to photonic crystals, and the term zones of inhomogeneous waves is proposed instead.     

Optics of globular photonic crystals

Recent experimental and theoretical results on the optical properties of globular photonic crystals coauthored by the author are presented. The dispersion relation for electromagnetic waves in a 1D photonic crystal that simulates the properties of a selected direction in the globular photonic crystal is calculated. The spectral ranges that are characterized by the anomalous slowing of electromagnetic waves in the photonic crystal and correspond to the stop-band edges are determined. A method for the measurement of the transmission and reflection spectra of the broadband radiation in photonic crystals is proposed. The method enables one to find the characteristics of the stop bands. The features of the secondary emission that emerges in opals due to the UV and visible excitation are reported. The conditions for the low-threshold lasing in opals filled with rareearth elements are presented. The experimental results on the induced-globular light scattering are demonstrated. Such a scattering implies the coherent excitation of vibrational states of the globules in a globular photonic crystal. A new phenomenon (slow light scattering) which involves the excitation of slow photons (slowtons) that correspond to the stop-band edges of the photonic crystal is observed. The conditions for the measurement of the slow light scattering in opals excited using the ruby and nitrogen lasers are experimentally determined. The experimental and theoretical results open up the prospects for low-threshold nonlinear optical processes in material media.     

Theory of chirped photonic crystals

In chirped photonic crystals, the structural parameters describing a unit cell are progressively varied from a unit cell to the nearest ones. Geometric and dielectric response functions can be affected by this modulation, but here we only investigate the effect of a long-range, slowly varying, modulation of the refractive index. The Bloch modes are modified by essentially being modulated by an envelope function which adapts to the long-range dielectric function perturbation. It is shown that this envelope function obeys a simple linear Schrödinger equation of classical (non-quantum) origin. Close to a band extremum, at a gap edge, the envelope functions can be interpreted as wave functions of particles, called “energy carriers”. These carriers have a mass and come as two species, referred to as “effective photons” (for positive band curvatures) or “photonic holes” (for negative band curvatures). The energy transfer through the chirped structure can be viewed as resulting from the migration of these particles under forces implied by the long-range dielectric function modulation.     

Nonlinear-optical properties of photonic crystals

The results of the experimental study of nonlinear-optical effects in photonic crystals, i.e., synthetic opal matrices and nanocomposites (matrices with voids filled with different nonlinear liquids) are presented. The following nonlinear-optical effects were observed under experimental conditions: the photonic flame effect (PFE), stimulated globular scattering (SGS), and stimulated Raman scattering (SRS). The dependence of these effects on the excitation conditions, nanocomposite refractive index contrast, and sample temperature was studied. PFE lines were detected in the Stokes spectral region. SGS spectra at the temperature of liquid nitrogen were studied.     

Control of photonic band gaps in one-dimensional photonic crystals

The photonic band gaps in one-dimensional photonic crystals (PCs) are theoretically investigated. A new method to broaden the photonic band gaps is introduced. Based on the similar method, a kind of photonic crystals is constructed to generate photonic band gaps with proportioned central frequencies. This technology can be used for designing nonlinear PCs for harmonic generation.     

Extension of photonic band gaps in one-dimensional photonic crystals

A method is proposed for extending photonic band gaps by constructing periodic structures from two or more consecutively located photonic crystals with different lattice constants or filling factors. The photonic band gaps with a maximum extension are predicted by superposing the photonic band gap maps on one another. It is demonstrated that both the lowest and higher order band gaps can be merged in photonic crystals with a high refractive index contrast.     

Nonreciprocal photonic band structure of low-symmetry magnetic photonic crystals

We propose multicomponent magnetic photonic crystals as a basis component for nonreciprocal optical elements. It is shown that introduction of three or more components may provide violation of mirror reflection symmetry, which is a necessary condition for obtaining nonreciprocity in the dispersion of the structure's eigenmodes. Numerical simulations confirm that nonreciprocity indeed develops in the form of nonreciprocal photonic band structure of three-component low-symmetry photonic crystal. We find that symmetry constraints produce fine structure in the nonreciprocity at high-symmetry points of the Brillouin zone.     

Entangling atoms in photonic crystals

We propose a method for entangling a system of two-level atoms in photonic crystals. The atoms are assumed to move in void regions of a photonic crystal. The interaction between the atoms is mediated either via a defect mode or via a resonant dipole-dipole interaction. We show that these interactions can produce pure entangled atomic states. We analyze the problem with parameters typical for currently existing photonic crystals and Rydberg atoms and we show that the atoms can emerge from photonic crystals in entangled states. Depending on the linear dimensions of the crystal we estimate that a pair of atoms entangled in a photonic crystal can be separated by tens of centimeters. Receive 11 June 1999 and Received in final form 4 October 1999     

Optical echo in photonic crystals

The dynamics of a photonic wavepacket in the effective oscillator potential is studied. The oscillator potential is constructed on the base of one-dimensional photonic crystal with a period of unit cell adiabatically varied in space. The structure has a locally equidistant discrete spectrum. This leads to an echo effect, i.e., the periodical reconstruction of the packet shape. The effect can be observed in the nonlinear response of the system. Numerical estimations for porous-silicon based structures are presented for a femtosecond Ti:sapphire laser pump. The text was submitted by the authors in English.