The main feature of PXs is the periodic modulation of such property dielectric constant along one, two or three directions of space see Fig. In a composite formed by two dielectrics, we will consider the scattering centre that in which light propagates more slowly, i. If the scattering centres are regularly arranged in a medium, light is coherently scattered.

In this case, interference will eventually cause that some frequencies will not be allowed to propagate, giving rise to forbidden and allowed bands. Under certain conditions that will be detailed later in this section, regions of frequency may appear that are forbidden regardless of the propagation direction in the PX see Fig. On the contrary, if the forbidden photonic band varies with the propagation direction in the PX, a photonic pseudogap is spoken of.

What is more, by introducing defects in the PX, we can introduce allowed energy levels in the gap, as occurs when a semiconductor is doped. All these facts permit to establish a parallelism between the formalism used for electrons in ordinary crystals and that for photons in PX. The eigenstates of this equation are also periodic functions with period R.

### Various photonic Crystals

The dispersion relationship derived, E k , will present a forbidden band for all energies E which have imaginary values. These equations show the parallelism between electrons in crystalline solids and photons in PX. In Fig. The energy dispersion relation for an electron in vacuum is parabolic with no gaps. When a periodic potential is present gaps open and electrons with energies therein have localized non-propagating wavefunctions as opposed to those of electrons in allowed bands which have extended propagating wavefunctions.

In a similar way, a periodic dielectric medium will present frequency regions where propagating photons are not allowed and will find it impossible to travel the crystal.

There are several structure types that have been constructed: [28]. One promising fabrication method for two-dimensionally periodic photonic crystals is a photonic-crystal fiber , such as a holey fiber. Using fiber draw techniques developed for communications fiber it meets these two requirements, and photonic crystal fibres are commercially available. Another promising method for developing two-dimensional photonic crystals is the so-called photonic crystal slab.

These structures consist of a slab of material—such as silicon —that can be patterned using techniques from the semiconductor industry. Such chips offer the potential to combine photonic processing with electronic processing on a single chip. For three dimensional photonic crystals, various techniques have been used—including photolithography and etching techniques similar to those used for integrated circuits. To avoid the complex machinery of nanotechnological methods , some alternate approaches involve growing photonic crystals from colloidal crystals as self-assembled structures.

## 1D Photonic Crystals: Principles and Applications in Silicon Photonics

Because the particles have a softer transparent rubber coating, the films can be stretched and molded, tuning the photonic bandgaps and producing striking structural color effects. The photonic band gap PBG is essentially the gap between the air-line and the dielectric-line in the dispersion relation of the PBG system. To design photonic crystal systems, it is essential to engineer the location and size of the bandgap by computational modeling using any of the following methods:.

Essentially, these methods solve for the frequencies normal modes of the photonic crystal for each value of the propagation direction given by the wave vector, or vice versa. The various lines in the band structure, correspond to the different cases of n , the band index. For an introduction to photonic band structure, see Joannopoulos.

The plane wave expansion method can be used to calculate the band structure using an eigen formulation of the Maxwell's equations, and thus solving for the eigen frequencies for each of the propagation directions, of the wave vectors. It directly solves for the dispersion diagram. Electric field strength values can also be calculated over the spatial domain of the problem using the eigen vectors of the same problem. For the picture shown to the right, corresponds to the band-structure of a 1D distributed Bragg reflector DBR with air-core interleaved with a dielectric material of relative permittivity For large unit cell models, the RBME method can reduce time for computing the band structure by up to two orders of magnitude.

Photonic crystals are attractive optical materials for controlling and manipulating light flow. One dimensional photonic crystals are already in widespread use, in the form of thin-film optics , with applications from low and high reflection coatings on lenses and mirrors to colour changing paints and inks. Higher-dimensional photonic crystals are of great interest for both fundamental and applied research, and the two dimensional ones are beginning to find commercial applications.

The first commercial products involving two-dimensionally periodic photonic crystals are already available in the form of photonic-crystal fibers , which use a microscale structure to confine light with radically different characteristics compared to conventional optical fiber for applications in nonlinear devices and guiding exotic wavelengths. The three-dimensional counterparts are still far from commercialization but may offer additional features such as optical nonlinearity required for the operation of optical transistors used in optical computers , when some technological aspects such as manufacturability and principal difficulties such as disorder are under control.

From Wikipedia, the free encyclopedia. Comparison of 1D, 2D and 3D photonic crystal structures from left to right, respectively. Play media. Animal coloration Animal reflectors Colloidal crystal Left-handed material Metamaterial Nanomaterials Nanotechnology Optical medium Photonic-crystal fiber Photonic metamaterials Structural coloration Superlens Superprism Thin-film optics Tunable metamaterials.

Optics and Lasers in Engineering.

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## Photonic Crystals

N; Kaplyanskii, A. A; Prokofiev, A. V; Samoilovich, L. A; Samoilovich, S. This course introduced light propagation in periodic systems, photonic crystals and band gaps, localized defect states, 3d fabrication technology, hybrid structures and index guiding, and photonic-crystal fibers, among other topics. MIT Photonic-Bands : MPB is a free program to compute the band structures dispersion relations and electromagnetic modes of periodic dielectric structures; it is designed for studying the photonic-crystal systems that are the focus of our research.

MPB's features include: fully-vectorial, 3D computations; a flexible user interface based upon the GNU Guile scripting language; output in HDF format; and iterative, "targeted" eigensolver methods to address very large problems by solving for only a few states near a specified frequency.

It is portable to most Unix-like systems, and parallel support is forthcoming. Research Projects and Results The Color of Shock Waves in Photonic Crystals : New physical effects occur when light interacts with a shock wave propagating through a one-dimensional photonic crystal. Light can also be slowed down by orders of magnitude. Resonant Cavities : By making point defects in a photonic crystal, light can be localized, trapped in the defect.

The frequency, symmetry, and other properties of the defect mode can be easily tuned to anything desired.