Photonic band gap materials
It has long been known by diamantists, and used to check for authenticity, that dipping a gem in a solution with a matching index of refraction turns it invisible. This is due to the fact that what makes a homogeneous object’s boundaries visible is the reflection and refraction of light therein. Therefore if light does not find difference in traversing a surface it will not be in any way scattered, as it is not in a homogeneous medium. The main feature of PXs is the periodic modulation of such property (dielectric constant) along one, two or three directions of space (see Fig. 1). In a composite formed by two dielectrics, we will consider the scattering centre that in which light propagates more slowly, i.e. that of higher e. 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. 1). In such case, this PX is said to present a full PBG. 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.
|Fig. 1.- Real space representation of a photonic crystal and reciprocal space regions for which propagation is shielded.
The Schrödinger equation for an electron of effective mass m in a crystal, in which the potential is V(r), can be written as:
where V(r) is a periodic function with the periodicity of the lattice, R:
The eigenstates of this equation are also periodic functions with period R. The dispersion relationship derived, E(k), will present a forbidden band for all energies E which have imaginary values. Similarly, in a medium in which a spatial modulation of the dielectric constant ε(r) exists, photon propagation is governed by the classical wave equation for the magnetic field H(r):
In a PX, ε(r) is a periodic function:
These equations show the parallelism between electrons in crystalline solids and photons in PX. In Fig. 2 it is shown how gaps are developed in both an electronic crystal and a PBG material.
|Fig. 2.- Energy dispersion relations for free electron and electron in a (1D) solid and for a free photon and a photon in a PX.
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. One important difference between electrons and photons rests on the different nature of their associated waves. Electrons are scalar waves, while photons are vectorial ones. This implies that, in the latter case, polarization must be taken into account. This finally results in much more restrictive conditions for gap appearance that is the case for electrons. On the contrary, the electron wave equation is not scalable, since an intrinsic length measure is associated to the electron (de Broglie wavelength) whereby the potential periodicity can be gauged. This restriction does not apply for photons. The photon wave equation is scalable, hence if a PX presents a given periodicity length, it will show photonic bands in certain range of frequency and a scaling if it will result in a new system with exactly the same band scheme only accordingly scaled. If we halve the size we double the energies. The parameters on which the optical features of a PX will depend are indicated below:
-The type of symmetry of the structure.
-Dielectric constant contrast (ε1/ ε2).
-Filling factor, that is, the ratio between the volume occupied by each dielectric with respect to the total volume of the composite.
-The topology, which can be either cermet: scattering centres are isolated from each other; or network: scattering centres are connected between them.
-The shape of the scattering centres.
All these factors determine the photonic band structure of the PX and, therefore, its optical properties. Economou and Sigalas have published a general discussion about topologies in PBG theory and the generalization to other classical waves. Recently, it has been shown that a slight modification of either the symmetry or the shape of the scatterer can enlarge the gap.
Fig. 3.- Real and reciprocal space representation of triangular and square twodimensional crystals to highlight the formation of a full bandgap and a pseudogap.
In Fig. 3 it can be seen that high symmetry favours the photonic gap appearance. In this case the higher symmetry of the hexagonal lattice is reflected by the fact that the Brillouin zone is closer to circular than the square lattice: distances in reciprocal space to the principal points of the irreducible zone are similar. It is obvious that a high dielectric constant contrast produces that the optical bands significantly depart from the free photon behaviour (ω=ck) producing wider gaps at the Bragg planes (Brillouin zone edges) in reciprocal space.
The concept of PBG is deeply rooted on that of Bragg diffraction. By virtue of coherent scattering each set of crystallographic planes may give rise to an X-ray diffraction peak at a certain frequency related to the interplanar distance. X-ray diffraction follows Bragg law:
2d(hkl)Sin Θ=mλ; m =1,2,3,
where d(hkl) is the distance between atomic crystalline planes labelled by the Miller indices (hkl), q is the angle of the incident radiation, m is the diffraction order and l is the X-ray wavelength. As a consequence of the destructive interference, X-ray photons in the Bragg diffraction peak are not allowed to propagate through the crystal and are reflected. This effect reveals the absence of photonic states for the selected direction with the frequency determined by Bragg law. Bragg diffraction peaks, in ordinary crystals, appear at the X-ray region since the lattice parameters are of the order of several Angstroms. An entirely similar effect takes place in PXs. Due to the existence of crystalline planes in the PX, some frequency regions will be diffracted according to the Bragg law for the optical region:
where λc is the wavelength of the EM wave, d(hkl) the interplanar distance for the (hkl) crystallographic direction, áeñ the average dielectric constant of the PX
and q(hkl) the angle between the incident radiation and the normal to the set of crystalline planes determined by the (hkl) indices. An important difference between X-ray diffraction in solids and diffraction in PXs is the bandwidth of the Bragg peaks. X-ray diffraction peaks are extremely narrow (Δλ/λ» 10-6) and mainly instrumentally broadened. In PXs, the diffraction condition for a given direction of the wave vector, is met for a larger range of frequencies (Δλ/λ » 10-2). This originates mainly in the different contrast of dielectric constants occurring for these two well apart wavelengths: indices of refraction for X-ray radiation scarcely differ from unity whereas at optical wavelengths they are rather larger. Eventually, Bragg peaks in PXs become so broad that can overlap others coming from different crystallographic planes. This is schematically depicted in Fig. 3 where we show that this occurs in a hexagonal symmetry as opposed to the square symmetry in which the gaps do not overlap. Consequently, it could be possible to find a certain frequency region where the photon is not allowed to propagate regardless of direction. A material with these properties is called a PX with a full band gap.