The well known Bragg low refers to the following simple equation:
This equation was used for the first time in order to explain the reflection of X-rays on different crystallographic planes of atomic crystals. In Equation 59, λ is the wavelength of X-ray, d is the distance between the crastallographic planes, θ is the incidence angle of the rays relative to the normal to the surface and n is an integer number.
The Bragg law can be easily obtained geometrically. Lets assume an ideal case that two parallel and in phase X-rays are incident on a crystal with the distance between the planes equal to d. Also, let us assume that one of the rays is hitting an atom from the first atomic plane and consequently is reflected, whereas the second ray is penetrating slightly the crystal and is reflected by an atom from the second plane of the crystal surface (see Figure 5.1). Thus, the second ray has to travel a longer path AB+BC in order that the two rays to stay parallel also after reflection. More than that, if the rays have to remain in phase also after reflection, then the additional path traveled by the second ray should be a multiple of the wavelength of the rays. In other words, the following relation should be satisfied
Because AB=BC, Eq. 60 can be written as
Figure 5.1: Schematic representation of X-ray reflection from a set of crystallographic planes.
Additionally from the triangle ABZ one can easily obtain that
By substituting Eq. 62 in Eq. 61 the Bragg low presented in Eq. 59 is obtained.
The Bragg low, according to Eq. 59, says that the rays with a certain wavelength λ will be reflected only when the incident angle will be strictly equal to a certain angle θ. In reality, however, not only the rays which strictly have the incident angle &thata; will satisfy the Bragg law. The rays having the incident angle θ+/-δ will be reflected by the crystal as well. The value of δ is determined by the degree of scattering of X-rays by the atoms of the crystal. Thus, the Bragg law can be written as
A crystal, however, has a multitude of parallel planes with different distances between them. Each of these planes will reflect the X-rays according to the Bragg low. If a crystal is rotated in a flux of X-rays, every time the Bragg low is satisfied for a definite set of planes the rays will be reflected. A schematic representation of X-ray reflection from two sets of planes of a crystal is presented in Figure 5.2.
Figure 5.2: Reflection of X-rays from two sets of planes a and b.
The Bragg low is not true only for X-rays but for all electromagnetic waves including the visible light. However, visible light is not reflected by atomic crystals according to the Bragg low. The reason is that the Bragg low works perfectly only when the wavelength of the incident electromagnetic waves and the dimensions of the scatters, i.e. atoms, have the same order of magnitude. This condition is true for atoms and X-rays, however it is not true for atoms and visible light. The visible light has much bigger wavelengths (~500 nm) as compared with the dimension of the atoms (~1 Ĺ). Therefore, much bigger scatterers are required in order to observe Bragg reflection in the visible region.
Before starting to build a periodic structure with bigger scatterers or "atoms" it is worth to have a look around in nature, where probably such kind of new crystals are already present. Indeed, by using electronic microscopes it is possible to see that the play of colors in opals is due to their structure, which is nothing else than a periodical arrangement of small spheres forming a "crystal" of submicronic spheres (see Figure 5.3a).
Figure 5.3: The structure of a) an opal and b) a wing of a butterfly, detected by an electronic microscope.
A similar periodic structure was observed on some species of butterflies with lively colored wings (see Figure 5.3b). Thus, the nature confirms the assumption that it is possible to control the light by the Bragg law using periodic structures with the dimensions of the "atoms" in the range of the used wavelength. Such kinds of periodic structures are called photonic crystals.
The scientific potential of photonic crystal (PC) structures was first underlined in 1987 by Eli Yablonovitch . In 1991, Yablonovitch et al. produced the first artificial photonic crystal by mechanically drilling millimeter wide holes into a block of material with a refractive index of 3.6. This structure, known as 'Yablonovite', exhibits a 3D photonic band gap, i.e. reflects the microwaves with wavelengths within the photonic band gap independent from the incident angle.
However, in order to make photonic crystals for visible light and thus to be able to design devices compatible with the already existing optoelectronic ones, it is necessary to go to shorter dimensions. This means that the size of dielectric particles constituting the PC should be in the sub-micrometer range. In a good approximation the dielectric 'atoms' should have dimensions equal to the wavelength of the light for which the PC is designed, divided by the refractive index of the medium. This means that the higher the refractive index of the material is, the smaller the constituent dielectric "atoms" should be. Taking into account that according to theoretical calculations a photonic band gap is easier to obtain in periodic dielectric structures with high refractive indexes it is obvious that challenging sub-µm technological problems are to be expected.
It took more than a decade to fabricate photonic crystals that work in the near-infrared (780-3000 nm) and visible (450-750 nm) regions of the spectrum. The main challenge has been to find suitable materials and processing techniques to fabricate structures that are made of dielectric “atoms” in the sub-micrometer scale. Different techniques were developed for producing 2D and 3D photonic crystals, such as:
Taking into account that most of the above enumerated techniques are quite expensive, the main advantage of the electrochemical etching technique is its cost-effective feature. On the other hand, in spite of the fact that it is possible to obtain 3D structures using electrochemistry, these structures are totally dependent on the crystallographic features of the substrate. For example, in the case of 3D Moldavite structures obtained by electrochemical etching of GaAs (see Chapter 3.6), the direction of pores is fixed along <111B> directions and can not be changed arbitrarily.
Fortunately 2D periodic lattices exhibit some of the useful properties of a truly 3D photonic crystals. In particular, these structures can block certain wavelengths of light at any angle in the plane of the device. The 2D porous structures, described in previous chapters, are suitable candidates for photonic crystal applications assuming that the necessary uniformity is provided.