Bragg equation, light diffraction

Artificial photonic crystals are structures built so that they contain diffracting centres separated by distances that are of the order of the wavelength of light. The interaction with light can be understood in terms of the Bragg equation. However, the terminology employed to describe diffraction in artificial photonic crystals is that of semiconductor physics. The transition from a diffraction description to a physical description can be illustrated with respect to a one-dimensional photonic crystal. 

Thus we see that a particle would be scattered by a crystal only when a diffraction equation similar to the Bragg equation for x-rays is satisfied. The wave length of light is replaced by the expression... 

This Chapter is concerned with methods for obtaining the relative phase angles for each Bragg reflection so that the correct electron-density map can be calculated and, from it, the correct molecular structure determined. When scattered light is recombined by a lens, as described in Chapters 3 and 6, the relationships between the phases of the various diffracted beams are preserved. In X-ray diffraction experiments, however, only the intensities of the Bragg reflections are measured, and information on the relative phases is lost. An attempt is maxle to remedy this situation by deriving relative phases by one of the methods to be described in this Chapter. Then Equation 6.3 (Chapter 6) is used to obtain the electron-density map. Peaks in this map represent atomic positions. 

A crystallographic plane (hkl) is represented as a light spot of constructive interference when the Bragg conditions (Equation 2.3) are satisfied. Such diffraction spots of various crystallographic planes in a crystal form a three-dimensional array that is the reciprocal lattice of crystal. The reciprocal lattice is particularly useful for understanding a diffraction pattern of crystalline solids. Figure 2.7 shows a plane of a reciprocal lattice in which an individual spot (a lattice point) represents crystallographic planes with Miller indices (hkl). 

However the de Vries equation is not valid in the region of A , which corresponds to a total reflection of circularly polarized light having the same sense as the helical pitch. This is often referred to as Bragg reflection, by analogy with X-ray diffraction, but only first order reflections are allowed for nor-... 

This equation is known as Bragg s law. For a given wavelength of light incident on atoms arranged in layers, we can measure the angle that produces constructive interference (which appears as a bright spot on the X-ray diffraction pattern) and then calculate d, the distance between the atomic layers ... 

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