Diffraction of a plane wave

Figure 1.1 Diffraction of a plane wave off successive crystal planes. Strong diffraction results when the angles of incidence and diffraction,, are equal and the path difference AOB between the two beams is equal to , an integral number of wavelengths. Hence the Bragg law, n =2rfsin...

If we imagine the diffraction of a plane wave from epilayers we see that there will in general be differences of diffraction angle between die layer and the substrate, whether these are caused by tilt or mismatch f Double or multiple peaks will therefore arise in the rocking curve. Peaks may be broadened... 

In the kinematical theory, we consider the diffraction of a plane wave (of wavelength X) incident upon a three-dimensional lattice array of identical scattering points, each of which consists of a group of atoms and acts as the center of a spherical scattered wave. Our problem is to find the combined effect of the scattered waves at a point outside the crystal, at a distance from the crystal that is large compared with its linear dimensions. In developing the theory, we make several important assumptions ... 

The fractional energy loss per transit due to diffraction of a plane wave reflected back and forth between the two plane mirrors is approximately given by... 

A Gaussian beam is a modified plane wave whose amplitude decreases, not necessarily monotonically, as one moves radially away from the optical axis. The simplest, or fundamental, Gaussian beam has an exp(—p /w ) radial dependence, where p is the radial distance from the optical axis and w is the 1/e radius of the electromagnetic field. The phase of a Gaussian beam also differs from that of a plane wave due to diffraction effects, as we will show subsequently. 

Figure 1.2 Scattering of a plane wave by a one dimensional chain of atoms. Wave front and wave vectors of different orders are given. Dashed lines indicate directions of incident and scattered wave propagation. The labeled orders of diffraction refer to the directions where intensity maxima occur due to constructive interference of the scattered waves.

Bartlma, F., and K. Schroder. 1986. The diffraction of a plane detonation wave at a convex corner. Combustion Flame 66 237-48. 

For our first example the diffraction losses of the plane FPI are about 5 x 10 and therefore completely negligible, whereas for the second example they reach 25% and may already exceed the gain for many laser transitions. This means that a plane wave would not reach threshold in such a resonator. However, these high diffraction losses cause nonnegligible distortions of a plane wave and the amplitude A(x, y) is no longer constant across the mirror surface (Sect. 5.2.2), but decreases towards the mirror edges. This decreases the diffraction losses, which become, for example, /Diffr 0 01 for N > 20. 

In order to estimate the magnitude of diffraction losses let us make use of a simple example. A plane wave incident onto a mirror with diameter la exhibits, after being reflected, a spatial intensity distribution that is determined by diffraction and that is completely equivalent to the intensity distribution of a plane wave passing through an aperture with diameter la (Fig. 5.5). The central diffraction maximum at 0 = 0 lies between the two first minima at d = Xlla (for circular apertures a factor 1.2 has to be included, see, e.g., [306]). About 16 % of the total intensity transmitted through the aperture is diffracted into higher orders with 0 > X/la. Because of diffraction the outer part of the reflected wave misses the second mirror M2 and is therefore lost. This example demonstrates that the diffraction losses depend on the values of a, d, X, and on the amplitude distribution A x,y) of the incident wave across the mirror surface. The influence of diffraction losses can be characterized by the dimensionless Fresnel number... 

In order to estimate the magnitude of diffraction losses let us make use of a simple example. A plane wave incident onto a mirror with diameter 2a exhibits, after being reflected, a spatial intensity distribution which is determined by diffraction and which is completely equivalent to the intensity distribution of a plane wave passing through an aperture with diameter 2a (Fig.5.5). The central diffraction maximum at = 0 lies between the two... 

A plane wave would not reach threshold in such a resonator. However, these high diffraction losses cause nonnegligible distortions of a plane wave and the amplitude A(x,y) is no longer constant across the mirror surface (see Sect.5.4), but decreases towards the mirror edges. This decreases the diffraction losses. 

Fig.5.4. The diffraction of a plane incident wave at successive apertures is equivalent to the diffraction by successive reflections in a plane mirror resonator...

Both ultrasonic and radiographic techniques have shown appHcations which ate useful in determining residual stresses (27,28,33,34). Ultrasonic techniques use the acoustoelastic effect where the ultrasonic wave velocity changes with stress. The x-ray diffraction (xrd) method uses Bragg s law of diffraction of crystallographic planes to experimentally determine the strain in a material. The result is used to calculate the stress. As of this writing, whereas xrd equipment has been developed to where the technique may be conveniently appHed in the field, convenient ultrasonic stress measurement equipment has not. This latter technique has shown an abiHty to differentiate between stress reHeved and nonstress reHeved welds in laboratory experiments. 

Rgure 3 Experimental and calculated results (a) for epitaxial Cu on Ni (001). The solid lines represent experimental data at the Cu coverage indicated and the dashed lines represent single-scattering cluster calculations assuming a plane wave final state for the Cu IMM Auger electron A schematic representation lb) of the Ni (010) plane with 1-5 monolayers of Cu on top. The arrows indicate directions in which forward scattering events should produce diffraction peaks in (a). 

In this chapter we introduce high resolution diffraction studies of materials, beginning from the response of a perfect crystal to a plane wave, namely the Bragg law and rocking curves. We compare X-rays with electrons and neutrons for materials characterisation, and we compare X-rays with other surface analytic techniques. We discuss the definition and purpose of high resolution X-ray diffraction and topographic methods. We also give the basic theory required for initial use of the techniques. 

Better conventional collimation will not do, except for the largest synchrotron radiation installations to obtain snb-arc-second collimation in the laboratory would require a collimator some 100m long with a sealed-tube source, and at this distance the intensity would be impracticably low. The problem is solved by the use of a beam conditioner, which is a further diffracting system before the specimen The measnred rocking cnrve is then the correlation of the plane wave rocking cnrves of the beam conditioner and the specimen crystals, from which most of the diffracting characteristics of the specimen crystal may be deduced. 

If the specimen crystal is curved, there will be a range of positions where the diffraction conditions are satisfied even for a plane wave. The rocking curve is broadened. It is simple to reduce the effect of curvature by reducing the collimator aperture. For semiconductor crystals it is good practice never to mn rocking curves with a collimator size above 1 mm, and 0.5 mm is preferable. Curved specimens are common if a mismatched epilayer forms coherently on a substrate, then the substrate will bow to reduce the elastic strain. The effect is geometric and independent of the diffraction geometiy. Table 2.1 illustrates this effect. 

What can we gness about the solution It shonld clearly be a wave equation. We expect (from knowledge of Bragg diffraction) it to be a plane wave, or sums of plane waves. We nse capital K g and K j, for wavevectors inside the crystal to distinguish them from k g and k j, ontside the ciystal. Inside the crystal the allowed wavevectors should satisfy conservation of momentnm, that is Kg+h=Kh... 

Figure 14. (a) Schematic of Huygens Principle showing construction of a new wave front 2 from the preceeding wave front S 0>) plane wave front incident on a slit of width b (c) diffraction for case where b < (d)... 

---