In this chapter we deal with the basic phenomena of the specular X-ray reflection (SXR) from samples with perfectly flat interfaces. We start with the Maxwell equations from which we derive the wave equation, dispersion relation and the plane wave solutions. By presenting these basic and well-know results we establish a general formalism which we will keep in the whole work for more sophisticated theoretical approaches. In the second part of the general introduction we discuss the refractive index of X-rays as a material parameter determining all the scattering experiments.
In the next part we study the theories applicable to the calculation of the specular reflectivity amplitude. We apply the boundary conditions for an interface between two layers and formulate two basic laws of optics, the law of reflection and the Snell's law. We represent them graphically by means of the Ewald construction. We derive the formulae for the Fresnel coefficients. The small negative value of the susceptibility causes the main difference of X-ray optics with respect to the usual (visible) one: total external reflection of X-rays occurs for all materials, the Fresnel coefficients fall down quickly above the critical angle and therefore the reflected intensity is measurable only for small angles of incidence (measured with respect to the sample surface).
Then we formulate the dynamical theory of X-ray reflection from layered structures by means of both the matrix and recurrent formalisms. We provide formulae also for the amplitudes of the wavefields inside the multilayer, from which we will make use later in the distorted-wave Born approximation. The main advantage`s of the matrix formalism introduced in the present chapter will be demonstrated mainly in the study of the reflectivity by multilayered gratings, where we can generalize the Fresnel coefficients, originally defined for the reflection from smooth interfaces. We find that for small reflected intensities the dynamical theory can be replaced by the single-reflection approximation. This approximation calculates the amplitude of the reflected wave as a sum of contributions of the waves scattered once at each interface, so that it calculates the reflectivity at each interface dynamically, but neglects the multiple scattering between different interfaces.
In the next section we solve the wave equation by means of the kinematical theory usual in X-ray diffraction. Since we calculate reflectivity from a homogeneous and laterally large multilayer, we cannot use the Fraunhofer approximation and therefore we employ a different calculation approach. This stationary phase method transforms the volume kinematical diffraction integral (equivalent to the Huygens principle known in optics) into the path integral along the classical path of the reflected beam in the sample. The kinematical theory is equivalent to the first Born approximation employing the vacuum plane waves as the eigenstates. It does not contain the effect of refraction, which, however, is of major importance in X-ray reflectivity. This is different to the X-ray diffraction from multilayers, where the refraction causes only a small shift in the Bragg position.
In the final part of this chapter we compare these theories by means of both the analytical treatment and numerical simulations. We perform the comparison for two kinds of samples: a periodic multilayer and a Fibonacci multilayer. Firstly, we calculate the reflectivity from a simple periodic multilayer. By using a quite unusual approach of describing the periodic layer sequence by the terminology of the physics of quasicrystals we prepare the next step of calculating the reflectivity from a Fibonacci multilayer. The layer sequence of this multilayer is quasiperiodic and an analytical description of the reflectivity profile is not easy. Therefore we employ an approach we developed earlier for the diffraction on Fibonacci multilayers and we show that we are able to calculate the Fourier transform of the refractive index distribution and to find an analytical formula for the peak positions.