In this chapter we will deal with X-ray scattering from multilayer gratings (see Fig. 5.1) in reflection geometry. This means that we study the intensity distribution near the origin of the reciprocal space for grazing incidence of the incoming wave. The main part of this work is devoted to laterally periodic gratings. The generalization to other gratings with a discrete Fourier transform of the lateral structure (e.g. quasiperiodic gratings) is also briefly discussed. We further use the name of a multilayer grating (MLG) for laterally periodic multilayer gratings, if it is not explicitly stated otherwise.
Our aim is to develop and compare different theories suitable for this calculation. This comprises the kinematical theory, the distorted-wave Born approximation and the dynamical scattering theory. We point out different approximations involved in these theories as well as their common features.
The principal characteristic of a multilayer gratings, the one-dimensional lateral periodicity, is involved in all the three treatments by means of the Fourier transform of the susceptibility. We solve the wave equation for this case of lateral periodicity. Thus we can formulate the presented theories in the way of the well-known X-ray diffraction theories for crystals, which use a similar idea to solve the wave equation in the crystal with a three-dimensional translation symmetry.
In the first section, we review the work published by other authors on the topic of this chapter. Then we deal with the basic features of the gratings and their reciprocal lattice. We introduce the notation used afterwards.
In the following sections we present the theories, starting by the kinematical theory. We solve the kinematical diffraction integral by means of the stationary phase method similarly to our procedure for the reflection from planar multilayers. The kinematical theory includes only single-scattering processes and it does not include the refraction and absorption effects. Nevertheless, the results obtained show the general features of the scattering from gratings: one incoming plane wave is spread into a fan of plane waves. That is a diffraction process, based on specular reflection. The lateral components of the wave vectors of the scattered (diffracted) waves differ from that of the incoming wave by the one-dimensional reciprocal grating vector. Therefrom follows the famous grating formula. We represent the scattering process in reciprocal space by means of the conventional Ewald construction. We show a way to generalize the kinematical Fresnel reflection coefficients.
The distorted-wave Born approximation (DWBA) as a more elaborate perturbation method is treated afterwards. DWBA has been used in the X-ray reflectivity from planar multilayers for the calculation of the diffuse scattering from rough interfaces. It was believed until now that the DWBA is valid only for small perturbing potentials, e.g. small interface profile displacements (small roughnesses). However, we show the applicability of this method to the calculation of the reflectivity from gratings too. Therefore the main point of our discussion of the DWBA concerns the range of its validity.
The dynamical theory of reflection from lamellar multilayer gratings is treated by many authors. The review is presented in the following section. In this work, we develop a dynamical scattering theory of X-ray reflection from gratings, taking pattern from the Darwin-Laue formulation of the dynamical theory of X-ray diffraction from crystals. Our formulation is known in optics as the modal eigenvalue method: we find a plane wave solution of the wave equation as the one-dimensional Bloch waves, we solve the eigenvalue problem and apply the boundary conditions at the MLG interfaces. This will be performed by the matrix formalism. The main advantage of our presentation of the theory in this work is that we keep the same notation as for the reflection from planar multilayers, which depicts the links to the reflectivity from both structure types. As a particular result, this will allow us to introduce a matrix generalizing the matrix of the Fresnel coefficients.
This fully dynamical theory is used for numerical calculations. However, the equations of this rigorous theory are rather cumbersome for being discussed in a qualitative way. Therefore we formulate multiple-beam approximations of the dynamical theory. These approximation are employed in the regions where the dynamical effects prevail. Our new approach consists of introducing the two-beam approximation, similarly to the two-beam case of X-ray dynamical diffraction by crystals. We discuss also non-coplanar scattering in the geometry where the incident wave falls (nearly) parallel to the wires.
The discussion of these theories, together with numerical simulations and their comparison is presented in the subsequent section. As a particular example we take a surface grating with a wire-to-period ratio 0.5 and a period of 8000 A, which was a characteristic sample of the series of the samples we worked with. We show that the odd order truncation rods are strong, and most parts of their profile can be well explained by both the dynamical theory and the DWBA. In contrary, the even truncation rods are forbidden by all the single-scattering theories, the calculation needs to deal with a multiple scattering and thus only the dynamical treatment is appropriate. We use the numerical simulation to determine the number of interacting wavefields needed in the multiple-beam approximations in order to obtain the scattered intensities with a sufficient precision.
Our theories and their formalism for the periodic grating can be applied for certain aperiodic gratings as well, mainly to the gratings where the lateral profile has a discrete Fourier transform. In particular, we study qualitatively the expected reflectivity map of a quasiperiodic Fibonacci grating. We briefly discuss how to use the theories in order to calculate the reflectivity by gratings with non-rectangular wire shapes (trapezoidal gratings).
The calculations above considered perfect multilayer gratings. However, as it was the case of the planar multilayers, the interfaces between the materials can be imperfect. We distinguish between the side wall roughness of the grating shape and the interface roughness of the horizontal interfaces. We use our matrix formalism of the dynamical theory and we show how these roughnesses influence the scattered intensity. Thus our approach is a generalization of the reflection from rough planar multilayers; however, for the case of gratings, we have not found a correct matrix treatment in the published literature where mostly the roughness is involved by supposing the usual exponential kinematical damping factors.
In the final part of this chapter we present a measurement and fit of the structural parameters of a multilayer grating with three-and-a-half partially etched periods.