Progress in the theory of X-ray scattering from multilayer structures

D. G. Stearns 

Lawrence Livermore National Laboratory, P.O. Box 808, Livermore  CA 

The development of multilayer (ML) ray imaging in advanced applications
such as extreme ultraviolet lithography (EUVL) and x-ray telescopes for
astronomy. The performance of these coatings is limited by the nonideal
structure of the layer boundaries. Specifically, intermixing at the
boundaries decreases the Fresnel reflectance, which results in increased
x-ray absorption in the coating and lower throughput for the optical
system. Roughness at the boundaries scatters radiation out of the
specular mode, resulting in lower throughput and decreased image
contrast. To understand and control the performance of x-ray ML
coatings, it is important to develop a theory that describes the
interaction of x-rays with realistic ML structures under conditions of
practical interest. We have previously developed a theory of x-ray
scattering [1] that, while useful in elucidating much of the important
physical behavior, has limited applicability to quantitative modeling
owing to the restrictions imposed by several simplifying approximations. 
In this paper we extend the previous theory by removing most of these
approximations, and thereby present a practical theoretical tool for
modeling the performance of realistic coatings. 

The key new developments are summarized as follows. The theory is
reformulated within the "distorted-wave Born approximation"[2], where
the zeroth order solution is Fresnel reflection, thereby removing all
limitations on the incident and scattering angles. The description of
the layer boundaries includes interfacial roughness represented by the
power spectral density (PSD) and a diffuse region of intermixing of
arbitrary spatial profile. The PSD of the interface is allowed to vary
within the ML stack according to a growth model that includes
replication of the roughness of the underlying layer and random
roughness introduced by the film growth process. This approach naturally
accounts for the partial correlation of the interfacial roughness. The
incident specular field is treated dynamically, where the Fresnel
coefficients are appropriately modified to account for the roughness and
mixing at the interfaces. Similarly, the propagation of the scattered
field is treated dynamically, that is, we include all multiple
reflections of the outgoing radiation. This is particularly important
for cases in which the scattered mode satisfies a Bragg condition, as
occurs for small angle scattering in an imaging application. 

The only significant approximation that remains in the theory is the
assumption of "small roughness", expressed as 4k_o^2 s^2  cos^2 theta_o  
<< 1, where k_o  and theta_o  are the wavenumber and angle of the
incident field, and  s  is the root-mean-square value of the interfacial
roughness. This criteria is equivalent to a requirement that the total
nonspecular scattering be small compared to the incident field, a
condition satisfied by definition for high-performance optical coatings.
For example, for ML coatings designed to operate near normal incidence
at a wavelength of 13 nm, the theory will be valid for interfacial
roughness  s <= 0.5 nm. 

As application of the theory, we will present calculations of
nonspecular scattering within the Bragg condition for typical normal
incidence configurations found in imaging applications. These results
will show interesting new interference effects associated with the
multiple reflection of the scattered field

[1]  D. G. Stearns, J. Appl. Phys. 65 , 491 (1989);D. G. Stearns, 
J. Appl. Phys. 71, 4286 (1992)
[2]  S. K. Sinha et al., Phys. Rev. B  38 , 2297 (1988)