Analytic theory of soft x-ray diffraction by lamellar multilayer gratings

I V Kozhevnikov,K.-J Boller,R Van Der Meer,F Bijkerk,H M J Bastiaens

doi:10.1364/oe.19.009172

Abstract

An analytic theory describing soft x-ray diffraction by Lamellar Multilayer Gratings (LMG) has been developed. The theory is derived from a coupled waves approach for LMGs operating in the single-order regime, where an incident plane wave can only excite a single diffraction order. The results from calculations based on these very simple analytic expressions are demonstrated to be in excellent agreement with those obtained using the rigorous coupled-waves approach. The conditions for maximum reflectivity and diffraction efficiency are deduced and discussed. A brief investigation into p-polarized radiation diffraction is also performed.

Highlights

Lamellar Multilayer Gratings (LMG) offer increased spectral and angular resolution in the soft x-ray (SXR) region compared to conventional multilayer mirrors (MM)
We developed an analytical theory for the reflection and diffraction of soft x-rays (SXR) by Lamellar Multilayer Gratings (LMGs) operating in the single-order regime
Analytical expressions for diffraction efficiencies were first derived for LMGs in specular reflectance, i.e. expressions for the zeroth-order diffraction efficiency

Summary

Introduction

Lamellar Multilayer Gratings (LMG) offer increased spectral and angular resolution in the soft x-ray (SXR) region compared to conventional multilayer mirrors (MM). Diffraction of SXR from LMGs has been analyzed by time-consuming numerical simulations using complex rigorous theories [2,7,8]. The optimization of LMG performance, in terms of resolution and reflectivity, is very difficult, as many of these time-consuming simulations need to be performed It is even more difficult for the inverse problem of obtaining multilayer structural data, such as intermixing or roughness, from reflectivity measurements. We will derive analytic expressions for the diffraction efficiency of SXR radiation by LMGs operating in the single-order regime. We will demonstrate that these expressions are in excellent agreement with results from rigorous diffraction theories This analytical description of the optical properties of LMGs makes earlier, complex, theories redundant.

Differential equations of the coupled waves approach

B B b2

Analytic solution for higher-order diffraction efficiencies

Diffraction of p-polarized radiation by LMG

Findings

Conclusions