Nano-optical theory of planar x-ray waveguides

X-ray waveguides are routinely used at synchrotron light sources in imaging setups and as a platform for experiments with quantum emitters, providing nanometer-sized confinement - even x-ray optics on a chip has been showcased. X-ray waveguides are weakly guiding and experience significant material absorption, such that the established waveguide theory is not immediately applicable. Here, a general self-contained nano-optical theory of planar waveguides is derived, which is appropriate for hard x-ray energies. Solutions of the electromagnetic fields and its Green's functions are derived in detail. Asymptotic expansions into resonant and non-resonant modes are derived, which are particularly useful in the presence of strong material absorption. A method to reliably find the resonant modes of x-ray waveguide structures is presented. Based on the general theory, certain common experimental geometries, namely evanescent coupling in grazing-incidence, front-coupling in forward-incidence and radiation from buried emitters, are discussed in more detail. Complementing the analytic discussion, numerical tools are provided and applied to quantitatively extract the main figures of merit. The theory provides an analytic foundation for the interpretation of past and future experiments and, combined with the numerical tools, will facilitate the computer-aided design of x-ray waveguides.