Analytical computation of the off-axis effective area of grazing incidence X-ray mirrors

Aims: Focusing mirrors for X-ray telescopes in grazing incidence, introduced in the 70s, are characterized in terms of their performance by their imaging quality and effective area, which in turn determines their sensitivity. Even though the on-axis effective area is assumed in general to characterize the collecting power of an X-ray optic, the telescope capability of imaging extended X-ray sources is also determined by the variation in its effective area with the off-axis angle. The effective area, in general, decreases as the X-ray source moves off-axis, causing a loss of sensitivity in the peripheral regions of the telescope's field of view.
Methods: The complex task of designing optics for future X-ray telescopes entails detailed computations of both imaging quality and effective area on- and off-axis. Because of their apparent complexity, both aspects have been, so far, treated by using ray-tracing routines aimed at simulating the interaction of X-ray photons with the reflecting surfaces of a given focusing system. Although this approach has been widely exploited and proven to be effective, it would also be attractive to regard the same problem from an analytical viewpoint, to assess an optical design of an X-ray optical module with a simpler calculation than a ray-tracing routine. This would also improve the efficiency of optimization tasks when designing the X-ray optical modules. In this paper, we thereby focused on developing analytical solutions to compute the off-axis effective area of double-reflection X-ray mirrors.
Results: We have developed useful analytical formulae for the off-axis effective area of a double-reflection mirror in the double cone approximation, requiring only an integration and the standard routines to calculate the X-ray coating reflectivity for a given incidence angle. The computation is easily applicable also to Wolter-I mirrors (such as those of NeXT, NuSTAR, HEXIT-SAT, IXO) and the approximation improves as the f-number of the mirror increases. Algebraic expressions are provided for the mirror geometric area, as a function of the off-axis angle. Finally, the results of the analytical computations presented here are validated by comparison with the corresponding predictions of a ray-tracing code.