Aberrations of diffractive-refractive optics:: Bragg-case sagittal focusing of multiple parabolic elements

The diffractive-refractive optical device consisting of four crystals in (+, -, -, +) setting with longitudinal parabolic grooves has a geometrical aberration which influences the achievable focus size. This aberration is discussed analytically by using a new, more precise formula for the calculation of focusing distance, which respects the finite distance between optical elements. The calculation of the intensity distribution surrounding the focus is illustrated by a ray-tracing method based on the dynamical theory of diffraction. It demonstrates an achievable focus size. Finally we discuss that this aberration may be suppressed by the slight narrowing of the groove profile. In particular, the parameter a in the equation of parabola has to slightly grow with x. A practical application may require an ultra--precise fabrication of the grooves.