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We present a formalism of x-ray dynamical diffraction from volume diffractiveoptics with large numerical aperture and high aspect ratio, in an analogy tothe Takagi-Taupin equations for strained single crystals. We derive a set ofbasic equations for dynamical diffraction from volume diffractive optics, whichenable us to study the focusing property of these optics with various gratingprofiles. We study volume diffractive optics that satisfy the Bragg conditionto various degrees, namely flat, tilted and wedged geometries, and derive thecurved geometries required for ultimate focusing. We show that the curvedgeometries satisfy the Bragg condition everywhere and phase requirement forpoint focusing, and effectively focus hard x-rays to a scale close to thewavelength.

Takagi-Taupin Description of X-ray Dynamical Diffraction from Diffractive Optics with Large Numerical Aperture