Focusing performance of a bent polychromator crystal with coincidence of the polychromatic and dynamical focusing effects Symmetric transmission geometry R oo. C o q p 2qB source detector Polychromatic focusing requires p fixed , q varied , crystal curvature adapted to each value of q

Phase factor Flat crystal propagator Crystal propagator as solution of Takagi-Taupin equations for a monochromatic point-source on the entrance surface Phase factor Flat crystal propagator With In the bent crystal case, it is not convenient to use the Fourier transformation for propagation from the source to the detector, because the propagator is not a function of the single variable

Cylindrical aberration: geometric rays tangent to caustic . aberration starts with term of order 4 in u

Integral expression of recorded intensity calculated numerically Dynamical focusing explained by using = Transmittance of cylindrical (convergent or divergent) lenses Paraxial focal distance

. of same sign if as usual , unfavourable for good focusing : Dynamical focusing condition according to a Aberration coefficient dependent of crystal parameters, not on curvature Effect of anomalous absorption . of same sign if as usual , unfavourable for good focusing : convergent and strongly absorbed divergent and weakly absorbed example of opposite case : the 220 reflection of calcite CaCO3

Si 111, 20 Kev, thickness 200 mm , paraxial focal distance = 2997 m Intensity at image center as function of distance q

Paraxial focal distance 500 mm Effective focal distance 750 mm Si 111 , 20 Kev thickness 200 mm intensity profiles at the paraxial focal distance (solid curve) and at the effective focal distance (dashed curve) Paraxial focal distance 500 mm Effective focal distance 750 mm

Discussion of results Coincidence of polychromatic and dynamical focusing: The crystal-to-focus distances predicted by the present approach are much smaller, by a factor larger than 2, than those predicted by the former approach (Mocella & al, 2008), in which the crystal propgator was considered as nearly independent on l .This corresponds much better to measurements performed on ID24. The present semi-analytical approach for dynamical focusing of an incident monochromatic cylindrical wave is found to be in agreement with calculations based on a numerical solution of the Takagi-Taupin equation by Nesterets & Wilkins (2008)