# Tests for Optical elements¶

## Reflection from asymmetrically cut crystal¶

A monochromatic beam of *E* = 2840 eV is diffracted by a Si111 crystal with
θ_{B}= 44.132 deg and α = -20.054 deg, which results in *b* = -0.453.

The phase space volume in the beam before and after the crystal is indeed
invariant, whereas the linear and angular sizes have scaled with the ratio
equal to *b* or 1/*b*. The product FWHM(z)×FWHM(z’) = 135 nmrad in both cases.
Notice also the deviation of the diffracted beam direction from the exact
departure angle 2θ_{B} seen on the exit screen which is put normally to
2θ_{B} (i.e. z’=0 at the exact departure at 2θ_{B}).
The ray-traced deviation -106 µrad is close to the calculated refractive shift
-105 µrad.

## Reflectivity from highly asymmetric crystal in backscattering¶

A test for the optical scheme from [ShvydkoPRL2006], Fig 3.

The blue curve was ray-traced by xrt. The dotted curve is a reconstruction from Fig. 4 of [ShvydkoPRL2006]. The dispersion rate was calculated from a linear fit to the energy-angle line (orange) and is similar to that reported in [ShvydkoPRL2006] as 8.5 µrad/meV.

[ShvydkoPRL2006] | (1, 2, 3) Yu. V. Shvyd’ko, M. Lerche, U. Kuetgens, H. D. Rüter,
A. Alatas, and J. Zhao, Phys. Rev. Lett. 97 (2006) 235502
(DOI: 10.1103/PhysRevLett.97.235502). |