Focusing X-Ray Optics for Astronomy

Figure 19

The left panel shows a ray (solid line) arriving at an interface between two media. The clear region “1” is a metal and the shaded region “2” is the vacuum, which has a larger index of refraction in the X-ray band. The dash-dot line is a continuation of the direction of the incident ray. We assume the interface should have been perpendicular to the incoming ray where it would have had no effect upon its direction. However, because of a slope error in the optic the interface is tilted at a finite angle of 𝜃₁. The angle of the refracted ray (dash-dot line) is 𝜃₂ so the error in the direction of the refracted ray is 𝜃₁−𝜃₂. Applying Snell’s law relating the indices of refraction to the ray’s directions the right panel calculates 𝜃₂. All angles are small so the angle and its Sin are essentially the same. The error is proportional to the difference in the indices of refraction. In the X-ray band the difference between beryllium and vacuum is of the order 10⁻⁶ to 10⁻⁵. Therefore the error is very small.