Spherical Mirrors — Question 9

Question 9

A tilted parallel bundle maps direction to focal-plane height

A narrow paraxial parallel bundle reaches a concave mirror with small signed incident slope alpha relative to +x. Use a flat reference plane at the vertex. At reflection height h, its outgoing slope measured away from the mirror is $u=\alpha-h/f$.

1. Find its height in a plane a distance f in front of the mirror and show independence from h.

2. For f=0.75 m and angular separation delta alpha=0.004 rad between two bundles, find their focal-plane separation.

Question 9 - Solution

A tilted parallel bundle maps direction to focal-plane height

(a) Method and calculation

$$y(s)=h+s(\alpha-h/f).$$$$\boxed{y(f)=f\alpha}\quad\text{independent of entrance height }h.$$

(b) Interpretation and further calculation

$$\Delta y=f\,\Delta\alpha.$$$$\boxed{\Delta y=0.75(0.004)=0.0030\,\mathrm m=3.0\,\mathrm{mm}}.$$

Physical check

This first-order mapping is the basis of imaging distant angular features; off-axis aberrations are outside the supplied paraxial model.