Spherical Mirrors — Question 3

Question 3

A measured sag determines spherical curvature

A spherical mirror cap has aperture radius a and sag s, the axial depth between the vertex and the rim plane. The parent sphere has radius magnitude R, with zero less than s less than R.

1. Derive R from a and s without a small-angle approximation.

2. For a=30 mm and s=0.75 mm, find R and the magnitude of its paraxial focal length.

Question 3 - Solution

A measured sag determines spherical curvature

(a) Method and calculation

$$(R-s)^2+a^2=R^2\Rightarrow2Rs=a^2+s^2.$$$$\boxed{R=(a^2+s^2)/(2s)}.$$

(b) Interpretation and further calculation

$$\boxed{R=(30^2+0.75^2)/(1.50)=600.375\,\mathrm{mm}}.$$$$\boxed{|f|=R/2=300.1875\,\mathrm{mm}}.$$

Physical check

The common approximation R approximately equals a squared over 2s neglects the small correction s/2.