Spherical Mirrors — Question 5

Question 5

A longitudinal-focus tolerance limits the usable spherical aperture

For a concave sphere, use the exact axial intercept $q(\theta)=R[1-1/(2\cos\theta)]$ and paraxial focus f=R/2. The aperture radius is $a=R\sin\theta_{\max}$. Require every admitted ray to satisfy $(f-q)/f\le\epsilon$.

1. Derive an exact bound on a/R.

2. Give its leading small-epsilon approximation and evaluate the exact bound for epsilon=0.01.

Question 5 - Solution

A longitudinal-focus tolerance limits the usable spherical aperture

(a) Method and calculation

$$\frac{f-q}{f}=\sec\theta-1\le\epsilon\Rightarrow\cos\theta\ge(1+\epsilon)^{-1}.$$$$\boxed{a/R\le\sqrt{1-(1+\epsilon)^{-2}}}.$$

(b) Interpretation and further calculation

$$(1+\epsilon)^{-2}\simeq1-2\epsilon\Rightarrow\boxed{a/R\lesssim\sqrt{2\epsilon}}.$$$$\epsilon=0.01:\quad\boxed{a/R\le0.140371}.$$

Physical check

The tolerance concerns longitudinal focal shift, not diffraction or transverse spot size.