Spherical Mirrors — Question 6

Question 6

Transverse spherical aberration grows cubically with aperture height

At the paraxial focal plane of a concave sphere, an exact ray at polar angle theta has signed height $y_F=R\sin\theta(\cos\theta-1)/\cos2\theta$. Use small-angle expansions and ray height h=R sin(theta). Neglect diffraction.

1. Find the leading transverse error in terms of h and R.

2. If the aperture radius is halved, find the leading change in edge-ray blur radius and the collected power for uniform illumination.

Question 6 - Solution

Transverse spherical aberration grows cubically with aperture height

(a) Method and calculation

$$\sin\theta\simeq\theta,\quad\cos\theta-1\simeq-\theta^2/2,\quad\cos2\theta\simeq1.$$$$\boxed{y_F\simeq-R\theta^3/2\simeq-h^3/(2R^2)}.$$

(b) Interpretation and further calculation

$$\boxed{b_{\mathrm{edge}}(a/2)/b_{\mathrm{edge}}(a)\simeq(1/2)^3=1/8}.$$$$P\propto\pi a^2\Rightarrow\boxed{P(a/2)/P(a)=1/4}.$$

Physical check

Stopping down sharply reduces geometrical spherical aberration, at the cost of collected light; diffraction eventually matters for small apertures.