Spherical Mirrors — Question 8

Question 8

Placing a source at the paraxial focus makes outgoing slopes vanish

A paraxial source is on the axis a positive distance p in front of a concave mirror of radius R. A ray hits at height h. To first order, its incident angle is h/p and the normal tilt is h/R. Measure outgoing slope relative to distance increasing away from the mirror.

1. Use reflection to obtain the outgoing slope and find the source distance producing collimation.

2. If p is slightly displaced from f=R/2 by delta, derive the first-order residual slope.

Question 8 - Solution

Placing a source at the paraxial focus makes outgoing slopes vanish

(a) Method and calculation

$$\boxed{u_{\mathrm{out}}=h/p-2h/R=h(1/p-1/f)}.$$$$u_{\mathrm{out}}=0\ \text{for all }h\Rightarrow\boxed{p=f=R/2}.$$

(b) Interpretation and further calculation

$$p=f+\delta:\quad1/p\simeq1/f-\delta/f^2.$$$$\boxed{u_{\mathrm{out}}\simeq-h\delta/f^2}.$$

Physical check

A source farther than the focus gives a negative slope for an upper ray, so the reflected bundle converges.