Three Proofs — Question 1

Question 1

The paraxial mirror equation follows by adding the two ray angles

A real axial object and its real image are distances p and q in front of a concave spherical mirror of radius R. A ray reflects at small height h. Neglect surface sag to first order. Define small positive angles alpha=h/p, beta=h/q, and normal tilt gamma=h/R.

1. Use the reflection relation alpha+beta=2 gamma to derive the mirror equation.

2. Take the distant-object limit and identify where the paraxial approximation entered the argument.

Question 1 - Solution

The paraxial mirror equation follows by adding the two ray angles

(a) Method and calculation

$$\alpha+\beta=2\gamma\Rightarrow h/p+h/q=2h/R.$$$$h\ne0:\quad\boxed{1/p+1/q=2/R}.$$

(b) Interpretation and further calculation

$$p\to\infty:\quad\boxed{q=f=R/2,\quad1/p+1/q=1/f}.$$$$\tan\alpha\simeq\alpha,\quad\tan\beta\simeq\beta,\quad\sin\gamma\simeq\gamma\quad(|h/p|,|h/q|,|h/R|\ll1).$$

Physical check

The axial ray alone gives no focusing condition; requiring neighboring nonzero-height rays to share a focus supplies the equation.