Three Proofs — Question 10

Question 10

Stationary optical path gives an independent proof of spherical refraction

An axial real object at x=-p and real image at x=q are separated by a spherical interface with vertex x=0 and radius R positive. Indices before and after are n1 and n2. At small ray height h, surface sag is $z(h)\simeq h^2/(2R)$. Expand optical path through order h squared.

1. Expand the two geometric path lengths and their index-weighted sum.

2. Impose stationarity for neighboring paraxial rays and recover the spherical refraction relation.

Question 10 - Solution

Stationary optical path gives an independent proof of spherical refraction

(a) Method and calculation

$$\ell_1=\sqrt{(p+z)^2+h^2}\simeq p+z+h^2/(2p),\quad\ell_2\simeq q-z+h^2/(2q).$$$$\mathcal L=n_1\ell_1+n_2\ell_2\simeq n_1p+n_2q+\frac{h^2}{2}\left[\frac{n_1}{p}+\frac{n_2}{q}+\frac{n_1-n_2}{R}\right].$$

(b) Interpretation and further calculation

$$\frac{d\mathcal L}{dh}\simeq h\left[\frac{n_1}{p}+\frac{n_2}{q}+\frac{n_1-n_2}{R}\right]=0\quad\text{for neighboring }h\ne0.$$$$\boxed{\frac{n_1}{p}+\frac{n_2}{q}=\frac{n_2-n_1}{R}}.$$

Physical check

The surface sag is second order in height and must be retained in this optical-path proof, even though the ray-angle proof neglects it at first order.