Three Proofs — Question 3

Question 3

Linearized refraction at the local normal yields the spherical imaging equation

A spherical interface has vertex x=0, center at x=R, incident index n1 and outgoing index n2. An axial object is at x=-p and its image at x=q. For a ray at height h, use signed slopes $\theta_i\simeq h/p$, $\theta_t\simeq-h/q$, and local normal slope $\theta_n\simeq-h/R$.

1. Apply paraxial Snells law measured relative to the local normal.

2. Rearrange to obtain the imaging relation and check its plane-interface limit.

Question 3 - Solution

Linearized refraction at the local normal yields the spherical imaging equation

(a) Method and calculation

$$n_1(\theta_i-\theta_n)=n_2(\theta_t-\theta_n).$$$$n_1(h/p+h/R)=n_2(-h/q+h/R).$$

(b) Interpretation and further calculation

$$\boxed{n_1/p+n_2/q=(n_2-n_1)/R}.$$$$R\to\infty\Rightarrow\boxed{q=-(n_2/n_1)p}.$$

Physical check

All three angles use the same laboratory sign convention; treating the normal tilt as an unsigned angle can reverse the surface-power sign.