Spherical Refracting Surfaces — Question 6

Question 6

A converging bundle aimed at the curvature center keeps its target

Use $n_1/p+n_2/q=(n_2-n_1)/R$ and $m=-n_1q/(n_2p)$ for paraxial refraction. Here p is positive on the incident side, q on the outgoing side, and R when the center lies on the outgoing side. A spherical interface has R positive and center C on its outgoing side. Before reaching the surface, incident rays converge toward C. The indices n1 and n2 are arbitrary positive values.

1. Assign p and solve for the image distance.

2. Find magnification and explain why the result also follows without a paraxial approximation for the axial target.

Question 6 - Solution

A converging bundle aimed at the curvature center keeps its target

(a) Method and calculation

$$\boxed{p=-R},\quad-\frac{n_1}{R}+\frac{n_2}{q}=\frac{n_2-n_1}{R}.$$$$\boxed{q=R}.$$

(b) Interpretation and further calculation

$$\boxed{m=-\frac{n_1R}{n_2(-R)}=n_1/n_2}.$$$$\theta_i=0\Rightarrow n_1\sin\theta_i=n_2\sin\theta_t=0\Rightarrow\boxed{\theta_t=0}.$$

Physical check

All axial-target rays are normal to the spherical surface and still meet at C. The transverse magnification describes neighboring off-axis points and need not be one.