Spherical Refracting Surfaces — Question 7

Question 7

The front and back focal distances of one refracting surface are unequal

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. Let n2 be greater than n1 and R be positive. Define the front focal distance F1 as the real object distance giving parallel output, and F2 as the real image distance for parallel input.

1. Derive F1 and F2.

2. Find their ratio and evaluate both for n1=1.00, n2=1.60, and R=12 cm.

Question 7 - Solution

The front and back focal distances of one refracting surface are unequal

(a) Method and calculation

$$q\to\infty:\quad\boxed{F_1=n_1R/(n_2-n_1)}.$$$$p\to\infty:\quad\boxed{F_2=n_2R/(n_2-n_1)}.$$

(b) Interpretation and further calculation

$$\boxed{F_2/F_1=n_2/n_1}.$$$$\boxed{F_1=20\,\mathrm{cm},\quad F_2=32\,\mathrm{cm}}.$$

Physical check

The index-weighted focal distances are equal: F1/n1 equals F2/n2. A single unqualified focal length would hide this asymmetry.