Optical Instruments — Question 10

Question 10

An eye pupil smaller than the exit pupil discards objective area

An ideal afocal telescope has objective diameter 60 mm and angular magnification magnitude 10. The centered eye pupil diameter is 3 mm. Assume uniform pupil illumination and no other loss or vignetting.

1. Find the exit-pupil diameter and the fraction of its light admitted by the eye.

2. Find the effective diameter of the objective region whose rays can reach the eye.

Question 10 - Solution

An eye pupil smaller than the exit pupil discards objective area

(a) Method and calculation

$$\boxed{D_{\mathrm{exit}}=D_o/|M|=60/10=6\,\mathrm{mm}}.$$$$\boxed{P_{\mathrm{eye}}/P_{\mathrm{exit}}=(D_{\mathrm{eye}}/D_{\mathrm{exit}})^2=(3/6)^2=1/4}.$$

(b) Interpretation and further calculation

$$D_{o,\mathrm{effective}}=|M|D_{\mathrm{eye}}.$$$$\boxed{D_{o,\mathrm{effective}}=30\,\mathrm{mm}}.$$

Physical check

Increasing objective diameter beyond the pupil-matched value does not deliver all of the extra collected light through this fixed centered eye pupil.