Two Types of Image — Question 4

Question 4

Defocus spreads a converging cone into a calculable blur disk

A circular ray bundle leaves an aperture of radius a at x=0 and converges to an on-axis real image at x=q, with q positive. Neglect diffraction. A perpendicular screen is placed at x=q+delta.

1. Derive the geometrical radius of the blur disk on the displaced screen.

2. For a=5 mm, q=200 mm, and a permitted blur diameter of 0.10 mm, find the allowed magnitude of delta.

Question 4 - Solution

Defocus spreads a converging cone into a calculable blur disk

(a) Method and calculation

$$r(x)=a|1-x/q|\Rightarrow\boxed{r_{\mathrm{blur}}=a|\delta|/q}.$$$$\boxed{b=2r_{\mathrm{blur}}=2a|\delta|/q}.$$

(b) Interpretation and further calculation

$$b\le b_{\max}\Rightarrow |\delta|\le b_{\max}q/(2a).$$$$\boxed{|\delta|\le(0.10)(200)/(2\cdot5)=2.0\,\mathrm{mm}}.$$

Physical check

This symmetric near-focus tolerance describes geometrical blur; diffraction is explicitly excluded.