Two Types of Image — Question 8

Question 8

Covering part of an ideal aperture changes brightness rather than image position

An ideal imaging system sends every unblocked ray from one object point to the same real image point. Its circular entrance aperture has radius a and uniform incident irradiance. An opaque stop blocks the central disk of radius a/2. Neglect diffraction and vignetting.

1. Find the transmitted power fraction.

2. Use a generic ray equation $y=y_0(1-x/q)$ to show whether the surviving annular rays change the image coordinate.

Question 8 - Solution

Covering part of an ideal aperture changes brightness rather than image position

(a) Method and calculation

$$A_{\mathrm{open}}=\pi a^2-\pi(a/2)^2=3\pi a^2/4.$$$$\boxed{P_{\mathrm{open}}/P_{\mathrm{full}}=3/4}.$$

(b) Interpretation and further calculation

$$y(q)=y_0(1-q/q)=0\quad\text{for every surviving}\ y_0.$$$$\boxed{x_I=q\ \text{unchanged}}.$$

Physical check

Each remaining ray still carries information about the same object point; losing part of the aperture does not remove a corresponding part of the ideal image.