Diffraction by a Circular Aperture — Question 2

Question 2

The Airy diameter depends on focal ratio rather than focal length alone

For an ideal circular aperture, the focal-plane first dark radius is $r_1=1.22\lambda f/D$. Define focal ratio $N=f/D$. A lens operates at $N=8.0$ and $\lambda=500\,\mathrm{nm}$; detector pixels are $2.0\,\mu\mathrm m$ wide.

1. Find the diameter of the central Airy disk and its width in pixels.

2. If focal length and aperture diameter both double, determine the changes in angular resolution and physical Airy diameter.

Question 2 - Solution

The Airy diameter depends on focal ratio rather than focal length alone

(a) Method and calculation

$$\boxed{d_A=2r_1=2.44\lambda N=9.76\,\mu\mathrm m}.$$$$\boxed{d_A/p=9.76/2.0=4.88\ \text{pixels}}.$$

(b) Interpretation and further calculation

$$f\to2f,\ D\to2D:\quad\boxed{\theta_R\to\theta_R/2}.$$$$N\to N\Longrightarrow\boxed{d_A\ \text{unchanged}}.$$

Physical check

Equal physical spot sizes at the image plane can correspond to different angular resolutions when focal length changes.