Diffraction by a Circular Aperture — Question 1

Question 1

Rayleigh resolution sets a required collecting diameter

Two equal incoherent point sources are separated transversely by $s=0.20\,\mathrm m$ at range $R=20.0\,\mathrm{km}$. An ideal unobstructed circular aperture observes them at $\lambda=550\,\mathrm{nm}$. Use the Rayleigh criterion $\theta_R=1.22\lambda/D$ and small source angles.

1. Find the minimum aperture diameter under this criterion.

2. If the wavelength becomes $825\,\mathrm{nm}$ with the same aperture, find the minimum resolvable transverse separation at that range.

Question 1 - Solution

Rayleigh resolution sets a required collecting diameter

(a) Method and calculation

$$\theta_s\simeq s/R=1.00\times10^{-5}\,\mathrm{rad};\quad\theta_R\leq\theta_s.$$$$\boxed{D_{\min}=1.22\lambda/\theta_s=0.0671\,\mathrm m}.$$

(b) Interpretation and further calculation

$$s_R=R(1.22\lambda^\prime/D).$$$$\boxed{s_R^\prime=(825/550)(0.20\,\mathrm m)=0.300\,\mathrm m}.$$

Physical check

The criterion applies to ideal optics and equal incoherent sources; it is a stated resolution convention rather than a universal detection threshold.