AcademyDiffraction
Academy
Circular Apertures and Resolution
Level 1 - Physics topic page in Diffraction.
Principle
A circular aperture forms an Airy pattern rather than a perfect point image. The finite aperture limits angular resolution, so two point sources are just resolved when one Airy maximum falls near the other's first minimum.
Notation
\(D\)
aperture diameter
\(\mathrm{m}\)
\(\lambda\)
wavelength
\(\mathrm{m}\)
\(\theta_R\)
Rayleigh angular resolution
\(\mathrm{rad}\)
\(L\)
distance to sources or screen
\(\mathrm{m}\)
\(s\)
minimum resolvable separation
\(\mathrm{m}\)
Method
Derivation 1: First Airy minimum
For a circular aperture, the first dark ring occurs at an angle set by wavelength divided by aperture diameter.
First dark ring
\[\theta_1\approx1.22\frac{\lambda}{D}\]
Rayleigh criterion
\[\theta_R\approx1.22\frac{\lambda}{D}\]
Derivation 2: Convert angular to linear resolution
For small angles, angular separation corresponds to linear separation divided by distance.
Small-angle geometry
\[\theta\approx\frac{s}{L}\]
Linear resolution
\[s_{\min}\approx1.22\frac{\lambda L}{D}\]
Derivation 3: Compare apertures
Larger apertures produce narrower diffraction patterns and better resolution.
Resolution scaling
\[\theta_R\propto\frac{1}{D}\]
Wavelength scaling
\[\theta_R\propto\lambda\]
Rules
Rayleigh criterion
\[\theta_R\approx1.22\frac{\lambda}{D}\]
Linear separation
\[s_{\min}\approx L\theta_R\]
Diffraction-limited separation
\[s_{\min}\approx1.22\frac{\lambda L}{D}\]
Examples
Question
Find the angular resolution of a
\[0.10\,\mathrm{m}\]
telescope aperture using \[550\,\mathrm{nm}\]
light.Answer
\[\theta_R=1.22\frac{550\times10^{-9}}{0.10}=6.7\times10^{-6}\,\mathrm{rad}\]
Checks
- Larger aperture means smaller diffraction blur.
- Shorter wavelength gives better diffraction-limited resolution.
- Rayleigh resolution is an angular criterion.
- Magnifying a blurred diffraction image does not create new resolved detail.