Calculate Resolving Power (Rayleigh)

Angular resolution from the Rayleigh criterion

Rayleigh Calculator (JavaScript)

Rayleigh criterion

For circular apertures, θ = 1.22·λ/D. Smaller angles mean better resolving power.

Result

Example calculations

Example 1: 100-mm telescope

Given: λ = 550 nm, D = 100 mm

\[\theta=1.22\cdot\frac{550\cdot10^{-9}}{0.1}=6.71\cdot10^{-6}\,rad\approx1.38''\]

Result: θ ≈ 1.38 arcseconds

Example 2: Required diameter

Given: λ = 550 nm, θ = 0.7''

\[D=1.22\cdot\frac{\lambda}{\theta}\approx0.198\,m\]

Result: D ≈ 198 mm

Example 3: Wavelength influence

Rule: shorter λ → smaller θ

\[\theta\propto\lambda\]

Interpretation: Blue light provides better theoretical resolution than red light.

Formulas and comprehensive description

The Rayleigh criterion gives the minimum angular separation of two point sources that can still be distinguished. It describes the diffraction limit of an ideal optical system with a circular aperture. In real systems, aberrations, atmospheric seeing, and detector sampling further affect practical performance.

Rayleigh formula
\[\theta=1.22\cdot\frac{\lambda}{D}\]
Wavelength
\[\lambda=\frac{\theta\cdot D}{1.22}\]
Diameter
\[D=1.22\cdot\frac{\lambda}{\theta}\]
Conversion
\[1\,rad=206265''\]
Note
A larger aperture improves resolving power (smaller θ). In astronomy, atmospheric conditions often limit resolution more than diffraction itself.

Description

What is resolving power?

The resolving power of an optical system describes its ability to image two closely spaced object points as distinct entities. Even with perfectly ground lenses or mirrors, this ability to distinguish between points is fundamentally limited because light is diffracted at the circular opening (aperture). Consequently, each object point does not appear as an ideal point, but rather as a small diffraction pattern – the so-called Airy disk.

The Rayleigh criterion

According to the Rayleigh criterion, two point sources are considered just resolvable when the central maximum of one Airy disk falls upon the first minimum of the other. This yields the smallest resolvable angular separation:

\[\theta = 1{,}22\cdot\frac{\lambda}{D}\]
  • θ – minimum angular separation (in radians; expressed here in arcseconds)
  • λ – wavelength of the light
  • D – diameter of the opening (aperture)
  • 1.22 – constant derived from the first zero of the Bessel function for circular apertures

A smaller angle θ means better resolving power. It therefore improves with a larger aperture diameter and a shorter wavelength – blue light allows for finer resolution than red light.

Applications
  • Sizing of astronomical telescopes
  • Resolution limits of microscopes and camera lenses
  • Assessment of the human eye (pupil diameter acting as the aperture)
  • Antenna and radar technology involving electromagnetic waves
Is this page helpful?            
Thank you for your feedback!

Sorry about that

How can we improve it?