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Small-angle Rayleigh criterion Calculator

Calculates the minimum angular separation at which two point sources can be resolved by a circular aperture.

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Angular Resolution

Formula first

Overview

This equation defines the diffraction limit of an optical system, such as a telescope or microscope, based on the wavelength of light and the diameter of the aperture. It assumes the small-angle approximation, where the sine of the angle is approximately equal to the angle in radians. When two point sources are separated by this angle, the central maximum of one diffraction pattern falls on the first minimum of the other.

Symbols

Variables

= Angular Resolution, = Wavelength, d = Aperture Diameter

Angular Resolution
rad
Wavelength
Aperture Diameter

Apply it well

When To Use

When to use: Use this when determining the theoretical resolution limit of an optical instrument like a telescope or camera lens.

Why it matters: It establishes the fundamental physical limit on how much detail can be captured by an optical system, regardless of how perfect the lenses are.

Avoid these traps

Common Mistakes

  • Forgetting to convert units so that wavelength and diameter match.
  • Assuming the formula applies to non-circular apertures (which have different constants).
  • Confusing the angular resolution with linear resolution.

One free problem

Practice Problem

A telescope has an aperture diameter of 2.0 meters. If it is observing light with a wavelength of 500 nanometers, what is the minimum angular resolution in radians?

Aperture Diameter2 m
Wavelength5e-7 m

Solve for: theta

Hint: Ensure the wavelength is converted to meters (500 nm = 500 x 10^-9 m).

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Young, H. D., & Freedman, R. A. (2020). University Physics with Modern Physics (15th ed.). Pearson.
  2. Halliday, D., Resnick, R., & Walker, J. (2014). Fundamentals of Physics (10th ed.). Wiley.
  3. OpenStax University Physics Volume 3, Diffraction and the Rayleigh Criterion, accessed 2026-04-09
  4. Hecht, Eugene (2017). Optics (5th ed.). Pearson. ISBN 978-0134267727
  5. NIST CODATA Recommended Values of the Fundamental Physical Constants (2018)
  6. IUPAC Gold Book: Resolution (optics)
  7. Hecht, Eugene. Optics.
  8. Born, Max, and Emil Wolf. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light.