Free Angular Resolution Calculator

Understanding Angular Resolution and the Rayleigh Criterion

The Angular Resolution Calculator — also called a Rayleigh Criterion Calculator or Diffraction Limit Calculator — determines the smallest angular separation an optical instrument can theoretically resolve. This metric, known as angular resolution, defines the finest detail a telescope, microscope, or camera lens can distinguish. By entering the light wavelength and the aperture diameter, you immediately obtain the diffraction‑limited resolution predicted by Lord Rayleigh’s classic principle.

What Angular Resolution Quantifies

Angular resolution measures the capacity of a lens or mirror to separate two point sources that lie at a small angular distance. The lower the angular resolution value, the greater the ability to see fine detail. For telescopes, this means resolving close binary stars; for microscopes, it enables viewing tiny cellular structures; for cameras, it directly relates to image sharpness. The fundamental barrier is set by diffraction — hence the Diffraction Limit Calculator is essential for any optical design.

The Rayleigh Criterion Formula

Lord Rayleigh’s criterion states that two point sources are considered resolved when the peak of one diffraction pattern falls exactly at the first dark ring of the other. For a circular aperture, this condition leads to the well‑known expression:

$$\theta = 1.22 \, \frac{\lambda}{d}$$

where:

• $\theta$ = angular resolution (radians)
• $\lambda$ = wavelength of the light
• $d$ = diameter of the aperture (same length unit as $\lambda$)

The constant 1.22 stems from the first zero of the Airy disk. Although the formula was originally derived for diffraction gratings, this Rayleigh Criterion Calculator applies it universally to any optical system with a circular aperture.

Practical Examples: Human Eye vs. Hubble Space Telescope

The human pupil adapts to lighting conditions: it is about 2 mm in bright daylight and expands to 8 mm in darkness. Using green light ($\lambda = 550\ \text{nm}$), the daytime angular resolution is:

$$\theta \approx 1.22 \times \frac{550 \times 10^{-9}}{0.002} \approx 3.35 \times 10^{-4}\ \text{rad} \approx 0.02^{\circ}$$

At night, with an 8 mm pupil, the resolution improves accordingly. By comparison, the Hubble Space Telescope’s 2.4 m mirror provides:

$$\theta \approx 1.22 \times \frac{550 \times 10^{-9}}{2.4} \approx 2.8 \times 10^{-7}\ \text{rad} \approx 1.6 \times 10^{-5}\,^{\circ}$$

This is about 125 times finer than the human eye’s daytime performance — a dramatic illustration of how aperture diameter directly enhances resolving power. Such calculations are fundamental for astronomers selecting telescope optics and engineers designing high‑resolution imaging systems.

How to Use This Optical Resolution Calculator

Enter the wavelength of light (in any convenient unit) and the aperture diameter. The tool instantly computes the angular resolution in radians or degrees. It works as a Telescope Resolution Calculator, Lens Resolution Calculator, and general Optical Resolution Calculator for any diffraction‑limited scenario. Because the Rayleigh criterion represents the theoretical best case, the result tells you the highest resolution your system can achieve under ideal conditions. Use it to benchmark a microscope objective, evaluate camera lens performance, or plan an astronomy setup — all based on a single, reliable formula.