Angular Resolution Calculator

Created by Gabriela Diaz
Last updated: Nov 21, 2022

Use this angular resolution calculator to determine a lens' angular resolution. The angular resolution of an optical instrument, such as a telescope, a microscope, or even the human eye, measures its ability to resolve fine details on an object or to distinguish between nearby objects. The term is used in various fields, including astronomy, engineering, and photography.

If you'd like to know:

  • What does angular resolution measure?;
  • The angular resolution formula or Rayleigh criterion;
  • How to calculate the angular resolution yourself; and
  • What the angular resolution of the Hubble Space Telescope is, and how it compares to the angular resolution of the human eye.

We invite you to read on! 🔭

What is angular resolution?

Angular resolution is the ability of an image-forming instrument, such as a telescope, to resolve fine details on an object or to distinguish between two very closely spaced objects.

The angular resolution of a telescope indicates how well we can see a star, the Moon, or any other celestial object and how much we can expect to distinguish between two adjacent objects. The angular resolution is limited by the diameter of the lens and the wavelength of light used. As you might imagine, the bigger the diameter of the lens of the telescope or any other instrument, the higher its resolution for the same wavelength.

💡 What is a lens? Take a look at the lensmaker equation calculator to read about this!

How to calculate the angular resolution – Angular resolution formula

To determine the angular resolution, we use the Rayleigh criterion. This one is represented by the following angular resolution formula:

θ=1.22λd\small \theta = 1.22 \, \cfrac{\lambda}{d}

where:

  • θ\theta – Angular resolution, expressed in radians;
  • λ\lambda – Wavelength of the light; and
  • dd – Diameter of the lens aperture.

The angular resolution is measured in terms of an angle in radians; the smaller its value, the greater the instrument's resolution. Also, notice from the formula that the bigger the lens's diameter, the higher the resolution for the same wavelength. In the case of radio wavelengths (the longest wavelengths in the electromagnetic spectrum), we need bigger lenses to get a good angular resolution.

Let's now explore how to calculate angular resolution with an example! We can use this formula to estimate the angular resolution of the Hubble Space Telescope. Its mirror lens has a diameter of 2.4 m (7.8 ft), and for wavelengths of around 500 nm, the angular resolution of the Hubble is:

θ=1.22λdθ=1.22500 nm2.4 mθ=1.22500×109 m2.4 mθ=0.000000254 radθ=0.254×106 rad\small \begin{split} \theta &= 1.22 \, \cfrac{\lambda}{d} \\[1em] \theta &= 1.22 \, \cfrac{500 \text{ nm}}{2.4 \text{ m}} \\[1em] \theta &= 1.22 \, \cfrac{500 \times 10 ^{-9}\text{ m}}{2.4 \text{ m}} \\[1.2em] \theta &= 0.000000254 \text{ rad} \\[1em] \theta &= 0.254 \times 10^{-6} \text{ rad} \end{split}

And what about the human eye? 👁 We can do a similar calculation using the same wavelength and considering an average eye's pupil diameter of 4.2 mm:

θ=1.22λdθ=1.22500 nm4.2 mmθ=0.0001452 radθ=145.2×106 rad\small \begin{split} \theta &= 1.22 \, \cfrac{\lambda}{d} \\[1em] \theta &= 1.22 \, \cfrac{500 \text{ nm}}{4.2 \text{ mm}} \\[1.1em] \theta &= 0.0001452 \text{ rad} \\[1em] \theta &= 145.2 \times 10^{-6} \text{ rad} \end{split}

What a difference! If we'd like to compare these two, consider a hypothetical distance of 100 km. At this distance, we humans can distinguish objects that are 14.52 m apart. However, with the Hubble Telescope, which has a finer resolution, we can differentiate between objects up to 0.0254 m (25.4 mm) apart!

(Please keep in mind this is just a hypothetical example. The Hubble is used for further distanced objects, e.g., it has discovered one of the farthest stars, Earendel, 12.9 billion years light ways from us).

💡 You can learn more about lenses at our focal length calculator or with the thin lens equation calculator!

Using the angular resolution calculator

Use this tool to find the angular resolution of the lens you're studying. By simply entering the Wavelength of the light and the Aperture diameter of the lens, the angular resolution calculator will display the corresponding Resolution.

Gabriela Diaz
Wavelength
nm
Aperture diameter
mm
Resolution
deg
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