Angular Resolution Calculator

Angular resolution is the smallest angle at which an optical device can distinguish two separate points. It determines the maximum sharpness of telescopes, microscopes, cameras and even the human eye. This calculator finds this limit based on the wavelength of light and the aperture size. The calculation can also be reversed: if any two of the three quantities (wavelength, aperture, angle) are entered, then the remaining quantity is calculated.

What to consider when measuring angular resolution:

When light passes through an aperture it spreads out, which is called diffraction. Individual points of light are not perfect dots but small fuzzy discs. When the points get close together these discs overlap. If the separation gets below a certain value then the device can no longer distinguish them and treats them as one point.

A smaller angular resolution is better, since it means the instrument can see finer detail. Large mirrors and short wavelengths reduce this limit; that's why research telescopes are so big and electron microscopes have higher resolution than optical microscopes.

The formula for angular resolution is:

The usual result is called the Rayleigh criterion after Lord Rayleigh. For a circular aperture it can be expressed as follows:

where θ is the angular resolution in radians, λ is the wavelength of light and D is the aperture. The factor 1.22 comes from the first dark ring that appears in the diffraction pattern of a circular opening, or Airy disk. For a narrow slit the factor is 1.00, and for a more accurate Sparrow criterion about 0.947. Depending on the optical system, the appropriate criterion can be chosen.

As an example, consider green light with a pupil diameter of 2 mm, which is approximately the size of the human eye in daylight. If λ is close to 550 nm and D is 0.002 m, then we get:

This corresponds to about 0.019 degrees or approximately 69 arcseconds, which is nearly a minute of arc. A typical value often used for visual acuity is one minute of arc, and this result is consistent with that.

The table below shows each symbol and its example values.

Telescopes - The Human Eye and the Hubble Space Telescope

Since the resolution only depends on the wavelength and aperture size, a larger reflector is better. The mirror diameter of Hubble Space Telescope is 2.4 meters while the pupil diameter of human eye is 2 millimeters.

The ratio of aperture is the result of 2.4 divided by 0.002 and thus 1,200 times larger. This allows the Hubble telescope to see details that are about 1,200 times finer than with the human eye in daylight. Its resolution limit is approximately 0.06 arcseconds. When you open the telescope setting, you can read off two practical values for each aperture: The empirical resolution used to separate close double star systems in visible light (Dawes Limit), and the maximum magnification at which the image is still sharp enough.

Microscopes and numerical aperture

In microscopy, the simple diameter is not used but rather the numerical aperture. The numerical aperture is defined as NA = n*sin(θ), where n is the index of refraction of the medium and θ is the half-angle of the cone of light collected by the lens. The smallest resolvable distance in a microscope also follows this same diffraction law.

The higher the numerical aperture and the shorter the wavelength, the smaller d is. This explains why oil-immersion objectives and blue light can produce the sharpest optical images. If you open your microscope setup and enter the numerical aperture, you can directly calculate this distance.

Resolution of details in the distance

The angular resolution is an angle but usually you also need to know the corresponding size. By multiplying the angle in radians by the distance to the object, you get the smallest resolvable distance.

This helps photographers judge whether they can see two lines on a distant sign as separate lines. It is also useful for astronomers to calculate the actual distance between two stars that are known to be at a certain separation from each other. If you enter the distance to the object, then the Distance Plane will do this calculation for you.

Applications of angular resolution:

This concept extends to the entire optical field. Telescope designers determine the size of a mirror based on the characteristics of binary star systems or planets they want to resolve. Microscope manufacturers strive for higher aperture numbers in order to observe smaller structures. Camera technicians find a balance between f-stop and diffraction blur. Even spy satellites and radio telescopes follow these same laws, meaning larger optical elements or shorter wavelengths are required to produce clearer images on the ground.

This calculator is for educational and planning purposes. In reality the resolution will be further reduced by imperfections in the optical elements, atmospheric effects, and sensor limitations so the diffraction limit is only a result under ideal conditions and not a guaranteed specification.