# How to Calculate Angular Resolution

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Angular resolution, also known as the Rayleigh criterion and spatial resolution, is the minimum angular distance between two distant objects that an instrument can discern resolvable detail. As an example, if a person holds two pens 10cm apart and stands 2m from you, you can discern there are two pencils. As the other person moves away, the pencils appear to move closer together or the angular separation decreases. The calculation of this angle is very important in optics. This angle represents the resolving power and precision of optical instruments such as your eye, a camera, and even a microscope.

Write down:

\sin{A}=1.220\frac{W}{D}

This formula is known as the angular resolution formula and is the mathematical representation of the Rayleigh criterion. The Rayleigh criterion basically says that two different points are resolved when the diffraction maximum of one image coincides with the first minimum diffraction of a second image. If the distance is greater, the two points are resolved and if it is smaller they are not resolved.

Calculate the wavelength of the light waves used to focus the image. This number is represented by W in the angular resolution formula. For example, say you are using yellow light. The wavelength for yellow light is about 577nm. This number can be looked up. To get a more precise answer you will need to know the frequency of the light you are using and the speed of light. The wavelength equation is:

\text{wavelength}(W)=\frac{\text{speed of light}(c)}{\text{frequency}(f)}

Find the value of the entrance pupil diameter (D) or the diameter of the lens aperture (D) of the imaging system you are using. For telescopes and most other optical instruments, the diameter of the aperture can be found in the user manual or you can contact the manufacturer who will be able to tell you the correct value.

Rewrite the formula substituting the wavelength (W) value and the diameter (D) value you just found.

Ensure your wavelength and diameter have been converted into the same units of measure. For example, if your wavelength is in meters then your diameter needs to be converted to meters or vice versa.

Manipulate the formula to solve for A by dividing both sides of the equation by sin. The manipulated formula should appear as the following:

A=\arcsin{1.220\frac{W}{D}}

Use your calculator to do the math to find out what the angular resolution (A) is equal to. The units of the wavelength and the diameter cancel so the answer is expressed in radians. For astronomy purposes, you can convert radians to seconds of arc.

#### Things You'll Need

• Wavelength in question
• Diameter of the lens aperture or objective
• Calculator

#### Tips

• If your wavelength is extremely small in comparison to your diameter you can eliminate the sin function in the angular resolution formula making it much easier to solve. In this case, the formula will be A = W/D.