Alexander Fufaev
My name is Alexander FufaeV and here I will explain the following topic:

Rayleigh Criterion: At What Minimum Distance Can Two Objects Be Distinguished?

Formula

Formula: Rayleigh Criterion (Angular Resolution)
Minimum distance between two still distinguishable objects (Rayleigh criterion)
What do the formula symbols mean?

Minimum distance

Unit
Minimum distance between two still distinguishable objects. If the objects are closer than \( d_{\text{min}} \), then the objects can no longer be perceived as two objects.

With this formula you can estimate how good, for example, a telescope can distinguish two nearby objects on the moon.

Wavelength

Unit
Wavelength of light falling on the lens from the two objects under consideration.

Refractive Index

Unit
Refractive index of the medium (e.g. air) between the viewed objects and the lens.

Angle

Unit
Half opening angle on the object side.
Explanation

Video

The ability of an optical device (telescope, microscope) to perceive two objects that are close to each other as separate is referred to as the resolving power of this device.

The resolution is limited by the diffraction of the light waves emitted by the two objects. In principle, the resolving power cannot be better than the wavelength \(\lambda\) of the diffracted light.

Example

If you illuminate two objects placed next to each other with blue light (approx. 450 nm) so that the blue light is diffracted at the two objects, they must not be closer than 450 nm from each other so that they can still be perceived separately.

The Rayleigh criterion specifies the minimum distance \(d_{\text{min}} \) between two planets at which the planets can still be perceived as two separate objects through the telescope. This distance depends on the refractive index \( \class{brown}{n} \) of the medium between the telescope and the objects, but also on the wavelength \( \class{blue}{\lambda} \) of the light used for the observation. The opening angle \( \varphi \) of the telescope also has an influence on the resolution limit. The minimum distance between two objects can be calculated using the following formula:

Formula anchor
fufaev.org Minimum Distance Between Two Distinguishable Objects (Rayleigh Criterion)
Minimum Distance Between Two Distinguishable Objects (Rayleigh Criterion)