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

Important Formula

Formula: Rayleigh Criterion (Angular Resolution)

$$d_{\text{min}} ~=~ 0.61 \,\cdot\, \frac{\class{blue}{\lambda}}{\class{brown}{n} \, \sin(\class{gray}{\varphi})}$$ $$d_{\text{min}} ~=~ 0.61 \,\cdot\, \frac{\class{blue}{\lambda}}{\class{brown}{n} \, \sin(\class{gray}{\varphi})}$$ $$\class{blue}{\lambda} ~=~ \frac{d_{\text{min}} \, \class{brown}{n} \, \sin(\class{gray}{\varphi})}{0.61}$$ $$\class{brown}{n} ~=~ 0.61 \,\cdot\, \frac{\class{blue}{\lambda}}{d_{\text{min}} \, \sin(\class{gray}{\varphi})}$$ $$\class{gray}{\varphi} ~=~ \arcsin\left(\frac{0.61 \class{blue}{\lambda}}{\class{brown}{n} \, d_{\text{min}}}\right)$$

Rearrange formula

What do the formula symbols mean?

Minimum distance

$$ d_{\text{min}} $$ Unit $$ \mathrm{m} $$

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

$$ \lambda $$ Unit $$ \mathrm{m} $$

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

Refractive Index

$$ n $$ Unit $$ - $$

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

Angle

$$ \varphi $$ Unit $$ - $$

Half opening angle on the object side.

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:

$$ \begin{align} d_{\text{min}} ~=~ 0.61 \,\cdot\, \frac{\class{blue}{\lambda}}{\class{brown}{n} \, \sin(\class{gray}{\varphi})} \end{align} $$

Minimum Distance Between Two Distinguishable Objects (Rayleigh Criterion)