Resolving power of Microscopes and Astronomical telescopes

To understand the resolving power of telescopes and microscopes, let us first understand the meaning of resolving power. The resolving power is an optical instrument’s capacity to make individually identifiable figures of two objects put at a little angular distance from each other. If the two objects are positioned near each other being out of the resolving power limit of the optical device, then the images of the two things will look like a single object. The resolving power is not definable by any unit as it has none. It is a thing having zero dimension. 

For example, you want to see stars with the help of a telescope.
But whenever you see the stars they appear hazy and blurred, or it may be that they are very close to each other. So this ability of an optical device to show them as two separate stars is called resolving power.

The resolving power of a telescope

A telescope is a device that is used to see faraway objects. It has lenses and twisted mirrors combined to see objects that are far-off. It is used to see various kinds of objects of the planetary system. A simple standard telescope has a large aperture, an objective lens with high focal length, and an eyepiece with a lower focal length, and a smaller aperture. Then an eye piece consists of two lenses at a distance from each other. The lens towards the objective is called the ‘field lens’, and the other nearer to the eye is called the ‘eye lens’. If two far-off stars are very near to one another, they provide angular separation of a very low level. It depends on the telescope’s resolving power to make us see stars as separate from each other. 

The resolving power of a telescope is given as

Resolving power= 1/Δθ= d/1.22 λ

Where, 

θ = Angular separation between the two objects just resolved

d = diameter of the lens

And, λ = wavelength of light

So the telescopes having a larger d or large diameter of the lens have a better resolving power. It also depends on light’s wavelength.

While using a telescope, the closest objects like individual stars and the galaxy’s binary stars get subtended due to the presence of smaller angles on the telescopes. And to resolve this problem, larger apertures are needed. A high diameter can lead to better resolution. The superb quality of optical telescopes have mirror diameters that are large, being around 10m to get the perfect resolution. To reduce the resolving power, wavelengths need to be larger. And so, for the microscope and telescopes that detect radio emissions from the sky, big-sized mirrors are needed. The simpler telescopes have an objective type of lens which has a big aperture, focal length of high power, and an eyepiece having a tiny aperture and focal length of low power. The telescope’s resolving power means the ability of it to create separate and individual figures of two sources very closely spaced or located. Each of the sources have plane waves that cross through the aperture which have a diffraction pattern. An overlapping diffraction pattern is created by the two source sets that form an upper limit theoretically for resolving power.

The resolving power of a microscope

A microscope is an instrument that is used to look at microscopic specimens which are very small or tiny to be visible by the naked eye. It consists of two kinds of lenses, namely, ‘objective lens’ and eyepiece or ‘ocular lens’. 

So, its resolving power can be given as:

Resolving power = 1/Δd=2a/λ

Where,

a=numerical aperture

λ =wavelength

The aperture numerically is given on the body of the lens, which can be obtained from the given formula:

a=n sinβ

Here

n=medium’s refractive index

And 2β=angle, formed at a particular point when straight lines from its extremities are joined at that point by the diameter of the focus’s objective lens.

This means that the resolving power tends to be high if the numerical aperture of the lens is high. So, by exchanging air with a medium having a higher index of refraction(n), the microscope’s resolving power can be made better.

The resolving power for a microscope is the inverse of the distance between two objects that can be resolved, or, in other words, it is the reciprocal of the distance present between two objects that are just resolved. The microscope’s resolution is different from magnification. The latter, in the field of microscopic view, is the smallest distance between two different points, which can be recognised as far entities. If the points are nearer than the resolution, their positions will not be accurate. The microscopes do also provide high magnification; but if the lens is of poor quality, then it will result in poorer resolution, thus degrading the quality of the image.

Limit of resolution

By limit of resolution, we mean the linear distance between two objects that are resolved. It is the same as resolving power. The reciprocal of the limit of resolution provides us the resolving power, which means that a lower value of the limit of resolution would result in a higher level of resolving power of the lens.

Diffraction limit

A point object is when clicked through a circular opening or an aperture, a pattern of diffraction is got instead of a point image. If we can compare the object size to light’s wavelength, the pattern of diffraction is more clearly visible. The pattern of diffraction looks like concentric rings, getting fainter as one proceeds further from the centre. These are named ‘Airy discs’ or ‘Airy patterns’. When the concentric rings keep fading out as we move further away from the centre, there is a spot where the Airy discs or patterns cannot be distinguished from each other. And this is what is the Diffraction Limit.

Rayleigh’s criterion

Rayleigh’s criterion will give us the distance between two objects that has been resolved. According to it, the two images are said to be resolved when the diffraction pattern of one image is over the first minimum diffraction of the second image.

The first minimum diffraction pattern is observed at θ = 1.22λ/D

Rayleigh’s criterion gives us the formula of minimum angular distance between two objects which are resolvable as, θ = 1.22λ/D, where D is the circular aperture, and λ is the wavelength.

Conclusion

The resolving power of an optical instrument like a telescope or microscope is its ability to create separate images of two objects placed close by. The resolving power of a microscope and an astronomical telescope is the inverse of angular separation or distance between the objects, which can be resolved when viewed via an optical instrument. Sometimes, when we look up at the sky, we can see a group of binary stars which appear as one star when seen by the naked eye, but when done so by using optical instruments their pictures can be resolved clearly.