Resolving Power of Optical Instruments

Resolving Power of Optical Instruments: Look at the night sky and you may find stars twinkling in the sky. Are these stars really point-sized? Now, look at your hands. Can you see the skin cells that cover your fingers? Clearly, our eyes are not all-powerful. They need some help sometimes. Optical instruments play a huge role in science. From the depths of space to the layers of hair, we need such instruments to view things that our eyes cannot see. One crucial aspect of optical instruments is their resolving power. 

The resolving power of a device is its ability to distinguish between objects that are close together. Devices like microscopes and telescopes have a combination of lenses that  are used to form images of different objects. When a point object is observed through a circular opening, the image obtained is not a mere point.

Together, the series of these concentric rings (that fade as one moves away from the centre) form an Airy pattern. This is especially true when the size of the object is comparable to the wavelength of light. Let us learn in detail about the resolving powers of various optical instruments and how to obtain the equation for resolving power of optical instruments.

What is Resolving Power?

The resolving power of an optical instrument is defined as the power of the optical device to distinguish between two objects that are close together and produce distinct images of the two objects. In simple words, resolving power varies inversely with the distance between the two objects to be resolved when viewed from an optical instrument. Thus, the images of two close-lying objects should appear distinct and separate when viewed from the device. Consider a telescope is used to view two stars placed closely; the telescope’s resolving power will depend on its ability to resolve the images of the two stars.

What is Resolution?

The resolving power of a device depends on its resolution. The resolution of an optical instrument is defined as the shortest distance between two points on a specimen that can be distinguished by the observer’s eye or the camera’s lens installed in the optical device. In simple terms, it’s the least resolvable distance. Scientifically, resolution can be defined as the precision with which an optical device can record or measure a specimen’s variables under observation. This term is often ignored when we deal with microscopes or telescopes.

The diffraction or aberration limits the resolution of such an instrument and often leads to blurring of the image. In geometrical optics, the problem of aberration can be solved by increasing the optical quality of the optical device. Diffraction is attributed to the wave nature of light. It depends on the aperture of the measuring instrument, and a circular aperture of the lens resembles that of a single slit experiment. Light interferes with itself when it passes through such a lens and creates a ring-shaped diffraction pattern called the Airy pattern. Such a pattern is obtained only when the wavefront of the light source is spherical or plane at the aperture’s exit point.

Resolution vs Magnification

Resolution is the ability to distinctly realise two objects placed closely, while magnification is the property of enlarging a given object to see its details properly. Magnification magnifies the object’s size while resolution distinguishes between two objects. The magnification and resolution of an optical instrument are inversely related to each other. When magnification increases, the resolution decreases and vice versa.

Rayleigh’s Criterion

The minimum distance between two point sources so that their images are distinguishable when viewed from an optical device is calculated using Rayleigh’s criterion.

When two objects placed at a distance from each other are separated by an angular separation \(\theta\), the diffraction patterns of the two objects will overlap each other. The two objects can be resolved and seen as distinct if the central maxima of the two diffraction patterns do not overlap. When one central maximum falls onto the first minimum of the other diffraction pattern, the two images are considered resolved. Still, they would appear as one when the two central maxima’s overlap. This defines Rayleigh’s resolution criterion. Mathematically, it can be given as:

\(\sin \theta = \frac{{1.22\lambda }}{D}\)
Where,
\(\theta\): the angular separation measured in radians
\(\lambda\): the wavelength of light
\(D\): the diameter of the aperture (aperture of our eye, the pupil; or aperture in telescope or camera).

Resolving Power of a Telescope

Telescopes make viewing objects like binary stars, individual stars, distant galaxies, quasars, and planets very easy. The angle subtended by these celestial objects on telescopes is quite small. Therefore, we need large apertures to improve the resolving power of a telescope. We can prove this using Rayleigh’s criterion. The resolving power for a telescope is inversely proportional to the angular separation between two distant objects. Rayleigh’s formula is generally used for evaluating the resolving power of the telescope.
According to Rayleigh’s resolving power of optical instruments formula, the angular separation between two distant objects can be given as:

\(\Delta \theta = 1.22\frac{\lambda }{d}\)
The resolving power is the reciprocal of this angular separation:
Resolving Power \( = \frac{1}{{\Delta \theta }} = \frac{1}{{1.22\frac{\lambda }{d}}} = \frac{d}{{1.22\lambda }}\)
Thus, the higher will be the value of \(d\) or aperture diameter, the better will be the resolution of the given telescope. The desired level of resolution can be obtained using astronomical telescopes that usually have mirrors of large diameters, often \(10\;\rm{m}\). Using large wavelengths will reduce the telescope’s resolving power, which is why radio and microwave telescopes require larger mirrors.

Resolving Power of Optical Microscope

The resolving power is inversely proportional to the distance between the two objects when it comes to microscopes. In 1873, Ernst Abbe gave the formula called Abbe’s criterion. Using Abbe’s criterion, a microscope’s resolving power can be easily evaluated. According to his criterion, the resolution \(\Delta d\) (smallest resolvable distance between two objects) of a microscope varies with the angular aperture as:
\(\Delta d = \frac{\lambda }{{2n\sin \theta }}\)

Note: Here, we measure the resolution in terms of distance and not the angular separation as was done previously. Mathematically, resolving power of optical microscope can be evaluated as:
Resolving Power \( = \frac{1}{{\Delta d}} = \frac{{2n\sin \theta }}{\lambda }\)
Here,
\(n\): is the refractive index of the medium between object and aperture
\(\theta\): is the half-angle of the lens, which depends on the focal length and diameter of the lens.
\(\lambda\): wavelength of light
The numerical aperture \((NA)\) of the microscope is the product \(n \sin \theta\)
For better resolution:

1. \( \sin \theta\) should be large, and it can be achieved by keeping the specimen close to the objective lens.
2. A medium of a high refractive index may be used, and that is why we use oil immersion microscopes because oil has a higher refractive index than air.
3. Use X-rays and gamma rays because they have a smaller wavelength.

Summary

The resolving power of an optical instrument is defined as the power of the optical device to distinguish between two objects that are close together and produce distinct images of the two objects.
The resolution of an optical instrument is defined as the shortest distance between two points on a specimen that can be distinguished by the observer’s eye or the camera’s lens installed in the optical device.
The minimum angular separation between two point sources so that their images are distinguishable when viewed from an optical device is calculated using Rayleigh’s criterion. Mathematically, it can be given as \(\sin \theta = \frac{{1.22\lambda }}{D}\).

Resolving power of a telescope \( = \frac{d}{{1.22\lambda }}\)
Resolving power of a resolving power of optical microscope \( = \frac{{2n\sin \theta }}{\lambda }\)

FAQs on Resolving Power of Optical Instruments

Following are the frequently asked questions on the resolving power of optical instruments:

Q.1. Define resolution.
Ans: Resolution can be defined as the precision with which an optical device can record or measure a specimen’s variables under observation.

Q.2. Give the expression for Rayleigh’s criterion of resolution.
Ans: According to Rayleigh’s criterion:
\(\sin \theta = \frac{{1.22\lambda }}{D}\).
Where, \(\theta\): the angular separation between the objects measured in radians, \(\lambda\): the wavelength of light and \(D\): the diameter of the aperture.

Q.3. What is the resolving power of an optical device?
Ans: The resolving power of an optical instrument is defined as the power of the optical device to distinguish between two objects that are close together and produce distinct images of the two objects.

Q.4. What is the resolving power of a telescope?
Ans: Resolving power of a telescope \( = \frac{d}{{1.22\lambda }}\).

Q.5. How can we improve the resolution of a resolving power of optical microscope?
Ans: Resolution can be improved:
1. By keeping the specimen close to the objective lens.
2. By using a medium of higher refractive index.
3. By using light of a smaller wavelength.

Learn All the Concepts on Microscopes

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