Resolving Power Of Microscope And Telescope

Resolving Power Of Microscope And Telescope

Edited By Vishal kumar | Updated on Jul 02, 2025 06:05 PM IST

The resolving power of microscopes and telescopes is a critical concept in optics, determining their ability to distinguish fine details and separate closely spaced objects. For microscopes, high resolving power is essential in fields such as biology and materials science, allowing scientists to observe cellular structures and nano materials with clarity. In astronomy, the resolving power of telescopes enables the detailed observation of distant celestial bodies, revealing features of planets, stars, and galaxies. In everyday life, the principles of resolving power are applied in devices like cameras and binoculars, enhancing our ability to capture and appreciate the intricate details of our surroundings. This article explores the factors influencing the resolving power of these instruments and their practical significance.

This Story also Contains
  1. Resolving Power of Optical Instruments
  2. Solved Examples Based on Resolving Power of Optical Instruments
  3. Summary
Resolving Power Of Microscope And Telescope
Resolving Power Of Microscope And Telescope

Resolving Power of Optical Instruments

The resolving power of an optical instrument is its ability to resolve or separate the images of two nearby point objects so that they can be distinctly seen. In reference to a microscope, the minimum distance between two lines at which they are just distinct is called the resolving limit (RL) and its reciprocal is called Resolving power (RP).

Resolving Power of Microscope

The resolving power of a microscope is a measure of its ability to distinguish between two points that are close together. It is determined by the wavelength of light used and the numerical aperture of the microscope lens. A higher resolving power allows scientists and researchers to observe fine details and structures in biological specimens and materials, revealing insights at the cellular and even molecular levels. This capability is crucial in fields like biology, medicine, and materials science, where understanding minute details can lead to significant discoveries and advancements.

In a microscope, the minimum distance between two lines at which they are just distinct is called the Resolving limit (RL) and its reciprocal is called Resolving power (RP)

R.L. $=\frac{\lambda}{2 \mu \sin \theta}$ and R.P. $=\frac{2 \mu \sin \theta}{\lambda} \Rightarrow R . P . \propto \frac{1}{\lambda}$

$\lambda=$ Wavelength of light used to illuminate the object,

$\mu=$ The refractive index of the medium between object and objective,

$\theta=$ Half angle of the cone of light from the point object

Rayleigh's criterion for the diffraction limit to resolution states that two images are just resolvable when the centre of the diffraction pattern of one is directly over the first minimum of the diffraction pattern of the other. We can use Rayleigh’s criterion to determine the resolving power.

$\Delta \theta=1.22 \frac{\lambda}{d}$
Resolving power $=\frac{1}{\Delta \theta}=\frac{d}{1.22 \lambda}$

Thus, the higher the diameter d, the better the resolution. The best astronomical optical telescopes have mirror diameters as large as 10m to achieve the best resolution. Also, larger wavelengths reduce the resolving power and consequently, radio and microwave telescopes need larger mirrors.

Therefore, from the above expression, we can see that

  • As the R.P. is directly proportional to the refractive index (n), The R.P. will increase when n increases.
  • As the R.P is inversely proportional to the wavelength (λ), So R.P will decrease when λ increases.
  • When the diameter of the objective is increased, θ increases. Hence, sinθ also increases.
  • As the R.P is directly proportional to the sinθ, The R.P will increase when the diameter of the objective increases.
  • As the R.P. is independent of the focal length of the lens, R.P. will remain unchanged when the focal length increases.
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Resolving Power of Telescope

The resolving power of a telescope is its ability to distinguish between two closely spaced objects in the sky, such as stars or planetary details. It depends on the diameter of the telescope's aperture and the wavelength of light being observed. A telescope with high resolving power can reveal fine details of distant celestial bodies, such as the surface features of planets, the structure of galaxies, and the separation of binary star systems.

In telescopes, very close objects such as binary stars or individual stars of galaxies subtend very small angles on the telescope. To resolve them we need very large apertures. The resolving power of a telescope is defined as the reciprocal of the smallest angle subtended at the objective lens of the telescope by two point objects which can be just distinguished as separate. We can use Rayleigh’s to determine the resolving power. The angular separation between two objects must be

$\begin{gathered}\Delta \theta=1.22 \frac{\lambda}{d} \\ \text { Resolving power }=\frac{1}{\Delta \theta}=\frac{d}{1.22 \lambda}\end{gathered}$

where,

$\lambda=$ Wavelength of light used to illuminate the object

d = is the critical width of the rectangular slit for just the resolution of two slits or objects.

$\theta=$ Half angle of the cone of light from the point object

Thus, the higher the diameter d, the better the resolution. The best astronomical optical telescopes have mirror diameters as large as 10m to achieve the best resolution. Also, larger wavelengths reduce the resolving power and consequently, radio and microwave telescopes need larger mirrors.

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Solved Examples Based on Resolving Power of Optical Instruments

Example 1: The value of the numerical aperture of the objective lens of a microscope is 1.25. If light of wavelength 5000 A is used, the minimum separation between two points, to be seen as distinct, will be :

1) $0.24 \mu \mathrm{m}$
2) $0.38 \mu \mathrm{m}$
3) $0.12 \mu \mathrm{m}$
4) $0.48 \mu \mathrm{m}$

Solution:

Resolving power of the microscope

$
R=\frac{2 \mu \sin \Theta}{\lambda}
$
$\mu=$ Refractive index
$\lambda=$ wavelength of light used
The minimum distance between two lines $\varepsilon=0.61 \times \frac{\lambda}{N . A}$.
where N.A is a numerical aperture
$
\begin{aligned}
& \Rightarrow \varepsilon=0.61 \times \frac{5000 \times 10^{-10}}{1.25} \\
& \Rightarrow \varepsilon=0.24 \mu \mathrm{m}
\end{aligned}
$

Hence, the answer is the option (1).

Example 2: If $R_{p=}$ Resolving power of optical instruments and $R_l=$ Resolving limit of optical instruments then $R_p$ and $R_l$ are related as

1) $R_p \propto R_l$
2) $R_p \alpha \frac{1}{R_l}$
3) $R_p \propto R_l^3$
4) $R_p \propto R_l^2$

Solution:

Resolving power of optical instruments

The resolving power of an optical instrument is its ability to resolve or separate the images of two nearby point objects so that they can be distinctly seen.

About a microscope, the minimum distance between two lines at which they are just distinct is called the resolving limit (RL), and its reciprocal is called Resolving power (RP).

I.e $R_p \alpha \frac{1}{R_l}$

Hence, the answer is the option (2).

Example 3: The wavelengths of light used in an optical instrument are $\lambda_1=4000 A^0$ and $\lambda_2=5000 A^0$ then the ratio of their respective resolving powers (corresponding to $\lambda_1$ and $\lambda_2$) is

1) 16:25

2) 9:1

3) 4:5

4) 5:4

Solution:

Resolving Power $\alpha 1 / \lambda$
Given,
$
\begin{aligned}
& \lambda_1=4000^{\circ} \mathrm{A} \\
& \lambda_2=5000^{\circ} \mathrm{A}
\end{aligned}
$
$
\begin{aligned}
& R P_{\lambda 1} / R P_{\lambda 2}=\lambda 2 / \lambda 1 \\
& \Rightarrow R P_{\lambda 1} / R P_{\lambda 2}=5000 / 4000=5 / 4 \\
& \Rightarrow R P_{\lambda 1}: R P_{\lambda 2}=5: 4
\end{aligned}
$

Hence, the answer is the option (4).

Example 4: Assuming the human pupil to have a radius of 0.25 cm and a comfortable viewing distance of 25 cm, the minimum separation between two objects that the human eye can resolve at 500 nm wavelength is :

1) $1 \mu \mathrm{m}$
2) $30 \mu \mathrm{m}$
3) $100 \mu \mathrm{m}$
4) $300 \mu \mathrm{m}$

Solution:

The radius of human solution pupil=0.25cm or diameter =0.5cm=5x10-3m

$
\begin{aligned}
& \lambda=500 \mathrm{~mm}=5 \times 10^{-7} \mathrm{~m} \\
& \text { Since } \sin \theta=\frac{1.22 \lambda}{d}=\frac{1.22 \times 5 \times 10^{-7}}{5 \times 10^{-3}}=1.22 \times 10^{-4}
\end{aligned}
$

The distance of comfortable viewing $=25 \mathrm{~cm}$
Let $x$ be the minimum separation between the two objects that the human eye can resolve them
$
\begin{aligned}
& \quad \sin \theta=\tan \theta=\frac{x}{0} \text { or } 0 \tan \theta=x \\
& x=(25 \mathrm{~cm}) \times 1.22 \times 10^{-4} \\
& =3 \times 10^{-5} \mathrm{~m}=30 \mu \mathrm{m}
\end{aligned}
$

Hence, the answer is the option (2).

Example 5: Two point white dots are 1 mm apart on black paper. They are viewed by an eye with a pupil diameter of 3 mm. Approximately, what is the maximum distance at which these dots can be resolved by the eye? [ Take wavelength of light = 500 mm]

1) 6 mm
2) 3 mm
3) 5 mm
4) 1 mm

Solution:

Radius of diffraction disc

$
R=1.22 \frac{\lambda D}{b}
$
wherein
$D=$ Distance of the screen from the hole
$b=$ slit width
Resolution limit $=\frac{1.22 \lambda}{d}$
Again resolution limit
$
\begin{aligned}
& \text { Again resolution limit }=\sin \Theta=\Theta=\frac{y}{D} \\
& \therefore \quad \frac{y}{D}=\frac{1.22 \lambda}{d}
\end{aligned}
$
or $D=\frac{\lambda d}{1.22 \lambda}$
$
\xrightarrow[D]{\stackrel{1}{\longleftrightarrow}}{ }^{\uparrow}{ }^y \text { or } \quad D=\frac{\left(10^3\right) \times\left(3 \times 10^{-3}\right)}{(1.22) \times\left(5 \times 10^{-7}\right)}=\frac{30}{6.1} \approx 5 \mathrm{~mm}
$

Hence, the answer is the option (3).

Summary

The resolving power of optical instruments like microscopes and telescopes is crucial for distinguishing fine details in closely spaced objects. For microscopes, high resolving power is essential in fields like biology and materials science to observe minute structures clearly. In telescopes, resolving power allows detailed observation of distant celestial bodies, aiding in astronomical studies. Factors influencing resolving power include the wavelength of light and the aperture size of the instrument.

Frequently Asked Questions (FAQs)

1. What is resolving power in optics?
Resolving power is the ability of an optical instrument, like a microscope or telescope, to distinguish between two closely spaced objects or points. It determines the level of detail an instrument can reveal. Higher resolving power means the instrument can distinguish finer details.
2. How does wavelength affect the resolving power of a microscope?
Wavelength is inversely proportional to resolving power. Shorter wavelengths of light allow for higher resolving power, meaning finer details can be observed. This is why electron microscopes, which use electron beams with very short wavelengths, can achieve much higher resolution than light microscopes.
3. What is the Rayleigh criterion?
The Rayleigh criterion is a standard used to determine the theoretical resolving power of an optical instrument. It states that two point sources are just resolvable when the center of the diffraction pattern of one falls on the first minimum of the diffraction pattern of the other. This creates a dip in intensity between the two peaks of about 26%.
4. How does aperture size affect resolving power?
Aperture size is directly proportional to resolving power. A larger aperture allows more light to enter the optical system, reducing diffraction effects and improving resolution. This is why larger telescope mirrors or microscope objective lenses generally provide better resolution.
5. Why can't we infinitely increase resolving power by using shorter wavelengths?
While shorter wavelengths do increase resolving power, practical limitations exist. For light microscopes, the shortest usable wavelengths are in the ultraviolet range. Beyond this, special optics are needed. Additionally, very short wavelengths like X-rays are difficult to focus using conventional lenses.
6. What is the formula for the resolving power of a microscope?
The formula for the resolving power of a microscope is d = λ / (2 * NA), where d is the minimum resolvable distance, λ is the wavelength of light used, and NA is the numerical aperture of the objective lens. This formula shows how wavelength and NA affect resolution.
7. What is the Abbe diffraction limit?
The Abbe diffraction limit, formulated by Ernst Abbe, describes the fundamental maximum resolution of a microscope. It states that the smallest resolvable distance is approximately half the wavelength of the light used, divided by the numerical aperture of the objective lens. This limit arises from the wave nature of light and diffraction effects.
8. What is the difference between Rayleigh and Sparrow resolution criteria?
Both Rayleigh and Sparrow criteria define resolution limits, but slightly differently. The Rayleigh criterion requires a 26% dip in intensity between two point sources, while the Sparrow criterion only requires that the combined intensity distribution of two point sources shows a flat top rather than a single peak. The Sparrow criterion generally allows for slightly higher theoretical resolution.
9. What is the difference between resolution and magnification?
Resolution refers to the ability to distinguish fine details, while magnification is simply making an image appear larger. High magnification without good resolution will result in a blurry, enlarged image. Resolution determines the useful limit of magnification.
10. What is the relationship between f-number and resolving power in photography?
The f-number is inversely related to resolving power in photography. A lower f-number (larger aperture) allows more light and generally provides better resolution due to reduced diffraction effects. However, this also reduces depth of field, which can affect overall image sharpness.
11. How does the use of synthetic aperture techniques improve resolving power in radar systems?
Synthetic aperture radar (SAR) improves resolving power by simulating a much larger antenna aperture. As the radar platform moves, it takes multiple measurements that are combined to create a high-resolution image. This technique allows for much higher resolution than would be possible with a static antenna of the same physical size.
12. How does numerical aperture (NA) relate to resolving power in microscopy?
Numerical aperture is directly related to resolving power in microscopy. Higher NA values indicate better light-gathering ability and higher resolving power. The formula for resolution includes NA in the denominator, so increasing NA improves resolution.
13. How does immersion oil improve resolving power in microscopy?
Immersion oil increases the refractive index between the objective lens and the specimen, allowing for a higher numerical aperture. This higher NA leads to improved resolving power according to the resolution formula. Immersion oil also reduces light loss due to reflection at the air-glass interface.
14. Why do astronomers prefer larger telescope mirrors?
Larger telescope mirrors improve resolving power by increasing the aperture of the telescope. According to the Rayleigh criterion, the angular resolution of a telescope is proportional to the wavelength of light and inversely proportional to the diameter of the aperture. Larger mirrors can resolve finer angular details in celestial objects.
15. What limits the resolving power of the human eye?
The resolving power of the human eye is limited by several factors: the size of the pupil (acting as the aperture), the wavelength of visible light, and the spacing of photoreceptors in the retina. The eye's resolving power is typically about 1 arcminute, or about 0.017 degrees.
16. How does atmospheric turbulence affect telescope resolving power?
Atmospheric turbulence, often called "seeing," limits the effective resolving power of ground-based telescopes. It causes rapid variations in the refractive index of air, leading to distortions in the wavefront of light from celestial objects. This is why space telescopes can achieve higher resolution despite having smaller mirrors than some ground-based telescopes.
17. What is the diffraction limit?
The diffraction limit is the fundamental maximum resolving power of an optical system due to the wave nature of light. It represents the smallest angular separation at which two point sources can be distinguished. The diffraction limit depends on the wavelength of light and the aperture size of the optical system.
18. How does adaptive optics improve telescope resolving power?
Adaptive optics systems use deformable mirrors to correct for atmospheric distortions in real-time. By measuring the wavefront distortions and rapidly adjusting the mirror shape, these systems can significantly improve the effective resolving power of ground-based telescopes, approaching their theoretical diffraction limit.
19. What is super-resolution microscopy?
Super-resolution microscopy refers to various techniques that allow optical microscopes to exceed the diffraction limit and achieve resolution beyond what is theoretically possible with conventional light microscopy. These methods often involve clever illumination strategies or the use of fluorescent markers to distinguish nearby molecules.
20. How does electron microscopy achieve higher resolution than light microscopy?
Electron microscopes use beams of electrons instead of light. Electrons have much shorter wavelengths than visible light, allowing for much higher resolving power according to the resolution formula. This enables electron microscopes to resolve details at the atomic scale, far beyond the capabilities of light microscopes.
21. How does pixel size affect the resolving power of digital imaging systems?
Pixel size in digital imaging systems can limit resolving power if it's larger than the optical resolution of the system. The Nyquist criterion states that to fully capture the resolution of an optical system, the pixel size should be at least half the size of the smallest resolvable feature. Smaller pixels generally allow for higher resolution, up to the optical limit.
22. How does phase contrast microscopy improve image resolution?
Phase contrast microscopy doesn't directly improve resolving power, but it enhances the visibility of transparent specimens by converting phase shifts in light passing through the sample into amplitude changes. This makes previously invisible structures visible, effectively improving the practical resolution of the microscope for certain types of samples.
23. What is the role of coherence in determining resolving power?
Coherence in light sources can affect resolving power. Highly coherent light, like laser light, can produce sharper diffraction patterns, potentially improving resolution. However, it can also introduce interference artifacts. In some cases, partially coherent illumination provides an optimal balance between resolution and image quality.
24. How does the Strehl ratio relate to resolving power?
The Strehl ratio is a measure of the quality of an optical system's performance compared to a perfect system. A higher Strehl ratio indicates better optical quality and typically correlates with better resolving power. A Strehl ratio of 1 represents a perfect optical system, while 0.8 is often considered diffraction-limited performance.
25. What is the significance of the Airy disk in understanding resolving power?
The Airy disk is the central bright spot in the diffraction pattern produced by a circular aperture. Its size is crucial in determining resolving power, as two point sources are considered resolvable when their Airy disks are sufficiently separated. The Rayleigh criterion is based on the separation of these Airy disks.
26. How does polarization affect resolving power?
Polarization itself doesn't directly affect resolving power, but polarization techniques can be used to enhance contrast and reduce unwanted scattering, which can improve the effective resolution of an optical system. Some super-resolution techniques, like STORM microscopy, use polarization effects to achieve higher resolution.
27. What is the impact of lens aberrations on resolving power?
Lens aberrations, such as spherical aberration, coma, and astigmatism, can significantly reduce the effective resolving power of an optical system. These aberrations cause light rays to focus imperfectly, blurring the image and limiting the ability to distinguish fine details. High-quality, well-corrected lenses are crucial for achieving optimal resolving power.
28. How does the concept of depth of field relate to resolving power?
Depth of field and resolving power are related but distinct concepts. A narrow depth of field (achieved with a large aperture) can provide high resolving power in the focal plane but poor resolution outside it. Conversely, a wide depth of field (small aperture) provides more consistent resolution throughout the image but may reduce overall resolving power due to diffraction effects.
29. What is the difference between angular and linear resolution?
Angular resolution refers to the ability to distinguish between two points separated by a small angle, typically used in astronomy. Linear resolution refers to the ability to distinguish between two points separated by a small distance, often used in microscopy. Both are measures of resolving power but apply to different scales and contexts.
30. How does sample preparation affect the achievable resolution in microscopy?
Sample preparation can significantly impact achievable resolution. Proper fixation, staining, and mounting techniques can enhance contrast and preserve fine structures, allowing for better resolution. Conversely, poor preparation can introduce artifacts, obscure details, or damage the sample, limiting the effective resolution regardless of the microscope's capabilities.
31. What is the role of contrast in determining effective resolving power?
Contrast is crucial for effective resolution. Even if an optical system can theoretically resolve fine details, these details may not be visible if there isn't sufficient contrast between the object and its background. Techniques that enhance contrast, such as phase contrast or differential interference contrast microscopy, can improve the practical resolving power of a system.
32. How does the Nyquist-Shannon sampling theorem apply to resolving power in digital imaging?
The Nyquist-Shannon sampling theorem states that to accurately represent a signal, it must be sampled at least twice the highest frequency present. In digital imaging, this means the pixel size should be at least half the size of the smallest detail to be resolved. Oversampling (using smaller pixels) can improve the digital representation of the image but won't improve optical resolution beyond the diffraction limit.
33. What is the relationship between resolving power and the point spread function (PSF)?
The point spread function describes how a point source of light is spread out by an optical system. The width of the PSF is directly related to the system's resolving power. A narrower PSF indicates better resolving power, as it means the system can more accurately reproduce point sources without blurring them together.
34. How does the use of multiple apertures, as in interferometry, affect resolving power?
Multiple apertures, as used in interferometry, can significantly improve resolving power. By combining light from multiple telescopes or microscope objectives, the effective aperture size is increased. This allows for much higher angular resolution than would be possible with a single aperture of the same size as any individual element.
35. What is the impact of noise on resolving power in imaging systems?
Noise can significantly degrade the effective resolving power of an imaging system. Even if the optical resolution is high, noise can obscure fine details and make it difficult to distinguish between closely spaced objects. Signal-to-noise ratio is therefore an important consideration in practical applications of high-resolution imaging.
36. How does the concept of modulation transfer function (MTF) relate to resolving power?
The modulation transfer function describes how well an optical system preserves contrast across different spatial frequencies. It's closely related to resolving power, as a system with a higher MTF at high spatial frequencies will be able to resolve finer details. The MTF provides a more complete picture of system performance than a single resolution value.
37. How does the use of structured illumination affect resolving power in microscopy?
Structured illumination microscopy (SIM) can improve resolving power by projecting a known pattern onto the sample. By analyzing how this pattern is distorted by the sample's structure, information about details smaller than the diffraction limit can be computationally extracted, effectively doubling the resolution compared to conventional microscopy.
38. What is the relationship between resolving power and the Abbe sine condition?
The Abbe sine condition is a design principle for optical systems that helps minimize off-axis aberrations. Lenses that satisfy this condition can maintain good resolving power across the entire field of view, not just at the center. This is particularly important for wide-field microscopy and high-quality camera lenses.
39. How does the use of confocal techniques affect resolving power in microscopy?
Confocal microscopy improves resolving power, particularly in the axial (depth) direction, by using a pinhole to reject out-of-focus light. This allows for better contrast and the ability to create sharp 3D images. While it doesn't overcome the diffraction limit, it can provide better practical resolution, especially for thick samples.
40. What is the impact of chromatic aberration on resolving power?
Chromatic aberration occurs when different wavelengths of light focus at different points, leading to color fringing and reduced sharpness. This can significantly degrade resolving power, especially in systems using white light. Achromatic and apochromatic lenses are designed to minimize this effect and maintain high resolution across a range of wavelengths.
41. How does the concept of Fourier optics relate to understanding resolving power?
Fourier optics provides a powerful framework for understanding resolving power by treating image formation as a process of spatial frequency filtering. In this view, the optical system acts as a low-pass filter, with the cutoff frequency determined by the system's resolving power. This approach helps in analyzing complex optical systems and in developing super-resolution techniques.
42. What is the role of deconvolution in improving effective resolving power?
Deconvolution is a computational technique that can improve the effective resolving power of an imaging system by reversing the blurring effects of the point spread function. While it doesn't increase the fundamental optical resolution, it can significantly enhance image clarity and the ability to distinguish fine details, especially in 3D microscopy.
43. How does the use of near-field techniques affect resolving power?
Near-field optical techniques, such as near-field scanning optical microscopy (NSOM), can achieve resolution beyond the diffraction limit by exploiting evanescent waves very close to the sample surface. These techniques can achieve resolutions on the order of tens of nanometers, far surpassing conventional far-field microscopy.
44. What is the significance of the Dawes limit in astronomical observations?
The Dawes limit is an empirical measure of the resolving power of a telescope, specifically for separating close double stars. It's given by 4.56/D arcseconds, where D is the telescope aperture in inches. While similar to the Rayleigh criterion, the Dawes limit is based on human visual perception and is often used as a practical guide for amateur astronomers.
45. What is the relationship between resolving power and the Nyquist frequency in imaging?
The Nyquist frequency is the highest spatial frequency that can be accurately represented in a digital image, equal to half the sampling frequency. For optimal resolving power, the optical system should be designed so that its resolution limit matches the Nyquist frequency of the detector. This ensures that the digital image captures all the detail resolved by the optics without introducing aliasing artifacts.
46. How does the use of quantum entanglement in imaging potentially affect resolving power?
Quantum entanglement-enhanced imaging techniques, still largely in the research phase, have the potential to surpass classical resolution limits. By using entangled photons, these methods can achieve higher sensitivity and potentially overcome certain aspects of the diffraction limit, promising significant improvements in resolving power for specific applications.
47. What is the impact of optical coherence tomography (OCT) on axial resolving power?
Optical coherence tomography significantly
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