Resolving Power and Rayleigh Criterion

1. From Point Image to Diffraction Pattern

In ray optics, we assume a point object forms a point image. However, wave optics shows us this is an idealization. When light passes through a circular aperture (like the pupil of your eye, or the objective lens of a telescope or microscope), diffraction occurs. The resulting pattern isn't a simple spot but a bright central circle, known as the Airy disc, surrounded by faint concentric rings.

The size of this Airy disc is critical. For a circular aperture of diameter $D$, the first dark ring (the first minimum) appears at an angle $\theta$ from the center, given by:

$\theta =1.22\frac{\lambda }{D}$

Where $\lambda$ is the wavelength of the light. Notice the factor of 1.22, which arises from the circular geometry of the aperture, making it different from the formula for a single slit ($\sin \theta =\lambda /a$).

The following video provides an excellent visual introduction to this problem, starting with the car headlight example and explaining how we arrive at a criterion to tell if two objects are distinguishable.

2. The Rayleigh Criterion: Defining the Limit

As you saw in the video, Lord Rayleigh proposed a clear and widely accepted standard to define when two objects are "just resolved."

Rayleigh's Criterion: Two images are just resolvable when the center of the diffraction pattern (central maximum) of one image falls directly on the first minimum of the diffraction pattern of the other.

From this criterion, the minimum angular separation, ${\theta }_{min}$, between two point objects that can just be distinguished is equal to the angle of the first minimum of the Airy pattern.

${\theta }_{min}=1.22\frac{\lambda }{D}(\text{in radians})$

This angle ${\theta }_{min}$ is known as the limit of resolution. A smaller limit of resolution means the instrument can distinguish between objects that are closer together, which means it has better resolving capabilities.

3. Resolving Power

The term that quantifies this capability is resolving power (RP). It is defined as the reciprocal of the limit of resolution.

$\text{Resolving Power (RP)}=\frac{1}{\text{Limit of Resolution}}$

• A small limit of resolution means high resolving power.
• A large limit of resolution means low resolving power.

Our goal in designing optical instruments is often to maximize their resolving power. Let's see how this applies to telescopes and microscopes. For a more detailed textual reference on these concepts, you can consult the following resource.

4. Application to Optical Instruments

For your NEET exam, you need to be proficient in applying these concepts to telescopes and microscopes.

A. Resolving Power of a Telescope

Telescopes are used to view objects that are very far away. Therefore, we are concerned with their ability to resolve two objects with a very small angular separation.

• Limit of Resolution (Angular): This is the minimum angular separation ${\theta }_{min}$.
  ${\theta }_{min}=1.22\frac{\lambda }{D}$
• Resolving Power:
  $R{P}_{\text{telescope}}=\frac{1}{{\theta }_{min}}=\frac{D}{1.22\lambda }$

Here, $D$ is the diameter of the telescope's objective lens or mirror.

Key Implications for Telescopes:

1. Larger Aperture is Better: Resolving power is directly proportional to the diameter $D$. This is the primary reason why research telescopes are built with massive mirrors—to collect more light and, crucially, to resolve finer details.
2. Shorter Wavelength is Better: Resolving power is inversely proportional to the wavelength $\lambda$. A telescope can resolve objects better when observing in blue or ultraviolet light compared to red or infrared light.

Let's see this in action with a worked example.

B. Resolving Power of a Microscope

Microscopes are used to view objects that are very close to the lens. Therefore, we are concerned with their ability to resolve two points separated by a small linear distance.

• Limit of Resolution (Linear): This is the minimum distance $d_{min}$ between two points that can be seen as separate. The formula, derived from the Rayleigh criterion, is:
  $d_{min}=\frac{1.22\lambda }{2n\sin \beta }$
• Resolving Power:
  $R{P}_{\text{microscope}}=\frac{1}{d_{min}}=\frac{2n\sin \beta }{1.22\lambda }$

Let's break down the terms in the denominator:

• $\lambda$: Wavelength of light used.
• $n$: Refractive index of the medium between the objective lens and the object.
• $\beta$: The half-angle of the cone of light that can enter the lens from the object.
• Numerical Aperture (NA): The term $n\sin \beta$ is called the Numerical Aperture. A higher NA means the lens can gather light from a wider angle, leading to better resolution.
  $NA=n\sin \beta$
  So, the formulas can be written as: $d_{min}=\frac{0.61\lambda }{NA}$ and $R{P}_{\text{microscope}}=\frac{NA}{0.61\lambda }$ (using $1.22/2=0.61$).

Key Implications for Microscopes:

1. Use Oil Immersion: To increase resolving power, we can increase the refractive index $n$. This is done by placing a drop of special oil ($n\approx 1.5$) between the objective lens and the slide. This increases the NA.
2. Use Shorter Wavelengths: Just like with telescopes, using shorter wavelength light (e.g., blue or UV light) improves resolution. This is the principle behind UV microscopes. Electron microscopes take this to the extreme by using electrons, which have a much smaller effective wavelength than visible light, allowing for vastly superior resolution.

The following video provides a NEET-centric walkthrough of these formulas and solves a typical problem.

Conclusion

Today we confronted a fundamental limit of optics. Because of the wave nature of light, diffraction prevents us from seeing infinitely fine detail. We quantified this limit using the Rayleigh criterion and derived the formulas for the resolving power of telescopes and microscopes, your key tools for solving NEET problems on this topic.

Key Takeaways:

• Diffraction limits resolution: Point objects are imaged as Airy discs, not points.
• Rayleigh Criterion: Two objects are "just resolved" when the central maximum of one pattern lies on the first minimum of the other.
• Limit of Resolution (Telescope): Minimum resolvable angle is ${\theta }_{min}=1.22\lambda /D$. Resolving power increases with a larger objective diameter ($D$).
• Limit of Resolution (Microscope): Minimum resolvable distance is $d_{min}=0.61\lambda /NA$. Resolving power increases with a higher numerical aperture ($NA$) and by using shorter wavelength light ($\lambda$).

Preview of the Next Lesson:
We have now covered the wave phenomena of interference and diffraction. This concludes our study of how light intensity is redistributed in space. In our next lesson, we will investigate a completely different property of light waves that cannot be explained by treating them as simple scalar waves: polarization. We will learn what it means for light to be polarized and explore the methods used to produce and analyze polarized light.