RESOLVING POWER OF A MICROSCOPE

Optics & Optical Instruments | Detailed Derivation

1. Introduction and Definitions

In optical instruments, the primary function is often thought to be magnification. However, mere magnification is useless if the image lacks detail. The ability of an instrument to show two closely spaced objects as separate and distinct is called its Resolving Power.

Limit of Resolution \( d \):
The minimum linear distance between two point objects such that they are just resolved (seen as separate) by the optical system.

Resolving Power (R.P.):
It is defined as the reciprocal of the limit of resolution. It quantifies the ability of the microscope to distinguish fine details. \[ \text{Resolving Power} = \frac{1}{\text{Limit of Resolution}} = \frac{1}{d} \]

2. Principle: Rayleigh's Criterion

According to wave optics, the image of a point object formed by a lens is not a sharp point but a diffraction pattern consisting of a central bright spot (Airy Disc) surrounded by concentric dark and bright rings.

Lord Rayleigh's Criterion for Resolution states:

• Two point sources are just resolved if the central maximum of the diffraction pattern of one source falls exactly on the first minimum of the diffraction pattern of the other.
• If the distance is smaller than this, they appear as a single blur (Unresolved).
• If the distance is larger, they are clearly distinct (Well Resolved).

3. Mathematical Derivation

Consider two point objects \( O \) and \( O' \) separated by a small distance \( d \). They are placed just outside the focus of the microscope objective lens. The medium between the object and the lens has a refractive index \( n \).

Geometry of the Setup

• Let \( A \) and \( B \) represent the aperture (diameter) of the objective lens.
• \( O \) and \( O' \) are the two self-luminous point objects separated by distance \( d \).
• \( \theta \) (or \( \alpha \)) is the semi-vertical angle of the cone of light entering the objective from the object.

Path Difference Calculation

Light rays from \( O \) and \( O' \) travel through the lens to form diffraction patterns \( I \) and \( I' \). According to Rayleigh's criterion, for the images to be just resolved, the position of the central maximum of \( I' \) must coincide with the first minimum of \( I \).

Consider the extreme rays traveling from \( O \) and \( O' \) to the edge of the lens aperture \( A \) and \( B \). The path difference \( \Delta \) between the rays reaching the lens is:

\[ \Delta = O'A - OA \]

From the geometry (dropping a perpendicular from \( O' \) to \( OA \)), the extra path length traveled is:

\[ \Delta \approx d \sin \theta \]

However, we must account for the path difference on the other side of the lens (image space). A rigorous treatment using the properties of the Airy disc shows that the condition for the first minimum of a circular aperture is:

\[ \text{Path Difference} = 1.22 \lambda \]

(Note: For a slit, it is \( \lambda \), but for a circular aperture, the factor 1.22 arises from Bessel functions).

Equating the optical path difference (taking refractive index \( n \) into account):

\[ 2 n d \sin \theta = 1.22 \lambda \]

Why the factor 2? It accounts for the path difference introduced due to the extreme rays from both sides of the aperture relative to the central axis.

Final Formula for Limit of Resolution

Rearranging the equation for \( d \):

\[ d = \frac{1.22 \lambda}{2 n \sin \theta} \]

Where:

• \( d \): Limit of resolution (minimum separation).
• \( \lambda \): Wavelength of light used.
• \( n \): Refractive index of the medium between object and objective.
• \( \theta \): Semi-vertical angle of the acceptance cone.

4. Numerical Aperture (N.A.)

The term \( n \sin \theta \) is a crucial parameter in microscopy known as the Numerical Aperture (N.A.). It represents the light-gathering ability of the objective lens.

\[ \text{N.A.} = n \sin \theta \]

Substituting this into our formulas, we get the standard expressions used in microscopy:

Limit of Resolution: \[ d = \frac{1.22 \lambda}{2 (\text{N.A.})} \]
Resolving Power: \[ \text{R.P.} = \frac{1}{d} = \frac{2 (\text{N.A.})}{1.22 \lambda} \]

5. Factors Affecting Resolving Power

To obtain a high resolving power (i.e., to see smaller details, minimize \( d \)), we can manipulate the following factors:

1. Wavelength (\( \lambda \))

Since \( d \propto \lambda \), using a shorter wavelength decreases \( d \) and increases resolving power.

• Blue light resolves better than red light.
• UV Microscopes use ultraviolet light to achieve higher resolution than visible light microscopes.
• Electron Microscopes use electron beams with extremely small de-Broglie wavelengths (\( \lambda \approx 0.005 \) nm), offering resolving power thousands of times greater than optical microscopes.

2. Refractive Index (\( n \)) - Oil Immersion

Since \( \text{R.P.} \propto n \), increasing the refractive index of the medium between the object and the objective lens improves resolution.

• Dry Objective: Air is the medium (\( n \approx 1 \)). Maximum \( \sin \theta \approx 0.95 \), so N.A. \( < 1 \).
• Oil Immersion Objective: Cedar wood oil (\( n \approx 1.5 \)) is placed between the slide and the lens. This increases N.A. to approx 1.4 or 1.5, significantly improving resolving power.

3. Semi-vertical Angle (\( \theta \))

Increasing the aperture of the lens increases \( \theta \). However, \( \sin \theta \) cannot exceed 1 (\( \theta \) cannot exceed 90°). In practice, the maximum semi-angle is around 70°-75°.

6. Conclusion

The resolving power of a microscope is fundamentally limited by the diffraction of light. It is distinct from magnification; you can magnify a blurry image indefinitely, but you will not see more detail without increasing the resolving power. High-resolution microscopy relies on high Numerical Aperture lenses (often using oil immersion) and shorter wavelengths of illumination.