RESOLVING POWER & DIFFRACTION LIMITS

From Circular Aperture to Microscope Resolution

1. Diffraction by a Circular Aperture (Foundation)

For a circular aperture of radius \( a \), the Fraunhofer diffraction intensity is given by:

\[ I(\theta) = I_0 \left[ \frac{2J_1(v)}{v} \right]^2 \]

where the dimensionless parameter \( v \) is:

\[ v = ka \sin \theta = \frac{2\pi}{\lambda} a \sin \theta \]

Here, \( J_1(v) \) is the first-order Bessel function of the first kind.

Condition for the First Minimum

The first diffraction minimum (dark ring) occurs when:

\[ J_1(v) = 0 \]

The first zero of the Bessel function is mathematically determined as:

\[ v = 3.832 \]

Substituting this back into the expression for \( v \):

\[ k a \sin \theta = 3.832 \] \[ \frac{2\pi}{\lambda} a \sin \theta = 3.832 \] \[ \sin \theta = \frac{3.832}{2\pi} \frac{\lambda}{a} \] \[ \sin \theta = 1.22 \frac{\lambda}{2a} \]

Since the diameter of the aperture is \( D = 2a \), we get:

\[ \sin \theta = 1.22 \frac{\lambda}{D} \]

For small angles (paraxial approximation), \( \sin \theta \approx \theta \):

\[ \theta = 1.22 \frac{\lambda}{D} \]

This \( \theta \) represents the angular radius of the central bright spot, known as the Airy Disk.

2. Rayleigh Criterion for Resolution

Two incoherent point sources are said to be just resolved when the central maximum of the diffraction pattern of one source falls exactly on the first minimum of the diffraction pattern of the other.

Thus, the minimum angular separation required for resolution is:

\[ \Delta \theta_{\min} = 1.22 \frac{\lambda}{D} \]

This defines the fundamental limit of resolution for any optical system with a circular aperture (e.g., telescopes, cameras).

3. Applying to a Microscope

Consider two object points \( P \) and \( Q \) separated by a linear distance \( y \). Let the lens aperture be \( D \).

From the geometry of the image formation (Fig 4.16), for small angles:

\[ \sin \theta' \approx \tan \theta' = \frac{y'}{D/2} \] \[ \sin \theta' = \frac{2y'}{D} \]

Using the condition for just resolution derived earlier:

\[ \sin \theta' = 1.22 \frac{\lambda'}{D} \]

where \( \lambda' = \lambda_0 / n' \) is the wavelength inside the image medium (refractive index \( n' \)).

Equating the two expressions:

\[ \frac{2y'}{D} = 1.22 \frac{\lambda_0}{n' D} \]

Canceling \( D \) from both sides:

\[ y' = 0.61 \frac{\lambda_0}{n'} \]

4. Relating Object and Image Distances

We use the Sine Law (Abbe Sine Condition) which relates object space and image space:

\[ n' y' \sin f = n y \sin i \]

Assuming the paraxial approximation where \( \sin f \approx 1 \) (or using magnification ratios):

\[ n' y' = n y \sin i \]

Substitute the value of \( y' \) we found in step 3:

\[ n' \left( 0.61 \frac{\lambda_0}{n'} \right) = n y \sin i \] \[ 0.61 \lambda_0 = n y \sin i \]

Solving for the object separation \( y \):

\[ y = \frac{0.61 \lambda_0}{n \sin i} \]

5. Numerical Aperture (NA)

The term \( n \sin i \) is defined as the Numerical Aperture (NA) of the microscope objective. It characterizes the range of angles over which the system can accept light.

\[ \text{NA} = n \sin i \]

Thus, the final expression for the minimum resolvable distance (limit of resolution) is:

\[ \Delta y_{\min} = \frac{0.61 \lambda_0}{\text{NA}} \]

6. Resolving Power

The Resolving Power (RP) is defined as the reciprocal of the minimum resolvable distance:

\[ \text{RP} = \frac{1}{\Delta y_{\min}} = \frac{\text{NA}}{0.61 \lambda_0} \]

7. Physical Meaning & Implications

This result encodes the fundamental physics of diffraction:

• Larger Aperture: Increases \( \sin i \), increasing NA and thus Resolution.
• Refractive Index: Using a medium with higher \( n \) (like oil) increases NA.
• Shorter Wavelength: Decreasing \( \lambda \) improves resolution directly.

This confirms that resolution is fundamentally a wave interference limit:

\[ \text{Resolution} \propto \frac{\lambda}{\text{Aperture Size}} \]

8. Why Oil Immersion Works

In a dry objective (air), \( n = 1 \). In an oil immersion objective, transparent oil is used with \( n \approx 1.5 \).

\[ \text{NA} \to 1.5 \sin i \]

This effectively increases the numerical aperture and improves the resolution by approximately 50% compared to air.

9. Comparison with Electron Microscope

The formula \( \Delta y_{\min} \propto \lambda \) explains why electron microscopes are so powerful.

For electrons accelerated at 100 keV, the de-Broglie wavelength is:

\[ \lambda \approx 0.003 \text{ nm} \]

This is thousands of times smaller than visible light (\( \sim 500 \text{ nm} \)). Consequently, the diffraction limit becomes extremely small, allowing electron microscopes to achieve atomic resolution.