FRAUNHOFER DIFFRACTION BY IDENTICAL APERTURES

# FRAUNHOFER DIFFRACTION BY IDENTICAL APERTURES

From Two Slits to Diffraction Grating | Detailed Derivation

1. Diffraction by N Identical Apertures

1.1 Starting from the Diffraction Integral

From the general Fraunhofer diffraction integral, the field at a point \( P \) is:

\[ u(P) = C \iint P(\xi,\eta) e^{ik(l\xi + m\eta)} d\xi d\eta \]

where \( l = \sin\theta_x \) and \( m = \sin\theta_y \). This represents the Fourier transform of the aperture function.

1.2 N Apertures Summation

Let there be \( N \) identical apertures centered at \( O_1, O_2, \dots O_N \). The total field is the sum of integrals over each aperture:

\[ u(P) = C \sum_{n=1}^{N} \iint_{S_n} e^{ik(l\xi + m\eta)} d\xi d\eta \]

By shifting coordinates such that \( \xi = \xi' + \xi_n \) and \( \eta = \eta' + \eta_n \), we can separate the integral into a single-aperture term \( u_s(P) \) and a phase summation term:

\[ u(P) = u_s(P) \sum_{n=1}^{N} e^{ik(l\xi_n + m\eta_n)} \]

1.3 Intensity Distribution

The intensity \( I = |u(P)|^2 \) becomes:

\[ I = I_s \left| \sum_{n=1}^{N} e^{ik(l\xi_n + m\eta_n)} \right|^2 \]

Physical Interpretation:
Total Intensity = (Diffraction Envelope of Single Aperture) \( \times \) (Interference Factor of N Sources)

2. Two-Slit Diffraction

Consider two long narrow slits separated by distance \( d \). The centers are at \( \xi_1 = 0 \) and \( \xi_2 = d \).

The field becomes:

\[ u(P) = u_s(P) [1 + e^{ikld}] \]

Defining the phase difference \( \delta = k d \sin\theta = \frac{2\pi}{\lambda} d \sin\theta \):

\[ I = I_s |1 + e^{i\delta}|^2 = I_s (2 + 2\cos\delta) = I_s (4\cos^2(\delta/2)) \]

Effect of Finite Slit Width

If each slit has width \( a \), the single slit diffraction intensity is \( I_s = I_0 (\frac{\sin\beta}{\beta})^2 \) where \( \beta = \frac{\pi a \sin\theta}{\lambda} \). The total intensity is:

\[ I = I_0 \left( \frac{\sin\beta}{\beta} \right)^2 \cos^2 \left( \frac{\pi d \sin\theta}{\lambda} \right) \]

Missing Orders

If an interference maximum coincides with a diffraction minimum, that order disappears. This happens when:

\[ \frac{d}{a} = \frac{n}{m} \]

3. Diffraction Grating (N Slits)

For \( N \) slits separated by \( d \), the interference term is a geometric series sum:

\[ \sum_{n=0}^{N-1} e^{iny} = \frac{1 - e^{iNy}}{1 - e^{iy}} \]

where \( y = \frac{2\pi d \sin\theta}{\lambda} \).

The intensity becomes:

\[ I = I_s \frac{\sin^2(Ny/2)}{\sin^2(y/2)} \]

Principal Maxima (Grating Equation)

Maxima occur when the denominator is zero (\( y = 2m\pi \)). This leads to the famous Grating Equation:

\[ d \sin\theta = m\lambda \]

The intensity of principal maxima scales as \( N^2 \), making them extremely sharp for large \( N \).

Dispersive and Resolving Power

Dispersive Power:

\[ \frac{d\theta}{d\lambda} = \frac{m}{d \cos\theta} \]

Resolving Power:

\[ R = \frac{\lambda}{\Delta\lambda} = mN \]

This shows that resolution depends linearly on the number of illuminated slits \( N \).

4. Physical Insight

The transition from a single slit to a grating reveals a fundamental wave property: constructive phase alignment narrows the intensity peaks. As \( N \) increases:

• Intensity increases (\( \propto N^2 \)).
• Width of peaks decreases (\( \propto 1/N \)).
• Resolution improves (\( \propto N \)).