Question

Write two differences in the patterns of double-slit interference experiment and single-slit diffraction experiment. Light waves from two pinholes illuminated by two sodium lamps do not produce interference patterns. Explain why.

Hint:

Remember: \begin{itemize} \item Double-slit: Interference + diffraction effects \item Single-slit: Pure diffraction pattern \item Coherence time $\tau_c = \frac{\lambda^2}{c\Delta\lambda}$ for quasi-monochromatic light \end{itemize}

Answer 1

Part 1: Pattern Differences

Double-Slit Interference Pattern	Single-Slit Diffraction Pattern
1. Equally spaced bright and dark fringes	1. Central bright fringe is twice as wide as other fringes
2. All bright fringes have equal intensity	2. Intensity decreases rapidly for higher-order fringes
3. Fringe position: \( y_n = \dfrac{nD\lambda}{d} \)	3. Minima position: \( y_n = \dfrac{nD\lambda}{a} \)

Part 2: Explanation for Lack of Interference from Two Sodium Lamps

Step 1: Coherence Requirement

• Interference necessitates a stable phase relationship between light sources (coherence).
• Two independent light sources (like sodium lamps) cannot maintain a fixed phase difference.

Step 2: Observational Aspects

• Each sodium lamp emits light due to unpredictable atomic transitions.
• The phase difference between independent sources shifts erratically approximately \(10^8\) times per second.

Step 3: Mathematical Rationale

Total intensity of interference:

\[ I_{\text{total}} = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\Delta \phi) \]

For incoherent sources, the phase difference \( \Delta \phi \) varies randomly, leading to:

\[ \langle \cos(\Delta \phi) \rangle = 0 \Rightarrow I_{\text{total}} = I_1 + I_2 \]

Consequently, no consistent interference pattern is observed.