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 the 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: Why Two Sodium Lamps Don't Produce Interference

Step 1: Coherence Requirement

• Interference requires a constant phase relationship between sources (i.e., coherence)
• Two independent light sources (like sodium lamps) cannot maintain a fixed phase difference

Step 2: Practical Observations

• Each sodium lamp emits light due to random atomic transitions
• The phase difference between independent sources fluctuates around \(10^8\) times per second

Step 3: Mathematical Justification

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, so:

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

Hence, no sustained interference pattern is observed.