Question

What are coherent sources ? Why they are necessary for observing stable interference pattern ? Draw a graph showing the variation of intensity of light with the position on the screen in Young’s double-slit experiment.

Hint:

Two independent light sources (like two separate bulbs) can never be coherent because atomic emission processes in independent sources are completely random and uncorrelated.

Answer 1

Concept:
• Coherent sources are two independent or derived sources of light that emit light waves having the same frequency, same wavelength, and maintain a constant phase difference over time.

• Resultant intensity at any point due to superposition of two waves is $I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos \phi$.

Step 1: Definition of Coherent Sources
Two light sources are said to be coherent if they emit light waves of the same frequency, same wavelength, identical waveform, and have zero or a constant phase difference between them over time.

Step 2: Need for Coherent Sources for Stable Interference
The intensity at a point on the screen due to two overlapping light waves is given by:
\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos \phi \]
If sources are incoherent, the phase difference $\phi$ changes randomly and extremely rapidly with time (on the order of $10^{-8}$ s).
As a result, the time-average value of $\cos \phi$ over human observation time becomes zero:
\[ \langle \cos \phi \rangle = 0 \implies I_{avg} = I_1 + I_2 \]
Thus, interference term vanishes, and uniform illumination is observed on the screen without distinct bright and dark fringes.
Coherent sources ensure that phase difference $\phi$ remains constant with time at every point, giving a time-independent, stationary, and stable interference pattern with sharp bright and dark fringes.

Step 3: Graph of Intensity Distribution
In Young's double-slit experiment using monochromatic light where both slits have equal intensity $I_0$:
Maximum intensity at bright fringes: $I_{max} = 4 I_0$ at path difference $\Delta x = n\lambda$.
Minimum intensity at dark fringes: $I_{min} = 0$ at path difference $\Delta x = \left(n + \frac{1}{2}\right)\lambda$.
All bright fringes have equal maximum intensity $4I_0$, and fringe width $\beta = \frac{\lambda D}{d}$ is uniform.

Step 4: Conclusion
Coherent sources maintain a constant phase difference required for stable, non-time-varying constructive and destructive interference fringes on the screen.