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Question #43722

Difficulty: 8/10 | 🟢 Free

The Question

A monochromatic light source emits a wave whose phase $\Phi(t)$ undergoes a random walk. The source is coupled to a radioactive decay process with a mean decay rate $R$ (in $\text{s}^{-1}$ ). Each decay event instantaneously shifts the phase of the emitted light by either $+\phi_0$ or $-\phi_0$ (in radians) with equal probability. The number of decays in any time interval follows a Poisson distribution.

The light is sent into a Mach-Zehnder interferometer with a path length difference $\Delta L$ (in $\text{m}$ ). The visibility of the resulting interference fringes is defined as $V = |\langle e^{i\Delta \Phi} \rangle|$ , where $\Delta \Phi$ is the random phase difference accumulated over the time delay between the two interfering wave packets.

Assuming $c$ (in $\text{m/s}$ ) is the speed of light in vacuum, which of the following expressions gives the exact visibility $V$ of the interference fringes?

Options

A.
$\exp\left[ -\frac{R \Delta L}{c} (1 - \cos \phi_0) \right]$
B.
$\exp\left[ -\frac{R \Delta L}{2c} \phi_0^2 \right]$
C.
$\exp\left[ -\frac{R \Delta L}{c} (1 - \sin \phi_0) \right]$
D.
$\cos\left( \phi_0 \sqrt{\frac{R \Delta L}{c}} \right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Optical Path Difference|Statistical Nature of Decay|Mean vs. RMS Values

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to bridge statistical mechanics (Poisson processes) with wave optics (coherence).

A common trap is to assume the phase fluctuations follow a continuous Gaussian distribution, leading students to calculate the variance $\langle \Delta \Phi^2 \rangle = \mu \phi_0^2$ and incorrectly apply the Gaussian characteristic function $V = e^{-\langle \Delta \Phi^2 \rangle / 2}$ (Option B).

Another trap is treating the RMS phase fluctuation as a static phase difference and substituting it directly into a cosine function (Option D).

The correct approach requires recognizing the discrete nature of the random walk and averaging the exact characteristic function over the Poisson distribution of emission events.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Optical Path Difference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free