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Question #43588

Difficulty: 8/10 | 🟢 Free

The Question

A monochromatic plane wave of wavelength $\lambda$ (in vacuum) is incident normally on a single slit of width $a$ and thickness $t$ . The slit is filled with a transparent medium whose refractive index varies linearly across the slit width as $n(x) = n_0 + \gamma x$ , where $x$ is the transverse coordinate from the center of the slit ( $-a/2 \le x \le a/2$ ) and $\gamma$ is a positive constant such that $\gamma t < 1$ . Assuming the thin phase screen approximation (ray bending inside the slit is negligible), what is the angular position $\theta$ of the central principal maximum in the Fraunhofer diffraction pattern observed on a distant screen?

Options

A.
$\theta = \sin^{-1}(\gamma t)$
B.
$\theta = 0$
C.
$\theta = \sin^{-1}\left(\frac{\gamma t}{n_0}\right)$
D.
$\theta = \sin^{-1}\left(\gamma t + \frac{\lambda}{a}\right)$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Diffraction Width|Refractive Index Relativity|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the Huygens-Fresnel principle in the presence of a non-uniform medium gradient.

A common trap is to assume that the central maximum always occurs at $\theta = 0$ by incorrectly using the average refractive index $n_{avg} = n_0$ , ignoring the phase modulation across the aperture.

Students might also incorrectly apply Snell's law at the exit interface, leading to a division by $n_0$ , or confuse the phase gradient shift with the standard diffraction width $\lambda/a$ .

The correct approach requires recognizing that the linear refractive index gradient acts as a phase prism, tilting the emerging wavefront and shifting the entire diffraction pattern by an angle determined by the optical path length gradient.

Quick Facts About This Question

Question Information

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