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Question #43594

Difficulty: 8/10 | 🟢 Free

The Question

An unpolarized light ray is launched from air ( $n=1$ ) into a transparent medium with a gradient refractive index $n(z) = \sqrt{3 - \alpha z}$ at $z=0$ . The ray is launched at an angle $\theta_0$ with the $z$ -axis. At a depth $z=d$ , the refractive index is $n(d) = \sqrt{2}$ . At this depth, there is a horizontal interface separating the gradient medium from a uniform cladding of refractive index $n_2 = 1$ . It is desired that the ray undergoes Total Internal Reflection (TIR) at $z=d$ , AND that the reflected ray is completely plane-polarized. Which of the following represents the required launch angle $\theta_0$ ?

Options

A.
$\arcsin\left(\sqrt{\frac{2}{3}}\right)$
B.
$\arcsin\left(\frac{1}{\sqrt{3}}\right)$
C.
$\frac{\pi}{2}$
D.
No such angle exists

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Brewster's Law|Total Internal Reflection (TIR)|Medium Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the conceptual boundary between Brewster's Law and Total Internal Reflection (TIR) in a gradient-index medium.

Students often fall into the trap of blindly applying Snell's law to trace the Brewster angle back to the launch angle, yielding the trap answer $\theta_0 = \arcsin(\sqrt{2/3})$ .

However, they fail to verify the physical consistency of the requested conditions.

For any interface where $n_1 > n_2$ , Brewster's angle $i_p = \arctan(n_2/n_1)$ is always strictly less than the critical angle $\theta_c = \arcsin(n_2/n_1)$ .

Therefore, a ray at Brewster's angle will always partially transmit and cannot undergo TIR.

Additionally, TIR reflects both s- and p-polarizations completely, making it impossible to produce completely plane-polarized light from an unpolarized source via TIR.

The conditions are mutually exclusive.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Wave Optics - Class 12
Concept	Brewster's Law
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free