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Question #43109

Difficulty: 8/10 | 🟢 Free

The Question

A graded-index optical fiber is modeled as a 2D dielectric slab occupying the region $0 \le y \le d$ , where $d = 5 \text{ cm}$ . The refractive index of the core depends on the transverse coordinate $y$ and the wavelength $\lambda$ as: $n(y, \lambda) = A + \frac{B}{\lambda^2} - k y$ where $A = 1.6$ , $B = 2.5 \times 10^{-14} \text{ m}^2$ , and $k = 4 \text{ m}^{-1}$ . The region $y > d$ is a uniform cladding with refractive index $n_c = 1.5$ . A narrow beam of white light enters the core at the origin $(0,0)$ from air ( $n_{air} = 1$ ). The beam is incident on the entry face $x=0$ at an angle $\theta_i = \sin^{-1}(0.8)$ with respect to the normal (x-axis), directed towards the $+y$ region. Which of the following represents the range of wavelengths $\lambda$ that are completely confined within the core (i.e., do not leak into the cladding)?

Options

A.
$\lambda \le 500 \text{ nm}$
B.
$\lambda \ge 500 \text{ nm}$
C.
$\lambda \le 250 \text{ nm}$
D.
$\lambda \ge 250 \text{ nm}$

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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

Spectrum Order and Frequency/Wavelength|Total Internal Reflection (TIR)|Non-Uniform Dielectric Filling

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of chromatic dispersion and spatial gradient-index (GRIN) optics under the Eikonal approximation.

The primary trap lies in the misapplication of Snell's Law at the entry face and within the stratified medium.

Students often incorrectly assume the internal ray angle is simply the incident angle, or they fail to construct the invariant $n(y) \cos \theta(y) = \sqrt{n_0^2 - \sin^2 \theta_i}$ .

A secondary trap is attempting to integrate the exact ray trajectory $y(x)$ instead of recognizing that the confinement condition depends solely on the invariant exceeding the cladding refractive index ( $C \ge n_c$ ).

By coupling the spatial TIR condition with Cauchy's equation for dispersion, the problem forces students to evaluate a multivariable boundary condition without getting lost in unnecessary trajectory calculus.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Spectrum Order and Frequency/Wavelength
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free