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Question #43133

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The Question

A cylindrical graded-index (GRIN) lens of length $L$ has a refractive index profile given by $n(r) = n_0 \left(1 - \frac{1}{2}\alpha^2 r^2\right)$ for paraxial rays, where $r$ is the radial distance from the central optical axis ( $z$ -axis). The lens is immersed in a uniform transparent medium of refractive index $n_1$ . By Fermat's principle, the optical path length in this medium acts analogously to an effective potential field, causing paraxial rays to follow oscillatory trajectories. A point object is placed on the optical axis at a distance $u$ from the front flat face of the lens, and its image is formed at a distance $v$ from the rear flat face. If the imaging condition is expressed in the form of Newton's lens equation $x x' = C$ , where $x$ and $x'$ are the distances of the object and image from their respective principal focal points, what is the value of the constant $C$ ?

Options

A.
$\frac{n_1^2}{n_0^2 \alpha^2 \sin^2(\alpha L)}$
B.
$\frac{1}{\alpha^2 \sin^2(\alpha L)}$
C.
$\frac{n_1^2}{n_0^2 \alpha^4 L^2}$
D.
$\frac{n_0^2}{n_1^2 \alpha^2 \tan^2(\alpha L)}$

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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

Newton’s Form of Lens Equation|Potential vs Field Relationship|Calculus (Gradient)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from standard thin-lens ray optics to Hamiltonian optics using the Eikonal equation in a gradient-index (GRIN) medium.

The primary trap lies in treating the GRIN medium purely by its internal ray trajectory (ignoring the boundary refraction matrices dictated by Snell's Law at the flat interfaces, leading to Option B) or by incorrectly applying the thin-lens approximation (integrating the optical power linearly, leading to Option C).

By framing the optical path length as a potential field, students must rigorously derive the ABCD system matrix and extract Newton's lens equation constants, realizing that the effective focal length squared ( $1/C^2$ ) is heavily modified by the external medium's refractive index.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Newton’s Form of Lens Equation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free