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Question #43127

Difficulty: 8/10 | 🟢 Free

The Question

A wide parallel beam of monochromatic light of wavelength $\lambda$ is incident normally on the flat surface of a plano-convex glass lens. The lens has a refractive index $n$ and the curved surface has a radius of curvature $R$ . The paraxial focal length of the lens is $f$ . Due to spherical aberration, the wavefront emerging from the lens is not perfectly spherical. According to the Rayleigh quarter-wave criterion, the lens will produce a nearly perfect focus if the magnitude of the optical path difference between the secondary wavelet arriving from the edge of the aperture (radius $a$ ) and the wavelet arriving from the center, evaluated at the paraxial focus, is equal to $\lambda/4$ . Assuming $a \ll f$ , which of the following expressions gives the maximum aperture radius $a$ that satisfies this criterion?

Options

A.
$\left( \frac{2 \lambda f^3 (n-1)^2}{n^2} \right)^{1/4}$
B.
$\left( \frac{\lambda f^3 (n-1)^2}{2 n^2} \right)^{1/4}$
C.
$\left( \frac{2 \lambda f^3 n^2}{(n-1)^2} \right)^{1/4}$
D.
$\left( \frac{\lambda f^3 n^2}{2 (n-1)^2} \right)^{1/4}$

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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

Spherical Surface Refraction|Impact Parameter Geometry|Superposition Principle|Vector Algebra

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to move beyond the paraxial approximation and apply the Huygens-Fresnel superposition principle to aberrated wavefronts.

A common trap is attempting to trace the exact refracted geometric rays (which miss the paraxial focus entirely) instead of evaluating the phase accumulation of secondary wavelets traveling from the exit pupil to the focal point.

Furthermore, students often incorrectly calculate the phase shift by only considering the lens thickness $n \Delta(r)$ while ignoring the geometric path difference in air.

By forcing a 4th-order binomial expansion of the optical path length, the problem exposes the physical origin of spherical aberration and its destructive interference effect at the focal point.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Spherical Surface Refraction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free