Two rays traveling parallel to the principal axis strike a...

Two rays traveling parallel to the principal axis strike a large plano-convex lens having a refractive index of 1.60 (Fig. P36.45). If the convex face is spherical, a ray near the edge does not pass through the focal point (spherical aberration occurs). Assume this face has a radius of curvature of 20.0 $\mathrm{cm}$ and the two rays are adius of $h_{1}=0.500 \mathrm{cm}$ and $h_{2}=12.0 \mathrm{cm}$ from the principal axis. Find the difference $\Delta x$ in the positions where each crosses the principal axis.

Key Concepts

Plano-Convex Lens Geometry

A plano-convex lens is characterized by one flat surface and one convex (spherical) surface. This design is widely used to collect and focus light, but the spherical side can introduce aberrations, particularly when light strikes far from the principal axis, leading to variations in the effective focal point for different rays.

Refraction at Spherical Surfaces

Refraction at spherical surfaces involves the bending of light as it passes across an interface with varying curvature and different refractive indices. In lenses with spherical surfaces, the curvature plays a crucial role in focusing light, and deviations from ideal behavior can lead to differences in focal positions, as observed in spherical aberration.

Paraxial Approximation

The paraxial approximation assumes that rays make small angles with respect to the optical axis, allowing simplifications in ray tracing and lens equations. This approximation is valid only for rays close to the axis and does not account for aberrations, such as spherical aberration, that occur when rays are far from it.

Spherical Aberration

Spherical aberration is an optical imperfection that occurs when light rays passing through a spherical surface of a lens or mirror do not converge at a single focal point. This is more pronounced with rays that strike the lens far from the optical axis (marginal rays), causing blurred or imperfect image formation compared to the ideal spot produced under the paraxial approximation.

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