Refraction at Spherical Surfaces and by Lenses

Refraction at a Spherical Surface

When light passes from one transparent medium to another through a spherical interface, the surface is curved. However, at any tiny point on the surface, the curvature is negligible, so the laws of refraction (Snell’s law) apply locally. The normal at the point of incidence is the line joining that point to the centre of curvature of the spherical surface. This normal is perpendicular to the tangent plane at that point.

Sign Convention (for spherical surfaces and lenses)

We use the Cartesian sign convention:

• Distances measured from the pole (vertex) of the surface in the direction of incident light are positive.
• Distances measured opposite to the incident light are negative.
• Heights above the principal axis are positive; below are negative.

Formula for a Single Spherical Surface

Consider a spherical surface of radius of curvature $R$, separating two media of refractive indices $n_1$ (incident side) and $n_2$ (refracted side). An object is placed at a distance $u$ from the pole. The image forms at a distance $v$ from the pole. The relation is:

• $n_1$: refractive index of the medium where the object lies.
• $n_2$: refractive index of the medium where the image forms.
• $u$: object distance (from pole).
• $v$: image distance (from pole).
• $R$: radius of curvature of the spherical surface (positive if centre of curvature is on the side of the outgoing light).

This formula is derived using Snell’s law and small-angle approximations (paraxial rays). It works for both convex and concave surfaces, provided the sign convention is followed.

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Refraction by a Thin Lens

A thin lens is a transparent medium bounded by two spherical surfaces, at least one of which is curved. The lens is called thin because its thickness is negligible compared to the radii of curvature and object/image distances.

Lens Maker’s Formula

To find the focal length of a thin lens, we apply the single-surface formula twice — once for each surface. Let:

• $R_1$ = radius of curvature of the first surface (the one light hits first).
• $R_2$ = radius of curvature of the second surface.
• $n$ = refractive index of the lens material.
• $n_1$ = refractive index of the surrounding medium (usually air, $n_1 = 1$).

After combining the two equations (and using the fact that the image from the first surface acts as the object for the second), we get the lens maker’s formula:

• $f$: focal length of the lens (positive for converging, negative for diverging).
• $n$: refractive index of the lens material relative to the surrounding medium.
• $R_1$, $R_2$: radii of curvature of the two surfaces, with sign according to the convention. …