Lenses

1. Introduction

A lens is a piece of transparent refracting material (such as glass or a diamond) bounded by two surfaces, of which at least one is curved (usually spherical). Light passing through a lens is refracted twice, once at each surface, and the combined refraction makes the lens either converge or diverge the light. Lenses are the working part of the eye, spectacles, cameras, microscopes, telescopes, and projectors, so understanding how they form images is central to geometrical optics.

Lenses are divided into two classes:

1. Convex lens (converging lens): thick at the centre and thin at the edges. A parallel beam of light is brought together (converged) to a real focus after passing through it.
2. Concave lens (diverging lens): thin at the centre and thick at the edges. A parallel beam of light is spread out (diverged) and appears to come from a virtual focus behind the lens.

This chapter covers the properties of spherical lenses, the lens formula relating object distance, image distance and focal length, the idea of visual angle and angular magnification, and the derivation of the lens maker's formula.

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2. Spherical Lenses and Their Properties

A spherical lens is a lens whose refracting surfaces are parts of spheres. The common types formed by combining spherical surfaces are the biconvex, plano-convex and concavo-convex (converging types), and the biconcave, plano-concave and convexo-concave (diverging types).

2.1 Important terms

To describe a spherical lens we use the following definitions.

1. Centres of curvature ($C_1$ and $C_2$): the centres of the two spheres to which the two lens surfaces belong. A lens has two centres of curvature, one for each surface.
2. Radii of curvature ($R_1$ and $R_2$): the radii of these two spheres. $R_1$ is the radius of the surface facing the object and $R_2$ is the radius of the second surface.
3. Principal axis: the straight line passing through the two centres of curvature $C_1$ and $C_2$.
4. Optical centre ($C$ or $O$): the central point of the lens on the principal axis. A ray directed towards the optical centre passes through the lens undeviated (its emergent path is parallel to the incident path).
5. Aperture: the effective diameter of the circular boundary of the lens through which refraction takes place.
6. Principal focus ($F$): for a convex lens, the point on the principal axis where a beam parallel to the axis converges after refraction (a real focus). For a concave lens, the point from which such a beam appears to diverge (a virtual focus). A lens has two principal foci, one on each side, because light can fall on either face.
7. Focal length ($f$): the distance from the optical centre to the principal focus.

2.2 Properties of spherical lenses

1. A convex lens converges a parallel beam of light to a real principal focus; a concave lens diverges a parallel beam so that it appears to come from a virtual principal focus.
2. Every lens has two principal foci placed symmetrically on the two sides, and (for a lens in a single medium such as air) the two focal lengths are equal.
3. A ray passing through the optical centre goes straight through without lateral deviation.
4. A ray travelling parallel to the principal axis passes (or appears to pass) through the principal focus after refraction.
5. A ray passing through (or directed towards) the principal focus emerges parallel to the principal axis.
6. Convex lenses can form both real and virtual images depending on the object position, while a concave lens (used alone) always forms a virtual, erect and diminished image.
7. The image formed by refraction through a lens depends only on the focal length and the object distance, which are linked by the lens formula (Section 3).