Lenses commonly are glass or plastic materials with two surfaces through which light passes. They produce images that may be magnifications or reductions of a subject. A lens surface may be flat, concave or convex. In a converging lens, light rays that pass through the lens converge at a focal point at its opposite side. In a diverging lens, rays emerge from the lens as though they came from a focal point at its opposite side. In each case, the distance between the focal point and the vertical axis of the lens is the focal length.
The surfaces of a converging lens are convex to the outside and their focal lengths are positive. A lens surface that is concave to the outside is a diverging lens with a negative focal length. The radius of curvature of a lens is the radius of an imaginary sphere from which the lens surface was cut.
The optical property of a lens material is its refractive index. First defined by Dutch mathematician Willebrord Snellius in the 17th century, it is the ratio of the speed of light in a vacuum to the speed of light in the material in question. The refractive index of water is 1.33 and for glass it is 1.5.
The optical axis of a lens is a straight line passing through the midpoint of each lens surface. A thin lens has a negligible thickness at this point compared to its focal length. The lens bends, or refracts, light instantaneously at a central plane that is vertical to the optical axis. The reciprocal of the focal length of a thin lens is equal to the refractive index multiplied by the sum of the radius of curvature of each lens surface.
In reality, the calculations of a lens system’s focal length must take into consideration the distance between the lens surfaces. Principal planes are mathematical concepts of surfaces within the lens system where refraction happens. The simplest system has a front principal plane where a light beam hits the lens and a back principal plane where it emerges. These planes are perpendicular to the optical axis and cross the axis at principal points, also called nodal points.
Gullstrand’s equation, named after 20th-century Swedish ophthalmologist Allvar Gullstrand, is a mathematical formula to calculate the focal length of a thick lens. It relates the focal length of the entire lens to the lens thickness at the optical axis, the lens material's refractive index and the focal length of each surface. The individual surface focal length is the distance between each focal point and its nodal point within the lens.