Lens

A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses, usually arranged along a common axis. Lenses are made from materials such as glass or plastic and are ground, polished, or molded to the required shape. A lens can focus light to form an image, unlike a prism, which refracts light without focusing. Devices that similarly focus or disperse waves and radiation other than visible light are also called "lenses", such as microwave lenses, electron lenses, acoustic lenses, or explosive lenses.
Lenses are used in various imaging devices such as telescopes, binoculars, and cameras. They are also used as visual aids in glasses to correct defects of vision such as myopia and hypermetropia.

History

The word lens comes from Lens, the Latin name of the lentil, because a double-convex lens is lentil-shaped. The lentil also gives its name to a geometric figure.
Some scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia. The so-called Nimrud lens is a rock crystal artifact dated to the 7th century BCE which may or may not have been used as a magnifying glass, or a burning glass. Others have suggested that certain Egyptian hieroglyphs depict "simple glass meniscal lenses".
The oldest certain reference to the use of a lens as a burning-glass is from Aristophanes' play The Clouds.
Pliny the Elder confirms that burning-glasses were known in the Roman period.
Pliny arguably referred to the use of a corrective lens when he mentions that Nero was said to watch the gladiatorial games using an emerald wrote that "to those who see through water, all things are far larger; letters, however minute and obscure, are perceived larger and clearer through a glass sphere filled with water; fruits seem more shapely in form than they are, if they float in glass."
Ptolemy wrote a book on Optics, which however survives only in the Latin translation of an incomplete and very poor Arabic translation. The book was, however, received by medieval scholars in the Islamic world, and commented upon by Ibn Sahl, who was in turn improved upon by Alhazen. The Arabic translation of Ptolemy's Optics became available in Latin translation in the 12th century.
Between the 11th and 13th century "reading stones" were used. These were primitive plano-convex lenses initially made by cutting a glass sphere in half. The medieval rock crystal Visby lenses may or may not have been intended for use as burning glasses.
Spectacles were invented as an improvement of the "reading stones" of the high medieval period in Northern Italy in the second half of the 13th century. This was the start of the optical industry of grinding and polishing lenses for spectacles, first in Venice and Florence in the late 13th century, and later in the spectacle-making centres in both the Netherlands and Germany.
Spectacle makers created improved types of lenses for the correction of vision based more on empirical knowledge gained from observing the effects of the lenses. The practical development and experimentation with lenses led to the invention of the compound optical microscope around 1595, and the refracting telescope in 1608, both of which appeared in the spectacle-making centres in the Netherlands.
With the invention of the telescope and microscope there was a great deal of experimentation with lens shapes in the 17th and early 18th centuries by those trying to correct chromatic errors seen in lenses. Opticians tried to construct lenses of varying forms of curvature, wrongly assuming errors arose from defects in the spherical figure of their surfaces. Optical theory on refraction and experimentation was showing no single-element lens could bring all colours to a focus. This led to the invention of the compound achromatic lens by Chester Moore Hall in England in 1733, an invention also claimed by fellow Englishman John Dollond in a 1758 patent.
Developments in transatlantic commerce were the impetus for the construction of modern lighthouses in the 18th century, which utilize a combination of elevated sightlines, lighting sources, and lenses to provide navigational aid overseas. With maximal distance of visibility needed in lighthouses, conventional convex lenses would need to be significantly sized which would negatively affect the development of lighthouses in terms of cost, design, and implementation. Fresnel lens were developed that considered these constraints by featuring less material through their concentric annular sectioning. They were first fully implemented into a lighthouse in 1823.

Construction of simple lenses

Most lenses are spherical lenses: their two surfaces are parts of the surfaces of spheres. Each surface can be convex, concave, or planar. The line joining the centres of the spheres making up the lens surfaces is called the axis of the lens. Typically the lens axis passes through the physical centre of the lens, because of the way they are manufactured. Lenses may be cut or ground after manufacturing to give them a different shape or size. The lens axis may then not pass through the physical centre of the lens.
Toric or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes. They have a different focal power in different meridians. This forms an astigmatic lens. An example is eyeglass lenses that are used to correct astigmatism in someone's eye.

Types of simple lenses

Lenses are classified by the curvature of the two optical surfaces. A lens is biconvex if both surfaces are convex. If both surfaces have the same radius of curvature, the lens is equiconvex. A lens with two concave surfaces is biconcave. If one of the surfaces is flat, the lens is plano-convex or plano-concave depending on the curvature of the other surface. A lens with one convex and one concave side is convex-concave or meniscus. Convex-concave lenses are most commonly used in corrective lenses, since the shape minimizes some aberrations.
For a biconvex or plano-convex lens in a lower-index medium, a collimated beam of light passing through the lens converges to a spot behind the lens. In this case, the lens is called a positive or converging lens. For a thin lens in air, the distance from the lens to the spot is the focal length of the lens, which is commonly represented by in diagrams and equations. An extended hemispherical lens is a special type of plano-convex lens, in which the lens's curved surface is a full hemisphere and the lens is much thicker than the radius of curvature.
Another extreme case of a thick convex lens is a ball lens, whose shape is completely round. When used in novelty photography it is often called a "lensball". A ball-shaped lens has the advantage of being omnidirectional, but for most optical glass types, its focal point lies close to the ball's surface. Because of the ball's curvature extremes compared to the lens size, optical aberration is much worse than thin lenses, with the notable exception of chromatic aberration.
For a biconcave or plano-concave lens in a lower-index medium, a collimated beam of light passing through the lens is diverged ; the lens is thus called a negative or diverging lens. The beam, after passing through the lens, appears to emanate from a particular point on the axis in front of the lens. For a thin lens in air, the distance from this point to the lens is the focal length, though it is negative with respect to the focal length of a converging lens.
The behavior reverses when a lens is placed in a medium with higher refractive index than the material of the lens. In this case a biconvex or plano-convex lens diverges light, and a biconcave or plano-concave one converges it.
Convex-concave lenses can be either positive or negative, depending on the relative curvatures of the two surfaces. A negative meniscus lens has a steeper concave surface and is thinner at the centre than at the periphery. Conversely, a positive meniscus lens has a steeper convex surface and is thicker at the centre than at the periphery.
An ideal thin lens with two surfaces of equal curvature would have zero optical power, meaning that it would neither converge nor diverge light. All real lenses have a nonzero thickness, however, which makes a real lens with identical curved surfaces slightly positive. To obtain exactly zero optical power, a meniscus lens must have slightly unequal curvatures to account for the effect of the lens' thickness.

For a spherical surface

For a single refraction for a circular boundary, the relation between object and its image in the paraxial approximation is given by
where is the radius of the spherical surface, is the refractive index of the material of the surface, is the refractive index of medium, is the on-axis object distance from the line perpendicular to the axis toward the refraction point on the surface, and is the on-axis image distance from the line. Due to paraxial approximation where the line of h is close to the vertex of the spherical surface meeting the optical axis on the left, and are also considered distances with respect to the vertex.
Moving toward the right infinity leads to the first or object focal length for the spherical surface. Similarly, toward the left infinity leads to the second or image focal length.
Applying this equation on the two spherical surfaces of a lens and approximating the lens thickness to zero leads to the lensmaker's formula.

Derivation

Applying Snell's law on the spherical surface,
Also in the diagram,, and using small angle approximation and eliminating,, and,