Lens

Light being refracted by a spherical glass container full of water. Roger Bacon, 13th century

Lens for LSST, a planned sky surveying telescope

The word lens comes from lēns, the Latin name of the lentil (a seed of a lentil plant), because a double-convex lens is lentil-shaped. The lentil also gives its name to a geometric figure.[lower-alpha 1]

Some scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia.[1] 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.[2][3][4] Others have suggested that certain Egyptian hieroglyphs depict "simple glass meniscal lenses".[5]

The oldest certain reference to the use of lenses is from Aristophanes' play The Clouds (424 BCE) mentioning a burning-glass.[6] Pliny the Elder (1st century) confirms that burning-glasses were known in the Roman period.[7] Pliny also has the earliest known reference to the use of a corrective lens when he mentions that Nero was said to watch the gladiatorial games using an emerald (presumably concave to correct for nearsightedness, though the reference is vague).[8] Both Pliny and Seneca the Younger (3 BC–65 AD) described the magnifying effect of a glass globe filled with water.

Ptolemy (2nd century) 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 (10th century), who was in turn improved upon by Alhazen (Book of Optics, 11th century). The Arabic translation of Ptolemy's Optics became available in Latin translation in the 12th century (Eugenius of Palermo 1154). Between the 11th and 13th century "reading stones" were invented. These were primitive plano-convex lenses initially made by cutting a glass sphere in half. The medieval (11th or 12th century) rock crystal Visby lenses may or may not have been intended for use as burning glasses.[9]

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.[10] 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,[11] and later in the spectacle-making centres in both the Netherlands and Germany.[12] 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 (probably without the knowledge of the rudimentary optical theory of the day).[13][14] 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.[15][16]

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.[17] 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.

Most lenses are spherical lenses: their two surfaces are parts of the surfaces of spheres. Each surface can be convex (bulging outwards from the lens), concave (depressed into the lens), or planar (flat). 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

Types of lenses

Lenses are classified by the curvature of the two optical surfaces. A lens is biconvex (or double convex, or just convex) 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 (or just concave). 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. It is this type of lens that is most commonly used in corrective lenses, since its shape minimizes some aberrations.

If the lens is biconvex or plano-convex, a collimated beam of light passing through the lens converges to a spot (a focus) 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 f 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.

Biconvex lens

If the lens is biconcave or plano-concave, a collimated beam of light passing through the lens is diverged (spread); 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.

Biconcave lens

Meniscus lenses: negative (top) and positive (bottom)

Convex-concave (meniscus) 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 (with a shorter radius than the convex surface) and is thinner at the centre than at the periphery. Conversely, a positive meniscus lens has a steeper convex surface (with a shorter radius than the concave 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

Simulation of refraction at spherical surface at Desmos

For a single refraction for a circular boundary, the relation between object and image is given by[18][19]

$${\displaystyle {\frac {n_{2}}{v}}+{\frac {n_{1}}{u}}={\frac {n_{2}-n_{1}}{R}}}$$

where R is the radius of the spherical surface, n₂ is the refractive index of the surface, and n₁ is the refractive index of medium.

Applying this on the two spherical surfaces of a thin lens leads to the lens maker's formula.

Derivation

The four cases of spherical refraction

Applying Snell's law on the spherical surface, ${\displaystyle n_{1}\sin i=n_{2}\sin r\,.}$

Also in the diagram,

$${\displaystyle {\begin{aligned}\tan(i-\theta )&={\frac {h}{u}}\\\tan(\theta -r)&={\frac {h}{v}}\\\sin \theta &={\frac {h}{R}}\end{aligned}}}$$

Using small angle approximation and eliminating i, r, and θ,

$${\displaystyle {\frac {n_{2}}{v}}+{\frac {n_{1}}{u}}={\frac {n_{2}-n_{1}}{R}}\,.}$$

Lensmaker's equation

The position of the focus of a spherical lens depends on the radii of curvature of the two facets.

The focal length of a lens in air can be calculated from the lensmaker's equation:[20]

$${\displaystyle {\frac {1}{\ f\ }}=(n-1)\left[{\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}+{\frac {\ (n-1)\ d\ }{n\ R_{1}R_{2}}}\right]\ ,}$$

where

• f  is the focal length of the lens;
• n is the refractive index of the lens material;
• R₁ is the (signed, see below) radius of curvature of the lens surface closer to the light source;
• R₂ is the radius of curvature of the lens surface farther from the light source; and
• d is the thickness of the lens (the distance along the lens axis between the two surface vertices).

The focal length f is positive for converging lenses, and negative for diverging lenses. The reciprocal of the focal length, f⁻¹, is the optical power of the lens. If the focal length is in metres, this gives the optical power in dioptres (inverse metres).

Lenses have the same focal length when light travels from the back to the front as when light goes from the front to the back. Other properties of the lens, such as the aberrations are not the same in both directions.

Thin lens approximation

If d is small compared to R₁ and R₂ then the thin lens approximation can be made. For a lens in air, f  is then given by[21]

$${\displaystyle {\frac {1}{\ f\ }}\approx \left(n-1\right)\left[{\frac {1}{\ R_{1}\ }}-{\frac {1}{\ R_{2}\ }}\right]\,.}$$

As mentioned above, a positive or converging lens in air focuses a collimated beam travelling along the lens axis to a spot (known as the focal point) at a distance f from the lens. Conversely, a point source of light placed at the focal point is converted into a collimated beam by the lens. These two cases are examples of image formation in lenses. In the former case, an object at an infinite distance (as represented by a collimated beam of waves) is focused to an image at the focal point of the lens. In the latter, an object at the focal length distance from the lens is imaged at infinity. The plane perpendicular to the lens axis situated at a distance f from the lens is called the focal plane.

If the distances from the object to the lens and from the lens to the image are S₁ and S respectively, for a lens of negligible thickness (thin lens), in air, the distances are related by the thin lens formula:[22][23][24]

$${\displaystyle {1 \over f}={1 \over S_{1}}+{1 \over S_{2}}\,.}$$

This can also be put into the "Newtonian" form:[25]

$${\displaystyle f^{2}=x_{1}x_{2}\,,}$$

where ${\displaystyle x_{1}=S_{1}-f}$ and ${\displaystyle x_{2}=S_{2}-f\,.}$

A camera lens forms a real image of a distant object.

Therefore, if an object is placed at a distance S₁ > f from a positive lens of focal length f, we will find an image distance S₂ according to this formula. If a screen is placed at a distance S₂ on the opposite side of the lens, an image is formed on it. This sort of image, which can be projected onto a screen or image sensor, is known as a real image. This is the principle of the camera, and also of the human eye, in which the retina serves as the image sensor.

The focusing adjustment of a camera adjusts S₂, as using an image distance different from that required by this formula produces a defocused (fuzzy) image for an object at a distance of S₁ from the camera. Put another way, modifying S₂ causes objects at a different S₁ to come into perfect focus.

Virtual image formation using a positive lens as a magnifying glass.[26]

In some cases S₂ is negative, indicating that the image is formed on the opposite side of the lens from where those rays are being considered. Since the diverging light rays emanating from the lens never come into focus, and those rays are not physically present at the point where they appear to form an image, this is called a virtual image. Unlike real images, a virtual image cannot be projected on a screen, but appears to an observer looking through the lens as if it were a real object at the location of that virtual image. Likewise, it appears to a subsequent lens as if it were an object at that location, so that second lens could again focus that light into a real image, S₁ then being measured from the virtual image location behind the first lens to the second lens. This is exactly what the eye does when looking through a magnifying glass. The magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by the lens of the eye to create a real image on the retina.

A negative lens produces a demagnified virtual image.

A Barlow lens (B) reimages a virtual object (focus of red ray path) into a magnified real image (green rays at focus)

Using a positive lens of focal length f, a virtual image results when S₁ < f, the lens thus being used as a magnifying glass (rather than if S₁ ≫ f as for a camera). Using a negative lens (f < 0) with a real object (S₁ > 0) can only produce a virtual image (S₂ < 0), according to the above formula. It is also possible for the object distance S₁ to be negative, in which case the lens sees a so-called virtual object. This happens when the lens is inserted into a converging beam (being focused by a previous lens) before the location of its real image. In that case even a negative lens can project a real image, as is done by a Barlow lens.

Real image of a lamp is projected onto a screen (inverted). Reflections of the lamp from both surfaces of the biconvex lens are visible.

A convex lens (f ≪ S₁) forming a real, inverted image (as the image formed by the objective lens of a telescope or binoculars) rather than the upright, virtual image as seen in a magnifying glass(f > S₁). This real image may also be viewed when put on a screen.

For a thin lens, the distances S₁ and S₂ are measured from the object and image to the position of the lens, as described above. When the thickness of the lens is not much smaller than S₁ and S₂ or there are multiple lens elements (a compound lens), one must instead measure from the object and image to the principal planes of the lens. If distances S₁ or S₂ pass through a medium other than air or vacuum a more complicated analysis is required.

Magnification

The linear magnification of an imaging system using a single lens is given by

$${\displaystyle M=-{\frac {S_{2}}{S_{1}}}={\frac {f}{f-S_{1}}}\,,}$$

where M is the magnification factor defined as the ratio of the size of an image compared to the size of the object. The sign convention here dictates that if M is negative, as it is for real images, the image is upside-down with respect to the object. For virtual images M is positive, so the image is upright.

This magnification formula provides two easy ways to distinguish converging (f > 0) and diverging (f < 0) lenses: For an object very close to the lens (0 < S₁ < |f|), a converging lens would form a magnified (bigger) virtual image, whereas a diverging lens would form a demagnified (smaller) image; For an object very far from the lens (S₁ > |f| > 0), a converging lens would form an inverted image, whereas a diverging lens would form an upright image.

Linear magnification M is not always the most useful measure of magnifying power. For instance, when characterizing a visual telescope or binoculars that produce only a virtual image, one would be more concerned with the angular magnification—which expresses how much larger a distant object appears through the telescope compared to the naked eye. In the case of a camera one would quote the plate scale, which compares the apparent (angular) size of a distant object to the size of the real image produced at the focus. The plate scale is the reciprocal of the focal length of the camera lens; lenses are categorized as long-focus lenses or wide-angle lenses according to their focal lengths.

Using an inappropriate measurement of magnification can be formally correct but yield a meaningless number. For instance, using a magnifying glass of 5 cm focal length, held 20 cm from the eye and 5 cm from the object, produces a virtual image at infinity of infinite linear size: M = ∞. But the angular magnification is 5, meaning that the object appears 5 times larger to the eye than without the lens. When taking a picture of the moon using a camera with a 50 mm lens, one is not concerned with the linear magnification M ≈ −50 mm / 380000 km = −1.3×10⁻¹⁰. Rather, the plate scale of the camera is about 1°/mm, from which one can conclude that the 0.5 mm image on the film corresponds to an angular size of the moon seen from earth of about 0.5°.

In the extreme case where an object is an infinite distance away, S₁ = ∞, S₂ = f and M = −f/∞ = 0, indicating that the object would be imaged to a single point in the focal plane. In fact, the diameter of the projected spot is not actually zero, since diffraction places a lower limit on the size of the point spread function. This is called the diffraction limit.

Images of black letters in a thin convex lens of focal length f are shown in red. Selected rays are shown for letters E, I and K in blue, green and orange, respectively. E (at 2f) has an equal-size, real and inverted image; I (at f) has its image at infinity; and K (at f/2) has a double-size, virtual and upright image.

Lenses do not form perfect images, and a lens always introduces some degree of distortion or aberration that makes the image an imperfect replica of the object. Careful design of the lens system for a particular application minimizes the aberration. Several types of aberration affect image quality, including spherical aberration, coma, and chromatic aberration.

Simple lenses are subject to the optical aberrations discussed above. In many cases these aberrations can be compensated for to a great extent by using a combination of simple lenses with complementary aberrations. A compound lens is a collection of simple lenses of different shapes and made of materials of different refractive indices, arranged one after the other with a common axis.

The simplest case is where lenses are placed in contact: if the lenses of focal lengths f₁ and f₂ are "thin", the combined focal length f of the lenses is given by

$${\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}\,.}$$

Since 1/f is the power of a lens, it can be seen that the powers of thin lenses in contact are additive.

If two thin lenses are separated in air by some distance d, the focal length for the combined system is given by

$${\displaystyle {\frac {1}{f}}={\frac {1}{f_{1}}}+{\frac {1}{f_{2}}}-{\frac {d}{f_{1}f_{2}}}\,.}$$

The distance from the front focal point of the combined lenses to the first lens is called the front focal length (FFL):[27]

$${\displaystyle {\text{FFL}}={\frac {f_{1}(f_{2}-d)}{(f_{1}+f_{2})-d}},.}$$

Similarly, the distance from the second lens to the rear focal point of the combined system is the back focal length (BFL):

$${\displaystyle {\text{BFL}}={\frac {f_{2}(d-f_{1})}{d-(f_{1}+f_{2})}}\,.}$$

As d tends to zero, the focal lengths tend to the value of f given for thin lenses in contact.

If the separation distance is equal to the sum of the focal lengths (d = f₁ + f₂), the FFL and BFL are infinite. This corresponds to a pair of lenses that transform a parallel (collimated) beam into another collimated beam. This type of system is called an afocal system, since it produces no net convergence or divergence of the beam. Two lenses at this separation form the simplest type of optical telescope. Although the system does not alter the divergence of a collimated beam, it does alter the width of the beam. The magnification of such a telescope is given by

$${\displaystyle M=-{\frac {f_{2}}{f_{1}}}\,,}$$

which is the ratio of the output beam width to the input beam width. Note the sign convention: a telescope with two convex lenses (f₁ > 0, f₂ > 0) produces a negative magnification, indicating an inverted image. A convex plus a concave lens (f₁ > 0 > f₂) produces a positive magnification and the image is upright. For further information on simple optical telescopes, see Refracting telescope § Refracting telescope designs.

An aspheric biconvex lens.

Cylindrical lenses have curvature along only one axis. They are used to focus light into a line, or to convert the elliptical light from a laser diode into a round beam. They are also used in motion picture anamorphic lenses.

Aspheric lenses have at least one surface that is neither spherical nor cylindrical. The more complicated shapes allow such lenses to form images with less aberration than standard simple lenses, but they are more difficult and expensive to produce. These were formerly complex to make and often extremely expensive, but advances in technology have greatly reduced the manufacturing cost for such lenses.

Close-up view of a flat Fresnel lens.

A Fresnel lens has its optical surface broken up into narrow rings, allowing the lens to be much thinner and lighter than conventional lenses. Durable Fresnel lenses can be molded from plastic and are inexpensive.

Lenticular lenses are arrays of microlenses that are used in lenticular printing to make images that have an illusion of depth or that change when viewed from different angles.

Bifocal lens has two or more, or a graduated, focal lengths ground into the lens.

A gradient index lens has flat optical surfaces, but has a radial or axial variation in index of refraction that causes light passing through the lens to be focused.

An axicon has a conical optical surface. It images a point source into a line along the optic axis, or transforms a laser beam into a ring.[28]

Diffractive optical elements can function as lenses.

Superlenses are made from negative index metamaterials and claim to produce images at spatial resolutions exceeding the diffraction limit.[29] The first superlenses were made in 2004 using such a metamaterial for microwaves.[29] Improved versions have been made by other researchers.[30][31] As of 2014 the superlens has not yet been demonstrated at visible or near-infrared wavelengths.[32]

A prototype flat ultrathin lens, with no curvature has been developed.[33]

A watch with a plano-convex lens over the date indicator

A single convex lens mounted in a frame with a handle or stand is a magnifying glass.

Lenses are used as prosthetics for the correction of refractive errors such as myopia, hypermetropia, presbyopia, and astigmatism. (See corrective lens, contact lens, eyeglasses, intraocular lens.) Most lenses used for other purposes have strict axial symmetry; eyeglass lenses are only approximately symmetric. They are usually shaped to fit in a roughly oval, not circular, frame; the optical centres are placed over the eyeballs; their curvature may not be axially symmetric to correct for astigmatism. Sunglasses' lenses are designed to attenuate light; sunglass lenses that also correct visual impairments can be custom made.

Other uses are in imaging systems such as monoculars, binoculars, telescopes, microscopes, cameras and projectors. Some of these instruments produce a virtual image when applied to the human eye; others produce a real image that can be captured on photographic film or an optical sensor, or can be viewed on a screen. In these devices lenses are sometimes paired up with curved mirrors to make a catadioptric system where the lens's spherical aberration corrects the opposite aberration in the mirror (such as Schmidt and meniscus correctors).

Convex lenses produce an image of an object at infinity at their focus; if the sun is imaged, much of the visible and infrared light incident on the lens is concentrated into the small image. A large lens creates enough intensity to burn a flammable object at the focal point. Since ignition can be achieved even with a poorly made lens, lenses have been used as burning-glasses for at least 2400 years.[6] A modern application is the use of relatively large lenses to concentrate solar energy on relatively small photovoltaic cells, harvesting more energy without the need to use larger and more expensive cells.

Radio astronomy and radar systems often use dielectric lenses, commonly called a lens antenna to refract electromagnetic radiation into a collector antenna.

Lenses can become scratched and abraded. Abrasion-resistant coatings are available to help control this.[34]

• Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. ISBN 978-0-201-11609-0. Chapters 5 & 6.
• Hecht, Eugene (2002). Optics (4th ed.). Addison Wesley. ISBN 978-0-321-18878-6.
• Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. SPIE. ISBN 978-0-8194-5294-8.

Wikimedia Commons has media related to Lenses.

• A chapter from an online textbook on refraction and lenses Archived 17 December 2009 at the Wayback Machine
• Thin Spherical Lenses (.pdf) on Project PHYSNET.
• Lens article at digitalartform.com
• Article on Ancient Egyptian lenses
• FDTD Animation of Electromagnetic Propagation through Convex Lens (on- and off-axis) Video on YouTube
• The Use of Magnifying Lenses in the Classical World
• Henker, Otto (1911). "Lens" . Encyclopædia Britannica. Vol. 16 (11th ed.). pp. 421–427. (with 21 diagrams)