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 (elements), 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
[edit]This section needs expansion with: history after 1758. You can help by adding to it. (January 2012) |
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.[a]
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][verification needed]
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.
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.[18]
Construction of simple lenses
[edit]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
[edit]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. 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 (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.
For a biconcave or plano-concave lens in a lower-index medium, 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.
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 (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 (also equal in the sign) would have zero optical power (as its focal length becomes infinity as shown in the lensmaker's equation), 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
[edit]For a single refraction for a circular boundary, the relation between object and its image in the paraxial approximation is given by[19][20]
where R is the radius of the spherical surface, n2 is the refractive index of the material of the surface, n1 is the refractive index of medium (the medium other than the spherical surface material), is the on-axis (on the optical axis) object distance from the line perpendicular to the axis toward the refraction point on the surface (which height is h), 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 .[21]
Applying this equation on the two spherical surfaces of a lens and approximating the lens thickness to zero (so a thin lens) leads to the lensmaker's formula.
Derivation
[edit]Applying Snell's law on the spherical surface,
Also in the diagram,, and using small angle approximation (paraxial approximation) and eliminating i, r, and θ,
Lensmaker's equation
[edit]The (effective) focal length of a spherical lens in air or vacuum for paraxial rays can be calculated from the lensmaker's equation:[22][23]
where
- 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 with respect to the principal planes of the lens, and the locations of the principal planes and with respect to the respective lens vertices are given by the following formulas, where it is a positive value if it is right to the respective vertex.[23]
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.
Sign convention for radii of curvature R1 and R2
[edit]The signs of the lens' radii of curvature indicate whether the corresponding surfaces are convex or concave. The sign convention used to represent this varies[citation needed], but in this article a positive R indicates a surface's center of curvature is further along in the direction of the ray travel (right, in the accompanying diagrams), while negative R means that rays reaching the surface have already passed the center of curvature. Consequently, for external lens surfaces as diagrammed above, R1 > 0 and R2 < 0 indicate convex surfaces (used to converge light in a positive lens), while R1 < 0 and R2 > 0 indicate concave surfaces. The reciprocal of the radius of curvature is called the curvature. A flat surface has zero curvature, and its radius of curvature is infinite.
Sign convention for other parameters
[edit]Parameter | Meaning | + Sign | - Sign |
---|---|---|---|
so | The distance between an object and a lens. | Real object | Virtual object |
si | The distance between an image and a lens. | Real image | Virtual image |
f | The focal length of a lens. | Conversing lens | Diverging lens |
yo | The height of an object from the optical axis. | Erect object | Inverted object |
yi | The height of an image from the optical axis | Erect image | Inverted image |
MT | The transverse magnification in imaging (= the ratio of yi to yo). | Erect image | Inverted image |
This convention seems to be mainly used for this article, although there is another convention such as Cartesian sign convention requiring different lens equation forms.
Thin lens approximation
[edit]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[25]
Derivation
[edit]The spherical thin lens equation in paraxial approximation is derived here with respect to the right figure.[25] The 1st spherical lens surface (which meets the optical axis at as its vertex) images an on-axis object point O to the virtual image I', which can be described by the following equation,For the imaging by second lens surface, by taking the above sign convention, andAdding these two equations yields,For the thin lens approximation where , the 2nd term of the RHS (Right Hand Side) is gone, so
The focal length of the thin lens is found by limiting ,
So, the Gaussian thin lens equation is
For the thin lens in air or vacuum where can be assumed, becomes
where the subscript of 2 in is dropped.
Imaging properties
[edit]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.
Lens equation
[edit]For paraxial rays, if the distances from an object to a spherical thin lens (a lens of negligible thickness) and from the lens to the image are S1 and S2 respectively, the distances are related by the (Gaussian) thin lens formula:[26][27][28]
The right figure shows how the image of an object point can be found by using three rays; the first ray parallelly incident on the lens and refracted toward the second focal point of it, the second ray crossing the optical center of the lens (so its direction does not change), and the third tray toward the first focal point and refracted to the direction parallel to the optical axis. This is a simple ray tracing method easily used. Two rays among the three are sufficient to locate the image point. By moving the object along the optical axis, it is shown that the second ray determines the image size while other rays help to locate the image location.
The lens equation can also be put into the "Newtonian" form:[24]
where and is positive if it is left to the front focal point , and is positive if it is right to the rear focal point . Because is positive, an object point and the corresponding imaging point made by a lens are always in opposite sides with respect to their respective focal points. ( and are either positive or negative.)
This Newtonian form of the lens equation can be derived by using a similarity between triangles P1PO1F1 and L3L2F1 and another similarity between triangles L1L2F2 and P2P02F2 in the right figure. The similarities give the following equations and combining these results gives the Newtonian form of the lens equation.
The above equations also hold for thick lenses (including compound lenses made by multiple lens elements) in air or vacuum (which refractive index can be treated as 1) if , , and are with respect to the principal planes of the lens ( is the effective focal length in this case).[23] If distances S1 or S2 pass through a medium other than air or vacuum a more complicated analysis is required.
If an object is placed at a distance S1 > f from a positive lens of focal length f, we will find an image at a 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.
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.
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.
For a given lens with the focal length f, the minimum distance between an object and the real image is 4f (S1 = S2 = 2f). This is derived by letting L = S1 + S2, expressing S2 in terms of S1 by the lens equation (or expressing S1 in terms of S2), and equating the derivative of L with respect to S1 (or S2) to zero. (Note that L has no limit in increasing so its extremum is only the minimum, at which the derivate of L is zero.)