Lens (optics)

Optical lenses is referred to as transparent optical components with two refractive surfaces, of which at least one surface is convex or concave. Lenses can be used individually or in combination for several optical imaging.

The shapes of the lens surfaces are formed spherical, as a rule. These lenses are easier to manufacture than Aspheric lenses, which have due to specially shaped lens surfaces in lower aberrations. Lenses with a weak curvature - ie a long focal length - can be made ​​thinner than strongly curved, thick lenses. The so-called thin lens has further advantages in theory - especially in optical radiation - treatment. It only has a principal plane, which also can be considered for symmetrical lenses in its median plane, which simplifies the specification or determination of the focal plane to a good approximation.

• 4.1 Refraction at a single spherical interface
• 4.2 refraction at a lens
• 4.3 Multiple or compound lenses

• 8.1 See also
• 8.2 Literature
• 8.3 References and notes
• 8.4 External links

Materials, preparation and properties

Lenses for use in the visible spectral range are made of optical glass or plastics, such as polycarbonates, polymethyl methacrylates or cyclo-olefin (co) polymers. Furthermore, in contrast to this, the use of amorphous materials can be crystalline, such as calcium fluoride or sapphire.

Blanks for glass lenses are primarily manufactured by hot pressing or casting. They are ground and polished to the optically active faces and centered by means of grinding to the same extent. At lower quality requirements the pressed at high temperatures blanks can be used directly. Plastic lenses produced by injection molding or injection compression molding.

With the help of the geometrical sizes Diameter, lens radii, center thickness, supplemented with manufacturing tolerances ( eg passport fault tolerance respectively. Average wavefront error ) and the material properties of refractive index, Abbe number and stress birefringence, supplemented by material tolerances (eg homogeneity ), the optical properties a spherical lens fully described. The most important characteristic of a lens for their imaging function is the focal length (unit: meter ), ie the distance between the focal point and focal plane and main levels. The reciprocal of the focal length is as refractive index (unit: diopter ) specified. The diameter of the usable surface of a lens is called opening or aperture.

An important property of a lens is the principle of the reversal of the optical path: If an incident from one side of the light ray is traced along its path, as opposed to an incident light beam is passed through this path reversed.

History

The first ideas of using curved surfaces to increase back to the Islamic scientists and naturalists Abu Ali al -Hasan ibn al - Haitham ( 964-1040 ). After the translation of his works on optics, these were taken up again in the 12th century by monks and the reading stone constructed a überhalbkugelige plano-convex lens with which it was possible to view enlarged font. These consisted mostly of beryl, which, as the word goes back glasses. End of the 13th century collection lenses were used in reading glasses to correct presbyopia or far - then for the first time. The first optical cameras with lenses were the microscope and the telescope, which were invented in the late 16th century or early 17th century.

Various lens shapes

Spherical Lenses

The simplest of the two lenses optically active surfaces being spherical. That is, they are surface sections of a sphere. Therefore, one can characterize these surfaces with their radius of curvature R.

We distinguish:

• Collecting lens having two convex surfaces, or with a convex and a flat surface at least in the middle, in the area of the optical axis, is thicker than at the edge; a beam parallel to the optical axis of incident light beams is ideally collected at a point behind the lens, the focal point or focus F. Your focal length f is positive.
• Divergent lenses or with two concave surfaces having a concave and a plane surface, or at least thicker at the edge than in the middle; a bundle of parallel rays incident runs so apart, as if it came from a point on the incident side of the light behind the lens. The focal length is negative.

In addition, there are lenses having a concave and a convex surface ( meniscus lens ); Such lenses are often used for correction of aberrations (see below) in optical systems with multiple lenses. They are converging lenses, if the convex surface is more curved, or concave lenses, if the concave surface is more curved.

A device with two plane and parallel optical surfaces is called planar plate or plane-parallel plate.

For the calculation according to the rules of geometric optics, the light in the direction of successive radii R1 and R2 are called ( with R3 and R4 etc. ) according to DIN 1335. The associated sign does not distinguish directly between convex and concave surface. The radius of a surface is defined as positive when the light passes through its center of curvature as the surface later. In reverse order of the radius is defined as negative. In graphs, the light is conventionally from the left ( or above).

For the three surfaces convex, flat (plane) or concave, the following signs arise:

• Convex surface ( it is curved outward ) R1 (R1 > 0), or R2 ( R2 is <0).
• Plane surface ( Its curvature is zero): R = ± ∞.
• Concave surface ( it is curved inwards ): -R1 (R1 < 0), or R2 (R2 > 0).

The line passing through the centers of curvature line is referred to as an optical axis O. If one of the two lens surfaces flat, so the optical axis is perpendicular to it.

Spherical lenses inherently lead to aberrations, because the focus of the marginal rays does not coincide with the focal point on internal radiation, and possibly also depends on the wavelength of light. To reduce these errors, lens systems ( anastigmats, Cooke triplet, Tessar and others) are used, which largely compensate for the error.

Aspherical lenses

Aspheric lenses are suitable to reduce aberrations caused by spherical lenses. You are rotationally symmetrical, but on average, not circular. The greater number of design parameters, as compared to the spherical lens makes it possible to reduce aberrations. In conventional optical systems aberrations can be corrected by a suitable combination of several spherical lenses made of materials with different refractive indices. Even through the use of an aspherical surface can be in such a system, 2-3 lenses reduce spherical. Disadvantage of aspherical lenses are their relatively high production costs.

An alternative for aspheric lenses are gradient lenses, in which the refractive index varies spatially.

Ideal lens

For two restricted purposes, there are lens shapes that have no aberrations:

• Exact bundles parallel to the optical axis of the incident light into one point: One possibility is that the incident light-facing surface of the lens is flat and the side facing away from the form of a hyperboloid. For the half opening angle of the hyperboloid belonging Asymptotenkegels must apply, with the refractive index of the lens material. The incident rays are just bundled in one of the two Hyperbelbrennpunkte - in that with the greater distance from the vertex of the lens.
• DC long optical path for all the rays that originate at a point on the optical axis to the common image point: The planar surface of the lens is replaced by a sphere to that point and the hyperbolic surface by a Cartesian oval. The mapping is done according to the Fermat principle. For the case that the adjacent points of the original image to be mapped uniformly onto adjacent pixels of the image, such considerations are even more complex.

Astigmatic lenses

Astigmatic lenses have different focal lengths in the wide two mutually perpendicular radial directions. Limiting case is the cylindrical lens in one of the two directions of a plane-parallel surface contours and in its typical form is actually a cylinder section: a cylindrical and a planar surface. It bundles parallel incident light to a focal line.

Astigmatic lenses are used, inter alia, in the following cases:

• Lenses, the astigmatism of the eye lens offset ( a cylindrical surface, the other often spherical to additionally compensate for nearsightedness or farsightedness ),
• Widescreen cinema projectors and movie cameras contain a cylindrical lens in order to save space depict the image in widescreen format to the normal aspect ratio of the movie and in the projection to equalize again ( anamorphic ) ( Cinemascope, Total Vision and similar ),
• Collimating the radiation from laser diodes, which have a non kreissymetrische divergence due to their structure.

Elastic lenses

The term " elastic lens " denotes an optical member, which changes its optical power by the deformation of an elastic solid. It will be apparent from the functional principle of the following benefits:

• The shape of the interface is freely selectable ( spherical, aspherical )
• The size of the power change is the use of rubber materials is very large ( about 15 diopters )
• The deformation can be done very quickly

Focal length and principal planes

The space used for an optical imaging property of a refractive lens depends on the refractive index of its material and the shape of its boundary surfaces. Both together expresses the focal length. In addition, two so-called principal planes are indicated, one each against Board and an image side as a reference plane for the Board or against the image-side focal length. But the two different focal lengths only when the optical medium of the lens with the front is not the same on the lens. In the general case there is air on both sides.

Both the focal lengths and the principal planes are ideal sizes that arise when working on the concept of paraxial optics. Within this concept, they can be made ​​of the material and the geometric properties specify theoretically, that is calculated. The refraction is examined separately in each of the two boundary surfaces. Subsequently the results and the mutual position of the surfaces to equations for the size of the focal length and the position of the principal planes are combined.

Refraction at a single spherical interface

The focal lengths of a single spherical interface are in the Abbe invariant, a fundamental equation of paraxial optics, with included. One of the two focal lengths is the focal length, if the other is located at infinity, is collected from the parallel incident light at the focal point.

If the average length at infinity, then to and from the Abbe invariant

Is:

In the reverse direction of the beam, the cutting width is at infinity, is to be, and is from the Abbe invariant:

The main plane passing through the apex of the spherical surface.

Refraction at a lens

In a lens the refraction takes place at two generally spherical interfaces. The common focal length can be found under the following guidelines:

• The mapping of the image-side focal point of the first surface by the second surface is the image-side focal point of the lens, since all incident parallel rays pass through both the one and the other point (red continuous line in the illustration ).
• The extension of a paraxial incident beam intersects the broken passing through the lens beam in the image-side principal plane of the lens (dotted line in the illustration ). This is based on the definition of the principal planes, the magnification between them is 1.

A basic connection in the optical image is contained in the angular relationship:

This allows the point P found by which the red polyline must lead.

The equation for the image-side focal length of the lens is the focal length and the two surfaces and their mutual distance:

The refractive index before and after the lens is the same and amounts to. The refractive index of the lens material. The focal lengths of a surface are derived and loud above: ,, . This information is the final result for the focal lengths:

The focal lengths of the lens material are functions ( ) and the lens geometry ( radii of the boundary surfaces and thickness).

If the lens is relatively thin ( in the thin lens is by definition ) to the above equation reduces to

With the specifications above, the position of the principal planes is determined.

The removal of the image-side principal plane from the apex ( in the illustration ) the image-side surface is

The same applies to the object side:

When the lens is relatively thin (), these distances are equal to zero. The main planes remain on the apexes of the surfaces.

The figure shows the results after the above expressions for the focal lengths of the surfaces have been used ( equations (3) and (2 ), with and ).

The positions of the principal planes such as the focal lengths of the lens material functions ( ) and the lens geometry ( the radii of the boundary surfaces and thickness).

Multiple or compound lenses

Optical systems such as microscopes, telescopes, and lenses contain multiple lenses. You can be assigned as a unit common focal lengths and principal planes.

To reduce aberrations, even theoretically possible as a single lens components of several lenses are often combined. When two contacting surfaces have the same curvature, these two single lenses can be cemented together. If the individual lenses are thin, is also the distance between them is small, so that the combination itself as a thin lens can be treated.

Aberrations

The laws of ray optics are sufficiently accurate only for thin lenses. With increasing lens curvature and thickness take some aberrations - including lens aberrations or aberrations ( aberrations ) called - appreciable size. These deviations from the ideal optical imaging through a lens or lens system cause a blurry or distorted image of the subject or object being photographed.

The most important aberrations are

The former error is usually caused by the spherical lens cut and the refraction of light, which separates white light into its spectral colors. Both can be largely off (see Achromat and Apochromat ) by combining two or more lenses.

On the other hand require astigmatism, coma and distortion more complicated measures, such as aspherical cut shapes, the combination of several lens groups (eg Anastigmat lenses, wide-angle lenses ) or simply the restriction to near-axis rays, eg by reducing the aperture or a smaller field of view.

Surface coating

In a real lens, a portion of the light at the surface is always reflected. At an air - glass interface, this is approximately 4 % of the incident intensity (with a refractive index of the glass of N = 1.5) and thus at a lens about 8%, and it can also cause multiple reflections from the lens surfaces. In the optical assemblies are made ​​up of several Lisen, such as lenses, the reflection loss would increase quadratically with the number of lens surfaces. Furthermore, multiple reflected at the interfaces light in addition to the useful signal from the system leak and lead to distortions of the image. To avoid this, the lens surfaces are usually provided with an anti -reflective coating, one also speaks of surface finish. Avoidance or reduction of the effect described is achieved by destructive interference of the reflected beams in the anti-reflection coatings. (see also: the use of thin layers in the optical system )

Other lens types

• Liquid lens
• Fourier lens
• Fresnel lens: Used to collimating a beam of light, such as in overhead projector or the lighthouse. Also use their headlights principle. Also magnifiers can be run as a Fresnel lens.
• Gas lens: the influence by the density of optical density of gases is used to focus high power laser radiation. For this purpose, the light is passed through a rotating pipe, which takes the gas moved along the Bernoulli effect due to lens characteristics.
• Gravitational lens: Can be formed by a massive astronomical object such as a black hole. By distant galaxies are distorted as arcs or multiple points Chance.
• Diffractive optical elements ( DO lens ) are sporadically used in lenses for SLR cameras.

Due to the wave nature of matter can also be operated with particle optics. An application happens in the electron microscope, where specially arranged electric fields and magnetic fields to focus electrons and distracted. The same happens in particle accelerators in nuclear and high energy physics.

Further reading and sources