# Lens (optics)

Simple biconvex lens (converging lens)

In optics, lenses are defined as transparent panes, of which at least one of each two surfaces is rotationally symmetrical - mostly spherical or spherical - curved. Light passing through is refracted at the surfaces and deflected towards the center of the light beam (collected, converging lens ) or scattered outwards ( diffusing lens ). A convex surface collects the light , a concave surface diffuses it.

The Mangin mirror is a combination of a lens and a mirror . The second surface is mirrored, which reflects the light. The additional deflection (collecting / scattering) by mirroring corresponds to the breaking on the non-mirrored second surface.

Individual lenses are combined with one another to correct image errors . Often two or more lenses are cemented together (the corresponding contact points have the same curvature), so that lens groups that are to be treated like individual lenses are created.

## history

### Antiquity

According to the archaeologists George Sines and Yannis A. Sakellarakis, many man-made lenses from antiquity, which were made of crystal (mostly quartz ), have already been discovered , but due to a lack of written sources it is not known whether these were used as visual aids or simply as magnifying glasses for making a fire were used. The oldest such artifact is the so-called Nimrud lens , which comes from Assyria in the 7th century BC. During archaeological excavations, Egyptian wall paintings from the 8th century BC were also discovered, which may represent the magnifying property of lenses, but this interpretation is controversial.

The oldest clear written description of burning glasses we have is the play The Clouds by the Greek poet Aristophanes , written in 423 BC. Was first performed. Pliny the Elder reports that Emperor Nero used an emerald to correct his nearsightedness , through which he could watch the gladiatorial games in the arena from his box. Both Pliny and Seneca described the phenomenon whereby objects viewed through glass spheres filled with water appear enlarged.

However, most ancient philosophers did not assume that light from objects falls into the eye, but instead followed the teaching of Empedocles from the fifth century BC , according to which the eye would actively fix and scan objects, so that there is still no adequate understanding refractive optics could be developed. Although Euclid did not set up his own theory of light, he criticized the teaching of Empedocles with the question of how the eye could almost directly reach the distant stars, and developed the first useful principles of geometrical optics for natural vision by using straight lines between the eye and Object went out. This was true in the ancient painting already realistic three-dimensional, with vanishing points working perspective along with mathematically exact foreshortening possible, which was used in the Greek set design for the theater and in the Roman wall painting, for analysis and description of the broken lenses light was good for them But not theory.

Although Lucretius subsequently developed in his work De rerum natura , which 55 BC BC appeared, a light particle theory independent of the human eye, but could not establish itself with it before the end of antiquity. In the first century AD, Heron of Alexandria studied reflection on the basis of simple Euclidean optics and, based on this, Claudius Ptolemy measured the exact refractive index of various transparent materials such as water, various crystals and glass, the latter also in the form of curved lenses. Although Ptolemy established a clear connection between the angle of refraction and the degree of curvature in this way, he could not theoretically explain his empirical measurement results, since he too was based on the teaching of Empedocles of the actively scanning eye. However, he was the first to expand the supposedly thin scanning beam of the eye to a conical viewing angle of the entire field of view , which he recognized as an independent factor of optics and perspective and which later became important in the form of the image angle determined by the recording format and focal length in the broken optics of lenses.

### middle Ages

Around 1050, the Vikings buried a treasure on Gotland , under which the rimmed, richly decorated, aspherical so-called Visby lenses , made from rock crystal , whose age has not yet been determined and which have a processing comparable to that of industrially manufactured high-precision lenses in the mid-20th century - and picture quality. It is believed that the Visby lenses could have come from Byzantium via trade links between the Varangians . Rodenstock made replicas of the Visby lenses in 1989.

Modern optics begins with the Arab philosopher al-Kindī , who in the 9th century developed the theory that is valid today, according to which the eye does not scan objects but, conversely, the light falls on the eye. Building on this, the Persian mathematician Ibn Sahl discovered Snellius' law of refraction in the 10th century , which for the first time enabled the exact calculation of the focal point and the lens shape required for a certain optical function.

Another student of al-Kindi was Alhazen , who in the 11th century finally summarized in his seven-volume treasure trove of optics all the ancient Greco-Roman as well as more recent Arab insights into optics and also Euclid's simple geometric optics with al-Kindi's theory of the incident light rays combined. By translating this basic work into Latin as De aspectibus or Perspectiva from the middle of the 13th century, medieval Europe first learned about the theory of incident light rays and the exact calculation of optical lenses.

After the translation of Alhazen's work, the content was taken up anew by European monks (among the first was the Franciscan Roger Bacon , who understood the light reflected by the objects under the designation species as an inherent force) and constructed the reading stone , an over-hemispherical one Plano-convex lens with which it was possible to look at writing enlarged. This lens was mostly made of beryl , from which the word glasses goes back. At the end of the 13th century, collecting lenses were first used in reading glasses to correct farsightedness or presbyopia . The center of this lens production was initially Italy, later also France and Holland.

The first optical apparatus to combine several lenses in a row were the microscope and telescope , which were invented in the late 16th and early 17th centuries, respectively.

## Basic properties

Radii of a converging lens: + R 1 (R 1 > 0); −R 2 (R 2 <0)
Radii of a divergent lens: -R 1 (R 1 <0); + R 2 (R 2 > 0)

Thin spherical lenses can be described by the following geometric and material properties:

• the diameter of the lens
• the radii of curvature of the entrance surface and the exit surface and${\ displaystyle \ textstyle R_ {1}}$${\ displaystyle \ textstyle R_ {2}}$
• the refractive index of the lens material.${\ displaystyle \ textstyle n '}$
From these, in connection with the refractive index of the surrounding material, the focal length and the refractive index can be derived as the most important optical properties:${\ displaystyle \ textstyle n}$ ${\ displaystyle \ textstyle f}$ ${\ displaystyle \ textstyle D}$
${\ displaystyle D = {\ frac {1} {f}} = {\ frac {n'-n} {n}} \ left ({\ frac {1} {R_ {1}}} - {\ frac { 1} {R_ {2}}} \ right)}$ .

This is the so-called lens grinder formula , which is a good approximation for thin lenses (i.e. the thickness of the lens is much smaller than both spherical radii). The exact variant below also takes the lens thickness into account, if this can no longer be neglected.

Thick lenses - these are in particular lenses that have a finite thickness at their thinnest point - also require the following information:

• the thickness of the lens in the middle ${\ displaystyle \ textstyle d}$
A thick lens has a different focal length than a thin lens with otherwise the same parameters; Furthermore, there are two main planes that are no longer superimposed, as the beam offset can no longer be neglected when passing through the lens (not axially parallel):
${\ displaystyle D = {\ frac {1} {f}} = {\ frac {n'-n} {n}} \ left ({\ frac {1} {R_ {1}}} - {\ frac { 1} {R_ {2}}} \ right) + {\ frac {(n'-n) ^ {2} \, d} {n \, n '\, R_ {1} \, R_ {2}} }}$.
If and denote the refractive power of the front and rear surfaces of the lens, the total refractive power of the lens can be expressed as${\ displaystyle D_ {1} = {\ frac {n'-n} {n}} \ cdot {\ frac {1} {R_ {1}}}}$${\ displaystyle D_ {2} = - {\ frac {n'-n} {n}} \ cdot {\ frac {1} {R_ {2}}}}$
${\ displaystyle D = D_ {1} + D_ {2} - {\ frac {n} {n '}} dD_ {1} D_ {2}}$
write what is known as the Gullstrand formula, especially in ophthalmic optics.

Furthermore, the outer appearance of the lens results from the radii of curvature, i. H. whether it is a (bi) concave or (bi) convex lens or one of the other shapes.

More detailed considerations lead to the subject of the inevitable imaging errors and other errors due to errors and inaccuracies in production (material errors, tolerances in grinding, assembly errors).

## Manufacture and materials used

Lenses for use in the visible spectral range are made from optical glasses or plastics such as polycarbonates , polymethyl methacrylates or cyclo-olefin (co) polymers . Furthermore, in contrast to these amorphous materials , the use of crystalline materials is also possible, such as calcium fluoride or sapphire .

Blanks for glass lenses are manufactured differently depending on the size and quality requirements:

• small, low requirements:
• Manufacture the lenses directly by hot pressing
• There are inhomogeneities in the refractive index, which are also anisotropic, due to mechanical stresses
• can be reduced by subsequent tempering
• higher requirements (precision optics):
• Archetypes: Glass blocks are poured and slowly cooled. Cooling time for coarse cooling: a few days, fine cooling: several weeks to a few months
• Cut-off grinding: Glass blocks are cut up by cut-off grinding: using circular milling in cylinders, these are then further cut into slices.

This is followed by grinding and polishing :

• Rough grinding by means of milling (remaining allowance: 100 to 200 µm)
• Fine grinding / lapping using diamond grains (remaining allowance: around 1 µm)
• Polishing with polishing red or cerium oxide (roughness: <λ / 10)
• Centering (grinding the edge to determine the optical axis)
• (In the case of aspherical lenses manufactured by molding , hot forming follows here)

If the quality requirements are lower, the blanks pressed at high temperatures can be used directly. Plastic lenses can be manufactured by injection molding or injection compression molding as well as by classic grinding and polishing.

With the aid of the geometrical sizes diameter lens radius, center thickness, supplemented with manufacturing tolerances (z. B. pass fault tolerance including average wavefront error), and the material properties of refractive index , Abbe number and stress birefringence , supplemented by material tolerances (z. B. homogeneity ), the optical Properties of a spherical lens fully described. The most important parameter of a lens for its imaging function is the focal length (unit: meter), i. H. the distance between the focal point or focal plane and main planes . The reciprocal of the focal length is given as the refractive index (unit: diopters ). The diameter of the usable area of ​​a lens is called the opening or aperture .

An important property of all systems that can be described by beam optics is the principle of reversing the light path: If a light beam incident from one side is followed along its path, a light beam incident in the opposite direction will follow this path in exactly the opposite way.

## Different lens shapes

### Spherical lenses

With the simplest lenses, the two optically active surfaces are spherical. That is, they are sections of the surface of a sphere. One differentiates:

Designation of lenses according to their refractive power or the curvature of their surfaces. The convex-concave lens is mirror-inverted here.
• Converging lenses with two convex surfaces or with one convex and one flat surface, at least in the middle, in the area of ​​the optical axis, thicker than at the edge; a bundle ofincident light rays parallel to the optical axis is ideallycollectedat a point behind the lens, the focal point or focus F. Their focal length f is positive.
• Diverging lenses ( diverging lenses ) with two concave surfaces ( biconcave ) or with one concave and one flat surface ( plano-concave ), at least thicker at the edge than in the middle; a bundle of incident parallel rays diverges behind the lens as if it came from a point on the incident side of the light. The focal length is negative.

In both groups there are lenses that have both a concave and a convex surface. Such lenses are often used to correct aberrations in optical systems with multiple lenses. They are converging lenses if the convex surface is more curved or diverging lenses if the concave surface is more curved. Originally only the former meniscus lenses were called (from the Greek. Μηνίσκος mēnískos, half-moon ), while the latter is now considered negative menisci are called.

A component with two plane parallel optically active surfaces and is called plane-parallel plate or plane parallel plate .

For calculation according to the rules of geometrical optics , the radii following one another in the direction of light are designated by R 1 and R 2 (with R 3 and R 4 ) according to DIN 1335 . The associated sign does not differentiate directly between convex and concave surfaces. The radius of a surface is defined positively if the light first passes the surface, then its center of curvature. If the order is reversed, the radius is defined negatively. In graphic representations, the light comes conventionally from the left (or from above).

The following signs result for the three surfaces convex, plane (plane) or concave:

• Convex surface (it is curved outwards): + R 1 (R 1  > 0) or −R 2 (R 2  <0).
• Flat surface (its curvature is zero): R = ± .
• Concave surface (it is curved inwards): −R 1 (R 1  <0) or + R 2 (R 2  > 0).

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

Due to their principle, spherical lenses lead to spherical aberration because the focal point of the marginal rays does not coincide with the focal point of the rays close to the axis, possibly also depending on the wavelength of the light. In order to reduce these errors, lens systems ( Anastigmate , Cooke triplet , Tessar ) are used, which largely compensate for the errors.

### Aspherical lenses

Aspherical lenses have more degrees of freedom in design and allow better correction of an optical system than a spherical lens. Many aspheres only deviate slightly from a spherical surface. On the other hand, there are also free-form lenses with complex, non-rotationally symmetrical surfaces. The disadvantages of aspherical lenses are increased manufacturing costs and a lower surface quality. A typical effect are grooves (which you can always clearly see in the bokeh) that arise either during the grinding itself or during the manufacture of the pressing tool.

Another category are gradient lenses in which the refractive index constantly changes spatially. Here, light is not only refracted at interfaces, but also in the glass itself. With them, effects similar to those with aspheres can be achieved.

### Ideal lens

For two limited purposes, there are lens shapes that have no aberration for monochromatic light.

• Precise bundling of incident light parallel to the optical axis at one point:
One possibility is that the surface of the lens facing the incident light is flat and the opposite side has the shape of a hyperboloid . For half the opening angle of the asymptotic cone belonging to the hyperboloid must apply, with the refractive index of the lens material. The incident rays are bundled in one of the two
hyperbolic focal points - the one with the greater distance from the vertex of the lens.${\ displaystyle \ alpha}$${\ displaystyle n \ cdot \ cos \ alpha = 1}$${\ displaystyle n}$
• Optical path of the same length for all rays that originate in a point on the optical axis up to the common image point:
The plane surface of the lens is replaced by a sphere around this point and the hyperbolic surface by a Cartesian oval . The mapping is done according to Fermat's principle . In the event that neighboring points of the original image are to be mapped uniformly onto neighboring points of the image, such considerations are even more complex.

### Astigmatic lenses

Cylinder lens as the borderline case of an astigmatic lens
A: converging lens, B: diverging lens

Astigmatic lenses have focal lengths of different sizes in two radial directions that are perpendicular to one another. The limit case is the cylinder lens , which has plane-parallel surface contours in one of the two directions and in its typical shape is actually a cylinder section: a cylindrical and a plane surface. It bundles parallel incident light on a focal line.

Astigmatic lenses are used in the following cases:

### Elastic lenses

Elastic lens refers to a lens that changes the refractive power by deforming an elastic solid. The functional principle results in the following advantages:

• The shape of the interface can be freely selected (spherical, aspherical).
• The size of the change in the refractive power is very large when using rubber materials (approx. 15 D).
• The deformation can take place very quickly.

The eye makes use of this principle of action, but is also occasionally used in technology.

## Focal length and main planes

Refraction at a spherical interface: Abbe's invariant

The refractive property of a lens used for optical imaging depends on the refractive index of its material and on the shape of its interfaces. Both together express the focal length. In addition, two main planes must be specified, one on the object and one on the image as a reference plane for the object and the image-side focal length. The two focal lengths only differ if the optical medium in front of the lens is not identical to that after the lens.

Both the focal lengths and the main planes are ideal sizes that result when working according to the concept of paraxial optics . Within this concept, they can be theoretically specified, i.e. calculated, from the material and geometric properties. The refraction is examined separately at each of the two interfaces. Then the results and the mutual position of the surfaces are combined to form equations for the size of the focal lengths and the position of the main planes.

### Refraction at a single spherical interface

Focal length f 'at a spherical interface

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

If the back focal length is at infinity, the Abbe invariant becomes to and from ${\ displaystyle s}$${\ displaystyle s'}$${\ displaystyle f '}$

${\ displaystyle n \ left ({\ frac {1} {r}} - {\ frac {1} {s}} \ right) = n '\ left ({\ frac {1} {r}} - {\ frac {1} {s'}} \ right)}$

becomes:

${\ displaystyle f '= r {\ frac {n'} {n'-n}}}$ .

If the direction of the beam is reversed, the back focal length is at infinity, becomes , and the Abbe invariant becomes: ${\ displaystyle s'}$${\ displaystyle s}$${\ displaystyle f}$

${\ displaystyle f = r {\ frac {n} {n-n '}}}$.

The main plane goes through the vertex of the spherical surface. ${\ displaystyle S}$

### Refraction at a lens

Focal point and main plane on the image side for two surfaces

In the case of a lens, the refraction takes place at two, usually spherical, boundary surfaces. The common focal length can be found by observing the following specifications:

• The imaging of the image-side focal point of the first surface by the second surface is the image-side focal point of the lens, because all incident parallel rays pass both the one and the other point (red line in the adjacent figure).
• The extension of an axially parallel incident ray intersects with the refracted ray passing through the lens in the main plane of the lens on the image side (broken line in the adjacent figure). This is based on the definition of the main levels that the image scale between them is 1.

A basic connection in optical imaging is contained in the angle ratio : ${\ displaystyle \ gamma '}$

${\ displaystyle \ gamma '_ {2} = {\ frac {\ tan \ sigma' _ {2}} {\ tan \ sigma _ {2}}} = {\ frac {t} {f '_ {2} }}}$ .

This makes it possible to find point P through which the red line must lead.

The equation for the image-side focal length of the lens reads with the focal lengths and the two surfaces and their mutual distance : ${\ displaystyle f '_ {2}, f_ {2}}$${\ displaystyle f '_ {1}}$${\ displaystyle d}$

${\ displaystyle f '= {\ frac {f' _ {1} f '_ {2}} {d-f' _ {1} -f_ {2}}}}$ .

The refractive index before and after the lens is the same and is equal to . The refractive index of the lens material is   . The focal lengths of an area derived above and are as follows:   ,   ,   . With this information, the final result for the focal lengths is: ${\ displaystyle n}$${\ displaystyle n '}$${\ displaystyle f_ {1} '= r_ {1} {\ frac {n'} {n'-n}}}$${\ displaystyle f_ {2} '= r_ {2} {\ frac {n} {n-n'}} = - r_ {2} {\ frac {n} {n'-n}}}$${\ displaystyle f_ {2} = r_ {2} {\ frac {n '} {n'-n}}}$

${\ displaystyle {\ frac {1} {f '}} = {\ frac {1} {f}} = {\ frac {n'-n} {n}} \ left ({\ frac {1} {r_ {1}}} - {\ frac {1} {r_ {2}}} \ right) + {\ frac {(n'-n) ^ {2} d} {n'nr_ {1} r_ {2} }}}$ .

The focal lengths are functions of the lens material ( ) and the lens geometry (radii of the interfaces and thickness). ${\ displaystyle n '/ n}$

Lens, general: equations and a.
Lens designed for focal length / s (1) and position of the main planes (3) and (2) : Calculation results for focal lengths and position of the main planes

If the lens is relatively thin (  when the thin lens is by definition ), the above equation is shortened to ${\ displaystyle d \ ll r_ {1}, r_ {2}}$${\ displaystyle d = 0}$

${\ displaystyle {\ frac {1} {f}} = {\ frac {1} {f '}} = {\ frac {n'-n} {n}} \ left ({\ frac {1} {r_ {1}}} - {\ frac {1} {r_ {2}}} \ right)}$ .

The position of the main levels is also determined with the above specifications.

The distance of the main plane on the image side from the vertex (  in the adjacent figure) of the image-side surface is ${\ displaystyle H '}$${\ displaystyle H_ {2}}$${\ displaystyle S_ {2}}$

${\ displaystyle {\ overline {H_ {2} H '}} = {\ frac {f' _ {2} d} {d-f '_ {1} -f_ {2}}}}$ .

The same applies to the item page:

${\ displaystyle {\ overline {H_ {1} H}} = {\ frac {-f_ {1} d} {d-f '_ {1} -f_ {2}}}}$ .

If the lens is relatively thin ( ) these distances become zero. The main planes remain on the vertices of the faces. ${\ displaystyle d \ ll f '_ {1}, f_ {2}}$

The figure opposite shows the results after the above expressions have been used for the focal lengths of the surfaces (equations (3) and (2); with   and ). ${\ displaystyle n = 1}$${\ displaystyle n '= n}$

Like the focal lengths, the positions of the main planes are functions of the lens material ( ) and the lens geometry (radii of the interfaces and thickness). ${\ displaystyle n '/ n}$

### Multiple and compound lenses

Optical systems such as microscopes , telescopes, and objectives contain multiple lenses. An equivalent focal length and main plane can be assigned to them as a unit . The calculation of the focal length and main planes can be carried out very efficiently using the matrix optics, assuming the paraxial approximation .

In order to reduce imaging errors, components that are theoretically conceivable as individual lenses are often composed of several lenses. If two contact surfaces have the same curvature, these two individual lenses can be cemented together. When the individual lenses are thin, the distance between them is also small, so that the combination itself can be treated like a thin lens.

## Image errors

Deviations from the optical image of an ideal lens or lens system result in a blurred or distorted image of the object being depicted.

The most important imaging errors are

1. the spherical aberration and the chromatic aberration
2. the astigmatism and the coma
3. the lens deflection for lens sizes over about 60 cm.
4. the curvature of field and the distortion .

The first-mentioned defects are caused by the usually spherical lens cut and the dispersion of the glass. Both can be reduced by combining two or more lenses ( see achromat and apochromat ).

On the other hand, astigmatism, coma and distortions require more complicated measures, such as aspherical cuts, the combination of several lens groups ( anastigmat optics, wide-angle lenses ) or simply the restriction to rays close to the axis by reducing the aperture or a smaller field of view .

## Surface finishing

With a real lens, part of the light is always reflected on the surface . In the case of an air-glass interface ( refractive index of the glass: n = 1.5) this is about 4 percent of the incident intensity, i. H. with a lens about 8 percent. In optical assemblies that are made up of several lenses, such as objectives , the losses continue to increase almost linearly. The scattering losses of a five-element lens would increase to 34 percent and that of a ten-element lens to 56 percent.

Furthermore, light that is reflected multiple times at the interfaces can exit the system in addition to the useful signal and lead to falsification of the image. To avoid this, the lens surfaces are usually provided with an anti-reflective coating, which is also known as surface coating . The avoidance or reduction of the effects described is achieved by destructive interference of the reflected rays in the anti-reflective layers. (See also: Application of thin layers in optics .)

## Special lens types and lens effects

Not only transparent components with refractive surfaces can create lens effects - i.e. the collection or dispersion of radiation . How to use electron specially arranged electric and magnetic fields to electron focusing. The same thing happens in particle accelerators in nuclear and high energy physics . A gravitational lens is used when a massive astronomical object, such as a black hole , causes lens effects. As a result, distant galaxies are occasionally distorted as arcs of a circle or into several points.

Commons : Lenses  - album with pictures, videos and audio files

## literature

• Wolfgang Demtröder : Experimental Physics. Volume 2: Electricity and Optics. 2nd, revised and expanded edition. corrected reprint. Springer, Berlin a. a. 2002, ISBN 3-540-65196-9 .
• Heinz Haferkorn: Optics. Physical-technical basics and applications. 4th, revised and expanded edition. Wiley-VCH, Weinheim 2003, ISBN 3-527-40372-8 .
• Miles V. Klein, Thomas E. Furtak: Optics. Springer, Berlin a. a. 1988, ISBN 3-540-18911-4 .
• Eugene Hecht: optics. 7th edition. De Gruyter, Berlin et al. 2018, ISBN 978-3-11-052664-6 .