# Aspherical lens

An aspherical lens is a lens with at least one refractive surface that deviates from a spherical or plane shape . The largely freely formable surface can avoid or reduce imaging errors that are unavoidable with spherical lenses. Specifically, the spherical aberration can be completely corrected. However, the production of an aspherical surface is generally much more complex than that of a spherical surface.

## shape

Arrow height with an aspherical lens

The shape of rotationally symmetrical aspherical surfaces is usually given as a conic section ( circle , ellipse , parabola , hyperbola ) plus a correction polynomial for deformations of a higher order. Non-rotationally symmetrical aspherical surfaces can be off-axis sections of such conic sections, but also optical surfaces that are freely defined in all directions (free-form aspheres).

${\ displaystyle z (r) = {\ frac {\ rho r ^ {2}} {1 + {\ sqrt {1- (1 + k) (\ rho r) ^ {2}}}}} + \ sum _ {i = 2} ^ {n} A _ {\ mathrm {2i}} \ cdot r ^ {2i} + \ sum _ {i = 1} ^ {m} A _ {\ mathrm {2i + 1}} \ cdot | r | ^ {2i + 1}}$

Formula according to DIN ISO 10110 Optics and Photonics - Creation of drawings for optical elements and systems , part 12 Aspherical surfaces with:

• ${\ displaystyle z}$= Arrow height
• ${\ displaystyle r}$= Distance perpendicular to the axis ( height of incidence )
• ${\ displaystyle \ rho}$= Vertex curvature (vertex radius )${\ displaystyle R = 1 / \ rho}$
• ${\ displaystyle k}$= conical constant
• ${\ displaystyle A_ {2i}}$ = even coefficients of the correction polynomial
• ${\ displaystyle A_ {2i + 1}}$ = odd coefficients of the correction polynomial (rarely used)
• ${\ displaystyle \ max (2n, 2m + 1)}$= Degree of the correction polynomial (usually is )${\ displaystyle m

The paraxial behavior of the aspherical surface is only determined by the curvature of the apex . ${\ displaystyle \ rho}$

Special cases of aspherical lenses are the cylindrical lens (constant radius of curvature in a cutting plane, infinite radius of curvature in the cutting plane perpendicular to it) and the toric lens (two different radii of curvature in two mutually perpendicular cutting planes).

### Calculation on a plano-convex lens

To the cutting length s in an optical image having a plano-convex aspherical lens having the principal plane H (green), the focal point F (red), the refractive index n = 1.5 and the radius of curvature R at a given incidence height H .

Using a plano-convex lens , the shape of the corresponding aspherical surface can be illustrated relatively easily. If one looks at an optical image from infinity with parallel, monochromatic light through such a lens with the radius of curvature at the height of incidence , the situation shown in the figure on the right results. ${\ displaystyle R}$ ${\ displaystyle H}$

In order to calculate the aspherical surface, rays of light can be considered which, with the height of incidence parallel to the optical axis , fall on the flat lens surface on the object side. These are not broken when entering the optically denser medium of the lens material with the refractive index , since they strike perpendicularly . On the image side, these rays form the angle to the surface perpendicular of the lens in the lens and outside the lens the angle . These angles behave as described by Snell's law of refraction . The following relationships apply: ${\ displaystyle H}$ ${\ displaystyle n}$${\ displaystyle \ alpha}$${\ displaystyle \ beta}$

${\ displaystyle \ sin (\ alpha) = {\ frac {H} {R}}}$
${\ displaystyle \ sin (\ beta) = {\ frac {n \ cdot H} {R}}}$

These rays then intersect the optical axis at the angle

${\ displaystyle \ gamma = \ beta - \ alpha}$

For paraxial rays ( ) is an image-side gives back focus respectively focal length of: ${\ displaystyle H \ rightarrow 0}$ ${\ displaystyle s_ {0}}$${\ displaystyle f}$

${\ displaystyle f = s_ {0} = R_ {0} \ cdot \ left ({\ frac {n} {n-1}} - 1 \ right)}$,

where the radius is at the vertex of the lens on the optical axis. ${\ displaystyle R_ {0}}$

The arrow height , measured from the main plane of the lens, can then be determined iteratively as a function of the height of incidence with the help of some auxiliary variables starting from and in steps of : ${\ displaystyle z}$${\ displaystyle H}$${\ displaystyle H_ {0} = 0}$${\ displaystyle \ Delta _ {0} = 0}$${\ displaystyle \ Delta H}$

${\ displaystyle H_ {i} = H_ {i-1} + \ Delta H}$
${\ displaystyle z_ {i} = \ Delta _ {i-1} + R_ {i-1} - {\ sqrt {R_ {i-1} ^ {2} -H_ {i} ^ {2}}}}$
${\ displaystyle \ gamma _ {i} = \ arctan {\ frac {H_ {i}} {f + z_ {i}}}}$
${\ displaystyle R_ {i} = {\ sqrt {\ left ({{\ frac {n \ cdot H_ {i}} {\ sin {\ gamma _ {i}}}} - f-z_ {i}} \ right) ^ {2} + H_ {i} ^ {2}}}}$
${\ displaystyle \ alpha _ {i} = \ arcsin {\ frac {H_ {i}} {R_ {i}}}}$
${\ displaystyle \ beta _ {i} = \ arcsin {\ frac {n \ cdot H_ {i}} {R_ {i}}}}$

The following applies to the back focal length from the vertex of the sphere with the radius on the optical axis: ${\ displaystyle s_ {i}}$${\ displaystyle R_ {i}}$

${\ displaystyle s_ {i} = {\ frac {n \ cdot H_ {i}} {\ sin \ gamma _ {i}}} - R_ {i}}$

Finally, the vertex distance from the main plane results from the difference between this focal length and the focal length for paraxial rays : ${\ displaystyle \ Delta _ {i}}$ ${\ displaystyle s_ {0}}$

${\ displaystyle \ Delta _ {i} = s_ {i} -s_ {0}}$

#### example

Design of a plano-convex, spherical lens with the heights of incidence H in steps of ten up to ± 90 with a refractive index of 1.5, a constant radius of curvature of 100 and a focal length of 200. With increasing height of incidence, the focal length measured by the main plane H continues to decrease, and Incident rays with heights of incidence of ± 70 and greater amounts are even totally reflected within the lens and therefore do not even contribute to the optical imaging. Only rays close to the axis intersect the optical axis in the vicinity of the focal point F.
Design of a plano-convex, aspherical lens with the heights of incidence H in steps of ten up to ± 100 according to the example table with a refractive index of 1.5, a radius of curvature at the vertex on the optical axis of 100 and a focal length of 200. The same results from the for all heights of incidence Main plane H measured back focal length, and all refracted rays intersect the optical axis at the focal point F.

The following table shows some example values ​​calculated in this way for , and the unitless linear dimensions and . As the height of incidence increases, the radii of curvature become larger and larger and both the centers and vertices of the corresponding circles move further and further away from the main plane on the object side. ${\ displaystyle n = 1 {,} 5}$${\ displaystyle R_ {0} = 100}$${\ displaystyle f = s_ {0} = 200}$

Incidence height
${\ displaystyle H}$
Arrow height
${\ displaystyle z}$
${\ displaystyle R}$
Vertex
distance
${\ displaystyle \ Delta}$
Angle in °
${\ displaystyle \ alpha}$
Angle in °
${\ displaystyle \ beta}$
Angle in °
${\ displaystyle \ gamma}$
0 0.0 100.0 0.0 0.0 0.0 0.0
10 0.5 101.1 0.0 5.7 8.5 2.9
20th 2.0 104.4 0.1 11.0 16.7 5.7
30th 4.5 109.7 0.3 15.9 24.2 8.3
40 7.8 116.7 0.8 20.0 30.9 10.9
50 12.0 125.2 1.6 23.5 36.8 13.3
60 16.9 134.8 2.8 26.4 41.9 15.5
70 22.4 145.3 4.5 28.8 46.3 17.5
80 28.5 156.6 6.5 30.7 50.0 19.3
90 34.9 168.5 8.9 32.3 53.2 21.0
100 41.8 180.8 11.6 33.6 56.0 22.5
110 48.9 193.6 14.6 34.6 58.5 23.8
120 56.3 206.6 17.9 35.5 60.6 25.1
130 63.9 219.9 21.4 36.2 62.5 26.2
140 71.7 233.4 25.0 36.9 64.1 27.3
150 79.6 247.1 28.9 37.4 65.6 28.2
160 87.7 260.9 32.9 37.8 66.9 29.1
170 95.8 274.9 37.0 38.2 68.1 29.9
180 104.1 288.9 41.2 38.5 69.2 30.6
190 112.4 303.0 45.5 38.8 70.1 31.3
200 120.9 317.3 49.9 39.1 71.0 31.9

Up to a height of incidence of 140, the convex surface of this lens according to DIN ISO 10110-12 (see above) without further aspherical parameters in the higher links corresponds relatively precisely to the relationship for a hyperboloid with the conical constant k = -2:

${\ displaystyle z (H) = {\ frac {H ^ {2}} {R_ {0} \ left (1 + {\ sqrt {1+ \ left ({\ frac {H} {R_ {0}}} \ right) ^ {2}}} \ right)}} = {\ frac {H ^ {2}} {R_ {0} + {\ sqrt {R_ {0} ^ {2} + H ^ {2}} }}}}$

## Applications

• Aspherical condenser lenses are used to focus light in projectors and spotlights and enable a higher light yield here , since the aperture can be enlarged without the spherical aberration disturbing.
• Aspherical spectacle lenses : the deviation from the spherical shape enables flatter, thinner, lighter and optically better spectacle lenses, especially for farsighted people ( hyperopes ), to be produced. In addition, with multifocal varifocals, varifocals are possible in which the lenses are not rotationally symmetrical and which compensate for the poorer accommodation in old age.
• High-quality eyepieces , especially wide-angle eyepieces for telescopes and binoculars with angles of view of up to 70 °, consist of up to 8 lenses, some of which are cemented together , and are sometimes provided with an aspherical surface.
• Zoom - (vario) optics with variable focal length , e.g. B. Photo lenses . These are all the more difficult to calculate and produce, the larger their focal length range, because the correction of the aberrations must be made as a compromise for all adjustable focal lengths. Such systems therefore often have many lenses, sometimes more than 15, and sometimes they can only be implemented using aspheres with acceptable aberrations. It can also make economic sense to use aspheres for simpler objectives, since these can be manufactured relatively inexpensively by molding (see below), and accordingly fewer lenses are needed to correct the errors sufficiently.
• Photo lenses with high speed or wide-angle lenses with a particularly large angle of view . If the aperture or the angle of view of the lenses is made very large, the aberrations increase significantly and require a high level of correction effort. Aspherical surfaces are helpful in correcting the errors well and at the same time preventing the number of lenses and the size and weight of the lens from increasing excessively.
• the aspherical correction plate of the Schmidt telescope . It almost completely eliminates the spherical aberration of the main mirror , which otherwise reduces the resolution or the image field.
• Focusing lenses for diode laser radiation can be aspherical to cope with the large apertures. Gradient lenses are an alternative .
• Used in exterior mirrors of cars, especially on the driver's side.

## Manufacturing

Aspherical surfaces can be made by a number of processes:

### grind

Grinding is the oldest, but also the most complex process for producing aspherical glass lenses. There have been camera lenses with such lenses for several decades, but until the molding process was ready for series production, they were limited to particularly high-quality and expensive lenses. Since the year 2000, machine technology based on CNC controls has developed so much that today (as of 2013) the use of CNC machines for the production of aspheres is common practice. CNC machining also enables quartz crystals or optics with large diameters to be machined, which cannot be produced at all or not in the required quality by means of molding.

### Impression

This method, which is cost-effective for series production, is often used for camera lenses, condenser lenses and for laser pick-up optics, for example in DVD players .

• Aspherical lenses made of plastic can be produced very inexpensively by molding. However, their dimensional accuracy and constancy of their properties are not good enough for camera lenses.
• A plastic layer with an aspherical surface can be pressed onto a spherical glass lens. The quality of such an element is sufficient for photo lenses of medium quality.
• For high-quality photo lenses, a glass lens is manufactured directly with a molded aspherical surface (molding). For this, however, you need suitable glasses with a transformation temperature that is not too high , because the material of the press die has only limited temperature resistance. You cannot use every optical glass .

### Magnetorheological polishing

As a magnetorheological polishing ( English Magneto Rheological Finishing , MRF) refers to a polishing method of optical components such as lenses. The method can also be used for local correction.

### Ion-beam figuring

Ion-beam figuring (also called ion-milling ) is a surface processing process in which the material is removed by means of an ion beam , a sandblaster at the atomic level , so to speak .

### Mechanical tension

The optics can be deformed during grinding by the application of force; it is then ground spherically. The spherical surface is demolded after releasing the tension and thus results in the asphere. An example of this is the Schmidt plate , which is deformed by a negative pressure and then ground flat on one side.

Alternatively, a spherical surface can be deformed into an asphere by the application of force.

## measurement

Metrology plays a decisive role in the manufacture of aspherical lenses. A distinction is made between different measuring tasks depending on the manufacturing process and processing status:

• Shape of the asphere
• Surface shape deviation
• Slope error
• Center thickness
• roughness

A distinction is made between tactile, i.e. touching, and non-contact measurement methods. The decision as to which process to use depends not only on the accuracy but also on the processing status.

### Tactile survey

Tactile measurements are carried out between two grinding steps in order to control the shape of the asphere and to adapt the subsequent work step. A section over the surface is measured with a so-called button. Due to the rotational symmetry of the lens, the combination of several of these profiles provides information about their shape. The disadvantage of tactile methods is the possible damage to the lens surface by the button.

### Optical / contactless measurement

Interferometers are used when sensitive or completely polished surfaces are measured. By superimposing a reference beam on the beam reflected from the surface to be measured, error maps, so-called interferograms, are created, which represent a full-surface deviation of the surface shape.

#### Computer generated hologram (CGH)

Computer-generated holograms (CGHs) represent a method for interferometric determination of the deviation of the lens from the target geometry. These generate an aspherical wavefront in the target shape and thus enable the determination of deviations of the lens from the target shape in an interference image. CGHs have to be specially manufactured for each test item and are therefore only economical for series production.

#### Interferometric measurement

Another possibility is the interferometric measurement of aspheres in partial areas, with minimal deviations from the best-fit sphere, and then combining the partial measurements into a full-area interferogram. These are very flexible compared to CGHs and are also suitable for the production of prototypes and small series.

## Individual evidence

1. Bernhard Braunecker, Rüdiger Hentschel, Hans J. Tiziani (eds.): Advanced Optics Using Aspherical Elements (=  SPIE PM . Volume 173 ). SPIE Press, 2008, ISBN 978-0-8194-6749-2 , pp. 53 .
2. Gérard R. Lemaitre: TRSS: A Three Reflection Sky Survey at Dome C with active optics modified-Rumsey telescope . In: Focus . tape 56 , p. 56 ( iap.fr [PDF; 1.4 MB ; accessed on February 23, 2012]).
3. Out-of-round perfection - a comparison of tactile measuring methods. asphericon GmbH, July 31, 2017, accessed on November 13, 2017 .
4. Unround in Perfection - Interferometric Measurement of Aspheres. asphericon GmbH, August 29, 2017, accessed on November 13, 2017 .