## Lenses | <<< | >>> |

Definition: transparent optical devices affecting the wavefront curvature of light

German: Linsen

An optical lens consists of a transparent medium, where light enters on one side and exits on the opposite side. Its purpose is to modify the wavefront curvature of the light, which implies that light is focused or defocused. Some typical examples:

- A collimated beam, having approximately flat wavefronts, is converted into a beam where the wavefronts are curved such that the beam converges to a focus.
The lens then acts as a
*focusing lens*– see Figure 1 (a). - The same kind of lens can convert a diverging beam into a collimated beam; it then acts as a
*collimating lens*. This can also be seen in Figure 1 (a) if you consider the incident beam to come from the right side. - Other lenses, having concave surfaces, can make a collimated or convergent beam divergent – see Figure 1 (b). Such lenses could also be used to collimate a beam which is originally convergent.

Although the change in beam radius is often regarded as a the actual function of a lens, its essential function is the change in wavefront curvature, which is the actual reason for changes in beam radius during propagation after the lens. (Note that the optical energy always propagates in a direction which is perpendicular to the wavefronts.) This is illustrated in Figure 2.

## Physical Origin of Wavefront Changes

For most lenses, the produced wavefront changes arise from the curvature of at least one of the surfaces. For the typical biconvex lens (i.e., a lens with two convex surfaces) as seen in Figure 2, the optical phase delay for light getting through the lens near its center is larger than for light propagating further away from the center, where the lens is thinner. This is because the refractive index of the lens material is larger than that of the surrounding medium (normally air). The radially varying phase delay directly implies the change in curvature of the wavefronts.

An alternative physical explanation is refraction at the lens surfaces. Particularly for thick lenses (see below), a detailed calculation based on refraction is more accurate than a calculation based on a radially varying phase delay, which ignores possible changes in beam size within the lens.

There are also *gradient index lenses* (*GRIN lenses*), where the refractive index systematically varies within the lens material.
For a focusing GRIN lens, the refractive index is highest near the center and lower outside; there should be an approximately parabolic change in refractive index with increasing radial position.
The surfaces of a GRIN lens are normally flat, so that it looks like an ordinary plate or (more often) like a cylindrical rod.

## The Focal Length

The focal length *f* of a focusing lens is the distance from the lens to a focus behind it, if the lens is hit by a collimated beam (see Figure 1 (a)).
For a defocusing lens, the focal length is negative: it is minus the distance to the virtual focus (see Figure 1 (b)).

The *dioptric power* (or *focusing power*) of a lens is the inverse of the focal length.

Ordinary lenses as used in laser technology have focal lengths somewhere between 10 mm and many meters. Tiny aspheric lenses (see below) can easily reach a few millimeters, sometimes even well below 1 mm.

An ideal lens with a given focal length *f* creates a radially varying phase delay for a laser beam according to the following equation:

This formula ignores the constant part of the phase change as well as aberrations.

## Lensmaker's Equation

The following equation, called *lensmaker's equation*, allows one to calculate the focal length of lens made of a material with refractive index *n* and with curvature radii *R*_{1} and *R*_{2} on the two surfaces:

The curvature radii are taken positive for convex surfaces and negative for concave surfaces. The last term is relevant only for thick lenses (see below) with substantial curvature on both sides. The equation holds for paraxial rays, not too far from the symmetry axis, and assumes that the ambient medium has a refractive index close to 1 (as is the case for air).

Note that different sign conventions are used in the literature. For example, a common convention takes the radius for the second interface as positive if the surface is concave. This is opposite to the convention used above.

## Thin and Thick Lenses

In many practical cases, a lens is so thin that the beam radius does not change appreciably within the lens. This is often the case for lenses with weak surface curvatures (i.e., large curvature radii). The third term in lensmaker's equation (see above) is then negligible, and by dropping it one obtains the simplified thin lens equation.

Thick lenses are often encountered when a large focusing power is required.
The thickness *d* (the distance between the lens surfaces as measured on the axis) then also has a significant influence on the focal length, as can be seen in lensmaker's equation.
Note also that the exact definition of the position of a thick lens, and thus also of its focal length, is not obvious for a thick lens, at least when it is asymmetric.

The distinction between thin and thick lenses is mostly a matter of approximations in calculations, rather than of practical use of lenses.

## The Lens Equation

If a divergent (rather than collimated) beam hits a focusing lens, the distance *b* from the lens to the focus becomes larger than the focal length *f* (see Figure 3).
This can be calculated with the *lens equation*

where *a* is the distance from the original focus to the lens.
This shows that *b ≈ f* if *a >> f*, but *b > f* otherwise.
That relation can be intuitively understood: a focusing power 1 / *a* would be required to collimate the incident beam (i.e. to remove its beam divergence), so that only a focusing power 1 / *f −* 1 / *a* is left for focusing.

If *a ≤ f*, the equation cannot be fulfilled: the lens can then not focus the beam.

Note that the lens equation applies for rays, assuming that the paraxial approximation is valid, i.e., all angles relative to the beam axis remain small.

## Numerical Aperture and f Number of a Lens

The numerical aperture (NA) of a lens is defined as the sine of the angle of the marginal ray coming from the focal point, multiplied with the refractive index of the medium from which the input beam comes. The NA of a lens (and not its focal length) is what limits the size of a beam waist which can be formed with that lens. Lenses with rather high NA (of the order of 0.5 to 0.9) are required e.g. for players and recorders of data storage media such as CDs, DVDs and Blu-ray Discs. In a microscope, the NA limits the image resolution obtainable.

High-NA lenses are also required for collimating laser beams which originate from small apertures. For example, this is the case for low-power single-mode laser diodes. When a lens with too low NA is used, the resulting collimated beam can be distorted (aberrated) or even truncated.

Obviously, a high-NA lens has to be relatively large if it has a large focal length.

The nominal numerical aperture of a lens may be smaller than what would be possible geometrically based on the open aperture, because operation in the very peripheral region may lead to excessive aberrations.

For camera lenses, an *f number* is often specified.
For example, a f/4 lens is one where the open aperture has a diameter of one quarter of the focal length.
(Note that *f*, but not the *f number* is the focal length!)
Assuming that the lens can be used up to its edge, this implies a numerical aperture of sin(1 / 4) ≈ 0.247 – in practice probably a bit less.

## Biconvex, Plano-convex, Biconcave, Plano-Concave and Meniscus Lenses

The focusing lenses shown in the figures above are all biconvex, i.e., convex on both sides. Plano-convex lenses are plane on one side and convex on the other one. Also, it is possible to make biconvex lenses with different curvature radii on both sides. Similarly, defocusing lenses can be biconcave or plano-concave.

According to lensmaker's equation (see above), a certain dioptric power can be achieved with different lens designs. However, they differ in terms of aberrations (imaging errors). For imaging a small spot to a spot of equal size, the symmetric biconvex lens is best suited. For an asymmetric application, such as focusing a collimating beam or collimating a strongly divergent beam, a plano-convex lens can be more appropriate. It should be oriented such that the curved surface is on the side of the collimated beam. Both lens surfaces then contribute to the focusing action.

*Meniscus lenses* are convex–concave, i.e., convex on one side and concave on the other side.
The contributions of both sides to the dioptric power cancel each other partially; overall the lens can be positive (focusing) or negative (defocusing).
Meniscus lenses are often used as *corrective lenses* in objectives: their main function is to correct for image aberrations.
They are also useful for condensers in illuminating systems.

*Doublet lenses* are made by bonding two lenses together, which consist of different materials.
Most common are *achromatic doublets* (see below).

## Cylindrical and Astigmatic Lenses

It is possible to have the curvature of a lens surface only in the horizontal direction, for example, but not in the vertical direction.
That *cylindrical lens* will then focus or defocus only in the horizontal direction while not affecting the wavefront curvature in the vertical direction.

Cylindrical lenses can be used to obtain a beam focus of elliptical size, or to generate or compensate astigmatism of a beam or an optical system, and is relatively difficult to fabricate.

If there is a curvature in both directions, but not of equal strength, one has an *astigmatic lens*.
It can be used, for example, to correct astigmatism from other sources.

## Aberrations Caused by Lenses

Lenses cause various types of aberrations (image degradations):

- Most lenses have spherical surfaces, simply because these are the easiest to fabricate.
However, a spherical surface more or less deviates from the ideal one, and this leads to image aberrations (particularly in peripheral regions) or to the deterioration of laser beam quality.
These are called
*spherical aberrations*. Aspheric lenses (see below) can have strongly reduced spherical aberrations. - When a collimated beam hits a lens with some angle against its symmetry axis, the resulting focused spot will be somewhat deformed.
That type of imaging error is caused
*coma*. It can be minimized by using an combination of curvature radii on both sides which is appropriate for the application. *Chromatic aberrations*arise from the chromatic dispersion of the material from which the lens is made. A typical consequence is that the focal length is somewhat wavelength-dependent, so that white light cannot be perfectly focused: different wavelength components have their focus at different points. Achromatic lenses (see below) exhibit strongly reduced chromatic aberrations.- When a laser beam with too large beam radius hits a lens, its beam profile can be truncated at the edges.
This can lead to substantial beam distortions.
Such
*aperture diffraction*also occurs in imaging applications; the finite size of a lens can limit the image resolution of an optical system. However, the image quality is not necessarily limited by diffraction, if the used optical components and the optical design are not of a very high quality.

Such imaging errors can often be substantially reduced by suitably combining several lenses. This is the reason why objectives usually consist of a substantial number of lenses.

## Aspheric Lenses

Although spherical aberrations can be largely compensated by suitably combining several lenses, it is sometimes preferable to use *aspheric lenses*, where the surface shape deviates from a spherical one.
It is then possible to obtain a good imaging quality (with low spherical aberrations) already with a single lens, or with fewer lenses in an objective.
However, aspheric lenses are more difficult to fabricate and thus more expensive.

## Achromatic Lenses

The most common approach for obtaining an *achromatic lens*, i.e., a lens with strongly reduced chromatic aberrations (see above), is to bond two lenses together, which consists of different materials (see the right side of Figure 4).
For example, one can combine a low-index crown glass of biconvex shape with a high-index flint glass of plano-concave shape to obtain such an *achromatic doublet*.
The curvature radii at the bonding interface are calculated for minimum chromatic dispersion, and must of course be precisely equal.

## Multiple Element Lenses

In many situations, it is difficult to accomplish various goals concerning properties like minimized aberrations with a single lens. It can then be better to use multiple element lenses, i.e., combinations of lenses. One particularly often uses lens doublets (consisting of two lenses) and triplets (with three lenses), but there are devices containing even more lenses. (In the previous paragraph, achromatic doublets have been mentioned already.)

The single lenses can be either cemented together or mounted with some air space in between. In any case, the multiple element lens is used like a single optical element.

## Coatings for Lens Surfaces

Many lenses have anti-reflection coatings on their surfaces, which substantially reduce the reflections caused by the refractive index change at the surface. Note, however, that this works only in a limited wavelength range. There is a trade-off between a high suppression of reflections and a broad operation bandwidth.

There are also *abrasion-resistant coatings*, making lenses more robust.

## Applications of Optical Lenses

The applications of lenses are manifold:

- Single lenses are commonly used as correction glasses, which can to some extent compensate visual impairments.
- Single lenses can be used for magnifying images, but multiple element lenses (objectives) are more common for such purposes. Examples are photographic objectives, microscope objectives and projection objectives.
- In laser technology, lenses are often used for collimating or focusing of laser beams. Particularly for large diffraction-limited beams focused with high-NA lenses, the achieved focal spot sizes can be very small (potentially with a beam waist radius below 1 μm).
- Lenses can also be used for mode formation in laser resonators, although curved mirrors are more common for that purpose. Disadvantages of lenses, when comparing with curved mirrors, are the reflection losses and chromatic aberrations. On the other hand, they allow for tight focusing without astigmatism. Such considerations are also relevant, for example, when using lenses (or mirrors) for focusing ultrashort pulses of light.

See also: focal length, refraction, numerical aperture, anti-reflection coatings, chromatic aberrations, achromatic optics, thermal lensing

and other articles in the categories general optics, photonic devices

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