Lenses

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 f of a thin 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)).

For a thick lens (see below), the situation is more complicated. It generally has two principle planes, and the focal length for each side is the distance between that point and the corresponding focal point. If the medium surrounding the lens is the same on both sides, the two focal length values are identical. See the article on focal length for more details.

The dioptric power (or focusing power) of a lens is the inverse of the focal length, or the refractive index of the surrounding medium divided by the focal length.

Ordinary lenses as used in laser technology have focal lengths somewhere between 5 mm and many meters. Ball lenses and tiny aspheric lenses (see below) can easily be smaller, 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.

For more details, see the article on focal length.

Lensmaker's Equation

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. There are many other equations in optics where simplified versions are available for thin lenses, or which hold only for thin lenses. The criteria for a lens being thin can be different in such equations.

Thick lenses are often encountered when a large focusing power is required, and for compound lenses. 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.

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.

If a collimated Gaussian beam with beam radius w₀ hits a focusing lens with focal length f, the beam radius of the beam waist (focus) after the lens can be calculated with the equation

where it is assumed that the beam radius at the focus is much smaller than the initial beam radius w₀. (This condition is violated for beams with a too small incident radius; the focus is then larger than according to the given equation.) Also, it is assumed that the beam radius is significantly larger than the wavelength λ, so that the paraxial approximation is valid.

Calculator for Beam Waist Radius

Wavelength:
Focal length:
Initial beam radius:					(collimated input beam)
Beam quality factor:					(1 for Gaussian beams)
Waist radius:				calc

Enter input values with units, where appropriate. After you have modified some inputs, click the "calc" button to recalculate the output.

The numerical aperture (NA) of a focusing or collimation 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.

Note that the definition of numerical aperture is not based on the focal length for lenses or objectives which are designed for imaging from some object plane to an image plane. Here, the angular opening is considered from a point in the object plane. See the article on numerical aperture for more details.

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.

For camera lenses (photographic objectives), a minimum f-number (or range of f-numbers) is often specified. For example, a f/4 lens is one where the maximum open aperture (more precisely: the maximum entrance pupil) has a diameter of one quarter of the focal length. (Note that f, but not the f-number is the focal length!) This implies a numerical aperture of sin(1 / 4) ≈ 0.247, but the f-number specification is more common for photographic objectives.

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, see below). 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. (The ideal solution would be an asymmetric lens with optimized curvature radii on both sides.) 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. See Figure 5 for examples of typically used configurations. The exactly optimum configuration may e.g. slightly deviate from a plano-convex lens in the case was beam focusing, but the plano-convex solution is usually close enough.

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.

Ball lenses have the shape of a sphere. Similarly, there are rod lenses with a cylinder shape.

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

There are also Fresnel lenses, which can be made much thinner, but can hardly provide high optical performance.

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. There are types of lenses with surface shapes which are optimized for certain applications. See the article on spherical aberrations for more details.
• For incidence away from normal direction, one obtains astigmatism and coma. Also, lenses generally exhibit field curvature: focal points for incidence from different directions lie on a curve rather than a plane.
• Further, there are image distortions, e.g. in the form of barrel, pincushion or mustache distortions.
• 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 either by optimizing single lenses or by suitably combining several lenses. This is the reason why photographic objectives, for example, usually consist of a substantial number of 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.

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.

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 form, or to generate or compensate astigmatism of a beam or an optical system.

If there is a curvature in both directions, but not of equal strength, one has an astigmatic lens. It can be used, for example, for vision correction.

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. Most of them can be called objectives, for example photographic objectives or projection objectives. (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 (air-spaced lens systems). In any case, the multiple element lens is used like a single optical element, e.g. as a photographic objective.

Lenses are often fabricated from pre-fabricated lens blanks which are somewhat larger. With suitable grinding and polishing techniques, one removes part of the material to obtain the finally required shape and surface quality. Finally, one often applies an anti-reflection coating.

After the process and frequently also during the fabrication process, sophisticated optical characterization techniques (e.g. involving interferometers) can be applied for monitoring fabrication errors which can e.g. result in wavefront errors.

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.

Optical lenses and lens systems are also often named according to their optical function in an instrument. Some examples for such terms:

• Ophthalmic lenses are used for vision correction, compensating visual impairments like myopia, hyperopia and astigmatism.
• Focusing lenses are used for focusing laser beams, for example. 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).
• Collimation lenses are used for collimating light beams.
• Scanning lenses (e.g. in the form of f–theta lenses and telecentric lenses) are required for scanning beam directions e.g. in laser material processing and laser displays.
• Condensers somehow condition illumination light, for example in a projector.
• Objectives receive light from some object in an imaging system. More specifically, they can be photographic objectives, telescope objectives, microscope objectives etc. Some lenses are used in combination with photographic objectives, e.g. for turning an objective into a macro objective (Figure 6).
• Field lenses are used to enlarge the field of view e.g. of telescopes.
• 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.

As lenses have only relatively few essential parameters (focal length, numerical aperture, suitable wavelength range), one frequently uses standard lenses, many of which are available from stock. However, one can also obtain custom lenses fabricated according to given specifications. For example, that way one can obtain lenses not only with unusual optical performance figures (e.g. aspheric lenses), but also with special geometrical features, optical materials etc.

There are some kind of specialty lenses with unusual properties. For example, axicons, having a surface with a conical shape, are sometimes considered as specialty lenses.

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