Definition: transparent optical devices affecting the wavefront curvature of light
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More specific terms: ball lenses, microlenses, achromatic lenses, aspheric aspheric, cylindrical lenses, rod lenses, diffractive lenses, ocular lenses, scanning lenses, zoom lenses, f–theta lenses, telecentric lenses, objective lenses, ocular lenses, field lenses, Fresnel lenses, gradient-index lenses, Kerr lens
Categories: general optics, photonic devices
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For purchasing lenses, use the RP Photonics Buyer's Guide – an expert-curated directory for finding all relevant suppliers, which also offers advanced purchasing assistance.An optical lens consists of a transparent medium (in most cases an optical glass or polymer), where light enters on one side and exits on the opposite side. Often, but not always, it has at least one curved surface. Its purpose is to modify the wavefront curvature of the light, which implies that light is focused or defocused. A focusing and defocusing lenses are also sometimes called to be converging or diverging lenses, respectively, although it seems more natural to use those adjectives for beams rather than lenses. Also, lenses are sometimes called positive (when focusing) or negative (defocusing).
Strictly speaking, a lens is always a single optical element, and this article treats mainly those. However, systems containing multiple lenses in a common housing are also often called lenses; we treat them in the article on objectives, or more specifically e.g. under photographic objectives.
Some typical examples of using an optical lens:
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: Between the lens and the beam focus, the light converges because of the wavefront curvature, and after the focus it diverges due to the wavefront curvature in the opposite direction.
The red and blue colors visualize the strength and sign of the electric field in one moment of time. The assumed wavelength is much larger than in reality.
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.
Another approach is based on photonic metasurfaces, which allow one to realize arbitrary transverse profiles of phase changes, and thus also to easily avoid certain optical aberrations. Moreover, such optical devices can be flat and extremely thin.
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 from 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.
The focal length of a lens is usually a fixed value, but there are also tunable lenses, e.g. employing a deformable droplet of a liquid. Further, one can make multiple element lenses with moving elements which are called varifocal lenses, or zoom lenses if they preserve focus.
A lens with a given focal length ($f$) creates a radially varying phase delay for a laser beam according to the following equation:
$$\Delta \varphi (r) = - \frac{\pi }{{\lambda f}}{r^2}$$This formula ignores the constant part of the phase change as well as aberrations. Note that depending on the function of the lens – for example, focusing collimated input beams or refocusing divergent light –, higher-order terms in the phase profile may be required to avoid optical aberrations.
For more details, see the article on focal length.
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:
$$\frac{1}{f} = (n - 1)\left( {\frac{1}{{{R_1}}} + \frac{1}{{{R_2}}}} \right) - \frac{{{{(n - 1)}^2}\;d}}{{n\;{R_1}\;{R_2}}}$$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.
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
$$\frac{1}{a} + \frac{1}{b} = \frac{1}{f}$$where ($a$) is the distance from the original focus to the lens. This shows that ($b \approx f$) if ($a \gg f$), 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 ($a \le f$), 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 ($w_0$) 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
$${w_{\rm{f}}} = \frac{{\lambda f}}{{\pi {w_0}}}$$where it is assumed that the beam radius at the focus is much smaller than the initial beam radius ($w_0$). (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 ($\lambda$), so that the paraxial approximation is valid.
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). Positive meniscus lenses may be used for beam focusing, but they are often used only 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.
In many technical applications, it is important that a lens is correctly centered, i.e., that its center is exactly placed on the intended beam path. If this is not the case, a beam which would ideally go through the center of the lens will be somewhat offset from that, and as a consequence it will be somewhat deflected. The centration error is usually specified as the angular deviation of such a beam.
Centration is of course most critical for lenses with short focal length in the sense that a small mismatch of center position causes a strong beam deviation.
Lenses cause various types of aberrations (image degradations):
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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 the near infrared, traditional dielectric materials can well be used for lenses. For longer wavelength in the mid-infrared, however, special infrared optics materials need to be used. For example, one can use semiconductors like silicon and germanium.
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 of such terms:
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 geometric 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.
For each of his four feature-length collaborations with writer-director Robert Eggers, cinematographer Jarin Blaschke has collaborated closely with Panavision, delving into the lens inventory and working closely with the Special Optics team to assemble the packages he would use to frame the projects’ period settings. Taking place in s New England, The Witch () found Blaschke choosing Cooke Speed Panchro Series II lenses, which he complemented with Bausch & Lomb Super Baltars and Panavision Super Speeds, framing for the 1.66:1 aspect ratio. For The Lighthouse (), which takes audiences to the year , Blaschke and Eggers elected to use the 1.19:1 aspect ratio and selected Baltar lenses and custom Petzval optics designed by Dan Sasaki, Panavision’s Senior Vice President of Optical Engineering and Lens Strategy. With The Northman (), the filmmakers embraced a wider 2.00:1 aspect ratio for an epic tale set in the time of the Vikings, circa A.D. 900; Blaschke’s primary lenses for the movie were Primo spherical primes modified by the Special Optics team. Most recently, for the s-set Nosferatu (), the filmmakers returned to the 1.66:1 aspect ratio, and Blaschke’s lens package combined Baltars with custom optics inspired by vintage photographic lenses.
In this video, which took place at Panavision’s headquarters in Woodland Hills, California, Blaschke and Sasaki dive deep into all four of those projects, detailing the cinematographer’s meticulous testing processes, digging into a variety of optical designs, analyzing test footage and final frames, and much more.
Pop-up definitions appear onscreen throughout the conversation to provide additional context or information related to topics and terms that Blaschke and Sasaki mention. For ease of reference, you can also find those definitions — and their time codes — below.
02:46 – Thorium is a radioactive lead isotope present in thoriated glass, which has a high refractive index and was used in a variety of vintage lenses.
02:54 – Double-Gauss lens designs incorporate symmetrical groupings of positive and negative elements. Reverse-telephoto (aka inverted telephoto or retrofocus) lenses use a net-negative objective to increase the back focal length while achieving a wide field of view.
03:23 – Chromatic aberration can be either lateral, manifesting as color fringing, or axial, which occurs when a lens does not focus all colors (i.e., wavelengths) of light onto the same focal plane.
03:32 – Substrate refers to the base material that makes up an optical component. Different substrates have unique refractive indices and dispersion qualities.
04:10 – Spherical aberration occurs when light rays passing through a lens converge at different focal points, resulting in a blurred or soft-focus image.
06:37 – Dagor, Artar, and Protar lenses were developed for still photography in the late s.
08:26 – Orthochromatic film was primarily sensitive to blue and green wavelengths of light, but not red. The Lighthouse was shot with panchromatic black-and-white film, which is sensitive to a broader spectrum of light.
09:44 – In a reflex motion-picture film camera, the spinning mirror that allows the operator to view through the lens sits between the film plane and the lens’ rear element, in some cases requiring a vintage lens’ housing to be shaved-down to clear the mirror.
10:24 – Named after its inventor, Joseph Petzval, the Petzval was the first mathematically designed lens. One of the byproducts of its design is a swirly bokeh, which results from astigmatism induced by its positively biased power.
11:23 – A doppel (from the German word for “double”) is a symmetrical lens with two identical optical groups to improve aberration correction.
13:39 – Adapted from the principles of Galileo’s early telescope, a Galilean lens can be as simple as a two-lens system comprising one convex (positive) element and one concave (negative) element.
16:42 – Examples of transverse aberrations — which cause image points to be displaced horizontally — include lateral chromatic aberration and geometric distortion.
18:01 – MTF (modulation transfer function) is used to measure a lens’ ability to maintain contrast and reproduce fine details. The higher a lens’ MTF, the sharper the image it produces will appear.
20:21 – A triplet is a three-element optical system. An achromat typically comprises a positive and a negative element to correct two wavelengths of light and reduce chromatic aberration.
22:10 – Created by photographers Ansel Adams and Fred Archer, the Zone System is a technique of visualizing and controlling brightness and contrast by dividing an image into “zones” ranging from pure black to pure white.
23:19 – From the late 19th and early 20th centuries, Eidoscope lenses produced a soft-focus effect well-suited to portraiture while capturing straight lines in a rectilinear fashion, without geometric distortion.
32:00 – A focus conjugate refers to the ratio between an object and its photographed image.
36:21 – The human eye’s photopic response is its “day vision,” controlled by the eye’s color-sensitive cones. The eye’s scotopic response kicks in under low light, when the eye’s rods take priority over the cones, resulting in a perceptually monochromatic image.
42:12 – “That theater” refers to the Tak Miyagishima Theater inside Panavision Woodland Hills. The theater is named after legendary Panavision design engineer Takuo “Tak” Miyagishima.
44:04 – Thomas “Tommy” Rose is an image technology specialist in Panavision’s Special Optics department. He’s also a cinematographer and served as director of photography for this video.
47:46 – The emperor in question was Emperor Hirohito, who was born in , the year after the Heliar lens design was introduced.
48:36 – An anastigmat is a type of lens designed specifically to correct astigmatism.
48:46 – Developed circa , the Plasmat lens design incorporates six elements symmetrically about the aperture and produces high sharpness with excellent control over spherical and chromatic aberrations.
49:14 – Shout-out to Panavision Special Optics lens technician Rudy Samkow.
51:40 – A spherical glass element features a surface that represents a section of a sphere, while an aspheric element has varied curvature across its surface. A single aspheric element can take the place of multiple spherical elements, resulting in better low-light and wide-angle performance.
52:22 – Merit function is used to evaluate and optimize multiple performance criteria, quantifying how well a lens minimizes aberrations, reduces distortion, improves sharpness, and more.
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