Laser Optics

Two-Lens Beam Expander
To change the diameter of a collimated laser beam, a simple type of Galilean telescope is used. Although a laser beam expander is nominally used on-axis, however if coma can be corrected, the system will be relatively insensitive to errors in alignment. For two-lens beam expander, two design parameters (i.e. the shapes of the two lenses) are available once the powers have been chosen. We might expect to be able to correct both the spherical aberration coefficient and coma coefficient.

Three-Lens Beam Expander
If a beam expander is composed of a plus-plus positive group. Then the coma is reduced. It can do better if the positive group is made from a plus-minus combination, in which the spherical aberration is correctable. To reduce the coma still further, we can split the two positive lenses, with a significant separation between them. It might be possible for the positive spherical aberration of the middle lens to correct the marginal ray height at the final lens.

F-Theta Lenses
Laser-printing systems generally require lenses with the property that the image size is proportional to the angle of the input beam, rather than to the tangent of this angle (which is common case when designing photographic lenses). These lenses are known as F-theta lenses. They always have an external pupil (the scanning mirror).

F-theta lenses often resemble one half of a double-Guass lens. In practice, a doublet does not provide enough parameters for an adequate F-theta lens, but a triplet would.

Lenses for Optical Disks
In optical disk players, a high-resolution lens is needed in order to read the data encoded on the disk. This lens must accurately follow the track that is recorded on the disk and therefore must be very lightweight. In practice the lenses are quite thick, and a useful improvement is obtained by the use of two aspherics.

Examples, the three-glass design by Sugiyama has a numerical aperture (NA) of 0.22 and the single aspheric lens by Arai et al has aspherics on both surfaces and a NA of 0.50. In laser physics, numerical aperture is defined slightly different, NA = n sinθ, where n is the index of refraction of the medium in which the lens is working (1.0 for air, 1.33 for pure water, and up to 1.56 for oils) and θ is the divergence of the beam. A Gaussian beam is formed as the lowest-order transverse mode in a stable laser resonator with spherical mirrors. The parameter of Gaussian beams are usually given at the 1/e fraction of maximum amplitude. The divergence of the beam is the far-field angle between the propagation direction and the distance from the beam axis for which the irradiance drops to 1/e^2 times the wavefront total irradiance.

Note: in most areas of optics, especially in microscope, θ is the half-angle of the maximum cone of light that can enter or exit the lens. In general, this is the angle of the real marginal ray in the system. In photography, instead of NA, angular acceptance of a lens is used. f/# = f/D, is the ratio of the focal length to the diameter of the entrance pupil. For small NA, f/# ~= 1 / 2NA. The f/# describes the light-gathering ability of the lens in the case where the marginal ray before (or after) the lens is collimated.

Sugiyama optical disk lens

Arai optical disk lens

Laser Diode Collimators
Since laser diodes produce a diverging beam, it is always necessary for the beam to be collimated. Collimators for laser diodes need to operate at high numerical apertures (about 0.60) to be compatible with the laser diode itself. In addition to the basic lens design (spherical aberration and coma), it is necessary to correct for the unequal divergence of the laser beam in the planes parallel and perpendicular to the emitting junction. Typically the beam widths in the two perpendicular directions are in the ration of about 2:1. Often this has to corrected so that the output beam from the collimator has a circular cross section rather than an elliptical cross section. This is commonly achieved by means of prisms arranged so that the beam diameter is increased in one direction.

High Dynamic Range Digital Pictures

Fujifilm F100fd - Auto mode, Multi-photometry, ISO 800


Fujifilm F100fd - Manual mode, Multi-photometry, ISO 800, Dynamic Range 400%


Hydra HDR software, using 4x pictures: [+2, +1, 0, -1], Fujifilm F100fd - Auto mode, Multi-photometry, ISO 800

Plastic Optics

Acrylic is known for its very good clarity and excellent transmission properties throughout the visible portion of the spectrum. This material is very hard and has very good mechanical stability. It has an index @588nm of 1.49, and an Abbe number of 55.3. A acrylic lens, when paired with an appropriate styrene lens offers an effective achromatic solution.

Polystyrene is generally less expensive than acrylic. It has a higher index @588nm of 1.59 and lower Abbe number (30.87) than acrylic. The material also tends to absorb somewhat in the deep blue spectrum. Polystyrene has a lower resistance to UV than acrylic and is more easily scratched than acrylic.

Polycarbonate is similar to polystyrene in index (@588nm, 1.586) and Abbe number (29.9). It is known for its very high impact resistance and for the ability to perform over a wide range of temperatures (-137 to +124 C). Because of its high ductility, polycarbonate is not easily machine than acrylic.

Four- and above component Lenses

Wide-angle lenses for rangefinder camera (Biogon)
Early wide-angle lenses were often derived from standard 50mm lenses. Biogon designed for Contax at Zeiss was derived from Sonnar. Because it was for rangefinder camera. Therefore the back focus is very small.

Split Triple
Splitting an element into two elements allows the aberrations of the two to be substantially reduced from that of the original element. In the Cooke triplet, the outer crowns (rear) are often split in order to reduce the zonal spherical aberration and to thus allow the speed of the lens to be increased. Note that, as the speed is increased, the angular field coverage is reduced. The split rear-crown form is often used in camera lenses for 8mm, 16mm, small format TV, and CCD cameras, typically at the speed of about f/2.

Double-Guass (Biotar)
The Biotar or double-Guass type is a descendant of the double-meniscus anastigmat lenses. One of the many variants of the double-meniscus form consisted of outer positive singlets and inner connected negative doublets, in a symmetrical construction. These lenses had the speed and angular coverage typical of their genre, i.e., good angular coverage at a quite modest aperture. A departure from symmetry allowed the speed of the lens to be increased to f/2, and a tremendously useful and powerful design form was born. It can be made into a moderately wide-angle lens, an enlarger lens, a high-resolution objective, or a lens of extremely high speed.

The basic six-element version with the positive front element is meniscus in shape, usually of a lanthanum flint glass with an index of about 1.7 and a V value of about 48. The second element tends to be meniscus, although a plano-convex or a mild biconvex form is not unusual; its index is typically a bit lower and its V value higher than the first element. The third element usually has an index close to that of the second element, and it is a dense flint from along the glass line. The fourth element is usually biconcave and made of a glass-line flint with a slightly higher index than the third element. The fifth and sixth elements are ordinarily both biconvex and of lanthanum flint glass.

Splitting the rear singlet (seven-element) of the Biotar is an excellent way to improve on the basic configuration. It is a commonly used technique to allow an increased speed, and is often seen in faster 35mm camera lenses. For eight-element Biotar, both outer elements are split and the capability for a reasonable state of correction is achieved using only medium-high-index glass.

Double-Guass (planar)
The first double-Guass lenses were quite thin, but in 1896, Paul Rudolph described the Zeiss planar, which used thick meniscus components, with an aperture of f/4.5. The planar also uses buried surfaces, in which there is a difference of dispersion (to correct chromatic aberration) but no significant difference in refractive index. Although it is not desirable in itself, it does provide a simple way of correcting chromatic aberration without affecting the monochromatic aberrations. The planar was not immediately developed into a high-aperture lens. H.W.Lee of Taylor Hobson designed the first high-aperture double-Guass in 1920. This lens has near front-back symmetry. It uses much thicker meniscus components, which are helpful in reducing the Petzval sum and the oblique spherical aberration. At f/2 it has aberrations that are not much worse than the planar at f/4.

These lenses performed very well as normal and medium-long focus lenses for small and medium format cameras. A classic Planar design is the 105mm f2.5 Nikkor produced from 1971 to 2006.

Lenses for compact point-and-shoot camera
Modern compact 35mm cameras use lenses with focal length of 35mm. It is basically a triplet with a meniscus field flattener.

Three- and Four-component Lenses

Triplet
The simplest design that is reasonable for 35mm cameras is the triple configuration. Once the glasses are chosen, there are just enough variables (three powers, three shapes, and two separations) to correct all seven Seidel aberrations, as well as the focal length. The basic method used to flatten the Petzval field curvature in all anastigmats is the longitudinal separation of positive power from negative power. Increasing the separation lowers the relative ray height on the negative, thus will reduce the (negative) power contribution. The result is an effective net positive power without the undesirable excess of inward Petzval curvature. All anastigmats make use of this principle. Cooke triplet can be viewed as the trunk of the family tree of the airspaced anastigmats, while older anastigmats such as the Dagor, Sonnar, and the Biotar (double-Guass) types use thick meniscus components to separate positive convex surfaces from negative concave surfaces.
The choice of glass types is an important degree of freedom and has a significant effect on the characteristics of the triplet design. The glass for the positive elements should be a dense barium or lanthanum crown type; an index of 1.6 or more is almost essential. A triplet using an ordinary low-index crown glass (or acrylic plastic) for the positive elements is possible, but the result is poor unless the aperture or field (or both) is small. The other important factor in glass choice is the use of the relative difference in V values between the crown and flint elements as a means to adjust the vertex length of the triplet to its optimum value.

Tessar
The Tessar, invented in 1902 by Paul Rudolph, is similar to the triplet. The difference is the cemented doublet replacing the rear singlet. Although Tessar is a modification of the Cooke triplet, it is actually a descendant of the double-meniscus anastigmat form. The Tessar performs better than the triple in COM3, AST3, PETZ3, and DIST3. The substitution of a new achromat doublet for a crown in the triplet is the equivalent of using high-index, high-V-value glass, and allows the possibility of utilizing the cemented surface of the doublet to control coma and oblique spherical aberration.

The Tessar design patent was held by Zeiss for two decades, and licensed to Rollei in Germany, Bausch & Lomb in the United States and to Krauss in France.
Sonnar
Sonnar was introduced by Bertele (Zeiss) in 1931, at a time when antireflection coatings were not yet available. It can thought of as a triplet derivative with a meniscus component in the front air space. If the meniscus is made thick, it can have a field-flattening effect of its own. The Sonnar form makes use of this to eliminate the need for the center flint negative element. These lenses begin to look like the front half of a Biotar/double-Guass combined with the rear of a triple or Tessar. Given that the transmission of a single air-glass surface is typically about 95%, a lens with six air-glass surfaces can have a transmission of 0.95^6 = 0.73.

The Sonnar has proven incompatible in shorter focal lengths with SLR cameras due to the space taken up by an SLR's mirror. Therefore, the Sonnar type had a brief vogue as 35mm camera lenses of normal and long focal length, but they have been largely superseded by other constructions (e.g. double-Guass).

Two-component Lenses

Petzval lenses
When longitudinal chromatic aberration has been corrected, we normally find that the focal positions for three wavelengths still do not coincide. This effect is known as secondary spectrum and is due to the nonlinearity of the curve of refractive index as a function of wavelength. It stems from a difference between the nonlinearity of the positive (crown) element and the negative (flint) element of an achromat. The basic Petzval lens consists of two positive components, spaced apart so that the astigmatism is controlled to be either zero or slightly positive. Usually the two components are achromats, typically doublets. Secondary spectrum is reduced in Petzval lenses because the second doublet operates at a lower aperture than the first.
The original Petzval lens was designed by Joseph Petzvel of Slovakia in 1839 as a portrait lens for the Daguerrotype camera and, at a speed of about f/3.5. Its purpose was to reduce the extremely long exposure times of the daguerreotype camera from about 30 minutes to about 30 seconds, using a wide aperture. The Petzval lens was optimized for large apertures and indeed could only operate at one fixed aperture. Petzval lenses were the mainstay of camera and projection lenses for almost 40 years during the dawn of photography. They are very rare now.

Telephoto lens
When moderately long focal lengths are required, design similar to normal 50mm local length systems are often use. The basic power arrangement of a positive component followed by a negative component, can produce a compact system with an effective focal length F which is longer than the overall length L of the lens. The ratio of L/F is called the telephoto ratio, and a lens for which this ratio is less than unity is classified as a telephoto lens. By definition, many camera lenses which are sold as telephoto lenses are simply long-focal-length lenses and are not true telephotos. Since the system is unsymmetrical, each component must be individually achromatized if both axial and lateral color are to be corrected. The aperture stop is usually at the front member or part way toward the rear. Since a telephotos lens covers only a relatively small angular field, coma, distortion, and lateral color (which in many lenses are reduced by an approximate symmetry about the stop) are not as troublesome as they would be with a wider field. Simple telephoto lenses are formed by two cemented or closely airspaced achromatic doublets. Additional degree of freedom can be gained by splitting the doublets into widely airspaced components.
Telephoto lenses

Long-focal-length lenses

Wide-angle lenses for SLR camera
In modern SLR camera, a back focus of about 38mm to 40mm is required for mirror mechanism. This means the earlier forms of wide-angle lens are not acceptable, and the inverted telephoto construction is necessary. The arrangement of a negative component following a positive component has a back focal length which is longer than the effective focal length. Although the lens is much larger than the Biogon, correction for field curvature is very good because of the separated negative power at the front of the lens. The simplest fully corrected form is a pair of achromatic components. For the negative front component, the crown glasses (used for the negative elements) should be high-index and the flint glasses (in the positive elements) should be low-index; this reduces the overcorrected Petzval contribution from this component. For the positive rear component, the ordinary glasses should be low-index crowns and high-index flint. This combination increases its negative, inward-curving contribution to the Petzval sum. Usually the aperture stop is at the rear component; the natural shape for the front negative achromat is then that of a meniscus, concave toward the stop.

Double-meniscus anastigmat (Hypergon, Topogon, Dagor, Dogmar)
The Hypergon lens consists of two identical meniscus elements, symmetrical about a central stop. The concave and convex radii differ by less than 0.7 percent so that the Petzval contributions of the convex surfaces are almost completely offset by the Petzval of the concave surfaces. The astigmatism is controlled by the distance of the lens from the stop, and the symmetrical construction almost completely eliminates the coma, distortion, and the lateral color. The lens covers an astonishing field of 135 degree. Rapid rectilinear lens is a double meniscus system in which two achromatized meniscus lenses are arranged symmetrically on either side of the aperture stop, reducing or eliminating distortion, coma and lateral color.
The obvious way to improve the Hypergon is to add negative flint elements to correct the spherical aberration and axial chromatic aberration. The Topogon covers a field of about 100 degree at a speed of f/6.3, using dense barium crown and extra-dense flint glasses, retaining the symmetrical construction and the strong meniscus configuration for all the elements.

The Dagor combines both the old achromat and the new achromat into a cemented triplet construction. If we visualize the central negative element of the triplet split into two parts, then the outer high-index element and the outer part of the middle (medium-index) element can be seen to make up a new achromat. The inner low-index crown element and the other part of the middle element make up the old achromat. The symmetrical construction about the stop minimizes the coma and distortion, while the spacing from the stop and the cemented surfaces control the astigmatism.

The Dogmar lens is also a member of the double-meniscus family. It can be realized if one considers each half to be a triplet with a center air lens. The Dogmar form is used as an excellent general-purpose camera lens, and its symmetry and stability of correction make it eminently suitable for an enlarger lens.

Third-order Aberrations and Lenses Components

The third-order (Seidel) aberration coefficients form the basis of lens design.

Seven Seidel coefficients (effectively wavefront aberration terms)
SA3- spherical aberration (S1)
COM3 - coma (S2)
AST3 - astigmatism (S3)
PETZ3 - Petzval (field curvature) (S4)
DIST3 - distortion (S5)
LCOLOR - lateral color (C1)
TCOLOR - longitudinal color (C2)

For lenses at higher numerical apertures and field sizes, third-order aberrations are insufficient for describing all of the aberrations. High order aberrations are induced because a surface is operating in an aberrated beam. Fifth-order and even the seventh-order coefficients need to include in the design.

Fifth-order aberrations
SA5 - spherical aberration
COM5 - coma
AST5 - astigmatism
PETZ5 - Petzval
DIST5 - distortion
TOBSA - tangential oblique spherical aberration
SOBSA - sagittal oblique spherical aberration
LCOMA - Elliptical coma

Singlets
plano-convex, plano-concave, equi-convex, or equi-concave. Most are made of BK7 lenses may have large aberrations. They are mostly used when apertures are quite small or when aberrations are not important.

Field flatteners
Although plano-concave singlets for field flattener do not correct astigmatism, the improvement in field curvature is significant.

Doublet and flattener


Meniscus singlets
The earliest anastigmats flattened the field by using a think meniscus, which separated the positive and negative outer surfaces. Approximating an aplanatic condition on the front surface and a rear surface that is almost concentric with the marginal ray when used in the converging beam after a doublet. The combination has significant better resolution than the equivalent doublet. However, meniscus lens is not color corrected. At the time there were no antireflection coatings, designers tried to minimize the number of air-glass surfaces. For this reason, most of the early anastigmats were limited to two cemented meniscus components.

Doublet and meniscus


Doublets and triplets
Spherical aberration, coma, and chromatic aberration are corrected.

Cemented triplets
Symmetrical triplets have zero coma, distortion, and lateral color at unit magnification. When it is used at infinity conjugates, the aberrations may be small enough for some applications.

Single lens for disposable camera
They use a single plastic lens, which can be aspheric, and a film (ASA 400-800) that is curved around a cylindrical surface. Although it is possible to correct coma, the astigmatism is then quite large. Moreover, field curvature, axial and lateral color are not corrected, but at an aperture of f/16 the performance is just acceptable.