optics

The optical image

An optical image may be regarded as the apparent reproduction of an object by a lens or mirror system, employing light as a carrier. An entire image is generally produced simultaneously, as by the lens in a camera, but images may also be generated sequentially by point-by-point scanning, as in a television system or in the radio transmission of pictures across long distances in space. Nevertheless, the final detector of all images is invariably the human eye, and, whatever means is used to transmit and control the light, the final image must either be produced simultaneously or scanned so rapidly that the observer’s persistence of vision will give him the mental impression of a complete image covering a finite field of view. For this to be effective the image must be repeated (as in motion pictures) or scanned (as in television) at least 40 times a second to eliminate flicker or any appearance of intermittency.

Historical background

To the ancients, the processes of image formation were full of mystery. Indeed, for a long time there was a great discussion as to whether, in vision, something moved from the object to the eye or whether something reached out from the eye to the object. By the beginning of the 17th century, however, it was known that rays of light travel in straight lines, and in 1604 Johannes Kepler, a German astronomer, published a book on optics in which he postulated that an extended object could be regarded as a multitude of separate points, each point emitting rays of light in all directions. Some of these rays would enter a lens, by which they would be bent around and made to converge to a point, the “image” of the object point whence the rays originated. The lens of the eye was not different from other lenses, and it formed an image of external objects on the retina, producing the sensation of vision.

There are two main types of image to be considered: real and virtual. A real image is formed outside the system, where the emerging rays actually cross; such an image can be caught on a screen or piece of film and is the kind of image formed by a slide projector or in a camera. A virtual image, on the other hand, is formed inside an instrument at the point where diverging rays would cross if they were extended backward into the instrument. Such an image is formed in a microscope or telescope and can be seen by looking into the eyepiece.

Kepler’s concept of an image as being formed by the crossing of rays was limited in that it took no account of possible unsharpness caused by aberrations, diffraction, or even defocussing. In 1957 the Italian physicist Vasco Ronchi went the other way and defined an image as any recognizable nonuniformity in the light distribution over a surface such as a screen or film; the sharper the image, the greater the degree of nonuniformity. Today, the concept of an image often departs from Kepler’s idea that an extended object can be regarded as innumerable separate points of light, and it is sometimes more convenient to regard an image as being composed of overlapping patterns of varying frequencies and contrasts; hence, the quality of a lens can be expressed by a graph connecting the spatial frequency of a parallel line object with the contrast in the image. This concept is investigated fully under Optics and information theory below.

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Optics had progressed rapidly by the early years of the 19th century. Lenses of moderately good quality were being made for telescopes and microscopes, and in 1841 the great mathematician Carl Friedrich Gauss published his classical book on geometrical optics. In it he expounded the concept of the focal length and cardinal points of a lens system and developed formulas for calculating the position and size of the image formed by a lens of given focal length. Between 1852 and 1856 Gauss’s theory was extended to the calculation of the five principal aberrations of a lens (see below Lens aberrations), thus laying the foundation for the formal procedures of lens design that were used for the next 100 years. Since about 1960, however, lens design has been almost entirely computerized, and the old methods of designing lenses by hand on a desk calculator are rapidly disappearing.

By the end of the 19th century numerous other workers had entered the field of geometrical optics, notably an English physicist, Lord Rayleigh (John William Strutt), and a German physicist, Ernst Karl Abbe. It is impossible to list all their accomplishments here. Since 1940 there has been a great resurgence in optics on the basis of information and communication theory, which is treated at length below.

The pinhole camera

An excellent example of the working of the wavelet theory is found in the well-known pinhole camera. If the pinhole is large, the diverging geometrical pencil of rays leads to a blurred image, because each point in the object will be projected as a finite circular patch of light on the film. The spreading of the light at the boundary of a large pinhole by diffraction is slight. If the pinhole is made extremely small, however, the geometrical patch then becomes small, but the diffraction spreading is now great, leading once more to a blurred picture. There are thus two opposing effects present, and at the optimum hole size the two effects are just equal. This occurs when the hole diameter is equal to the square root of twice the wavelength (λ) times the distance (f) between the pinhole and film—i.e., Square root of√2λ f. For f = 100 millimetres and λ = 0.0005 millimetre, the optimum hole size becomes 0.32 millimetre. This is not very exact, and a 0.4-millimetre hole would probably be just as good in practice. A pinhole, like a camera lens, can be regarded as having an f-number, which is the ratio of focal length to aperture. In this example, the f-number is 100/0.32 = 310, designated f/310. Modern camera lenses have much greater apertures, in order to achieve light-gathering power, of around f/1.2–f/5.6.

Reflection

The use of polished mirrors for reflecting light has been known for thousands of years, and concave mirrors have long been used to form real images of distant objects. Indeed, Isaac Newton greatly preferred the use of a mirror as a telescope objective to the poor-quality lenses available in his time. Because there is no limit to the possible size of a mirror, all large telescopes today are of this type.

When a ray of light is reflected at a polished surface, the angle of reflection between ray and normal (the line at right angles to the surface) is exactly equal to the angle of incidence. It can be seen that a convex mirror forms a virtual image of a distant object, whereas a concave mirror forms a real image. A plane mirror forms a virtual image of near objects, as in the familiar looking glass. Plane mirrors are often used in instruments to bend a beam of light into a different direction.

The law of refraction

When a ray of light meets the surface of separation between two transparent media, it is sharply bent or refracted. Because rays are really only directions and have no physical existence, the passage of light waves through a surface must be considered if refraction is to be understood. Refraction effects are based on the fact that light travels more slowly in a denser medium. The ratio of the velocity of light in air to its velocity in the medium is called the refractive index of the medium for light of a particular colour or wavelength. The refractive index is higher for blue light than for light at the red end of the spectrum.

In Figure 1, AA′ represents a plane wave of light at the instant that A′ meets the plane refracting surface A′B separating two media having refractive indices n and n′, respectively. During the time taken by the light to travel from A to B in material n, light travels from A′ to B′ in material of refractive index n′, forming the new wave BB′ in the second material, proceeding in direction BC. Hence, the relationship n′/n = AB/A′B ′ can be obtained; and dividing numerator and denominator by BA′ gives

The angles I and I′ are called the angle of incidence and angle of refraction between the refracting surface and the incident and refracted waves, respectively.

Returning now to the convention of considering the movement of light in terms of rays because entering and emerging rays are always perpendicular to the light waves they represent, angles I and I′ also denote the angles between the entering and emerging rays and the normal (perpendicular) line to the refracting surface at B.

Equation (1), known as the law of refraction, is generally written: n′ sin I′ = n sin I.

Dispersion

The difference between the refractive indices of a transparent material for a specific blue light and a specific red light is known as the dispersion of the material. The usual choices of blue and red lights are the so-called “F” and “C” lines of hydrogen in the solar spectrum, named by Fraunhofer, with wavelengths 4861 and 6563 angstroms (the angstrom unit, abbreviated Å, is 10⁻⁸ centimetre), respectively. It is generally more significant, however, to compare the dispersion with the mean refractive index of the material for some intermediate colour such as the sodium “D” Fraunhofer line of wavelength 5893 angstroms. The dispersive power (w) of the material is then defined as the ratio of the difference between the “F” and “C” indices and the “D” index reduced by 1, or,

Hundreds of different types of optical glass are currently available from manufacturers. These may be represented graphically on a plot of mean refractive index against dispersive power (Figure 2).

At first lenses were made from selected pieces of window glass or the glass used to make blown tableware. In the early 1800s, the manufacture of clear glass that was intended specifically for lenses began in Europe. The glass was slowly stirred in the molten state to remove striations and irregularities, and then the whole mass was cooled and broken up into suitable pieces for lens making. Subsequently, the pieces were placed in molds of the approximate size of the lens, slowly remelted to shape, and carefully annealed; i.e., allowed to cool slowly under controlled conditions to reduce strains and imperfections. Various chemicals were added in the molten state to vary the properties of the glass: addition of lead oxide, for example, was found to raise both the refractive index and the dispersive power. In 1884 it was discovered that barium oxide had the effect of raising the refractive index without increasing the dispersion, a property that proved to be of the greatest value in the design of photographic lenses known as anastigmats (lenses devoid of astigmatic aberration). In 1938 a further major improvement was achieved by the use of various rare-earth elements, and since 1950 lanthanum glass has been commonly used in high-quality photographic lenses.

The cost of optical glass varies considerably, depending on the type of glass, the precision with which the optical properties are maintained, the freedom from internal striae and strain, the number of bubbles, and the colour of the glass. Many common types of optical glass are now available in quite large pieces, but as the specifications of the glass become more stringent the cost rises and the range of available sizes becomes limited. In a small lens such as a microscope objective or a telescope eyepiece, the cost of the glass is insignificant, but in large lenses in which every millimetre of thickness may represent an additional pound in weight, the cost of the glass can be very high indeed.

Lenses can be molded successfully of various types of plastic material, polymethyl methacrylate being the most usual. Even multi-element plastic lenses have been manufactured for low-cost cameras, the negative (concave) elements being made of a high-dispersion plastic such as styrene.

Total internal reflection

When a ray of light emerges obliquely from glass into air, the angle of refraction between ray and normal is greater than the angle of incidence inside the glass, and at a sufficiently high obliquity the angle of refraction can actually reach 90°. In this case the emerging ray travels along the glass surface, and the sine of the angle of incidence inside the glass, known as the critical angle, is then equal to the reciprocal of the refractive index of the material. At angles of incidence greater than the critical angle, the ray never emerges, and total internal reflection occurs, for there is no measurable loss if the glass surface is perfectly clean. Dirt or dust on the surface can cause a small loss of energy by scattering some light into the air.

Light is totally internally reflected in many types of reflecting prism and in fibre optics, in which long fibres of high-index glass clad with a thin layer of lower index glass are assembled side-by-side in precise order. The light admitted into one end of each fibre is transmitted along it without loss by thousands of successive internal reflections at the interlayer between the glass and the cladding. Hence, an image projected upon one end of the bundle will be dissected and transmitted to the other end, where it can be examined through a magnifier or photographed. Many modern medical instruments, such as cystoscopes and bronchoscopes, depend for their action on this principle. Single thick fibres (actually glass rods) are sometimes used to transmit light around corners to an otherwise inaccessible location.

Graphical ray tracing

In 1621 Willebrord Snell, a professor of mathematics at Leiden, discovered a simple graphical procedure for determining the direction of the refracted ray at a surface when the incident ray is given. The mathematical form of the law of refraction, equation (1) above, was announced by the French mathematician René Descartes some 16 years later.

Snell’s construction is as follows: The line AP in Figure 3A represents a ray incident upon a refracting surface at P, the normal at P being PN. If the incident and refracted rays are extended to intersect any line SS parallel to the normal, the lengths PQ and PR along the rays will be proportional to the refractive indices n and n′. Hence, if PQ and the indices are known, PR can be found and the refracted ray drawn in.

A convenient modification of Snell’s construction can readily be used to trace the path of a ray through a complete lens. In Figure 3B, the incident ray BP strikes a refracting surface at P. The normal to the surface is PC. At any convenient place on the page two concentric circles are drawn about a point O with radii proportional to the refractive indices n and n′, respectively. A line OE is now drawn parallel to the incident ray BP extending as far as the circle representing the refractive index n of the medium containing the incident ray. From E a line is drawn parallel to the normal PC extending to F on the circle representing the refractive index n′. The line OF then represents the direction of the desired refracted ray, which may be drawn in at PB′. This process is repeated successively for all the surfaces in a lens. If a mirror is involved, the reflected ray may be found by drawing the normal line EF across the circle diagram to the incident-index circle on the other side.

Trigonometrical ray tracing

No graphical construction can possibly be adequate to determine the aberration residual of a corrected lens, and for this an accurate trigonometrical computation must be made and carried out to six or seven decimal places, the angles being determined to single seconds of arc or less. There are many procedures for calculating the path of a ray through a system of spherical refracting or reflecting surfaces, the following being typical: The diagram in Figure 4 represents a ray lying in the meridian plane, defined as the plane containing the lens axis and the object point. A ray in this plane is defined by its slope angle, U, and by the length of the perpendicular, Q, drawn from the vertex (A) of the surface on to the ray. By drawing a line parallel to the incident ray through the centre of curvature C, to divide Q into two parts at N, the relation is stated as AN = r sin U, and NM = r sin I. Hence

From this the first ray-tracing equation can be derived,

Applying the law of refraction, equation (2), gives the second equation

Because the angle PCA = U + I = U′ + I′, the slope of the refracted ray can be written asand, lastly, by adding primes to equation (2),

Having found the Q′ of the refracted ray, transfer to the next surface can be performed byin which d is the axial distance from the first to the second refracting surface. After performing this calculation for all the surfaces in succession, the longitudinal distance from the last surface to the intersection point of the emergent ray with the lens axis is found by

Corresponding but much more complicated formulas are available for tracing a skew ray, that is, a ray that does not lie in the meridian plane but travels at an angle to it. After refraction at a surface, a skew ray intersects the meridian plane again at what is called the diapoint. By tracing the paths of a great many (100 or more) meridional and skew rays through a lens, with the help of an electronic computer, and plotting the assemblage of points at which all these rays pierce the focal plane after emerging from the lens, a close approximation to the appearance of a star image can be constructed, and a good idea of the expected performance of a lens can be obtained.

Magnification: the optical invariant

It is frequently as important to determine the size of an image as it is to determine its location. To obtain an expression for the magnification—that is, the ratio of the size of an image to the size of the object—the following process may be used: If an object point B lies to one side of the lens axis at a transverse distance h from it, and the image point B′ is at a transverse distance h′, then B, B′, and the centre of curvature of the surface, C, lie on a straight line called the auxiliary axis. Then, by simple proportion, Hence, and the product (hnu) is invariant for all the spaces between the lens surfaces, including the object and image spaces, for any lens system of any degree of complexity. This theorem has been named after the French scientist Joseph-Louis Lagrange, although it is sometimes called the Smith-Helmholtz theorem, after Robert Smith, an English scientist, and Hermann Helmholtz, a German scientist; the product (hnu) is often known as the optical invariant. As it is easy to determine the quantities h, n, and u for the original object, it is only necessary to calculate u′ by tracing a paraxial ray in order to find the image height h′ for any lens. If the lens is used in air, as most lenses are, the refractive indices are both unity, and the magnification becomes merely m = u/u′.

The Gauss theory of lenses

In 1841 Gauss published a now famous treatise on optics in which he demonstrated that, so far as paraxial rays are concerned, a lens of any degree of complexity can be replaced by two principal, or nodal, points and two focal points, the distances from the principal points to their respective focal points being the focal lengths of the lens, and, furthermore, that the two focal lengths are equal to one another when the refractive indices of object and image spaces are equal, as when a lens is used in air.

The principal and focal points may be defined as follows: Figure 5 shows a lens system of any construction, with a bundle of rays entering from the left in a direction parallel to the lens axis. After refraction by the lens each ray will cross the axis at some point, and the entering and emerging portions of each ray are then extended until they intersect at a point such as Q. The locus of all the points Q is a surface of revolution about the lens axis known as the equivalent refracting locus of the lens. The point where this locus crosses the axis is called the principal point, P₂, and the central portion of the locus in the neighbourhood of the axis, which is virtually a plane perpendicular to the axis, is called the principal plane. The point where the emerging paraxial ray crosses the axis is called the focal point F₂, the distance from P₂ to F₂ being the (posterior) focal length f′. A similar situation exists for a parallel beam of light entering from the right, giving the anterior principal point P₁, the anterior focal point F₁, and the front focal length f. For a lens in air it can be shown that the two focal lengths are equal in magnitude but opposite in direction—i.e., if F₂ is to the right of P₂, then F₁ must lie to the left of P₁, as in the case of an ordinary positive lens (one that gives a real image). In a negative lens (one that gives a virtual image), F₂ lies to the left of P₂, and the posterior focal length f′ is negative.

The relation between the distances of object and image from a lens can be easily stated if the positions of the two principal points and the two focal points are known. (In using these expressions, distances are considered positive or negative depending on whether they are measured to the right or to the left from their respective origins.) For a lens in air: (a) If the conjugate distances measured from the respective focal points are x and x′, and if m is the image magnification (height of image divided by height of object), then m = -x′/f′ = f′/x and xx′ = −f′². (b) If the conjugate distances measured from the respective principal points are p and p′ and if m is the image magnification, then m = p′/p and 1/p′ = 1/p + 1/f′. The Lagrange equation (7) requires modification for a distant object because in that case the object height h is infinite, and the slope angle u is zero. If the off-axis distance h is divided by the object distance L, and u is multiplied by L, equation (7) becomes h′ = (n/n′)f′ϕ, in which ϕ is the angle in radians subtended by the distant object at the lens. This formula provides a means for defining focal length and for measuring the focal length of an unknown lens.

The thin lens

In a thin lens such as a spectacle, the two principal planes coincide within the lens, and then the conjugate distances p and p′ in the formula above become the distances of object and image from the lens itself.

The focal length of a thin lens can be computed by applying the surface-conjugate formula (6) to the two surfaces in succession, writing the l of the first surface as infinity and the l of the second surface equal to the l′ of the first surface. When this is done, the lens power (P) becomes

Chromatic aberration

Because the refractive index of glass varies with wavelength, every property of a lens that depends on its refractive index also varies with wavelength, including the focal length, the image distance, and the image magnification. The change of image distance with wavelength is known as chromatic aberration, and the variation of magnification with wavelength is known as chromatic difference of magnification, or lateral colour. Chromatic aberration can be eliminated by combining a strong lens of low-dispersion glass (crown) with a weaker lens made of high-dispersion (flint) glass. Such a combination is said to be achromatic. This method of removing chromatic aberration was discovered in 1729 by Chester Hall, an English inventor, and it was exploited vigorously in the late 18th century in numerous small telescopes. Chromatic variation of magnification can be eliminated by achromatizing all the components of a system or by making the system symmetrical about a central diaphragm. Both chromatic aberration and lateral colour are corrected in every high-grade optical system.

Longitudinal magnification

If an object is moved through a short distance δp along the axis, then the corresponding image shift δp′ is related to the object movement by the longitudinal magnification (m). Succinctly,in which m is the lateral magnification. The fact that the longitudinal magnification is equal to the square of the transverse magnification means that m is always positive; hence, if the object is moved from left to right, the image must also move from left to right. Also, if m is large, then m is very large, which explains why the depth of field (δp) of a microscope is extremely small. On the other hand, if m is small, less than one as in a camera, then m is very small, and all objects within a considerable range of distances (δp) appear substantially in focus.

Image of a tilted plane

If a lens is used to form an image of a plane object that is tilted relative to the lens axis, then the image will also be tilted in such a way that the plane of the object, the plane of the image, and the median plane of the lens all meet. This construction can be derived by the use of the lateral and longitudinal magnification relations just established above. With a tilted object the magnification at any point is given by the ratio of the distances of image and object from the lens at that point in the image, and, consequently, m varies progressively from one end of the image to the other. This arrangement is frequently used in view cameras equipped with “swings” to increase depth of field and in enlargers to rectify the convergence of parallel lines caused by tilting the camera, for example, in photographing tall buildings. The rule finds extensive application in photogrammetry and in the making of maps from aerial photographs.

System components

An optical system consists of a succession of elements, which may include lenses, mirrors, light sources, detectors, projection screens, reflecting prisms, dispersing devices, filters and thin films, and fibre-optics bundles.

Lenses

All optical systems have an aperture stop somewhere in the system to limit the diameter of the beams of light passing through the system from an object point. By analogy with the human eye, this limiting aperture stop is called the iris of the system, its images in the object and image spaces being called the entrance pupil and exit pupil, respectively. In most photographic lenses the iris is inside the objective, and it is often adjustable in diameter to control the image illumination and the depth of field. In telescope and microscope systems the cylindrical mount of the objective lens is generally the limiting aperture or iris of the system; its image, formed behind the eyepiece where the observer’s eye must be located to see the whole area being observed, called the field, is then the exit pupil.

The pupils of a lens system can be regarded as the common bases of oblique beams passing through the system from all points in an extended object. In most systems, however, the mounts of some of the lens elements cut into the oblique beams and prevent the beams from being perfectly circular, and the pupils are then not fully filled with light. This effect is known as vignetting and leads to a reduction in illumination in the outer parts of the field of view.

A common feature of many optical systems is a relay lens, which may be introduced to invert an image or to extend the length of the system, as in a military periscope. An example of the use of a relay lens is found in the common rifle sight shown diagrammatically in Figure 6. Here the front lens A is the objective, forming an inverted image of the target on the cross wire or reticle at B. The light then proceeds to the relay lens C, which forms a second image, now erect, at D. Beyond this image is the eyepiece E to render the light parallel so that the image may be seen sharply by the observer. Unfortunately, the oblique beam from the objective will usually miss the relay lens, and so a field lens must be inserted at or near the first image B to bend the oblique beams around and redirect them toward the relay lens. The power of the field lens is chosen so that it will form an image of the objective lens aperture on the relay lens aperture. The iris and entrance pupil of this system coincide at the objective; there is an internal pupil at the relay lens, and the exit pupil lies beyond the eyepiece as shown in Figure 6.

Brian J. Thompson The Editors of Encyclopaedia Britannica

Reflecting prisms

Reflecting prisms are pieces of glass bounded by plane surfaces set at carefully specified angles. Some of these surfaces transmit light, some reflect light, while some serve both functions in succession. A prism is thus an assembly of plane reflectors at relatively fixed angles, which are traversed in succession by a beam of light.

The simplest prism is a triangular block of glass with two faces at right angles and one at an angle of 45°. The face at 45° deflects a beam of light through a right angle. The common Porro prism used in a pair of binoculars contains four 45° reflecting surfaces, two to reverse the beam direction in the vertical plane and two in the horizontal plane (Figure 7). These reflecting faces could be replaced by pieces of mirror mounted on a metal frame, but it is hard to hold mirrors rigidly and harder still to keep them clean. Some microscopes are equipped with a 45° deflection prism behind the eyepiece; this prism may provide two or three reflections depending on the type of image inversion or left-for-right reversal required.

Prisms containing a semireflecting, semitransmitting surface are known as beam splitters and as such have many uses. An important application is found in some colour television cameras, in which the light from the lens is divided by two beam splitters in succession to form red, green, and blue images on the faces of three image tubes in the camera.

Fibre-optics bundles

As noted earlier, a thin rod or fibre of glass or other transparent material transmits light by repeated internal reflections, even when the rod is somewhat curved. An ordered bundle of rods or fibres is thus capable of taking an image projected upon one end of the bundle and reproducing it at the other end. A fibre-optics bundle can be fused together into a rigid channel, or it may be left flexible, only the ends being rigidly fastened together. Because a fibre bundle is exceedingly delicate, it must be handled with care; breaking a fibre would cause a black dot to appear in the reproduced image.

Rudolf Kingslake

Nonclassical imaging systems

Besides the familiar optical systems cited above, there are many nonclassical optical elements that are used to a limited extent for special purposes. The most familiar of these is the aspheric (nonspherical) surface. Because plane and spherical surfaces are the easiest to generate accurately on glass, most lenses contain only such surfaces. It is occasionally necessary, however, to use some other axially symmetric surface on a lens or mirror, generally to correct a particular aberration. An example is the parabolic surface used for the primary mirror of a large astronomical telescope; another is the elliptic surface molded on the front of the little solid glass reflector units used on highway signs.

Another commonly used optical surface is the side of a cylinder. Such surfaces have power only in the meridian perpendicular to the cylinder axis. Cylindrical lenses are therefore used wherever it is desired to vary the magnification from one meridian to a perpendicular meridian. Cylindrical surfaces are employed in the anamorphic lenses used in some wide-screen motion-picture systems to compress the image horizontally in the camera and stretch it back to its original shape in the projected image.

To correct astigmatism in the eye, many spectacles are made with toric surfaces—i.e., with a stronger curvature in one meridian than in the perpendicular meridian, like the bowl of a teaspoon. These surfaces are generated and polished by special machines and are made by the million every year.

Another nonclassical optical system is the bifocal or trifocal spectacle lens. They are made either by forming two or three separate surfaces on a single piece of glass or obtaining additional power by fusing a piece of high-index glass on to the front of the main lens and then polishing a single spherical surface over both glasses.

Two French scientists, Georges-Louis Buffon and Augustin-Jean Fresnel, in the 18th century suggested forming a lens in concentric rings to save weight, each ring being a portion of what would normally be a continuous spherical surface but flattened out. On a large scale, Fresnel lenses have been used in lighthouses, floodlights, and traffic signals, and as cylindrical ship’s lanterns. With fine steps a few thousandths of an inch wide, molded plastic Fresnel lenses are often used as condensers in overhead projectors and in cameras as a field lens in contact with a ground-glass viewing screen.

Lenses have occasionally been made with one surface taking the form of a flattened cone. Such lenses produce a long, linear image of a point source, lying along the lens axis; for this reason they are commonly referred to as axicons. They have been used to produce a straight line of light in space for aligning machines and shafting, but since about 1965 the beam from a gas laser has generally been used instead.

Seidel sums

If a lens were perfect and the object were a single point of monochromatic light, then, as noted above, the light wave emerging from the lens would be a portion of a sphere centred about the ideal image point, lying in the paraxial image plane at a height above the axis given by the Lagrange theorem. In practice, however, this condition is most unlikely to occur; it is much more probable that the emerging wave will depart slightly from a perfect sphere, the departure varying from point to point over the lens aperture. This departure is extremely small, being of the order of the wavelength of light that is only half a micron, so it would be impossible to show this departure on a drawing. It can be represented mathematically, however, in the following way: The coordinates of a point in the exit-pupil aperture will be represented by x₀ and y₀, the y₀ coordinate lying in the meridian plane containing the object point and the lens axis. The departure of the wave from the ideal sphere is generally called OPD, meaning optical path difference. It can be shown that OPD is related to x₀ and y₀ by five constants S₁ through S₅, and the quantity h′_o,Each of these five terms is considered to be a separate “aberration,” the coefficients S₁,…S₅, being called Seidel sums after the 19th-century German scientist L.P. Seidel, who identified the imperfections. These aberrations are respectively spherical, coma, astigmatism, Petzval field curvature, and distortion. The symbol h′₀ refers to the height of the final image point above the lens axis, and hence it defines the obliquity of the beam.

The five Seidel sums can be calculated by tracing a paraxial ray from object to image through the lens and by tracing also a paraxial principal ray from the centre of the aperture stop outward in both directions toward the object and image, respectively. The angle of incidence i and the ray slope angle u of each of these paraxial rays at each surface are then listed and inserted into the following expressions for the five sums. The angle u′₀ represents the final emerging slope of the paraxial ray.

The calculation starts by determining the radius A of the exit pupil by A = Square root of√x₀² + y₀² and also the quantity K at each surface by

The corresponding K_pr for the paraxial principal ray is also determined at each surface. Then, the five aberrations may be written

To interpret these aberrations, the simplest procedure is to find the components x′, y′ of the displacement of a ray from the Lagrangian image point in the paraxial focal plane, by differentiating the OPD expression given above. The partial derivatives ∂OPD/∂x₀ and ∂OPD/∂y₀ represent respectively the components of the slope of the wave relative to the reference sphere at any particular point (x₀, y₀). Hence, because a ray is always perpendicular to the wave, the ray displacements in the focal plane can be found byin which f is the focal length of the lens. The aggregation of rays striking the focal plane will indicate the kind of image that is characteristic of each aberration.

This procedure will be applied to each of the five aberration terms separately, assuming that all the other aberrations are absent. Obviously, in a perfect lens x′ and y′ are zero because OPD is zero. It must be remembered, however, that by using rays instead of waves, all fine-structure effects caused by diffraction will be lost, and only the macroscopic image structure will be retained.

Spherical aberration

The first term in the OPD expression is OPD = S₁(x₀² + y₀²)². Hence

These displacements can both be eliminated simultaneously by applying a longitudinal shift L to the focal plane. This changes x′ by −Lx₀/f and y′ by −Ly₀/f; hence, if L is made equal to 4 f ²A²S₁, both ray displacements vanish. The aberration, therefore, represents a condition in which each zone of the lens has a different focus along the axis, the shift of focus from the paraxial image being proportional to A². This is known as spherical aberration (see Figure 8).

Coma

The S₂ term in the OPD expression represents the aberration called coma, in which the image of a point has the appearance of a comet. The x′ and y′ components are as follows:

When this aberration is present, each circular zone of the lens forms a small ringlike image in the focal plane, the rings formed by successive concentric zones of the lens fitting into two straight envelope lines at 60° to each other (Figure 8). Because the brightness of this image is greatest at the tip, coma tends to form a one-sided haze on images in the outer parts of the field.

Astigmatism

If only the S₃ term is present, thenFor any one zone of the lens, x′ and y′ describe a vertical ellipse with major axis three times the minor axis. The images formed by all the smaller zones of the lens fit into this ellipse and fill it out with a uniform intensity of light. If the image plane is moved along the axis by a distance L, as in focussing a camera, then, at L = 2f ²h₀′²S₃, the ellipse shrinks to a radial focal line (R). Twice this displacement yields a circle; three times this L gives a tangential focal line (T), which is followed by an ellipse with its major axis in the x direction, as in Figure 8, bottom. The usual effect of astigmatism in an image is the appearance of radial or tangential blurring in the outer parts of the field.

Petzval curvature

For the S₄ term taken alone,

The image of a point is now a small circle that contracts to a point at a new focus situated at a longitudinal distance L = 2f ²h₀′²S₄ from the paraxial image. As the longitudinal displacement of the focus is proportional to the square of the image height h₀′, this aberration represents a pure field curvature without any accompanying loss of definition (all lines remain sharp). It is named after the Hungarian mathematician József Petzval, who studied its properties in the early 1840s. The effect of Petzval curvature can be somewhat offset by the deliberate introduction of sufficient overcorrected astigmatism, as was done in all the pre-anastigmat photographic objectives. This added astigmatism is, of course, undesirable, and in order to design an anastigmat lens having a flat field free from astigmatism, it is necessary to reduce the Petzval sum S₄ drastically.

For a succession of thin lenses (1, 2, 3,…etc.) in a system, the Petzval sum becomes simply 1/f₁n₁ + 1/f₂n₂ + 1/f₃n₃ +…etc., in which f is the focal length of each element and n is its refractive index. Therefore, to reduce the sum and minimize this aberration, relatively strong negative elements of low-index glass can be combined with positive elements of high-index glass. The positive and negative elements must be axially separated to provide the lens with a useful amount of positive power. The introduction of high-index barium crown glass with a low dispersive power in the 1880s initiated the development of anastigmat lenses.

Distortion

For the S₅ aberration,

When this aberration is present, the entire image point is displaced toward or away from the axis by an amount proportional to the third power of the transverse distance h₀′ of the image from the axis. This leads to the formation of an image of a square that is either a barrel-shaped or a cushion-shaped figure.

It is to be noted that the five Seidel aberrations represent the largest and most conspicuous defects that can arise in an uncorrected optical system. Even in the best lenses in which these five aberrations have been perfectly corrected for one zone of the lens and for one point in the field, however, there will exist small residuals of these aberrations and of many other higher order aberrations also, which are significantly different from the classical types just described. The typical aberration figures shown in Figure 8 are, of course, grossly exaggerated, and actually it requires some magnification of a star image to render these appearances clearly visible. Nevertheless, they are important enough to require drastic reduction in high-quality lenses intended to make sharp negatives capable of considerable enlargement.

General relations

All photometric concepts are based on the idea of a standard candle, lamps having accurately known candle power being obtainable from the various national standards laboratories. The ratio of the candle power of a source to its area is called the luminance of the source; luminances range from about 2,000 candles per square millimetre at the surface of the Sun down to about 3 × 10⁻⁶ candle per square centimetre (3 × 10⁻⁶ stilb) for the luminous paint on a watch dial. Ordinary outdoor scenes in daylight have an average luminance of several hundred candles per square foot. The quantity of light flux flowing out from a source is measured in lumens, the lumen being defined as the amount of flux radiated by a small “point” source of one candle power into a cone having a solid angle of one steradian. When light falls upon a surface it produces illumination (i.e., illuminance), the usual measure of illuminance being the foot-candle, which is one lumen falling on each square foot of receiving surface.

It is often important to be able to calculate the brightness of an image formed by an optical system, because photographic emulsions and other light receptors cannot respond satisfactorily if the light level is too low. The problem is to relate the luminance of an object with the illuminance in the image, knowing the transmittance and aperture of the optical system. A small area A of a plane object having a luminance of B candles per square unit will have a normal intensity of AB candles. This source radiates light into a cone of semi-angle U, limited, for example, by the rim of a lens. The light flux (F) entering the cone can be found by integration to be

If the object luminance is expressed as B_L lamberts, the lambert being an alternative luminance unit equal to 1/π (i.e., 0.32) candle per unit area, the flux (F) isbecause there are π times as many lamberts in a given luminance as there are candles per unit area.

A fraction t of this flux finds its way to the image, t being the lens transmittance, generally about 0.8 or 0.9 but less if a mirror is involved. The area of the image is Am², in which m, the magnification, is given by

Hence, the image illuminance (E) is

The image illuminance thus depends only on the luminance of the source and the cone angle of the beam proceeding from the lens to the image. This is a basic and most important relation underlying all calculations of image illuminance.

It is often more convenient to convert the angle U′ into other better known quantities, such as the f-number of the lens and the image magnification. The relation here is

The f-number of the lens is defined as the ratio of the focal length to the diameter of the entrance pupil; m is the image magnification; and m_p is the pupil magnification—i.e., the diameter of the exit pupil divided by the diameter of the entrance pupil. Combining equations (8) and (9) gives

As an example in the use of this relation, if it is supposed that an f/2 lens is being used to project an image of a cathode-ray tube at five times magnification, the tube luminance being 5,000 foot-lamberts (1.7 candles per square centimetre), the lens transmittance is 0.8, and the pupil magnification is unity. Then the image illuminance will be

The image is very much less bright than the object, a fact that becomes clear to anyone attempting to provide a bright projected image in a large auditorium.

Distribution of illumination over an image

So far only the illumination at the centre of an image has been considered, but the distribution of illumination over a wide field is often important. In the absence of any lens, the small plane source already considered radiates in a direction inclined at an angle ϕ to the axis with an intensity AB cos ϕ. This light has to travel farther than the axial light to reach a screen, and then it strikes the screen at another angle ϕ. The net result is that the oblique illumination on the screen is smaller than the axial illumination by the factor cos⁴ ϕ.

The same law can be applied to determine the oblique illumination due to a lens, assuming a uniform extended diffusing source of light on the other side of the lens. In this case, however, the exit pupil will not in general be a perfect circle because of possible distortion of the iris by that part of the optical system lying between the iris and the image. Also, any mechanical vignetting in the lens will make the aperture noncircular and reduce still further the oblique illumination. In a camera this reduction in oblique illumination results in darkened corners of the picture, but, if the reduction in brightness is gradual, it is not likely to be detected because the eye adapts quickly to changing brightness as the eyes scan over the picture area. Indeed, a 50 percent drop in brightness between the centre and corners of an ordinary picture is scarcely detectable.

Visual brightness

The apparent brightness of things seen by the eye follows the same laws as any other imaging system, because the apparent brightness is measured by the illuminance in the image that is projected on the retina. The angle U′ in equation (8) inside the eye is determined by the size of the pupil of the eye, which varies from about one millimetre to about eight millimetres, depending on the brightness of the environment. Apart from this variation, retinal illuminance is directly proportional to object luminance, and objects having the same luminance appear equally bright, no matter at what distance they are observed.

From this argument, it is clear that no visual instrument, such as a telescope, can possibly make anything appear brighter than when viewed directly. To be sure, a telescope having a large objective lens accepts more light from an object in proportion to the area of the lens aperture, but it magnifies the image area in the same proportion; so the increased light is spread over an increased area of the retina, and the illuminance remains unchanged. Actually, the telescopic view is always dimmer than the direct view because of light losses in the telescope due to glass absorption and surface reflections and because the exit pupil of the telescope may be smaller than the pupil of the eye, thus reducing the angle U′.

The case of a star being observed through a telescope is quite different, because no degree of magnification can possibly make a star appear as anything other than a point of light. Hence, star images appear brighter in proportion to the area of the telescope objective (assuming that the exit pupil is larger than the eye pupil), and the visibility of a star against the sky background is thus improved in proportion to the square of the diameter of the telescope objective lens.