Astronomical Optics Part 4: Optical Aberrations Chromatic Aberrations Longitudinal Chromatic Aberration Lateral Chromatic Aberration Achromatic Optics Monochromatic Aberrations Defocus Spherical Aberration Coma Distortion Field Curvature Astigmatism Aberration Balancing Spherical Aberration of the Exit Pupil The Observer's Eye Minimizing Optical Errors Appendix 1: Graphing Aberrations Wavefronts & Lenses Spot Diagrams Field Height Plots Ray Intercept Curves Appendix 2: Third Order Analysis The Characteristic Function The Seidel Sums The Seidel Errors in Reflecting Telescopes In their basic design, optical systems are held to the standard of first order or Gaussian optics: a monochromatic "point" light source located at infinity and centered on the optical axis will appear as a "point" image at the center of a focal plane that is flat and perpendicular to the optical axis. This standard is then extended off axis to include the image of any point visible anywhere within the telescope image area or eyepiece field of view. Any departure from this optical perfection is an aberration. The most important of these were identified and analyzed in the mid 19th century — empirically by the Hungarian optician Joseph Petzval, and theoretically by the German mathematician Philipp Ludwig von Seidel (pronounced ZY·dul). They are usually called the Seidel errors, and in optical systems that are symmetrical around an axis of rotation that is identical with the optical axis, the Seidel errors are significant because they have both the greatest impact on image quality and the greatest utility as guides to improving an optical design. The five Seidel errors, in traditional order, are: (1) spherical aberration, (2) coma, (3) astigmatism, (4) field curvature and (5) distortion. Two types of first order (6,7) chromatic aberration (caused when the image is not monochromatic) are consistently included among the important aberrations; and (8) spherical aberration of the exit pupil is a flaw often encountered in wide angle eyepieces. Aberrations can be analyzed and classified in several ways. As shown in the chart (left), a simple distinction is between aberrations that fail to produce a point image for every object point in a common focal plane (termed errors of focus), or aberrations that fail to produce an equal image scale either at the image point or across the entire image area (termed errors of magnification). Astigmatism combines features of focus and magnification error and is intimately associated with both field curvature and distortion. All three tend to become more pronounced in short focal ratio objectives, long focal length (low magnification) eyepieces and wide field eyepieces (60° or greater apparent field of view). Aberrations may appear in light rays entering the lens close to and parallel to the optical axis (termed paraxial rays); or they may appear only in light rays entering the system near the edge of the objective or at an angle to the optical axis (off axis or abaxial rays). The abaxial errors become more significant as the distance between the object image and the center of the field of view (of the objective and/or the eyepiece) increases. The starting assumptions are that all the refracting or reflecting surfaces are either flat or are spherical surfaces of rotation around an optical axis, the aperture and field stops are circular and perpendicular to the optical axis, and all stops and surfaces are centered on the optical axis. These assumptions mean that the system is perfectly collimated. Collimation is the procedure of aligning the objective optics with the eyepiece optics, focal reducer or camera focal plane so that they share a single optical axis. Apart from atmospheric turbulence, aberrations in astronomical optics may appear at any of three points — the telescope optics (objective lens, or primary and secondary mirrors), the eyepiece and related optics (such as a field flattener, focal extender or focal reducer), and the observer's eye. Severe misalignment between the observer's eye and the instrument optical axis are common, and can produce aberrations in the center of the field. Practical experience indicates that the eyepiece or photographic lens generally dominates the contribution of aberrations in telescope systems, and that thermal turbulence in the instrument and atmosphere limit the magnification and visual or photographic resolution to "what the seeing will allow." Optical Parameters & Units Discussion of aberrations depends on a few technical parameters and terminology, summarized in the diagram (below) and reproduced from the page Telescope & Eyepiece Combined. See there for further explanation. Aberrations are calculated Each of the eight aberrations is illustrated below with a diagram that simulates its "pure" appearance in a telescope field of view. These are not photographs of actual aberrations and are intended only to aid visual recognition. They do not characterize the aberrations as they appear in all situations; typically, two or more aberrations will affect an image at the same time and will appear in different proportions at different field heights within the image. Chromatic Aberrations Chromatic aberration is a failure to focus or magnify the image equally across all spectral wavelengths or spectral hues ("colors"). It arises because optical materials slow the long (red) wavelengths of light into a less extreme angle of refraction than the short (violet) wavelengths, visibly spreading out the spectral hues in an optical effect called dispersion. Chromatic aberrations are considered the most objectionable (visually most confusing or distracting) aberrations in an optical system, and designs to minimize chromatic errors appeared very early. Pursuit of chromatically better corrected optical designs was a principal reason for the development of new types of optical glasses in the 19th century. Chromatic aberration appears in two forms. Longitudinal chromatic aberration (above, left) is an error of focus; it occurs when different wavelengths of light from a single object point are refracted into separate focal planes at different focal lengths along the optical axis. Lateral chromatic aberration (above, right) is an error of magnification, which occurs because different wavelengths of light from an off axis object point are dispersed at different powers from opposite sides of the aperture. Longitudinal Chromatic Aberration Longitudinal or "axial" chromatic aberration focuses different spectral hues at different distances along the optical axis, with "violet" wavelengths focused at a point closer to the lens than "red" wavelengths. The separation between these extreme spectral focal points defines the secondary spectrum of the focused light. 2Δƒ = fe / Ve This spectral separation causes the "best focus" bright point images to appear tinged with green and surrounded by red fringes, and edges between light and dark areas to be fringed with red or violet color. A ray tracing (geometrical) analysis of longitudinal chromatic aberration (diagram, above) shows that the refracted light does not define a single focal plane. Instead the focal point defined by yellow, which creates the visually brightest image, is usually chosen as the best focus. At the "yellow" focus both the red and violet wavelengths are located outside the apparent edge of an object image, where they mix to create a red violet fringe, while the blue, green, yellow and orange wavelengths mix within the object image to produce (for a white object) a white color tinged with green. From this best focus, the intrafocal image presents only red at the outside fringe and produces a central image tinged with blue, while the extrafocal image presents an outside fringe of blue violet and a central image tinged with red. As an axial optical error, longitudinal chromatic aberration affects all parts of the image roughly equally. Lateral Chromatic Aberration Lateral or transverse chromatic aberration appears in off axis image areas. It occurs in images brought into perfect refractive focus because the lateral spectrum produced by dispersion is of different widths or magnifications in light arriving from opposite sides of the aperture. 2Δh' = (1/2Ve)Do This produces radial color within the meridional plane, particularly for highly contrasted points and edges, as shown in exaggerated scale in the diagram (below). Because the spectra are overlapped in reverse order, mixing complementary hues, this light mixture mostly produces a "white" focal image. However light from the opposite side of the aperture arrives at a longer focal length and is therefore dispersed at a slightly higher magnification, so that it extends beyond the focal "white" image on both the "red" and "violet" ends. Typically the "violet" end is magnified to a greater extent so that it appears larger than the "red" end nearer the optical axis. The spread in dispersion is always a small fraction of the refractive angle, so lateral color in well corrected achromat optics is usually at a scale similar to the Airy disk diameter and is only visible at high magnifications and sharp edges. Lateral color is affected by the location of stops or silhouetted objects such as etched or wire reticules. In optics where the aperture stop (in objectives) or field stop (in eyepieces) is in front of the lenses, the "blue" fringe is located on the opposite side of the image from the center of the field; these are termed overcorrected. In systems where the aperture or field stop is inside or behind the lenses, the "blue" end of the spectrum is located on the same side of the image as the center of the field; these are termed undercorrected. The entire image area will show lateral color at the image of the edge of the field stop; in overcorrected systems the edge of the field will be tinged with blue, or tinged with red in undercorrected systems. (The color near to the optical axis does not appear because it overlaps with dispersed light from the inner parts of the image field.) In optimally corrected systems the field stop will appear fringed with yellow green. As an abaxial optical error, lateral chromatic aberration increases as the angular height (β) or field height (h) increases. Achromatic Optics The method for minimizing chromatic aberration in refractive optics is to use two lenses whose combined refraction and dispersion powers are manipulated to "fold" the secondary spectrum back onto itself, so that the extreme ends of the spectrum focus in the same focal plane. Seventeenth and 18th century refractors were built to extremely slow focal ratios (ƒ/100 and longer) primarily to minimize axial color. The Huygens eyepiece, a combination of two lenses that minimizes lateral color through the spacing of two positive lenses, was developed late in the 17th century. The strategy to eliminate lateral color by means of a doublet composed of a positive and negative lens was developed independently in 18th century England by Chester Moore Hall and John Dolland for eyepiece optics, and brought to a high refinement in large aperture telescope objectives by Joseph von Fraunhofer and Carl Steinheil. These doublet lenses are termed achromatic. Dollond's son Peter conceived the addition of a third lens so that the middle of the achromatic spectrum is folded on itself again, resulting in an improved achromatic system, termed apochromatic (diagram, below). Note that in relative terms, the reductions in chromatic error are roughly equivalent — about 80% compression of the secondary spectrum — at each step. But in absolute terms (the amount of defocus along the optical axis) there is a gross improvement from an uncorrected to achromatic system, but a much less dramatic improvement from achromatic to apochromatic. In naked eye experience, fringing is a strong signal of luminance intensity, as dispersions appear more colorful ("prismatic") in bright lights. But when the orange/blue contrast fringes light areas against a dark background, it has a strong spatial or depth effect on the image that is distracting and confusing (see image, above). Both forms of chromatic aberration are absent in nondiffracting (smooth mirror) reflective optics. Monochromatic Aberrations After the chromatic aberrations are the five most serious monochromatic aberrations, all errors of refraction rather than dispersion. These affect the image focus or magnification even when the image is monochromatic — transmitted in a single wavelength of light. First in importance among these are spherical aberration and coma. Defocus, Decenter & Tilt Three forms of optical misalignment affect the first order or Gaussian image point. • Defocus is the longitudinal displacement of the effective focal plane from the first order or Gaussian focal point. This causes a radial expansion of the image point on the effective image plane, decreasing the luminance of each point and causing point images to overlap. • Decenter is the displacement of an optical element so that its center is no longer on the optical axis. • Tilt is the rotation of an optical element around an axis that passes through the center of the element and is perpendicular to the optical axis. Collimation is the process of adjustment that eliminates tilt and decenter from the optical system. Spherical Aberration Spherical aberration (diagram left) is a failure of focus, caused when rays parallel to the optical axis have different focal lengths at different aperture heights s. It produces a roughly uniform, concentric defocus around all the bright elements in all parts of an image, and a blurring in extended and low contrast detail, preventing any part from coming into sharp focus. The aberration is uniform across the entire image area. The analysis of spherical aberration is related to its appearance in the spherical reflection of collimated rays (diagram, below). Each aperture height s defines a concentric area of the objective with a specific (and different) focal length. Undercorrection or negative spherical aberration occurs when rays incident at a large aperture height (reflected from the near edge of the objective) have a shorter focal length than the central rays; this is characteristic of spherical mirrors and converging lenses. Overcorrection or positive spherical aberration occurs when the peripheral rays have a longer focal length than the central rays, which is chatacteristic of ellipsoidal mirrors and spherical diverging lenses. The terms probably arise from the traditional method of figuring a Newtonian mirror, which is first ground out to an approximately spherical figure with the desired focal length, then polished or "corrected" into a paraboloid figure by flattening the outer surfaces. However the terms apply equally to lenses or mirrors of any conic shape. Because the light cones of the rays cross through each other as they come in and out of focus, spherical aberration produces some defocus at every point along the optical axis. The best available focus, in which the rays that appear brightest to the eye (hues cyan to orange) are contained within the smallest area, is usually (in undercorrection) extrafocal from the circle of least confusion, where the diameter of all the intersecting light cones is at a minimum. As the diagram (above) shows, at an extrafocal position (with the eyepiece focal plane closer to the observer than the least circle of confusion), the core of the image displays more of the peripheral rays; at an intrafocal position (the least circle of confusion closer to the observer than the eyepiece focal plane) it displays more of the central rays. In many instances, spherical aberration produces a "best focus" or "least confused" image of a star that is only marginally larger than the Airy disk produced by diffraction: most of the out of focus light appears as a brightening of the rings around the disk. For this reason, spherical error does not significantly degrade the resolution power of a telescope in double stars of similar magnitude, but brightening of the rings does impair resolution for binaries of unequal magnitude, as the rings of the brighter star will mask the fainter. Spherical aberration also significantly impairs the resolution of low contrast detail, for example in planetary and deep sky objects. Coma Coma (diagram, right) is a failure to magnify equally off axis light rays passing through the objective or eyepiece. Unlike spherical aberration, the effect is not constant within the same concentric aperture height height (s); instead the magnification also varies with field height (h). It produces elongated, wedge shaped "tails" that extend radially from point images or luminance edges in off axis light. The aberration occurs because light from opposite sides of the objective are reflected or refracted at different powers, due to the different angles of incidence at the optical surface. The severity of the aberration increases with field height (h) of the object point, with the more distorted rays being incident at a higher aperture height (s) of the objective. Coma can also appear in on axis object images if the optical system is not correctly collimated. Coma is inherent in parabolic mirrors, because parallel but off axis rays of light do not strike opposite sides of the mirror at equal angles of reflection: on one side of the mirror the angle become decreases it on the opposite side; in effect, it is as if one side of the mirror becomes flatter, and the other side more curved (diagram below, left). A similar effect occurs as an imbalanced refractive power on opposite sides of a lens, due to the different angles of incidence of parallel but abaxial rays entering the lens on opposite sides (diagram below, right). In external coma, the more common form, this produces a greater magnification of star images as the field height of the incident rays increases, causing a series of images to appear both larger in diameter and at a greater distance from the center of the field: the tails of these comatic blurs point toward the field edge. In internal coma, the tails point toward the field center. As shown in the illustration (above), coma also becomes more extreme as the field height increases: point images near the edge of the field are more strongly blurred than images near the center, and the comatic blurring increases with the larger angular heights contained in low power images. The apex of the coma is defined by the rays passing closest to the principal point of the lens (red lines, above). The radial length of the coma, aligned to the axis of the aberration, constitutes the tangential aberration, and the diameter of the widest part the sagittal aberration. Their lengths are related in the ratio 3:2, and are proportional to the angular height of the point image times the square of one half the focal ratio: tan(β)·(D/2ƒo)2. Most of the aberrated image is contained within the brightest "V" rim of the figure, expanding as a 60° wedge from the point formed by rays passing through the center of the objective. In modern optics, coma principally results when the eyepiece and/or objective are not perfectly collimated (their optical axes are exactly aligned), or when the focal ratio is fast (the focal length is a small multiple of the aperture, typically below ƒ/5). After the chromatic aberrations, spherical aberration and coma were the optical errors next successfully tackled in eyepiece designs. To identify this achievement, the 19th century German optician Ernst Abbe introduced the term aplanatic to describe eyepieces free of both aberrations. Aplanatic optics obey the constant ƒe = h'/sine(β) for all collimated rays, known as the sine condition (diagram, above). This is equivalent to the condition that the light rays are refracted to the focal point along paths of equal length, as if perpendicular to the surface of a spherical "principal plane" (shown as a circular arc in the diagram) whose radius is the focal length (ƒe). With the sine condition satisfied, distortion, astigmatism and field curvature remained the aberrations constraining the width of field (maximum angular height β) in astronomical optics. Distortion Distortion is caused by a failure to magnify the image equally as the field height increases. Distortion is characterized in two ways. In positive or rectilinear distortion, the magnification increases toward the edge of the field: this causes straight lines to appear curved outward, even when the image is stationary. In negative or angular magnification distortion, the magnification decreases toward the edge of the field: this causes straight lines to appear projected onto a spherical surface curved toward the viewer, which is especially noticeable when the optical axis is moved — the image then appears to "roll away" near the field edge. The sign of the weight assigned to distortion can be remembered by visualizing the effect on a square placed with one corner on the optical axis, so that the two sides and diagonal originating from the corner are radial lines from the center of the image. Since the diagonal is longer than a side, it has a greater field height and will show a greater alteration by the distortion. • If the distortion coefficient is positive (rectilinear distortion), then the diagonal is made longer and the far corner of the square becomes pointed (less than 90°); hence it is sometimes called pincushion distortion. • If the coefficient is negative, the diagonal is made shorter and the far corner becomes larger or rounded (greater than 90°); this is sometimes called barrel distortion. The two types of distortion cannot be eliminated at the same time: given an ocular field height (h') and angular height at the exit pupil (β), both in radians, then rectilinear distortion will be zero when the image is scaled as h' = ƒe·tan(β); but angular magnification distortion will be zero when the image is scaled as h' = ƒe·(β). Visually, rectilinear distortion is the effect produced when a plane image is projected onto a spherical surface, and angular magnification distortion is the effect when a spherical image is projected onto a plane surface. Geometrically, making them equal amounts to making both the side and diagonal of a square the same length. Obviously, then, either positive or negative distortion must be allowed if it is necessary to reduce the other. Some amount of positive (rectilinear) distortion, which causes straight lines to appear bent toward the edge of the field, is usually introduced to minimize or eliminate the negative (angular magnification) distortion, which causes circular objects (such as planets) to appear squashed or stretched out near the edge of field. A common marketing tout for wide field eyepieces is: "Our eyepieces are absolutely free of angular magnification distortion!" Unfortunately, this is achieved by significantly increasing the rectilinear distortion. In addition, increased astigmatism is often the result of minimizing distortion in pursuit of a "flat" field: the magnification error is treated as separate sagittal and tangential components, which allows them to be manipulated separately. In 19th century and early 20th century eyepiece designs, especially for military sighting optics, elimination of rectilinear distortion was critical for use of the optics to measure field angles as indicators of distance. In astronomical optics, elimination of angular magnification distortion is usually critical, as the "roll off" of the image as the telescope is moved is considered unesthetic and distracting. Field Curvature Field curvature (called Petzval curvature in the technical literature) is a failure to focus the entire image on a single plane perpendicular to the optical axis. Instead the focal "plane" is a paraboloid surface resembling a bowl or meniscus. This produces a characteristic inability to focus the center and edges of the field at the same time. Field curvature is directly related to astigmatism, as the average of the sagittal and tangential curvatures (see below, and Appendix 2). Almost all astronomical objectives and eyepieces present some combination of curvature of field, astigmatism or distortion, especially wide field optics. The curved principal focal plane is termed the Petzval surface, and the amount of curvature is indicated by the best fitting spherical radius of the curvature, the Petzval number. A smaller Petzval number indicates a shorter radius and therefore a greater degree of curvature in the focal plane. Nearly all telescope objectives produce an image with positive curvature, in which the edges of the field are focused at a shorter focal length than the center of the field (the bowl is turned toward the objective and away from the observer). However, in eyepieces, where the focus is at the image plane, positive curvature means the bowl is facing toward the observer. Thus, if the objective has a positive curvature (as most telescope objectives do), and the eyepiece has a negative curvature, then the two focal surfaces will curve in the same direction. significantly near the edge of the field, especially in low power eyepieces or low focal ratio objectives. Positive curvature in the objective image causes objects at the edge of the field to lie at an intrafocal position in relation to a focused field center in the eyepiece focal surface. In an SCT, once the field center is brought into focus, the focusing knob must be turned clockwise to bring the edges into focus. In negative curvature, the edges of the field are focused farther from the objective than the center (the focal bowl faces the viewer), and the edges are brought into focus by a counterclockwise rotation of the focusing knob. A classic Schmidt Cassegrain camera produces a large negative curvature of field, which is compensated by using a photosensitive surface with a matching convex surface. The eyepiece tangential and sagittal astigmatic curves can be manipulated to counteract some field curvature, usually at the cost of introducing astigmatism. Studies of eyepiece/objective combinations by Rutten & Van Venrooij (1988) indicate that eyepiece astigmatism is responsible for most of the astigmatism in any objective/eyepiece combination, and that eyepiece astigmatism overwhelms the coma produced by a Newtonian reflector, particularly toward the edge of the field. Astigmatism Finally, astigmatism occurs when light rays from perpendicular cross sections of the image cone do not have the same focal distance along the optical axis. It is therefore an error both of focus and of magnification. Astigmatism is pervasive in optical systems, and occurs in all abaxial light passed through any refracting lens. It is the most difficult aberration to correct. Astigmatism produces two focal points for every abaxial image point. These are described in terms of two planes governed by the position and field height of the off axis object (diagram, below). The first plane contains both the optical axis and the principal ray from the off axis object, and divides the focal cone in half along the optical axis. This is the meridional plane, and the fan of rays within it defines the tangential focus. The second plane also contains the principal ray of the image point but is perpendicular to the tangential plane; it only intersects the optical axis at the principal point. The fan of rays within this plane defines the sagittal focus. Geometrically, the difference between the two components of astigmatism is easiest to visualize as separate diagrams for an object located directly below the optical axis, in the direction indicated by the white arrow. In this diagram, the path of the sagittal rays appears in the view from above (yellow lines) and the tangential rays in the view from the side (green lines). Note that within each plane, the opposing plane is perpendicular and appears as a single line because it is viewed edge on. Viewed from above, the sagittal plane is symmetrical around the optical axis. Although the sagittal rays strike the lens at equal angles to the surface normal on opposite sides of the lens, the abaxial tilt in relation to the optical axis effectively thickens the lens: the rays must travel a longer distance within the lens before reaching the opposite side, and therefore meet the lens at a point where its curvature in relation to the optical axis is less extreme and its refractive power is therefore reduced. This causes the sagittal rays focus long as a very thin line or ellipse perpendicular to the optical axis, which contains the aberration in the tangential rays (diagram above, left). Viewed from the side, the tangential plane is symmetrical around the principal ray of the object. As a result, the tangential rays strike the lens at different angles to the surface normal on opposite sides of the lens, changing its effective figure and producing different focal lengths in the opposing rays in a manner similar to coma. This causes the rays to focus short as a very thin line or ellipse radial to the optical axis, which contains all the aberration in the sagittal rays (diagram above, right). The "best focus" is located approximately midway between the sagittal and tangential focal points, and produces an approximately circular and defocused image of a point source which contains the tangential and sagittal aberrations in equal proportions. The complexity of astigmatism is that the sagittal aberration is governed by a symmetrical change in focal length, while the tangential astigmatism is governed by an asymmetrical difference in focal length. Astigmatism therefore combines errors in focus and magnification: the sagittal and tangential images come into focus at unequal distances from the optical axis, so they are at different powers (magnifications). The two components of astigmatism change in focal length, and typically diverge from each other, as as a function of the square of the angular height of the image (β2). As a result they define two separate, bowl shaped (rather than flat) focal surfaces, that exactly coincide or intersect at the optical axis. In field height diagrams they are represented as a radius cross section through each bowl, which produces two separate curves for the sagittal (S) and tangential (T) surfaces (diagrams, below). In these diagrams, the vertical scale is the field height of the focused rays and the horizontal scale is the difference in focus diopter adjustment from the paraxial focus. Note that either or both forms of astigmatism can define a positive or negative curvature, and that either type of astigmatism can be more extreme (produce a larger defocus) than the other. In the geometrical diagram (above), the tangential rays focus in front of, and the sagittal rays focus behind, the focal point defined by paraxial rays; but the tangential rays can also focus behind the sagittal rays, and both can focus in front of or behind the focal point of paraxial rays. The field curvature plots (above) show the amount of tangential and sagittal astigmatism in the illustrative lens designs, where the horizontal scale is diopters of focus, the vertical scale is field angle, and the objective is on the left. This illustrates that the tangential astigmatism is much more variable than the sagittal: in uncorrected systems, the tangential curve is always farther from the Petzval surface than the sagittal curve. The visual effects of astigmatism can be reduced by manipulating the sagittal and tangential surfaces to counterbalance each other. In the simplest strategy to do this, the tangential surface is made positive and the sagittal surface negative (or vice versa), so that their average is close to zero at all field heights (middle two diagrams, above). In another strategy, the outer "rim" of either the sagittal or tangential astigmatic "bowl" is turned backward so that passes back through the curvature of the opposing dimension, producing zero astigmatism at some constant radius (usually around 70% of the maximum field height) from the optical center and minimizing the divergence between the two dimensions as much as possible. An eyepiece with minimal or no astigmatism is called anastigmatic. When the concentric smearing of sagittal astigmatism is combined with coma, it creates the effect photographers call sagittal coma flare, which induces a "flying bird" appearance to star images at increasing field height, shown as both ray spot and the visually more accurate point spread functions in the diagram (below). Seidel astigmatism is not the same as the common astigmatism that affects the human eye, which occurs even for paraxial light and is caused by the cornea and/or lens having different curvatures in different cross sections. Aberration Balancing In most modern eyepiece designs, distortion is primarily a difficulty in wide field or ultra wide field eyepieces, with maximum angular heights (β) of 30° or greater. In eyepieces with apparent maximum angular heights of 20° or less, the difference between an angle and its tangent is visually so small (as a difference in h') that distortion does not become noticeable (chart, below). Systems that have minimized both angular magnification and rectilinear distortion are termed orthoscopic: specifically they satisfy the magnification constant Me = h'/tan(β). When distortion occurs, the orthoscopic formula alters to: Me = h'/[tan(β)·(1+E·tan(β)3)] where E is the coefficient of distortion. This formula shows that distortion increases as the cube of the angular height (β3), as shown in the chart (above). aberration vs. aperture vs. field angle spherical (longitudinal) D2 . spherical (lateral) D3 . coma D2 β field curvature (longitudinal) . β2 field curvature (transverse) D β2 astigmatism (tangential) . β2 astigmatism (sagittal) D β2 distortion (percent) . β2 longitudinal chromatic D . lateral chromatic . β Since the tangent that defines distortion and the sine that defines coma produce different ratios, an eyepiece cannot be both orthoscopic and aplanatic. However, as already mentioned, the differences among the ratios defined by the field height (h'), orthoscopic (tangent β) and aplanatic (sine β) quantities diverge noticeably for values of β greater than about 20° (chart, above). Spherical Aberration of the Exit Pupil Seidel errors can apply to the image formed within the eyepiece just as in the objective, and these contribute to the aberrations already in the objective image. However, spherical aberration presents itself visually not as an aberration of focus but as an aberration of position, known as spherical aberration of the exit pupil or "kidney bean blackout" of the image field. This situation arises because the eyepiece is overcorrected: it focuses the peripheral rays at an image plane in front of the central rays, as shown below. The eye is optimally positioned with the exit pupil centered within the pupillary aperture, which forms the entrance pupil of the eye (upper diagram). So long as the eye is fixated at the center of the field, and the pupil is placed at the focal plane of the central and near central rays (red and yellow lines in the diagram), the peripheral rays (blue lines in the diagram) fall on the sides of the retina where imaging resolution is very poor, and the lack of focus is not noticeable. However, if the eye is moved forward and rotated to admit the peripheral rays from one side of the image field in order to view that side of the field distinctly (lower diagram), the peripheral and near central rays from the opposite side are blocked by the foreshortened pupil or the wall of the eye, producing a mobile, kidney shaped "blackout" on that side of the visual field. Spherical aberration of the exit pupil only makes obvious a more basic constraint: the portion of the total visual field of the eye that is capable of forming a distinct image is relatively small. Extreme detail can only be imaged at the fovea, which subtends about 2° at the fixation point, and then only for light that is above the mesopic threshold (roughly, above 0.1 candelas per square meter). The parafoveal area, about 20° wide, is the image width commonly preferred by observers who are asked to adjust an extended object image to the largest size that still lets them view the object as a whole, when fixated at its center. This implies a physical diameter to distance ratio of about 1:3 — equivalent to viewing an audio compact disc (12cm diameter) at a distance of 36cm from the eye. The smallest apparent field offered today in commercial astronomical eyepieces is about 40°, twice the parafoveal image area and a diameter/distance ratio of 1:1.4 (a CD viewed from 16 cm). Many types of eyepieces offer apparent fields extended up to 60° (approximately 1 radian, or 57.3°), a diameter/distance ratio of 1:1.15. At this extreme width, the parafoveal area can be directed to one side or the other of the available field without overlapping the central image area, and when the eye is fixated to one side of the image area the opposite side becomes completely indistinct. It's worth mention here that there are at least three distinct forms of "shadowing" or partial obscuration of the eyepiece field (diagram, below). Spherical aberration of the exit pupil produces a crescent or oblate shadow, mobile within the field, that appears both detached from the edge of the field and surrounded at its edges by smeared out star images. A similar type of shadow, appearing as a crescent edge that occludes the field from one side, occurs because a fully flat exit pupil is not completely encircled by the eye pupil (the eye's optical axis is displaced to one side, or the eye is turned sharply to one side). Finally, if the pupil is smaller than the central shadow of the secondary mirror within the exit pupil, this appears as a circular shadow in the center of the field, which can entirely black out the field in extreme cases. This is most common when using a low magnification ocular under daylight illumination, which causes the pupil to contract to a diameter of about one or two millimeters. Spherical aberration of the exit pupil almost exclusively afflicts eyepiece designs with fields larger than 60° — wide angle or super wide angle eyepieces. Given the limitations of the eye, and the fact that spherical aberration will black out a large part of the eyepiece field, rendering it useless, it is more practical to use eyepieces that present the true field of view as an apparent field no wider than 60°, then choose an eyepiece focal length (magnification) that fits an extended object entirely within the eyepiece's true field of view. The Observer's Eye For many observers, the most significant source of optical aberrations is not the astronomical objective or eyepiece but their own eye(s), so these deserve mention as a source of optical errors. Unfortunately no conclusive description can be given for eyes in general or for an individual eye specifically, because the eye adjusts to the visual stimulus by changing the shape of the lens (accommodation), changing the diameter of the pupil, and/or changing the response sensitivity of the photoreceptor cells and neural networks in the retina (adaptation). As an added complication, the resolution of the eye is not a single fixed value because it depends on the specific type of discrimination being measured. For example, discriminating different shapes, such as block letters, can be done down to about 5 arcminutes; discriminating a grating of alternating sinusoidal black/white lines from a flat gray area can be achieved down to about 2 arcminutes; separating two closely spaced points can be done down to 1 arcminute; recognizing a sideways misalignment between the ends of two parallel straight lines (Vernier discrimination) can be done down to 10 arcseconds; and recognizing a spatial separation between the edges of two objects at different distances can be achieved down to a binocular difference of 5 arcseconds. As a benchmark, we can assume that the resolution limit for line spacings at mesopic levels of luminance, viewed within the 2° to 4° visual area of the fovea around the fixation point, is approximately 1 arcminute, or 1/3438th (0.0003) of a radian. This quickly declines to 0.002 radian (~7 arcminutes) at a field angle of 10° and to 0.005 radians (~17 arcminutes) at a field angle of 30°. Performance becomes worse as the luminance falls to scotopic (starlight) levels, increasing the effects of abaxial aberrations such as astigmatism and chromatic aberration, which contribute 1 to 3 diopters of defocus at a 30° field angle in a normal eye. This is in addition to the eye's negative field curvature, which conforms to the spherical surface of the retina. (Astigmatic eyes will contribute more defocus, and nearer the fovea.) ***check*** Defined as a point spread function, the average eye reaches its peak optical quality at a pupil diameter of about 3.5mm, where the Airy disk projected on the retina is about 7.6 micrometers wide. At this peak, the Airy disk stimulates around 10 foveal cones (spaced at around 2.3 cones per micrometer). Below this diameter, the Airy disk enlarges quickly, to about 0.000022 mm at 1mm (~90 cones); above this diameter, the geometrical spreading of the spot due to abaxial aberrations expands it to 0.000028 mm at 6mm diameter (~150 cones). Note that the telescope exit pupil serves to stop down the aperture of the eye in exactly the same way as the iris. The eye suffers from significant longitudinal chromatic aberration, and the secondary spectrum has a spread of about 2.5 focus diopters: from yellow (peak sensitivity) red is focused at +0.5 diopters and violet at Ð2 diopters. This is suppressed by the blue filtering effects of the yellowed lens and maculate pigmentation, and by neural processing. Lateral chromatic aberration is also present, especially when the exit pupil of a telescope and the pupil of the eye are not aligned exactly, but the added chromatic aberration is only around 0.04% per millimeter of displacement. Minimizing the Optical Errors These eight optical defects are related mathematically (see Appendix, below) in ways that make it impractical if not impossible to eliminate them all in an astronomical eyepiece or objective, especially when the goal is a wide field design or the focal ratio (ƒ/D) is small. It is therefore necessary to balance one aberration against the others in optical designs in order to produce the best optical performance. Because the abaxial errors — lateral chromatic aberration, coma, astigmatism, distortion, curvature of field and spherical aberration of the exit pupil — become more pronounced as the field height (h) or angular height (β) of the light source increases, the most distorted part of any astronomical image is toward the periphery of the image area, and the best corrected part of the image is at center of the image, around the optical axis. The two chromatic aberrations, spherical aberration and coma are generally considered the most objectionable image defects, and all are consistently almost entirely removed in modern telescope designs. In systems where the defects are unavoidable — astigmatism in a Ritchey-Chrétien reflector or coma in a Newtonian or Dall-Kirkham reflector — the field of view is limited (due to a large relative aperture) to the area around the optical axis where the aberrations are so small that they are below the visual or photographic angle of resolution. Field Size & Collimation Aberrations are a function of the focal ratio, the angular width of the apparent field of view (in the eyepiece), and the magnification, or angular width of the true field of view. Since Galileo's time, optical defects in a telescope have commonly been minimized by stopping down the telescope aperture, a practice used by early 20th century visual astronomers such as Percival Lowell to reduce the effects of atmospheric turbulence. In photographic cameras with very short focal ratios, even those with 12 or more spherical and aspheric lenses, stopping down is the function of the diaphragm or field stop, which limits light to the area around the optical axis of the system, minimizing off axis aberrations. Even in these systems, abaxial aberrations can still be evident in point source images located toward the edge of the field of view and photographed in dark environments when the diaphragm must be fully opened. Wide field eyepieces with long focal length (low magnification) create the most serious demands on an optical system and are most likely to show some combination of coma, astigmatism, distortion and field curvature at field widths greater than 60°. In the same way, objectives with a fast relative aperture are especially susceptible to aberrations, particularly coma, astigmatism and distortion. Aberrations can also be produced by optical systems that are out of collimation or alignment to a single shared optical axis. Collimation includes the viewer's eye, and aberrations can emerge when the optical axis of the eye does not coincide with the optical axis of the exit pupil. These appear in the centered disk of a moderately bright star, and also in the form of a slightly defocused star image. A third source of aberrations is atmospheric turbulence, which can produce a dizzying sequence of defocused, comatic, astigmatic, and displaced (distorted) aberrations in a perfectly focused star image. Under poor seeing these follow one another so quickly that the image degrades into a boiling ball of light, fraught with rapidly shifting and dissolving, very thin and very crisp diffraction shadows. Focus and Resolution Nearly all modern optical systems introduce some form of optical aberration into their images. The key issue is whether these are noticeable or bothersome — if they have any impact on the intended use of the telescope. Photographic imaging generally requires more stringent performance than visual systems. The eye's retinal resolution is roughly 2 arcminutes, and taking the Airy disk diameter as a standard unit of angular error within an optical system, aberrations in the objective become noticeable at roughly 2 diameters in photography and 3 diameters in naked eye observing. Defocus is the most primitive or basic form of optical aberration, and precise focus is critical to good optical performance. In many systems, defocus also projects underlying optical errors more clearly, so that a star image just out of focus may appear comatic or astigmatic. In reflector telescopes, the shadow of the secondary mirror produces a "donut" shaped ring of light in a defocused star image. The inner and outer borders of this figure will be chromatically tinged with a slight residue of lateral chromatism. These fringes reverse position inside and outside of best focus: in undercorrected systems, the extrafocal image will appear red around the interior and blue around the outside, reversing position in the intrafocal image. Appendix 1: Graphing Aberrations It may be useful to review the mathematical analysis of the Seidel (third order monochromatic) aberrations, as this can clarify how they are produced by an optical surface. Wavefronts & Lenses Eyepiece lenses are constructed as solids of rotation, which means their refracting surfaces are defined geometrically around an axis of rotation. Most are spherical surfaces, although aspheric lenses use surfaces such as ellipsoids, paraboloids or hyperboloids (respectively an ellipse, parabola or hyperbola rotated around its major axis). In the thin lens model the radius of the Petzval curvature (ρ) measured from the focal point ƒ is equal to: ρ = –n·ƒ' A positive lens produces a surface concave to the left (diagram, above), and a negative lens produces a surface concave to the right. Spot Diagrams The spot diagram is commonly used in optical design to assess image quality. It is created by tracing a large number (usually around 100) of rays from a point source that are incident in a regular or geometric pattern across the entire aperture. The spot diagram plots the location of these rays where they intersect the image plane. The diagram (below) shows the spot diagrams for "pure" forms of spherical aberration, coma and astigmatism, in focal planes that bracket the optimal focal position. Usually several aberrations affect the spot at once. For that reason, the spot at best focus is assessed in relation to a criterion spot diameter, which indicates the maximum spot dimension that is acceptable for a specific application. In visual optics the criterion can be as small as the diameter of the Airy disk (equivalent to the Rayleigh 1/4λ criterion); for photographic applications the criterion can be larger, on the order of 0.025 mm. The point spread function is the wave diffraction equivalent of the spot diagram. An example point spread function is illustrated here. More complex to calculate, it is used to provide the most accurate and detailed quantitative assessment of optical quality. Field Height Plots Only two meridonial paraxial rays are necessary for the analysis: the axial ray which intersects the aperture at the same field height as the point source, and the principal ray through the principal point at the aperture stop (diagram, below). More. More. Ray Intercept Curves Finally, aberration diagrams can also be based on the trigonometry of meridional rays. These are known as a ray intercept curve or H'–tan U' curve. The diagram (left) shows the basic logic of this kind of graph. Each meridional ray is plotted as the field height h' of its exit from the last refracting surface and the tangent of its angle u' to a reference plane that is perpendicular to the optical axis and placed some arbitrary small distance behind the optimal focal point. The inset diagram shows that a perfectly formed image will have More. Ray intercept curves also allow interpretation of the effects of refocusing the image, shown as the slope of the line or curve relative to the horizontal (x) axis. Refocusing rotates the curve around its central point, either counterclockwise (for intrafocal adjustment) or clockwise (for extrafocal adjustment). In the example (diagram at right, top), the curve for the undercorrected zonal spherical aberration is presented, with dashed lines showing the amount of aberration at the optimal focus (that is, the focus that most reduces the aberration in the central part of the image), and dotted lines showing the amount of aberration at the focus that reduces the total aberration. The second diagram (left, bottom) shows the effect of refocusing the image intrafocally to the optimal criterion — the dashed line is now parallel to the x axis. Most of the rays are refocused to within the criterion error. Note however that the amount of defocus in the total light is larger, since the slopes for the optimal and total refocusing in the upper diagram are not equal. The ray intercept curves show that coma cannot be reduced or eliminated by refocusing: focus only determines which part of the image will be most out of focus. Refocusing can significantly reduce spherical aberration, and either tangential or sagittal astigmatism (at the expense of the other). Refocusing cannot eliminate axial or lateral chromatic aberration, although it does determine which part of the spectrum will appear in focus. Appendix 2: Third Order Analysis It may be useful to outline the mathematical analysis of the Seidel (third order monochromatic) aberrations, as this can clarify how they are produced and interrelated. Only two paraxial rays from an object point O are necessary for the analysis: the principal ray through the principal point at the aperture stop, and a marginal ray that intersects the aperture space at point A. Both rays converge after refraction to the image point I (diagram, below). The principal ray defines the meridional plane, which always includes the object point O, the principal point P, the Gaussian image point I, and the optical axis. Not shown in the diagram is the sagittal plane, which is perpendicular to the meridional plane and includes the principal ray (points O, P and I) instead of the optical axis. Because all lenses are surfaces of rotation around the optical axis, the direction in which the object point is off axis doesn't matter — only its off axis radial distance h has any importance. So for convenience the analysis framework is arranged so that the y axis in object space and the y' axis in image space lie in the meridional plane: y = h. The aperture space uses a different convention for the incidence point A: its location is defined by s, the distance from the principal point, and θ, the angle measured at P between A and the meridional plane. This angle is expressed in radians: for rays in the meridional plane θ = 0 or pi; for rays in the sagittal plane θ = 1/2pi or 3/2pi. All points A that are not within the meridional plane are skew rays, and can represent any light ray emanating from O that is incident on the entire surface of the aperture. For astronomical objects (at optical infinity) the skew rays are parallel to the principal ray and both the meridional and sagittal planes. The significance of using a principal and a marginal ray is that aberrations are created primarily by light rays at a distance from the optical axis: either rays that are incident near the margin of the lens, and those incident near the margin of the image. Now consider the implication of various values of h, s and θ: • The significance of h is that it determines the angular height α at the principal point, which describes the abaxial angle of the column of light originating at h. For the object distances and true fields of view typical in astronomical work, h can be considered identical with either angle α expressed in radians or the tangent of α. Since angle α is equal on both sides of the lens, h also determines the field height of the point in the image plane. The off axis distance h shifts the angle of incidence of all rays originating at O in the same direction and by an equal amount (the angle α). In the sagittal plane, the angles of incidence for matching rays on opposite sides of the principal ray remain equal. But because the meridional plane contains the optical axis, rays above the optical axis are inclined away from the axis while rays below the axis are inclined toward it, so that the angles of incidence for matching rays on opposite sides of the refracting surface can be quite unequal. • The significance of s is that it defines the distance of the incidence point A from the vertex of the refracting surface at P. This indicates greater curvature in the lens surface: as the marginal ray and principal ray diverge by a larger distance (and s increases), the refractive power of the lens at point A must also increase in order to make the rays converge again at the image point I. This refraction is characterized as a zone that is concentric around the principal point at a constant radius s. For all equal values of s the refracting power of the optics will be equal, and as the value of s increases the concentric refracting power also increases. • The significance of θ is that its sine and cosine represent the relative proportion in the refractive effect at A that can be attributed to the sagittal tilt or the meridional tilt induced by h. Thus, cos(θ) is 1 or –1 when the ray is in the meridional plane and 0 when the ray is in the sagittal plane; and sin(θ) is –1 or 1 when the ray is in the sagittal plane and 0 when it is in the meridional plane. (Mnemonic: sine is sagittal.) In other words, the sine and cosine of θ define the X,Y coordinate locations of all points on a circle that is concentric around P and has a unit radius s. The Characteristic Function An intuitive analytic formulation of optical aberrations was developed by William Rowan Hamilton in 1833 and is known as the characteristic function. All light rays from the object point, regardless of their incidence point in the aperture area, must focus at a single point in the image plane, located by the principal ray on the y' axis at a height proportional to h. Any nonzero value of x' is by definition an aberration; any value of y' that is not proportional to h is also an aberration. The characteristic function identifies the origin and form of the departure from the Gaussian point focus x' and y' as the weighted sum of different combinations of the normalized field height of the object point (h, where the field limit equals 1.0), the normalized aperture height of any incident ray (s, where the aperture stop equals 1.0), and the aperture azimuth angle of any incident ray (θ, where the meridional plane is θ = 0° or 180°). The parameters h and θ define the type of aberration; s shows whether and how the effect varies over the aperture area. In this form the Seidel errors are technically known as third order aberrations because their mathematical definition requires terms for field height (h) and aperture height (s) whose exponents sum to three — h3, hs2, h2s or s3. (Fifth order and seventh order aberrations are sometimes analyzed in complex or asymmetrical optical systems.) Relative Orientation Gaussian Spherical Aberration Coma Astigmatism Curvature Distortion y' meridional A1·s·cos(θ)+A2·h +B1·s3·cos(θ) +B2·s2·h·(2+cos(2θ)) +3B3·s·h2·cos(θ) +B4·s·h2·cos(θ) +B5·h3 x' sagittal A1·s·sin(θ) +B1·s3·sin(θ) +B2·s2·h·sin(2θ) +B3·s·h2·sin(θ) +B4·s·h2·sin(θ) . The size or magnitude of each aberration is indicated by the subscripted weights A1, A2 and B1 to B5, and these weighted aberration terms add together to indicate the total aberration in the point image. Because the principal ray passes through the origin of the aperture coordinate system, s equals zero. All the aberration terms drop out by multiplication with 0, and only the Gaussian value A2 dependent on h remains: the field height of the image point I is linearly proportional to the angular height of the object point O. Any deviation from this point by the marginal ray image point, and the terms that contribute to this deviation, define the type and degree of aberration. Gaussian. The Gaussian parameters for x' and y' in A1 do not contain h: therefore there is no noticeable difference in image focus or image quality between on axis (h = 0) and off axis (h > 0) image points. The refractive power is linearly related to s, and there is no difference in effect between the meridional and sagittal planes. These are the characteristics of Gaussian or "perfect" optics. The radial term B5 dependent on h does not drop out for the principal ray and is independent of the aperture incidence point A of any marginal ray. This reflects the nonlinear increase in magnification for objects closer than –2ƒ to the aperture plane. The Gaussian sagittal (x') and tangential (y') terms are nonzero primarily in cases of defocus or longitudinal chromatic aberration. In those cases the size of the aberration is dependent only on the refractive power s, and produces a circular enlargement of the image that is concentric around the principal image point. Spherical aberration. This is the only third order aberration independent of the object angular height h: it can therefore appear even in on axis image points and affects all image points equally. Thus, the aberrated rays are symmetrically distributed around the image point (the parameter forms are identical in the x' and y' equations). The aberration increases in proportion to the cube of the aperture height (s3). If we assume that the aperture area is equally illuminated, then the exponent means that most of the visible aberration is generated by the areas with large s that are near the marginal circumference of the aperture, and this marginal effect is much more pronounced for s3 than for s2. That interpretation always assumes that the focal plane is located at the Gaussian image point. As illustrated above, moving the focal plane along the optical axis will bring the rays from different aperture heights into focus, so that for example the visible aberration can be made to arise from the central aperture area at the marginal ray focal point. In other words, refocusing will change the size and even algebraic sign of the aberration weights A and B. Coma. The complexity of the coma aberration is evident from its parameters in the characteristic equation. We see first that coma is related to both h and s: coma becomes linearly larger as field angle h increases, and increases with the refracting power s2 toward the marginal circumference of the optics. Thus coma appears in off axis image points, and proportionately most of the visible aberration arises from the marginal rays. However, the aperture orientation θ is doubled for both the x' and y' dimensions, which means that the comatic image results from the superposition of light rays from opposite sides of the aperture area and concentric around the principal point P, as shown in the diagram (right). In addition, the tangential contribution is shifted by a factor of 2, regardless of the value of θ or s, which means the center of the circular aberration is shifted radially away from the principal ray image point I by a distance equal to twice its diameter. (Because θ always has a unit diameter, the coordinate locations of B and D will be x' = –1 and 1, A and C will be located at y' = 3 and 1.) The last two factors produce the distinctive shape of the comatic aberration, which consists of circular focal rings nested within a 60° angle expanding from the principal ray image point. The aberration is brightest at the point and along the sides, producing a "webbed V" shape. The exponential terms s2 and s3 indicate that spherical aberration and coma are much more strongly affected by the shape of the refractive surface than by the off axis angle of the incident rays. That means both aberrations are always present in single lenses with spherical surfaces. The aberrations must be corrected by making a single lens or reflecting surface aspheric — either paraboloid or hyperboloid — or by the use of compound lenses with different radii whose combined effect approximates an aspheric power. Astigmatism. Like coma, astigmatism is a complex aberration with asymmetrical contributions to the x' and y' dimensions. The tangential weight is 3 times the size of the sagittal, which means they focus at different locations; and these are in relation to the focal surface defined by the Petzval curvature — the astigmatic tangential focal surface is 3 times farther from the Petzval surface than the sagittal focal surface. Petzval curvature. The curvature parameter is identical in form to the astigmatic, dependent on both aperture zone s and the field angle h2. Unlike astigmatism, the x' and y' contributions are identical. Petzval curvature defines a paraboloid focal surface whose vertex is at the principal focal point that becomes more extreme with the field angle. Distortion. This aberration is a tangential (radial) displacement of the image as field height increases; it has no sagittal component. It increases as the cube of the field angle. The Seidel Errors in Reflecting Telescopes Listed below are the calculations for the five Seidel aberrations, as derived from basic calculations and as given by Nicklas (1994). These reveal the dependence between the Abbe invariant (Q) and spherical aberration (I), spherical aberration and coma (II), and finally the dependences among astigmatism (III), field curvature (IV) and distortion (V), which are all three affected by the Petzval sum (Σ) and the position of the aperture stop (p). Each surface of the optical system contributes an additive portion of the total aberration, and is denoted by a subscript: h1 denotes the first surface, and hν denotes each of the ν subsequent surfaces. Notation nν – refractive index of air or glass hν – intersect height of refracted ray on lens surface sν – distance from surface intersect to optical axis measured, from surface vertex along optical axis uν – angle of refracted ray to optical axis rν – radius of curvature of refracting surface dν – focal length of lens dEP – distance from surface vertex to aperture stop x signifies a quantity before refraction (to the left of refracting surface) x' signifies a quantity after refraction (to the right of refracting surface)