Part 2: Telescope & Eyepiece Combined
This page introduces the optical principles necessary to understand the design and performance of astronomical telescope systems the telescope and eyepiece used as a visual instrument with the eye included as a third component. It is one of series: the previous page explained basic optics; subsequent pages discuss optical aberrations, eyepiece designs and evaluating eyepieces.
Included at the end of each page is a list of Further Reading that identifies the sources used and points to additional information available online.
Telescope & Eyepiece Combined
To begin, let's consider the basic functions, parts and terminology that characterize the Keplerian telescope an "inverting" objective combined with an eyepiece. On this page, diagrams schematically represent the telescope objective as a single refracting lens, but it may (without changing the optical characteristics) be a dioptric system (comprising one or more lenses), a catoptric system (one or more mirrors), or a catadioptric system (a combination of mirrors and lenses). In all cases the objective is combined with an eyepiece, used as a magnifier to inspect the detailed content of the objective image.
Basic Telescope Functions & Attributes
An astronomical telescope has three basic functions:
(1) Light grasp. Our ability to see very faint (low luminance) objects is limited by the area of the pupil opening of the eye, which admits only a small amount of light. The telescope admits a column of light whose cross sectional area is many times larger than the pupil, increasing the total illuminance (light content) of the image. Implicit is the complementary function of beam compression, which means that the light gathered by the large aperture is concentrated into a much smaller diameter beam that can completely pass through the relatively tiny pupil of the eye or fit within a limited area of film or photosensor chip.
(2) Angular resolution. Our ability to see objects that are small or at great distances is limited by the eye's nominal angular resolution which can be from about one to over six arcminutes, depending on the brightness and contrast of the image. A telescope transmits the image of a distant object through a very wide pupil, producing diffraction limited resolution on the order of one arcsecond. Necessary for the visual use of that resolution is the complementary function of magnification of the telescopic image, which means the arcsecond details of the image are enlarged to match the arcminute resolution of the eye. This magnified image remains sufficiently illuminated because of the light grasp of the telescope.
(3) Pointing. The eye as a mobile receptor can look in any direction and remain fixed on a moving single object. The telescope must be able to match this capability so that its optical axis can be aligned in any specific direction and the alignment sustained for extended observation or long exposure photography. For telescopes on the surface of the rotating earth, this implies a mounting with two axes of alignment and computer regulated sidereal motors capable of tracking any point on the celestial sphere.
Note that light grasp, beam concentration and magnification can be entirely analyzed in terms of geometric light rays, but angular or linear resolution must be analyzed in terms of physical light wavefronts.
These generic telescope functions light grasp (with beam compression), resolution (with magnification), and pointing (with tracking) may be adapted in different ways for solar astronomy, spectroscopy, wide field photography, transit measurements or field portability, but the tradeoffs among them become most evident in telescopes used for visual astronomy.
Three Fundamental Optical Attributes. A slightly different way to think about the three basic telescope functions is to consider only the optical qualities of the instrument. These, again, consist of just three attributes:
(1) Aperture - Aperture determines several important but very different optical and physical attributes of the telescope: light grasp, the relationship of light to a standard sensor, for example the stellar limit magnitude as perceived by the visual observer; the angular or image resolution, the smallest angular interval that can be imaged above the diffraction limits of the instrument; the aperture seeing, the relative effect of atmospheric turbulence on the telescopic image; and, to a less precise degree, the instrument seeing (partly determined by the mass and thickness of the objective), and the total mass or weight of the instrument, which affects its portability and the type of mount that can be used.
(2) Focal Length - Optically, focal length primarily determines, and substantially limits, the magnification of the instrument. A longer focal length increases the baseline magnification by means of the same projective geometry that increases image dimensions as a projection screen is moved farther from an imaging pinhole: for astrophotography this is the sole source of image magnification at the focal plane. As the magnification in visual astronomy is modified by the choice of eyepiece, the focal length sets the upper and lower bounds of magnification possible within the range of commercially available eyepieces (generally, from about 3mm to 40mm). Focal length also determines the dimensions of the telescope (separate from the mass or weight), which can significantly affect viewer ergonomics unless the optical path is folded as in the various cassegrain designs.
(3) Relative Aperture (Focal Ratio) - Relative aperture has the specific quality of affecting the image brightness or relative illuminance of the image, in the same angular geometry that causes the page of a book to be more brightly illuminated the closer it is placed to a window. Shorter (lower numerical value) relative apertures are considered "faster" optical systems, concentrating more light per square millimeter at the image plane. This does not affect the visual limit magnitude because the greater concentration of light is extended over a wider field of view, and aperture limits how much light is available to illuminate this wider field. Relative aperture also determines the linear scale of the focal plane, a basic parameter in astrophotography.
These three fundamental optical attributes are sometimes combined in the metric of the exit pupil, which can be understood as either the expression of the effective relative aperture as the ratio of aperture to system magnification, or as the expression of effective relative aperture as the ratio of dioptric magnification to objective relative aperture. In both cases, exit pupil derives the same quantity from the ratio of two system attributes. Here effective relative aperture means that the exit pupil defines the angular width of a virtual entrance pupil (aperture) as it would appear from a focal length of a single objective lens or mirror that delivered the total system (objective plus eyepiece) magnification. As the magnification increases the effective focal length increases, and the effective relative aperture of the system numerically increases, delivering less light to the image as the price of increased magnification.
The exit pupil is a "scale free" optical parameter, in the sense that it provides no information about the actual (physical) dimensions of the aperture, focal length or focal ratio of the telescope itself. Instead, it has a relatively limited interpretation as proportional to the relative brightness of the image compared to the maximum image brightness permitted by the diameter of the observer's eye pupil, and as proportional to the angular size of the image diffraction limits of the instrument. (The exit pupil does not apply when imaging is done at the objective focal plane.)
Resolution vs. Light Grasp. Magnification, resolution, light grasp and limit magnitude (the faintest star visible with the aperture for a given eye pupil diameter and sky brightness) all increase with larger aperture. However, object magnification increases as the aperture diameter (Do), resolution increases as the reciprocal of aperture diameter (1/Do), light grasp as the area of aperture (Do2) and limit magnitude as the logarithm of aperture diameter (log[D]). These different mathematical definitions indicate that the factors will not change in equal proportion as aperture increases. Does this affect the optimal choice of aperture?
To evaluate this issue, we can examine a range of apertures suitable for astronomical observing, from the minimum aperture that might be found useful ("min" aperture, here assumed to be a 40 mm binocular) up to the largest telescope most astronomers could afford or find practical to use ("max" aperture, here assumed to be 20 inches or 500mm). Then we calculate the changes in resolution and light grasp as the aperture increases across the feasible range (that is, as a proportion of the 460 mm between 40 mm and 500 mm, starting at 0 or 40 mm up to 1.0 or 500 mm). The diagram (below right) shows the relative change in light grasp and resolution (y axis) plotted along this increasing proportional aperture (x axis).
Across this range, the resolution limit at first decreases dramatically with aperture but then begins to level out, so that 75% of the total possible gain in resolution is achieved with only 20% of the maximum aperture (130 mm or about 5 inches) and 93% of the possible gain is achieved at half the maximum aperture increase (270 mm or 10.6 inches). As a result, when the range of feasible apertures is large, better resolution is a weak incentive to increase aperture above this 50% threshold. (Magnification increases linearly, along the dotted diagonal in the diagram.)
In contrast, light grasp at first increases only slowly less than 30% of the total possible gain in light grasp is achieved with half the maximum aperture increase then accelerates across the largest apertures: 75% of the maximum possible gain is only achieved with 85% of the maximum affordable or practicable aperture increase (430 mm or 17 inches). Relative changes in aperture have greater impact on light grasp (the curve adopts a steeper slope) in the upper half of the aperture range.
Although not shown in the diagram, the most important gains in limit magnitude are also obtained across the smaller apertures. Nearly 50% of the total possible gain in limit magnitude is achieved with just 20% of the aperture increase, and 75% of the gain is obtained with 50% of the aperture increase. (The limit magnitude of the 500 mm aperture is 16.1, of the 250 mm is 14.6, and of the 40 mm is 10.6.)
Finally, the diagonal in the diagram does not convey the full picture of magnification increase with larger aperture. Paul Couteau (1976) suggests that the minimum magnification necessary for measuring double stars is double the "resolution magnification", usually said to be equal to the aperture (M = Do in millimeters). J.B. Sidgwick (1938) suggests that the maximum "practical magnification" for visual double star astronomy is found as M = 28·√Do. For M = Do the maximum useable exceeds the minimum necessary up to Do = 780 mm, but for Couteau's criterion M = 2Do the crossover is at ~195 mm, and for M = 2.5Do it is at ~125 mm. Thus, smaller telescopes allow more useable magnification to be wrung out of the aperture.
These cost/benefit calculations divide the range of affordable or practical aperture into two regimes. The utility of relatively small telescopes is primarily in their resolution and stellar limit magnitude, and any increase in aperture within this range increases resolvable detail proportionately more than light grasp. Above roughly the half aperture point, increases in aperture deliver proportionately far more gains in light grasp than in resolution. As a result:
Double star, lunar and planetary astronomers generally prefer high optical quality telescopes on the "resolution" side of their affordable or practicable aperture choices. The primary image criterion is the resolution of very small angular detail and high contrast transfer (a central obstruction ratio ≦ 0.3); the comparative lack of light grasp is offset by the intrinsic brightness of the targets and the adequate stellar limit magnitude. Smaller apertures are also less sensitive to atmospheric turbulence, making full use of the aperture's resolution and magnification potential, and they permit longer focal length for the aperture (N ≥ 10) which increases magnification and minimizes off axis aberrations.
Deep sky astronomers generally prefer instruments in the "light grasp" half of the regime despite the resolution problems that may arise from mirror mass (flexure, cool down, collimation) and the sensitivity of larger apertures to atmospheric turbulence. These instruments are usually manufactured with a small relative aperture (N ≦ 6) that produces a brighter visual image and wider field of view, most effective for extended targets with a low surface brightness, despite the off axis aberrations and collimation problems that may result. Shorter focal length also makes the instrument physically more compact and easier to transport to dark sky locations.
Astrophotographers and visual generalists generally prefer telescopes configured to an intermediate relative aperture (N = 6 to 10) since this (with corrector optics if necessary) produces a relatively flat imaging field across an image area large enough to cover a CCD chip. Focal ratio constrains the aperture to sizes not much larger than 300 mm and very often less, as atmospheric turbulence discourages the use of large aperture. Extended exposures and software image stacking can compensate for restricted light grasp, and "lucky imaging" with rapid image capture can overcome aperture and atmospheric seeing.
Technology complicates the simple magnification/light grasp tradeoff. In amateur equipment, electronic imaging with video or infrared sensors can substantially boost the light grasp of small instruments. In research telescopy, the largest 8 to 10 meter terrestrial instruments (Mauna Kea, La Palma, Cerro Paranal) have the large feasible aperture of light grasp instruments, and their aperture is often applied to infrared imaging (where the large wavelength reduces resolution) and to spectroscopic studies of faint objects such as exoplanets and distant galaxies. Astrometric and survey tasks are generally handled by smaller aperture instruments; the 2.4 meter, /24 Hubble Space Telescope is well below the 5 meter half aperture mark, making it more suitable as a resolution rather than light grasp instrument, especially as it suffers no degradation from atmospheric turbulence. (The famous "Deep Field" and "eXtreme Deep Field" images from the Hubble telescope required cumulative exposures of approximately 6 days and 23 days to compensate for the relatively small aperture; they were touted to reveal the high resolution "actual forms and shapes" of the earliest galaxies.) However, the largest telescopes are now equipped with adaptive optics, and relatively small telescopes can be yoked as interferometric arrays, boosting their resolution and useful magnification.
Afocal Systems. For visual use, a key feature of astronomical telescopes is that the combination of telescope objective and eyepiece creates an afocal optical system. The objective focuses a beam of parallel light rays from an "infinite" (far distant) object or object space (the celestial sphere) as a point on the image plane at the apex of a converging light cone. The eyepiece receives the light rays diverging from the opposite side of the image plane and focuses them a second time into a much smaller exiting beam of collimated light rays.
As a result, the telescope and eyepiece combination does not have a single focal point or focal length, although the separate objective and eyepiece do.
The afocal output provides a comfortable visual inspection of the telescope image. The eye is structured so that small muscles can cause the eye lens to bulge, reducing its focal length to focus the diverging light rays from nearby objects. Infinitely distant objects enter the eye as parallel light rays, and to focus these the focal eye muscles become completely relaxed. Because a telescope produces an exit beam of parallel light rays from every object point, the eye can view the telescope image in a completely relaxed state, allowing extended viewing without strain. When the telescope is used with a camera or other recording instrument, the afocal output is not required and the telescope is typically used without an eyepiece.
Stops, Pupils, Windows & Baffles
The light grasp and beam compression functions of a telescope are defined by stops, which are physical obstructions within the system that limit the aperture or image area. The optical lenses and mirrors of the system create images of these obstructions, called pupils and windows, which can be viewed from either the object or observer end of the telescope (diagram, below). Baffles do not affect either the aperture or image area of a telescope system: they are positioned outside the path of the imaged light in order to minimize internal reflections.
The diagram (below) introduces the basic terminology. Note that object space refers to the physical location of whatever is being observed with the telescope system, and image space refers to any point within the system where a real and in focus image is formed.
The optical axis is the axis of rotation for all refracting or reflecting surfaces; mirrors, lenses, stops and baffles are centered on the axis and perpendicular to it. The exceptions, not discussed here, include systems with redirecting optics such as diagonal mirrors or prisms, or with tilted objectives (such as the the Schiefspiegler telescope design).
Axial light rays are emitted from a point on the optical axis; if the point is sufficiently far away (at optical infinity), the light rays reach the telescope as a collimated beam of rays parallel to each other and to the optical axis.
Stops. A stop is a physical obstruction that limits the amount of light entering or leaving the telescope system.
The aperture stop is the obstruction that limits axial light rays entering the telescope system: its interior diameter defines the aperture (Do) of the telescope. In astronomical telescopes this is either the mounting of the refracting objective lens or catadioptric corrector plate, or the circumference of a catoptric reflecting mirror. The aperture stop defines the principal point of the objective; its position affects the appearance of aberrations in the image created at the objective focal surface.
The field stop is the obstruction that limits the area of the image or the amount of light exiting the telescope system. It is usually coincident with the objective focal plane and defines the image diameter (do), and is either a diaphragm within the eyepiece barrel or a field lens lock nut doubling as the field stop, or the edges of a camera CCD chip or film holder. The field stop controls the apparent field of view (the angle of marginal rays passing through the eyepiece or onto the CCD sensor), and it limits the appearance in the image of aberrations that increase with field height.
The linear diameter of the aperture stop defines the clear aperture which determines the angular resolution of the objective. For reflecting telescopes with a central obstruction (secondary mirror support), the area of the secondary obstruction (Ds) must be subtracted from the area of the aperture stop to compute the effective aperture (Deff) that determines the light grasp of the system.
Pupils. A pupil is an image of the aperture stop, viewed from a point on the optical axis at either end of the telescope.
The entrance pupil is the image of the telescope aperture stop, illuminated from the image space and viewed from the telescope object space.
The exit pupil (symbol de) is the image of the telescope aperture stop produced by the objective and eyepiece combination, illuminated from the object space and viewed from the observer end. (See also the discussion of the exit pupil here.)
Pupils are images of the the same telescope obstruction, so the magnification of the telescope system (Mt, diagram above) is equivalent to the diameter of the entrance pupil divided by the diameter of the exit pupil:
Mt = Do/de.
In the United Kingdom the exit pupil is sometimes called the Ramsden disc. This is the specific plane cross section of the telescope's afocal output at the eye point, where the the exit pupil has the smallest diameter. When the observer's eye pupil coincides with the exit pupil or Ramsden disc, all the light focused through the exit window (image area) will enter the eye, provided the eye pupil is large enough.
Windows. A window is an image of the field stop (the image area), viewed from a point on the optical axis at either end of the telescope.
The entrance window is the virtual image of the eyepiece or camera field stop, illuminated from the image space and viewed from the object end of the telescope.
Baffles. A baffle is an interior obstruction, constructed as a diaphragm or tube, that blocks stray light and internal reflections from entering the image area and introducing glare or reducing contrast in the image. Baffles do not affect the aperture, image dimensions or appearance of aberrations. Because they are not stops, baffle edges are always just outside the circumference of the light cone concentrated by the objective.
Finally, note that the telescope, as an afocal system, produces a collimated beam of light exactly like the collimated beam of light that enters it except that it has a much reduced diameter. The diagram illustrates this beam compression function as a collimated beam of axial rays, but an identical beam compression and exiting collimated beam is produced in the collimated rays arising from any off axis point.
Focal Length & Field of View
The angular resolution and magnification functions of the telescope system are defined by the diameters and optical powers of the telescope objective and eyepiece. These must be explored by tracing the path of geometric rays through the telescope system, and measuring their location at the objective, the image plane, and exiting the eyepiece.
First, some definitions. The axial rays (green lines in diagram, above) originate from the unique object point on the optical axis and, for objects at infinity, arrive as a collimated beam parallel to the optical axis and perpendicular to the entrance pupil (equivalent to a flat wavefront perpendicular to the optical axis and parallel to the entrance pupil). Collimated axial rays converge at the objective focal point (o) where the optical axis intersects the focal plane, and exit the eyepiece as a compressed and collimated beam centered on the optical axis. Oblique rays (red or blue lines in diagram, above) are rays that are not parallel to the optical axis; they focus at a point off the optical axis, and produce oblique exit beams from the eyepiece. Marginal rays are those axial and oblique rays that pass the edge of the aperture stop: these are used to specify the effect of stops on the image and the size of the unvignetted image area.
All linear or area measurements are conventionally (and for calculation purposes conveniently) expressed in millimeters, and angles are expressed in either radians or (for very small angles) as tangents or sines.
The aperture stop fixes the location of the objective entrance pupil (Do). This defines a plane perpendicular to the optical axis of the objective. The objective back principal plane is the principal plane of the objective (the plane intersecting the vertex of a positive mirror, or the back principal plane of an objective lens) independent of the location of the entrance pupil.
The intersection of the entrance pupil with the optical axis is the principal point of the objective. A principal ray or chief ray is any geometric ray that passes through this principal point.
The objective focal length o is distance along the optical axis from the back principal plane to the objective focal point, defined as the convergence point of axial rays from the object space. (In Cassegrain telescopes, this is not the telescope focal length, only the primary mirror focal length.)
The convergence point of collimated beams exiting the eyepiece is the exit pupil, and the intersection of the exit pupil with the optical axis is the eye point or optimal location of the observer's eye pupil.
The location of points in the object space and of conjugate points within the telescope system are measured by a system of angles and corresponding physical distances measured in relation to the optical axis:
The object height h is the distance of an object point from the optical axis as measured in units of the physical object space.
The field angle α of an object point is the angle, measured at the objective principal point, between the optical axis and the principal ray from any point in the object space. Field angles are used in place of the object height h to locate points in an inaccessible or infinitely far object space (the celestial sphere), and can be defined as either the angle itself when the angle is not large (less than ~3°) or as the radian or tangent equivalent of the angle.
The aperture height y is the point at which any incident ray, regardless of its field angle, intercepts the entrance pupil. It is measured as the distance of the incident point from the optical axis on the plane of the entrance pupil.
The corresponding image ray angle u' is the angle between a geometric ray and the optical axis, measured at the image plane and in the plane that contains the object, incident and image points and the objective principal point. For axial rays, this plane will always contain the optical axis. For oblique rays, the unique plane that contains the optical axis is called the meridional plane, and the plane perpendicular to it that passes through the object, principal, incident and image points is the sagittal plane.
The objective focal ratio or relative aperture (No) is the objective focal length divided by the aperture: No = o/Do. This determines the angle of convergence of marginal rays onto the image plane (proportional to the tangent of u') and the widest true field of view possible with the objective.
The image height h' is the physical distance from the optical axis to an image point measured on the objective image plane. It is calculated from the field angle as:
h' = tan(α)·o or h' = o·(α/57.3).
The image radius is measured as the most oblique chief ray that can pass the field stop. Measured as physical units (millimeters) on the image plane, this is the image height h' of the field stop; twice this width is (for a circular exit window) the image diameter. Measured as a visual angle at the focal point, this is the field angle α of marginal rays, and twice this width is the projected true field of view (TFOV) the maximum angular width of sky focused by the objective within the area of the field stop.
The projection angle β is the angle measured at the eye point between the optical axis and the central ray of a pencil of light originating in a specific object point.
The ray paths that define the points and angles listed above can be depicted as lying within the plane of a single cross sectional diagram (for example, in the diagram above). However, it is often important (especially in the analysis of optical aberrations) to specify the radial or "clock face" orientation of this plane: this is the azimuth angle of an incident (θ) or image (θ') point. Azimuth angle is defined as the clockwise rotational angle between a reference plane and the plane that contains both the optical axis and the relevant points or rays. The reference plane contains the optical axis and is oriented to intercept a point located at vertical ("12 o'clock") in object space in relation to the position of the observer.
These heights and angles are the fundamental units used to calculated the path of light from the entrance pupil or objective to the image plane.
The magnification of the telescope system is equivalent to the ratio between the field and projection angles (diagram, above):
Mt = β/α
where both angles are expressed in radians and are measured from the optical axis to a chief ray that defines the image radius (at the inside edge of the field stop).
Finally, the eyepiece is two or more refracting lenses, mounted as a unit, that can be conveniently placed at or removed from a position immediately behind the objective focal plane. The eyepiece functions as a magnifier that expands the objective image so that it can be seen, enlarged and in focus, by the observer's relaxed eye.
The eye point is the location on the optical axis where the convergent pencils of light from the eyepiece are focused.
The distance from the eyepiece field stop (focal plane) forward to the eyepiece front principal plane is the eyepiece focal length (e); the location of the eyepiece front principal plane is calculated from the separate focal lengths of the eyepiece components. Equivalently, the effective focal length is the distance from the eye point backward to the location of a plane perpendicular to the optical axis at which the projection angle (β) of the image radius intersects the physical height of the field stop (dashed green line in diagram, above).
The exit pupil is located within a plane perpendicular to the optical axis at the eye point and is the smallest diameter area that passes all the light transmitted by the eyepiece. The exit pupil contains the smallest and brightest image produced by the eyepiece. It is the point at which the entire eyepiece image area (field stop circumference) can pass the observer's eye pupil: at any longitudinal displacement of the observer's eye pupil, the exit pupil functions as a stop that can create a keyhole view or reduced area view of the projected image.
The distance from the exit pupil to the last air boundary of the eyepiece is termed the eye relief (ER) or eye distance, often equivalent to the back focal length (BFL). Astigmatic eyeglass wearers generally prefer eyepieces with a large eye relief.
When attached to a telescope, the eyepiece does not project a conjugate image point. Light from any point in the object space exits the eyepiece as a bundle (or pencil) of parallel rays gathered across the entire aperture. Because the rays are parallel, they are afocal: they do not form a unique focal surface.
Because the exit rays from the eyepiece are parallel from any specific point in the image field, they create the illusion that the eyepiece image arrives from infinitely far away. In that case, the parallel rays within each pencil can be made to converge to a focus by the optical power of the eye when the eye is fully relaxed, just as it is when viewing a far landscape or the night sky. (Note that if the image is projected onto a screen at some larger distance from the eyepiece, for example to view sunspots or a planetary transit, the eyepiece is placed in the extrafocal position and pencils exiting the eyepiece converge into a real image at a finite distance.)
This section illustrates the most common optical formats found in astronomical telescopes. Excluding the eyepiece from consideration, the historical order of innovation has been from dioptric systems (comprising one or several lenses), to catoptric systems (one or more mirrors), to modern catadioptric systems (a combination of mirrors and lenses).
Refractors (Dioptric Systems)
History. The refractor is the oldest telescope format, dating from the spyglass designs invented by Hans Lipperhey (1608) and first used for astronomical research by Galileo Galilei (1609). This became the aerial telescope used in the 17th century by Huygens, Jan Heweliusz (Hevelius) and Gian Domenico Cassini. Refractors built in the 17th century minimized aberrations and increased magnification by utilizing extremely long focal lengths that were cumbersome to mount and point: the "Cavendish" telescope owned after 1691 by the Royal Society of London had an 8" (20 cm) lens and a focal length of 122 feet (/186). A 1688 engraving of the Paris Observatory shows observers on the ground using aerial telescopes supported on a tower or from the roof of a building four stories high. By the turn of the 18th century, smaller spyglasses were common among the affluent, and in the last half of the 18th century John Dollond produced a number of small tripod mounted, /22 achromatic refractors with apertures under 6 cm.
Early refractors provided good light transmission but were hampered by the manufacturing difficulties of pouring large, chemically homogenous, strain free and bubble free glass blanks, the fabrication difficulty of grinding the glass to the required figure, the theoretical problem of designing lens combinations to eliminate optical aberrations (chromatic and spherical aberrations in particular), and the mechanical challenge of mounting the instrument so that it could be conveniently pointed at any part of the sky and then driven to hold the object stationary in the eyepiece field of view.
These four difficulties were overcome in the early 19th century in two 24 cm /18 achromatic refractors with optics, German equatorial mounts (the first of their kind) and clockwork sidereal drives designed by Joseph von Fraunhofer. These were the Dorpat Refractor installed in Tartu Observatory (Estonia) in 1824 and a twin instrument installed in Berlin Observatory in 1835. Within a dozen years three more large refractors were built: a 34 cm /23 achromat by Cauchoix was installed at Markree Castle, Ireland (1834), and two 38 cm /18 achromats at Pulkovo (1839) and Harvard College Observatory (1846). Along with a 38 cm /23 achromat installed at Meudon, Paris in 1855, these were the largest refractors in the world at the mid 19th century (chart, right).
The last half of the 19th century became the great age of refractors, led by the optical firms of Merz & Mahler (Germany), H. Grubb (Ireland) and Alvan Clark & Sons (USA) and culminating in the major instruments (all with apertures greater than 65 cm and focal ratios between /12 and /21) installed after 1870 at the U.S. Naval Observatory (Washington, DC), Charlottesville (Virginia), Vienna, Pulkovo, Nice, Meudon (Paris), Greenwich, Treptow (Berlin) and Potsdam. However, subsequent to the installation of two large reflectors at Mt. Wilson Observatory in 1908 and 1917, the pace of refractor construction slowed and refractors played a steadily diminishing role in astronomical research. They are still the instrument of choice for many amateur visual astronomers and astrophotographers.
The fundamental historical constraint on refractors is dimensional in both focal length and aperture. Without mirrors, the focal length must be straight from objective to eyepiece, and the requirement to minimize chromatic aberrations forced the focal ratio to be large.
In addition, the objective can only be supported around its perimeter, but the maximum size of glass lenses that will not deform under their own weight is around 1 meter. As a result, the largest operational refractors ever made are the 91 cm /19.3 Lick Telescope and the 102 cm /18.6 Yerkes Telescope (image, left), both manufactured by Alvan Clark & Sons for observatories completed in the 1890s. Clark aspired to build a 152cm (60") refractor, but James Keeler reported clear flexure effects in the Yerkes images, indicating that refractor glass aperture had reached its useful limit.
Even at their "modern" relative aperture of /19, the Lick and Yerkes telescopes are enormous for their aperture. Thus, for visual use of the Yerkes at a 30° zenith angle, the observer may need to be positioned 20 or more feet above the floor, and the sidereal drive will move the eyepiece end of the telescope about 1.5 inches every minute, requiring continuous adjustment of observing position.
The early advantages of refractors were in the relative simplicity of figuring their spherical surfaces and the low maintenance required to use them; the transmission of thick glass lenses is somewhat better than the reflectance of speculum mirrors, but the glass does not tarnish. Once chromatic aberration, spherical aberration and coma were minimized through the use of achromatic objectives and sophisticated optical fabrication (including very specific glass formulations for refraction and dispersion and hand correction of lens surfaces), refractors were manufactured in the 19th century to very high optical standards. These instruments opened the door to fundamental research in lunar, solar, planetary and double star astronomy, as well as the earliest applications of photography in astrometrics, spectroscopy and wide field surveys.
Ray tracing a refractor is a straightforward application of the principles outlined in the previous page. The entrance pupil defines the front principal plane and the principal point for all chief rays. The back principal plane emits all rays at the refracted angle. The focal length is positive and the image orientation is negative and rotated (reduced, inverted and reverted).
Refractor Design Principles
Refractor objectives can be constructed from one to four separate lenses or elements, each potentially having distinct surface curvatures and made of a very specific type of glass with different refractive and dispersive attributes.
Achromats. The historically earliest form of refractor a single biconvex crown lens produces an image with severe longitudinal chromatic aberration due to dispersion of the refracted wavelengths.
In an achromatic doublet, the negative flint lens produces dispersion in the opposite direction, which is manipulated to match and therefore cancel the dispersion in the positive lens at the desired focal length. The negative flint does not have sufficient refractive power to undo the converging effect of the positive lens, but it does increase the focal length and focal ratio of the combination.
Unfortunately the dispersion of the crown and flint glasses, if matched at two widely separated wavelengths, will not match at every other wavelength, which is quantified as the partial dispersion of the glass. This produces a spherochromatism in which the "green" wavelengths have a longer focal length than the "red" or "violet" focal lengths: the intrafocal image is surrounded by magenta light, and the extrafocal image by green light.
Apochromats. In an apochromatic triplet, two crown lenses are combined to produce a dispersion that closely matches the dispersion of a specific flint glass. Ernst Abbe defined a true apochromat as an objective corrected parfocally for three widely spaced wavelengths and corrected for spherical aberration and coma for two widely separated wavelengths. In fact, this rigorous definition is not met in modern "apochromatic" lenses: the variation in focus with wavelength (spherochromatism) can only be corrected with a very long focal length, large air spaces, aspheric lens surfaces or a Petzval design. Instead, spherochromatism is minimized as far as feasible with some combination of those techniques, and coma is corrected at one wavelength, which effectively minimizes it at all visible wavelengths.
Design Process. The constraints on an optical design are the clear aperture, the focal length (which also yields the focal ratio), the edge thickness of the positive element, the center thickness of the negative element, and the glass specification for both elements. Then the four radii of curvature and interlens spacing are determined analytically.
Glass selection is often the critical design choice, limited by the optical and fabrication attributes of the glass, the availability of material as glass blanks of the necessary size and quality, and cost. The positive crown element must have a higher index of dispersion than the flint, and the flint a higher index of refraction; the partial dispersions, or changes in dispersion across the spectrum, must be as nearly equal as possible.
In an achromatic doublet, spherical aberration is controlled by the ratio between the radii of curvature of the interior surfaces, r2/r3, and to a lesser extent the interlens spacing; coma is controlled primarily by the radius of curvature of the front surface (r1). The relative spherical aberration of the red and blue light is affected by the power of the front lens, and field curvature is proportional to focal length. An aplanatic system (free of spherical aberration and coma) is possible by making r2 = r3 with a substantial air space between elements, but this reduces the available field of view and increases the possibility of decentering of the lenses and sensitivity to miscollimation.
The dominant design feature of a refractor is its relative aperture (No) or /ratio. A larger focal ratio (longer focal length at a constant aperture) increases the system magnification and because it is produced by a lower power objective (with less surface curvature) it minimizes optical aberrations, including chromatic aberrations. Classical achromats in apertures up to 280 mm are available today in focal ratios as short as /10, and 19th century refractors were manufactured at focal ratios of /12 to /26, where the refractor's visual performance is excellent in bright target observations (solar, planetary and double star astronomy).
Chromatic aberration in achromats is sometimes evaluated by the CA ratio, defined as:
RCA = N/D
where N is the relative aperture (/ratio) and D is aperture in inches. Chromatic aberration is usually deemed "objectionable" when RCA < 5.0 (the Conrady standard for achromatism) or 3.0 (the standard proposed by Sidgwick), but this can be interpreted with equal validity as an observer characteristic some people are very distracted by chromatic aberration, while others don't notice it. In addition, observers who concentrate on bright targets such as the planets and the Moon will notice much more chromatic aberration than observers of deep sky objects and faint double stars.
In the interest of a wider field of view (for photographic use) and portability (shorter tube length), most refractors manufactured today have much smaller focal ratios, usually from /8 down to /5. Certain types of optical glass, denoted ED or extra low dispersion glass, allow greater accuracy in the design of refractive surfaces that reduce both coma and chromatic aberration. Achromats made with an ED glass element are sometimes marketed as ED achromats or semi apochromats, for example by Sky-Watcher.
To manage the intrusion of optical aberrations at the necessary higher powers, a third lens is used in an apochromatic design. These are often marketed as APO or APO triplet designs, for example by TEC, Astro-Physics, Takahashi and Sky-Watcher. A further refinement is to use a fourth lens to form a crown front lens with an ED triplet, sometimes called a super apochromat, for example in the TSA refractors by Takahashi.
Reflectors (Catoptric Systems)
History. The reflecting telescope is also a 17th century conception, originating in three different sources: the "back focus" designs of Mersenne (1636) and Cassegrain (1672) which were actually not built in any numbers until the 20th century and the prototype "side focus" reflector designed and built by Isaac Newton (1668). These designs consist of one primary mirror to gather and focus light and one secondary mirror to direct the converging light cone to an observing location that does not obstruct the aperture.
Although John Hadley built a 6" (15 cm) /10 Newtonian reflector presented to the Royal Society in 1721, the first reflector used for significant astronomical research was the 6.2" (20 cm) /13 "7 foot" reflector "with a most capital speculum [mirror]" made in 1774. Herschel discovered hundreds of double stars and the planet Uranus with this telescope, and he made over 60 similar instruments for English and Continental astronomers and royalty. In 1783 he constructed an 18.5" [46 cm] /13 "20 foot" reflector, mounted in a large timber altazimuth structure, with which he discovered hundreds of nebulae and star clusters; it was rebuilt and used by his son John Herschel. These instruments initiated an almost continuous period of two centuries in which reflectors have been the leading astronomical telescope for deep space astronomy.
The historical constraints on Newtonian reflectors were reflectivity and mass. Telescope mirrors in the 18th and 19th centuries were made of polished speculum, a heavy and bright alloy of copper and tin that could achieve a freshly polished reflectivity of ~65% and rapidly tarnished to less. A two mirror telescope would have a transmittance of only 40%, so Herschel used his large telescopes as a single mirror, with the observing location off axis. Since polishing a metal mirror effectively refigures it, polishing was a time consuming process requiring optical testing. Most focal ratios were limited to /10 or longer, where a spherical figure becomes optically equivalent to a paraboloid.
The speculum mirror, metal mirror cell and metal tube of a large 18th century reflector were extremely massive, which limited these telescopes to huge and cumbersome altazimuth mounts. The largest 19th century reflector, the Earl of Rosse 72" (182 cm) /9 "Leviathan of Parsonstown" completed in 1845 at Birr Castle, Ireland, weighed 12 tons and was mounted between two buttressed masonry walls 40 feet high and over 6 feet thick. Two mirrors were made, weighing 3 tons each, so that one could be repolished while the other was used in the telescope.
The method to plate cheaper metal with a thin film of decorative silver, after a patent by Varnish & Mellish, was displayed at the London Great Exhibition of 1851 and refined by the German chemist Justus von Liebig. The technique was applied to glass telescope mirrors by Carl von Steinheil in 1856 and independently by Léon Foucault in 1857. Because silver was a far more reflective (~98%) and durable surface than speculum, and resilvering a glass mirror did not alter the optical figure, large aperture two mirror telescope designs (including Cassegrain variants) became practicable for the first time. (Three large Cassegrain designs with speculum mirrors were attempted in 1835 [Grubb], 1845 [Nasmyth] and c.1870 [the "Melbourne" telescope], but none were successful for deep sky astronomy.)
Equally important, the knife edge test that Foucault devised in 1859 allowed mirrors to be figured as true aspheric surfaces and at much shorter focal ratios. Faster focal ratios, lighter glass mirrors and folded (Cassegrain) optical designs permitted more compact and less massive truss optical tube assemblies that could be mounted on accurate equatorial mounts, as pioneered by William Lassell. All these trends made reflecting telescopes far more productive and allowed their use specifically for deep space astrophotography and spectrography.
The first large telescope using a silvered glass mirror was an 80 cm /8 Newtonian telescope built by Foucault (1862). The 36" (91 cm) /5.8 Crossley reflector (1886), which R.N. Wilson called "the first modern reflector", had a Calver mirror commissioned by A.A. Common in 1871 and used by him to take the first photograph of the Orion Nebula. Common sold the telescope to Edward Crossley in 1885, who donated it to Lick Observatory in 1895. There it was rebuilt and used by James Keeler in a photographic survey of nebulae and galaxies that established modern astrophysics.
The Mt. Wilson 60" (152 cm) /5 primary mirror telescope (1908; image, right) one of the most productive telescopes in the history of astronomy and described by R.N. Wilson as "arguably the greatest relative advance in astronomical observing potential ever achieved" was fabricated by George Ritchey from a glass blank supplied by George Ellery Hale, and with experience gleaned from building a 60 cm (24") /3.9 telescope at Yerkes observatory in 1901. Ritchey designed the open tube and fork mount so that by substituting secondary mirrors of different figure and orientation, the focal ratio could be extended up to /30 in Cassegrain or Coudé configurations and light could be directed to five different locations for visual, photographic, spectrographic and solar astronomy. In all configurations, the optics were diffraction limited.
The 60" and the 100" (254 cm) /5.1 primary "Hooker" reflector (also fabricated by Ritchey) installed at Mt. Wilson in 1917 conclusively demonstrated the many advantages of the large aperture glass mirror reflector. A three decade period of consolidation and improvement in reflecting telescopes of smaller aperture and shorter focal ratio followed before completion in 1949 of the 200" (508 cm) /3.3 primary "Hale" telescope at Mt. Palomar, California (graph, above). This incorporated a Ritchey-Chrétien optical design and numerous innovations in fabrication and mount a Surrier truss optical tube assembly, a ribbed back Pyrex mirror, and the iconic "Horseshoe" mount adaptation of the English or Yoke mount used for the Hooker telescope. A second, four decade period of consolidation (excluding the Russian BTA-6 telescope, which was apparently never fully functional) preceded construction of the twin 1000 cm /1.75 primary "Keck" telescopes on Mauna Kea, Hawaii in 1993 and 1996, and the 1040 cm /1.6 primary "Gran Telescopio Canarias" at La Palma (Canary Islands) in 2009. Today the majority of personal telescopes of all sizes, and all large research telescopes (many of them now funded and operated by multinational consortia or partnerships) are reflectors.
Tilted Component Telescopes (TCTs). The practice of viewing the reflected telescope image from an off axis position at the side of the entrance pupil was used by William Herschel with both his 18.5 inch /13 "20 foot" and his rarely used 126 cm /9.5 "40 foot" reflectors, in order to avoid the light loss that would be introduced by a speculum secondary mirror at the cost of "miscollimation" coma in the telescope image. Even in the speculum era, it was never the predominant configuration of a small reflector. Later designs, culminating in the Schiefspiegler of Anton Kutter, were similarly designed to eliminate the secondary mirror obstruction. These designs are rarely built or used today and not considered here.
Newtonian Design Principles
The Newtonian reflector is the most reductive telescope design, in the sense that there is only one surface of positive optical power. (The secondary mirror is flat and therefore in the ideal case contributes nothing to the quality of the final image.) This greatly reduces the optical issues.
Tube Length. The first requirement is that the front edge of the telescope tube should be located at the primary focal point. In other words, the distance of the secondary mirror from the front edge of the telescope tube should be equal to the distance from the secondary mirror to the prime focus imaged through the focuser.
Secondary Offset. The next requirement is the secondary mirror offset. In smaller focal ratios especially, the secondary mirror must be offset slightly from a centered position on the optical axis. The offset must be both by a small distance away from the focuser, and by an equal distance toward the primary mirror.
As illustrated in the diagram (below), an elliptical secondary placed with its center on the optical axis does not evenly reflect the axial light cone from the primary mirror, because the front (bottom) side of the mirror intercepts the axial light cone at a larger diameter than the back (top) side. This is not significant to the image of on axis objects, since the mirror still reflects the entire light cone.
The problem arises in the off axis areas of the image: one side of the abaxial light cone can be reflected to the image plane, but the opposite side is lost. This both decenters and reduces the diameter of the fully illuminated field. In addition, the diagonal position makes visual collimation more difficult, because the diagonal will be correctly aligned when the primary mirror appears to be off center within it.
The solution is to move the mirror by an equal distance slightly downward (away from the focuser) and toward the primary mirror (to compensate for the decentering effect of the downward shift). In this position the mirror can reflect the abaxial light cone equally in all field orientations, and the fully illuminated field is both centered on the optical axis and physically larger.
Given a working distance distance w, tube exterior radius r and secondary mirror minor axis Ds, the necessary offset Os is calculated as:
Os = [(Do Ds) / 4*(orw)]*Ds
Thus a 254 mm, /4 primary with a Ds = 82 mm secondary mirror, mounted in a 305 mm diameter tube to produce a 100 mm working distance should be offset by:
Os = [(254 82) / 4*(1016152.5100)]*82 = [172/(4*(763.5))]*82 = 4.62 mm
A related formula can be used to evaluate the effect of the secondary diameter on the physical diameter of the fully illuminated field. First calculate the two ratios, Dz = Ds/Do and wz = (w+r)/o; then:
FI = [(Dz wz)*Do]/ (1 wz)
which yields, for the system described, a fully illuminated field of
FI = [(0.323 0.248)*254]/(10.248) = 0.075*254/0.752 = 25.3 mm
Aberrations. The major uncorrected aberrations in a Newtonian reflector are coma and field curvature, which become visually negligible at focal ratios above /10 but pronounced at focal ratios below /6.
Usually an added optical component is mounted in front of the eyepiece or camera to minimize these aberrations at low focal ratios. A commercial coma corrector is usually a pair of compound lenses (e.g., a negative doublet in front of a positive doublet) mounted so that they can be inserted into a focuser drawtube with the objective focus located in the space between the two groups; the corrector can accept an eyepiece or camera adapter at the output end. The front negative element introduces negative coma (comatic tail pointing toward the optical axis) that largely but not entirely compensates for the positive coma produced by the primary mirror. These also slightly increase the eye relief, reduce field curvature and astigmatism, and enlarge the image by about 1.1 times.
Collimation tolerances become much more stringent in Newtonian telescopes with focal ratios below /6, and collimation of a Newtonian telescope requires multiple adjustments to both primary and secondary mirrors. The task is made easier by the use of a Cheshire eyepiece. The normal procedure is to first center the circular outline of the secondary mirror within the circular aperture of the focuser drawtube; then to center the image of the primary mirror within the outline of the secondary mirror, and finally to center the image of the secondary mirror within the outline of the primary mirror.
Cassegrain Design Principles
The Cassegrain format now dominates in large observatory and satellite instruments and commercial instruments manufactured for the amateur astronomy market. These instruments have great flexibility and can be specialized for visual astronomy or astrophotography.
In the Cassegrain design, light is focused by a concave (positive) primary mirror and intercepted by a convex (negative) secondary mirror placed close to and in front of the primary focal point (diagram below, top). The secondary magnifies the primary light cone to a longer focus and narrower focal angle sufficient to pass through a hole in the primary mirror. The objective effective focal length is equivalent to extending this narrower light from the secondary focal point until it matches the diameter of the entrance pupil (diagram below, bottom).
Three parameters are fixed arbitrarily at the outset in a Cassegrain design:
(1) the clear aperture D1 of the positive primary mirror, which determines the overall dimensions of the optical tube assembly and the telescope's light grasp and resolution;
(2) the system effective focal length , which determines the objective magnification and relative aperture, and the diameter of the useable image area; and
(3) the amount of back clearance b necessary for the mounting and adjustment of an eyepiece focuser or camera adapter at the visual back a distance that must include the thickness of the mirror glass, mirror cell and back plate (proportionately smaller in larger aperture telescopes).
In most designs these parameters have no limiting effect on the others given a few obvious constraints e.g., the back clearance should not require a visual back opening larger than the secondary mirror diameter, the secondary magnification should not be so low that it produces a large secondary obstruction, etc. Aperture, effective focal length and back clearance are determined solely by the usage requirements and construction details of the system.
The effective focal length is usually chosen to produce a relative aperture no less than Nt = ~12 and sometimes as much as Nt = 20 to 30. (Below Nt = 12 the magnification and contrast potential of the system is wasted; above Nt = 30 the image area becomes unacceptably small.) (Note however that commercial Schmidt Cassegrains are commonly designed at around Nt = 10 and Ritchey Chrétien Cassegrains as low as Nt = 8.)
The principal design decision is how to divide the necessary beam compression between the primary focal length and secondary magnification. To optimize optical quality while minimizing manufacturing costs and producing a sufficiently short optical tube assembly, the primary relative aperture is usually chosen between N1 = 2 to 4. Then the required secondary magnification (M2) is:
Nt = /D1 and N1 = 1 / D1
M2 = /1 = Nt /N1
Nt = N1M2.
The same system relative aperture is produced by increasing the primary focal length while reducing the secondary magnification, or vice versa. The preference is for greater secondary magnification, as this reduces the secondary obstruction from roughly 43% (at M2=2) to roughly 18% (M2=6). (A smaller secondary obstruction significantly improves image contrast for visual astronomy but increases the difficulty of baffling unwanted light from the focal surface; a larger obstruction is acceptable in instruments designed for astrophotography.) The drawbacks are that higher magnification reduces the useful field of view and increases image aberrations and field curvature.
Once M2 is fixed, and given the required value of b (in optimal designs b = D1 approximately), the remaining dimensions (as labeled in the diagram) are:
f = (1+b)/(M2+1)
d = 1f
BFL = d+b
At this point it is useful to confirm both the minimum diameter of the secondary mirror that will reflect all axial light in the primary light cone:
D2 = (D1BFL)/ = (D1f)/1
and the minimum diameter of the secondary mirror that will reflect all light (axial and oblique) in the primary light cone:
D2 = [D1BFL/]+Ad/
where A is the diameter of the opening in the visual back. The secondary diameter should be kept to a minimum consistent with the desired magnification, size of fully illuminated field and effective baffling, usually around 25% of the clear aperture.
The remaining focal lengths, intermirror spacing and radii of curvature of the Cassegrain system are:
1 = d+(BFL/M2)
2 = (d+b)/(M21)
r1 = 21
r2 = 22 .
The field curvature (Σ) is:
1/Σ = 2/r12/r2
greater curvature arising predominantly from a shorter primary focal length. Both curvature and coma become inconsequential for visual use, due to the small marginal image radius, calculated as:
y = arctan(A/o).
For photographic work a field flattener is often employed. This is usually a doublet lens consisting of a field positive semiconvex lens cemented to a negative semiconvex lens. This creates a strongly negative field curvature which counteracts the positive field curvature inherent in the Cassegrain design with a small focal ratio primary mirror. Since a field flattener will also contribute some astigmatism or coma, the field flattener must usually be designed to work with a specific Cassegrain design and the amount of coma or astigmatism the system produces. The recently introduced EdgeHD instruments from Celestron have a field flattener built into the system, as do the Dall Kirkham designs from Planewave and the corrected Mewlon (Dall Kirkham) designs from Takahashi.
Refractor/Reflector Hybrids (Catadioptric Systems)
Catadioptric systems originate in instruments designed c.1930 for wide field (survey) astrophotography. They consist of a Newtonian or Cassegrain format reflector with one or more refracting elements added. The refracting component is either one or more corrector plates located at the aperture stop, or a compound lens located just before the prime focus as a coma reducer or field flattener.
The purpose of the correctors in all cases is to improve the off axis optical quality of the telescope, particularly coma and curvature of field, which is commonly desired in order to shorten the focal ratio and increase the width of field. Thanks to modern methods of robotic optical manufacturing, catadioptric designs are today one of the most popular formats for amateur telescopes.
Catadioptric systems have become a popular design, with refractors and Dobsonians, in amateur (commercial) telescopes. All are modifications of either the Maksutov or Schmidt camera designs, with Maksutovs dominating in the smaller apertures, including spotting scopes. Both have the advantage of a closed optical tube assembly that provides dust protection for the primary mirror and a secondary support without the vanes that produce diffraction spikes in star images. The relatively short OTA makes these instruments relatively light weight and easily portable, and unlike Newtonian designs they do not require ladders or platforms for viewing near the zenith with large apertures. The main drawback is their long cooldown times: the closed tube inhibits cooling convection currents (especially in the Maksutov designs with their thick corrector lens), so that mirror seeing can persist for 2 or more hours, even in small (180 mm or less) apertures.
The optical design details of commercial Schmidt Cassegrain (SCT) designs are proprietary, but several sources (including Rutten & van Venrooij and Smith, Ceragioli & Berry) have published analytic deconstructions.
Field of Full Illumination
Not all the light incident on the objective contributes to the image at the objective focal surface. Physical telescope construction means that some part of the off axis collimated rays passing through the entrance pupil will be blocked from reaching the focal plane of the objective, reducing image illuminance as field angle increases.
The limitation of field illumination as a function of field angle or field height is vignetting, and it divides the image area into two parts: the central field of full illumination in which an object is imaged with all the light that passes through the entrance pupil, and the concentric vignetted periphery out to the field stop where the image illuminance is reduced as field angle increases.
While a dimming of the image at the field circumference of 50% or more typically goes completely unnoticed in visual astronomy, it becomes a hazard in variable star astronomy (where comparison stars in the same field are used to judge the variable star's magnitude) and a blemish in nearly all photographic applications.
The field of full illumination is always calculated as a function of the field angle of the off axis rays and the radius of a limiting field angle or vignetting obstruction internal to the telescope. This can be either the desired widest angular width or linear dimension of full illumination produced at the field stop (which determines the radius dimensions necessary for all secondary mirrors, stops and baffles), or the secondary mirror minor axis (which controls the effective field of full illumination), or the internal diameter of a refractor or Cassegrain stop or baffle (in refractors, usually the last stop before the focal surface).
The metric in vignetting calculations is the maximum field angle of a bundle of off axis rays at which no vignetting occurs. The diagram (above) summarizes the dimensions that are necessary for the calculation in the three basic telescope formats.
(1) The first limiting angle on the field of full illumination is the shadow cast on the objective diameter by the edge of the dew shield or tube opening (in a refractor or catoptric reflector) or the dew shield, corrector plate or lens mounting (in the case of a catadioptric Maksutov Newtonian, Maksutov Cassegrain or Schmidt Cassegrain):
tan(φ1) = ±0.5(CDo) / t.
Most refractors are designed with retractable dew shields that vignette at field angles of 3° or more. For a typical (well designed) Newtonian telescope, where the tube diameter is ~50 mm larger than the mirror (to minimize the effect of tube currents) and t = o (to minimize off axis aberrations), this angle is ±25/o or ±0.7° in an /8 254 mm system. For an /8 254 mm Maksutov Newtonian with a corrector aperture of 300 mm the angle is ±0.65°. For an /10 254 mm Schmidt Cassegrain with an /2 primary mirror that is ~6 mm larger than the corrector aperture stop, the angle is ±3/(2·Dos) or ±0.45°.
(2) The second limitation (in a reflector) is the minor axis (Ds) of the diagonal secondary mirror in relation to the diameter of the light cone converging from the primary mirror. This in turn is a function of the distance of the secondary mirror surface in front of the primary mirror focal point, which in a Newtonian reflector depends on the primary mirror relative aperture, the tube diameter and the desired "working distance" (w) outside the tube, and in a Cassegrain depends on the secondary magnification. The generic formula is:
tan(φ2) = ±0.5[Ds (s/No)] / (os).
For an /8 254 mm Newtonian reflector with a 20% secondary obstruction, 25 mm tube clearance and a 120 mm working distance (s = 272 mm), this angle is ±0.5[51(272/8)] / 1760 or ±0.28°. For a Schmidt Cassegrain with a 30% central obstruction, an /2 primary mirror and s =130 mm, this angle is ±0.5[76(130/2)] / 378 or ±0.83° in a 254 mm system.
(3) The third limitation is the internal diameter of the last smallest obstruction before the focal surface, which may be the last tube baffle (in a refractor), the cylindrical or conical primary mirror baffle (in a Cassegrain), the opening in the visual back, or the internal diameter of the focuser drawtube (symbolized in the diagram above as opening z in relation to distance w):
tan(φ3) = ±0.5[z (w/No)] / (ow).
For an /8 254 mm refractor or Newtonian reflector with a 120 mm working distance (w = 120 mm) and a drawtube internal diameter of 51 mm, this angle is ±0.5[51(120/8)] / 1912 or ±0.54°. For an /10 254 mm Schmidt Cassegrain with a 42 mm central baffle opening at w = 360 mm, this angle is ±0.5[42(360/10)] / 2180 or ±0.08°.
At each obstruction, for any angle greater than φ, the amount of the light reduction increases in approximate proportion to the ratio cosine(φ)/pi. Vignetting effects can be calculated separately and are linearly subtractive. In all cases, the very small foreshortening of the entrance pupil or objective optical surface that occurs in off axis rays is negligible in comparison to the effect of internal obstructions.
The calculations given above, although generic, illustrate the relative constraints on field size in the different telescope designs:
Refractors are vignetted only by the drawtube or last tube baffle which often has a diameter close to the widest eyepiece field stop to minimize off axis aberrations. Newtonian reflectors are constrained by the size of the secondary obstruction, which can be made larger to provide a wider field of full illumination at the cost of slightly reduced image contrast. All Cassegrain designs are significantly limited by the front opening of the primary baffle and in some cases by the opening in the visual back; at large effective focal ratios (typically > 18), depending on the diameter and magnification of the secondary mirror, no baffling solution may be possible that blocks all stray light without vignetting. In no design is the tube opening the most significant vignetting obstruction.
Newtonian Reflector. Because the principal constraint on the field illumination and image contrast in a Newtonian reflecting telescope is the secondary mirror minor semiaxis (Ds), the size and position of the secondary becomes a critical issue in the telescope performance.
The formula for the proportion of full illumination (I) at linear field height h' is:
Ih' = [A/90] - [h'No(2/s2/o)sin(A)/pi]
where o is the effective focal length of the objective, N is the relative aperture of the objective, and S is the distance from the secondary mirror (diagonal) to the focal plane. The angle A is defined as:
Cassegrain. Vignetting and baffling in a Cassegrain design are complicated by the fact that off axis illumination or "stray light" can enter the focal plane directly, producing contrast loss and ghost reflections. Eliminating this requires adjustment of three factors: the size of the secondary mirror, the desired angular diameter of the image area (physical diameter of the field stop), and the secondary magnification (distance from secondary to focal plane). The diagram (below) illustrates some of the complications.
The dotted lines show the path of a column of collimated rays, from entrance pupil to convergence at the focal plane; the lines are continued through the secondary mirror to show the convergence of rays at the primary mirror focal point (1).
The yellow rays outline a bundle of rays fully illuminating the primary mirror that originate in a source located off axis by an angle α = 1°. These are rays not intercepted by the front edge of the telescope tube. They are reflected from the primary mirror and produce an angular displacement θ from the collimated focal point. This displacement can be enough to cause some part of the light cone to strike the secondary baffle or to miss the secondary mirror entirely. In addition, the converging light cone is reflected from the secondary mirror at an angle typically greater than 2θ how much greater depends on the magnification of the secondary and this can deflect the light cone so that a significant part of it is obstructed by the the primary baffle or the central opening in the primary mirror.
Working from the other direction, the orange rays outline a light cone emerging from a point inside the field stop that just avoid the primary baffle and strike the outer edge of the secondary mirror. It turns out that these rays must be reflected from points inside the full aperture of the primary mirror (reducing the effective aperture), and in addition are limited to a field angle of less than 0.4°. Thus the diameters of the primary baffle and secondary mirror limit both the usable angular field and the aperture of the primary mirror.
Finally, the green ray shows that these restrictions on collimated rays are still insufficient to eliminate off axis illumination or "stray light", which is not blocked by either the secondary mirror or the primary baffle and can reach the image area inside the field stop. Stray light illuminates the image area, reducing image contrast and causing a variety of ghost reflections.
To eliminate stray light the secondary obstruction must be made a larger proportion of the primary aperture, the primary baffle must be extended further toward the secondary mirror, or the usable field must be made smaller. The diagram shows that extending the baffle will block the orange rays, further reducing the area of the primary mirror that can illuminate the off axis areas of the image; enlarging the secondary obstruction reduces the contrast in the image.
The standard remedy is to increase the diameter of the secondary mirror and opening in the visual back, which increases the angular diameter of the usable image area at the cost of a slight reduction in image contrast. Rutten & van Venrooij also suggest a conical shape in the primary and secondary baffles, so that the flared edges of the secondary baffle will block light that might be admitted by the primary baffle. However the secondary mirror diameter can be minimzed and create the possibility of stray light, provided that a small angular field is acceptable, planets are viewed only on axis, and stray light from the Moon is accepted as unavoidable.
Modern eyepieces typically consist of the few components, shown in the cutaway diagram (below) for the 28 mm RKE eyepiece used as the example of eyepiece optical design. This illustrates the conventional eyepiece terminology and basic features.
First, an eyepiece consists of a few simple mechanical components, shown above with underlined labels.
The mount is a turned metal or delrin cylinder with a threaded and centered cylindrical interior chamber, the lens cell. Lenses are placed inside the cell, separated from each other as necessary by metal or plastic spacer rings and secured at the open end by a threaded lock ring that requires a spanner wrench to tighten or remove. The mount is finished on the telescope side as a flat shoulder which fits snugly against the eyepiece holder and aligns the eyepiece to the optical axis. The mount exterior is typically embossed with the eyepiece type, focal length (in millimeters) and manufacturer's brand, and may be fitted with a fixed or adjustable eye rest or distinctive but basically decorative plastic housings.
The barrel is a threaded metal tube usually made of nickel plated steel, brass or aluminum, used to secure the eyepiece in the focuser or eyepiece holder. The barrels of all modern astronomical eyepieces manufactured in the United States have either a 1.25 inch (31.5 mm) or 2 inch (50.5 mm) nominal exterior diameter; many older eyepieces some eyepieces manufactured in Japan have a 0.965 inch (24.5 mm) nominal diameter. The interior surfaces of the barrel should be blackened (or made of delrin or black anodized metal) to minimize scattered light. High quality eyepieces are also manufactured with internal threads at the interior field end so that threaded filters can be mounted in the eyepiece.
The undercut is a continuous flat notch in the eyepiece barrel, located to fit over the compression ring of the telescope eyepiece drawtube. It is designed to prevent the eyepiece from falling out of the focuser when the telescope orientation places the eyepiece in a downward direction. Undercuts are a necessity in modern wide angle or super wide angle eyepieces, which in the longest focal lengths can weigh 2 or more pounds; but in the traditional standard field eyepieces that weigh only a few ounces undercuts are an annoying superfluity.
The field stop is a rigid diaphragm or beveled ring fixed on the interior of the eyepiece barrel. In shorter focal length eyepieces it may be defined by the interior edge of the field lens lock ring. It is located at the eyepiece focal plane and provides both a crisp boundary to the field of view and a mask to eliminate optical aberrations or vignetted image areas at wide field angles. It can also be used to occlude a bright star or planet so that a faint companion or satellite is easier to see.
Positive eyepieces place the focal surface and field stop in front of the field lens, outside the lens assembly. Negative eyepieces (such as the Huygenian) place the focal surface somewhere between the field lens and eye lens.
The optical components of the eyepiece include the following:
A lens is a refracting piece of glass. The individual component lenses of an eyepiece are called elements; a compound lens is two or more elements cemented together with balsam or resin, which is then considered a single lens. (The eyepiece in the diagram is a three element, two lens design.)
The lens at the telescope end of the eyepiece is the field lens; the lens at the observer's end is the eye lens. Lenses in the middle of the eyepiece have no designation.
The opening in the field stop that limits the image to be magnified by the eyepiece is the entrance window. The smallest area that contains all the focused light from the eyepiece is the exit pupil (de). The eye receives the widest, brightest and least distorted view of the telescope image when the eye pupil and exit pupil are exactly coincident and concentric. The exit pupil can be viewed and measured as a small circle of light that appears just above the eye lens when the telescope is turned toward the daytime sky.
The focal length (EFL) of the eyepiece is measured from the eyepiece focal plane to the eyepiece principal plane, and is always stated in millimeters.
The visual extent of the image as it appears through the eyepiece is characterized by the apparent field of view and by the image magnification (illustration, below). These determine the true field of view (TFOV) by their ratio: TFOV = AFOV/Mt.
The eyepiece image radius (β) is the angle between the optical axis and the image of the field stop, as measured from the eye point (the location of the exit pupil on the optical axis).
Twice this projection angle (2β) is the apparent field of view (AFOV) produced by the eyepiece. The AFOV, as the image the eyepiece creates of its own field stop, never changes regardless of which telescope, barlow lens, field flattener or coma corrector the eyepiece may be used with.
The eyepiece true field of view (TFOV) is the angular width of the image content, as measured in the naked eye image space (for example, on the celestial sphere).
The illustration suggests that a small AFOV at low magnification can produce a true field similar to a large AFOV at high magnification. But most of the change in TFOV is a function of magnification, not apparent field of view. The AFOV in most commercial eyepiece designs varies from about 50° to 80°, a ratio of 1.6:1; most visual astronomers use a limited selection within that range. Commercial eyepiece focal lengths vary from about 40 mm to 3.5 mm, a ratio of 11.5:1. Most of the practical variation in TFOV is due to the difference in eyepiece focal length.
The eye relief is the available distance between the outer surface of the eye lens and the pupil of the observer's eye. This distance must comfortably accommodate the space requirements of the cornea of the eye, the observer's eyelashes, contact lenses or eyeglasses (necessary to correct astigmatism in the eye). The actual eye relief can also be reduced by a bulky or protruding eyepiece mount.
Some observers will fit eyepieces (especially those with large eye relief) with a latex or plastic eye cup or eye guard that provides support for the observer's eye (or eyeglasses, if those are worn while observing), steadies the alignment and distance of the eye to the eyepiece, and eliminates stray light (image, right). Many high quality eyepieces come equipped with eye guards, and in some brands these are adjustable.
Lenses are usually modified with optical coatings to minimize reflections from the lens surfaces. Eyepieces are multicoated if two or more coating layers have been applied to the lens surfaces; they are fully coated if all lens surfaces, including cemented surfaces, have at least one coating.
Optical testing is a complicated topic, but amounts to
To greatly simplify, the evaluation of optics can be broken into three levels of analysis:
Figure The large scale shape of the mirror as a figure of revolution around the optical axis, and whether this overall shape conforms to the abstract ideal shape required by the optical design. For example, the design may call for a parabolic mirror, but the mirror may have a spherical figure.
Ripple Medium scale deviations from the figure, generally not large. These can be thought of as local area errors in the grinding of the mirror to its figure. Zone is a ripple of constant figure at a uniform radius. Roughness is the periodic or random error in figure caused by the grinding mechanism, especially when excessive mechanical pressure or unsuitable grit compounds are used.
Polish Fine scale deviations from the figure approaching a wavelength of light in diameter that can be assessed as the scratch/dig rating of the surface.
Optical Test Criteria
Rayleigh, etc.Peak to valley wavefront error (PV). Strehl, Maréchal. Root mean square (RMS).
A variety of artificial or "bench test" methods are available:
Foucault Knife Edge
Lyot The easiest and the most sensitive is the star test, which involves a careful study of the image of an out of focus star or the pinpoint reflection of the Sun or a laser from a small round object such as a ball bearing or paperweight observed from a long distance.
According to W.T. Welford (1969), the star test is capable of identifying 1/20 wavefront errors in slowly varying (slope) aberrations and 1/60 wavefront error in minutely varying (zone edge) imperfections, so even excellent optics may display "defects" in a star test.
Well cooled system. Well collimated. Good seeing. Bright star. Near zenith. Centered in field. High magnification (≤ 0.5 exit pupil). No prism diagonal. Orthoscopic or plossl eyepiece. Dark green or yellow filter for refractors.
Method is to defocus the star on opposite sides of focus and look for differences between the images. Ideally the two will appear exactly the same, although this is possible only in refractors: reflectors will include the shadow from the secondary obstruction that is weaker (or absent) in the extrafocal image.
(1) In focus. Examine diffraction artifact. Check collimation; rate seeing.
(2) Defocus to 1 or 2 rings, look for differences in the relative brightness of the rings, any lopsidedness in the rings, or symmetrical changes in the shape of the image, or a greater focus travel distance on one side of focus.
(3) Defocus to 5 or 6 rings, look for specifically at the edge ring (turned down edge), and any difference between in and out focus in terms of the homogeneity of brightness. Intrafocal, outer disk brighter, sharper than inner, extrafocal, inner disk is brighter, sharper ... is undercorrected spherical aberration. Intrafocal, inner disk is brighter than outer, overcorrection. In apochromatic refractor, color should be the same; achromat will show blue in extrafocal.
(4) Defocus to 10 rings, look for specific rings that vary noticeably in brightness, thickness, evenness and sharpness: this indicates zonal defects. Roughness will appear as broken or bumpy interference rings, and both roughness and incomplete polish will scatter light outside the unfocused disk. Mirror currents will produce a plume that rises from the center of the image; tube currents will produce disruption across the top edge of the image; atmospheric currents are more visible in the extrafocal position.
Astronomical Optics, Part 1: Basic Optics - an overview of basic optics.
Astronomical Optics, Part 2: Telescope & Eyepiece Combined - the design parameters of astronomical telescopes and eyepieces, separately and combined as a system.
Astronomical Optics, Part 3: The Astronomical Image - analysis of the image produced by a telescope and the eye that receives it.
Astronomical Optics, Part 4: Optical Aberrations - an in depth review of optical aberrations in astronomical optics.
Astronomical Optics, Part 5: Eyepiece Designs - an illustrated overview of historically important eyepiece designs.
Astronomical Optics, Part 6: Evaluating Eyepieces - methods to test eyepieces, and results from my collection.
Survey of Achromats (General Considerations & Cemented Doublets) by Roger Ceragioli - excellent history and optical analysis of the achromatic objective.
Survey of Apochromats (General Considerations & Doublets) by Roger Ceragioli - excellent history and optical analysis of the apochromatic objective.
Amateur Astronomer's Handbook by J.B. Sidgwick - excellent basic chapters in optics, light gathering, resolution, magnification and more.
Telescopic Limiting Magnitudes by Bradley Schaefer - an attempt to predict telescopic limiting magnitudes using visual data, star color, age of observer, sky brightness and other factors.
Human Eye from Handbook of Optical Systems, Vol. 4: Survey of Optical Instruments by Herbert Gross, Fritz Blechinger & Bertram Achtner (eds.). (Berlin, DR: Wiley-VCH, 2008).
Visual Acuity - Summary of the various methods used to test the human visus.
Average optical performance of the human eye as a function of age in a normal population by A Guirao, C González, M Redondo, E Geraghty, S Norrby and P Artal. Investigative Ophthalmology & Visual Science Jan. 1999, pp. 203-213.
The N.A.A. Telescope Calculator - handy and accurate calculator page for most optical parameters of a telescope/eyepiece combination.
Eyepiece Focal Length Measurement - Jim Easterbrook explains how to measure eyepiece focal lengths in grating projections.
The Collimation - an excellent optical discussion by Thierry Legault
What Is a MTF Curve? - Basics of the MTF.
Startest Images Gallery by Markus Ludes - An instructive gallery of star test images on telescopes various formats and optical quality.
Last revised 11/26/13 ©2013 Bruce MacEvoy