## Astronomical Optics## Part 1: Basic OpticsThis page describes the optical principles necessary to understand the design and function of telescopes and astronomical eyepieces. Subsequent pages discuss the telescope & eyepiece combined, eyepiece 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 to develop the pages and background information available online. ## Physical & Geometric OpticsLight propagates in the form of oscillations in an electromagnetic field, which expand from a point light source as evenly spaced and concentric The radiation of light through space can be represented in two ways: (1) as actual In astronomical applications, light sources are so distant that the concentric wavefronts become a series of equally spaced parallel planes across the width of any practical telescope aperture. To illustrate: across the aperture of a 1 meter (39.4") telescope, light rays from a single point on the Moon, the closest astronomical object at 384,403 kilometers, diverge from parallel by no more than 1/384403000 of a radian or 0.0000026 millimeter, which is 0.0047 or 1/200 a wavelength of "green" light. Since the fabrication limits of the highest quality astronomical optics are around ## Wavelength & FrequencyThe distance between identical points on two adjacent wavefronts of light is the
Thus the frequency of "green" light at 550 nm is Light wavelengths are commonly measured in angstroms (10 ## RefractionRefraction, or the Wavefronts of light have a uniform speed
This is 1.00029 for air; 1.3333 for water; and anywhere from 1.4 to 2.0 for optical glasses. When light crosses the boundary between materials of different refractive index, the average direction of the wavefront will be deflected in a new direction. The geometrical analysis of this deflection is nicely summarized in a classic illustration made by Christiaan Huygens in 1678 (diagram, left). The physical wavefronts are indicated as parallel white bands, and the geometric light rays as parallel thin white lines perpendicular to the wavefronts. As the wavefronts of light Geometric rays are always (by definition) at right angles to the wavefronts they describe, so they create the right triangle This relationship is summarized as sine( or sine( where n are the refractive indices of the two media that form the refracting boundary, and _{2}θ and _{1}θ are the _{2}angle of incidence and angle of refraction. These angles are measured from a line normal (perpendicular) to the boundary surface of the two media at the incidence point of a light ray. Both light rays and the line normal must lie in a single plane, and the incident and refracted rays will be on opposite sides of the line normal.Snell's Law can be derived from the wavefront character of light, as the formation of a single wavefront from the coalescing circular wavefronts expanding from each incidence point along the surface area Transparent materials do not refract all wavelengths of light at the same angle. Instead, short wavelength "violet" light is refracted at a greater angle than long wavelength "red" light. This is the reason that prisms (or raindrops in the air) spread "white" light into the characteristic light spectrum. Variation in the refractive index across different wavelengths is called ## ReflectionReflection is the characteristic behavior of light that is incident on a smooth and opaque (reflective) material. In this situation the Law of Refraction must be qualified in three ways: • For angles of incidence equal to 0°, the ratio of the sines is zero and no refraction occurs: the light is slowed but its direction is not altered. • If the angle of incidence is greater than a
then the ray is n = 1.4 to 2.0)._{2}• When reflection occurs, either from reflecting surfaces such as mirrors or glass surfaces positioned to the incident light at angles greater than the critical angle, the
For all angles of incidence between the critical angle and 0°, significant reflection also occurs when the difference between the refractive indices of the two media is greater than about 0.25. This light energy is emitted by the surface at the reflected angle of incidence, again with the two angles on opposite sides of the line normal. ## InterferenceAlthough the optical behavior of materials can be described in terms of geometric rays, the The first is In this situation light composed of parallel wavefronts from a single light source encounters a thin opaque barrier divided by two parallel, closely spaced and very narrow slits. The slits allow a part of the wavefront to pass, but as it does so the wavefront expands concentrically from each slit aperture. The slits are positioned so that the oscillations of separate wavefronts must be either coincident or opposed. Where they coincide a wavefront The interference effects produced by physical obstructions are the origin of the diffraction artifact produced by a star or "point" light source, and the bright and dark pattern of speckles produced in the image of a star disrupted by atmospheric turbulence. The second way in which interference can arise is through the reflection of the wavefront from thin, parallel layers of transparent media. This was first investigated by Isaac Newton as the interference fringes that appear between two closely spaced sheets of glass; it commonly appears as As just explained, significant light is reflected from an optical boundary between media, such as air and glass, whose refractive indices differ by more than 0.25. This reflection can be minimized by layering the boundary with a third material that has a refractive index between the two materials. This creates two surfaces where reflection can occur, but these reflections can be used to cancel each other. Interference is governed by the spacing between the first and second reflective surfaces. If the spacing is an even number multiple of the light wavelength divided by 4 (e.g., 2/4 or ## ScatteringIf there are not just two edges or surfaces diffracting light, but hundreds or millions, then the cumulative effect is If a refracting or reflecting surface is not perfectly smooth at the atomic level, it creates random "error" in the angles of incidence across an incident wavefront. This scattering produces a random variation in the average angle of refraction or reflection that appears as a cloud of diffuse light around the image of a bright light source, such as the star Formalhaut (image, right). In lenses, scattering is minimized by the absence of scratches, rough spots, pits or small bubbles in the glass, imperfections described by a scratch/dig specification. These imperfections can be somewhat ameliorated by the vacuum application of optical coatings, but can only be minimized by careful glass manufacture, grinding and polishing. In high quality optics scatter is usually due to water condensation (dew), dirt or grease on optical surfaces, or to humidity, dust or exhaust pollution in the atmosphere. In both cases, the behavior of scattering depends on the size of the optical imperfection or atmospheric particles in relation to the dimensions of light wavelengths, so it is also best analyzed by physical optics. ## Ray Tracing a LensFurnished with the geometrical description of light, the Law of Refraction and information about the refraction and dispersion of optical materials, we can analyze the basic attributes of any optical system. Optics can be divided into two levels of geometrical analysis: • • First order analysis is developed from the simplified properties of This Exact analysis is possible by calculating the physical properties of light wavefronts, but the paraxial approximation is invaluable for its simplicity and power to describe the basic attributes of an optical system. ## Gaussian ConceptsIn the Gaussian analysis, the optical system is assumed to provide a perfect (distortion free and precisely focused) image at the optical axis: analysis is only used to define the The analysis builds on the fact that the behavior of an optical system can be diagrammed in relation to three pairs of The diagram (below) illustrates the key concepts and terminology in the first order analysis of a schematic biconvex lens surrounded by air. These concepts can also apply to single or compound lenses of any type, treated as a single optical unit or "black box". A few basic properties of the optical system are assumed to apply. All optical components are constructed as Lens surfaces are assumed to be (and in most commercial eyepieces and refractor objectives are) manufactured as sections of a sphere, defined by a Light rays arise from an The focal point can be located by means of Finally, all refracting optical systems are reversible: they can refract light passing through them from left to right or from right to left. This creates a focal point on each side of the lens. In a thick or compound element (consisting of two or more lenses) there are also two principal points and corresponding principal planes (diagram, above). The first principal plane, first principal point and first focal point are assigned to the surface where light enters the lens; the second principal plane, second principal point and second focal point are assigned to the surface where light exits the lens. This basic layout defines a number of related concepts, specific labels and symbols, also illustrated in the diagram (above): • • An • A • An • For elements notated by the same letter symbol before and after an optical change, all the elements on the image side — the heights or angles of rays after refraction or reflection by an optical surface — are denoted by an apostrophe (e.g., • An object ray intersects the first principal plane at a specific • The perpendicular distance from the optical axis to the most extreme object point is the • The • In most Gaussian equations the front and back focal lengths are assumed to be equal, and multiples of the focal length can be used to specify the location of images behind the lens and/or objects in front of the lens. In first order analysis all angles are measured in radians; for the very small slope angles produced by paraxial rays, the tangent and sine of an angle are equal to the angle itself. In astronomical optics most converging mirrors and a few wide angle eyepieces utilize ## Sign ConventionsIn order for Gaussian equations to produce correct numerical values in all variations, the algebraic sign of measured quantities used in the equations must follow arbitrary but specific • As explained above, light is represented as radiating left to right; the object is always to the left of the lens and the real image is always to the right. • A measurement reference point, usually a principal point of the optical system, is used as the origin of a Cartesian coordinate system. Lengths measured along the optical axis from left to right indicate the direction of incident light and are positive in sign; lengths measured right to left against the original direction of light are negative. Lengths measured upward from the optical axis are positive; lengths measured downward are negative. • Important exception: both focal lengths of a converging lens are positive, and both focal lengths of a diverging lens are negative. The single focal length • The angle of an oblique ray measured at its intersection with the image plane or optical axis, or its originating point in the object space, is positive if the oblique ray must be rotated clockwise to be made parallel with the optical axis, and negative if it must be rotated counterclockwise. (Rotation is always through the smaller angle to the optical axis.) • Angles of incidence and refraction are positive if they must be rotated clockwise to reach the line normal of the refracting surface, and negative otherwise. (Rotation is always through the smaller angle to the line normal.) • A radius of curvature and center of curvature are positive if they lie to the right of the surface described, and negative if they lie to the left. Application of these conventions is illustrated in the optical calculations below. ## Locating the Principal PlanesBefore developing the Gaussian formulas for optical systems, it will be helpful to illustrate how Snell's Law and basic trigonometry are applied to locate the second principal plane, second principal point and effective focal length of a symmetrical biconvex lens (diagram, below). In this example, the lens has a refractive index of n = 1.0. We follow the path of a single collimated light ray (parallel to the optical axis of the lens) that is incident on the front surface of the lens at an aperture height _{A}y from the optical axis.The lines normal to the front surface of the lens are produced as lines radiating from the second The angle of the line normal to the optical axis at the entrance point The ratio between the indices of refraction between the first and second materials (air and glass) is 1.0/1.6 = 0.625, so the sine of the refracted ray is 0.309·0.625 = 0.193, for an angle of refraction θ_{1} to the line normal (this permits us to see the silhouette of our head in an eyepiece lens). The rest of the light ray continues along the refracted path inside the lens.The next step is to calculate the thickness of the lens along the refracted light path. This is a function of the relative curvature of the front and back surfaces, the separation When the light ray reaches the back surface of the lens at point θ = –41° to the line normal outside the lens or –24° to the optical axis. _{4}On this oblique path the image ray continues until it intersects the optical axis at the Diagrammatically, the original light ray can be continued from point A plane through the point ## Image AttributesAt this point it is useful to introduce the terminology for the four attributes of an optical image. (1) An image is (2) An image is (3) An image is (4) Finally, an image can be As the diagram shows, nearly all astronomical telescopes ## Types of LensesThe possible combinations of spherical and plane surfaces that can be used to make a lens fall into six generic types of positive or negative lenses, shown in the diagram (below) with two indications of their relative refracting power: the effective focal length and a schematic tracing of two collimated rays. These illustrations show lenses of a typical optical glass (
The biconvex, positive meniscus and plano convex designs are The two principal points are shown by the red dots: even when symmetrical, "thick" lenses have two principal points for the refractive effect in opposite directions. In general, the principal points in a biconvex lens are both internal and spaced about 1/3 the distance from the front to back vertices. In a lens with a plane surface, one principal point is the plane vertex of the lens. In a meniscus lens, one or both principal points can be external to the lens. The biconvex and biconcave lenses have roughly twice the refractive power of the plane and meniscus forms. The meniscus lenses are most sensitive to changes in the lens thickness, while the plano convex and plano concave lenses are unaffected by variations in thickness. In asymmetrical positive lenses, the power of the lens is greater when the surface with the shorter radius (or the curved surface in plano lenses) is oriented toward the object. In asymmetrical negative lenses, the reverse holds: the power is greater when the greater power or curved surface is oriented away from the object. Used as single lenses, positive lenses always produce real, inverted, reverted images, while negative lenses produce Positive meniscus lenses are commonly used in eyeglasses to correct for farsightedness or difficulty focusing on objects at short distances from the eye ( ## Image Size & Location (Positive Lens)Now that the underlying logic of first order analysis has been illustrated and the concepts defined, we can turn to the basic formulas that result — first for positive or converging lenses, then for diverging or negative lenses. In the Gaussian model, the optical effect of a lens can be analyzed through the use of three analysis rays. The diagram below shows this analysis applied with two principal planes, which is done by disregarding the space between them. If it is acceptable to assume that the optical effect of the lens thickness (the distance between the front and back incidence points of a light ray) is inconsequential to the slope of the exiting image ray, then the lens can be modeled by a single principal plane located at the center of the lens, in what is called a 1/ where 1/ which becomes 1/ 1/ The "thin lens" analysis originated in 18th century optics in which the main applications were low power, long focal length spyglass lenses, simple eyepieces and very thin eyeglass lenses. It is applicable to any lens where the focal length is much larger than its maximum thickness. That criterion is ambiguous and only suggests how well a physical lens might be described by the idealized thin lens model: in particular, it implies that wide angle or short focal length (strongly curved) lenses cannot be analyzed in this way. Given that we have already located the principal points and focal points from the formula above, we are interested to find the location, size and orientation of the image formed by a specific object at a distance From a
For an object at an infinite optical distance — the distance at which large changes in the object distance do not affect the location of the focal point — the use of ray For an object at finite optical distance, all three rays will intersect at a single point that is not on the optical axis. Given the focal distances
• the image height Note the explicit symmetry in the procedure: an object height
Reasoning from the various congruent triangles created by the three rays, and given the object distance 1/ and the other parameters: [ [ [object distance] [focal distance] [object size] [image size] and several equations for the image magnification (
The
Refractive power can be expressed in a standardized measure, the diopter, which is power measured in meters (1/ As an illustration and check on your practice calculations, the relationships among these different measures are illustrated in the table below for a constant object size of 2.5 cm and a symmetrical biconvex lens with an effective focal length of 25 cm.
Again, note carefully the sign conventions necessary for consistent calculations: —for a converging lens both focal lengths are positive, for a diverging lens both focal lengths are negative; If only one principal plane is used in the ray tracing (the "thin lens" model), then all distances are measured from it. This analysis method indicates the changes in the image location, size and orientation produced by objects at different distances from the lens (diagram, above; note that Objects that are far distant enough to make the principal rays (a) When the object is between –50,000 (b) When the object is located exactly –2 (c) If the object is located between –1 (d) If the object is located exactly at –1 If the object is located at a distance less than –1 Note that it is not possible for an object at any distance to form an image between the lens and the second focal point (at a distance less than the effective focal length). ## Image Size & Location (Negative Lens)The diagram below shows the "thin lens" analysis for a negative (diverging) lens, again with two principal planes; the analysis is the same if the two planes are replaced by a single plane. The major difference from the positive lens analysis is that the effective focal length is measured From a
Given the first and second focal distances – x (always negative), the –ƒ' to image distancex' (always positive), and the image height h' (always erect) as shown in the diagram (above).A real image cannot be produced by the diverging refracted light rays (yellow lines): instead a The focal points both have negative distances from their respective principal planes, which means (according to the sign conventions and the thin lens formulas) that the image focal distance This analysis shows that a negative lens always produces an erect but virtual image when viewed from the side opposite the object. For an object at infinity, analysis ray ## Ray Tracing a Spherical MirrorIn a mirror there is only one optical surface, so analysis is simplified. The major variations are for objects at finite or optically infinite distances, for concave or convex mirrors, and for real and virtual images. The diagram (below) shows the analysis for a concave and a convex mirror with the object at a finite distance, and for a concave mirror with the object at infinite optical distance (the astronomical case). The sign conventions for a concave (converging) mirror are that the single focal point and image focal distance are positive, because they are measured in the direction that the light is traveling; the object distance is measured opposite the direction of the light and is therefore negative. For a convex mirror the focal point and image focal distance are negative, because they are opposite the direction of the reflected light. As before, object and image heights and image angles are negative when below the optical axis. The image size and location are defined as with lenses, by means of three analysis rays from a single off axis object point:
The first diagram (above) shows the ray tracing for a concave (positive, converging) mirror with an object point of height In contrast, an object placed The middle diagram (above) shows the ray tracing for a convex (negative, diverging) mirror, again with an object point at height The relationships between object distance, image distance and focal length are: 1/ 1/ where 1/
which means ray
with magnification
and for objects at infinite optical distance the magnification is infinitely small, all objects are reduced to points ( The last diagram (above) shows the ray tracing for a concave mirror with an object at infinite distance — the usual astronomical situation. In this case all rays from the object are parallel, so rays The image forms at the focal point
The average diameter of the Moon is approximately 30 arcminutes or 0.0087 radians; its disk will be 21.8 mm wide at the focal plane of a 250 mm The diagram also shows that as the aperture height
For a 250mm (10 inch) mirror figured to Single mirrors, like lenses, produce a curved focal surface whose radius ## Lens CombinationsAstronomical eyepieces are designed as two or more lenses or components mounted along a common optical axis. Their design and evaluation requires optical formulas that extend and generalize the analytical framework described above for a single lens. This section first outlines the traditional thin lens formulas, then more general analytical formulas. ## Thin Lens FormulasA number of optical formulas were derived in the 18th century to characterize the behavior of "thin lens" combinations calculated in relation to a single principal plane for each optical element. We start with this framework for its simplicity and to illustrate basic relationships between two lenses and their spacing as an optical system. The diagram (below) shows a simple two element eyepiece with an effective focal length of 30mm, analyzed by means of a single principal plane through the center of two positive lenses with focal lengths of 57mm (for the front or The design procedure is first to select two lenses of the appropriate aperture and power ( Once the interlens spacing is fixed, the front focal length ( The lens combination can be traced with a pair of collimated rays at the edges of the field stop, called Note the following: • In the thin lens (Gaussian) model, the focal length of a single spherical lens can be calculated from its index of refraction ( 1/ and the effective focal length of a compound lens (where
Recall that the focal length of a negative (diverging) element is negative. • The front focal length • The back focal length • The two effective focal lengths • The apparent field of view (the angular diameter of the field stop viewed from the eye point) is derived from the radius of the field stop (
• To maximize eye relief, the lens with the higher power is usually used as the eye lens. (To verify this, note that • A point discovered by Huygens, and implemented in the eyepiece design that bears his name, is that two lenses of the same refractive index but different powers produce the most achromatic image when
The Kellner modification of the Huygenian design handles chromatic aberration by using an achromatic doublet for the eye lens. The graph (below) shows the optical effect on this specific eyepiece when the focal length of either the front or back lens, or the distance between them, is reduced up to 25mm, while holding the other elements constant. (A field stop diameter of 19.5 mm is used to calculate AFOV.) As the focal length of either the field lens or eye lens is made shorter, the effect on Reducing the focal length of the field lens has a relatively small effect on the system The direction of these changes is the same in both lenses, and differs from the effect of the lens spacing only in the change in eye relief, so the eyepiece focal length decreases and the apparent field of view increases, and by a much greater proportion than the eye relief is reduced, when both lenses are made to a shorter focal length and are placed closer together. This also implies more noticeable aberrations, and a greater emphasis on the dispersion attributes of glasses used for the lenses. These examples reveal the underlying design principles of traditional astronomical eyepieces: • The function of the field lens is primarily to "stage" or prepare the image by partially correcting the divergence in the light rays (caused by the relative aperture or • The function of the eye lens is to bring these corrected rays into a much shorter focus at a much steeper angle to the optical axis, which produces a wider The significance of the exit pupil is that it is the point where the angular width of the image is at maximum and the projected light has the smallest diameter (obvious in the diagram). This compressed, wide angle beam can most easily pass the small entrance pupil aperture of the observer's eye and fill the wide area of the observer's retina. Both the field lens power and the interlens spacing can be manipulated to compensate for the very short eye relief that a high power eye lens will produce, to place the eye point and exit pupil where it can be comfortably examined by the observer. The final issue concerns apparent field of view (blue line), which in traditional eyepiece designs rarely exceeds ~45°. A high power eye lens or a very wide field design (when ## Thick Lens Optical AnalysisThe previous sections have explained how lenses can be analyzed using a paraxial approximation: the front and back surfaces of the lens within an extremely small radius of the optical axis are approximated as a single refractive principal plane, and this plane is extended over the entire diameter of the lens to calculate focal distances (power), image location and orientation, and magnification. This approximation assumes that surface curvature of the lens is relatively small, the power of the lens is weak, and therefore the thickness of the lens, like the thickness in a sheet of window glass, does not have a significant effect on image formation. For strongly curved (more powerful) lenses the paraxial approximation breaks down, and the analysis must be constructed on either two principal planes within each lens or optical element, or a strict trigonometric analysis of optical surfaces and distances. The earlier example omitted consideration of the lens thickness in order to illustrate how Snell's Law and simple trigonometry is applied to locate a principal plane. This section introduces calculations that include lens thickness ( Again, the sign conventions are positive for measurements in the direction of light (left to right) and for objects above the optical axis; negative against the light and below the optical axis. Focal lengths are positive for converging lenses and negative for diverging lenses. Angles are always expressed in radians, which is approximately the tangent for angles less than 20°. First, let's revisit the procedure outlined above for locating the focal point and second principal plane of a single converging lens. This can be generalized by analyzing an axial rather than collimated incident ray (diagram, below). The refracting power of lens in air results from the radius of curvature of the two faces of the lens, the thickness of the lens, and the index of refraction of the glass. Therefore measurement of the two surface curvatures ( For a single lens (where The step by step calculations proceed as follows:
t – vertex to vertex thickness of lens (positive, by sign convention)n_{0}, n_{2} – index of refraction for air ( = 1.0)n_{1} – index of refraction for lens materials – distance of axial point (negative, by sign convention)y – aperture height of incident axial ray (positive when above the optical axis, by sign convention)
[1. entry incidence slope] [2. entry refraction slope] [3. exit (image ray) aperture height] [4. exit (image ray) slope, given [5. back focal length] [6. effective focal length] [7. power of the lens] Once all calculations are completed, first the focal point is located by measuring the back focal length from the back vertex as before. Then the principal plane of the compound lens is located by measuring the effective focal length forward from the focal point. The power of the compound lens is determined as before from the effective focal length. This procedure can be repeated for a cemented compound lens consisting of two or three elements. Simply make the aperture height and incidence slope of the entry ray into the second (or subsequent) lenses equal to the exit ray aperture height and slope from the previous lens, and calculate steps 2-4 with the front and back curvatures of the second lens. (If the two lenses are separated by air, then a different calculation is necessary, as described below.) For example, in step 2 To identify the opposite focal point, the two radii of curvature are reversed in the formulas and the sign conventions will reverse the signs of the radii of curvature. ## Multiple Lens Optical AnalysisNext we generalize the "thin lens" formulas for two air spaced lenses or multiple element components treated as single lenses. This procedure departs from the trigonometric analysis given above in that the thickness of the optical components [single or compound lenses] is replaced by a pair of principal planes, located by a prior analysis of each of the components. These are specifically • Front and back focal lengths are equal: • All rays entering the first principal plane at aperture height • An axial object ray angle • An object point at The ray tracing requires only two analysis rays in a common meridional plane: (1) an axial (marginal) ray from the object at distance
and the corresponding image ray relationships for the system will be:
–s (note that u' and u'_{p} as diagrammed are negative by the sign conventions)As the diagram makes clear, the image angles have been transformed by the optical system such that the simple equalities
First, we can get the generalized form of the "thin lens" equations of the previous section, which will then apply to both thin or thick components. Given the unit magnification of the principal planes for a single component yields the basic relationships:
Substituting these into the thin lens formula (above) by replacing object and focal distances by ray angles (e.g., 1/
the transfer equations into the second component:
the exit ray from the second component:
and the effective focal length of the total system (components 1 and 2 combined at distance
Alternately, assume that
which is identical to the reciprocal of the focal length formula given above in the "thin lens" case:
If the effective focal distance, back focal distance and component spacing are known, the separate effective focal distances of the components can be found as:
These equations permit fast layout and approximate analysis of optical systems without the incremental calculation of ray traces through all refracting surfaces. Finally, all focal optical systems conform to four systematic proportionalities that equivalently define a quantity known as the
This quantity states the relationships between refractive power, object distance and image focal distance purely in terms of angles and relative heights. It can be reduced to an alternative statement of system magnification:
The invariant applies equally to single or compound lenses or multi element optical systems, which are treated as a "black box" and analytically bracketed by the entrance pupil and back principal plane. In analyses related to optical aberrations, the first principal plane is usually coincident with the ## Eyepiece Prescription DataIn most eyepieces, the thin lens formulas are insufficient to design an eyepiece or guide its manufacture. The total information required to specify an eyepiece optical design is called the The prescription data are given in tabular form (diagram, lower left): measurements begin at the objective focal plane and end at the eyepiece exit pupil. Note that the distance measurements are strictly sequential — Given the prescription data (available in eyepiece patent documents or optical references), an assigned system focal ratio and field diameter (equivalent to the interior diameter of the eyepiece field stop), ray tracing software can calculate the path of light from the objective focal plane through the eyepiece for any field height on the object focal plane. These rays are often shown in different colors for field heights equal to 0%, 70% and 100% of the field radius; the colors do not denote spectral frequencies but help to interpret the diagram visually. Conventionally, three rays are calculated as radiating from each field point (originating from opposite sides and the center of the objective aperture) so that their convergence exactly defines the exit pupil. Ray tracing allows calculation of the eyepiece principal planes, principal points, effective focal length, front and back focal length, eye relief, the angle of the apparent field of view, and the Petzval radius, which describes the field curvature in the focal plane of the eyepiece. (Note that a negative Petzval radius indicates a positive field curvature — that is, the center of the focal surface is closer to the field lens than the perimeter of the focal surface; a large Petzval radius describes a relatively small field curvature.) A single spectral frequency is necessary to compute these standard optical parameters; this is usually ## Optical MaterialsGlass is the generic material used to refract light in astronomical instruments. Glasses are amorphous (not crystalline) mixtures of fused silica (silicon oxide, SiO Glasses, plastics and other transparent materials bend all wavelengths of light through Optical materials are therefore identified according to those two attributes: (1) the Across the entire electromagnetic spectrum, most glasses and plastics have periodic Because each wavelength of light is refracted at a slightly different angle by the same lens, the index of refraction must be standardized on a specific wavelength of light, defined as one of the emission lines of a chemical element. By convention this is either the
and an important index of partial dispersion (
where the comparison wavelengths used (image, left) are the emission lines of hydrogen at The diagram (above) illustrates the relative effect of the refractive index and But the two metrics are not exactly proportional. When glasses are charted on Glasses on the left (dispersion of 55 or more at refractive index below 1.60 and 50 or more at refractive index above 1.60) are called The dots indicate specific glasses available from Schott, one of many major glass suppliers. They illustrate that the distribution of glasses is not random, but is strongly clustered around the righthand diagonal. Most of the variations occur within middle values of the indices of refraction and dispersion, straddling the boundary between crowns and flints. This permits the manufacture of compound lenses made of two or more glasses that produce the same refraction but different dispersions, or the same dispersion but different refractions, which is necessary to eliminate chromatic aberration in the focused image. Traditionally, crowns are "hard" glasses with higher melting temperatures, made with small quantities of potassium oxide, combined with oxides of other metals such as phosphorus, zinc, lanthanum or barium. Crowns made with boric oxide (borosilicate glasses, including Pyrex™ glass) or fused quartz have been especially important in the manufacture low expansion telescope mirrors. Glasses marketed as Flints are "soft" glasses with lower melting temperatures, historically made with varying quantities of lead oxide. One of the highest index flints ( (Historical note: the term In the late 20th century, a few hard plastics (acrylics, polycarbonates, urethane monomers) have been successfully used as optical materials in many applications, and lenses incorporating diffractive (ribbed or grooved) surfaces have been used as well. Plastics can be injection molded but are also relatively soft and easily scratched, and can deform under moderately high temperatures, which makes optical coatings impractical. However certain plastics have replaced Canadian balsam as cements commonly used to bond together the separate elements of a compound lens. ## Optical CoatingsAs explained above, a small fraction of the light directed through an optical system is not refracted by an optical surface but is reflected from it, and additional light is absorbed by the material and scattered by internal reflections at the second (exit) surface. In general, the transmission of a light ray through both surfaces of a single uncoated optical element of refractive index
For an angle of incidence of between 0° to ~30°, the two surfaces of a single optical glass element in air transmit about 92% ( Optical coatings are very thin layers of material vacuum deposited on an optical element to minimize or control the reflection or transmission properties of its surfaces. A variety of materials used for this purpose include fluorides and oxides of various metals, but the most commonly used material is Coatings manage unwanted reflections by virtue of their Light reflected by the internal surface of a 1/4 wavelength layer will be exactly 1/2 wavelength out of phase with light reflected by the external layer, resulting in destructive (wave canceling) interference of all reflected light. This effect is maximized when the refractive index of the layer (in air) is equal to the square root of the refractive index of the glass. Magnesium fluoride ( Due to its fixed thickness and refractive index, the wave canceling effect of a single layer optical coating becomes less effective at wavelengths longer or shorter than the In The commercial designations Aluminum mirrors are also coated with very thin layers of magnesium fluoride or silicon monoxide (SiO) to provide protection against oxidation and cleaning. Certain multilayer coatings can also increase the reflectivity of an aluminum coating from 88% up to 99% at certain wavelengths, but not more than ~95% as an average across the entire visible spectrum. ## Further ReadingAstronomical 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. Modern Optical Engineering by Warren Smith - a classic, authoritative and clearly written survey of optical principles and applications. Basic Optics and Optical Instruments by Fred A. Carson - a simplified but useful exposition of geometrical analysis. Ray Tracing of Thin Lenses by Darryl Meister - Clear and well organized explanation of thin lens ray tracing. Nature and Properties of Light by Linda Vandergriff - overview of modern optical techniques for the detection and manipulation of light. Basic Geometrical Optics by Leno Pedrotti - the basics of light reflection and refraction and the use of simple optical elements, such as mirrors, prisms, lenses, and fibers. Basic Physical Optics by Leno Pedrotti - the phenomena of light wave interference, diffraction, and polarization. Last revised 11/26/13 • ©2014 Bruce MacEvoy |