Exact optics II. Exploration of designs on- and off-axis

Transcription

1 Mon. Not. R. Astron. Soc. 342, (2003) Exact optics II. Exploration of designs on- and off-axis R. V. Willstrop and D. Lynden-Bell Institute of Astronomy, The Observatories, Madingley Road, Cambridge CB3 0HA Accepted 2003 January 15. Received 2003 January 10; in original form 2002 November 7 ABSTRACT The two-mirror telescopes/cameras that have no coma, no spherical aberration and arbitrarily fast focal ratios form a two-parameter (s, K) family of exact optical designs analytically derived earlier. We explore the full range of these designs in the s K plane. Here s is the mirror separation/focal length and K is the distance from secondary to focus/focal length. Besides perfect-focus analogues of most well-known telescope designs, there are families of spectrograph cameras, X-ray telescopes and solar furnaces. Systems with s = 2 have minimum astigmatism and have exceptionally good images. Key words: instrumentation: spectrographs telescopes. 1 INTRODUCTION The primary aberrations of optical systems are spherical aberration and coma. These are followed by astigmatism, field curvature and distortion. In Paper I of this series, Lynden-Bell (2002) solved analytically the equations that govern the shapes of pairs of mirrors giving systems with no spherical aberration and no coma. These designs give the mirror shapes in parametric forms which depend on two constants: s = mirror separation/focal length and K = distance from secondary to focus/focal length. A third trivial parameter is the scale of the whole design. That analysis, which is true for arbitrarily fast focal ratios and includes all possible solutions, involved only the perfect on-axis focus and invoked Abbe s sine condition to ensure no coma off-axis. Astigmatism and other aberrations that limit the field of good definition were not considered, nor was the full generality of the two-dimensional set of these exact systems explored as s and K were varied. The present paper completes that exploration by drawing the mirrors for wide ranges of s and K. It also determines the fields of view by tracing rays through the systems and measuring the off-axis images. The perfect-focus no-coma designs have primary mirrors given by parametric formulae giving R p (T ) and X p (T ), where R p is the radius from the axis to the mirror at position X p along the axis see Fig. 1. Likewise the secondary mirror is given parametrically by formulae giving R(T) and X(T). The function t(t) is defined by t(t ) = [T/(1 + T 2 )](1 K 1 η 1 T 2 η )/s, (1) where η = s/(1 s), (2) R p = 2bT/(1 + T 2 ), (3) X p = bs b[1 (st/t )(1 η 1 T 2 )]/(1 + T 2 ), (4) R = ρ2t/(1 + T 2 ), (5) X = ρ(1 T 2 )/(1 + T 2 ), (6) where ρ = bk 1 η 1 T 2 η /(1 tt). (7) The physical meaning of b is the effective focal length; this sets the scale of the whole system. T is tan (φ/2), where φ is the angle at which a ray initially parallel to the axis hits it at the final focus. Likewise such a ray makes an angle 2θ to the axis when it is between the primary and the secondary, and t(t) is tan θ expressed as a function of T see Fig. 1. The first few terms in the power series for the mirrors have been found (by DL-B) to be, with b = 1, for the primary mirror: X p = s K + [(K 1)/s](R p /2) 2 + (1/2!)(K/s)(R p /2) (1/3!)(K/s)(4 + 1/s)(R p /2) (1/4!)(K/s)[(10 + 1/s)(4 + 1/s) 12](R p /2) 8, (8) and for the corrector mirror: 2X c = 1 Q 1 R 2 c Q 2 R 4 c Q 3 R 6 c Q 4 R 8 c, (9) where X c = X/(2K ), R c = R/(2K ), and Q 1, Q 2, Q 3 and Q 4 are given by using intermediate quantities Q 0, Q 5 and Q 6 as below: Q 0 = (K + s 3/2)/s, (10) rvw@ast.cam.ac.uk (RVW); dlb@ast.cam.ac.uk (DL-B) Q 5 = [12(K + s) 13 1/s]/(3! s), (11) C 2003 RAS

2 34 R. V. Willstrop and D. Lynden-Bell Figure 1. (a) The simplest layout of a two-mirror system giving the notation. (b) Detail of the corrector mirror with the angles involved. Q 6 = [120(K + s) 114 5/s 2/s 2 ]/(4! s), (12) Q 1 = (K + s 1)/s, (13) Q 2 = Q 2 1 Q 0 = [K 2 + K (s 2) s/2 + 1]/s 2, (14) Q 3 = ( Q Q 1 Q 2 + 4Q 0 3Q 1 Q 0 Q 5 ), (15) Q 4 = Q Q2 1 Q 2 + Q 1 Q 3 Q 5 (5Q 1 8) Q 6 Q 0 [ (3Q1 4) 2 3Q Q 2 + 4Q 1 ]. (16) These expressions have been checked by comparison with the Zeiss convex sphere cardioid combination in which the cardioid has Q 1 = 3, Q 2 = 7, Q 3 = 33 and Q 4 = 195. Also they have been compared with power series determined by an iterative fitting to the exact mirror profiles of several of the systems described here, and the terms up to R 6 are found to agree to about one part in 10 5 or better. The coefficients of the R 8 terms disagreed by rather more, but this is to be expected since the iterative fitting takes account of higher-order terms while the mathematical power series expansion ignores higher-order terms completely. These terms may be compared with those in the general expansion of a conic. If we denote the pericentric distance by p, and put E = (1 e)/(1 + e) and Y = RE/p, then for a conic [ X = p 1 1 (1 e) ( 1 2 Y Y Y )] 128 Y 8 +. For n 2 the general term in the round brackets is (2n 3)Y 2n /(n!2 n ). (17) We use a ray-tracing subroutine that determines the position of the image with least rms spread so field curvature does not contribute. The image spreads found off-axis are generally proportional to the square of the angle off-axis, strongly suggesting that astigmatism dominates other defects. The principal exceptions to this are the case of s = 2, where Seidel astigmatism is zero (in the smallaperture, small-angle approximation) and some grazing-incidence systems. For normal-incidence systems it is found that the astigmatic circle of minimum confusion has an extreme diameter in arcseconds of Diam = (α 2 /F) s 2 /K, (18) where α is the off-axis angle in degrees and F is the f number (focal length/aperture). Petzval (1843) showed that when the tangential and sagittal image surfaces coincide with each other, i.e. in the absence of astigmatism, they also coincide with the Petzval surface, and the curvature of this surface is equal to twice the sum of the curvatures of the mirrors. If the tangential and sagittal image surfaces do not coincide, they are always on the same side of the Petzval surface, and the tangential image is three times as far as the sagittal image from the Petzval surface. The best images (circles of minimum confusion) lie midway between the tangential and sagittal image surfaces (Conrady 1929; Jenkins & White 1951). Using the R 2 coefficients in the power-series expansions of the mirrors, it is found that the Petzval curvature of the image surface is given by Ptz = (K 2 2K s + 1)/(sK). (19) Thus s = 2 systems with K = 1 ± 2 have Ptz = 0 and so a flat Petzval surface. This paper is organized as follows. Section 2 presents and explains a Road Map that shows how many known telescope designs are approximations to our perfect-focus systems. One new design for a spectrograph camera, s = 2, K = 2, illustrates the use of the Road Map. We draw rays both parallel to the axis, and then also, in a subsequent diagram, inclined rays. These diagrams show which parts of the primary and secondary mirrors are useful and which parts must be cut away to let the light through. The Road Map itself consists of four pages of optical designs showing how these change with s and K. Section 3 gives details of designs that are perfect-focus analogues of known designs, and also includes some new systems. Section 4 describes how the ray-tracing is computed when the mirrors are defined by parametric equations. The program deals with both on-axis and off-axis bundles of rays, but does not allow any misalignment of the mirrors to be included. Because one of the authors (RVW) had an up-and-running general-purpose ray-tracing program in which the radial profiles of mirrors could be defined by power series in R 2, and allowed misalignments of the mirrors and additional optical components (e.g. a cryostat window for a charge-coupled device detector) to be included, we have also used this. Power-law fits to the mirror shapes were used in the generalpurpose program to obtain the off-axis image spreads. Section 5 gives the conclusions.

3 Exploration of designs on- and off-axis 35 Figure 2. Cross-section of the mirrors of the s = 2, K = 2 system. Primary mirror continuous line; secondary mirror dashed line. The first sheet of the primary is the convex mirror between the cusps. Light from this enters the final focus, shown by the small circle on the primary, with φ< π/2 see Fig. 3. For light falling on the second sheet of the primary, see Section DIRECTORY OF THE PERFECT-FOCUS SYSTEMS 2.1 Explanation of the diagrams and notation Because of the many designs encompassed by the formulae of the last section as s and K are varied, we have drawn out a representative sample of them, which we present as a Road Map on four pages comprising Figs These are preceded by a key to the Road Map, showing which systems are found on which page of the map. So that readers can learn how to use both key and map, we first take them through one example: a spectrograph camera that occupies the position s = 2, K = 2 in the map. According to the key, this system is found upper-middle on the right of the second page of the map (i.e. Fig. 8). However, while readers are encouraged to find the relevant figure, it will as yet seem rather confusing. To elucidate it, let us start with a magnified version that shows only the mirrors as determined from our parametric formulae. 2.2 An example, s = 2, K = 2 Fig. 2 shows the cross-sections of the two mirrors as they intersect a plane that includes the optical axis. The primary mirror is shown as a continuous line, and the secondary mirror as a dashed line. The focal length is equal to one unit on either the X or the Y scales marked at the edge of the box. The focus is marked by a small circle, and in this example it lies on the surface of the primary mirror. Light approaches from the left see Fig. 2. The secondary mirror in this example (shown by the dashed line) is continuous, but the primary mirror has two sheets, separated by the cusps. To allow light to reach the part of the first sheet of the primary mirror near the focus, the long tapering second sheet must be removed, and the secondary mirror must have an axial perforation. The light reflected from the first sheet of the primary mirror falls on the part of the secondary mirror to the left of the focus, i.e. in negative X space see Fig. 3 and the Road Map. The second sheet of the primary gives rays that approach the final focus in the opposite direction to those from the first sheet of the primary, so the two sheets are not used in the same optical system. In this example it is physically impossible to use all of the second Figure 3. Ray paths initially parallel to the optical axis in the system shown in Fig. 2. Symbols used to identify points on the mirrors corresponding to φ/2 = 1.0, 2.0, 3.0: filled square; 0.5, 1.5, 2.5: five-pointed star; 0.05, 0.1, 0.15, etc.: circle with central dot; other points on the primary mirror: large dot; secondary: small dot. sheet of the primary mirror, but the mathematics is interesting. Light reflected from the outside of the tapering second sheet of the primary falls on the part of the secondary mirror in positive X space, and is reflected to the focus from the right. See the figures above and below the s = 2, K = 2 figure on the second page of the Road Map which show such rays. Rays of light that are rather close to the axis are reflected at extreme grazing incidence, and may impact the second sheet of the primary mirror a second time. If they are allowed to pass through it, and through the first sheet of the primary mirror also, they can reach the part of the secondary mirror in positive X space, and be reflected to the focus. All of these reflections are real, and result in a real image, not a virtual image. The path-lengths to the focus from a plane surface normal to the X axis are found to be equal to within less than of the focal length, from whatever direction they arrive at the focus. This error is compatible with the rounding errors in the calculations of the points at which the rays strike the mirrors. Using Fermat s principle, it is clear that the on-axis images are perfect. In Fig. 3 points on the mirrors are identified, which correspond to rays that reach the focus making angles with the X axis φ/2 = 0.01, 0.02,..., 3.14 radians. In this case the primary mirror is identified by larger dots, and the secondary mirror by smaller dots, for most of the points. For both mirrors every fifth point is emphasized by using a larger circle with a central dot, every fiftieth point by a five-pointed star, and readers with a magnifying glass will see that every hundredth point is a filled square. It is easier to follow these around the secondary mirror than around the primary; starting with φ/2 = 0at(X, Y ) = ( 2, 0) the increasing values of φ/2 go clockwise around the mirror. On the primary mirror, one starts at (X, Y ) = (0, 0) and goes to the cusp at Y =+1, which corresponds to φ/2 = π/4. Values of φ/2 between 1.37 and 1.77 are outside the diagram and are not plotted. Values of φ/2 greater than 1.57 have negative Y. The notations used in this figure are the same as those used in the Road Map and it is essentially an enlarged version of the s = 2, K = 2 entry on the second page of that map. Other figures of the Road Map should now be understood by analogy. 2.3 On-axis rays in that system Rays have been drawn in Fig. 3 for an approximately f /1 camera, showing the part of the secondary mirror that must be removed, and

4 36 R. V. Willstrop and D. Lynden-Bell the parts of the primary that do not contribute light to the final image. Because these rays are initially parallel to the optical axis, each is fully defined by its value of φ/2. The (X, Y) coordinates of the points on the mirrors corresponding to these values of φ/2 are stored while the mirror cross-sections are being computed, and when these are complete the corresponding points are joined by straight lines. The validity of this method is demonstrated by the constant path-lengths from a plane surface perpendicular to the optical axis to the focus, noted above. At the edge of Figs 2 and 3 are three marks to indicate the values of s, K and b. The small black square at the top right corner is to show that in this example the focal length b is positive. In some other designs it is necessary to set b negative, in which case the black square is at the bottom right corner. A negative b has the effect of left-to-right reversing the diagram and so enables us to keep the convention that light always comes in from the left even if the (b positive) design only works when light comes in from the right. The black triangle at the lower edge of these figures indicates the value of s on the outer scale, shown only on Fig. 3, which runs from 3 at the left margin to +3 at the right edge. The black triangle at the left margin shows the value of K on the outer scale, shown only in Fig. 3, which runs from 4 to+4. The edges of the figures in the Road Map likewise indicate ±3 and ±4 on imagined scales for the markers there. Values of s and K are also indicated in the outer margins of the Road Map. 2.4 Useful parts of mirrors and off-axis rays Fig. 4 shows, in the lower half, the mirrors and the paths of the rays for a camera with focal length 100 and aperture 125 (f /0.8), with incoming light collimated and inclined by 2. 5 to the optical axis. In practice, the central obstruction will be half of the diameter of the aperture, but, for a reason that will be explained below, we retain most of the rays for the present. The units of the dimensions are not decided at this stage, but we shall assume that they are millimetres. Across the upper part of this figure are four spot diagrams, showing at the left the distribution of the rays in the plane of the entrance aperture, and three images at the focus, with the minimum rms image spread: (1) in the direction perpendicular (Z) to the plane of the lower diagram; (2) in the plane of the lower diagram and perpendicular to the X axis (Y); and (3) at the position that minimizes the sum of these rms spreads. The last three are highly enlarged: the Y spread in (1) is arcsec and the Z spread in (2) is arcsec. The last image is arcsec. We have shown above that in perfect-focus systems the condition for zero astigmatism is that s approaches 2 for small aperture, and this condition applies for three previously known systems: the Couder (K = 0.385), Schmidt (K = 1.0) and Bowen two-sphere camera (K = 4.236). Other systems such as the Schwarzschild telescope, in which s = 1.25, have substantial astigmatism. It is therefore no surprise to find that in our s = 2, K = 2 example a moderate change in s alone, or in s and K together, is sufficient to remove the astigmatism entirely. When the primary mirror remains with its pole at X = 0.0, and the secondary mirror is moved from X = to , the images at 2. 5 off-axis are reduced to a total spread of arcsec ( µm for f = 100 mm), as shown in Fig. 5. Here the enlargement of the image spot diagrams is greater than in Fig. 4. The rms spread of these images is now arcsec ( µm), and the position of best focus is constant within f. The images computed and plotted thus far have depended only on the parametric equations of Paper I, but these equations do not allow for the inclusion of additional components such as a cryostat window, nor for the investigation of the effects of misalignments. While computing these spot diagrams we saved the (X, Y) coordinates of the points at which 27 of the rays meet the mirrors. Eighteen of these lie on the diameter in the X, Y plane, and nine on a radius normal to that plane. In the same program we then fit power series in Y 2 with up to 10 terms to the data saved. To be sure of a good fit from the centre to the edge of the mirrors, it is necessary for the rays also to be distributed from centre to edge, and this is the reason for adopting a central obscuration of 20 per cent above, rather than the realistic 50 per cent by radius. The power series are printed in the output of the program in such a format as to allow them to be copied directly to the input data file for a more versatile ray-tracing program. Because the present example is a fast (f /0.8) system with moderately deeply curved mirrors, we use the maximum number of terms, 10, in each power series. The largest surface height residual on either mirror is just under mm. The quality of fitting is very much better than Figure 4. Lower diagram: paths of initially parallel rays, 2. 5 off-axis in a spectrograph camera based on the s = 2, K = 2 system. Focal length 100, aperture 125, f /0.8. Upper diagrams: spot diagrams showing (left to right) the distribution of the rays on the entrance pupil, the image focused for minimum rms spread normal to the plane of the lower diagram, the minimum rms spread in the Y direction, and the minimum sum of rms spreads, here arcsec. Figure 5. Same as Fig. 4, with the system adjusted to eliminate astigmatism 2. 5 off-axis by setting s = K = All three images have rms spreads reduced to arcsec.

5 Exploration of designs on- and off-axis 37 this over most of the surfaces. As the shapes of the mirrors are determined by the values of s and K (which were chosen to eliminate astigmatism at an aperture of 125 mm and 2. 5 off-axis) and not by the positions of the rays used to provide data for the power-series fitting, we are free to sample the mirrors over a larger diameter, by adopting either a larger aperture or a larger angle off-axis. In this example we retained an aperture of 125 mm and only increased the angle to Key to the Road Map It is the responsibility of those entering previously unexplored territory to provide a map so that others may later visit it. Here we present a Road Map of the (s, K) plane, and because the Road Map covers four pages we start with a key map (Fig. 6). We have sampled the (s, K) plane at s = 2.0, 1.25, 0.5, 0.5, 1.25 and 2.0, and values of K from +3to 3 at intervals of 0.5, as well as at +0.1 and 0.1 and some other values. These are marked on the key map as +. Those that fit the regular grid are shown on the pages of the main map in some detail, but on a reduced scale. All the systems numbered on the key map are discussed in the appropriate subsections of Section 3, and are identified in Table 1. The shaded areas indicate those systems which cannot form real images. 2.6 The Road Map As indicated by the shading in the key, most of the first page of the Road Map (Fig. 7) is a barren area, occupied by systems that cannot form real images. For K = 0.1 and 0.5 each system is drawn twice, with focal length b =+1 above and b = 1 below. Careful comparisons of row 2 with row 3, and row 4 with row 5, will show that changing the sign of the focal length causes the system to be turned around, relative to the incident light, which always approaches from the left. In this part of the Road Map we find some real reflections of rays, some that are imaginary, and some that are doubly imaginary. Continuous straight lines show the incident light, and the real reflected rays. If these pass through the focus, a real image is formed there. Dashed lines show the imaginary backward extensions of reflected rays. If these meet the focus, then they appear to have come from a virtual image. Rays that are drawn dotted are those that have had two imaginary reflections, and are doubly imaginary rays. Although drawn through the point (0, 0), the focus that they define is a physical impossibility. Mathematically they are useful in a summation of the optical path-length for the application of Fermat s principle. Real and doubly imaginary rays (continuous and dotted lines) contribute positively to the optical path-length, while the imaginary rays (dashed) contribute negatively. Let us use s = 1.25, K = 0.5 as an example. With b = 1 (see column 2, row 3), the rays labelled φ/2 = 0.45 and 2.8 rad strike the gently concave first sheet of the primary mirror, and are reflected to the left, converging to the optical axis, towards the second mirror. These rays have passed twice through the second sheet of the primary mirror, which for these rays we ignore. At the second mirror they are reflected away from the focus, so we have two real reflections on each ray, giving a virtual focus. The third ray, identified by φ/2 = 1.45 rad, is reflected at the second sheet of the primary mirror near its cusp on the axis, and leaves the figure at the lower edge at X = 0.8. Clearly this will not meet the part of the second mirror that lies to the left, but its imaginary backward extension is reflected in the second sheet of the second mirror, and leaves the figure at the Figure 6. A key to the Road Map of the (s, K) plane in the following four figures. The parameter s is plotted with positive values to the right, and K is plotted with positive values upwards. The italic numbers indicate the scales of the axes. Plus signs (+) indicate the (s, K) values of systems that are shown on the main Road Map and/or discussed in detail in Section 3. The non-italic numbers (1 14) and brief names identify the latter systems and correspond to the numbers in Table 1 and the subsections in Section 3. The larger encircled numbers (1) (4) refer to the parts of the Road Map displayed in Figs The shaded areas indicate (s, K) values that do not form real images. For further details, see the text, Section 2.5. right edge near Y = 0.7. The backward extension of this ray, shown dotted, must be thought of as doubly imaginary. With b =+1 (see column 2, row 2), the first sheet of the primary mirror acts as a gently convex mirror, and lies entirely to the left of the first sheet of the secondary. The rays identified by φ/2 = 0.45 and 2.8 rad are now reflected away from the secondary, and their imaginary backward extensions are reflected away from the focus, so that only doubly imaginary rays, shown dotted, pass through the focus. On the other hand, the ray marked φ/2 = 1.45 now has two real reflections, in the second sheet of each mirror, and the imaginary ray arising from the reflection in the secondary passes through the first sheet of this mirror (which for this ray we ignore) on its way

6 38 R. V. Willstrop and D. Lynden-Bell Figure 7. Road Map, part (1): s < 0, K 0.1.

7 Exploration of designs on- and off-axis 39 Figure 8. Road Map, part (2): s > 0, K +0.5.

8 40 R. V. Willstrop and D. Lynden-Bell Figure 9. Road Map, part (3): s < 0, K 0.5.

9 Exploration of designs on- and off-axis 41 Figure 10. Road Map, part (4): s > 0, K +0.1.

10 42 R. V. Willstrop and D. Lynden-Bell Table 1. Useful perfect-focus two-mirror systems. No. System s K (1) Gregorian spectrograph camera (2) Solar furnace (3) Gregorian telescope (4) X-ray telescopes: Wolter type Wolter type Wolter type (5) Ritchey Chretien telescope (6) USNO Astrometric telescope (7) Zeiss sphere and cardioid (8) Sphere and trumpet (9) Wright (reflecting analogue) (10) Schwarzschild telescope (11) Bowen spherical mirror camera (12) A new spectrograph camera (13) Schmidt (reflecting analogue) (14) Couder telescope to the focus. Thus an observer looking at the secondary will see a virtual image from such rays. In the bottom row of the first page of the Road Map K = 0.1, and with s and b both negative all rays are able to reach the focus after two real reflections, but some parts of one or both mirrors may have to be removed to allow this. The primary mirror in each case consists of two sheets, of which the one nearest to the focus reflects all rays defined by 0.0 <φ/2 < 0.78 and 2.36 <φ/2 < π. The secondary mirror is a tear-drop shape, and these rays are reflected in the part of it that lies to the left of the focus. These rays arrive at the focus from all directions within a hemisphere, and the systems form effective solar furnaces. A particular example, s = 0.72, K = 0.2, is discussed in Section 3.2. Rays defined by 0.79 <φ/2 < 2.35 are reflected by the second sheet of the primary mirror, provided that the first sheet has been removed, and then by the inside surface of the elongated part of the secondary mirror, after first passing through it. This is practicable if, for example, the half of the primary mirror in positive Y space is used to feed the half of the secondary mirror in negative Y. There is however no obvious advantage, and construction would be difficult. On the second page of the Road Map (Fig. 8), s, K and b are everywhere positive, and most rays reach the focus after two real reflections, though in nearly every case after having to pass through a part of the secondary mirror in order to reach the primary, and in many cases through a part of the primary to reach the focus after reflection in the secondary. This is not a serious problem, because the parts of the mirrors that cause the obstruction can usually be removed without affecting the useful rays. When K =+1, however, the rays reflected by the primary mirror return to the part of the secondary through which they have already passed, so these systems are impracticable in the all-reflection form. If the primary mirror were replaced by an aspheric lens giving the same deflection (or retardation) as the mirror, we would obtain the Schmidt camera when s =+2, and the Wright camera when s =+1. A similar problem arises with s =+2 and K =+3, when after two reflections the light returns to, or very close to, the point on the primary mirror from which it was first reflected, and must now pass through to reach the focus. The systems that provide useful designs for large twomirror reflecting telescopes (with the secondary mirror substantially smaller than the primary) have K in the approximate range +0.2 to +0.5, and spectrograph cameras (secondary mirror larger than the primary) have K > 2s (Bowen and Zeiss systems) when the twicereflected light passes around the outside of the primary mirror or (the system used as an example in Sections 2.2 to 2.4 above) s = 2, K = 2, when the focus is accessible at a perforation in the centre of the primary mirror. On the third page of the Road Map (Fig. 9), s, K and b are everywhere negative, and again most rays can reach the focus after two real reflections. These now occur on opposite sides of the optical axis, so asymmetrical designs may be used to overcome problems with rays that are obstructed by part of a mirror. The Gregorian system (Gregory 1663; King 1955) is the only previously known form of telescope on this page of the Road Map. In the top row, K = 0.5, are more solar furnaces resembling the one discussed in Section 3.2 but these have larger secondary mirrors that would obstruct an undesirably large fraction of the aperture. At s = 2, K = 2, there is a second possible new spectrograph camera, but this has no advantage over s =+2, K =+2. It is discussed in Section 3.1. At s = 2, K = 3, light within the annulus defined by 0.29 < φ/2 < 0.41 can pass around the edge of the first sheet of the primary mirror to a focus behind it. More light could reach the focus if the first sheet of the primary mirror were trimmed to a slightly smaller diameter. In the part of the fourth page of the Road Map (Fig. 10) that is shaded in the key map (s +1, K 0), there are no real images, as on most of the first page. The only useful systems are those in the top row (K =+0.1) and in the left-hand column (s =+0.5). In the top row, the focus is so close to the secondary mirror that any detector will obstruct a significant fraction of the light approaching the secondary mirror, so the systems will be inefficient, although the secondary mirror is very much smaller than the primary and obstructs little of the incident light itself. In the systems in the lefthand column, only a limited number of rays can reach the focus after two real reflections, so they have annular apertures with rather large central obstruction. The only case of much interest is s =+0.5, K = 2.0, in which the primary mirror is exactly hemispherical. This is discussed in Section Catastrophes and mirror asymptotes The discussion of mirror singularities given in Paper I erroneously did not consider cusps on the secondary away from the focus. These occur when 1 s < 0. Fig. 11 is the corrected form of fig. 4 from Paper I, and predicts what singularities to expect at each s and K. For instance at s = 1.25 and K = 2.0 we should expect that the secondary mirror has a regular surface while the primary will have a regular first sheet up to the cusps that always exist at T = 1. These divide the first sheet from the second sheet but the latter should have an asymptotic spike along the negative X axis. The key tells us that s = 1.25, K = 2.0, is found in the upper middle row of the central column of the second page of the Road Map, and indeed we see a regular almost elliptical secondary surrounding a convex primary which beyond the cusps continues out of the diagram getting smaller in diameter as X gets larger. That it asymptotes to a spike on-axis rather than to a cylinder is not clear from the diagram, but Paper I assures us that it should be a spike on-axis. Other predictions of catastrophes can be checked by the reader. 3 INDIVIDUAL SYSTEMS In this section we describe each of the practical systems. The order we have chosen has s increasing from 2 to+2, i.e. from left to

11 Exploration of designs on- and off-axis 43 Figure 12. A solar furnace, s = 0.72, K = 0.20, b = 1, showing the whole system. The light comes to the focus over a hemisphere of directions. See Section 3.2 for details. Figure 11. The diagram illustrates by different shadings, which sometimes overlap, the parts of the s, K plane of optical designs in which the mirrors have different types of singular behaviour. For example, a design with s < 1 and K < 0 will have both a corrector mirror without singularities and a primary with a cusp on the axis of its second sheet. This is why such designs lie in regions in which the diagonal shading overlaps the dots in squares pattern. The values of η = s/(1 s) are given at the top for convenience. right across the Road Map, and for the few systems with the same values of s, with K decreasing, i.e. from top to bottom. This brings together the three forms of X-ray telescopes (s = 0.02), the two systems with spherical primary mirrors (s = 0.5), and four systems that are free of Seidel astigmatism as well as coma and spherical aberration (s = 2.0). 3.1 Gregorian spectrograph camera (s = 2.0, K = 2.0, b = 1) This system appears on the third page of the Road Map. It has been named the Gregorian spectrograph camera because, as in the Gregorian telescope, light rays cross the optical axis between the mirrors. The primary mirror must be perforated to allow a detector to be placed at the focus, which coincides with the pole of this mirror. The secondary mirror is approximately twice as large as the primary, so the system is unsuitable for a telescope. If it is used with a prism or a transmission grating, the secondary mirror must be perforated. A reflection grating could be mounted directly in front of the second mirror, but it would increase the obscuration. This system has substantial astigmatism. Because this system has both its speed and its angular field limited by the astigmatism, while its central obstruction is no smaller than that of the spectrograph camera described in Section 3.12 below, it is unlikely to be of use in astronomy. 3.2 Solar furnace (s = 0.72, K = 0.2, b = 1) This system uses an approximately hyperboloidal primary mirror to feed light into a tear-drop shaped secondary mirror see Fig. 12. The separation of the mirrors has been chosen to give the minimum tube length that is consistent with all rays from the primary mirror being able to pass the edge of the second mirror and then be reflected Figure 13. The solar furnace of Fig. 12 detail of the secondary mirror and rays near the focus. The parts of the secondary mirror forward of the focus can be removed see Fig. 12 and Section 3.2. in it to the focus see Fig. 13. It provides an aperture of 2f, and a central obstruction less than 4 per cent of the area, from which the solar energy is concentrated on a small crucible or other object over a solid angle of 2π steradians. Astigmatism is 3 arcmin at 15 arcmin off-axis, i.e. at the solar limb, so energy is concentrated into an area not very much larger than the geometrical (aberrationfree) image of the solar disc. 3.3 Gregorian telescope (s = 0.3, K = 0.35, b = 1) The classical Gregorian telescope was the first practical form of reflecting telescope to be proposed (Gregory 1663, see also King 1955). With a paraboloidal primary mirror and ellipsoidal secondary, both concave, it is not coma-free, but as it is usually constructed, with an aperture of f /20, coma is not obtrusive. This system is a coma-free Gregorian analogue of the Ritchey Chretien. Using the expression for the diameter of the circle of minimum confusion given in the Introduction (equation 18), we see that for equal final focal ratios and the same angle off-axis the Gregorian will have more astigmatism than the Ritchey Chretien in the ratio (s g 2)/(s rc 2) = ( 2.3)/( 1.7) = However,

12 44 R. V. Willstrop and D. Lynden-Bell Table 2. Wolter X-ray telescopes. Type Primary Secondary s K mirror mirror 1 Concave Concave paraboloid hyperboloid 2 Concave Convex paraboloid hyperboloid 3 Convex Concave paraboloid ellipsoid Gregorians are often made with longer final focal ratios to minimize the obstruction by the secondary mirror, which leads also to smaller fields of view. The net result is that astigmatism is not much better or worse than in the Ritchey Chretien. The Petzval sum is larger because both mirrors are concave, and the field curvature has the opposite sign to that of the Cassegrain or Ritchey Chretien. Shectman (1994) has taken advantage of this feature in the design of the Magellan telescopes to match the curvature of the focal surface of the telescope to that of a wide-field refracting collimator. 3.4 X-ray telescopes Wolter (1952) described three forms of pairs of grazing-incidence mirrors that can be used to focus X-rays, either in microscopy as he proposed or in astronomy. Wolter s designs used conic-section mirrors as described in Table 2. These designs were free of spherical aberration. While coma was not eliminated, it was small because the radiation approached the focus at small angles to the axis and the field of view was also small. The values of (s, K) given in the last two columns are examples of exact optics analogues of Wolter s three types, and are entirely free of both spherical aberration and coma. The three X-ray telescopes defined by the values of s and K above are shown in Fig. 14. The value of s = 0.02 was chosen to ensure that the nine rays drawn on each side of the axis are distinguishable over most of their paths. The angles of incidence shown here are not close enough to 90 to give efficient reflection of X-rays; smaller values of s eliminate this difficulty but are not so readily illustrated. A Wolter type 1 telescope with K = 1.05 has rays approaching the focus at φ = 2 sin 1 ( s) and angles of incidence measured from the normal (90 φ/4). This system can have both mirrors on the same substrate, but there must be a shallow channel a few millimetres wide between the reflecting surfaces. A polishing tool for either mirror will touch and damage the other surface if it encroaches on it during polishing. The mirrors in type 2 and type 3 systems are mechanically separate, so there is no danger of damaging one mirror while polishing the other, but it is likely to be more difficult to bring the mirrors into alignment and to keep them there. A recent paper by Thompson & Harvey (2000) describes the Solar X-ray Imager (SXI), which uses the Wolter type 1 design. It has a focal length of 660 mm and an aperture at the intersection of the two mirrors of 160 mm. This defines the inner edge of the aperture; the outer edge of the primary mirror has a diameter of mm and the obstruction ratio is Changing the mirrors from the classical paraboloid/hyperboloid form to two hyperboloids removes the coma, but makes only a small improvement in the off-axis images, where the principal contribution to image spread is curvature of field. The coma-free design has a small amount of spherical aberration, which results in ray-theoretical extreme image spread of 0.05 arcsec on the axis, where Wolter s form has no spherical aberration see Table 3. The exact optics version that has been evaluated Figure 14. X-ray telescopes: Wolter type 1 (top); Wolter type 2 (centre); Wolter type 3 (bottom). The grazing angles of incidence have been increased for clarity. See Section 3.4 for details. had the same relative aperture and obstruction as the coma-free form (focal length 100, outer aperture , obstruction ratio ). The mirror shapes were defined by s = , K = and are of course free of both spherical aberration and coma. 3.5 Ritchey Chretien telescope (s = 0.3, K = 0.35) This was designed to have the smallest spherical aberration and coma consistent with the use of conic-section mirrors (Chretien 1922). In the form in which it is normally specified, and working at about f /8, the axial image is certainly smaller than the atmospheric turbulence (say 0.5 arcsec) and may be diffraction-limited (say arcsec for a 4-m telescope). In the exact optics version the axial raytheoretical image spread is under 1 µas, whatever the focal ratio. At 20 arcmin off-axis, and at f /8, astigmatism causes a circle of minimum confusion about 1 arcsec in diameter, both in the original design using hyperboloidal mirrors and in the new theory developed in Paper I. 3.6 USNO Strand Astrometric telescope (s = 0.5, K = 0.5) This telescope uses a paraboloidal primary mirror and a flat secondary in a Cassegrain arrangement. It has been described by Strand (1962, 1963, 1964). Intended principally for astrometry, coma was limited by having the primary mirror work at about f /10 and by limiting the field of view to about 29 arcmin in diameter. The flat secondary mirror makes the design insensitive to misalignment. The image quality is shown in Fig. 15. The images are on-axis and 0. 08, and off-axis; those in the top row are individually focused and those in the lower row are on a flat focal surface, which

13 Exploration of designs on- and off-axis 45 Table 3. Ray-theoretical image spreads in Solar X-ray Imager on the plane through the axial focus (rms radius in arcsec). Degrees Wolter form Coma-free Exact optics off-axis a a This axial image spread results from a secondary mirror with eccentricity (equivalent to a conic constant e 2 of ), while Thompson and Harvey used , which gives arcsec. Figure 16. Spot diagrams to illustrate the performance of a coma-free form of astrometric telescope (f /6). Images are on-axis, and 0. 1, 0. 2, 0. 3 and 0. 4 off-axis. Rows 1 and 3: mirrors correctly aligned. Rows 2 and 4: secondary mirror displaced 3 mm laterally. Rows 1 and 2: each image at its best focus. Rows 3 and 4: all images at the best focus for the image 0. 3 off-axis. The circle is 1 arcsec in diameter, as in Fig. 15. See Section 3.6. systems would be able to maintain alignment to within this limit. The coma introduced (see the axial image, second row) is arcsec, approximately one-quarter of that at the edge of the field in the original design. Figure 15. Spot diagrams to illustrate the performance of the US Naval Observatory (USNO) Strand Astrometric telescope (f /10). Top row: each image at its best focus. Bottom row: all images at the focus of the image off-axis. Other images are on-axis, and and off-axis. The circle is 1 arcsec in diameter. is the best focus for the image off-axis. The circle is 1 arcsec in diameter. The system illustrated in the Road Map at s = 0.5, K = 0.5, has the focus coincident with the centre of the primary mirror. To make it more like a Cassegrain telescope, with the focus below the primary mirror, we readjust the values of s and K. Ifwekepts + K = 1.0, the secondary mirror would have no optical power and its centre would be flat; it is better to give this mirror a little convexity so that the surface slopes are minimized. We have used s = and K = With an equivalent focal length of 15.0 m, the separation of the mirrors is 6.89 m and the focus 8.09 m from the secondary, or 1.2 m below the primary. We have increased the aperture from 1.55 to 2.54 m, and the field of view from 29 to 48 arcmin. Spot diagrams are shown in Fig. 16 on the same scale as Fig. 15. Again the circle is 1 arcsec in diameter. The images are on-axis and 0. 1, 0. 2, 0. 3 and 0. 4 off-axis. In the upper two rows each image is individually focused to minimize the sum of the rms image spreads in the radial and tangential directions, while in the lower two rows they are at the best focus for the image 0. 3 off-axis. The top and third rows have the mirrors correctly aligned; in the second and bottom rows the secondary mirror has been moved laterally by 3 mm. We believe that modern servo- 3.7 Zeiss convex sphere and cardioid (s = 0.5, K = 2.0) This is the only exact optics system to have been described previously (Martin 1932). The primary mirror is convex. Over the hemisphere defined by values of φ/2 from π/4 to+π/4 its radius of curvature is equal to the focal length within less than one part in This variation arises from the limitations of our doubleprecision calculations and the iterative procedure used to find the points of impact of each ray on the mirror. Being spherical, the primary would be easy to construct, but the secondary mirror needs an aspheric depth of ( ) times the focal length for an f /1 system. The asphericity has the opposite sign to that of a paraboloid. The axial separation of the mirrors is only half of the focal length, and as a result there is a central obstruction of about 70 per cent of the diameter, or 50 per cent of the area, of the primary mirror see Fig. 17. Martin (1932) describes the use of this system in place of the more common condenser lens to provide a steeply convergent, hollow, cone of light for dark-field illumination in microscopy. In this application its aberrations are of no importance, and with an extended light source the secondary mirror might be allowed to remain spherical. If an attempt were made to use the system as a microscope objective, even with a secondary mirror having the correct asphericity computed here, the large obstruction would cause loss of resolution and contrast through its effects on the diffraction image. The principal aberration is Seidel astigmatism; a system in which the aperture of the primary mirror is equal to the focal length (f /1, nominally, but effectively f /1.4 because of the obstruction) gives a circle of minimum confusion just under 13 arcsec in diameter at 1 off-axis. The circles of minimum confusion lie on a spherical surface with a radius of the focal length, and the Petzval surface is more gently curved to a radius 2.04 F.

14 46 R. V. Willstrop and D. Lynden-Bell forms a circle of minimum confusion 18 arcsec in diameter. To put this in perspective, if the Ritchey Chretien were to be scaled to the same field and relative aperture as the Schwarzschild, its astigmatism would be even larger: 1 (8/3) [3/(2/3)] 2 = 54 arcsec. Figure 17. Zeiss convex sphere (primary) and cardioid (secondary). The principal aberration is astigmatism; the circle of minimum confusion is approximately 13 arcsec in diameter at 1 off-axis in an f /1 system. See Section 3.7 for details. 3.8 Concave sphere and trumpet (s = 0.5, K = 2.0) Lynden-Bell (2002) has proposed a corrector for a large, fast, concave spherical mirror consisting of a trumpet, reflecting on the outer surface. We have examined this system already (unpublished work) and find that, though it can be made free of spherical aberration and coma, the astigmatism is very large. It is of some interest because each trumpet is small, and a number of them could be used simultaneously to correct the images of several rather widely spaced point objects. The radius of curvature of the primary mirror is equal to the focal length, with a range of for all values of φ/2 from +π/4 to +3 π/4. The central obstruction is effectively 38 per cent by area, because, although the correcting trumpet itself is small compared with the spherical mirror, the thicker end of the trumpet, which corrects the rays from the outer zones of the spherical mirror, will itself obstruct light from zones at radii smaller than of the outer zone. However, a plane mirror at the backside of the trumpet could capture the remaining 38 per cent of the light and direct it to another focus. 3.9 Wright camera (s = 1.01, K = 1.0) The computer programs fail if s = 1.0 exactly, i.e. when η (see equation 2 above), so a value of 1.01 was used to draw the perfect systems. This system is not useful in practice, because the secondary mirror obstructs the light approaching the primary, but this poses no problem to the theoretical ray-tracing, and in practice the primary mirror can be replaced by a thin aspheric lens. At 1 off-axis, and with an aperture of f /4, the astigmatic image spread was arcsec. A system with a refracting corrector plate and oblate spheroid mirror was ray-traced by another program, which gave an image spread of arcsec Schwarzschild telescope (s = 1.25, K = 0.50) This form of telescope was proposed by Schwarzschild (1905). Two examples are recorded by Dimitroff & Baker (1945) as having been built. It is possible that experience with using these two instruments discouraged others from building any more, because although spherical aberration and coma are fully corrected, astigmatism is not. At the edge of a field 3 in diameter, in an f /3 camera, astigmatism 3.11 Bowen spectrograph camera (s = 2.0, K = 4.236) Bowen (1967) described a spectrograph camera that uses two spherical mirrors, and was adapted from a reflecting microscope objective designed by Burch (1947). The mirrors are concentric, so the system has no axis; therefore coma and astigmatism are absent. Spherical aberration is controlled by the ratio of the radii of curvature of the mirrors. At small relative apertures the radius of the larger mirror should be times the radius of the smaller, but large relative apertures have significant amounts of fifth-order spherical aberration, which can be balanced by a small amount of third-order aberration, which may be introduced by reducing the ratio of the radii of curvature. At f /1.1 the image spread arising from spherical aberration was rad. Away from the centre of the field the beams of light are decentred on the mirrors, and therefore form part of a faster system, so off-centre images are larger. The central obstruction caused by the primary mirror in the path of the light converging to the focus is about half of the diameter of the aperture. The perfect-focus form of this camera, like all other perfect-focus systems, is designed to have no spherical aberration of any form, and no coma. In addition, the condition for zero astigmatism approaches s = 2.0 when the aperture and the angle off-axis each approach zero. For useful apertures and fields of view, s should be significantly smaller than 2.0. This is illustrated in Table 4. The exact-optics form of this camera with a relative aperture of f /1.1 has the sagittal and tangential images coincident at 2 off-axis if s = and K = The image spread there is less than half of the minimum spread in the spherical mirror form, and does not reach rad until off-axis see Fig. 18. For use with modern detectors, such as charge-coupled devices (CCDs), these cameras need both a field-flattening lens very close to the detector, and a plane-parallel cryostat window, which will affect the spherical aberration of the images A flat-field spectrograph camera (s = 2.0, K = 2.0) This system, which was used as an example in Section 2, was chosen from among the quantized values in the Road Map. Here we start to fine-tune the values of s and K to find the optimum configuration and optical performance. In the Introduction we gave expressions for the astigmatic blur and the Petzval curvature in terms of the parameters s and K. When s = 2 the astigmatism is zero, and it can then easily be shown that if K = 1± 2 = 1± the Petzval curvature is also zero. These results are strictly true only in the limit as the numerical aperture and field both tend to zero. With finite aperture and field the higher-order (non- Seidel) astigmatism is not exactly zero, and some improvement in the images can be made by reintroducing a little Seidel astigmatism Table 4. Values of s that give zero astigmatism (central obstruction half of the aperture). Angle off- Aperture D/f axis (deg)