Transforming Newton To Cassegrain

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Transforming Newton Cassegrain
Fig. 1.4. (a) Facsimile of the Gregory telescope from "Optica Promota", 1663 (after Danjon and Couder [1.3]). (b) Raypath of the Gregory form (after King [1.1])

moderate telephoto effect (reduced length - see Chap. 2), a reasonable field, and an upright image, a useful advantage at that time. Gregory not only gave the correct geometrical forms for the two mirrors - parabola and ellipse - but also provided a stray-light baffle through the long tube separating the primary from the eyepiece (Fig. 1.4(a)). He also gave the correct formula for the equivalent focal length of the mirror combination. Gregory attempted to convert his admirable concept into practice with the help of the London opticians Reeves and Cox whom he commissioned to make a telescope of 6-foot focus [1.1][1.3]. Inevitably, if only because of the elliptical secondary, their efforts were a complete failure, and Gregory abandoned the attempt.

Newton was well informed of Gregory's proposals and attempts at manufacture. His determination to attempt to make a reflecting telescope was a result of his classic experiment in 1666 demonstrating that white light was composed of the colours of the spectrum which were refracted differently by a prism. This was, of course, not only a radical advance in the theory of light and vision but also provided the correct explanation of the chromatic aberration which had limited the performance of refractors since their invention nearly 60 years earlier. Unfortunately, Newton arrived too hastily at the conclusion that refraction and chromatic dispersion were linked by the same linear function for all refracting materials. Backed by his authority, this error delayed the invention of the achromatic objective for over 50 years. Nevertheless, the scientifically correct explanation of chromatic aberration was a huge step forward. Newton also showed [1.1] that the effect of spherical aberration in typical refracting telescopes was only about 1/1000 of that of chromatic aberration, thus correcting the illusion generated by Descartes concerning the need for aspheric lens surfaces to correct the observed defects. Furthermore, Newton's error led to the construction of the first reflecting telescope which could rival the better refractors.

Newton solved the front-view obstruction problem of the observer's head by the elegant device of his plane mirror at 45° (Fig. 1.5). This invention, which seems simple to us today, had eluded all his predecessors with the pos-

Telescope Invention

sible exception of Zucchi, whose ideas were too vaguely expressed to warrant a claim to its invention [1.3]. In fact, the Newton plane mirror was even more important than it seemed; for it was the only form proposed which had any chance of manufacturing success, since the manufacturing problems of the secondary mirrors of the Mersenne, Gregory or (later)Cassegrain solutions were insuperable for a considerable time to come (see Chap. 5 and RTO II, Chap. 1). Forewarned by the manufacturing problems of Gregory's over-ambitious plan, Newton himself made his first telescope in 1668 with a primary of only about 3.4 cm aperture and a focal length of about 16 cm. He used a plano-convex (Kepler) eyepiece giving a magnification of about 35 times. The "turned-down edge" of the primary was masked off by a diaphragm near the exit pupil behind the eyepiece. Newton claimed the performance was comparable with, or better than that of a Galileo-type refractor of 4-foot focal length, although light losses were inevitably much higher. No doubt the success of Newton's manufacture of the optics was due to his more scientific approach compared with the "spectacle lens quality" delivered by most opticians (he invented polishing on a pitch lap or was, at least, the first to publish it), and to the more modest size of the telescope. A duplicate instrument, apparently of better quality, was the one that became famous through its demonstration at the Royal Society.

This first phase in the development of the reflecting telescope was completed by the announcement by de Berce in 1672 of the invention by Cassegrain of the form that bears his name (Fig. 1.6). De Berce's description was very poor, in no way comparable with the excellent exposition by Gregory of his form 9 years earlier. Newton was forthright in his condemnations, above all in order to refute claims of priority on Cassegrain's part, which were anyway weakly based. His criticisms of the need to make a hyper-boloidal secondary and the difficulties of achieving that were justified; but his other criticisms had no validity. Both he and his contemporaries in England and elsewhere (e.g. Huygens) completely failed to appreciate the huge tele-photo advantages; also apparently its links with the proposals of Mersenne. The Cassegrain form simply completed the theoretical possibilities of the 2-mirror compound reflector as expounded by Mersenne and elaborated by Gregory, by using the convex secondary instead of the concave one in the Gregory-type "mixed" telescope of mirrors forming a real image observed by a refracting eyepiece.

The year 1672 marked the end of the first phase of telescope development and was followed by a period of consolidation. This first phase, lasting only some 64 years, had produced remarkable results. It was totally dominated by optics, the problem of getting a good image. Since planetary observation dominated the astronomical research of the time, the demands were high. Mechanical problems were also serious, but mainly on account of the length of telescopes required to reduce chromatic aberration to acceptable levels.The situation during this first phase regarding the optical development of refractors and reflectors was remarkable and can be summed up as follows:

\ Parabola

\ Parabola

Fig. 1.6. The Cassegrain reflecting telescope, 1672, (a) as drawn by de Berce. (b) Raypath of the Cassegrain form (after King [1.1])

Refractors: The optical theory of the refractor made no advance after Galileo up to the time of Newton, except for the positive Kepler eyepiece, because the physical origin of the dominant colour aberration was not understood. Newton explained chromatic aberration and laid the basis for the scientific theory of light propagation and colour vision, but - by a serious error of interpretation - blocked further development of the refractor for over 50 years. In spite of these failures to advance the theory, the refractor remained the only practical form available because it could be manufactured with sufficient precision to achieve imagery of the quality limit set by the chromatic aberration of long telescopes.

Reflectors: The situation here was exactly the opposite of that of refractors. The optical theory developed rapidly through the work, above all, of Descartes, Mersenne, Gregory, Newton and Cassegrain. It was so complete that no real further advance was made till 1905! However, even for minimum quality requirements, manufacture had proved impossible until Newton's first reflector in 1668. The reason for this is quite simple and lies in elementary optical theory (see RTO II, Chap. 1): the precision required for the same 'performance of a reflecting surface is about 4 times higher than that for a refracting surface. This increase of manufacturing precision compared with refracting surfaces was not possible even for spherical surfaces and out of the question for secondaries of compound telescopes, which required an as-pheric form. Primaries could remain spherical if the relative apertures were reasonably low. This, combined with the centering problem of 2-mirror telescopes (Chapters 3 and 5), is the reason why the Newton (or Herschel) forms were the only ones feasible in practice for about 180 years! A further central problem in the early development of the reflector was the poor reflectivity, and hence poor efficiency, of the speculum metal used and, of course, its further reduction due to tarnishing. For a given aperture, therefore, refractors had a huge advantage in light-gathering power.

1.2 Period 1672-1840

For the reflecting telescope to attempt to rival the long refractors of the day, the theoretical work described above had to be complemented by major improvement in practical manufacture. Realistically, the Newton optical geometry was pursued. In 1721, a Newton reflector of 6 inches aperture and 62 inches focus (f/10.3) (Fig. 1.7) was presented by John Hadley to the Royal Society of London [1.1]. This telescope, only about 6 feet in length, was tested and compared with the 123-foot focus refractor of Huygens with a comparable effective aperture of the order of 6 inches [1.3]. Inevitably, the refractor gave the brighter image, but Hadley's reflector gave comparable definition. The ease of manipulation greatly impressed Bradley and Pound during their comparative tests.

Hadley's work was the birth of the practical reflecting telescope. His workmanship, as shown in Fig. 1.7, was so beautiful that it would do credit to a modern amateur even if the design is not what one would attempt to produce today. He invented the autocollimation pinhole test at the centre of curvature [1.1], the first scientific test during manufacture of a primary mirror. Using this test, Hadley was able to estimate in a qualitative way the errors from a true sphere of the mirror during working. This was his great advance over his predecessors, who had effectively been "working blind" using methods associated more with spectacle lens manufacture, for which the quality requirements were much lower. Following Cartesian theory, Hadley attempted to "flatten" the outer parts of the mirror to produce a paraboloid. It is doubtful whether this attempt produced any improvement, since the difference between the spherical and paraboloidal form for a 6-inch aperture working at f/10.3 was negligible. Although, as has been stated above, the basic theory for perfect axial imagery in all the fundamental forms of the reflecting telescope was already laid down by 1672, this was not the case for a correct analysis of manufacturing tolerances. What was needed for this

Aberration Spherical Reflectors
Fig. 1.7. John Hadley's 6-inch, f/10.3 Newton reflector, 1721 (courtesy Royal Astronomical Society, through Peter Hingley)

in the case of primary mirrors was the series expansion of the circle which gives directly the difference from the parabola with the same vertex curvature (see § 3.1), together with an idea of what surface accuracy was required to give good imagery. In fact, with Descartes' analytical geometry combined with his own binomial theorem and "method of fluxions" (calculus), Newton possessed, in principle, the mathematical tools not only to derive the series expansion of the circle but also to deduce, by differentiation, the transverse image aberration corresponding to the difference between the sphere and the paraboloid. If Hadley had known this, he could have compared (perhaps very crudely) the values with the transverse errors shown by his pinhole test, although the interpretation would have been strictly geometrical, without knowledge of the decisive influence of the wave theory of light which was only clarified by Rayleigh about 150 years later. In any event, there is no evidence that an analysis of tolerances using a series expansion of the circle took place till long after Newton. The full understanding of the convergence limitations of the binomial theorem was published in 1715 but not really noticed or understood till about 1772.

The more generalized series expansion of a conic section, had it been known, would also have scientifically validated the concentration on the Newton form of the reflector: it would have shown (see Chap. 3) that the aspheric secondary mirrors in the form proposed by Cassegrain required a far greater difference from the sphere than the primaries.

A further notable advance in the manufacturing quality of reflectors was made by James Short, who dominated the scene both for quality and volume of output between 1733 and 1768. It was Short who raised the level of the reflector to that of a practical instrument surpassing the old "long" refractors. Most of his telescopes were in the Gregory form using a concave elliptical secondary. Short was therefore the first optician who mastered the manufacture of appreciably aspheric concave mirrors to adequate quality for the astronomical observations of the time; aspherics being essential for his telescopes because he used much steeper primaries than his predecessors, giving relative apertures between f/3 and f/8 at the Gregory focus. It should be remembered that, even at that time, the demands on resolution were high because of the emphasis on planetary observation. It is very important to note that Short nearly always made the Gregory and not the Cassegrain form. It is far easier to make a concave than a convex mirror (see Chap. 3 and RTO II, Chap. 1) because of the test procedures involved. As was unfortunately common at the time, Short gave no account of his manufacturing methods. It seems possible he invented the "overhang" method of aspherising [1.1]. In 1752 he made an 18-inch Gregory telescope for the King of Spain (price £1200) which remained for many years the largest reflector in the world. In 1749 he introduced a "universal" equatorial mount, but not in a very stable form. In all, according to Baxandall [1.1], he made well over a thousand telescopes.

As was so often the case in the fascinating history of the development of the telescope, the advance of the reflector to surpass the single lens objective refractor (above all because of compactness) was accompanied by a major breakthrough in the refractor. The development of the achromatic objective was delayed (see above) by Newton's false conclusion about the link between refraction and dispersion. This was first challenged in 1695 by David Gregory, Savilian professor of astronomy at Cambridge and nephew of James Gregory, on the basis of the supposed achromatism of the eye. In 1729, Chester Moor Hall worked out the basic theory of an achromatic doublet and had such objectives made by opticians in London, but they apparently failed to understand their full significance. About 1750, John Dollond became aware of this and also of a paper by Euler (1747, in ignorance of Hall's previous work) expounding the theory of an achromatic doublet. Dollond had previously believed Newton's statement but then convinced himself by experiments that an achromat was possible. A letter to Dollond from Klingenstierna in Uppsala confirmed that Newton's conclusions had been wrong. Dollond then succeeded in making his first achromatic objectives but suppressed all reference or credit to Hall, Euler or Klingenstierna. John Dollond's behaviour in this respect was far from admirable, his son Peter Dollond's subsequent behaviour was even worse (he not only denied all credit to Hall, but also blocked the rights of the opticians with whom Hall had worked - Bass, Bird and Ayscough - by a successful patent action) [1.1]. In 1790, sixty years after Hall's original invention, it became clear that Hall had established the theory not only of correction of primary chromatism but also of spherical aberration. It was Clairaut, finally, who put the theory into modern form (1761-1764) and established the normal doublet form (still known in France as "un Clairaut") also corrected for spherical aberration for one wavelength [1.1]. Clairaut also pointed out the inevitable residual "secondary spectrum". Furthermore, he investigated by ray-tracing the imagery of such objectives off-axis, the first time imagery in the field was considered, effectively discovering the aberrations of coma, astigmatism and field curvature (see Chap. 3).

A major limitation in the optimum manufacture of achromats at that time was the impossibility of correctly measuring the refractive indices for a specified colour. Another problem was the diameter limitation of glass of good quality, above all flint glass. Four inches was already a large blank. In the constant battle for supremacy between refractor and reflector, this was the principal factor limiting the development of the achromat before the improvements in glass manufacture by Fraunhofer and Guinand. In spite of this, the transmission of achromats was so much higher than the reflectivity of reflectors with two speculum mirrors that some of the best achromats of Peter Dollond (triplets giving reduced secondary spectrum) with apertures of 3.8 inches rivalled appreciably larger reflectors. In 1777, tests by Maskelyne with such a Dollond telescope resolved n Coronae Borealis with a separation of 0.9 arcsec.

In this period of intensive development of the refractor, two important advances were made by Jesse Ramsden, an excellent instrument maker. In 1775 he discovered what was later known as the "Ramsden disk", i.e. the exit pupil of a telescope (Fig. 1.8). Vague ideas of the exit pupil were certainly current after the introduction of the Kepler (positive) eyepiece, but Ramsden was the first to explain it correctly. He was also, perhaps, the first to make a Cassegrain telescope of "reasonable" quality. In fact, Ramsden's Cassegrain telescope was a special form which he called a catoptric (i.e. all reflecting) micrometer which used a split convex secondary whose tiltable halves formed the micrometer. However, the instrument was not successful, probably confirming that the manufacture of the Cassegrain secondary was beyond the technology of the time, above all from a test point of view (RTO II, Chap. 1).

Objective and entrance

Objective and entrance

Fig. 1.8. The "Ramsden disk" (exit pupil) explained by Ramsden in 1775 (after King [1.1])

1.3 William Herschel's telescopes

It is my feeling that William Herschel was the greatest astronomer of all time. But I would assert with more confidence that Herschel was the greatest telescope maker of all time. His work [1.1] produced the biggest quantum jump in light-gathering power, while maintaining or even improving resolution, that has occurred since Galileo. Figure 1.9 shows his portrait at the age of 46 in 1785. His achievements seem all the greater if one reflects that he only turned his attention to astronomy in a practical way at the age of 35.

Fig. 1.9. Sir William Herschel (1738-1822) painted by L.T. Abbot in 1785 (courtesy Deutsches Museum, Munich)
Spherical Aberrations
Fig. 1.10. William Herschel's "large" 20-foot focus telescope, aperture 18.8 inches (f/12.8), completed in 1784, (reproduced from an engraving of 1794, courtesy Science History Publications Ltd., Cambridge, England)

Herschel started off as an amateur in Bath with refractors, going up to 30 feet length with purchased objectives. Because of the inconvenience of the long, thin tubes, he borrowed a Gregory-type reflector. He found this so much better that he tried to purchase a 5 or 6-foot3 reflector, but its price was beyond his means. So he started to make reflectors himself and had made a 5 2-foot Gregory telescope before the end of 1773. However, alignment of this form gave him such trouble that he switched to the Newton form. He made a 7-foot telescope giving excellent performance on Saturn. By May 1776 he

3 At this stage of development, telescopes were still normally characterized by their focal lengths rather than their apertures.

was working on a 20-foot telescope. Herschel was the first to realize that the key to success was the production of a high quality spherical surface and that the difference between the sphere and parabola was minimal for his primaries which normally had relative apertures of f/12 - f/13. Apparently he was also the first to use, systematically, pitch polishers cut into squares. His discovery of Uranus in 1781 was made with an excellent 7-foot telescope with 6.2 inches aperture made in 1778.

As his projects became more ambitious in size, Herschel was forced to use a higher copper content in casting his speculum metal blanks to avoid fracture. Typically, for large blanks he used 73% copper and 27% tin. The lower tin content made the mirror more prone to tarnish.

His work became known to the Royal Society and the Greenwich Observatory. Comparisons with the best telescopes otherwise available were made in 1782 and revealed the marked superiority of Herschel's instruments. He then received an official salary as "Royal Astronomer" but augmented this, after the move to Datchet in 1782, by making and selling reflecting telescopes. In three years he made and sold about 60 telescopes, most of 7 to 10 feet in focal length. This was a prodigious feat in view of the extensive telescope development for his own work and the astronomical observations performed with them.

For his work on nebulae, starting about 1784 from Messier's catalogue, Herschel had available his "large" 20-foot telescope with an aperture of 18.8 inches (f/12.8). This Newton-type telescope (Fig. 1.10) was to prove to be his greatest achievement from the point of view of practical astronomical results. However, he had plans for an even larger telescope of similar altazimuth mounting, with an aperture of 48 inches. A blank cast in London in 1785 was polished and gave reasonable results. A second blank cracked in cooling (Her-schel did not understand the necessity of slow cooling in the casting furnace) and a third was given an even higher proportion of copper, a guarantee of rapid tarnishing. The focal length was 40 feet (f/10), a formidable mechanical undertaking. The giant telescope (Fig. 1.11) went into operation in 1789 and gave "pretty sharp images". The extra light-gathering power enabled Her-schel to discover two further satellites of Saturn, Enceladus and Mimas. But, compared with the 20-foot telescope, the 40-foot was a relative failure. The high content of copper in the mirror caused rapid tarnishing; but, above all, Herschel had reached a telescope size where mechanical problems became the limiting factor, rather than the problems of optical figuring, though these were also formidable. His simple support system, a radial iron ring, gave rise to considerable flexure problems when the telescope was used at appreciable zenith distance. This was exacerbated by the relatively thin mirror, although this had the advantage of reducing the thermal sensitivity. Herschel used the "front-view" form (Fig. 1.1) which bears his name, having already experimented with it with the 20-foot.

William Herschel Universo
Fig. 1.11. William Herschel's largest telescope: 4 feet in aperture, 40-foot focus (f/10), completed in 1789 (courtesy Deutsches Museum, Munich)

For freshly polished mirrors Herschel determined a reflectivity of 67%, justifying his use of the Herschel focus. The optical quality of his smaller telescopes must have been extraordinarily good. In well-conceived experiments using terrestrial objects he established that a telescope of 8.8 inches would show a "real disc", as distinct from the "spurious disc", of 0.25 arcsec. This suggests clearly that the telescope was diffraction limited,, a conclusion supported by his observations when the telescope was stopped down. His observational achievements indicate that his best telescopes, including the 20-foot, were "seeing limited" for the seeing at Slough in England at that time, a quality of resolution not significantly improved till the 20th century.

A plausible measure of the optical quality of Herschel's two largest telescopes can be derived from his use of the Herschel "front-view" focus position (Fig. 1.1). With a reasonable estimate of the eyepiece position relative to the tube axis, the tangential field coma (see Chap. 3) of the 20-foot was about 0.28 arcsec, that of the 40-foot about 0.58 arcsec. These amounts of coma were presumably smaller than, or at most comparable with, the best atmospheric seeing and other telescope errors, implying excellent quality even by modern standards (see RTO II, Chap. 3 and 4).

In a classic paper [1.9] presented to the Royal Society in 1799, Herschel analysed with great rigour "the power of penetration into space" of his telescopes, as distinct from their formal magnifying power. We shall consider his criterion further in RTO II, Chap. 4.

Herschel was so much in advance of his colleagues, both astronomically and technically, that a further 50-60 years were to pass before his achievements could be surpassed. To understand the further advances in the reflecting telescope, the theory of Chapters 2 and 3 is required. The modern historical development from about 1840 will be taken up again in Chap. 5.

2 Basic (Gaussian) optical theory of telescopes

"God invented the number, the Devil invented the sign" - remark by Christoph Kühne at Carl Zeiss, Oberkochen, about 1969, concerning the problems caused by the sign in geometrical and technical optics.

2.1 Basic function of a telescope

A telescope is a device whose basic technical purpose is very simple: it should provide a high quality image of distant objects, which may be point sources or extended objects. The telescope must be constructed in such a way that it can be pointed at the desired object field. Furthermore, if the object is moving, as in the astronomical case as a result of the earth's rotation, the telescope must provide the means to compensate this movement. So an astronomical telescope has three basic functions:

- High quality imaging

- Pointing

- Tracking

The first requirement is the main subject of this book, but the other two aspects are closely related to it and cannot be achieved without the telescope's image-forming property. Adequate tracking is also essential to high image quality and represents one of the most difficult technical requirements in modern telescopes. Until the invention of photography and its first applications to astronomy in the mid-nineteenth century, the only detector available to investigate the information contained in the image was the human eye. The normal device for doing this was the ocular, and the combined form of imaging device known as the refracting telescope (objective and ocular) is that normally explained in elementary textbooks. However, the basic form of a telescope is simply a device producing a real image for some detector, used either directly in the image plane or indirectly by a transfer system such as an ocular to the human eye or an instrument to a modern detector. The basic function is therefore that of a photographic camera, a device which produces a real image of distant objects for detection by a photographic emulsion. A modern astronomical telescope is simply a photographic camera with a huge aperture and focal length, but working with a modern detector and an angular field which is small compared with those of photographic objectives, even for cases of so-called "wide-field telescopes".

Even before the invention of photography, the use of the telescope in this basic mode was known in the form of the "camera obscura", in which the real image was projected on to a screen. In its original form, probably invented by della Porta [2.1][2.2] before the invention of the telescope, the image was produced by a pinhole camera. The system was used later, above all for solar projection, the pinhole being replaced by a telescope objective functioning exactly as a camera objective. The "camera obscura" had the disadvantages of light and resolution losses due to the remittance properties of the screen and, above all, the fact that the image size was too small because the focal length of the early objectives was fairly short. The combination forming the refracting telescope by the addition of an ocular was therefore a revolutionary advance as it provided substantial magnification, i.e. the equivalent of a "camera obscura" of much longer focal length. This can best be understood from the fundamental concept for the human eye of the minimum distance of distinct vision (dmin) or near point. This varies strongly with age, but the standard average value is normally taken to be 25 cm. This means that the minimum distance for observation of the image in the "camera obscura" will be about 25 cm, and may be more if the linear size of the image gives an angular field at this distance which is too large for vision in comfort, for example with a wide field, long focus objective. Considered as a telescope, the magnification of the "camera obscura" is then with the objective focal length f

which expresses the ratio of the angular size of the object or image, as seen from the objective, relative to that of the image on the screen seen by the observing eye. If the screen and eye are replaced by an ocular and eye to give a normal afocal refractor (see § 2.2.4), the angular magnification is the ratio of the focal lengths fco/fLular , whereby f'cuiar can easily be made as short as 10 mm or even less. With /'ocular = 10 mm , the length of the refractor for the same magnification is only about 1/25 of that of the equivalent "camera obscura". Furthermore, the eye behind the ocular is in a relaxed state focused on infinity rather than the state of maximum accommodation (ca. 25 cm) required for the "camera obscura". The role of the screen in the "camera obscura" is to scatter remitted light in such a way that sufficient light intensity reaches the eye pupil for the maximum semi-field angle observed. It plays the role of a light-inefficient "field lens", but with the advantage that the scattering properties of the screen are the same over a large area so that the position of the image on it is uncritical. A field lens would image the objective (entrance pupil) on to the eye pupil. This function is shown for a Kepler-type refractor in Figs. 1.8 and 2.8 and plays an important role in many telescope designs.

Before we consider the precise function of such telescope systems, it is necessary to review some of the basic properties of optical imaging systems in general.

2.2 The ideal optical system, geometrical optics and Gaussian optics

2.2.1 The ideal optical system and Gaussian concept

An excellent treatment of the basic theory of optical systems is given by Welford [2.3], to which the reader is referred for a full account. Here we will give a condensed version of the essential points.

An ideal optical system is assumed to have a unique axis of symmetry, building a so-called centered optical system. Most practical optical systems are conceptually centered systems, including telescope optical systems. However, there are important exceptions. The general theory of non-centered systems is very much more complex because of the vast increase in parametric freedom.

The essential theory of imagery by an ideal optical system was first laid down by Gauss in his classic work of 1841 [2.4]. He introduced the concept of principal planes as the equivalent of any centered optical system and the strict definition of focal length.

Fig. 2.1. The ideal optical system: the principal planes and unit magnification between them

In Fig. 2.1, let F P P' F' represent the axis of an ideal centered system S which may consist of any number and arrangement of centered elements. The word "centered" implies symmetry of all elements to the axis so that the axis cuts all elements normally. Let the ray r1 enter from the left at any arbitrary height above the axis and parallel to it, i.e. from the axial point of an infinitely distant object. This ray emerges at some height on the image side of the system S and crosses the axis at the point F . Let r2 be a ray at identical height, also parallel to the axis, but entering in reverse direction from the right. This ray emerges at some height on the left hand side of S and cuts the axis at F. Let ri cut r2 (its own projection with reversed direction) at PH, where the suffix H simply implies a higher point than P' in the plane of the figure. Similarly, r2 cuts r1 at Ph . The points P and P are then constructed by dropping perpendiculars from Ph and PH to the axis.

F and F are the focal points of the system; P and P are the principal points, the planes through PPh and P PH perpendicular to the axis the principal planes. Since r1 and r2 both pass through Ph and PH, these points must form object and image to each other, i.e. they are conjugate points.

Similarly, PPh and P PH are conjugate 'planes. Since, by the nature of the construction, PPh = P Ph, it follows that the magnification from one plane to the other is unity. Although, by definition, F and P are always in the "object space" and F and P are always in the "image space", the points F and P may lie to the left of the system; or F may lie to the right and P to the left. An analogous situation obtains for F and P.

The distance P F is the image-side focal length, denoted by f , while PF is the object-side focal length, denoted by f. We shall see that |f = |f | if the media (i.e. refractive indices) are the same in both object and image spaces.

The four points F, F , P, P along the axis completely define the properties of the ideal optical system. They lead at once to a construction enabling the complete geometrical determination of image formation, as shown in Fig. 2.2.

The rays r1 and r2 are constructed from the object point Ih above the axis and placed at a distance PI to the left of the object principal plane PhPPl, where the suffix L refers to the lower half of the principal plane. ri is parallel to the axis, passes from Ph to Ph with unit magnification and crosses the axis at F . The ray r2, now proceeding in the incident direction from left to right, which is defined as the normal direction of incident light, is drawn from Ih through F to Pl. After unit magnification transfer to PL, r2 must proceed parallel to the axis, since it passes through F, meeting r1 at IH and thereby defining the position and size of the image I Ih . For generality, |P F | has been made not equal to |PF|, implying that the object and image media are different, e.g. air and water. In telescope systems, this is very rarely the case.

A decision must now be taken on the signs of the geometrical distances involved. This is a matter of fundamental importance in geometrical optics and many sign conventions have been proposed and used in its long history. In accordance with Welford [2.3] and the modern trend in optical design in general, we shall use throughout this book a strict Cartesian system. Only in this way can complete generality of formulation be achieved, even though the price may be an apparent complication of certain simple cases. This is a price well worth paying to avoid ambiguities and errors.

Let the axial distances z and z be measured from F and F , so that z is negative, z positive. Similarly, measuring f and f from P and P respectively, f is negative and f is positive. Also n is positive, n is negative. Then n/n = —f/z = m (2.1)

Combining these two, we have the well-known formula zz' = ff' , (2.3)

Newton's conjugate distance equation.

Equations (2.1) and (2.2) also define the magnification m of the system. However, these simple relationships in z and z do not usually represent the most convenient form, since P and P are usually more convenient reference points for the object and image positions than F and F . Let PI = s, P I = s .As with z and z , s is negative in Fig. 2.2 and s is positive. Then s = z + f, s' = z' + f' (2.4)

Eliminating m gives

If the media in object and image space are identical, the usual case, then it will be shown that f = —f and (2.7) reduces to the well-known "lens formula"

From (2.5) we have for the magnification m:

The marked difference in form between Eqs. (2.3) and (2.7) arises from the fact that P and P are conjugate points whereas F and F are not conjugates.

Telescopes are usually used on very distant, or effectively infinitely distant objects. In this case s ^ ro and the above formulae for the magnification m become zero and hence meaningless if the system terminates at the first image plane I Ih in Fig. 2.2, the case of the photographic camera or normal modern telescope. We shall see in § 2.2.6 that the concept of magnification is replaced by the concept of scale.

Finally, we must introduce the concept of nodal points. In Fig. 2.2, the rays IhP and P Ih do not have the same angle to the axis because, for generality, the object and image spaces have been shown with different media whose refractive indices are n and n respectively. The consequence is that |f | and |f | are not numerically identical but are related by Eq. (2.20) below. If the media are identical, then any ray cutting the principal point P will leave P with the same angle to the axis. This is the general property of nodal points. They are the same as the principal points unless n = n. Since this case is very unusual in telescope optics, it will not be considered further here. But it should always be borne in mind for exceptional cases where an image is not formed in air.

2.2.2 Geometrical optics and geometrical wavefronts

We have used above the concept of a ray of light without defining its exact meaning. Everyday experience with shadows makes the ray concept plausible as the straight line light path from a point source to any point it illuminates in the same medium. The definition of a ray is closely linked to that of a geometrical wavefront. The geometrical wavefront is the surface of constant phase, or optical path, which was introduced by Fermat, and published in 1667. The wavefront is orthogonal to the rays which, to the approximation of geometrical optics, are the paths along which the radiation is propagated. The limitations of this interpretation, arising from the wave nature of light at boundaries, are the subject of physical optics and diffraction theory. Much of telescope theory can be handled by geometrical optics, though diffraction limitations are of great importance in defining the theoretical limits of resolution and the tolerancing of optical errors.

In Fig. 2.3, the point source I sends out the wavefront W in object space. This object geometrical wavefront is, by definition, exactly spherical with its centre at I. It advances with the speed of light in the object medium and strikes some centered optical system with principal planes at P and P . Suppose this system has sufficient optical power to form a point image I of the point object I. Then the optical system has transformed the incident

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