## Y

b) Transfer from surface v to surface (v +1)

V(v+1) = VV + dvuV c) Transfer from the last surface l to the image

Back focal distance

1 ui

Equivalent focal length (efl) f =--T

if the paraxial ray is traced for an object at infinity. The symbols are defined in Figs. 2.2 and 2.5 and the equations above, apart from dV which defines the axial distance from surface v to surface (v + 1).

2.2.4 The conventional telescope with an ocular

Although we shall afterwards be concerned only with the reflecting telescope in its various forms, it is instructive to consider the Gaussian optics of the conventional refractor with an ocular, for comparison. This is shown in Fig. 2.8. In normal use with an ocular, such a telescope is afocal both in the object and image spaces. Both these spaces are in the same medium, normally air.

The objective is represented by its two principal planes P1 and P1. For a classical "thin" doublet objective, P1 and P1 will be quite close together and the entrance pupil is also close to P1. For generality, we show it at E1, in front of the objective, which is also the aperture stop A. The field stop has the radius I1IL1 and defines the image size. Since the object is at infinity, we have a parallel incident beam, so that all incident rays for a given object field angle have the same angle upr to the axis, the angle of the principal ray through A. The image size (radius), I1IL1 = n is given by (2.31) as n = upr f1, where f1 is the focal length of the objective and upr is in radians.

P2 and P2 are the principal planes of the ocular. In normal adjustment, with a relaxed eye, the ocular will be focused such that the eye focuses on infinity, giving an afocal emergent beam which implies that all rays associated with a given image direction are parallel.

From the construction of Fig. 2.1 for the principal planes of a total optical system, we see that, in this afocal case, the rays r1 and r2, which are incident parallel to the axis, also emerge parallel to the axis. The principal planes of the total afocal system are therefore at infinity

It is important to remember that, in the Gaussian (paraxial) region, all angles are defined by the first term of the Maclaurin expansion of (2.14). There is then, by definition, no difference between the natural angle, the sine and the tangent. The principal planes are correctly represented as planes to this approximation. As soon as the next term in (2.13) or (2.14) is considered, this is no longer true. This will be encountered with the sine condition in Chap. 3.

The exit pupil of the afocal telescope is at ag , the paraxial image of A. If A were placed at P1, the exit pupil would shift slightly to bg. If A were placed at F1, the objective would be telecentric and the exit pupil would shift slightly in the other direction to cg. However, because of the afocal exit beam, the diameter of the exit pupil is independent of its position and is solely determined by the diameter of the axial exit beam. This is the property that was recognized by Ramsden, giving the name Ramsden disk to the exit pupil - see Fig. 1.8.

The essential property of such an afocal telescope is therefore beam compression: the incident parallel beam of diameter 2y1 is compressed into an emergent parallel beam of diameter 2y2. From Fig. 2.8 it is clear that the beam compression factor C is given by

if E1 and E2 express the diameters of entrance and exit pupils respectively, E2 being the diameter of the Ramsden disk.

In such a conventional telescope with an ocular, another essential property is the magnification m. For an infinitely distant object and image, m is defined simply as the viewing angle under which the image is viewed relative to the object:

Consider now the ray rp in Fig. 2.8 as one of the rays with angle upr of the incident beam forming the image point IL1 and also passing through the object focal point F1 of the objective. After refraction, rp must pass through IL1 parallel to the axis and is refracted by the ocular through the axial point cg, also with angle upr because of beam parallelism. Now the heights of rp at the objective and ocular are equal, so we have f\upr = f\Upr = f2Upr , (2.42)

since P2cg = f2, the focal length of the ocular. From (2.41) and (2.42)

showing that the magnification is negative and an image inversion has taken place in such a conventional, Kepler-type telescope with a positive ocular. Also

C = V1/V2 = P111 /l1 P2 = f1 /f2 = -f1 /f2 , (2.44)

or or

The beam compression ratio is therefore identical to the magnification and can, in practice, by measurement of the diameter of the Ramsden disk, give an accurate determination of m. This has important practical consequences in the visual use of telescopes in general. For example, an expensive 10 x 50 binocular designed for low light levels and bird-watching will have m =10 and an aperture |Ei| = 50 mm, giving an exit pupil diameter |E2| of 5 mm according to (2.40). If it is used in bright sunlight, the observer's eye pupil will shrink to 2 mm (or less). This effective exit pupil is projected back to the entrance pupil and diaphragms the objective to 20 mm (or less), 16% of its area, thereby wasting an expensive instrument.

The same danger exists in visual astronomical observation. With a field of faint stars, an eye pupil of 5 mm is quite possible; but if a bright object such as Jupiter or the moon is observed, the pupil will probably hardly exceed 1 mm diameter. If an ocular is used with a modern 3.5 m telescope on such an object, a magnification of 3500 is required to compress the beam down to 1 mm diameter for passage into the eye. Such a magnification is higher than is normally acceptable for atmospheric seeing conditions, since it increases the normal resolution of the eye of about 1 arcmin to about 1/60 arcsec, so a lower value is preferred. This gives better views but diaphragms off the telescope. William Herschel was well aware of these problems (see RTO II, Chap. 4) and accordingly used magnifications as high as possible with his larger telescopes.

The eye pupil problem is one of the main reasons why visual judgement of larger telescopes is notoriously dangerous and usually too optimistic. Oculars are no longer used for professional astronomical observation, but are often applied for technical purposes, above all during set-up. The focal length of the ocular to avoid diaphragming must be f2 < e2Ni , (2.46)

where E2 is the estimated eye pupil diameter and N1 the f/no of the objective or primary image-forming system.

The magnification and beam compression laws of (2.44) and (2.45) are related to thermodynamics in that they are concerned with the energy transfer conditions. We mentioned above in § 2.2.3 that the energy transferred is proportional to H2, where H is the Lagrange Invariant. Ignoring absorption or reflection losses, the capacity of a telescope to transfer energy is defined by the "Throughput" or "Light Transmission Power" as defined by Hansen [2.8][2.9][2.10]:

Here, n is the refractive index, y2 a measure of the area of the pupil, r/2 a measure of the area of the image, f1 the distance between them, the focal length, and k a constant. The LTP has the dimensions [L]2.

The Lagrange Invariant for the case of an afocal telescope can be deduced at once from (2.43) and (2.44) since

for a telescope in air. In the general case, with object and image media n and n , we have n upry2 = nupryi , (2.50)

in agreement with Eq. (2.30), there given in notation for a single-element transfer for object and image distances tending to infinity.

Above we have considered the normal case of the afocal telescope with the eye focused on infinity. Of course, it is possible, though more tiring for the observer, to focus the ocular further in, giving a diverging emergent beam. The final (virtual) image can then be observed at any desired distance down to the minimum distance of distinct vision, normally 25 cm. The angle uF of the ray rp after refraction by the ocular remains unchanged, but the other angles, including upr will change somewhat because of the divergence of the beam.

Alternatively, the ocular can be focused further out, giving a converging emergent beam. This can no longer be focused by the eye, but the technique is widely used by amateurs in order to project a real image, such as that of the sun, on to a screen. This case is instructive in comparison with the reflecting telescope forms treated in § 2.2.5 below. In Fig. 2.9, let the plane P1 represent an objective so thin that its principal planes fall together, and P2 a similar plane for the ocular. Let the latter be defocused outwards to produce a converging beam and a real image at I2. If the emergent ray r is projected to meet the projection of the incident ray r, parallel to the axis, the plane P where they intersect is the image-side principal plane of the total system, which is no longer at infinity. The effective focal length is f . Since it is measured from P to I2, it is negative. The focal length f will determine or

Fig. 2.9. Image principal plane in the defocused telescope of Fig. 2.8, producing a real image at I2

Fig. 2.9. Image principal plane in the defocused telescope of Fig. 2.8, producing a real image at I2

the scale (see § 2.2.6) of the real image I2. The smaller the defocus, the smaller will be ur2 and the larger the value of f , which can be made far larger than f1 = P1I1, the focal length of the objective. In other words, the defocused telescope becomes a strong telephoto objective, a property identical with that of reflecting telescopes of the Gregory and Cassegrain forms. We shall see that the case of Fig. 2.9 corresponds exactly to the Gregory telescope.

The telephoto effect of Fig. 2.9 arises from the fact that the defocused ocular largely neutralises the power of the objective. It is easily shown [2.3] that the total power K of two separated thin lenses in air is given by

where K1 and K2 are their powers and d the separation. In air, therefore

giving 1/f = 0 if d = f1 + f2, the afocal case of Fig. 2.8. If d is slightly increased, a small negative balance of power remains giving a long negative focal length. In the next section, we shall consider the more general form of Eq. (2.52), applicable to mirrors.

The object space principal plane P, corresponding to P for the image space in Fig. 2.9, may be constructed by tracing a ray parallel to the axis backwards into the ocular end of the system. P will lie to the left of P1 giving a positive value of f so that, in air, f = —f , as required by (2.21).

Finally, it should be mentioned that the defocused Galileo-type telescope, using a negative ocular inside the objective focus, is the similar equivalent of the Cassegrain reflector. In this case, f2 in (2.52) is negative. The properties of the Galileo-type refractor can be demonstrated by the analogous construction to Fig. 2.8. The length is more compact with d = f1 — f2 for the afocal case, but the exit pupil is virtual and on the opposite (objective) side of the ocular from the image space where the eye must be placed (see also Fig 1.3). It is therefore impossible to place the eye pupil at the exit pupil, giving a severe field limitation. The Kepler-type, giving an accessible exit pupil at the Ramsden disk, is therefore the definitive form for most purposes, though the Galileo-type still survives, because of its compactness, in opera glasses.

### 2.2.5 Basic forms of reflecting telescope

The modern astronomical telescope is a reflector, in its normal function a large photographic objective forming a real image for some detector. We are concerned here with the Gaussian properties of the basic forms of reflecting telescope.

2.2.5.1 Prime focus telescopes with a single powered mirror. The simplest forms are the Herschel and Newton forms, using only the prime focus of a concave primary mirror (Fig. 2.10). From the point of view of

Gaussian optics, these forms are identical. The optical power provided by the "thin" concave primary is the equivalent of a "thin" convex lens as objective. The pole of the primary embodies both principal points and is normally the aperture diaphragm and both entrance and exit pupils. In the Newton case, this implies that the flat must be large enough to accept the defined angular field of the detector. The sag of the mirror surface is meaningless in paraxial terms. In large modern telescopes the direct axial focus point I1 can be used directly without the Newton flat since obstruction problems are no longer serious. The inclination of the central incident beam in the Herschel case has no significance in Gaussian optics, although it is very important if the higher terms (aberrations) are considered.

Consider now (Table 2.1) the signs of the basic paraxial parameters in the ray-tracing scheme of Eqs. (2.36) to (2.38) for direct imagery by reflection of an object at infinity. Strict adherence to this sign convention is essential for the understanding and correct use not only of the Gaussian relationships but also the entire aberration theory of Chap. 3.

Surface |
Positive quantities |
Negative quantities |

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