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Fig. 2.5. Derivation of the Lagrange Invariant

Fig. 2.5. Derivation of the Lagrange Invariant or

Let the height of the ray defining the aperture be y. Because they are conjugate planes of unit magnification, the height y is the same at both principal planes. Then u = —y and u = — ■yr , (2.26)

ss bearing in mind the Cartesian sign convention for the angles. Combining with Eq. (2.25) we obtain nun = nun = H , (2.27)

H being the Lagrange Invariant. In a system in a vacuum (or air) with n = n, the invariant reduces to u' n' = un (2.28)

This relation is of great importance in practical work with telescopes. If, for example, a "focal reducer" is put into the emerging beam of a telescope to reduce the emerging f/no from f/8 to f/4, then u in (2.28) is doubled and the image size n must be halved. So long as normal (centered) optical systems are used, the Lagrange Invariant will always apply. The connection with thermodynamics is clear from the fact that the total flux collected by an optical system from a uniformly radiating object surface is proportional to H2, as we shall see below.

In telescope optics, afocal systems have great significance. An afocal system has an image or object, or both, at infinity. In the case of telescopes, the term normally implies that both are at infinity. If we use the forms of Eq. (2.26) for u and u to give y and y , and the equivalent forms n = uprs and n = uprs (2.29)

for n and n , where upr and u^r are the angles subtended by the object and image over the distances s and s respectively (see Fig. 2.6, where |s| and js'j tend to infinity for the afocal case, and also Eq. (2.50)), then substitution in (2.27) gives the afocal form of the Lagrange Invariant:

This will be demonstrated in the next section in the specific case of an afocal telescope. We shall see that an afocal telescope has its principal planes at infinity.

The suffix (pr) of upr and upr in Eqs. (2.29) and (2.30) actually refers to the principal ray, which is defined relative to the entrance and exit pupils of a system. Only in the afocal case, where all rays of the entrance and emergent beams are parallel, is it possible to ascribe the field angles upr and upr to the rays incident on P and emerging from P , except in the rare cases where the entrance and exit pupils are coincident with the principal planes. With the general definition of upr and upr related to the pupils, Eq. (2.29) is only true in the afocal case.

The entrance pupil of an optical system is some aperture which limits the diameter of the light beam entering the system for all field angles which the field stop allows to pass. The field stop limits the area of the object or image. Because of the axially symmetrical nature of centered optical systems, the limiting aperture is normally circular or an approximation thereto. Figure 2.6 shows this schematically. A is the aperture stop, E the entrance pupil,

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