## Lagrange Invariant

A fundamental law of Gaussian optics can be derived from the properties of A, B points and their conjugates A',B' (Fig. 1.16). This law which links the transverse magnification M = n'/n to the aperture angle ratio u/u', is the Lagrange invariant also known as Smith, Lagrange, or Helmholtz relationship.

Let us consider the case of a single diopter, Eq. (1.25a) may be written n1 n = n'1 n' ■ (1.30)

Now from the object point A, the ray of aperture angle u meets the x,y plane at the height x = uz and goes through the image point A' from the same height than in object space, then uz = u' z!, (1.31)

whatever the case of refractive or reflective diopter for the signs. Multiplying the above equations together, we obtain nun = n'u'n' ■ (1.32)

Considering a system with a next diopter with next medium of index n'', we could obtain in the same manner n''u'' n'' = n'u'n' and so on. The result applies to any intermediate space and to the last space - the final image space - of an optical system, hence nun = L-invariant, (1.33a)

and is called the Lagrange invariant.

With afocal systems, the aperture angles are u = u' = 0, the object and image locations are at infinity which leads to n ^ ±°° and n' ^ T°°, so the above formulation is indeterminate.

In the object space, the ray height at the optical surface is x = uz, and the field height is n = zq where q is the field angle; the substitution in (1.33a) of un by x q gives nxq = L-invariant, (1.33b)

which is the Lagrange invariant afocal form.

This invariance property entails the following consequences for focal and afocal systems having both ends in a same medium (n' = n) as in air.

• Focal ratios and linear fields of view: Consider focal reducer systems currently used in astronomy, which are instruments working at finite distances with a magnification M = n'/n such that M2 < 1.

^ If the output f-ratio is 1/M times faster than the input f-ratio, then the linear size of the output field of view is M times smaller than that of the input field.

• Afocal beams and angular fields of view: Consider an afocal system, such as a Mersenne telescope or a laser beam expander, we may define a number K = \x'/x\ expressing the diameter transformation of the conjugate beams.

^ If the diameter of the output beam is decreased by a factor 1/K, then the field angle of the output beam is increased by a factor K.

Typical examples of the Lagrange invariant consequence in astronomy are: the increased detection of extended objects by Courtes (1952) with focal reducers, and for the afocal case the well known Galilean form of refracting telescope (Fig. 1.17).

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