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Excerpt from Field Guide to Geometrical Optics
The linearity of paraxial optics provides a relationship between the heights and angles of any two rays propagating through the system. The Lagrange invariant ( or H) is formed with the paraxial marginal and chief rays:
This expression is invariant both on refraction and transfer, and it can be evaluated at any z in any optical space, and often allows for the completion of apparently partial information in an optical space by using the invariant formed in a different optical space. Many of the results obtained from raytrace derivations can also be simply obtained with the Lagrange invariant. The Lagrange invariant is particularly simple at images or objects (y = 0) and pupils (= 0):
If two rays other than the marginal and chief rays are used, the more general optical invariant I is formed.
Given two rays, a third ray can be formed as a linear combination of the two rays. The coefficients are the ratios of the pair-wise invariants of the values for the three rays at some initial z. The expressions are then valid at any z.
Changing the Lagrange invariant of a system scales the optical system. Doubling the invariant while maintaining the same object and image sizes and pupil diameters halves all of the axial distances (and the focal length).
The throughput, etendue or ΑΩ product in radiometry and radiative transfer are related to the square of the Lagrange invariant: