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2.9 Optical Invariant and Lagrange Invariant
ОглавлениеThe field angle, i.e. the angle of the chief ray and the marginal ray angles, will change as the rays propagate through an optical system. The relationship between these angles is inherently constrained by the magnification properties of the optical system in the paraxial approximation. The optical invariant is a parameter that, in the paraxial approximation, constrains the relationship between any two rays that propagate through an optical system. We now have two general rays as described by their ray vectors:
The optical invariant, O, is given by:
(2.4)
The optical invariant is, in the paraxial approximation, preserved on passage through an optical system. That is to say:
n′, h′, θ′, etc. are ray parameters following propagation.
Derivation of the above invariant is straightforward using matrix analysis.
Hence:
From (1.23) we know that the determinant of the matrix is given by the ratio of the refractive indices in the relevant media, so:
Finally we arrive at Eq. (2.5)
The optical invariant is a generalised constraint that relates system lateral and angular magnification and applies to any arbitrary pair of rays. A very specific descriptor is created when the ray pair consists of the chief ray and the marginal ray. This special case of the optical invariant is known as the Lagrange invariant. The Lagrange invariant, H is given by:
(2.6)
If we now simply evaluate H at the entrance and exit pupils where, by definition, hchief is zero, then the product nhmarginalθchief is constant. The Lagrange invariant then simply articulates the fact that the angular and lateral magnifications are inversely related. In fact, the Lagrange invariant captures a more fundamental constraint to an optical system. If the object plane is uniformly illuminated, then the total light flux emanating from the plane is proportional to the square of the maximum field angle. The proportion of that flux that is admitted by the entrance pupil is itself proportional to the square of the marginal ray height. Therefore, the total flux passing through an optical system is proportional to the square of the Lagrange invariant, H2. Thus the Lagrange invariant is an expression of the conservation of energy as light propagates through an optical system. This will become of paramount significance when, in later chapters, we consider source brightness or radiance and the impact of the optical system on optical flux flowing through it.