Introduction to Nonimaging Optics, 2nd Edition

18

Étendue in Phase Space

18.1 Étendue and the Point Characteristic Function

Here we derive the conservation of étendue from optical first principles, utilizing a reference wavefront from which we can calculate the optical path length to a given point P = (x₁, x₂, x₃). It is then possible to define a function S(P) = S(x₁, x₂, x₃) that gives the optical path length between the reference wavefront and any given point. The momentum or a light ray at point P is given by p = ∇S, where ∇ = (∂/∂x₁, ∂/∂x₂, ∂/∂x₃). Accordingly, if we now consider another point P*=(x1*,x2*,x3*), we have p* = ∇*S where ∇*=(∂/∂x1*,∂/∂x2*,∂/∂x3*).

Based on the definition of function S(P), we can now define the point characteristic function, V(P,P*)=V(x1,x2,x3,x1*