ABSTRACT
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 = (x 1, x 2, x 3). It is then possible to define a function S(P) = S(x 1, x 2, x 3) 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 ∇ = (∂/∂x 1, ∂/∂x 2, ∂/∂x 3). Accordingly, if we now consider another point P * = ( x 1 * , x 2 * , x 3 * ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315215501/e0671961-6ea5-4049-a9f3-afbe37d2d110/content/inequ18_1.tif"/> , we have p* = ∇*S where ∇ * = ( ∂ / ∂ x 1 * , ∂ / ∂ x 2 * , ∂ / ∂ x 3 * ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315215501/e0671961-6ea5-4049-a9f3-afbe37d2d110/content/inequ18_2.tif"/> .
