Fermat’s principle and conservation of 2D etendue

Juan Carlos Minano; Pablo Benitez

doi:10.1117/12.560754

29 September 2004 Fermat’s principle and conservation of 2D etendue

Juan Carlos Minano, Pablo Benitez

Author Affiliations +

Proceedings Volume 5529, Nonimaging Optics and Efficient Illumination Systems; (2004) https://doi.org/10.1117/12.560754
Event: Optical Science and Technology, the SPIE 49th Annual Meeting, 2004, Denver, Colorado, United States

Abstract

Application of the Stokes theorem to the conservation of the 2D etendue of a one-parameter bundle of rays leads to the Lagrange's integral invariant, one consequence of which establishes that the integral &sh;p∙dx between any two points is independent of the path of integration (p is the ray vector field and x is the vector position), and more generally, the integral &sh;p∙dx between two wavefronts is independent of the path of integration. This integral is called the optical path length. This is another way to see Fermat's principle. The conservation of 2D etendue is a property of any Hamiltonian system. Using the Hamiltonian formulation, there is no difference between the configuration variables x and their conjugates p. Thus an integral invariant &sh;x∙dp can also be established similar to the Lagrange invariant. We show how its application to simple cases leads to Cartesian-oval designs through an unconventional method. The 2D etendue conservation is connected with Levi-Civita's anormalita function and with the ray equation. In this connection we found that the equation p×(∇×p)=0 suffices for a vector field to be a ray vector field.

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Juan Carlos Minano and Pablo Benitez "Fermat’s principle and conservation of 2D etendue", Proc. SPIE 5529, Nonimaging Optics and Efficient Illumination Systems, (29 September 2004); https://doi.org/10.1117/12.560754

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