Abstract
In this contribution an alternative method to standard forward ray-tracing is briefly outlined. The method is based on a phase-space description of light propagating through an optical system. The propagation of light rays are governed by Hamilton’s equations. Conservation of energy and étendue for a beam of light, allow us to derive a Liouville’s equation for the energy propagation through an optical system. Liouville’s equation is solved numerically using an hp-adaptive scheme, which for a smooth refractive index field is energy conservative. A proper treatment of optical interfaces ensures that the scheme is energy conservative over the full domain.
Highlights
Standard forward ray-tracing methods for geometrical optics, based on Monte Carlo methods with bin counting, are in general not energy conservative and exhibit a slow convergence
A recently new approach is based on a phasespace description of light, where each light ray propagating through an optical system evolves according to an optical Hamiltonian [1, 2]
A light ray in phase-space is described by position q ∈ Rd and momentum p ∈ Rd coordinates, where the momentum is related to the direction of a light ray
Summary
Standard forward ray-tracing methods for geometrical optics, based on Monte Carlo methods with bin counting, are in general not energy conservative and exhibit a slow convergence. The Hamiltonian description of light allows us to apply Liouville’s theorem, where we consider a differential element of phase-space. In the context of optics such elements in phase-space are called étendue. It describes a beam of light instead of individual light rays. The basic-luminance is measured in lumen per projected area per steradian, and is a function of the distance along. This allows us to derive a Liouville’s equation for the basic-luminance, which reads. A proper treatment of the optical interfaces is needed, since Snell’s law and the law of specular reflection cause for non-local boundary conditions in phase-space, at which the momentum p is discontinuous
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