Abstract
Two basic problems in optics are presented. The solutions to both problems are formulated in terms of the associated ray mappings. An alternative formulation based on a weighted sum of the actions along the rays is derived. Existence of solutions is established via the Weighted Least Action Principle. Numerical methods for computing the ray mappings are discussed. Finally, we demonstrate the theoretical considerations by presenting complete solutions to a phase retrieval problem and to a specific beam shaping lens design.
Highlights
Ray mappings are fundamental objects in geometrical optics
The ray mapping condition for the beam shaping lens, that was derived in an earlier work by us, is Rubinstein et al Journal of Mathematics in Industry (2018) 8:6 presented in Sect
7 Discussion and conclusions The notion of ray mapping was presented in the context of two canonical optical problems
Summary
Ray mappings are fundamental objects in geometrical optics. Its Hamilton’s point eikonal function includes the information on all possible ray mappings induced by it. In many applications one looks for specific ray mappings that satisfy certain constraints. In the first case we search for all ray mappings associated with a single monochromatic beam whose intensity is known at two planes. This question, known as the phase from intensity problem, arises in many applications ranging from astronomy [18, 24] to ophthalmology [12]. Beam shaping has many applications ranging from solar energy to chip manufacturing and medical instruments [6]
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