The problem of calculation of the light field eikonal function providing focusing into a prescribed region is formulated as a variational problem and as a Monge-Kantorovich mass transportation problem. It is obtained that the cost function in the Monge-Kantorovich problem corresponds to the distance between a point of the source region (in which the eikonal function is defined) and a point of the target region. This result demonstrates that the sought-for eikonal function corresponds to a mapping, for which the total distance between the points of the original plane and the target region is minimized. The formalism proposed in the present work makes it possible to reduce the calculation of the eikonal function to a linear programming problem. Besides, the calculation of the "ray mapping" corresponding to the eikonal function is reduced to the solution of a linear assignment problem. The proposed approach is illustrated by examples of calculation of optical elements for focusing a circular beam into a rectangle and a beam of square section into a ring.
We propose a method for designing refractive optical elements for collimated beam shaping. In this method, the problem of finding a ray mapping is formulated as a linear assignment problem, which is a discrete version of the corresponding mass transportation problem. A method for reconstructing optical surfaces from a computed discrete ray mapping is proposed. The method is suitable for designing continuous piecewise-smooth optical surfaces. The design of refractive optical elements transforming beams with circular cross-section to variously shaped (rectangular, triangular, and cross-shaped) beams with plane wavefront is discussed. The presented numerical simulation results confirm high efficiency of the designed optical elements.
We consider a method for designing freeform mirrors generating prescribed irradiance distributions in the far field. The method is based on the formulation of the problem of calculating a ray mapping as a Monge–Kantorovich mass transportation problem and on the reduction of the latter problem to a linear assignment problem. As examples, we design freeform mirrors generating a uniform irradiance distribution in a rectangular region and a complex chessboard-shaped distribution. The mirror generating a rectangular irradiance distribution is fabricated and experimentally investigated. The experimental results are in good agreement with the numerical simulations and confirm the manufacturability of the mirrors designed using the considered method.
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