19 November 2001 Elliptic bundles in homogeneous refractive index media: toward the general solution
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Abstract
The first 3D ideal concentrators that were found composed solely of mirrors had the property of transmitting elliptic bundles. These are the flow line concentrator (FLC) and the cone concentrator (CC), designed in the late 1970s. More recently, the Lorentz geometry formalism was applied to the problem of finding elliptic bundles in a medium of homogeneous refractive index. In this approach, the edge rays of the elliptic bundle were identified with lightlike curves in Einstein's gravitational theory. A restrictive condition was imposed in this approach: the edge rays were forced to be geodesics of the Euclidean metric and of the Lorentz metric. This restriction provided a tool for getting results, and new elliptic bundles were found. Later, by application of a series expansion from the bundle defined at a reference surface ,it was proven that other solutions exist, and thus the condition imposed in the Lorentz geometry approach was shown to be too restrictive. However, it was not demonstrated that the reference surface approach is general either. A subset of the reference surface solutions was also recently found using the Poisson Bracket design method in curvilinear coordinates which provided a deeper insight in the properties of these bundles. In this paper we present a new approach that leads to the equations which must be fulfilled by all the possible elliptic bundles in an homogeneous medium. This approach is based on the application of the Poisson Bracket design method in Cartesian coordinates. The already-known elliptic bundles are identified as particular solutions of the general equations. The search of new solutions is open, and the condition that must be fulfilled by them is given.
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Pablo Benitez, Juan Carlos Minano, Manuel Gutierrez, "Elliptic bundles in homogeneous refractive index media: toward the general solution", Proc. SPIE 4446, Nonimaging Optics: Maximum Efficiency Light Transfer VI, (19 November 2001); doi: 10.1117/12.448814; https://doi.org/10.1117/12.448814
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