Poisson Bracket design method review.: Application to the elliptic bundles

Poisson bracket design method is, as yet, one of the few known 3D nonimaging concentrator design methods. In general, this method provides concentrators requiring variable refractive index media, which become quite impractical. Another fruitful 3D-concentrator study was done by application of the Lorentz geometry to the nonimaging theory. In this study some new 3D ideal concentrators (with constant refractive index) were found by the analysis of the certain sets of edge rays with some particular symmetries. These bundles were called elliptic bundles of rays. On the other hand, a new approach based on the concept of reference surface has been recently used to find new bundles of this type. In this paper we review Poisson bracket design method, reformulate it and apply it to the analysis of elliptic bundles. The basic equations have become fully symmetric after reformulation Moreover, the expression of the Hamiltonian function is of the same type as the expression of the first integral that contains the elliptic bundle. Ln this way one cannot distinguish, from the mathematical point of view which is the Hamiltonian function and which is the first integral function. The formulation must be done with some selected coordinates in which one of the coordinate lines are the flow lines of the elliptic bundle and the other two ones are normal to the flow lines and tangent to the symmetry planes of the cone of edge rays. All the elliptic bundles found with the Lorentz geometry method and the reference surface method are again found here excepting those that aren't symmetric with respect the coordinate surfaces. No new elliptic bundle has yet been found Nevertheless the new analysis procedure is more adapted to arbitrary orthogonal coordinate systems and to variable refractive index, and so it is expected it will provide new bundles.