Abstract
The (N -1)-dimensional Maxwell fish-eye is an optical system with an SO(N-1) manifest symmetry and a SO(N) hidden symmetry, but also an SO(N,1) potential group and the SO(N, 2) group of the (N-1)-dimensional Kepler and point rotor systems. The optical Hamiltonian is proportional to the Casimir invariant. We use a stereographic map extended to a canonical transformation between the two phase spaces of the rotor and the fish-eye. The groups permit a succint ‘4π’ wavization that shows that the constrained system can support only discrete light colors and that it unavoidably has chromatic dispersion. Elements of the potential group relate the fish-eye asymptotically to free propagation in homogenous media.
Two points \(\vec q_1\) and \(\vec q_2\) are conjugate when they are antiparallel and their magnitudes relate as g1g2 = p. They are the stereographic images of a pair of antipodal points on the sphere of radius p.
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Frank, A., Leyvraz, F., Wolf, K.B. (1981). Potential group in optics: The maxwell fish-eye system. In: Dodonov, V.V., Man'ko, V.I. (eds) Group Theoretical Methods in Physics. Lecture Notes in Physics, vol 382. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54040-7_95
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DOI: https://doi.org/10.1007/3-540-54040-7_95
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