Position and momentum bases on the sphere for the monochromatic Maxwell fish-eye

30th International Colloquium on Group Theoretical Methods in Physics

Salto-Alegre, Cristina^1 ; Wolf, Kurt Bernardo^1

Instituto de Ciencias Fisicas, Universidad Nacional Autonoma de Mexico, Av. Universidad s/n, Cuernavaca, Morelos

62251, Mexico^1

关键词: Finite dimensional systems;    Finite systems;    Monochromatic light;    Normal derivatives;    Optical medium;    Refractive-index distribution;    Spherical harmonics;    Stereographic projection;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/597/1/012067/pdf
DOI  :  10.1088/1742-6596/597/1/012067

来源: IOP

【 摘 要 】

The sphere is well understood manifold, as is the basis of spherical harmonics for functions thereof. A stereographic projection of the sphere realizes the Maxwell fish-eye, an optical medium whose refractive index distribution is such that light rays travel in circles. Definite angular momentum on the sphere corresponds with monochromatic light in the fish-eye. A contraction of these manifolds (as the radius of the sphere grows without bound) results in a plane medium of wave functions subject to the Helmholtz equation. In the latter, there is the continuous basis of 'momenta' (plane waves) and a denumerably infinite basis of 'positions' and 'normal derivatives' (Bessel ∼ J0(x) and ∼ J1(x)/x functions) for the Hilbert space of these wavefields. We present the pre-contraction of these bases to the finite-dimensional systems of fish-eye medium and the sphere. The 'momentum' basis is a subset of Sherman-Volobuyev functions; in this paper we propose new 'position' and 'normal derivative' bases for this finite system. The bases are not orthogonal, so their measure is non-local, and here we find their dual bases.