&lt;title&gt;Supersymmetric features of the Maxwell fish-eye lens&lt;/title&gt;

Rosu, H. C.; Reyes, M.; Wolf, Kurt Bernardo; Obregón, O.

doi:10.1117/12.231113

Cited by 2 publications

(2 citation statements)

References 7 publications

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“…Since this system is a stereographic projection of the sphere on a plane or higherdimensional flat manifold [5,6,7,8,9,10,11], [12,Sect. 6.4], the analysis remits us to the sphere, treated as in angular momentum theory through the well-known spherical harmonics, whose relevant properties are recalled in Sect.…”

Section: The Harmonic Basis On the Spherementioning

confidence: 99%

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

Salto-Alegre

Wolf

2015

J. Phys.: Conf. Ser.

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Abstract. 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.1. The harmonic basis on the sphere Phase space is a concept created in classical mechanics that applies as well in geometric optics [1]. In quantum mechanics and in wave optics, there are several approaches to phase space through the definition of the Wigner [2] and Wigner-like [3] quasidistribution functions. Those that follow Wigner's original formulation hinge on the Heisenberg-Weyl algebra of noncommuting operators of position and momentum; other formulations also rely on a supporting Lie algebra [4].In this work we start on a different path, based on the monochromatic Maxwell fish-eye wave-optical system, presenting what appear to be the natural wavefields of definite momentum and position. Since this system is a stereographic projection of the sphere on a plane or higherdimensional flat manifold [5,6,7,8,9,10,11],[12, Sect. 6.4], the analysis remits us to the sphere, treated as in angular momentum theory through the well-known spherical harmonics, whose relevant properties are recalled in Sect. 2, while the link to the fish-eye model is summarized in Sect. 3. In our case we have not a continuous, but a finite system, i.e., the Hilbert space of wavefields for which we provide the bases is finite-dimensional.The basis of functions of 'most definite momentum' are known as the Sherman-Volobuyev functions, of which we need only a finite number for fixed angular momentum in Sect. 4, because they correspond to monochromatic wavefields in the optical model. The basis of 'most definite position' presented in Sect. 5 actually involves two sub-bases, of positions and of normal

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Section: The Harmonic Basis On the Spherementioning

confidence: 99%

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

Salto-Alegre

Wolf

2015

J. Phys.: Conf. Ser.

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“…The N-dimensional Maxwell fish-eye is 'perfect' because it has a higher symmetry group SO( + N 1) as shown by Buchdahl [5], within a higher hidden symmetry SO( + N 1, 2) [6] that is conformal and canonical [7, chapter 6]. It can be thus related to the Kepler and Bohr systems that exhibit the same symmetry [8,9]; it has supersymmetric features [10], and its various versions also describe classical and quantum systems with position-dependent masses [11][12][13]. The bases of momentum and position introduced here can be plausibly corresponded with similar ones in those systems.…”

Section: Introductionmentioning

confidence: 99%

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

Salto-Alegre¹,

Wolf

2015

J. Phys. A: Math. Theor.

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In geometric optics the Maxwell fish-eye is a medium where light rays follow circles, while in scalar wave optics this medium can only 'trap' fields of certain discrete frequencies. In the monochromatic case characterized by a positive integer ℓ, there are + ℓ 2 1independent fields. We identify two bases of functions: one, known as the Sherman-Volobuyev functions, is characterized as of 'most definite' momenta; the other is new and composed of 'most definite' positions and normal derivatives for the fish-eye scalar wavefields. Their construction uses the stereographic projection of the sphere, and their identification is corroborated in the → ∞ ℓ contraction limit to the homogeneous Helmholtz medium.

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<title>Supersymmetric features of the Maxwell fish-eye lens</title>

Abstract: We provide a supersymmetric analysis of the Maxwell fisheye (MF) wave problem at zero energy. Working in the so-called R 0 = 0 sector, we obtain the corresponding superpartner (fermionic) MF effective potential within Witten's one-dimensional (radial) supersymmetric procedure.

Cited by 2 publications

References 7 publications

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

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

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

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