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

Salto-Alegre, Cristina; Wolf, Kurt Bernardo

doi:10.1088/1751-8113/48/19/195202

Cited by 3 publications

(4 citation statements)

References 26 publications

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“…it behaves in many respect as if it where an orthogonal basis). This result differs from other constructions like the one used in [9][10][11][12], where they introduce a non-orthogonal, discrete-continuous, overcomplete family of states defining a non-tight frame and requiring a dual frame to reconstruct any state in terms of them.…”

Section: Generalized Fourier Transformmentioning

confidence: 70%

“…An alternative and simpler description of momentum space for non-Abelian groups, which does not rely on Pontryagin duality theory and parallels the Abelian case, is given by the Sherman-Volobuyev construction [9,10]. There, an overcomplete (and non-orthogonal) basis in configuration space and its dual are given, whose labels, both discrete and continuous, play the role of a pair of dual 'momentum spaces' [11,12].…”

Section: Introductionmentioning

confidence: 99%

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$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

Guerrero

López-Ruiz

Aldaya

2020

J. Phys. A: Math. Theor.

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We perform the momentum-space quantization of a spin-less particle moving on the SU (2) group manifold, that is, the three-dimensional sphere S 3 , by using a non-canonical method entirely based on symmetry grounds. To achieve this task, non-standard (contact) symmetries are required as already shown in a previous article where the configuration-space quantization was given. The Hilbert space in the momentum space representation turns out to be made of a subset of (oscillatory) solutions of the Helmholtz equation in four dimensions. The most relevant result is the fact that both the scalar product and the generalized Fourier transform between configuration and momentum spaces deviate notably from the naively expected expressions, the former exhibiting now a non-trivial kernel, under a double integral, traced back to the non-trivial topology of the phase space, even though the momentum space as such is flat. In addition, momentum space itself appears directly as the carrier space of an irreducible representation of the symmetry group, and the Fourier transform as the unitary equivalence between two unitary irreducible representations.

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Section: Generalized Fourier Transformmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 99%

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

Guerrero

López-Ruiz

Aldaya

2020

J. Phys. A: Math. Theor.

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“…We propose to use these to generate bases of 'momentum' and of 'position' for wavefunctions on the sphere, supported by their appearance in the contraction limit ρ, → ∞, to be seen in Sect. 6, that mantains the ratio ρ/ = λ/2π = 1/k constant, so that the functions on the sphere and fish-eye plane come to be solutions of the Helmholtz equation of wavenumber k, plane waves and localized Bessel J 0 's, whose properties in this regard have been studied in [14] and [13].…”

Section: The Momentum Basismentioning

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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“…The 2D model of the latter has a continuous compact basis (a circle) of plane waves, and also an infinite discrete basis of a line of J 0 (z) and J 1 (z)/z Bessel functions, placed a half-wavelength apart [4], [11]. For the latter, the 1D case has been examined through its iso (2) Wigner function depicted on a cylinder [13].…”

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confidence: 99%

Royal road from geometric to discrete optics

Wolf

2015

Photon. Lett. Poland

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Abstract-There are well-trodden paths between geometric and wave optics while, with sampling and interpolation, a further bridge to discrete optics is usually traversed. Our previous work points to a Royal road to link the paraxial, metaxial, and global geometric with discrete models, where canonicity becomes unitarity and where phase spaces retain their meaning. This short review maps the terrain that has been traversed.The line of work of the Óptica Matemática projects centres on the applications of group theoretical methods to geometric, wave and discrete (mostly finite) optical models, although it still lies on the fringes of the mainstream applied research in Mexico. We use the fact that when two models exhibit the same underlying symmetry group, a correspondence can be established between them. In this contribution to the monographic Special Issue we give an overview of the common symmetries and distinct realizations in the regimes of geometric and discrete optics.The geometric model of optics assumes "screens" on which a manifold of rays is characterized by their position q and their momentum p (i.e. |p| = n sin θ, with the index of refraction n and angle θ to the screen normal, allowing for p є R 2 in the paraxial régime), which satisfy the basic Poisson brackets {q i , p j } = δ i,j . Optical transformations q → q′(q,p), p → p′(q,p) due to free flight, refracting surfaces, or transit through inhomogeneous media, can only be canonical, i.e., preserve the Poisson brackets [1, ch. 3]. Canonical transformations are reversible and, since no rays are gained nor lost, they conserve information; caustics are not singularities in a phase space. In the linear (paraxial) regime, these transformations form the real symplectic group Sp(2D,R) in D dimensions. Light fields are understood as generally complex functions ϱ(q,p), whose absolute square can represent intensity.The discrete model of optics that we use consists of finite pixelated screens, where each pixel contains a complex value of the discrete wavefunction that forms the image. The pixel address is given by the equally-spaced finite spectrum of a position operator Q, and a momentum operator is defined through P:= i[H,Q], with the Lie bracket (commutator) of the generator H of the FourierKravchuk transform [2] (i.e. of a harmonic oscillatorother models with infinite discrete screens use the * E-mail: bwolf@fis.unam.mx repulsive oscillator or the free system [3-4]). The resulting structure is the unitary Lie algebra su(2) for 1D screens, and su x (2)×su y (2) = so(4) for 2D N x ×N y screens. This Lie algebra exponentiates to the corresponding unitary Lie groups of transformations. An important property of the discrete model is that, when the number and density of pixels increases without bound, it contracts to the common Heisenberg-Weyl algebra and its quantum-mechanical or paraxial wave-optical rendering of operators and wavefields, which in turn come down to the classical or geometric model when the wavelength tends to zero.In 1D, this formalism leads to ...

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

Cited by 3 publications

References 26 publications

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

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

Royal road from geometric to discrete optics

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

Cited by 3 publications

References 26 publications

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S 3

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S 3

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

Royal road from geometric to discrete optics

Contact Info

Product

Resources

About

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³

$SU(2)$ -particle sigma model: momentum-space quantization of a particle on the sphere S ³