Radial quantum number of Laguerre-Gauss modes

Karimi, Ebrahim; Boyd, Robert W.; Hoz, P. de la; Guise, Hubert de; Řeháček, J.; Hradil, Z.; Aiello, Andrea; Leuchs, Gerd; Sánchez-Soto, L. L.

doi:10.1103/physreva.89.063813

Cited by 95 publications

(68 citation statements)

References 57 publications

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“…However, the radial index p has to be included in the mode summation, for generality and completeness of the modal decomposition [14]. Indeed, there has been a recent surge in interest in the radial term, and it has been sought to substantiate the role p can play in informatics [15][16][17]. Of course, l and p both feature in the detailed form of the LaguerreGaussian radial distribution:…”

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confidence: 99%

Quantum features in the orthogonality of optical modes for structured and plane-wave light

Andrews

Forbes

2018

Opt. Lett.

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A fundamental photon creation-annihilation commutation relation underpins the familiar quantum formulation of optics. However, an internal inconsistency becomes apparent in the pursuit of structured light applications. This requires the relationship between operator commutation and mode orthogonality to be recast in a form ensuring full consistency with the precepts of quantum theory. A suitable reformulation, shown to register correctly an intrinsic quantum uncertainty in the associated interactions, has special relevance to optical vortex physics-particularly with regard to information content-through its connection to the degrees of freedom in the associated radiation modes. In the field of quantum optics, much of the basic theory is conventionally cast in terms that are primarily designed to apply to a single mode of radiation-a single wavelength, direction, and polarization. In the most obvious extension of these principles-allowing for more numerous modes of radiationthere are usually significant intervals between the accommodated frequencies, as, e.g., in the case of frequency combs [1], or else there are substantial differences in the directions of propagation. At the heart of most of the quantum formalism, there is a simple and widely familiar boson commutation relation between photon creation and annihilation operators [2]. One physical interpretation is that a photon propagating in a given radiation mode in vacuum cannot spontaneously divert into another mode.In the sphere of optics, the creation-annihilation commutation relation is often presented in a mathematically concise form, neglecting polarization features that ought also, for generality, to be accommodated. Although these features are not commonly given attention, they supplement the wave vector information in uniquely defining each radiation mode. The commutation relation is specifically cast in terms of a binary basis, for although polarization measurements can be made at a spatially localized location with high fidelity, it is well understood that two linear polarization states whose electric vectors differ by only a few degrees cannot be regarded as orthogonal. However, for the directions of propagation associated with the wave vector, the assumption of an orthonormal basis is not so straightforward. Moreover the wave vector of light does not commonly engage with detection apparatusunless weak quadrupole attributes are engaged, which is rarely the case.The difficulties of assuming an infinitely sharp distinction between modes of similar wave vectors are thrown into sharp relief in the case of structured radiation [3], in which there can be local variation of wave vector direction within the confines of a single well-defined beam. An optical vortex, or "twisted light," constitutes a prime example, where the wave vector represents the normal to a wavefront of helicoidal form. Significantly, it has been shown that the twisted character extends down to the level of individual photons [4,5], such that any given photon in a structured beam may ...

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confidence: 99%

Quantum features in the orthogonality of optical modes for structured and plane-wave light

Andrews

Forbes

2018

Opt. Lett.

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“…Recently, however, it has been acknowledged that index p is more important than initially thought and, as a result, its role has seen a growth in interest. 25,26 The il e φ factor in the analytic signal for LG modes is consistent with the orthogonality of modes with different topological charge, reflected in the δ ′ ll factor in the photon commutation relation. It is the orthogonality of disparate modes -through either their radial or angular functions, or both -that offers the basis for determining the requisite information content in a detected beam.…”

Section: Orthogonal Modes and Quantum Commutationmentioning

confidence: 64%

Quantum issues with structured light

Williams

Andrews

2016

SPIE Proceedings

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Descriptions of optical beams with structured wavefronts or vector polarizations are widely cast in terms of classical field theory. The corresponding fully quantum counterparts often present new insights into what is physically observed, and they are especially of interest when tackling issues such as entanglement. Similarly, when determining angular momentum densities, it appears that the separate roles of photon spin and beam topological charge can only be satisfactorily addressed within a quantum framework. In some such respects, the quantum versions of theory might be considered to introduce an additional layer of complexity; in others, they can clearly and very substantially simplify the theoretical representation. At the photon level, the fully quantized descriptions of topologically structured and singular beams nonetheless raise important fundamental questions and puzzles, whose resolution continue to invite attention. Many of the mechanistic interpretations and predictions (those that appear to be supported by a true congruence between classic and quantum optical descriptions, essentially conflating electromagnetic field and state wavefunction concepts) can lead to theoretical pitfalls. This paper highlights some physical implications that emerge from a fully quantum treatment of theory.

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“…(17.3) depend on two indices, the azimuthal index ' and the radial index p. But there are also further subtleties involved in exploiting the radial distribution, related to the fact that the radial coordinate ρ ranges from 0 to 1, unlike the azimuthal coordinate ϕ, which ranges from 0 to 2π. Recently, Karimi et al [35] presented a theoretical analysis of the operator nature of the radial degree of freedom. Moreover, Karimi et al [36] have studied the dependence of Hong-Ou-Mandel interference on the transverse structure of the interfering photons.…”

Section: Fundamental Quantum Studies Of Structured Light Beamsmentioning

confidence: 99%