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

Abstract

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 ...

Symmetry and Quantum Features in Optical Vortices

“…Both E ⊥ and B then have components parallel to the beam axis, and a component pointing inwards. Their relative magnitudes are locally determined, at a radial displacement ρ from the beam axis, by an angle expressed as follows [42]:…”

“…inwards. Their relative magnitudes are locally determined, at a radial displacement  from the beam axis, by an angle expressed as follows [42]:…”

“…It has been shown that close to the beam axis the distinguishability scales as k 2 ℓ −1 , i.e., inversely proportional to the topological charge ℓ. However, for positions remote from the beam axis, the resolution limit scales with k −1 ℓ 2  −3 , quadratic in ℓ [42].…”

“…Quantum uncertainty also obviates the precise positional registration of any individual photon's angular momentum [41]. It is now appreciated that optical modes with marginally different directions of propagation cannot be discriminated with arbitrary precision [42]. The core of an optical vortex is not a precise singularity; it has an effectively finite width [43,44].…”

Kramers-Heisenberg dispersion formula for scattering of twisted light

“…In Eq. (4), C | | p is a normalization constant and L | | p is the generalized Laguerre polynomial of order p. The fundamental annihilation-creation operator commutation relation for twisted light takes the form [40] [a (η)…”