# Symmetry and Quantum Features in Optical Vortices

## Abstract

**:**

## 1. Introduction

_{z}, and energy density, w, must satisfy the condition

## 2. Electrodynamics and Quantum Features of Structured Light

#### 2.1. Field Operator Symmetries

**E**and

**B**respectively, for freely propagating light—although confined radiation may conform to mode-specific rules. For example, a linearly polarized standing wave in a laser cavity may have either even or odd spatial inversion symmetry, according to whether it supports an even or odd number of half-wavelengths between its end-mirrors.

_{2}has eigenvalues of ±1 denoting even or odd parity. Charge conjugation has little relevance for the interactions of structured light, and in considering the spatial parity of propagating light at a measurement instant in time we can focus on $\mathcal{P}$ alone.

**A**, it is most appropriate to develop a detailed theory in terms of transverse (solenoidal) electric and magnetic fields, ${E}^{\perp}$ and

**B**, respectively, developed as expansions in any complete set of electromagnetic modes [57].

**B**is itself pure solenoidal, irrespective of the gauge; the transverse label is unnecessary. These fields are rigorously related to the vector potential:

**A**, though they are not sufficient to fully define it (for it is a gauge-dependent quantity, and not an observable). It follows that the quantum field operators ${E}^{\perp}$ and

**B**have specific and opposite signatures for space and time parity: ${E}^{\perp}$ is odd and

**B**even with respect to inversion $\mathcal{P}$; both have the opposite signatures under $\mathcal{T}$. It is as a result of this difference that chiroptical effects (those that discriminate the handedness of chiral matter) are most commonly manifest in light-matter interactions entailing an interference of electric and magnetic coupling with light [55,58,59,60].

**E**, c

**B**) space [62,63], whose implementation is equivalent in physical effect to a local coordinate rotation of π/2 in physical 3D space, about an axis defined by the local Poynting vector. To such an end, it is possible to cast equations in terms of another adjunct vector field,

**C**(representing one of a potentially cascading sequence), through the relation $E\left(r,t\right)=-\nabla \times C\left(r,t\right)$ [64], using the non-zero curl as an alternative basis for representing electromagnetic properties [65]. However,

**C**and its kind again represent incompletely defined, gauge-dependent properties, and it emerges that all quantifiable expressions for all observables may correctly be secured without their involvement.

#### 2.2. Quantised Fields and Mode Expansions

**k**

_{Ω}(with wave number k

_{Ω}). Perhaps surprisingly, quantum aspects of radiation are quite commonly cast in terms of plane-wave descriptions, despite the intrinsic lack of physicality associated with their infinite transverse extent. However, for vortex structured beams such a representation is clearly inadmissible, and with a mapping ${\mathbb{R}}^{3}\to {\mathbb{Z}}^{2}\mathbb{R}$ the wave-vector space is typically redefined in terms of $k,\ell ,p$, signifying partition into axial and transverse (angular and radial) functions. Here, $\ell \mathrm{and}p$ are integers, the former designating a topological charge and the latter a secondary index denoting a form of radial distribution [67]; in the well-studied case of Laguerre-Gaussian optical beams, $\ell \mathrm{and}p$ signify the degree and order of an associated Laguerre polynomial ${P}_{\ell}^{p}$ that tempers a Gaussian radial distribution. For Bessel beams, $\ell $ denotes the order of a Bessel function of the first kind, and the secondary index is redundant [68].

**B**, which represent observables, can be cast as Hermitian quantum operators. Their explicit time dependence can be secured from the Heisenberg equations of motion, as indicated in the following. On expansion in terms of any appropriate and complete set of modes Ω, the operator representations of these operators, and the vector potential, at a position

**r**take the following forms, conveniently expressed in terms of cylindrical coordinates $r\equiv \left(z,\rho ,\varphi \right)$;

**e**

_{Ω}, the complex polarization vector, and a

_{Ω}, the corresponding photon annihilation operator; the counterpart term featuring the creation operator ${a}_{\mathsf{\Omega}}^{\u2020}$ is the Hermitian conjugate written as h.c. Also, ${\mathsf{\Xi}}_{\mathsf{\Omega}}$ is a mode normalization factor incorporating V

^{−1/2}, where V is the quantization volume V, such that the number of photons in this volume is given by ${a}_{\mathsf{\Omega}}^{\u2020}{a}_{\mathsf{\Omega}}$; ${f}_{\mathsf{\Omega}}$ is a dimensionless radial distribution function. The electric and magnetic fields both satisfy the time-independent Helmholtz equation:

_{Ω}= ω/c.

**B**. Under the inversion operation $\mathcal{P}$, which spatially inverts polarization, wave-vectors, and position vectors, the electric field has odd spatial parity, and is therefore formally representable as a polar vector; the magnetic field is represented by an axial vector (a pseudovector).

#### 2.3. Linear Momentum Density

_{0}is the vacuum permittivity; the second term is included to ensure Hermiticity of the operator, since the operators ${E}^{\perp}$ and

**B**do not commute [71]. Hence,

**P**is clearly odd in both time and space, as befits linear momentum. Its local direction serves to identify the local normal to the optical wavefront surface. The Poynting vector is not, however, anchored in the physical 3D space of linear dimensions—although, as an expression of linear momentum density, it can indeed be defined at any point or region within that space. Momentum cannot be graphically represented by a vector portrayed in physical space—only in reciprocal (momentum) space.

**B**is intrinsically transverse with respect to the local Poynting vector. For a given mode, $\left({E}^{\perp},B,P\right)$ represent a right-handed orthogonal set and their components may be used to define a locally orthogonal triad of axes, of Cartesian or cylindrical form, for example. In structured light the local Poynting vector is not simply identifiable with the direction of beam propagation [72]—its orientation may vary across the beam as illustrated in Figure 1. Exact calculations of the Poynting trajectories for both LG and Bessel beams in fact exhibit a subtle difference between surfaces of hyperboloid and cylindrical form [73]. Both ${E}^{\perp}$ 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]:

#### 2.4. Quantum Uncertainty

**k**and

**k**′ of the same magnitude k, across a range of radial distances $\rho ,\rho +\delta \rho $. It has been shown that close to the beam axis the distinguishability scales as k

^{2}$\ell $

^{−1}$\delta \rho $, i.e., inversely proportional to the topological charge $\ell $. However, for positions remote from the beam axis, the resolution limit scales with k

^{−1}$\ell $

^{2}ρ

^{−3}$\delta \rho $, quadratic in $\ell $ [42].

## 3. Angular Momentum Quantization

#### 3.1. Quantum Operators

**J**of any finite beam is given by the following volume integral:

**S**operator defined by Equation (13), and the radiation Hamiltonian. Any other polarization state may be resolved into a fractionally weighted superposition of these two circular polarizations, equivalent to a rotation of the basis space on the Poincaré sphere.

**J**=

**S**+

**L**, where

**L**is an orbital angular momentum (OAM) given by [26]:

**ε**denotes the Levi-Civita antisymmetric tensor, and repeated indices require summation:

#### 3.2. Conservation of Angular Momentum

## 4. Spatial Symmetry Aspects of Optical Vortices

#### 4.1. Polarisation States, Chirality, and Helicity

^{2}κ for monochromatic (not necessarily paraxial) light [115], the flow of chirality satisfies the following continuity equation [59,116,117,118]:

#### 4.2. Cylindrical and Rotational Symmetry

_{n}symmetry elements for all integers n, where the Schoenflies symbol S

_{n}signifies invariance under reflection with rotation about a perpendicular axis through an angle 2π/n. As is well known, the operation of spatial inversion represented by the parity operator $\mathcal{P}$ is exactly equivalent to mirror reflection in any plane, followed by π rotation about the normal to the chosen plane of reflection. (Regarding planar surfaces or interfaces, the sole requirement is a lack of reflection symmetry in any plane perpendicular to the surface).

_{R}, where ${z}_{\mathrm{R}}\approx {w}_{0}/\theta $. With the standard phasor decomposition of Equation (3), the local electric field at a position $r\equiv \left(z,\rho ,\varphi \right)$ in cylindrical coordinates can be represented as follows:

**θ**′ is the gradient of θ.

**B**. Field gradients prove to play a significant role in a wide range of quadrupole and higher multipole interactions [39,123,124,125,126,127,128,129,130,131,132]. In molecular or dielectric materials, and especially those with a chiral structure, electronic transitions may be mediated by more than one type of multipole, and through involvement of the azimuthal phasor in twisted light, the electric quadrupole form of coupling in particular introduces a range of new and interesting features. For example, it can introduce a capacity for discriminating physical media of left- or right-handed structure, as for example in angle-resolved Raman scattering of vortex light to resolve optically active molecular enantiomers [24,133,134].

**B**fields in each twisted mode, under $\mathcal{P}$, are those of the parent operators. The same applies, of course, to $\mathcal{C}$ and $\mathcal{T}$ parity signatures.

**θ**′

_{1}is a constant irrespective of the azimuthal angle ϕ. This signifies that its Cartesian components in the transverse plane change sign on π/2 rotation about the beam axis. As a property of the beam it has the properties of the irreducible representation (irrep) A

_{2}in the point group C

_{∞}

_{v}symmetry, being invariant under arbitrary rotations, and antisymmetric under reflections in any vertical plane [138].

**Φ**′

_{1}, whose complex value changes non-uniformly around any ring, so that C

_{∞}cylindrical rotation symmetry is absent. Consider rotation by π radians about the z axis: the real part of

**Φ**′

_{1}is symmetric under C

_{2}rotation in the (x, z) plane, and antisymmetric for C

_{2}rotation in the (y, z) plane; for the imaginary part of

**Φ**′

_{1}, the converse applies. These characteristics mark out Re(

**Φ**′

_{1}) as transforming under the irrep B

_{2}of the dihedral point group D

_{2}, whilst Im(

**Φ**′

_{1}) transforms under irrep B

_{3}.

**θ**′

_{2}is once again invariant under cylindrical rotations and antisymmetric under reflections; it, too, belongs to the irrep A

_{2}in the point group C

_{∞v}. Here, however, the symmetry representations under the operations of the D

_{2}point group are A for Re (

**Φ**′

_{2}) and B

_{1}for Im (

**Φ**′

_{2}). Note the difference in behaviour of both the real and imaginary parts, under C

_{2}(z) rotation, exhibited by these differing values of $\ell $.

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Phasor structure of an $\ell $ = 1, p = 0 LG beam, cycling as it propagates; hue denotes the phase, colour intensity the level of irradiance; the latter vanishes along the propagation axis (green arrow) at the phase singularity. The orientation of the local wave-vector (normal to the wavefront surface) is azimuth-dependent, as indicated at positions of opposite phase by red and blue arrows. (

**b**) $\ell $ = 3, p = 0 LG beam, showing the mutually orthogonal disposition of the cylindrical components of a local wave-vector.

**Figure 2.**(

**a**) Colour-continuum representation of the phasor Φ for a vortex at z = 0: (

**a**) topological charge $\ell $ = 1, where phase θ = ϕ; (

**b**) $\ell $ = 2, phase θ = 2ϕ (ϕ = 0 as usual designating the notional x-axis). The phase gradient field, indicated by yellow arrows (twice the magnitude in the latter case), is constant in θ. Any shift ϕ →ϕ + π imparts a change of sign to the phasor in (

**a**), as for all cases of odd $\ell $; the same shift in (

**b**) retains the sign, as is true for all even values of $\ell $. The vector field ${\Phi}^{\prime}$ of the phasor gradient varies non-uniformly in direction, its real and imaginary parts (respectively represented by blue and red arrows) having signs either parallel or antiparallel to the phase gradient, according to the quadrant in (

**a**), and the octant in (

**b**).

**Table 1.**Irreducible representations of the point group ${\mathrm{D}}_{\ell}$ and salient symmetry operations for the real and imaginary parts of the azimuthal phasor of a paraxial optical vortex, and of its vector gradient field for odd topological charge; m denotes any positive integer number of rotations.

${\mathbf{D}}_{\ell}(\mathbf{odd}\ell \ge 3)$ | E | $\ell {\mathbf{C}}_{\ell}^{\mathit{m}\le \ell -1}$ | $\ell {\mathbf{C}}_{2}$ | |
---|---|---|---|---|

A_{1} | +1 | +1 | +1 | $\mathrm{Im}{{\Phi}}_{\ell},\mathrm{Re}{{\mathbf{\Phi}}^{\prime}}_{\ell}$ |

A_{2} | +1 | +1 | −1 | $\mathrm{Re}{{\Phi}}_{\ell},\mathrm{Im}{{\mathbf{\Phi}}^{\prime}}_{\ell}$ |

**Table 2.**Irreducible representations for even topological charge $\ell \ge 4$; all other details as in Table 1.

${\mathbf{D}}_{\ell}(\mathbf{even}\ell \ge 3)$ | E | $\ell {\mathbf{C}}_{\ell}^{\mathit{m}\le \ell -1}$ | $\left(\ell /2\right){\mathbf{C}}_{2}^{\prime}$ | $\left(\ell /2\right){\mathbf{C}}_{2}^{\u2033}$ | |
---|---|---|---|---|---|

A_{1} | +1 | +1 | +1 | +1 | $\mathrm{Re}{{\Phi}}_{\ell}$ |

A_{2} | +1 | +1 | −1 | −1 | $\mathrm{Im}{{\Phi}}_{\ell}$ |

B_{1} | +1 | +1 | +1 | −1 | $\mathrm{Re}{{\mathbf{\Phi}}^{\prime}}_{\ell}$ |

B_{2} | +1 | +1 | −1 | +1 | $\mathrm{Im}{{\mathbf{\Phi}}^{\prime}}_{\ell}$ |

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Andrews, D.L.
Symmetry and Quantum Features in Optical Vortices. *Symmetry* **2021**, *13*, 1368.
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Andrews DL.
Symmetry and Quantum Features in Optical Vortices. *Symmetry*. 2021; 13(8):1368.
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2021. "Symmetry and Quantum Features in Optical Vortices" *Symmetry* 13, no. 8: 1368.
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