## Abstract

The insight that a beam of light can carry orbital angular momentum (AM) in its propagation direction came up in 1992 as a surprise. Nevertheless, the existence of momentum and AM of an electromagnetic field has been well known since the days of Maxwell. We compare the expressions for densities of AM in general three-dimensional modes and in paraxial modes. Despite their classical nature, these expressions have a suggestive quantum mechanical appearance, in terms of linear operators acting on mode functions. In addition, paraxial wave optics has several analogies with real quantum mechanics, both with the wave function of a free quantum particle and with a quantum harmonic oscillator. We discuss how these analogies can be applied.

This article is part of the themed issue ‘Optical orbital angular momentum’.

## 1. Introduction

The renewed interest in angular momentum (AM) of radiation fields that started 25 years ago did not arise from the discovery that optical AM exists. In fact, in the very first sentences of [1], it is stated that this is a well-known result of Maxwell’s theory, and explained in textbooks on electrodynamics. The novelty of this work consists of the explicit expressions for the density of orbital AM of Laguerre–Gaussian (LG) light beams. It was argued that these beams possess a well-defined component of orbital AM along the beam axis. By using laser beams, the effects of orbital AM transfer were expected to be readily accessible to experimental realization, in particular when cylindrical lenses are used to convert Hermite–Gaussian (HG) beams into LG ones. It was commonly known that the circular polarization of photons leads to a spin component in the propagation direction. This may be viewed as an example of the concept of helicity, well known in particle physics. However, the picture of photons carrying orbital AM in their propagation direction turned out to be intriguing, and a new field of research came up. This also stimulated the study of AM in general Maxwell fields, outside the restrictions of the paraxial approximation. In particular, the (im)possibility of a meaningful separation of optical AM into an orbital and a spin part was widely reconsidered [2–7].

In this paper, we compare the expressions for the densities of conserved quantities such as energy, momentum, spin and orbital AM in paraxial modes with the case of arbitrary modes in three dimensions. These quantities are expressed in terms of operators acting on mode functions. This gives the classical expressions for the densities a quantum mechanical appearance. The expressions simplify when the concept of the optical helicity is included in the discussion [8–10]. We consider eigenmodes of the various operators. In a classical description, the densities of these quantities as well as their integral values are always well determined, whether or not the mode is an eigenfunction of the operators. In the case of paraxial beams, we point out a general correspondence between solutions of the paraxial wave equation and wave functions of the quantum harmonic oscillator (HO) in two dimensions. This analogy explains the structure of standard basis sets of HG and LG modes, as well as the transformation between these sets. Moreover, owing to the simplicity of the dynamics of HOs, this correspondence allows simple analytical studies of light beams with special structures, such as vortices.

## 2. Modes and mode operators

A mode of the radiation field is a monochromatic solution of Maxwell’s equation in the absence of charges and currents. Because of the linearity of these equations, any solution can be expanded in terms of modes. As usual, we represent a mode in complex notation, so that the physical electric and magnetic fields are and , with frequency *ω*=*ck*. The fields can be expressed in terms of the transverse (divergence-free) complex vector potential **A**, so that
2.1Fields and potentials are transverse complex vector solutions of Helmholtz’s equation ∇^{2}**A**=−*k*^{2}**A**=−**∇**×(**∇**×**A**). Alternatively, the fields can be expressed in the electric vector potential **C**, so that [11]
2.2Note that the vector potentials are related as **∇**×**C**=−i*ω***A** or *c*^{2}**∇**×**A**=i*ω***C**. The corresponding real potentials are and .

### (a) Mode operators for angular momentum and helicity

In this paper, we shall make use of Hermitian linear operators acting on mode space. These operators are analogous to quantum operators on wave functions or spinors, although their significance is purely classical. Therefore, we avoid the use of as the quantum of AM. We denote mode operators by a caret. Vector operators analogous to momentum and orbital AM are
2.3The operator generates translations of the vector field. The separate operators and generate rotations of the **r**-dependence and the vector nature, which in general does not preserve the transversality of a mode [2,4]. Because a mode obeys Helmholtz’s equation, it is by definition an eigenfunction of , with eigenvalue *k*^{2}. Because modes are vector fields, mode operators can also have a matrix character. An example is the vector operator for spin. Its three components are 3×3 matrices acting on the vector, with matrix elements
2.4with *ϵ*_{ijk} the fully antisymmetric tensor of rank 3. The operator obeys the compact identity
2.5

Maxwell’s equations in free space are symmetric for interchanging electric and magnetic fields [4,12]. This duality symmetry can be expressed by the action of a dimensionless mode operator defined by 2.6It has the significance of a helicity operator that generates the corresponding symmetry transformation of the electromagnetic field [10,13–15]. This symmetry also suggests that fundamental conserved quantities of the electromagnetic field should be invariant under this symmetry transformation [16]. Because is the unit operator, the eigenvalues of are ±1. This implies that the two operators 2.7obey the identity . They are projection operators on the subspaces of eigenmodes of with eigenvalues ±1. These two subspaces are spanned by the plane-wave modes with right- or left-hand circular polarization. Application of the helicity operator on the fields gives the results 2.8

### (b) Density of energy, momentum and helicity

The expressions for the density of energy and momentum of the electromagnetic field are well known from Maxwell’s theory. A useful definition of the helicity density is [8,9,17,18]. These quantities are conserved, in the sense that the densities obey continuity equations, even in the presence of sources [15,19]. Here, we restrict ourselves to the case of a single mode, defined by the complex transverse vector potential **A**. Moreover, we consider only the expressions averaged over time, so that rapidly oscillating terms proportional to are omitted. In that case, the densities are conveniently expressed in terms of the helicity operator, with the result
2.9The density of AM is **j**=**r**×**p**. By using the commutator between the operators **r** and **∇**, this can be rewritten as
2.10The vector operator, generates rotations of the full mode.

Expressions for the total energy, momentum, AM and helicity of the mode are obtained by integrating these densities over space, so that , etc. This leads to the results
2.11in terms of the mode operators. In the case of *W*,**P** and **J** partial integration has been applied. These expressions have the flavour of quantum mechanical expectation values. This suggests that the classical mode functions **A**, or the electromagnetic field in general, may be viewed as a wave function of photons. In fact, this can be justified in some sense [20,21].

In a similar spirit, it is natural to distinguish eigenmodes of the helicity operator with opposite eigenvalues, which may be thought of as representing photons with opposite spin in the propagation direction. For a mode **A**_{±} with eigenvalue ±1, we find for the densities (2.9) and (2.10) the particularly simple form
2.12Note that, for helicity eigenmodes, the densities of energy and helicity are determined by the local strength of the mode function. The densities of momentum and AM are proportional to the cross product (2.5), which determines the local polarization, and has the nature of the spin density.

## 3. Exact eigenmodes of angular momentum

### (a) Spherical modes

Spherical modes arise in a natural way by considering the radiation fields emitted by oscillating multipoles [22]. Here we restrict ourselves to spherical modes without singularities. As our starting point, we take the scalar solutions of Helmholtz’s equation that arise when a plane wave is expanded in spherical harmonics *Y* _{lm}. So we introduce the scalar functions
3.1in terms of standard spherical coordinates *r*, *θ* and *ϕ*, with *C* an arbitrary constant and *j*_{l} the spherical Bessel functions, always with the argument *kr*. We denote the unit vectors in the direction of varying spherical coordinates as **e**_{r}=**r**/*r*, , **e**_{θ}=**e**_{ϕ}×**e**_{r}, and the derivatives with respect to these coordinates as ∂_{r}, ∂_{θ} and ∂_{ϕ}. These scalar functions can be transformed into transverse vector fields that solve Helmholtz’s equation by applying the vector operator . This produces the set of modes
3.2Here, we used that ∂_{ϕ}*Y* _{lm}=*imY* _{lm}. Because the operator commutes with the operator , the complex vector potential (3.2) solves Helmholtz’s equation. Moreover, it is obviously transverse, so that it indeed defines a mode. The index TE is an abbreviation for transverse electric, indicating that the electric field **E**_{klm,TE}=i*ω***A**_{klm,TE} of this mode is directed normal to **r**. The corresponding magnetic field **∇**×**A**_{klm,TE} is given by the expression
3.3

Another class of modes is generated by application of the helicity operator on the TE modes. If we define
3.4it is obvious from equation (2.8) that the corresponding electric and magnetic fields are
3.5In these transverse magnetic (TM) modes, it is the magnetic field that is normal to **r**. It is easy to check that the TE and the TM modes are normal to each other, in the sense that
3.6

The densities of energy, momentum, AM and helicity in each TM mode are obviously the same as in the corresponding TE mode. These densities can be directly evaluated from the general expressions (2.9). In the modes **A**_{klm,TE} and **A**_{klm,TM}, the densities of momentum and AM are
3.7in terms of the scalar function
3.8So in geographical language, on each sphere around the origin, the momentum density is directed eastwards, and the density of AM is directed northwards. The expression for the energy density *w* is directly found from the fields, but it is not very illuminating. It is relevant to note, however, that the densities of momentum and AM are not simply proportional to the energy density. The helicity density *h* is found to vanish in each of these modes.

Any vector operator acting on scalar functions of **r** obeys the standard commutation rules , with the orbital AM operator . This is equivalent to the identity . This means that the action of the components of the AM operator on the spherical modes is the same as the action of on the spherical harmonics *Y* _{lm}. With , one obtains
3.9for *τ*=TE or TM. Therefore, the modes **A**_{klm,τ} are eigenmodes of with eigenvalue *m*, and of with eigenvalue *l*(*l*+1).

The spherical modes are not eigenmodes of the helicity operator . A basis of eigenmodes of the helicity operator is obtained by application of the projection operators on the TE modes. The resulting modes,
3.10are orthogonal eigenmodes of with eigenvalues ±1. The modes **A**_{klm,±} are fully specified by the requirement that they are eigenmodes of the commuting operators , , and , with eigenvalues *k*^{2}, *l*(*l*+1), *m* and ±1.

### (b) Cylindrical modes

Cylindrical modes are obtained in a way analogous to spherical modes, when one starts with scalar solutions of Helmholtz’s equation in cylindrical coordinates *R*, *ϕ* and *z*, with , . As scalar functions we choose
3.11with . Here, *J*_{m} is a Bessel function with integer *m*, always with the argument *KR*. The functions (3.11) obey Helmholtz’s equation ∇^{2}*g*_{κKm}=−*k*^{2}*g*_{κKm}, with *k*^{2}=*κ*^{2}+*K*^{2}. Note that *K* is the wavenumber in the *xy*-plane, and *κ* is the component of the wavevector in the *z*-direction. Then, the vector functions
3.12have a vanishing divergence and obey Helmholtz’s equation, so that they define modes. These are generalizations of the Bessel beams, which have the remarkable property that they are free of diffraction [23,24]. However, as beams, they might be called unphysical, because the total power passing the *xy*-plane is infinite. Explicitly, one finds
3.13with *J*′_{m} the derivative of *J*_{m}. The electric field **E**_{κKm,TE}=i*ω***A**_{κKm,TE} is then restricted to the *xy*-plane, which is expressed by the index TE. The magnetic field **∇**×**A**_{κKm,TE} in this mode is
3.14The corresponding TM mode with magnetic field in the *xy*-plane is introduced by the definition , so that
3.15Again, the TE modes and the TM modes are orthogonal to each other.

When using the identities for Bessel functions,
3.16we find for the momentum density in both modes **A**_{κKm,τ}
3.17The AM density **j**=(*R***e**_{R}+*z***e**_{z})×**p** has a *z*-component
3.18The helicity density is found to be
3.19

Application of the projection operators on the modes **A**_{κKm,TE} produces cylindrical modes **A**_{κKm,±}, which are eigenmodes of the operators , , and , with eigenvalues *k*^{2}=*κ*^{2}+*K*^{2}, *κ*, *m* and ±1 [2].

## 4. Paraxial modes

### (a) Paraxial approximation

The paraxial approximation for the description of a radiation field applies when the wavevectors of the field fall within a cone with a small opening angle. This is commonly the case for laser beams. This small angle is of the order of the ratio of the components of the wavevector normal and parallel to the beam axis, which is used as a smallness parameter. To zeroth order, the complex vector potential **A** of a monochromatic light beam that freely propagates in the positive *z*-direction can then be written as the product of a plane wave and a slowly varying envelope **a** as
4.1where the vector **a** is restricted to the transverse (*xy*) plane. Here, **R**=(*x*,*y*) is the two-dimensional transverse component of the position vector **r**. In this approximation, the complex electric and magnetic fields can likewise be written as and . It follows from equation (2.1) that in the paraxial approximation the fields are related to **a** as
4.2Just as the envelope potential **a**, the fields **f** and **b** lie in the transverse plane. The vector potential is a quarter phase ahead of the electric field, whereas at each point, the magnetic field *c***b** is equal to the electric field **f** rotated over an angle *π*/2.

Because **A** is divergence-free, to first order it must have a small *z*-component, so that ∂_{R}⋅**a**+i*ka*_{z}=0, where we introduced the symbol ∂_{R}=(∂_{x},∂_{y}) for the two-dimensional gradient operator in the transverse plane. A similar argument holds for the fields **E** and **B**. So the fields **a**, **f** and **b** in each *xy*-plane determine the *z*-components
4.3

### (b) Helicity, momentum and angular momentum of paraxial beams

The helicity operator for a paraxial mode (4.1) can be expressed as . From the third equality of (2.9), one finds for the helicity density in this case
4.4We introduced the polarization parameter *σ* by the relation −i**e**_{z}⋅(**a***×**a**)=*σ***a***⋅**a**, so that *σ* measures the local polarization (or the spin per photon). At points of circular polarization, *σ*=±1, and *σ* vanishes for linear polarization. Note that the polarization may vary over the transverse plane.

Expressions for the densities of energy and momentum in a paraxial beam follow from the other equations (2.9). The density of energy and the *z*-component of momentum have a leading zeroth-order term that obeys the relation
4.5However, we are not interested in the AM arising from this photon momentum along the axis, but in the component *j*_{z} of the AM density in the propagation direction. This component arises from the transverse component **p**_{t}=(*p*_{x},*p*_{y}) of the momentum density, because
4.6According to equation (2.9), **p**_{t} originates from products of the *xy*-component of **A** and the *z*-component of , and vice versa. The transverse component of the momentum density is
4.7which is of first order. This result can be expressed in terms of the vector potential **a** and *a*_{z} by using equations (4.2) and (4.3). After some rewriting, one finds a separation in the form **p**_{t}=**p**_{phase}+**p**_{pol}, in a term arising from a phase gradient of the field, and a helicity-dependent term. These contributions are
4.8Note that the operator −i∂_{R} is the transverse component of the mode operator .

The AM density *j*_{z}=*l*+*s* of a paraxial beam is likewise separated into an orbital and a spin part. In cylindrical coordinates, the density of AM in the *z*-direction takes the form *j*_{z}=*l*+*s*, with
4.9

Equations (4.8) and (4.9) show that both the density of transverse momentum and of AM in the propagation direction can be separated into an orbital part, which is determined by the phase gradient of the mode, and a spin part, which contains the gradient of the helicity density. The spin contribution is significant where the helicity density varies appreciably. This can be roughly understood when one imagines the helicity density as consisting of local momentum circulations. As long as the density of circulations is uniform, the circulation of neighbouring points compensate each other. A net momentum density remains only owing to a gradient of the density of circulation, in a direction normal to this gradient. This picture is not unlike the net current arising from a non-uniform magnetization density. It is easy to check that , so that the spin per unit length of the beam is equal to the helicity per unit length.

Modes **a**_{±} with a uniform circular polarization *σ*=±1 are eigenmodes of as well as of the operator , both with eigenvalue ±1. Then, the helicity density is *h*=±*w*/*ω*, whereas the spin density is *s*=∓*R*∂_{R}*w*/(2*ω*).

In [1], expressions similar to equations (4.8) and (5.17) were derived for the special case of a LG beam with uniform polarization. Our results are valid for an arbitrary paraxial beam, also for a position-dependent polarization. They are derived from the general equations (2.9) in the special case of paraxial modes. The momentum and AM of paraxial beams that arise as superpositions of beams with different polarizations and different phase distributions have also been discussed in [25].

### (c) Paraxial wave equation

The paraxial approximation can be viewed as the lowest-order term of an expansion in the small paraxial parameter *δ*=1/(*kγ*_{0}), with *γ*_{0} the beam waist [26]. The zeroth order is given by equation (4.1) in terms of the envelope **a** that is restricted to the transverse plane. The two-dimensional vector **a** can be decomposed into two polarization components, which propagate independently, as described by a scalar wave equation. This equation follows from Helmholtz’s equation with the assumption that the transverse variation of the envelope function is small over a wavelength. In that case, **a** varies slowly with *z*, so that its second derivative with respect to *z* can be ignored. For a freely propagating paraxial beam, this leads to the scalar paraxial wave equation for each polarization component
4.10A paraxial beam with uniform polarization can be expressed by the two-dimensional vector **a**=**e***u*, where *u* is a solution of the paraxial wave equation, and **e** is a complex unit vector in the *xy*-plane. An arbitrary paraxial beam can always be decomposed into a superposition of two such beams with opposite uniform polarization.

It is easy to demonstrate that the energy, the orbital and spin AM and the helicity, all calculated per unit length of the beam, are invariant under free propagation [27]. The paraxial wave equation is exactly equivalent to the Schrödinger equation for the wave function of a freely moving quantum particle in a two-dimensional space. The propagation of the mode in the *z*-direction is equivalent to the time dependence of the wave function.

## 5. Paraxial beams and quantum harmonic oscillators

### (a) Hermite–Gaussian modes

An orthonormal basis of exact normalized solutions of the scalar paraxial wave equation (4.10) is given by the HG mode functions [28,29]. They are separable in Cartesian coordinates. For our purposes, it is convenient to express them in the form [30]
5.1The width *γ*, the Gouy phase *χ* and the radius of curvature *q* of the wavefronts are functions of *z*, and their *z* dependence is determined by the equalities
5.2The length *b* is the diffraction length (or the Rayleigh range, indicated as *z*_{R} in [28,29]). The Gouy phase increases by an amount *π* from to . At the focal plane *z*=0, the width *γ* takes the value . The functions *ψ*_{n}(*ξ*) for *n*=0,1,… are the real normalized eigenfunctions with eigenvalue of the dimensionless Hamiltonian
5.3for the quantum HO in one dimension. As is well known from basic quantum mechanics, their explicit expressions are
5.4with *H*_{n} the Hermite polynomials.

For a single HG mode with uniform polarization **e**, the transverse component of the momentum density can be evaluated from equation (4.8), where we substitute **a**=**e***u*_{nxny}. The only phase variation over the beam profile arises from the curvature radius *q* in equation (5.1), and we find for the momentum density arising from the phase gradient
5.5This is just the momentum density in the transverse direction which results from the curvature of the wavefronts. Before focus, for *z*<0, the momentum density points towards the axis, and after focus the momentum points outwards. The total transverse momentum integrated over the beam profile vanishes. Because this part of the momentum density is directed in the radial direction, the density of orbital AM vanishes in an HG mode.

### (b) Gaussian modes and ladder operators

The Gouy phase term in the HG modes (5.1) is proportional to the energy eigenvalue *n*_{x}+*n*_{y}+1 of the two-dimensional quantum HO. This allows us to express an arbitrary solution of the paraxial wave equation in terms of an arbitrary time-dependent solution of the Schrödinger equation
5.6of the two-dimensional HO, where we replace time by the Gouy phase *χ*. The wave functions are stationary solutions of this Schrödinger equation, and by taking linear combination of these, one obtains the most general solution *Ψ*(*ξ*,*η*,*χ*). We conclude that an arbitrary solution *Ψ*(*ξ*,*η*,*χ*) of the Schrödinger equation for the HO gives an arbitrary solution *u*(**R**,*z*) of (4.10), by the identification [31]
5.7with *ξ*=*x*/*γ*, *η*=*y*/*γ*, and where the parameters *γ*, *q* and *χ* are specified by equation (5.2) as functions of *z*.

The correspondence (5.7) is exact, and it works both ways. For a given HO wave function, we find a paraxial mode, after choosing a value for the Rayleigh range *b*, which is a measure of the focal region. The overlap of two modes is the same as the overlap of the two corresponding wave functions. In particular, a normalized mode *u* corresponds to a normalized wave function *Ψ*. Because the Gouy phase increases by an amount *π*, any mode function *u* from to can be mapped on half a cycle of the oscillator. The HG mode (5.1) corresponds to the stationary state of the HO, with *N*=*n*_{x}+*n*_{y}.

It is well known from elementary quantum mechanics that the eigenfunctions *ψ*_{n}(*ξ*) of the HO are connected by (lowering and raising) ladder operators of the form
5.8They obey the bosonic commutation rules , and the Hamiltonian (5.3) is equal to . From these rules, it can be shown that neighbouring eigenstates are connected by
5.9For the two-dimensional HO, expressions similar to (5.8) define the ladder operators and . The eigenstates *Ψ*_{nxny} of the HO can be reached from the ground state by repeated application of the raising operators and , so that
5.10

The correspondence (5.7) between paraxial modes and HO wave functions implies the existence of similar ladder operators and for the modes. As a function of *z*, the lowering operators take the form [30]
5.11and a similar expression holds for . For each value of *z*, the HG modes (5.1) are eigenmodes of the operators and , with eigenvalues *n*_{x} and *n*_{y}. It is important to note that the ladder operators do not describe the annihilation or creation of photons. They just link classical modes of different orders.

### (c) Laguerre–Gaussian modes

HO eigenstates with a circular nature can be obtained from the ground state by application of the circular raising operators
5.12Expressed in polar coordinates *ρ* and *ϕ*, these operators have an azimuthal dependence . The HO eigenstate
5.13is a linear combination of the eigenstates (5.10) with *n*_{x}+*n*_{y}=*N*=*n*_{+}+*n*_{−}. It has an AM *m*=*n*_{+}−*n*_{−}, expressed by the azimuthal dependence . The operator identity
5.14confirms this azimuthal dependence. The basis of stationary states (5.13) of the HO corresponds to the basis of LG paraxial modes. Commonly, these modes are labelled by the azimuthal mode index *m* and the index *p*=min(*n*_{+},*n*_{−}) for the radial dependence, so we indicate the LG modes as *u*_{pm}. When we use the same labels for the circular HO states, equation (5.13) leads to their explicit expression
5.15where is the generalized Laguerre polynomial of order *p* and degree |*m*|. The energy eigenvalue of the state (5.13) is *n*_{+}+*n*_{−}+1=2*p*+|*m*|+1. When applying the correspondence (5.7), this leads to the Gouy phase term for the LG modes *u*_{pm}. This is an eigenmode of the operator , with eigenvalue *m*.

For a single LG mode with uniform polarization **e**, the only phase variation over the beam profile arises from both the curvature radius *q* and the term in equation (5.15), and the momentum density arising from the phase gradient is
5.16The orbital part *l* of the AM density is found from equation (4.9), in the form
5.17In this case, the ratio of the density *l* of orbital AM and the density *w* of energy has the uniform value *m*/*ω*. One should note that the density *j*_{z} of AM as found in (3.18) has no uniform ratio with the density of energy or energy flux.

### (d) General basis sets of Gaussian modes

It is now rather obvious to find basis sets of generalized Gaussian modes. Basis sets of stationary states of the HO arise from the ground state by application of other pairs of bosonic raising operators
5.18The basis consists of the states
5.19and the corresponding paraxial modes are shape-invariant. The value of (*θ*,*ϕ*) determines a point on a sphere, and each point defines a different basis. These basis sets form a continuous transition between the HG modes for *θ*=*π*/2 (on the Equator), and the LG modes for *θ*=0 or *π* (on the poles). The sphere is analogous to the Bloch sphere for spins , and has been termed the Hermite–Laguerre sphere [32].

The notation in terms of the ladder operators demonstrates the perfect analogy between the various sets of the energy eigenfunctions of the two-dimensional HO and the various basis sets of Gaussian paraxial modes. The HG modes are analogous to the Cartesian set of HO eigenfunctions, which are just products of eigenfunctions of the one-dimensional HO in the *x*- and the *y*-direction. They have vanishing orbital AM. The LG modes are analogous to the HO eigenfunctions of energy and orbital AM. These functions factorize in cylindrical coordinates, with azimuthal dependence . The basis sets (5.19) of the HO provide a continuous transition between the HG and the LG sets. The transformation between these basis sets is identical for paraxial modes and for HO eigenfunctions. It is based on the relations (5.12) and (5.18) between the ladder operators. Substitution of these relations in the expressions (5.13) and (5.19) for the basis sets leads to explicit expressions for the transformations between basis sets. These transformations are identical for the HO states and the corresponding Gaussian modes. These transformations only couple modes where the sums of the mode indices are equal, so that *n*_{x}+*n*_{y}=*n*_{+}+*n*_{−}=*n*_{a}+*n*_{b}=*N*. For a given value of *N*, the basis sets contain *N*+1 states. Any linear combination of these HO states is stationary. For the corresponding paraxial mode, this means that the beam profile is invariant under propagation, apart from a scaling factor that is equal to *γ*(*z*). This algebraic technique of mode transformation is much simpler and more transparent than analytical methods [33,34].

The use of ladder operators is also useful in the case of astigmatic modes, as they arise in resonators with two astigmatic mirrors [35]. In that case, the corresponding HO states are no longer stationary, and the paraxial modes are not shape-invariant anymore. In the case of general astigmatism, the ellipses of constant intensity and of constant phase are not parallel [32,36]. These modes can carry orbital AM even in the fundamental mode, which contains no vortex.

The correspondence rule (5.7) combined with the algebra of ladder operators is quite convenient in many cases. As an example, we mention the dynamics of vortices in paraxial beams. For example, a vortex with charge 1 imprinted on a Gaussian beam at the point **R**_{0}=(*x*_{0},*y*_{0}) is expressed by multiplying the mode function in a transverse plane by *x*−*x*_{0}+i(*y*−*y*_{0}), or, in the HO language by *ξ*−*ξ*_{0}+i(*η*−*η*_{0}). This factor is equivalent to application of the operator , which, when acting on the ground state, gives a simple combination of circular states (5.13). The propagation of this imprinted vector in the picture of the HO can be expressed in analytical terms [31].

### (e) Gaussian modes with non-uniform polarization

When a beam with non-uniform polarization passes a polarizer, the mode profile of the outgoing beam depends on the setting of the polarizer. This means that the mode function **a** does not factorize into the form **e***u*(**R**,*z*), with a fixed polarization vector **e**. On the quantum level, this means that for each photon in the beam its polarization and its spatial degrees of freedom are entangled. Light beams with a non-uniform linear polarization and axial symmetry are widely studied. They can be generated by using liquid-crystal converters [37]. Another technique is based on spatially varying dielectric gratings [38,39]. As an example, we consider the superposition of two paraxial LG light beams with the same radial mode number *p* and opposite azimuthal mode numbers ±*m*, and with opposite circular polarizations . From the analogy (5.15) between LG modes and the HO states, it follows that such a mode pattern can be expressed as
5.20where the real function *F*_{mp}, which contains the Laguerre polynomial , does not depend on *ϕ*. The mode function (5.20) is the superposition of two components with *σ*=±1 and eigenvalue ∓*m* of the operator . The local polarization of the mode (5.20) is linear, in the direction . Because the polarization is linear, the density *h* of helicity and the density *s* of spin are zero. Along a circle around the beam axis, the polarization direction makes *m* full rotations in the positive direction. The mode function is real, apart from the phase variation owing to the curvature radius. The orbital part of the momentum density is therefore the same as in equation (5.5), whereas the density of orbital AM is zero. The directions of linear polarization as a function of *ϕ* are indicated by the black arrows in figure 1. For negative values of *m*, the polarization direction rotates in the negative direction along the circle. In the special case that *m*=1, the number of rotations is 1, and the pattern is invariant under rotation about the axis. Then, the polarization direction is always in the radial direction.

An interesting generalization is the case of a similar superposition of modes with opposite circular polarization, and *ϕ*-dependent phase terms with two arbitrary *m*-values. This gives a transverse mode function
5.21prepared in a single transverse plane, where now the azimuthal mode numbers *m* and *m*′ are arbitrary integer numbers. We omitted the *z*-dependence of the mode, because in the general case, the two terms will undergo different diffraction, so that, for different transverse planes, the radial mode functions will no longer be identical, and the combined mode will not be shape-invariant. When we extract a phase factor , the remaining real polarization vector is . The number of rotations of the polarization vector along a circle around the beam axis is now (*m*−*m*′)/2. This is a half-integer value when *m*−*m*′ is odd. The polarization pattern is illustrated in figure 2 for the cases that *m*−*m*′=±1. The overall phase factor indicates that the phase of the polarized field varies along the circle. The energy density for the beam profile (5.21) is found to be
5.22The densities of momentum and AM can be directly evaluated from equations (4.8) and (4.9). Because the density of helicity is zero for linear polarization, also **p**_{pol} and *s* vanish. The remaining densities of orbital momentum and AM are
5.23

In these examples, the local polarization is linear everywhere, with a direction that varies with the angular coordinate *ϕ*. There is also interest in beams where the polarization varies with the angular as well as the radial coordinate. When the full range of polarization states is covered, as represented by points on the Poincaré sphere, these beams have been termed Poincaré beams [40,41].

## 6. Conclusion

We have expressed the densities of momentum and AM of a mode of the electromagnetic field in terms of operators acting on mode functions, in particular the operator of helicity. We consider spherical and cylindrical modes, which are defined as eigenmodes of a set of mode operators, and we analyse and discuss explicit expressions for these densities. In the special case of paraxial modes, analogies with the quantum mechanical HO in two dimensions are helpful to identify various basis sets of Gaussian modes. LG modes are eigenmodes of the operator for the *z*-component of orbital AM. In this case, the density of orbital AM is proportional to the energy density of the mode. This suggests the picture of a local density of photons, each with a well-defined value of the *z*-component of orbital AM. In this sense, this quantity could be termed well defined [1]. However, this picture should be handled with care. The cylindrical modes are also eigenmodes of the *z*-component of the total AM (orbital plus spin), but the density of this AM component is not proportional to the energy density or to the energy flux density.

Finally, we recall that this paper is entirely based on classical Maxwell theory, even though the description uses mode operators that are reminiscent of quantum mechanics. There is no quantum indeterminacy here, and each physical quantity has a well-defined value for any mode. This is also true for the densities of the three components of spin and orbital AM and for their integral values, in spite of the fact that the corresponding operators are non-commuting. The suggestive expressions (2.11) for the integral values of conserved quantities are not taken as a starting point, but they originate from well-known expressions for the densities in terms of the Maxwell fields. In contrast, a quantum mechanical wave function in general does not define a local density of physical quantities. The ladder operators introduced in §5 do not refer to photons, but merely change the order of classical modes.

## Competing interests

The author declares that there are no competing interests.

## Funding

I received no funding for this study.

## Footnotes

One contribution of 14 to a theme issue ‘Optical orbital angular momentum’.

- Accepted July 13, 2016.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.