## Abstract

Exact solutions of Maxwell's equations representing light beams are explored. The solutions satisfy all of the physical requirements of causal propagation and of energy, momentum and angular momentum conservation. A set of solutions can be found from a proto-beam by an imaginary translation along the beam direction. The proto-beam is given analytically in terms of the Bessel functions *J*_{0}, *J*_{1} and the Lommel functions *U*_{0}, *U*_{1}, or equivalently in terms of products of the spherical Bessel functions and Legendre polynomials. The complex wavefunction has rings of zeros in the focal plane. Localization of the focal region is to within about one half of the vacuum wavelength.

## 1. Introduction

There exists a considerable literature on the physically and theoretically interesting focal region of beams (scalar and electromagnetic), reviewed in the texts by Born & Wolf [1], Stamnes [2] and Nye [3]. Here we concentrate on beam wavefunctions which satisfy what the author views as the non-negotiable requirements for physical beams: (i) the wavefunction must satisfy the relevant wave equation, for example, that derived from Maxwell's equations in the electromagnetic case, (ii) seven beam invariants must exist, corresponding to the laws of conservation of energy, momentum and angular momentum, and (iii) propagation behaviour must be causal: far from the focal region, on one side the beam must propagate purely in towards the focal region, on the other side it must propagate purely outwards.

There may also be a fourth requirement: (iv) the existence of zeros in the complex beam wavefunction in the focal region. The zeros and the associated phase singularities or dislocations (Nye & Berry [4]) have been conjectured to be universally present in the focal region on topological grounds ([5], §20.1.4). The topology of the isophase surfaces is further discussed in §4.

All of these four sets of requirements are met by the family of beams discussed here, which can be obtained by an imaginary shift along the beam axis from a proto-beam, the most tightly focused of the family. The proto-beam is a generalized Bessel beam (§2); it is expressed in terms of Lommel functions of two variables and (equivalently) in terms of Legendre polynomials and spherical Bessel functions (§3). The surfaces of constant phase and constant modulus are discussed in §4, the beam invariants in §§5 and 6. A detailed examination of an electromagnetic beam which is circularly polarized in the plane wave limit appears in §7, with emphasis on its polarization properties. Section 8 defines longitudinal and lateral measures of the focal region localization; for the scalar proto-beam these lengths are of the order of one half of the vacuum wavelength. Finally, §9 outlines the properties of the family of beams which are obtained by an imaginary shift along the beam axis.

For monochromatic beams in free space, in which the time dependence of the complex fields is contained in the factor e^{−iωt}, the quantum and linearized acoustic scalar amplitudes *ψ* satisfy the Helmholtz equation
** E** and

**of an electromagnetic wave satisfy the above equation. This follows from Maxwell's equations by expressing the magnetic and electric fields in terms of the vector and scalar potentials**

*B***and V,**

*A***then satisfy (1.1) (e.g. [6], pp. 218ff), and so do their derivatives such as**

*A***and**

*E***. In particular, the TM, TE, ‘LP’ and ‘CP’ beams have their vector potentials respectively proportional to [7]**

*B*## 2. Generalized Bessel beams

The Helmholtz equation (1.1) is separable in cylindrical coordinates (*ρ*, *ϕ*, *z*): it reads
*generalized Bessel beams* [8]
*f*(*q*) can be complex; it is constrained by the necessary finiteness of beam invariants (see §§5 and 6). In [5], §20.1.1, it is shown how *ψ*_{0}(** r**) is related to the time-harmonic version of Bateman's [9] integral solution of the wave equation,

*ρ*= 0) we get

*g*in (2.3) given by the axial value of the beam wavefunction.

There is a one-to-one correspondence between (2.3) and the *m* = 0 generalized Bessel beam solution (2.2). The zero-order Bessel function containing the square root can be rewritten by using Bessel's integral (3.6), which transforms the *m* = 0 version of (2.2) into
*m* = 0 generalized Bessel beams, the Bateman amplitude function *g* is given by
*f*(*q*):

## 3. A set of exact beam solutions

We shall look at a particular form of the *m* = 0 generalized Bessel beam
*z* (real or complex), once we have evaluated (3.1) for a particular *f*(*q*) we have automatically also found a one-parameter set of solutions
*z* by an imaginary distance was used by Deschamps [10] to obtain a beam-like wavefunction from the spherical wave *r*^{−1}e^{ikr}, namely *R*^{−1}e^{ikR}, *R*^{2} = *ρ*^{2} + (*z* − i*b*)^{2}. The Deschamps wavefunction is singular on the circle {*ρ* = *b*, *z* = 0}. Removal of the singularity is possible by combination with the complex-shifted spherically converging wave *R*^{−1}e^{−ikR}, but that introduces non-physical backward propagation far from the focal region. The wavefunctions considered here are forward-propagating, by construction. There is, however, some localized vortex backflow in the focal region associated with the zeros of the beam wavefunction.

Let us consider the proto-beam obtained by setting *f*(*q*) proportional to *q*, specifically
*k*^{2} normalizes the beam wavefunction to unity at the origin: *ψ*_{0}(0,0) = 1. In the focal plane *z* = 0 we have, on changing the variable of integration to *a* and focal length *f* is of the form (3.4), but with *kρ* replaced by *kρ*(*a*/*f*), as shown for example in [1], §8.8. The beam wavefunction has an infinite number of circles of zeros in the focal plane, as discussed in the Introduction.

The form of the wavefunction on the beam axis *ρ* = 0 is also easily obtained:
*F*, *G* are expressed in terms the Bessel functions *J*_{0} = *J*_{0}(*v*), *J*_{1} = *J*_{1}(*v*) and the Lommel functions of two variables ([12], and §16.5 of [11]) *U*_{0} = *U*_{0}(*u*,*v*), *U*_{1} = *U*_{1}(*u*,*v*). It is convenient to define the auxiliary variable *w* = (*u*^{2} + *v*^{2})/2*u*, and also *C* = cos *w*, *S* = sin *w*. Then
*z* = 0, *u* = *k*(*r *− *z*) → *kρ *= *v*, so *G *→ 0 and *F* → 2*J*_{1}(*v*)/*v*, in accord with (3.4). The correspondence of the axial value *ψ*_{0}(0,*z*) with (3.5) is considered below.

The defining equation (3.3) shows that the real and imaginary parts of *ψ*_{0} have opposite parity:
*F*, *G* in terms of the radial coordinate *r* and the axial coordinate *z*. Since *u* = *k*(*r* − *z*), *v* = *kρ*,
*z*, and also manifestly a solution of the Helmholtz equation (1.1) (here and below, *P _{n}*(cos

*θ*) =

*P*(

_{n}*z*/

*r*) are the Legendre polynomials,

*j*(

_{n}*kr*) are spherical Bessel functions):

The real part requires more study: using (3.13), we can rewrite *F*(*ρ*, *z*) as:

The first term is even in *z*. To show that the second term is also even we need to prove that *H*(*ρ*, *z*), the quantity in braces, is odd in *z*. We note first that it is zero in the focal plane, *H*(*ρ*, 0) = 0, from (A5) and (A6). On the beam axis *ρ* = 0 we can use (A4) to rewrite *H*(0, *z*) as
*z* > 0, *r* = *z* (when *ρ* = 0) so (3.16) reduces to 1 − cos *kz* − *kz* sin *kz*. For *z* < 0, *r* = −z, and (3.16) reduces to −1 + cos *kz* + *kz* sin *kz*. We have thus verified that the real part *F* of the beam wavefunction is even on the beam axis. For *G* → −2 cos *θ* cos *kr*/*kr*. The asymptotic form of *ψ*_{0}(0, *z*) for large *z* is therefore that of a spherically converging or diverging wave:
*π*, relative to the plane wave e^{ikz}, in going across the focal region, first noted by Gouy [13].

In appendix B we show that in the spherical coordinates *r*, *θ*,
*z* and a solution of the Helmholtz equation. *A*_{0} = 1, because the wavefunction is normalized to unity at the origin. It is interesting that, in terms of the spherical eigenfunctions of equation (1.1), the imaginary part of *ψ*_{0} consists of just one term, while the real part is an infinite sum over products of Legendre polynomials and spherical Bessels. The expansion coefficients for *n* = 1, 2, 3,…are found in appendix B:
*F*(*r*,*θ*) for large *kr* is found from *j*_{2n}(*kr*) → (−)^{n} sin *kr*/*kr*:
*θ*| is even about *θ* = *π*/2, and can therefore be expanded in the orthogonal set of even order Legendre polynomials. Equation (3.20) can be verified from
*kr* the proto-beam wavefunction takes the form of spherically converging and diverging waves for the incoming and outgoing parts, respectively:
*θ* factor ensures that (asymptotically) there is no incoming or outgoing wave perpendicular to the beam propagation direction. However, the leading term in the asymptotic expansion is zero on the focal plane *θ* = *π*/2, and does not correspond to (3.4) at large *kρ*.

## 4. Surfaces of constant phase and of constant modulus

The proto-beam wavefunction *ψ*_{0} has the imaginary part 2*P*_{1}(cos *θ*)*j*_{1}(*kr*), and thus the surfaces on which the phase of *ψ*_{0} is a multiple of *π* is given by the zeros of *j*_{1}(*kr*), namely the spheres on which tan *kr* = *kr*. We choose the phase to be zero at the origin. The phase will be +*π* on the hemisphere *kr*_{1} ≈ 4.4934, *z* > 0, and −*π* on the hemisphere *kr*_{1} ≈ 4.4934, *z* < 0. The phase will be ±2*π* on the hemispheres *kr*_{2} ≈ 7.7252, *z* > 0, *z* < 0, and so on. Asymptotically, the *kr* values at which the phase is ±*nπ* tend to (*n* + (1/2))π. The *P* = ±*nπ* isophase surfaces can meet at the focal plane *z* = 0, since they differ in phase by 2*nπ*. All other isophase surface pairs *P*, −*P* can meet only at the zeros of *ψ*_{0} in the focal plane, which occur at the zeros of *J*_{1}(*kρ*)/*kρ*, namely

The modulus |*ψ*_{0}| is zero on the focal plane circles just discussed, and nowhere else. On the beam axis, we have from (3.5) that

All the isophase surfaces meet the beam axis vertically. On the axis the phase *P*(*z*_{0}) is found from (3.5):
*G*(*ρ*, *z*) of *ψ* = *F* + i*G*. On an isophase surface which crosses the beam axis at *z*_{0}, tan *P*(*ρ*, *z*) = tan *P*(*z*_{0}). On expanding this equation in powers of *ρ*, we find that
*ψ*_{0}(*ρ*, *z*) rise from the beam axis in proportion to the square root of |*z* − *z*_{0}|.

## 5. Invariants of the scalar proto-beam

The simplest application of the scalar beam wavefunction is to a coherent beam of spinless atoms, for which there exist seven invariants, corresponding to the conservation of particles and of momentum and angular momentum. As in [14], we define the momentum density by
** p**/

*M*, where

*M*is the mass on one atom. The conservation of particles (the continuity equation) reads

*Ψ*is an energy eigenstate and develops in time according to

*Ψ*|

^{2}= |

*ψ*|

^{2}is independent of time, and therefore ∇ ·

**= 0. From the zero divergence of**

*p***it follows that**

*p**z*. For the ‘steady’ proto-beam,

*F*and

*G*analytically, but it is far easier to evaluate

*ρ*can be evaluated using the Hankel integral formula ([11], §14.4)

*N*′ is independent of the position along the beam axis for generalized Bessel beams, but it is not in general an invariant: [14] gives a counterexample.) For weakly focused beams we expect the ratio

For the scalar beam without azimuthal dependence there is only one other beam invariant apart from *f _{i}* of the external force is to be added to the right hand side of (5.12).)

As shown in [14], the momentum conservation law leads to three beam invariants, of which only one is non-zero when *ψ* is independent of the azimuthal angle, namely

The

The invariants *z* = 0 plane all the derivatives of *ψ*_{0} may be evaluated analytically, directly from the defining integral (3.3). We shall list all up to second order, as they will be needed later.

## 6. Invariants of electromagnetic beams derived from *ψ*_{0}(*ρ*, *z*)

The concept of cycle-averaged energy, momentum and angular momentum content per unit length of an electromagnetic beam was introduced by Van Enk & Nienhuis [15], and used by Barnett & Allen [16]. Let a bar denote cycle averaging (monochromatic beams are assumed); then
*τ _{ij}* is minus the usual Maxwell stress tensor (the sign change, introduced by Barnett [17], is made so the conservation equations take the same form),

^{2}

*r*to the cycle average of the two conservation equations (6.3), we obtain the beam invariants

We shall first consider the TM (transverse magnetic) beam with vector potential ** A** =

*k*

^{−1}

*E*

_{0}[0, 0,

*ψ*], and scalar potential satisfying the Lorenz condition (1.3). For monochromatic beams

*ψ*then satisfies the Helmholtz equation (1.1), and the fields are

*h*(

*κ*) = 2

*κ*/

*k*

^{2}, which gives us

*U*′ depends on

*z*; it follows from

**→**

*E***,**

*B***→ −**

*B***. Thus the energy and momentum densities and the stress tensor are the same for the TE and TM beams, and so are the values of**

*E*In the next section we shall be considering the *steady* ‘circularly polarized’ beam, for which Lekner [8], §5 shows that

For the proto-beam, with *h*(*κ*) = 2*κ*/*k*^{2}, we find

## 7. Polarization properties of a ‘CP’ beam

We shall look at two examples of ‘circularly polarized’ beams (the quotes indicate that the beams are only truly circularly polarized in the plane wave limit). We begin with the vector potential ([7], §4)
*ψ* → e^{ikz},

The beam derived from the vector potential *A*_{1} has *B*_{1} transverse, with different polarization properties to that of *E*_{1}. Also the electromagnetic energy and momentum densities defined in (6.2) oscillate in time, at angular frequency 2*ω* = 2*ck*. It is possible to construct *steady* beams, in which the complex fields satisfy ** E** = ±i

**. These steady beams have electromagnetic energy and momentum densities which do not oscillate in time ([20], §4). Further, the polarization properties of the electric and magnetic fields are the same. The vector potential of a circularly polarized steady beam is**

*B***= ∇ ×**

*B***is**

*A***= i**

*E***, as may be verified from the free space, time-harmonic version of Ampere's Law,**

*B***= (i/**

*E**k*)∇ ×

**. In the plane wave limit**

*B**ψ*→ e

^{ikz},

The invariants of the previous section were calculated for the steady ‘CP’ beam, for which we now examine the polarization properties. For linear polarization of the electric field, the real and imaginary parts of the complex vector ** E** =

*E*_{r}+ i

*E*_{i}are collinear:

**are perpendicular and equal in magnitude:**

*E**π/ω*at each point in space. Note that for a monochromatic beam the polarization is a function only of the position.

A measure of the *degree of linear polarization* used in [7] is the dimensionless ratio
** B**, which in the case of steady beams with

**= ±i**

*E***has the same degree of linear polarization as**

*B***, since then**

*E*It is clear from (7.6) that *Λ*(** r**) = 1 on curves where

*E*_{r},

*E*_{i}are collinear and the field is linearly polarized, while

*Λ*(

**) = 0 on curves where the conditions (7.7) for circular polarization are satisfied. The topology of these curves has been studied by Nye & Hajnal [21] and Berry & Dennis [22]. The form taken by**

*r**Λ*for a steady ‘CP’ beam based on a wavefunction, such as

*ψ*

_{0}, which does not depend on the azimuthal angle

*ϕ*is calculated in [7], §4. Let suffixes

*ρ*,

*z*denote differentiations. Then

*ψ*

_{0}have been evaluated analytically in §5.

Figures 2 and 3 show the degree of linear polarization in the focal plane of our ‘CP’ proto-beam. We see from the figures that there is perfect circular polarization on the beam axis, but that *Λ* increases from zero in the radial direction, and, remarkably, attains the value unity on circles surrounding the axis, at *k*ρ ≈ 4.827, 8.040, etc..Thus, in the focal plane, the central *circular* polarization is surrounded by rings of perfect *linear* polarization. The same effect was found in a previous study of polarization in strongly focused beams [7].

It is interesting to also show, in figure 3, the energy density and the axial component of the Poynting vector. For steady beams with ** E** = i

**the energy density and the Poynting vector are both time-independent, and are given by ([20], §4)**

*B**ψ*were given in (5.15)–(5.20). Near the origin the energy, energy flux density (normalized by the same factor to unity at the origin), and degree of linear polarization are, to order (

*kρ*)

^{2}and (

*kz*)

^{2},

## 8. Measures of the extent of a focal region

The proto-beam is the limiting form of a set of exact solutions of the Helmholtz equation; it is the one with the tightest focus. There are two lengths (in general) characterizing the extent of the focal region: the transverse localization length *L*_{t}, and the longitudinal localization length *L*_{l}. There are constraints on how we can define these lengths, because beam wavefunctions decay as the inverse of the distance from the focal region: see for example (3.17) and (3.22). We shall define the longitudinal and transverse extents of the focal region of a scalar beam by
*ρ* in the integrand of *L*_{t} comes from the volume integral in cylindrical coordinates,

We begin with the example of the fundamental Gaussian mode, characterized by two parameters *k* = *ω*/*c* and *b*:
*ψ*_{G} is an *approximate* solution of the Helmholtz equation, valid only when *kb*, and also when *ρ* > *b*. Note that there are no focal plane zeros of *ψ*_{G}.] For the Gaussian beam fundamental mode we find
*b* (which also goes by the names of *diffraction length* and *Rayleigh length*), while the transverse extent or *beam width* at focus is determined by (*b*/*k*)^{1/2}.

The proto-beam on the other hand, being a limiting form, has just the one length parameter, *k*^{−1}. Hence both the longitudinal and transverse extents of the focal region will be proportional to *k*^{−1}. We find, on using the axial form (4.1) and the focal plane form (3.4) of |*ψ*_{0}|^{2} that
*k*^{−1} = *c*/*ω* = λ_{0}/2*π*, where λ_{0} is the vacuum wavelength at angular frequency *ω*, we see that the scalar proto-beam is localized in the focal region both longitudinally and transversely to roughly one half of the vacuum wavelength: *L*_{l} ≈ 0.58λ_{0} and *L*_{t} ≈ 0.43λ_{0}.

Naturally there is interest in the tightest possible focus of light beams [23,24]. The tightness of focus depends on the polarization properties of the beam, with a radially polarized beam achieving localization to about 0.40λ_{0}, whereas a linearly polarized beam achieves localization to 0.51λ_{0}. The sharpness of focus of higher order radially polarized beams is discussed in [25], and limits of the effective focal volume in multiple-beam light microscopy in [26].

In relation to the localization problem, we note that the wavelength within a beam is not λ_{0}, except far from the focal region. For example, on the beam axis we can track the phase *P*(*z*) of the proto-beam: this is the phase of (3.5), namely
*local wavelength* can be defined by the large *N* limit of *z*) ≈ (3/2)λ_{0} near the centre of the focal region of the proto-beam. The longer wavelength can be seen in figure 1, and is of course related to the cumulative phase shift of *π* across the focal region, noted below (3.17).

For vector beams there is a set of isophase surfaces associated with each of the components of ** E** and

**. There can be up to six such sets, but for steady beams for which the complex fields are related by**

*B***= ±i**

*E***, there at most three sets of isophase surfaces. Each set of isophase surfaces has associated with it a local wavelength given by λ(**

*B***) = 2**

*r**π*/|∇

*P*|.

## 9. Discussion

We have explored the properties of a particular exact solution of the Helmholtz equation, constructed to represent scalar and vector beams. These beams have the desired properties of possessing finite beam invariants, and the causally necessary property of propagation, respectively, into and out of the focal region at infinity. Far from the focal region there are no backward-propagating waves, which can exist only in the presence of reflecting surfaces.

As mentioned in §3, from the scalar proto-beam and its various vector extensions, one can construct a new set of solutions by a translation along the beam axis by an imaginary distance: *z* → *z* − i*b*. All the solutions thus obtained are still solutions of the Helmholtz equation in cylindrical coordinates, since (2.1) is invariant to translations in *z*. In making this imaginary displacement, *r* is a non-negative physical distance, we have to decide which branch of the complex function *R* to choose. We shall take *R* is a complex coordinate rather than a distance. On the beam axis it becomes the complex-shifted longitudinal coordinate *z* − i*b*. At the origin *R* → −i*b*. When we wish to indicate a distance we shall write |*R*|.

The integral form of the beam wavefunction is
*b* > 0, the normalization is different from unity at the origin:
*ψ _{b}*(

*ρ*,

*z*), (now both complex) give

*R*→ −i

*b*, and setting

*P*

_{2n}(1) = 1, we have

The asymptotic form of *ψ _{b}*(

*ρ*,

*z*) is, from (3.22), proportional to |

*R*|

^{−1}e

^{−ik|R|}for the incoming part of the beam, and to |

*R*|

^{−1}e

^{ik|R|}for the outgoing part of the beam, as is required by causality. Other aspects of the complex-shifted beams are not explored here, as we wished to concentrate on the properties of the most tightly focused proto-beam.

## Competing interests

There are no competing interests.

## Funding

No external funding was provided.

## Acknowledgement

The paper has benefited from careful reading and constructive comments by the reviewers.

## Appendix A: *ψ*_{0} in terms of Lommel functions of two variables

We shall make use of the following relations for the Lommel functions of two variables *u*,*v*, with the auxiliary variable *w* = (*u*^{2} + *v*^{2})/2*u* ([11], §16.5):
*V*_{0}(*u*, 0) = 1, *V _{n}*(

*u*, 0) = 0 (

*n*≥ 1). Thus (A 3) implies

The proto-beam is given by (3.7), which we split into two integrals *Z* = e^{iϕ}. In the first integral, we use the Jacobi expansions
*U*_{0}(*u*,*v*), and its derivative ∂_{u}U_{0}(*u*,*v*), with *u* = *k*(*r *− *z*), *v* = *kρ*. The use of (Bertilone [27], eqn (21), [11], §16.52, (2))

## Appendix B: reduction of the real part of *ψ*_{0}

The wavefunction *ψ*_{0} is a non-singular solution of the Helmholtz equation (1.1) and can thus be expanded as a sum over products of Legendre polynomials and the non-singular spherical Bessels. To evaluate the coefficients *A** _{n}* in

*B*=

_{n}*A*

_{n}P_{2n}(0). The expansion in powers of

*kr*of both sides of (B 1) are known,

*kr*)

^{2n}in the two expressions on the right-hand side of (B 1) gives

*A*is multiplied by a matrix in echelon form: we can solve the

*n*= 0 equation to find

*A*

_{0}= 1, then solve the

*n*= 1 equation (which contains

*A*

_{0}and

*A*

_{1}) for

*A*

_{1}and so on. We find

*n*= 1, 2, 3,…)

- Received July 4, 2016.
- Accepted November 2, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.