Elegant Laguerre–Gaussian beams—formulation of exact vector solution

In photonic applications of optical beams, their transverse cross-section should be often narrow, with a diameter in their waist of the order of one wavelength or even less. Within this range, the paraxial approximation of beam fields is not valid and standard corrections by field expansions with respect to a small parameter are not efficient as well. Thus, still there is a need for more accurate beam field description. In this report, an exact vector solution for free-space propagation is given in terms of elegant Laguerre–Gaussian beams. The analysis starts from the known paraxial field approximation and next, through bidirectional field transformation and application of a Hertz potential leads to an exact vector solution. The role of the paraxial solution in construction of the exact solution is elucidated. The method works well not only in cases of free-space propagation but also in description of beam interactions with planar interfaces and multilayers.


Introduction
Although the notion of elegant or complex-valued Gaussian beams was introduced by Siegman a long time ago [1,2], they are still considered mostly as the scalar paraxial approximation of the Helmholtz equation solution. Their nonparaxial counterparts were mainly obtained in terms infinite series in powers of a small parameter yielding only successive corrections to the paraxial solutions [3,4]. On the other hand, the set of the elegant Laguerre-Gaussian (eLG) beams of arbitrary order constitute a complete and biorthogonal base for electromagnetic fields, carry finite energy per unit length along their propagation direction and are expressed entirely by elementary functions. Their attractive properties, closely related to orbital angular momentum of light [5,6], were demonstrated in the context of optical focusing, trapping and manipulations of nanoelements [7,8].
Several nonparaxial attempts to treat exactly the beam propagation problem, based on the grounds of a scalar wave equation or on a full set of Maxwell's equations, were already reported, for example in [9][10][11][12][13][14] to name a few. Moreover, other different techniques, those based on Bessel functions or on nondiffractive beams, were reported as well [15][16][17][18][19]. In addition, an independent technique, which is based on a bidirectional transformation, was also presented in the past in constructions exact solutions to the propagation problem of localized electromagnetic pulses or focus wave modes [20][21][22][23][24][25]. After introducing some modifications and extensions to basics [20,21] of this bidirectional technique, it is applied here in derivation of a new exact solution to the problem of beam propagation. It is stipulated that the beams are not accelerating. Note that the most reports mentioned above are of theoretical nature. Still, they show certain potential in considerations on recent progress in photonic technology [26][27][28][29][30].
In this paper, an exact bidirectional vortex beam solution is derived in an analytic closed form. Hertz potentials are created by direct use of eLG beams of arbitrary order. Starting points of this analysis are reports [31,32] on paraxial beams and [33,34] on nonparaxial beams. Beam propagation in free-space was considered in [31][32][33][34], meanwhile both cases of normal and oblique incidence upon planar interfaces or multi-layered structures were accounted for in [31][32][33]. The beams are scaled in the transverse and longitudinal dimensions. Scalar, threedimensional paraxial eLG beams are defined by applying complex derivatives [35] to the fundamental Gaussian. Their exact versions are obtained by creating separable beam solutions in the frame of bidirectional coordinates [36]. Next, from many available possibilities of using Hertz potentials [37], two of them, equal each other and oriented along the same direction of beam propagation, are chosen in construction exact vector components of the eLG beam.
Finally, the electromagnetic field of the exact vector eLG solution is obtained in a square-integrated form of finite power flow. The solution is complete and expressed only by standard elementary functions, without the need of referring to any approximation or infinite field expansion. Each one from its transverse magnetic (TM) or transverse electric (TE) solutions comprises two transverse and one longitudinal mutually orthogonal eLG beam ingredients. Their amplitudes are distinguished uniquely by the ratio of the transverse and longitudinal field scales. The solution is exact but still in phase front planes of the exact eLG beams they replicate the standard paraxial eLG beams. In the case of incidence on planar layered structures, the transverse field components satisfy a transmission matrix equation specified by Fresnel coefficients of the scattering structure. Thus, the solutions obtained are valid not only for the description of beam propagation in free-space, but also for the analysis of beam interactions with planar interfaces or multilayers, including both cases of normal and oblique beam incidence. To the best of the author's knowledge, the solution presented here is new.
The paper is organized as follows. After a short introduction given in section 1, coordinate scaled notations are specified in section 2. Scalar eLG paraxial beams are described in real and complex variables in sections 3 and 4, respectively. Scalar and vector exact eLG beams are defined by the bidirectional transformation in section 5 and by use of the Hertz potential in section 6, respectively. In section 7, the analysis of beam-interface interactions is presented and illustrated by results of numerical simulations. Finally, the main characteristics of the eLG beam solutions obtained are summarized in section 8.

Notation and scaling
In this analysis, the spatial variables x, y and their momentum counterparts k , x k y transverse to the z-axis of beam propagation direction are scaled by a transverse scaling parameter-a cross-section radius or half-width w w of a cylindrically symmetric beam placed at its waist plane = z 0: Similarly, using a diffraction length = z kw , D w 2 the spatial variables z, ct and their longitudinal counterparts-wave number k and frequency w = kc are scaled by z D : on the grounds of (2a). The scales w w and z D specify a paraxial parameter: small for wide (paraxial) beams and large for narrow (nonparaxial) beams. As was shown in [33,34], this parameter distinguishes between two beam field ingredients which appear in the exact vector beam field solution. Optical beams will be considered monochromatic µe ikct and propagating, unless otherwise stated, in a linear, transparent, isotropic and homogeneous medium specified by its characteristic admittance Y and impedance Z. However, after normalization of electromagnetic field: the beams will be understood as propagating in free-space. Note that the scaling changes a form of wave equations. For example, an amplitude U of a scalar beam field = -V Ue ikct will be governed by the scaled Helmholtz equation: In the following, the Cartesian (ˆˆˆ) e e e , , , x y z cylindrical r j (ˆˆˆ) e e e , , z and circular (ˆˆˆ) e e e , , R L z polarization frames will be interchangeably used.

Scalar paraxial eLG beams in real coordinates
Let us start with the eLG beam  G p l , as solution to the paraxial (Fock) equation: The beam is labeled by radial p and azimuthal ±l indices, where p and l are non-negative integers. The 'upper' and 'lower' signs in ±l indicate the right-handed and left-handed orbital angular momentum of beam vortices, respectively. This solution is expressed by the product: In spite of the presence in (6b) the beam complex radius scaled = + ( ) ( ) v z iz 1 1 2 the beam envelope depends only on the complex argument and so on. The beam complex radius ( ) v z is the only quantity in the beam field solution (6)-(9) which depends on z. It defines, through the relation changes of the beam radius w and the phase front curvature R appearing in the course of beam propagation along the z-axis.
Contrary to the vortex factor j  e , il the beam envelope Q p l , does not depend on signs of the azimuthal index ±l or of the azimuthal angle j  . The amplitude normalization of all the eLG functions is introduced by the condition = g 1 imposed on the Gaussian beam field amplitude at the centre of its waist plane.

Scalar paraxial eLG beams in complex coordinates
On the other hand, there exists another, although completely equivalent, definition of the eLG paraxial beams [31]. This definition seems to be more suitable in further analysis of the problem. Let us introduce new complex coordinates with single vortex factors f  e i and j  e i explicitly present in their definitions in the configuration and momentum domains, respectively: . y x In these new coordinates Helmholtz and Fock equations read: Note that the same notation is used for functions dependent on V + and Vinstead of x and y. The Gaussian beam The eLG beams of higher orders , , , 14 , , . 14 p l p l , 2 These definitions of the eLG beam  G p l , and its envelope = are equivalent to those given in (5) and (6).
create new eLG beams with their order increased to creates new eLG beam of the order + + p l 2 2and so on. Moreover, through the Fourier transform: in the momentum or spatially spectral domain is given by: , , , 17 , , , 17 and Gaussian beam = k -ĝ e .
v 2 2 Note that the definitions (17) are valid not only for positive but also for fractional or even negative values of the radial index p.
In addition to the creation new eLG beams of higher orders shown in (15), the multiplication them by vortex factors f  e i and j  e i 2 also creates new eLG beams but this time of the same order [31]. The definition (14) implies in the configuration domain: In the momentum domain, identities equivalent to (18) are obtained for the Fourier transformed eLG beams defined in (17). That implies substitution in (18) the terms  G p l , and f for  G p l , and j, respectively. The interesting case in (18) seems this where the radial index changes its sign. For example, for = p 0 and = l 1 the identities of (18a) read: These cases will be discussed more deeply in section 7. Note that identities (18) and (19) are equally valid for paraxial and nonparaxial eLG beams. The identities (18) and (19) describe the effects of interactions between eLG beam fields and additional vortices nested in them. They are particularly suitable in considerations on solutions of beam propagation in free-space as well as on beam interactions with interfaces or multilayers. Note that the identities (18) follow exactly the original relations (10) and (11) given in [34] in the momentum domain. Contrary to what I previously suggested in the Erratum to [34], all the relations, (10) and (11) given in [34] and (18a) and (18b) presented here, are valid and sufficient in derivations of the solutions considered in this paper for positive as well as for negative values of the beam field topological charges ±l.

Scalar exact eLG beams
The scalar eLG solution presented in section 4 is valid only for paraxial beams. But still, an exact version of this solution can be obtained as well by bidirectional modification of this solution. To this end, let us start, per analogy to the problem discussed in [36], with the transform of the coordinates z and ct into the new bidirectional coordinates z − and z + : The first definition in (20) together with a new definition of the complex radius squared of the beam field envelope: The new field solution is governed exactly, without any (paraxial) approximation, by the scalar wave and Fock equations: All of that results in the bidirectional exact extension of the previously paraxial approximation of the wave equation. The monochromatic field V expressed by (21) and the eLG beam field  G p l , expressed by (22) are now exact solutions to both wave equation (24) and Fock equation (25), respectively. Note that the ansatz  -+ z ct z z , , applied in equations (21)-(25) is in the order opposite to that of  + -z ct z z , , applied in the derivation of focus wave modes [20][21][22][23].
Let us look closer into the main features of this exact solution. A boundary value problem is assumed here at the moment = ct 0 and at the waist plane This condition is exactly the same in its form as that for the approximate paraxial solution. Moreover, for ¹ ct 0, although the beam phase front plane is placed, as it was in the paraxial case, at = -= -z z ct 0, the position of the beam waist plane now is not constant, but is moving backward to its new position at = + = + z z ct 0. Therefore, these two planes are moving in the opposite directions along the z-axis and the distance between them is always 2z. At the phase front plane defined by = -z 0 and = + z z 2 the beam radius (24) of the exact solution is given by to approximate Fock equation (13). In other words, at phase front planes of the beams, the exact scalar eLG solution reduces to the well known paraxial scalar eLG solution.

Vector exact eLG beams
In section 5, the derivation route from the paraxial scalar eLG beam solution to its exact scalar version was presented. However, the exact vector beam solution should satisfy a full set of Maxwell's equations:  In the circular frame (ˆˆˆ) e e e , , R L z , these compositions read [34]: with the magnetic field given from the duality principle. Application of the beam definitions presented in section 4 and the relations between transverse (polar) parts of the cylindrical and circular polarization frames:^^^^= field components. Meanwhile, both transverse and longitudinal field components possess the same topological charge equal ±l, the charges attributed only to the eLG beams in the transverse TM or TE fields equal  + ( ) l 1 and are larger or smaller by one than that of the longitudinal beam component. As expected, divergence of all these field solutions is null. In spite of the opposite sign in amplitudes in the TE solution, the difference between the eLG beams in the paraxial and nonparaxial parts of the solution reduces only to the increment of the radial index by one.
In solution (33), three mutually orthogonal exact scalar eLG beams of different mode orders contribute separately to the transverse and longitudinal components of the vector eLG beam. Positions of their phase front and waist planes are specified by zeroes of the bidirectional longitudinal coordinates z − and z + , respectively. The impact of beam propagation on the beam spatial shape is specified completely by values of the beam complex radius v dependent only on z + as shown in (27). The longitudinal TM field component - does not depend on the paraxial parameter f . On the contrary, each transverse field component of TM or TE polarization consists of two mutually independent-paraxial -ingredients, with their amplitudes specified by the paraxial parameter f and phase shifted by

= --
The solutions -  (18), are building blocks of paraxial, nonparaxial and longitudinal parts of the exact vector eLG beam solution, respectively. Note that, in spite of their names, these parts of the total exact solution are also exact. Owing to completeness and biorthogonal features of these beams, their compositions span the whole space of physically accessible modes of finite power flux for free-space propagation. Moreover, it appears that the same set of the eLG beams defines also normal modes of planar interfaces or multilayers of the hosting medium [31,32]. This will be shown in the next section.

ELG beams at planar interfaces
Let us consider now the canonical problem of beam interaction with a planar interface between two semi-infinite isotropic and homogeneous media. The interface is assumed transverse to the incident beam propagation direction along the beam axis and is specified in the momentum domain by Fresnel transmission t p and t s coefficients for TM and TE polarization, respectively. The parameters t p and t s are dispersive as they depend on the ratio h q q = / cos cos 2 1 of cosines of refraction and incidence angles θ 2 and θ 1 , respectively. Meanwhile the longitudinal field componentˆ˜( The second term of this matrix implies excitation of the eLG beam with polarization orthogonal to that of the incident eLG beam, with the beam amplitude multiplied by the factor h -( ) t t p s 1 2 and with the radial p and azimuthal l indices changed by 1 and ±2, respectively. This effect of the crosspolarization coupling between orthogonal transverse beam components disappears for normal incidence in the momentum domain, that is where h -= t t 0.

R L ,
The case of external reflection is considered here with the contrast of refraction indices equal to 1.5. The radius w w of the incident beam waist is equal to one wavelength. In this case, the field intensity modifications due to the dispersion of the coefficients t p and t s still are not strongly significant. For comparison with theoretical predictions and by analogy to the analysis presented in [31], a few examples of eLG beam field distribution are presented below.
Results of numerical simulation of transmitted fields excited at the interface by incidence of the beamê G L 0,1 are presented in figure 1 in the configuration domain. This incidence deserves attention because it shows reverse of the sense of phase rotation under the beam interaction with the interface.
The field intensity distributions of G 0,1 and 1, 1 Moreover, the phase changes by p are also present for the beam -G 1, 1 at points of the circle ofr equal close to 2, as shown in figure 1(d).
The analogical case of incidenceê G R 0,1 is presented in figure 2. Similarly to what was presented in figure 1, the field intensity distributions shown in figures 2(a) and (b) follow also the definitions (37) of G 0,1 and -G , 1,3 respectively. However, the sense of the phase rotation presented in  In the numerical simulations presented above, only one beam G 0,1 of circular polarization-left-handed or right-handed -is incident upon the interface and excites two transmitted field components of the circular polarization. However, on the grounds of the transmission matrix form (36)   and in the configuration domain: In other words, the beam fields of polar polarization fulfilled equation (40) are composed of the beam fields of the circular polarization fulfilled equation (38). One example of such composition is presented in figures 3 and 4. The beam fields are shown in the momentum domain for the incident beams rẽ G 0,1 and ĵẽ G 0,1 of radial and azimuthal polarization, respectively. Note that in figure 3(b) the amplitude of the eLG beam ĵẽ G 0,1 is less than 5×10 −34 . Thus, it approximates zero field in the momentum domain in a quite high precision. This numerical approximation of zero corresponds to the exact zero field component indicated by the upper non-diagonal corner of the transmission matrix r j ( ) T , defined in (39). Moreover, the speckle pattern vivid in figure 3(d) corresponds to the same zero field component. It seems that this purely numerical effect is reminiscent of speckle patterns formed during light propagation through disordered multiply scattering media [30]. Figure 3 shows that, for incidence of the beam rẽ G 0,1 of radial polarization, no beam ĵẽ G 0,1 of the orthogonal, azimuthal polarization is excited at the interface. Similarly, as is seen in figure 4, for incidence of the beam ĵẽ G 0,1 of azimuthal polarization, no beam rẽ G 0,1 of orthogonal, radial polarization is excited at the interface. Its absence is indicated by zero placed in the lower non-diagonal corner of r j The discussion presented in this section indicates that the numerical simulations eLG beam interactions with the dielectric interface confirm closely theoretical predictions based on the equations (14)- (19) and (36)- (42). These predictions are valid not only for normal incidence of the eLG beams, but also for their oblique incidence including the cases of Brewster, critical and grazing incidences. Under oblique incidence many interesting phenomena triggered by the cross-polarization coupling [31], like beam spatial reshaping, shifting, splitting and switching, appear [32,38]. Theoretical predictions of this coupling were subsequently followed by additional numerical simulations of beam propagation and refraction phenomena [39,40]. Analogical results can be also obtained for the exact eLG beams interacting , .
x y with any planar layered dielectric structure, provided that the transmission coefficients specific for such the structure are used in the equations (36), (38)-(40).

Conclusions
Scalar and vector representations of exact eLG solutions are presented. The solutions of both types are given in a form of normal modes for free-space beam propagation as well as for beam propagation in stratified media. Transverse TM and TE field components of the vector solution are composed of two still exact parts-paraxial and nonparaxial-distinguished in their amplitudes by the factors f −1 and f +1 , respectively, where the paraxial parameter f is the ratio between transverse and longitudinal scale parameters of the beams. Total field of the exact solution is defined, in spite of the additional vortex factor, by three independent scalar eLG beams being exact solutions of Helmholtz and Fock equations. At phase front planes of the vector solution, the paraxial and nonparaxial parts of the eLG beams replicate in their form the beam solutions known from conventional approximate analysis. It seems that the presented solution could be useful in analyses of narrow beam interactions with horizontally patterned photonic structures, especially with metasurfaces, metalenses and other components of flat optics [41][42][43][44][45][46]. , .
x y