Astigmatic Gaussian beams: exact solutions of the Helmholtz equation in free space

Exact solutions of the 3D Helmholtz equation in free space are presented in the form of a superposition of plane waves. The solutions asymptotically reduce to general astigmatic Gaussian beams and include no backward waves.


Introduction
Theoretical descriptions of localized wave propagation demanded in the theory of lasers, have been initially presented for the time-harmonic (monochromatic) regime by means of the parabolic-equation approach (see, e.g. [1][2][3][4][5][6][7]). The latter allows for various types of paraxial Gaussian beams which are approximate solutions of the Helmholtz equation const 0, 1 xx yy zz 2 that is widely accepted as a simple model for the Maxwell equations. Paraxial solutions were presented for axisymmetric and general astigmatic fundamental modes, and for a rich variety of respective higher-order modes. It was immediately understood that paraxial solutions do not exactly satisfy the Helmholtz equation. This was impressively demonstrated recently in [8] where paraxial solutions on the beam axis and in the focal plane were thoroughly studied in the case of axial symmetry. A challenging problem of derivation of exact solutions of the equation (1), asymptotically showing the same localized behaviour as the paraxial Gaussian beams, has been addressed by several researchers exclusively for the case of axial symmetry. To the best of our knowledge, the studies preceding [9] and [10] describe solutions that either do not satisfy equation (1) in the whole free space or involve a backward propagating wave. The most important example of a research of the first group is the theory of 'complex source' developed after pioneering papers [11,12] and [13] in both monochromatic and non-monochromatic cases. Singularities due to which these solutions do not satisfy the basic equation (1) in the whole space are described in detail in [14,15]. As an example of an exact solution satisfying (1) everywhere but involving a backward wave, we mention a nice simple solution found in [16]. In contrast to [16], a discussion of this solution given in [15] is not based on 'sinks and sources in complex space'. A more detailed discussion of the relevant literature can be found in [9,10] where the required axisymmetric solutions of the Helmholtz equation (1) in the whole space were presented as rather complicated expressions containing double integrals. The construction given in [10] rests on a tricky formula due to Bateman and seems to essentially employ the axial symmetry. The techniques of [9] is based on the classic expansion in plane waves and is generalizable to general astigmatic modes.
It is worthy of notice that simple exact localized non-time-harmonic solutions of the wave equation in the whole space-time, such as Gaussian beams and Gaussian packets, were presented quite a long time ago (see, e.g. pioneering papers [17,18] and an extensive review [19]). Relatively recently these results were extended to the general astigmatic case in [20].
In the current paper, we present a solution of the Helmholtz equation (1) in the whole free space, having a general astigmatic Gaussian-beam asymptotics and not involving a backward wave. We find it in the form of a classical superposition of plane waves, nontrivially generalizing the techniques developed in [9] for the axisymmetric case. An alternative approach to construction of axisymmetric exact solutions of the equation (1) in the whole space, having a Gaussian-beam behavior, was given recently in [10].
The paper is organized as follows. We start by summarizing the results of the approximate parabolicequation approach. Then we consider an exact solution presented by a superposition of plane waves with an arbitrary smooth weight function. We find the weight via matching these solutions in a far-field area where  ¥ kz . Finally we demonstrate that the related exact solution has the desired astigmatic Gaussian-beam asymptotic behaviour at large negative and moderate values of kz.
In what follows we deal with 2D vectors denoted by small bold letters (e.g. p, r) and with 3D vectors denoted by capital letters (e.g. R, N). The scalar product of 2D vectors f and g will be written as f g T , whereand T stand for complex conjugation and transposition, respectively. The 3D scalar product of F and G will be denoted by T . Throughout the paper I is standing for the imaginary part.

Paraxial fundamental astigmatic Gaussian modes
We shortly summarize the common approximate theory of monochromatic Gaussian beams based on the parabolic-equation method.

Paraxial wavefield at a moderate longitudinal distance
We suppress the time-dependence factor e − iωt and consider localized wave propagation along the z-axis. The standard parabolic-equation approach (which can be traced back to Leontovich and Fock, see [1]) starts by singling out the plane-wave oscillation along the preferred direction, i.e., via the substitution = = ( ) ( ) u e w w w x y z k , , , ; , 3 ikz followed by discarding the term w zz in the exact equation w xx +w yy +w zz +2ikw z =0. We arrive at the socalled parabolic equation The wavefield is assumed to be localized near the z-axis, that is, Equation (4) has a solution satisfying the paraxiality condition (5), T (see, e.g. [3,6,7]) where G( ) z is a symmetric matrix with a positive definite imaginary part, = ( ) x y r , T . We call a real symmetric 2×2 matrix A positive definite and write A>0 if r T Ar>0 for any  Î r 2 , r¹0, which is equivalent to the assertion that both eigenvalues of A are strictly positive. For definiteness, the square root in (6) is assumed to have positive imaginary part.
The resulting expression for approximate solution of the Helmholtz equation (1), is commonly known as the fundamental general astigmatic Gaussian mode [3,7,21]. It describes a beam running along the z-axis and Gaussian-localized at each cross-section with respect to the distance = + | | x y r

Paraxial far-field
It is well known (see, e.g. [19,22]) that at a large longitudinal distance, i.e., as  ¥ kz , the plane wave e ikz in (3) does not correctly describe the leading oscillation of the beam. There, the expression (7) requires modification and must be matched with spherical waves, outgoing and incoming, respectively, The patterns F ± (N) are smooth functions of angular variables where R=(x, y, z) T and = = + 2 . It is convenient to employ classical spherical coordinates which we introduce as follows: sin sin , cos , 13 sin cos , sin sin , cos . 14 T We will use notation for patterns F ± (N)=F ± (χ, j). It will be seen that patterns F ± are not negligibly small only in a neighborhood of the z-axis, where Consider the expression (7) for large values of | | z . Under condition (15), we have The identity (8) implies Under the assumption that (15) holds, the amplitude and the phase in (7) become: wherer is the following 2D vector: r sin cos , sin sin sin cos , sin . 20

T T
To summarize, we found that in the paraxial area described by (15), the wavefield (6) matches with outgoing spherical wave (10) as  ¥ kz and with incoming spherical wave (11) as kz→−¥, respectively, where the patterns are^Ĝ = - By paraxial fundamental astigmatic Gaussian beam we mean a matched asymptotic expression equal to incoming spherical wave (11), (22) for large negative z, to the parabolic-equation expression (7) for moderate z and to outgoing spherical wave (10), (21) for large positive z. The area where (7) is valid overlap with areas of validity of representations of the wavefield by spherical waves (see, e.g. [22]). Throughout the paraxial area, this matched wavefield propagates solely in the directions close to that of the positive z-axis, with no backward waves.
3. Gaussian localization of the far-field with respect to the angle χ. Large parameter Consider the paraxial expression for outgoing spherical wave in more detail. Let b 1 and b 2 be the eigenvalues of the positive definite matrix G -I 0 and . 23 1 2 Under the paraxiality condition (15), we obviously have for  ¥ kz :^ which shows that the pattern F + is strongly localized in the vicinity of the positive direction of the z-axis if Essentially, the parameter b k is the large parameter of the paraxial asymptotic theory describing astigmatic Gaussian beams. The wavefield (10) sufficiently differs from zero only for small χ (more precise, where ò is an arbitrarily small fixed positive number). Similarly, as kz→−¥.

Exact theory based on expansion in plane waves
Any solution of the Helmholtz equation (1) 4 can be presented as a superposition of plane waves: Here, á ñ R P , is a scalar product of a vector R=(x, y, z) T and a unit real 3D vector P, = | | P 1 2 , that parameterizes points of a unit sphere over which integration proceeds, and dS(P) stands for its area element. The weight function A(P) is an arbitrary generalized function.
Let us parameterize the integrand in (26) sin sin cos cos cos . 28 The desired weight function will be found by asymptotic matching of the integral (26) and the paraxial solution in the far-field zone. Consider the asymptotics of the integral (26) as  ¥ kR . We assume that the weight A(P) is infinitely differentiable on the unit sphere, which allows for application of the standard stationary-phase techniques (e.g. [24][25][26]). The integral asymptotically reduces to a sum of contributions of critical points of the phase á ñ R P , on the sphere . Such a calculation is described, e.g. in [9]. It proves that the phase has two critical points where N is given by (14). As  ¥ kR , the integral in equation ( are the contributions coming from ±N (see, e.g. [9,19]). For large kz, u + and u − describe outgoing and incoming spherical waves, respectively. For z>0, they are forward and backward propagating ones. Since we aim at finding a solution with no backward wave, the function A(N) must vanish where the projection of N on the direction of the z-axis is negative.

5.
Matching at  ¥ kz and small χ. The weight Following the approach developed earlier for the axisymmetric case in [9], we will match the functions given by equations (10), (21) and (29)