1. INTRODUCTION

In our previous work [1], we developed the idea of classifying correction methods for paraxial solutions by studying the angular plane wave spectrum of the solution of the Helmholtz equation obtained from the application of the full correction method of the paraxial solution. The plane wave spectra of both the paraxial and Helmholtz solutions were compared in terms of the degree of coincidence of their Taylor series at the axial direction. As we stated earlier, it is a necessary condition that two solutions of the Helmholtz equation behave paraxially equal if the Taylor series of their angular plane wave spectra match up to the third order. This condition also holds for paraxial plane wave spectra and the reconstructed solution in order to consider a method to be valid.

The problem comes from the misuse of notation made with $\alpha$. We recall that the axial direction is identified as $\alpha =0$. In fact, the slowly varying envelope approximation is equivalent to making the paraxial optics approximation of the trigonometric functions near the axial direction [1]. It follows that, as an approximation, we continued to use the parameter $\alpha$ to identify directions in space, but in the paraxial plane wave spectrum formulation $\alpha$ is not an angle.

The solution is to define the angular plane wave spectrum for paraxial solutions in order to compare paraxial plane wave spectra to Helmholtz plane wave spectra. Once this is done, the results concerning the correction methods considered in [1] are amended.

2. ANGULAR PLANE WAVE SPECTRUM OF PARAXIAL SOLUTIONS

Any solution of the paraxial wave equation proposed in Eq. (5) in [1],

Thus, we use this change of variable in Eq. (2) and we obtain the angular plane wave spectrum representation for paraxial solutions,

3. CLASSIFICATION OF CORRECTION METHODS

Let us apply the tools obtained to the correction methods considered in [1]: the method of Couture and Bélanger [2], and Wünsche’s methods $T_1$ and $T_2$ [3]. The procedures carried out in [1] are correct if we exchange the variable “${\alpha }^{\prime }$” for the variable $q$. Only the mapping functions for the constructed Helmholtz solution are modified.

A. Method of Couture and Bélanger

In this method, the evaluation of the Helmholtz wave on the axis must be the same as the paraxial wave field. Hence, following the procedure in [1], it is obtained for this method that

Now, we can reuse the variable $\alpha$ for the angular plane wave spectrum of the reconstructed solution,

Computing the Taylor series, we obtain the classification of this method,

The method scores $N=2$ in the general case. For solutions with ${\partial }_{\alpha }F(\alpha ,\beta ){|}_{\alpha =0}=0$, it scores $N=3$, which is the case of even angular plane wave spectra (for instance, fundamental Gaussian beams are in this group of solutions).

B. Transversal Wave Field Substitution Method

In this method, a wave field in a transverse plane to $z$ is used as a boundary condition to build the Helmholtz solution. Following the procedure in [1],

But we also know that Eq. (3) must be fulfilled, and then for the transversal wave field substitution method (Wünsche’s $T_1$ operator), and thus

Consequently, for this method,

Computing the Taylor series, the classification of the method is obtained,

On account of this last expression, the method classifies as $N=1$ in the general case. Only for solutions with at least a second-order zero at $\alpha =0$ in the angular plane wave spectrum function, the method can be taken as a valid one.

C. Wünsche’s ${\mathsf{T}}_{\mathsf{2}}$ Operator

In this method, the mapping function in between $q$ and ${\alpha }^{\prime }$ is the same as in the transversal wave field substitution method, that is, Eq. (12). Hence, following the procedure,

Therefore, the computation of the Taylor series at $\alpha =0$ yields

4. DISCUSSION AND CONCLUSIONS

We see that the methods corresponding to Wünsche’s operators share a common mapping function to recover the angle in the corrected solution. For the method of Couture and Bélanger, the mapping function is different. It follows that the mapping function from angles in the paraxial framework to angles in the Helmholtz plays a key role in reconstruction methods. It affects not only the transformation of the spectrum function but also the kernel of the integral representation, that is, the wave vector directions of plane waves.

Taking this paper together with article [1], a classification scheme for correction methods of time harmonic paraxial solutions is described. Wünsche’s $T_2$ operator gives rise to a method that exceeds the reference value to consider a method to be valid. The method of Couture and Bélanger is only appropriate for some solutions, and our classification states that it is adequate for fundamental Gaussian beams although the complex source solution presents problems [4]. The method that takes a paraxial wave field in a transverse plane does not lead to a valid method in general, because some restrictions must be overcome to reach the reference value in the classification scheme. This fact is in accordance to previous results [5]. Obviously, if the wave field is well defined in a transverse plane, this method must be used since, along with Sommerfeld radiation condition, it can be considered as a well-posed problem.

Last, we recall that the scoring in the classification lies in a necessary, but not sufficient, condition. Thus, the application of a correction method must be further studied to assure a correct behavior of the solution for directions quite different from the axial one.