On the validity of localized approximations for Bessel beams: All N-Bessel beams are identically equal to zero

Highlights

• Localized approximation has been used to evaluate BSCs of Bessel beams.

• N-beam procedure fails to provide a localized approximation for Bessel beams.

• Localized approximation should be used only for small axicon angles.

Abstract

Localized approximation procedures are efficient ways to evaluate beam shape coefficients of a laser beam. They are particularly useful when other methods are ineffective or inefficient. Several papers in the literature have reported the use of such procedures to evaluate the beam shape coefficients of Bessel beams. Relying on the concept of N-beams, it is demonstrated that care must be taken when constructing a localized approximation for a Bessel beam, namely a localized Bessel beam is satisfactorily close enough to the intended beam only when the axicon angle is small enough.

Introduction

When describing a laser beam for use in light scattering theories such as various Generalized Lorenz–Mie Theories (GLMTs), e.g. [1] or the Extended Boundary Condition Method (EBCM), e.g. [2], [3], in spherical coordinates, electric and magnetic fields are expanded in terms of a complete set of Vector Spherical Wave Functions (VSWFs). Expansion coefficients of such expansions may be expressed in terms of sub-coefficients named beam shape coefficients (BSCs) $g_{n,\mathit{TM}}^m$ and $g_{n,\mathit{TE}}^m$ [4], which reduce to simpler expressions for an on-axis axisymmetric beam, denoted as g_n and named special beam shape coefficients, and eventually to a simple constant phase term in the case of an on-axis plane wave [5]. Several methods have been designed to evaluate the BSCs, from quadratures as in the original formulation, e.g. [6], to the numerically most efficient localized approximations. A review on localized approximations is available from [7], to be complemented by [8], [9]. The most general and complete rigorous justification of localized approximations in spherical coordinates is available from [10] and, in the present paper for the sake of simplicity of terminology, is named the localized approximation, omitting other versions which are either very close to it or even equivalent [9]. In contrast with earlier versions which were devoted to Gaussian beams, the localized approximation is valid for “arbitrary shaped beams” which are specifically defined in Section 2. Now, the literature contains several papers which apply a version of the localized approximation to the case of Bessel beams. Unfortunately, Bessel beams are not within the class of “arbitrary shaped beams” considered in [10]. In the present paper, we show that localized approximations cannot be derived by this method for the case of Bessel beams. Relying on the concept of N-beams introduced in [10], our demonstration is of a very general nature and points to the essence of the problem. A companion paper will be devoted to specific Bessel beams [11].

The present paper is organized as follows. Section 2 recalls the procedure used in [10], introduces the definition of “arbitrary shaped beams”, the concept of N-beams, and recalls the main results, namely the way to be used to derive the localized approximation of a beam. Section 3 considers a general class of Bessel beams, points out that such beams are not within the class of “arbitrary shaped beams” previously defined, demonstrating that all N-beams of the Bessel type are identically zero, preventing us from designing a localized approximation for Bessel beams using this procedure, and constituting an alert concerning the use of localized approximations for Bessel beams. Section 4 is a discussion aiming to the interpretation of the result obtained in the previous section. Section 5 is a conclusion.