Evaluation Of Beam Shape Coefficients Research Articles

Evaluation of beam shape coefficients in T-matrix methods using a finite series technique: on blow-ups using hypergeometric functions and generalized Bessel polynomials

In T-matrix methods (generalized Lorenz–Mie theories or extended boundary condition method), beam shape coefficients encoding the shape of the illuminating structured beam have to be evaluated. This may be carried out by using the finite series technique, which, however, generates blow-ups when the partial wave order of the beam shape coefficients increases. Using hypergeometric functions and generalized Bessel polynomials, we demonstrate in the case of on-axis Gaussian beams that these blow-ups are genuine phenomena, not due to a lack of numerical precision, and we establish criteria to evaluate the critical partial wave order that implies blow-ups.

Angular spectrum decomposition method and quadrature method in the generalized Lorenz–Mie theory for evaluating the beam shape coefficients of TEM[formula omitted] doughnut beam

The evaluation of beam shape coefficients (BSCs) which encodes the description of the illuminating beam is an essential issue in light scattering theories of spherical particles, such as the generalized Lorenz–Mie theory (GLMT). Different methods have been developed in this context, including the traditional (quadratures, finites series, localized approximations), to be complemented by use of the angular spectrum decomposition (ASD). The present paper is devoted to a comprehensive study of the relationship between the traditional quadrature method and the ASD method to the evaluation of BSCs. We shall establish that evaluation of BSCs using the quadrature method can be modified into expressions in the spectral space, leading to the same results as those obtained using the ASD. These BSCs are afterwards approximated under the paraxial conditions, leading to the same results as these obtained by using the localized approximations. The TEM0l∗ doughnut beam is taken as an example of the shaped beam and the theoretical analysis is confirmed by numerical calculations.

Modified finite series technique for the evaluation of beam shape coefficients in the T-matrix methods for structured beams with application to Bessel beams

The finite series technique is a rigorous mathematical formulation to exactly calculate the beam shape coefficients gn,TMm and gn,TEm which describe an arbitrary structured beam within the framework of the T-matrix methods for structured beams, e.g. generalized Lorenz–Mie theory or Extended Boundary Condition Method. It allows for fast computation without having recourse to time-consuming quadratures with double or triple integrals. Here, a simpler version of the finite series – the Modified Finite Series – is presented in which gn,TMm and gn,TEm are obtained by using an arbitrary choice of the spherical angle θ in the development of the finite series technique. To illustrate it, a linearly polarized vector Bessel beam is described within this new formalism. We expect that the present method will be an important alternative tool in the field of light scattering and light-matter interaction within the optical domain.

Finite series expressions to evaluate the beam shape coefficients of a Laguerre-Gauss beam focused by a lens in an on-axis configuration

Pursuing a study concerning the evaluation of Beam Shape Coefficients (BSCs) of Laguerre-Gauss beams, after having dealt with the case of beams freely propagating, the present paper establishes finite series expressions to evaluate the BSCs of Laguerre-Gauss beams focused by a lens, which have the known advantage of speeding up computations with respect to the use of numerical quadrature techniques and which may deal with the limitations of localized approximations in the case of beams exhibiting topological charges and/or axicon angles.

On the beam shape coefficients of fundamental nondiffracting beams

In generalized Lorenz–Mie theory (GLMT) applied to structured beams the evaluation of beam shape coefficients (BSCs) constitutes a challenge. In this paper, we propose to calculate BSCs of fundamental nondiffracting beams, cosine beams, Bessel beams, Mathieu beams and parabolic beams. The aim of our method is the use of the Whittaker integral related to a scalar nondiffracting beam, and well known angular spectrum decomposition. Also, we exploit the relationship between solutions of scalar Helmholtz equation to expand the BSCs of nondiffracting beams in terms of higher order Bessel Beams (HOBB) BSCs. Some numerical simulations and discussions are given to validate our results.

Evaluation of beam shape coefficients of paraxial Laguerre–Gauss beam freely propagating by using three remodeling methods

In the framework of a series of papers devoted to warnings concerning the evaluation of beam shape coefficients (BSCs) of electromagnetic beams by using a localized approximation, a recent paper provided finite series explicit expressions to evaluate BSCs of paraxial Laguerre–Gauss freely propagating beam. In the present paper, these expressions are numerically implemented and finite series numerical results are compared with results obtained by using quadratures and a localized approximation. These three methods provide a remodeling of the beam, changing the non-Maxwellian paraxial Laguerre–Gauss beam to a beam which perfectly satisfies Maxwell’s equations. Comparisons between the three methods are discussed in terms of accuracy and computational time efficiency.

Intrinsic method for the evaluation of beam shape coefficients in spheroidal coordinates for oblique illumination

When dealing with T-matrix scattering, e.g. in the framework of Generalized Lorenz-Mie Theory (GLMT) or of Extended Boundary Condition Method (EBCM), the incident electromagnetic fields are expanded over a set of vectorial wave functions, with expansion coefficients expressed in terms of beam shape coefficients (BSCs). In the case of structured beams, the evaluation of these BSCs may constitute a challenging issue, particularly in the case of spheroidal coordinates. BSCs in spheroidal coordinates have been originally evaluated using an extrinsic method, i.e. they have been expressed in terms of more conventional BSCs in spherical coordinates. Later on, it has been possible to provide expressions to evaluate spheroidal BSCs using an intrinsic approach, i.e. by working only inside the spheroidal coordinate system used to deal with the scattering problem. Fairly recently, the intrinsic method for the evaluation of BSCs in spheroidal coordinates for on-axis standard configuration has been released in the archival literature. The present paper extends the intrinsic method to the case of oblique illumination. It also serves as a review devoted to GLMTs in spheroidal coordinates.

Intrinsic method for the evaluation of beam shape coefficients in spheroidal coordinates for on-axis standard configuration

The generalized Lorenz–Mie theory for spheroids requires the evaluation of beam shape coefficients in spheroidal coordinates. Up to now, this task has been performed by using an extrinsic method in which beam shape coefficients in spheroidal coordinates are expressed in terms of beam shape coefficients in spherical coordinates. For various reasons, in particular with the aim of developing localized beam models in spheroidal coordinates, it would be desirable to possess an intrinsic method as well, in which beam shape coefficients in spheroidal coordinates would be directly expressed in spheroidal coordinates. The aim of the present paper is to provide a solution to the evaluation of beam shape coefficients in spheroidal coordinates by using an intrinsic method, for the case of an on-axis standard configuration and, for the first time, to present numerical results for an intrinsic method in spheroidal coordinates.

Solution to the intrinsic method for the evaluation of beam shape coefficients in spheroidal coordinates

The generalized Lorenz–Mie theory for spheroids requires the evaluation of beam shape coefficients in spheroidal coordinates. Up to now, this task has been performed by using an extrinsic method in which beam shape coefficients in spheroidal coordinates are expressed in terms of beam shape coefficients in spherical coordinates. For various reasons, in particular with the aim of developing localized beam models in spheroidal coordinates, it would be desirable to possess an intrinsic method as well, in which beam shape coefficients in spheroidal coordinates would be directly expressed in spheroidal coordinates. The aim of the paper is to provide the first solution, up to now available, for the evaluation of beam shape coefficients in spheroidal coordinates by using an intrinsic method.

Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz–Mie theory for spheres

A so-called localized approximation, allowing one to speed up the evaluation of beam shape coefficients in the generalized Lorenz–Mie theory for spheres, has been previously introduced and, in the case of Gaussian beams, rigorously justified. We examine and demonstrate the validity of this approximation for arbitrary shaped beams.

Evaluation of laser-sheet beam shape coefficients in generalized Lorenz–Mie theory by use of a localized approximation

The use of laser sheets is of growing interest in many measurement techniques relying on light scattering. Device performances must often be analyzed with generalized Lorenz–Mie theory, which requires the evaluation of beam shape coefficients describing the incident beams. In the case of Gaussian beams these coefficients are most efficiently computed by a localized approximation. The localized approximation procedure for Gaussian beams is here successfully generalized to the case of laser sheets.