Transformations of spherical beam shape coefficients in generalized Lorenz–Mie theories through rotations of coordinate systems: III. Special values of Euler angles

This paper is Part III of a series of papers devoted to the transformation of beam shape coefficients under rotations of coordinate systems. These coefficients are required for the expanded description of laser beams, particularly for use in the framework of generalized Lorenz–Mie theories. In Part I of this series of papers, we presented a general formulation for the transformation of spherical beam shape coefficients through rotations of coordinate systems, under the form of a theorem of transformation. Part II was devoted to the special case of axisymmetric beams, more particularly of on-axis axisymmetric beams. With this Part III, we investigate simplifications of the general formulation for special values of the Euler angles defining the rotation. As in Part II, one of the aims is to uncover compact forms of formulae useful to speed-up computations.