Rayleigh limit of generalized Lorenz-Mie theory for on-axis beams and its relationship with the dipole theory of forces. Part II: Non-dark axisymmetric beams of the second kind and dark axisymmetric beams, including a review

•Optical forces exerted on-axis axisymmetric beams are studied.•The beams are non-dark of the second kind and dark beams.•The framework used is the Rayleigh limit of the generalized Lorenz-Mie theory.•The relationship with the traditional dipole theory of forces is discussed.•A mini-review of optical forces in the Rayleigh limit of GLMT is provided. A recent work has been devoted to the study of the Rayleigh limit of generalized Lorenz-Mie theory for on-axis beams in the case of non dark axisymmetric beams of the first kind. The present work complements this previous work by studying the case of non-dark axisymmetric beams of the second kind and of dark axisymmetric beams. This paper being presumably the last one of a series devoted to the Rayleigh limit of the generalized Lorenz-Mie theory in the case of lossless particles, it is complemented by a mini-review allowing one to gain an overview of the issue.