$4π$ Combination of Gaussian Laser Beams

This work is a generalization of our previous study [arXiv:2406.17944], which dealt with the coherent and incoherent combination of linearly polarized Gaussian laser beams operating in continuous-wave mode under a $2\pi$ focusing configuration. In the present work, the laser beams are focused within a solid angle that can reach $4\pi$ by using two opposing source planes, where all the lasers from both planes are focused onto the same point in the focal plane. Our results show that the focal spot area obtained with either one or two source planes remains unchanged, despite the fact that, in the two-source-plane configuration, the numerical aperture (NA) of the system is twice that of the $2\pi$ focusing scheme. However, the intensity at the focal point is four times greater than that obtained with a single source plane. In addition, the longitudinal resolution is significantly improved, yielding a focal volume smaller than $0.5 \lambda^3$ for a numerical aperture of NA = 0.895. For incoherent beam combination, the focal volume remains practically the same in both the $2\pi$ and $4\pi$ focusing configurations, while the intensity at the focal point is twice that obtained in the $2\pi$ case. Furthermore, in the incoherent configuration, we observe a depth of focus reaching up to 30,000 $\lambda$ for a system with a low numerical aperture.