  Improving three-dimensional (3D) localization precision is of paramount
importance for super-resolution imaging. By properly engineering the point
spread function (PSF), such as utilizing Laguerre-Gaussian (LG) modes and their
superposition, the ultimate limits of 3D localization precision can be
enhanced. However, achieving these limits is challenging, as it often involves
complicated detection strategies and practical limitations. In this work, we
rigorously derive the ultimate 3D localization limits of LG modes and their
superposition, specifically rotation modes, in the multi-parameter estimation
framework. Our findings reveal that a significant portion of the information
required for achieving 3D super-localization of LG modes can be obtained
through feasible intensity detection. Moreover, the 3D ultimate precision can
be achieved when the azimuthal index $l$ is zero. To provide a
proof-of-principle demonstration, we develop an iterative maximum likelihood
estimation (MLE) algorithm that converges to the 3D position of a point source,
considering the pixelation and detector noise. The experimental implementation
exhibits an improvement of up to two-fold in lateral localization precision and
up to twenty-fold in axial localization precision when using LG modes compared
to Gaussian mode. We also showcase the superior axial localization capability
of the rotation mode within the near-focus region, effectively overcoming the
limitations encountered by single LG modes. Notably, in the presence of
realistic aberration, the algorithm robustly achieves the Cram\'{e}r-Rao lower
bound. Our findings provide valuable insights for evaluating and optimizing the
achievable 3D localization precision, which will facilitate the advancements in
super-resolution microscopy.
