The generalized Lorenz-Mie theory is employed to calculate the force exerted on a sphere by focused Laguerre-Gaussian beams. The key parameters of the theory, namely, the multipole coefficients of the beams, are exactly derived from the beams' angular spectra in terms of some auxiliary coefficients. Several recurrence formulas, which can improve the calculation of the auxiliary coefficients and accordingly the force, are also derived. According to the calculated force, the trapping performances of the beams are investigated in the Mie regime. It is found that low(high)-azimuthal-order beams usually have advantages in the radial trapping of the high(low)-refractive-index sphere and the axial trapping of the low(high)-refractive-index sphere. The influences of the parameters of the beams, lens, and sphere on the trapping performance are also investigated.

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The rotation dynamics of particles trapped in a rotating beam is theoretically investigated. We find that there is a critical angular speed for the rotating beam. If the angular speed of the rotating beam is smaller than the critical value, the angular velocity of the trapped particle is nearly the same as that of the rotating beam, which is in accord with existing experimental observation. Otherwise, the angular velocity of the trapped particles will become periodic or quasi-periodic with time, depending on the beam polarization, which, to the best of our knowledge, has not been previously reported. Moreover, we also propose some methods to determine the ratio between the beam power and the maximal angular speed of the trapped particle, which can be used to estimate the minimum power required to rotate the particle at a given angular speed.

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