Transportation dynamics of dielectric particles with the gradient forces in the field of orthogonal standing laser waves

We develop the theory of transportation and localization of a transparent dielectric spherical particle with the gradient forces in the interference field of orthogonally directed standing laser waves $E_z (\cos kz)$ and $E_x (\cos kx)$. It is shown that, when the waves $E_z$ and $E_x$ are coherent, the interference radiation field contains two harmonic components with the periods $Λ_0=π/k$ and $Λ_Δ=π/(k \sin (π/4))$. The amplitudes of the gradient force components depend on the ratio of the particle radius $R$ to the modulation periods due to inhomogeneity of radiation in the particle volume and are given by the Bessel functions $J_{3/2} (2 πR/Λ_0)$ and $J_{3/2} (2 πR/Λ_Δ)$. We find the critical particle radii $R_0$ and $R_Δ=\sqrt{2} R_0$ defined by the Bessel functions zeros and corresponding to the vanishing components of the gradient forces. In particular, for the radiation with the wavelength $λ_0=1.064$ μm and a particle in water, the smallest critical radii are $R_0=0.286$ $μ$m and $0.492$ μm and $R_Δ=0.404$ $μ$m and $0.696$ $μ$m, respectively. For a number of special cases, we obtain the analytical solutions of the Newton equations and the particle trajectories that depend on the ratio of wave intensities and the particle radius. The results can be used to study the dynamics of the "optical assembly" of a two-dimensional particles matrix which behaves as a molecular crystal.