When atoms are exposed to a non-uniform laser field, there will be gradient potential due to the electric dipole interactions between the light field and the atoms, and therefore atoms will be subjected to the action of optical dipole forces. With the recent development of ultra-short laser technology and light manipulation techniques, new types of light fields with complex spatial structures are possible to be constructed. Meanwhile, the application of these new laser fields in optical tweezers to achieve special and accurate control of micro-particles becomes a hotspot in light-matter interactions. Compared with the fundamental transverse mode Gaussian beams, these laser fields, such as hollow Gaussian beam, Laguerre-Gaussian beam, Bessel-Gaussian beam, and Airy beam, have more complex field structures and special optical characteristics. Additionally, they can provide extensive controllable degrees of freedom and more atomic beam guidance pathways for laser manipulation. Our paper studies the electric dipole interactions between the femtosecond Laguerre-Gaussian laser pulses of high radial modes and the three-level atomic systems. The spatiotemporal distribution characteristics of optical potential traps and optical dipole forces exerted by different radial modes of the femtosecond Laguerre-Gaussian laser beams are analyzed. We also reveal the advantages of Laguerre-Gaussian laser beams with high radial modes in atom trapping and manipulation. This theoretical study is expected to give insight into the optical manipulation of micro-particles with structured optical laser fields and provide guidance for possible experimental studies.

The semi-classical theory is employed to study the interactions of optical dipole forces between the femtosecond Laguerre-Gaussian laser pulses with high radial modes and the cascade three-level atoms. The laser field is treated with the classical Maxwell's theory and the atoms are treated with the quantum mechanical density matrix theory. Based on the density matrix theory, the optical Bloch equations for a cascade three-level system are derived without rotating wave approximation, and the coupled optical Bloch equation is solved numerically by utilizing the self-consistent numerical scheme. The induced electric dipole moments are then calculated from the product matrix trace of the density matrix operator and the electric dipole moment operator. The optical potentials and optical dipole forces are simulated for different radial modes of the femtosecond Laguerre-Gaussian lasers. Without generality loss, the atomic sodium is taken as the prototype for the cascade three-level atomic model, and the transitions from the ground state 3s to 3p and from 3p to 4s excited states of the sodium atom in the visible and infrared light bands respectively are chosen.

When the three-level atomic systems are exposed to a Laguerre-Gaussian beam of the n-th radial mode, there will be n+1 optical potential wells/barriers formed for negative/positive laser field detuning. With the same peak intensity of the laser field, by increasing the radial mode number n, the depth of the main potential well/barrier remains constant, but the spatial range of the main potential well/barrier becomes narrower, and the optical dipole force becomes stronger due to the increasing radial gradient of the potential (Figs. 1 and 2). Therefore, the particles are bound in much narrower optical potentials induced by the Laguerre-Gaussian laser beams of the higher radial modes, which is more conducive to the accurate manipulation and capture of particles. The transverse dipole force exerted by a Laguerre-Gaussian beam of the n-th radial mode has 2nnodal circles in the radial direction, and the direction of the dipole force is opposite on both sides of a nodal circle due to the changing electric field gradient. Therefore, the atomic beam will be split and trapped in the optical potential wells at different positions (Figs. 3 and 4). When atoms are exposed to an ultra-short femtosecond laser field, the carrier-wave effect becomes important and the induced optical dipole force oscillates with a frequency two times the carrier-wave frequency of the laser field.

This study provides insight into the optical manipulation of micro-particles in structured optical beam fields. Our attention is paid to the influence of the radial mode of the Laguerre-Gaussian beams on the optical potential and optical dipole force. The higher radial mode leads to steeper optical potential and larger optical dipole force. The Laguerre-Gaussian beam of the n-th radial mode will generate n+1 optical potential wells/barriers under negative/positive laser field detuning, and the corresponding optical dipole force has 2n nodal circles in the radial direction with opposite signs on both sides of any nodal circle. Therefore, atoms can be split and bound in much narrower optical potentials induced by the Laguerre-Gaussian laser beams of higher radial modes, and the Laguerre-Gaussian laser beams of higher radial modes are beneficial to the accurate manipulation, capture, and steering of particles. In the time regime, the carrier-wave effect induced by the femtosecond laser pulse is important and the optical dipole force oscillates with a frequency two times the carrier-wave frequency of the laser field.