The electron dynamics and the Thomson backscattering of an electron moving in a combined field of a tightly focused Gaussian laser pulse and an external uniform magnetic field are investigated in detail. It is found that by considering the tightly focused Gaussian laser pulse, the electron can be pushed out from the laser pulse by the ponderomotive force, resulting in the symmetry breaking of the electron dynamics from the Gaussian envelope, which can dramatically enhance the radiation intensity. It is also found that by introducing an external magnetic field, the emergence of the cyclotron motion of the electron under the external magnetic field also breaks the symmetry of the electron dynamics and enhances the radiation. Especially in the resonance case, i.e., the cyclotron frequency of the electron is close to the laser frequency, the emission spectrum is further enhanced due to the great extension of the interaction time and the symmetry breaking by the beat wave between the helix motion and the cyclotron motion of the electron in the combined field, and a platform with high radiation intensity containing the THz band has also appeared.

The electron dynamics and the Thomson backscattering spectra for an electron accelerating in a tightly focused Gaussian laser pulse are first investigated in detail. It is found that for a tightly focused Gaussian laser pulse, the ponderomotive force introduced due to the non-uniform intensity distribution of the laser pulse has the tendency to push out the electron from the laser pulse, which leads to the trajectory symmetry-breaking of the electron and then the generation of the even-order harmonics at the same time. Further, for the tightly focused Gaussian laser pulse, changes in several laser parameters, such as the increase of the laser peak amplitude, lengthening of the pulse width, and decrease of the beam waist, lead earlier to the relative ejected position of the electron to the laser pulse, which causes the more obvious trajectory symmetry-breaking of the electron, and then the more intensive peak intensity of the even-order harmonics. It is different from the well-known results of the plane waves and the Gaussian laser pulse with uniform transverse intensity distribution and provides a possible way for the generation of the even-order harmonics in nonlinear Thomson backscattering.

In the Reply, we suppose that although the higher order correction terms of the combined field with a tightly focused Gaussian laser pulse and an external uniform magnetic field have certain influence on the dynamics and radiation spectrum of the electron, it does not affect the issues and concerns studied in our paper [EPL, 139 (2022) 14001]. Therefore, it is feasible for us to adopt the lowest order approximation for the tightly focused Gaussian laser pulse and it does not affect the main ideas and conclusions of our paper.

In a recent Comment, Chang et al. [Phys. Plasmas 29, 114703 (2022)] have studied the electron dynamics for an electron accelerating in a tightly focused Gaussian laser pulse with the higher order correction terms, and it is found that the initial position of the electron is the reason why the electron is pushed to the − z axis at the end. In this Reply, the electron dynamics and its nonlinear Thomson backscattering in a tightly focused Gaussian laser pulse with the higher order correction terms are presented, and it is found that the result is consistent with the lowest order approximation case in our paper [Hong et al., Phys. Plasmas 29, 043102 (2022)]. Meanwhile, it is also found that when the longitudinal deceleration effect of the ponderomotive force on the electron introduced by the falling part of the tightly focused laser pulse is greater than the longitudinal acceleration effect of the rising part, the electron that is initially stationary will be slightly pushed to the − z axis at the end, which can well explain the “return” phenomenon of the electron in the longitudinal direction both in our paper and in the Comment.