## 1 Introduction  
Since the advent of chirped pulse amplification in 1985 [1], there has been a major breakthrough in laser physics. Ultrashort laser pulses are currently the focus of attention due to their great potential for the study of ultrafast processes as the latest frontier of international laser science. As an important source of modulated X-rays, ultrashort ultra-intense laser pulses have also made important contributions to research in modern astrophysics [2, 3], medicine [4, 5, 6], nuclear physics [7, 8], and archaeology [9]. On the one hand, some researchers pioneered the study of charged particles moving in electromagnetic waves and constant magnetic fields, known as \"self-resonance\". They found that under certain conditions, the energy of a particle can increase indefinitely. In the late 1990s and 2000s, [PERSON], [PERSON], and [PERSON] proposed the theory that the spectrum of radiation emitted by electrons in laser fields and uniform magnetic fields [10, 11, 12]. Later, [PERSON] et al. found that backscattered electrons can achieve higher energies than forward-scattered electrons and analyzed the acceleration of electrons in magnetic fields induced by linearly polarized laser pulses. They found that the radiation occurs at the higher harmonics of the cyclotron frequency [13, 14]. Later, [PERSON] et al. found that an applied magnetic field could constrain the motion of electrons [15]. Later, [PERSON] et al. found that the electron retains the energy for larger distance even after passage of the laser pulse in a magnetic field [16]. Later, [PERSON] et al. found that by choosing the appropriate fields and initial momentum of electron, the high frequency part of the Thomson scattering spectra can reach the frequency range of soft X-ray [17]. In nonlinear Thomson scattering, on the other hand, different parameters of the incident laser pulse affect the trajectory and radiation properties of the electrons. For example: different pulse widths [14], different laser intensities [18, 19], different initial positions of the electrons [20], different beam waist radii [21, 22, 23]. Nonlinear Thomson scattering refers to the transfer and scattering of energy caused by the interaction of photons during the interaction between the electron and the intense laser in Figure 1. This interaction causes the electrons to be forced to vibrate. The vibrating electrons radiate electromagnetic waves to their surroundings. In this process, nonlinear effects cause the frequency, phase, and amplitude of the light to change, resulting in new properties and effects. In this scheme, Thomson scattering occurs in a strongly nonlinear regime, and electrons are directly accelerated and wiggled within an intense laser field. Indeed, in that regime, the term \(-\mathrm{e}\overrightarrow{\mathrm{v}}\times\overrightarrow{\mathrm{B}}\) of the Lorentz force--negligible at low intensity--becomes comparable to the \(-\mathrm{e}\overrightarrow{\mathrm{E}}\) component, and the motion is a nonlinear function of the driving field in addition to becoming relativistic. The electron motion is no longer harmonic, and the radiation emitted consists of high-order harmonics forming a broadband spectrum that extends up to the X-ray range. In nonlinear Thomson scattering, \"nonlinear\" refers to the situation in which the relationship between the intensity of light and the electric field in the scattering process does not satisfy linearity. In traditional scattering theory, the interaction between light and electrons is linear, i.e., the intensity of light is proportional to the electric field. However, in nonlinear Thomson scattering, the interaction between light and electrons is nonlinear, and there is a high degree of nonlinearity between the intensity of light and the electric field [24]. Previous work has focused on the study of stationary electrons subjected to tightly focused laser pulses [18]. The resulting electron radiation collimation is not high enough and the number of high harmonics is low.