## 1 Introduction  
Since the advent of the laser generator in 1960, in order to apply lasers in multiple fields, researchers have continuously explored and conducted studies to enhance the intensity and performance of lasers [1, 2, 3, 4]. After the emergence of chirped pulse amplification technology [5], ultrashort laser pulses have acquired the ability to generate attosecond X-rays, which has drawn significant attention and in-depth research from experts in the field of the interaction between high-energy lasers and microscopic particle-like substances. Different from femtosecond pulses that can only measure at the molecular and atomic scales, the attosecond-scale pulses brought by chirped pulse amplification technology are of great assistance in the study of high-energy electrons. The motion process undergone by high-energy electrons in a strong laser field exhibits relativistic features [6]. In nonlinear Thomson scattering (NTS) under this condition, parameters like the beam waist radius, pulse width, initial phase and polarization state of the incident laser pulse, along with the electron's initial speed, initial trajectory orientation,and starting coordinates, all lead to considerable regular variations in the motion trajectory and radiation characteristics of the electron. In recent years, NTS in the interaction between high-energy lasers and high-energy electrons has garnered numerous theoretical and experimental investigations [7]. Numerous studies have presented the characteristics of electron trajectories, spectral radiation characteristics, spatial radiation characteristics, etc., in the interaction between high-energy laser pulses and electrons. [PERSON] and [PERSON] et al. examined that the beam waist radius of attosecond-pulse high-energy lasers would exert a certain influence on the pulse shape and monochromaticity of the lasers [8]. [PERSON] et al. [9] and [PERSON] et al. [10] investigated the impact of pulse width and beam waist radius on the electron radiation distribution in the interaction between high-energy lasers and electrons in nonlinear Thomson scattering. [PERSON] et al. experimentally verified the theoretical predictions regarding the relativistic Thomson scattering of high-energy lasers and electrons [11]. Nevertheless, a large number of studies are concerned with nonlinear inverse Thomson scattering (NITS) radiation, and the X-rays produced by the mentioned methods are generally parallel to the optical axis [12; 13; 14; 15]. The intense radiation produced backward renders it challenging to install radiation detection instruments oriented in this direction for measurement, thereby causing certain difficulties in the practical application of these research findings. Meanwhile, it is necessary to utilize off-axis electrons within the range of radiation power to prevent damage to the lens caused by ray radiation [16; 17]. In this paper, we establish a model of the cross-collision between a tightly focused linearly polarized pulsed laser and a single relativistic electron within the framework of classical nonlinear Thomson scattering, generating observable transverse X-ray pulses. Through this mode of cross-collision, the nonlinear cross Thomson scattering (NCTS) radiation resulting from the cross-collision of electrons and laser beams can be generated. The beam waist radius of the pulsed laser is regarded as the control variable to investigate the trajectory of the electron, the spatial distribution of radiation, the temporal characteristics, and the spectrum characteristics. In the context of tight focusing, the electron is subject to certain spatial constraints during the interaction process in the spatial range. Therefore, in order to make the experiment more effective and rigorous, it is also necessary to control the parameter of the delay time of the laser pulse emission for research. To further optimize the experimental design, the influence of the laser pulse's arrival timing is examined systematically.