## 1 Introduction  
Since the first ruby laser [1] was invented in 1960, laser technology has advanced rapidly from a basic light source to an extreme physical property control tool. In the 1990s, the breakthrough of chirped Pulse Amplification (CPA) technology [2, 3, 4, 5, 6] increased the peak intensity of lasers, which gave rise to the widespread application of ultrashort (femtosecond to attosecond) pulses [7] and became the core means to study ultrafast processes such as chemical bond cleavage and biomolecular reactions [8]. To generate radiation with shorter wavelengths and higher energies, scientists have developed a variety of nonlinear techniques: dynamic compression based on molecular modulation [9], frequency shift control of stimulated Raman scattering [10], and nonlinear Thomson scattering. Moreover, it is important to note that relativistic electrons in a strong laser field are driven by a periodic ponderomotive force to generate high-order harmonics, and finally radiate X-rays [11, 12, 13, 14, 15, 16] and even gamma photons [17, 18, 19, 20, 21]. This kind of ultrashort and high-energy photon source not only makes it possible to directly observe the electron ionization dynamics at the atomic scale but also provides a revolutionary tool for pump-probe experiments and high-resolution imaging. Meanwhile, the latest research utilizes the characteristics of the quantized electromagnetic field coupled with Frenkel excitons in organic materials to form an exciton-polariton (EP) [22]. Currently, nonlinear Thomson scattering has become a key research direction in extreme optical physics and the design of compact particle accelerators due to its characteristics of efficiently generating narrow pulse width and high-brightness radiation in a strong field, continuously promoting multi-field innovations from basic quantum dynamics to medical imaging technology.  
It is well known that nonlinear Thomson scattering does not require complex accelerator facilities. It only needs the interaction between the laser and the gas target to generate a compact high-energy photon source, significantly reducing the experimental cost and scale. These characteristics make it have irreplaceable application potential in fields such as dynamic imaging, nuclear physics, and medical radiotherapy. Moreover, we only need to precisely manipulate the spatial angular distribution and spectral characteristics of radiation by regulating the laser polarization (linear, circular, or elliptical polarization), thereby achieving flexible customization of the light source parameters. For circular polarization, [PERSON] [23] and others studied the spatial radiation characteristics of electron Thomson scattering in a circularly polarized tightly focused laser beam and found that in a strong laser pulse, most of the radiation energy is decoupled. [PERSON] [24] and others studied the optimal waist radius of nonlinear Thomson scattering under the action of an external magnetic field. For the linearly polarized laser system, [PERSON] et al. [25] studied the emission spatial characteristics under different laser intensities and found changes in the position and number in the lobed radiation pattern. [PERSON] et al. [26] studied the analysis of the waist in the spatial emission characteristics of electrons driven by a strong linearly polarized laser pulse. However, there is currently a lack of systematic analyses of the spatial emission characteristics of electrons under the synergy of laser intensity and waist radius of linearly polarized laser pulses. This paper focuses on the parametric space of linearly polarized NTS. By constructing a three-dimensional electron dynamics model, we reveal the coupling regulation law of laser intensity and waist radius on radiation power, electron trajectory, and time spectrum, aiming to establish a universal framework for inverting laser parameters from radiation characteristics and provide theoretical support for the on-demand design of highly directional X/\(\gamma\)-ray sources.