## 1 Introduction  
In recent years, with the development of CPA [1, 2] (Chirped Pulse Amplification) technology, it has been able to generate ultrashort and ultra-intense laser pulses with peak power up to the petawatt (PW) level [3], which provides an essential tool for strong-field physics research [4]. The ultra-intense pulses generated by CPA can be used to study the nonlinear interaction of lasers with plasma [5], atoms, molecules, etc., and to promote the development of high harmonic generation [6, 7], attosecond pulse generation [4], etc. CPA is capable of generating ultrashort pulses at the femtosecond (fs) or even attosecond (as) level, which is widely used in the fields of ultrafast spectroscopy [8], chemical reaction dynamics, etc. In the future, it will continue to play a key role in the fields of particle acceleration, attosecond science [9], laser fusion, precision machining, ultrafast electron microscope [10], and medical science, etc. It will push forward the advancement of many disciplines. Previous studies have found that the carrier envelope phase (CEP) affects the phase matching condition. The CEP is the phase difference between the peak position of the carrier electric field and the peak position of the pulse envelope. Previous experimental studies have shown that isolated attosecond pulses can be generated by proper CEP modulation [11], which enables a more precise temporal control of high harmonic radiation on attosecond time scales and provides experimental support for attosecond pulse applications. For example, [PERSON] [6] theorized from a plasma perspective that by changing the initial phase CEP, which affects the spatiotemporal tuning of electron ionization and acceleration, the phase relationship between the driving laser and the resulting high harmonics can be tuned to optimize the efficiency of high harmonic generation. [PERSON]'s [12] team further revealed the above mechanism by obtaining isolated attosecond pulses by the CEP locking technique for the first time. [PERSON] et al. [13] found that the proton beam quality can be optimized by modulating the spectral phase of CEP in PW-class laser experiments. In addition, there are also [PERSON] [14] and [PERSON] [15] who further proved the phase matching mechanism of CEP by observing the CEP-dependent valence electron motions and recollisional phases in real time via the attosecond transient recording technique, respectively. We analyze this process through classical Thomson scattering theory, where an incident plane wave accelerates a free electron, resulting in a dipole radiation emission [16]. The earliest Thomson scattering theory was started in 1907 after the discovery of electrons by [PERSON], and has been strictly derived in modern electrodynamics textbooks [16; 17]; there is also the single-electron model, which is the study of linear and nonlinear scattering processes [18] focused on a single electron and light. Nonlinear Thomson scattering dynamics in high-intensity laser fields exhibit strong parametric dependencies. For relativistic electrons interacting with ultra-intense laser pulses (I \(>\) 10\({}^{18}\) W/cm\({}^{2}\)), the radiation spectra demonstrate pronounced nonlinear characteristics distinct from classical predictions [19]. This complexity arises from the non-perturbative nature of electron-light interactions at extreme field strengths. In attosecond \(\gamma\)-ray generation via aperiodic vortex-laser-driven nonlinear Thomson scattering, relativistic electron sheets impart both energy and angular momentum to \(\gamma\)-photons. Key radiation characteristics, including spectral density, photon energy, and brilliance, exhibit strong dependence on the electron ensemble's kinetic energy (\(\gamma\)-factor) [20] and spatial charge distribution. This momentum-energy dual transfer mechanism enables controlled \(\gamma\)-ray vorticity generation.