## 1 Introduction  
In recent years, laser techniques, such as chirped-pulse amplification (CPA) [1], have witnessed remarkable advancements. The duration of ultrashort ultra-strong laser pulses has now been reduced to the order of a femtosecond [2; 3] or even attosecond [4; 5; 6; 7], showing exceptional temporal resolution characteristics. When subjected to intense laser pulses, relativistic electrons experience scattering, leading to nonlinear Thomson scattering. This process can generate attosecond [8; 9] or even sub-attosecond [10] ultrashort X-ray pulses, highlighting the immense potential of laser technology in generating high-intensity, ultrashort pulses. As this technology continues to make groundbreaking advancements, ultrafast tools, such as ultrashort electronic pulses, ultrashort laser pulses, and ultrashort X-ray laser pulses, are increasingly finding application in diverse fields like medicine, biology, physics, and materials science. Especially in research that requires high-precision temporal resolutions, these tools play an indispensable role. Furthermore, they have propelled the progress of ultrafast electron diffraction, microscopy [11; 12; 13; 14; 15], femtosecond laser spectroscopy [16; 17], and ultrafast X-ray diffraction. In intense laser fields, free relativistic electrons exhibit nonlinear Thomson scattering, generating higher harmonics. This phenomenon holds considerable research significance. Currently, the pulse width of ultrashort laser pulses has been narrowed to the order of a single light period [18; 19], making the carrier phase's influence on electron motion within the electromagnetic field increasingly critical. Against this backdrop, this paper explores the effect of the initial phases of laser pulses on the characteristics of higher harmonic radiation. This line of research has attracted the attention of pioneers such as [PERSON] and [PERSON] since 1971 [20]. Subsequently, [PERSON] et al. [21] further expanded the research in this field by discussing in detail the influence of the initial phase of the laser field on electron Thomson scattering. In addition, [PERSON] et al. [22] investigated the effect of the initial phase on high-energy electron radiation. Their results demonstrate that the initial phase of the laser field has a significant effect on both the electron trajectories and radiation frequencies. Despite the significant progress achieved in exploring the impact of the initial phases of linearly polarized laser pulses on the electron radiation characteristics, the corresponding influence of the initial phases of circularly polarized laser pulses on these properties remains a relatively unexplored area. The intricacies of circular polarization, which involves the rotation of the electric field vector in a plane perpendicular to the direction of propagation, introduce unique complexities that demand further scrutiny. Recognizing this gap in the current research landscape, this paper endeavors to conduct a comprehensive theoretical investigation and numerical simulations, focusing specifically on circularly polarized, tightly focused laser pulses. By employing rigorous theoretical frameworks and advanced simulation techniques, we aim to elucidate the intricate effects of the initial phases of circularly polarized laser pulses on the electron radiation characteristics. The tightly focused nature of these laser pulses is particularly significant as it enables the generation of intense, localized electromagnetic fields that can significantly influence the electron dynamics. Moreover, by incorporating variations in the initial energy of the electrons as a parameter in our analysis, we strive to enrich the scope and depth of our research, thereby providing a more comprehensive understanding of the complex interplay between laser polarization, electron dynamics, and radiation characteristics.