## 3 Results and Discussion
Therefore, the multi-photon ionization cross-section under linearly polarized femtosecond laser is stronger than that under circularly polarized femtosecond laser, leading to the stronger ionization rate under linearly polarized femtosecond laser. As a result, the free electron density under linearly polarized femtosecond laser is higher than that under circularly polarized femtosecond laser. The damage threshold under linear polarization is lower than that under circular polarization. To observe the phenomenon more intuitively, we use a microscope to observe the damaged morphology of the film under linearly polarized and circularly polarized femtosecond lasers, as shown in Figure 4. It is found that the damage  
Figure 3: Damage probability with laser fluence at 1-on-1. under the linearly polarized laser is more evident than that under the circularly polarized laser, which better presents the conclusion in Figure 3; that is, the damage threshold under circular polarization is higher than that under linear polarization. Secondly, the damage threshold of the Ta\({}_{2}\)O\({}_{5}\)/SiO\({}_{2}\) multilayer film under linearly polarized and circularly polarized femtosecond lasers was measured at S-on-1 (S = 2, 5, and 10). Figure 5 shows the damage probability of linearly polarized and circularly polarized femtosecond lasers with the femtosecond laser fluence at S-on-1 (S = 2, 5, and 10). According to Figure 5, the damage thresholds (S = 2, 5, and 10) under linearly polarized 1 femtosecond laser are 1.56 1.44, and 1.42 J/cm\({}^{2}\), respectively; the damage thresholds (S = 2, 5, and 10) under circularly polarized femtosecond lasers are 1.61, 1.50, and 1.47 J/cm\({}^{2}\), respectively. It is found that the damage thresholds of linearly polarized and circularly polarized femtosecond laser decreased with an increase in the number of laser irradiations. Figure 6 shows the damage morphology of the film under linearly polarized and circularly polarized femtosecond at 5-on-1. It is found that the damage of the film under linearly polarized and circularly polarized femtosecond for 5-on-1, which better verifies the conclusion in Figure 5. [PERSON] et al. conducted laser damage testing in multilayer mirrors under 1-on-1 and S-on-1 modes [27], and their results showed that the damage threshold of 1-on-1 mode was much higher than the S-on-1 mode. At the S-on-1 mode, the main factor influencing the damage threshold is the accumulation of irreversible laser-induced defects and native defects, as shown in the case of circular polarization 5-on-1 from Figure 6. [PERSON] et al. used single-shot mode and multi-shot mode of laser damage characteristics of silica at 1064 and 355 nm [28], their results showed that the damage threshold of silica decreased with an increase in shot number, and the damage threshold decreased by 35% (1064 nm) and by 20% (355 nm) after 100 shots. The result revealed a systematic fatigue effect of material in shot number. In terms of damage threshold and damage morphology, repeated laser pulses have a cumulative effect on the damage of the Ta\({}_{2}\)O\({}_{5}\)/SiO\({}_{2}\) multilayer film, and the cumulative effect becomes more obvious with more radiation times. Therefore, as the number of radiation increases from 2 to 10, the damage threshold of the film decreases. Figure 4: Typical damage morphology under linearly and circularly polarized lasers at 1-on-1. Figure 5: Damage probability of linearly and circularly polarized lasers under 2-on-1 (**a**), 5-on-1 (**b**), and 10-on-1 (**c**).