3. Discussion

There are two reasons for the relationship between the temporal coherence and the LIDT of fused silica. Firstly, the LTCLs have many temporal spiked structures, so the intensity is much higher than the average intensity (as shown as Figure 1(c)), and these spikes make a significant contribution to the LIDTs of the fused silica. Secondly, the enhanced self-focusing effect of the LTCL causes the decrease of the LIDTs of the fused silica.

The intensity probability distributions of the temporal spikes for these different bandwidths all satisfy the negative exponential distribution^[ Reference Zhao, Ji, Liu, Gao, Rao, Cui, Feng, Li, Shi, Shan, Ma and Sui^8], which means that the highest intensity of the temporal spikes is consistent for each different bandwidth pulse. With respect to the relationship between the LIDTs of the MLM pulse and the highest spike intensity reported by the pioneering group^[ Reference Smith and Do^13], the LIDTs should be the same for each bandwidth laser and about 1/10 that of the SLM pulse laser. However, in our experiments, the LIDTs of each LTCL did not drop dramatically in that way, and the LIDTs of the fused silica vary with the bandwidth. This implies that the intensity of temporal spikes is not directly correlated with LIDTs. The LIDTs of the LTCLs are not simply affected by spike intensity exclusively, and there should be different physical mechanisms corresponding to different coherent light-induced fused silica bulk damages.

The temporal spiked structures affect the LIDTs of fused silica through the accumulation effects by the irradiation of the LTCLs. These temporal spikes structures could be considered as a series of pulse trains, and the multi-pulse accumulation effect could reduce LIDTs^[ Reference Natoli, Bertussi and Commandré^18] by laser-induced defects^[ Reference Mero, Clapp, Jasapara, Rudolph, Ristau, Starke, Krueger, Martin and Kautek^19] and the band-gap^[ Reference Gao, Xie and Xu^20]. Because the duration of the spikes and the bandwidth are Fourier transform pairs, the wider bandwidth, the smaller duration of spikes and, therefore, the number of the spikes would increase as the bandwidth increases at the same duration of the incident pulse. This means that the repetition frequency of spikes at the same pulse duration becomes higher. As a consequence, it caused the LIDTs of fused silica to drop more significantly^[ Reference Nguyen, Emmert, Patel, Menoni and Rudolph^21]. This verifies the previous speculation on the reason for multi-mode lasers decreasing the LIDT of SiO₂^[ Reference Chmel^22].

Meanwhile, the difference in the self-focusing effect during the propagation in the fused silica of each beam also influences the LIDT test results. In order to analyze the physical mechanism, the expression of the LTCL was established according to its instantaneous broadband and random phase distribution properties. Since the spectral intensity distribution and the temporal intensity signal are Fourier transform pairs, according to the relevant spectral parameter and the phase random distribution characteristics of the LTCL, the temporal signal of the LTCL can be obtained by Fourier transform. Combining the nonlinear Schrödinger equation with the dispersion term (Equation (1))^[ Reference Diddams, Eaton, Zozulya and Clement^23], the characteristics of the self-focusing effect for each bandwidth could be distinguished clearly by a step-by-step Fourier algorithm:

(1) $$\begin{align} 2i{k}_0\frac{\partial A}{\partial z}+{\nabla}_{\perp}^2A-{k}_0{\beta}_2\frac{\partial^2A}{\partial {t}^2}+{k}_0^2\frac{n_2}{n_0}{\left|A\right|}^2A = 0,\end{align}$$

where ${k}_0 = 2{n}_0\pi /\lambda$ is the propagation wave vector, ${n}_0$ is the linear index of refraction, ${n}_2$ is the nonlinear index of refraction, $\lambda$ is the central wavelength of the incident pulse, ${\beta}_2$ is the group velocity dispersion coefficient, ${\nabla}_{\perp}^2 = {\partial}^2/\partial {x}^2+{\partial}^2/\partial {y}^2$ and $t$ is in the frame moving at the group velocity. The second, third and fourth terms on the left-hand side of Equation (1) represent spatial diffraction, temporal dispersion and nonlinear effects, respectively. Figure 2 shows the spatial energy distribution of each incident laser with the same parameters (incident energy: 2 mJ; pulse duration: 3 ns; focal point diameter: 0.135 mm) transmitted at a distance of 0.009 m (the Rayleigh length) obtained by simulation. The self-focusing effect of the LTCL is higher than that of the traditional SLM laser under the same incidence parameters, and it becomes enhanced with the bandwidth increasing for the LTCL. The analysis agrees with the experimental result of LIDTs.

Figure 2. Simulation results of the self-focusing effect for each bandwidth.

Figure 3. The results of z-scans of the SLM laser and the LTCLs with each bandwidth. The curves are the simulation results, and the dots correspond to the experimental results.

In order to validate the analysis results, we conducted two experiments through different methods. Firstly, we applied a closed aperture (CA) z-scan to compare the self-focusing effect of each bandwidth light. Since the nonlinear phase variation under nanosecond laser irradiation at the fundamental frequency is too small for fused silica, a thick fused silica (5 mm) was chosen as the test sample, and two photodiodes were utilized to observe the variation of transmittance^[ Reference Olivier, Billard and Akhouayri^24]. Figure 3 shows the experimental curves of the CA z-scan for the fused silica under the SLM laser and each different bandwidth of the LTCL excitation. The experimental data are fitted according to Equation (2)^{[Reference Zhao and Palffy-Muhoray25]}:

(2) $$\begin{align}T(z) = 1-\frac{4\Delta {\phi}_0\left(z/{z}_0\right)}{\left(1+{z}^2/{z}_0^2\right)+\left(9+{z}^2/{z}_0^2\right)},\end{align}$$

where $\Delta {\phi}_0 = {k}_0{n}_2{I}_0{L}_{\textrm{eff}}$ is the phase change of the incident beam due to nonlinear refraction, ${I}_0$ , ${L}_{\mathrm{eff}}$ and ${z}_0$ are intensity at the focus point, effective interaction length and Rayleigh diffraction length, respectively, and z is the position of the test sample from the focus. According to the relationship between the phase variation and the nonlinear self-focusing effect ( $\Delta {\phi}_0 = {k}_0{n}_2{I}_0{L}_{\mathrm{eff}}$ ), it could be concluded that the larger nonlinear self-focusing effect ( ${n}_2$ ), the more significant the phase variation ( $\Delta {\phi}_0$ ) under the same incident intensity ( ${I}_0$ ). Meanwhile, based on the relationship between the peak–valley difference of the CA z-scan and the laser phase variation ( $\Delta {T}_{\mathrm{P}-\mathrm{V}}\approx 0.406{\left(1-S\right)}^{0.25}\left|\Delta \phi \right|$ , where $S$ is the transmittance of the CA), it could be found that the more significant phase variation ( $\Delta {\phi}_0$ ) is, the larger the peak–valley difference ( $\Delta {T}_{\mathrm{P}-\mathrm{V}}$ ) under the same incident intensity ( ${I}_0$ ), that is, the peak–valley difference ( $\Delta {T}_{\mathrm{P}-\mathrm{V}}$ ) would become larger as the self-focusing effect ( ${n}_2$ ) was enhanced. Therefore, according to the CA z-scan test result (as shown in Figure 3), the nonlinear self-focusing effect of the LTCL is stronger than that of the SLM pulse laser, and the nonlinear self-focusing effect increases with the wider bandwidth of the LTCLs. Under the same incident intensity conditions, the LTCL has a stronger self-focusing effect and more significant phase variation during the transmission, which produces more serious self-focusing filamentation damage, which ultimately makes its LIDT lower than that of the SLM pulse laser.

Furthermore, to analyze the effect of bandwidth on the damage, we measured the laser-induced damage transient processes by the time-resolved method. The temporal waveform variation of the fused silica damage process was recorded by an InGaAs photodiode (D2) in the transmittance direction. The experimental results are presented in Figure 4. There are two temporal waveform variations of the laser-induced damage process. Previously, Shen et al.^[ Reference Shen, Chambonneau, Cheng, Xu and Jiang^26] clearly obtained the process of the filamentary damage by the pump–probe technique, and interpreted it with the moving breakdown model. The filamentary damage of fused silica has two characteristic morphologies. The filamentary tail caused by the self-focusing effect occurs at first. Next, head damage is formed by energy accumulation. These damage processes could also be obtained by the time-resolved method of transmitted light (as shown in Figure 4(a)). In the process of laser fluence absorption by the fused silica, when the laser phase change induces self-focusing filamentation damage, then the laser energy will induce a rapid decrease in the transmission signal by a huge absorption and scattering due to the damage, which is recorded by the photodiode in the transmission direction (transmission signal) as the damage time ( ${t}_{\mathrm{d}}$ )^[ Reference Xu, Emmert and Rudolph^27]. Then, as the filamentation damage absorbs the laser energy, it develops towards the input surface, and eventually a micro-explosion occurs at the filamentation head. At this time, a great deal of laser energy is scattered, causing a small rise in the transmission signal (recorded by the InGaAs photodiode (D2)). The micro-explosion produces a large head damage, resulting in absorption and scattering of the subsequent pulse. It leads to a second drop in the transmission signal after a small rise until the end of the pulse. Because the damage in the nanosecond regime is intrinsically stochastic, the test results in Figure 4(b) are the closest to the average of results obtained by each incident laser irradiation with the same fluence (156.73 J/cm²) for 10 repetitions. According to the damage time ( ${t}_{\mathrm{d}}$ ), the incubation fluence ( ${F}_{\mathrm{d}}$ ) filamentation damage for each incident laser could be calculated as follows:

(3) $$\begin{align}{F}_{\mathrm{d}} = {I} {\cdot} {\int}_{\!\!\!\kern-3pt{-}\infty}^{t_{\mathrm{d}}}f(t)\mathrm{d}t.\end{align}$$

Here, $I$ is the incident peak intensity, ${t}_{\mathrm{d}}$ is the time of laser-induced damage and $f(t)$ represents a function of the temporal waveform (these incident lasers are all flat-topped square wave with a pulse width of 3 ns, so $f(t)$ is a rectangular function). The total fluence could be expressed as ${{F}_{\mathrm{tot}} = {F}_{\mathrm{d}}+{F}_{\mathrm{exp}}}$ ^[ Reference Lamaignère, Diaz, Chambonneau, Grua, Natoli and Rullier^28], where ${F}_{\mathrm{exp}}$ is the expansion fluence after the damage occurs ( ${t}_{\mathrm{d}}$ ) at the end of pulse. It could be observed that under the same incident fluence condition (156.73 J/cm²), the damage times ( ${t}_{\mathrm{d}}$ ) are 1.20 ns (narrow band), 0.77 ns (3 nm), 0.54 ns (10 nm) and 0.33 ns (15 nm), respectively. The damage of the LTCL occurs before the SLM pulse laser under the same incident fluence. According to Equation (3), it can be concluded that the ${F}_{\mathrm{d}}$ of the LTCL is smaller than that of the SLM pulse laser, and for the LTCL, ${F}_{\mathrm{d}}$ decreases with increasing bandwidth. The relationship between the ${F}_{\mathrm{d}}$ for each laser is consistent with the simulated beam evolution of the transmission process, as mentioned above, that is, the self-focusing effect of the LTCL is stronger than that of the SLM pulse laser, which made the LTCL induce a rapid phase evolution and form filamentation damage quickly with a low ${F}_{\mathrm{d}}$ , and for the LTCL, the broader bandwidth, the faster the nonlinear self-focusing evolution process and the lower ${F}_{\mathrm{d}}$ . It is worth noting that since each laser-induced damage is at the rising edge of the temporal waveform, after the normalization, the rising edge of the pulse with smaller ${t}_{\mathrm{d}}$ and an earlier decrease would appear steeper.

Figure 4. The results of transmission signal variation by the time-resolved test method: (a) schematic diagram of the filamentary damage; (b) the test result of the SLM laser and the LTCLs with each bandwidth.

There are two reasons why the self-focusing effect of the LTCL is stronger than that of the SLM laser. First of all, the LTCLs increased the self-focusing effects due to their own properties of temporal spikes, phase random distribution, etc. Due to the Kerr effect, a nonlinear refractive index was induced during the interaction of the intense laser with the fused silica; therefore, the material’s integral refractive index consists of both linear and nonlinear components:

(4) $$\begin{align}n = {n}_0+\Delta n = {n}_0+I\times {n}_2,\end{align}$$

where $n$ is the integral refractive index of the fused silica, ${n}_0$ is the linear refractive index, $\Delta n$ is the refractive index variation induced by the nonlinear effect and ${n}_2$ is the nonlinear refractive index. Since the LTCLs are composed of a large number of ultrashort pulses of high intensity, they induce a larger overall nonlinear effect than the SLM pulse laser, which results in a larger phase variation during the transmission of the incident laser, eventually inducing a stronger self-focusing effect. Secondly, the accumulation effect of multiple pulses caused by the spike structure would also change the nonlinear self-focusing effect of the fused silica^[ Reference Melloni, Frasca, Garavaglia, Tonini and Martinelli^29].

The damage morphologies of fused silica also reveal the physical mechanisms of laser-induced damage for each incident laser (as shown in Figure 5). There is also randomness in the damage morphology for the nanosecond laser, and thus the experimental results show the closest to the average damage size with the same incident fluence for 10 repetitions. The LTCLs have a shorter L length (the distance from the input surface to the head of damage) in the direction of beam propagation (the horizontal direction in the pictures) than that of the SLM laser at the same incident intensity (I = 57.86 GW/cm²). The self-focusing filamentation damage consists of the filamentation and micro-explosion damage at the head. The filamentation damage occurs firstly due to the self-focusing effect, and then the filamentation damage prevents most of the laser energy from continuing forward. The laser energy accumulates at the head of the filamentation, which eventually induces the micro-explosion of the head^[ Reference Shen, Chambonneau, Cheng, Xu and Jiang^26]. Micro-explosions are formed by the laser transmitted over a certain distance, where the accumulated phase variation causes the pulse to undergo self-focusing filamentation damage, which is ultimately induced by energy accumulation. Therefore, the position of the micro-explosion is mainly determined by the self-focusing effect of the incident laser. Under the same incident intensity, the stronger the self-focusing effect, the greater the phase variation of the incident laser and the shorter the distance required for self-focusing into the filamentation damage, which ultimately induced the micro-explosion of the head closer to the input surface. Therefore, as shown in Figure 5, under the same incident intensity (I = 57.86 GW/cm²), an LTCL with a stronger self-focusing effect induced in the micro-explosion position of filamentation damage would be closer to the input surface than the SLM pulse laser. For the LTCL, the larger bandwidth, the closer the micro-explosion position to the input surface. For the LTCLs, the explosion size of the damage head (the vertical direction in Figure 5(b)) increased with the bandwidth increasing, since the expansion fluence ( ${F}_{\mathrm{exp}}$ ) was increased with bandwidth according to the relationship between ${F}_{\mathrm{d}}$ and ${F}_{\mathrm{exp}}$ , as mentioned above.

Figure 5. The damage morphologies of the fused silica for each incident laser: (a) integral filamentation damage captured by the CCD camera; (b) head damage morphologies of filamentation damage captured by the microscope.

Although the LIDT of the LTCL is lower than that of the SLM pulse laser, the LTCL has a single spatial mode and will remain a good spatial coherence in the propagation. The phases at different spatial positions of the beam are consistent (without self-focusing effect). It is beneficial to the amplification and propagation, avoiding the divergence problem of spatial multimode lasers. To eliminate crossbeam scattering, beam smoothing methods such as induced spatial incoherence (ISI) and continuous phase plate (CPP) are used. They induce the low-temporal coherence into space and achieve a good beam smoothing effect. Moreover, recently experimental results also confirmed that low-coherence light has a significant effect on suppressing LPI processes^[ Reference Wang, An, Fang, Xiong, Xie, Wang, He, Jia, Wang, Zheng, Xia, Feng, Shi, Wang, Sun, Gao and Fu^30]. Therefore, research on LTCL-induced optical component damage would be an important contribution to the development of laser inertial confinement fusion.