2.1 STPCL-Mm numerical model

STPCL-Mm has multiple transverse and longitudinal mode laser light fields featuring spatiotemporal coupling and three-dimensional (3D) spatiotemporal particle characteristics. We started with the spatial field correlations (mutual intensity)^[ Reference Wolf^25, Reference Chriki, Mahler, Tradonsky, Pal, Friesem and Davidson^26]: $J\left({x}_1,{x}_2;t\right)=\left\langle E\left({x}_1;t\right){E}^{\ast}\left({x}_2;t\right)\right\rangle \Delta \tau$ , where ${x}_1$ and ${x}_2$ are two-dimensional (2D) transverse coordinates, $E\left(x;t\right)$ is the electric field at point $x$ and time $t$ and $\left\langle\cdot \right\rangle \Delta \tau$ denotes temporal averaging. Based on the mode superposition model, at a specific moment, the total electric field distribution of the STPCL-Mm can be expressed as the product of the spatial and temporal components^[ Reference Chriki, Mahler, Tradonsky, Pal, Friesem and Davidson^26, Reference Siegman^27]:

(1) $$\begin{align}{E}_\mathrm{nf}\left(x,y,t\right)&=\sum \limits_{q=1}^Q\sum \limits_{n=1}^N\sum \limits_{m=1}^MH\left({\nu}_0\pm q\Delta {\nu}_\mathrm{FSR}\right) \notag\\ &\quad \times F\left[{E}_{mn}^\mathrm{T}(t)\right]{e}^{i2\pi \left({\nu}_0\pm q{\nu}_\mathrm{FSR}\right){t}} \notag\\ &\quad \times \sqrt{C_{mn}}{u}_{\mathrm{nf}- mn}\left(x,u\right){e}^{i{\varphi}_{\mathrm{nf}- mn}\left(x,y\right)}.\end{align}$$

Here, $m$ and $n$ are transverse-mode numbers, while $q$ is the longitudinal mode number. The expression $H\left({\nu}_0\pm q\Delta {\nu}_\mathrm{FSR}\right)\times F\left[{E}_{mn}^\mathrm{T}(t)\right]$ corresponds to the temporal spectral distribution, and ${e}^{i2\pi \left({\nu}_0\pm q{\nu}_\mathrm{FSR}\right){t}}$ signifies the inverse Fourier transform operator to obtain the distribution of the temporal pulse. Here, $H\left({\nu}_0\pm q\Delta {\nu}_\mathrm{FSR}\right)$ is the actual spectral distribution, which is regulated by the longitudinal modulus q; $F\left[\cdot\right]$ is the Fourier transform operator. ${E}_{mn}^\mathrm{T}(t)={G}_{mn}(t)\times iF\left[{e}^{i{\varphi}_{mn}\left(\nu \right)}\right]$ , where ${G}_{mn}(t)$ is a square-shaped super-Gaussian pulse: ${G}_{mn}(t)={e}^{-{\left({t}^2/{t}_1^2\right)}^8}$ ; ${t}_1$ = 3 ns denotes the pulse width and t is time; ${\varphi}_{mn}\left(\nu \right)$ follow a uniform probability over the interval [0, 2 $\pi$ ], which varies between different transverse modes indicated by m and n; ${C}_{mn}$ is the spectral density ratio of the mnth order mode, as listed in Tables 1 and 2 in the Appendix; ${u}_{\mathrm{nf}- mn}\left(x,y\right)$ and ${\varphi}_{\mathrm{nf}- mn}\left(x,y\right)$ denote the spatial amplitude distribution and spatial phase of the mnth transverse mode, respectively, where ${u}_{\mathrm{nf}- mn}\left(x,y\right)={u}_{\mathrm{nf}-m}(x){u}_{\mathrm{nf}-n}(y)$ , ${u}_{\mathrm{nf}-m}(x)={\left(\frac{2}{\pi {R}_{00}^2}\right)}^{\frac{1}{4}}\frac{1}{{\left({2}^mm!\right)}^{\frac{1}{2}}}\times {H}_m\left(\frac{x\sqrt{2}}{R_{00}}\right)\exp \left(-\frac{x^2}{R_{00}^2}\right)$ ; ${R}_{00}$ is the radius of TEM₀₀. Equation (1) reveals a pronounced spatiotemporal coupling effect.

The laser near-field distribution of STPCL-Mm is the same as that of the coherent light flat-top super-Gaussian near-field distribution in the SG II coherent laser driver^[ Reference Chen, Zhang, Zhang, Tang, Wang and Zhu^28], adopting a laser near-field aperture of 310 mm. Under nanosecond pulse widths, all of the $\mathrm{TEM}_{mnq}$ modes compete without phase locking and have different initial phases. The spectral profiles of $\mathrm{TEM}_{mn}$ is assumed to be the same as that of $\mathrm{TEM}_{m^{\prime }{n}^{\prime }}$ , differing only in the spectral density proportion ${C}_{mn}$ . This mathematical construction corresponds exactly to the output mode of the 4F cavity^[ Reference Cao, Chriki, Bittner, Friesem and Davidson^29– Reference Sun, Fu, Huang, Liu, Z and Chen^35]. Corresponding to the actual parameters of the designed self-reproducing multi-mode laser resonator, the parameter sets for ${C}_{mn}$ were established, with the maximum transverse-mode orders set to ${m}_\mathrm{max}={n}_\mathrm{max}=33$ , as shown in Tables 1 and 2 in the Appendix^[ Reference Zhu^36], and with the frequency interval between transverse modes (with the same q) set to 40 kHz, whereas the longitudinal mode interval was configured to 130 MHz, with the spectral profile set to eighth-order super-Gaussian.

To systematically reveal the characteristics of the multi-transverse mode and multi-longitudinal mode laser light fields, we analyzed the near-field and far-field distributions of the multi-mode and multi-wavelength beams, as shown in Figures 1 and 2. Figure 1 shows the near-field characteristics of beams with different transverse and longitudinal mode orders, demonstrating the near-field evolution of STPCL-Mm. The 3D light-field distribution of STPCL-Mm exhibited distinct spatiotemporal scattered particle characteristics. The longitudinal mode numbers govern the temporal coherence length, whereas the transverse-mode numbers determine the spatial distribution^[ Reference Zhang, Zhang, Zhang, Zhang, Xu, Zhou and Zhu^37, Reference Xu, Zhang, Zhang, Zhang, Zhang, Zhou and Zhu^38]. During consecutive time slices, substructures with different spatial particle distributions and dynamic spatial reconfigurations were observed. This competition between space and time patterns resulted in spatiotemporal decoherence characteristics, as shown in Figure 1(a). When the number of transverse modes was maintained at a certain value and the number of longitudinal modes increased, spatiotemporal particles exhibited a phenomenon of time shortening, as shown in Figure 1(b), with a decrease in temporal coherence. When the number of longitudinal modes was maintained at a certain value while the number of transverse modes was increased, the spatial particles exhibited the phenomenon of more spatial particles and the spatial coherence increased, as shown in Figure 1(c). The second-harmonic emission supports many more transverse modes than the fundamental emission, and exhibits lower spatial coherence^[ Reference Liew, Knitter, Weiler, Monjardin-Lopez, Ramme, Redding, Choma and Cao^39]. In applications such as parallel imaging and projection display, the trade-off between the efficiency of second-harmonic emission and the spatial coherence of the light source is a key consideration. Figure 2 shows the far-field characteristics of the beams with different transverse and longitudinal mode orders. It presents the far-field evolution of STPCL-Mm. The evolution patterns of the different transverse modes were similar to those of the near-field. For multiple transverse and longitudinal modes, the spatial distribution of the far-field changed continuously over time. Within a certain time integral, the focal spot was extremely smooth, thereby achieving internal beam smoothing and reducing the complexity of the laser-driving system.

Figure 1

Near-field 3D spatiotemporal light-field distributions: (a) and (b) have a maximum transverse-mode order of 18, with longitudinal mode numbers of 468 and 3740, respectively, and spectral bandwidths of 0.25 and 2 nm, respectively; (c) and (d) have a maximum transverse-mode order of 33, with longitudinal mode numbers of 468 and 3740, respectively, and spectral bandwidths of 0.25 and 2 nm, respectively.

Figure 2

Far-field 3D spatiotemporal light-field distributions: (a) and (b) have a maximum transverse-mode order of 18, with longitudinal mode numbers of 468 and 3740, respectively, and spectral bandwidths of 0.25 and 2 nm, respectively; (c) and (d) have a maximum transverse-mode order of 33, with longitudinal mode numbers of 468 and 3740, respectively, and spectral bandwidths of 0.25 and 2 nm, respectively.

2.2 Uniform irradiation of STPCL-Mm with response to spatiotemporal coherence

To verify the compatibility of the STPCL-Mm with the traditional laser-driven architecture, we evaluated the beam smoothing effect of a CPP originally optimized for coherent lasers. For multi-transverse and multi-longitudinal mode operations, the far-field spatial distribution changed continuously over time, thereby achieving intrinsic beam smoothing and significantly reducing the complexity of the laser-driving system. Figure 3 illustrates the far-field evolution of the STPCL-Mm equipped with a CPP.

Figure 3

Three-dimensional instantaneous and time-integrated far-field distribution of STPCL-Mm equipped with a CPP. The maximum transverse-mode order was 33, the longitudinal mode number was 374 and the spectral bandwidth was 0.02 nm.

To evaluate the potential of STPCL-Mm for uniform irradiation applications, we analyzed the RMSs of the focal spot irradiated directly by STPCL-Mm, STPCL-Mm passing through a CPP and the focal spot irradiated by a coherent laser with the 2D-SSD and CPP. For comparison, the CPP designed for the coherent laser was also utilized for the STPCL-Mm. The spectra were matched in width and configured to an eighth-order hyper-Gaussian profile, with full widths at half maximum (FWHMs) of 1, 0.5 and 0.1 nm at 351 nm, respectively. The phase of the 2D-SSD can be expressed as follows:

(2) $$\begin{align}{\phi}_\mathrm{ssd}&={\delta}_x\mathit{\sin}\left(2{\pi \nu}_{mx}\left(t+\frac{\Delta {\theta}_x}{\Delta \lambda}\frac{\lambda }{c}x\right)\right)\notag\\ &\quad + {\delta}_y\mathit{\sin}\left(2{\pi \nu}_{my}\left(t+\frac{\Delta {\theta}_y}{\Delta \lambda}\frac{\lambda }{c}y\right)\right).\end{align}$$

Here, ${\nu}_{mx}$ =10.4 GHz, ${\nu}_{my}$ =3.3 GHz and ${\delta}_x$ =1.96, ${\delta}_y$ =6.15 are the modulation frequency and modulation depth, respectively; $\frac{\Delta {\theta}_x}{\Delta \lambda}\frac{\lambda }{c}$ = 0.3 ps/mm, $\frac{\Delta {\theta}_y}{\Delta \lambda}\frac{\lambda }{c}$ = 1.13 ps/mm are the variation in phase across the beam due to the angular grating dispersion. The FWHM of the spectrum of the SSD was approximately 0.1 nm. Figure 4 shows the RMSs of the uniform irradiation for different spatiotemporal coherence values over time after integration for 2 ns. Figure 4(a) shows that the RMSs of the focal spot irradiated directly by STPCL-Mm were influenced by temporal coherence and were entirely independent of spatial coherence. With a 1 nm bandwidth, the RMS decreased to below 2 $\%$ after integration for 2 ns, indicating that direct irradiation by STPCL-Mm can enhance the beam smoothing effect. Furthermore, this approach simplifies the complexity of the ICF laser-driver system and offers significant advantages for practical applications. Figure 4(b) shows that the decline rates of the RMSs of uniform irradiation for STPCL-Mm passing through the CPP were influenced by both temporal coherences, whereas the minimum final RMS values were only determined by the spatial coherence in the same spectral width. With a consistent bandwidth of 0.1 nm, the decline rate of the RMSs for the 32 TDL STPCL-Mm with a CPP was faster than that of coherent lasers with a CPP and SSD, with final values (2 ns integration) of 6.6 $\%$ and 7.0 $\%$ , respectively. The minimum RMS of the 32 TDL STPCL-Mm with a 0.1 nm bandwidth was slightly lower than that of the 16 TDL (with ${m}_\mathrm{max}$ = ${n}_\mathrm{max}$ = 18) STPCL-Mm with a 1 nm bandwidth after integration for 1 ns. We attribute this to the higher line density in the spectrum of the 32 TDL STPCL-Mm. Each transverse mode had a distinct frequency ${\nu}_{qmn}$ and could alter the fine structure of the spectrum. Therefore, both the line density and bandwidth of the spectrum were important factors for determining the beam smoothness. These results show that STPCL-Mm achieves good uniform irradiation performance with a narrow bandwidth, which is particularly advantageous for high-efficiency third-harmonic generation. Thus, STPCL-Mm demonstrates significant potential as an ICF laser driver. Figure 3 shows the far-field evolution of the STPCL-Mm through the CPP. To verify the compatibility of the STPCL-Mm with the traditional laser-driving architecture, we evaluated the beam smoothing effect of a CPP originally optimized for coherent lasers. In contrast, for multi-transverse-mode and multi-longitudinal-mode operations, the spatial distribution of the far-field changed over time, achieving intrinsic beam smoothing and significantly reducing the complexity of the laser-driving system.

Figure 4

RMS variation of the target surface with integration time for (a) STPCL-Mm and (b) STPCL-Mm with CPP. (c) RMS variation with the ratio of integration time to coherence time.

Figure 5

(a) Acceptance angle and (b) acceptance bandwidth of type-I third-harmonic generation for KDP (1.5 GW/cm ${}^2$ ).

To reflect the impact of temporal coherence on the beam smoothing performance, we normalized the total integration time using the coherence time as a reference, as shown in Figure 4(c). The three descending curves of the RMSs for direct irradiation by STPCL-Mm, shown in Figure 4(a), were combined into a single curve. This indicated that the RMSs for direct irradiation by STPCL-Mm were solely dependent on the ratio of integration time to coherence time ( $\mathrm{RI2C}$ ) (effective irrelevant speckle pattern ${N}_\mathrm{e}$ ) and were not influenced by spatial coherence. The RMSs coincided with the curve $1/\sqrt{\mathrm{RI2C}}$ . In contrast, the six declining RMS curves of the STPCL-Mm with CPP shown in Figure 4(b) converged into two curves, which depended only on spatial coherence. This suggests that a decrease in spatial coherence is advantageous only when it is affected by a CPP. The rate of decline in the RMSs of the focal spot irradiated by a coherent laser with 2D-SSD and CPP differed from that of STPCL-Mm with the CPP. We attributed this discrepancy to the different spectral shapes of 2D-SSD and STPCL-Mm despite sharing the same bandwidth^[ Reference Xu, Zhang, Zhang, Zhang, Zhang, Zhou and Zhu^38, Reference Rothenberg^40, Reference He, Li, Chai and Wang^41]. Consequently, the declining trend of the RMSs was influenced by the spectral shape and fine structure, and the value of the RMSs depended on RI2C, which was determined by the coherence time and spectral bandwidth. When RI2C exceeded 1000, the rate of decline in the RMSs slowed, providing a reference value for the degree of temporal coherence necessary to achieve uniform irradiation over a specified integration time. The final RMS value of the STPCL-Mm with the CPP was influenced by the inevitable low-frequency modulation of the CPP^[ Reference Siegman^27]. Lower spatial coherence resulted in a lower RMS associated with the low-frequency modulation of the CPP. In summary, we present the relationship between spatiotemporal coherence and the RMSs of uniform irradiation, offering reference values for both temporal and spatial coherence to achieve uniform irradiation.

2.3 Third-harmonic generation efficiency of STPCL-Mm

In traditional coherent third-harmonic generation for collinear phase matching, the relationship among the phase mismatch, divergence angle and bandwidth can be expressed as follows:

(3) $$\begin{align}\Delta k=\frac{\partial \left(\Delta k\right)}{\partial \theta}\Delta \theta +\frac{\partial \left(\Delta k\right)}{\partial \lambda}\Delta \lambda .\end{align}$$

Figure 5 illustrates the relationships among the third-harmonic generation efficiency of potassium dihydrogen phosphate (KDP) (peak power density: 1.5 GW/cm ${}^2$ ), divergence angle and bandwidth. Notably, achieving a third-harmonic conversion efficiency exceeding 50 $\%$ requires that the fundamental laser’s bandwidth be less than 0.75 nm and the divergence angle be less than 650 μrad for the traditional collinear phase-matched third-harmonic generation scheme. When the near-field beam aperture is 310 mm, the divergence angle corresponding to 32 TDL is approximately 145 μrad, which is significantly less than 650 μrad. The fundamental-frequency bandwidth of 0.75 nm corresponds to a triple-frequency bandwidth of approximately 0.25 nm. According to Figure 5(b), approximately 7% of the RMS can be achieved within 0.5 ns through the combination of STPCL-Mm and CPP irradiation, while approximately 2.5 $\%$ of the RMS can be obtained through direct irradiation of STPCL-Mm. Therefore, STPCL-Mm demonstrates a superior advantage in achieving beam uniformity and smoothing within a relatively narrow bandwidth ( $\leqslant$ 1 nm). Moreover, if the bandwidth of STPCL-Mm is further increased (1–5 nm), the conversion efficiency will inevitably decrease. A balance must be evaluated between bandwidth and efficiency. Furthermore, considering the overall demand for large-scale drives in the future, the requirements for ultrahigh bandwidths ( $\ge$ 5 nm) and shorter wavelengths have been prioritized. The efficient generation of high-order harmonics with large bandwidths is a significant challenge that may necessitate major design changes.

3.1 Seed source for STPCL-Mm

As the seed source for generating the STPCL-Mm, the 4F imaging laser resonator is illustrated in Figure 7. The 4F imaging resonator shares a common configuration with the spatial filters used in ICF laser drivers. Owing to its significantly larger Fresnel number compared with conventional resonators^[ Reference Chriki, Mahler, Tradonsky, Pal, Friesem and Davidson^26, Reference Cao, Chriki, Bittner, Friesem and Davidson^29, Reference Nixon, Redding, Friesem, Cao and Davidson^30, Reference Zhang, Zhang, Tao, Zhang, Wei, Yang and Zhu^32], it demonstrates enhanced resistance to diffraction effects. This characteristic enables it to accommodate a large number of high-order transverse modes. Consequently, the number of high-order transverse modes in the 4F imaging resonator exceeds that achievable in multi-mode transverse fiber laser cavities. From an imaging perspective, laser oscillation can occur between points A and ${\mathrm{A}}^{\prime }$ . All imaging systems operate as low-pass filters, and their cutoff frequency is determined by the aperture stop of the system. Therefore, adjusting the aperture in the spectral plane enables precise control over the spatial coherence. In 4F imaging resonators, the frequency interval of longitudinal modes is governed by the cavity length (i.e., the optical path length between points A and ${\mathrm{A}}^{\prime }$ ). However, practical imaging systems are subject to aberrations that induce slight variations in the optical paths of the object points at different heights, resulting in minor deviations in the longitudinal mode interval. The number of longitudinal modes is ultimately determined by both the longitudinal mode interval and the emission spectrum of the gain medium.

3.2 Amplification routes for STPCL-Mm

To achieve laser amplification within different wavelength bandwidth ranges, we have proposed two amplification technical routes, as illustrated in Figure 6. The first amplification technology route employs neodymium-doped glass (Nd:glass) amplification, which supports a bandwidth amplification of 13 nm. However, the amplification efficiency of different wavelengths of light in Nd:glass varies, resulting in a narrowed gain range. To obtain a specific spectral shape, a birefringent filter composed of a quartz crystal and two polarizers is generally used to adjust the spectral gain during the amplification process^[ Reference Gao, Cui, Ji, Rao, Zhao, Li, Liu, Feng, Xia, Liu, Shi, Du, Liu, Li, Wang, Zhang, Shan, Hua, Ma, Sun and Chen^16], which leads to a certain shaping loss. For STPCL-Mm amplification with larger bandwidth (exceeding 13 nm), the second approach is to combine Nd:glass amplification with optical parametric amplification (OPA). This hybrid method effectively overcomes the narrow gain limitation of the Nd:glass amplifier and enhances the amplification efficiency of STPCL-Mm^[ Reference Dorrer, Spilatro, Herman, Borger and Hill^42, Reference Ekanayake, Spilatro, Bolognesi, Herman, Hill and Dorrer^43].

The reduced spatial coherence of the partially spatially coherent light results in an enlarged far-field focal spot. Consequently, the aperture size of the spatial filters should be adaptively adjusted based on the level of spatial coherence, necessitating modifications to the spatial filter design. Unlike traditional coherent laser systems, the initial stage spatial filter no longer requires vacuum pumping, and in some configurations, the filter itself can even be omitted.

The reduced spatial coherence (e.g., 32 TDL) significantly diminishes the impact of phase distortions and thermal distortions on the optical components, rendering these effects negligible. Consequently, relaxed manufacturing tolerances for optical components lower the system construction costs while enhancing the repetition rate of the laser driver.

During transmission and amplification, lasers passing through damaged regions or contaminants (e.g., dust) can induce modulation. In particular, under high-intensity conditions, these modulations can lead to nonlinear growth and small-scale self-focusing (SSSF), which are currently the primary factors limiting the load capacity of high-power laser drivers. Our research group has studied the influence of spatial coherence on the suppression of SSSF in the Nd:glass gain medium, and derived the empirical formula for the SSSF modulation unstable gain coefficient in partially coherent spatial light beams. In addition, we established the numerical relationship between the B-integral suppression and spatial coherence, providing a theoretical basis for enhancing the load capacity of partially coherent laser drivers^[ Reference Siegman^27].

3.3 Efficiency of third-harmonic generation of STPCL-Mm

Two third-harmonic conversion schemes are presented for STPCL-Mm with varying spectral bandwidths, as shown in Figure 6. One is the crystal cascading scheme, and the other is the method of generating third-harmonic light by utilizing angular dispersion, which is proposed by the Laser Energy Laboratory (LLE). Studies have indicated that the third-harmonic conversion efficiency in Scheme II is comparable to the performance of conventional narrowband third-harmonic generation methods^[ Reference Zhang, Li, Cheung, Wong and Li^21, Reference Kupka^22].

For the narrow bandwidth STPCL-Mm (with a bandwidth of less than 1 nm) scheme, non-collinear phase matching between the small narrowband 1 $\omega$ light beam and the coherent laser 2 $\omega$ light beam is achieved, enabling a third-harmonic generation conversion with an output bandwidth of less than 1 nm. In contrast, for the broadband bandwidth STPCL-Mm (1–5 nm), the angular scattering broadband STPCL-Mm 1 $\omega$ beam and the coherent laser 2 $\omega$ beam are combined through non-collinear sum-frequency generation to produce a broadband 3 $\omega$ beam. Research has shown that the third-harmonic conversion efficiency of Scheme II is comparable to that of the traditional narrowband third-harmonic generation method^[ Reference Zhang, Li, Cheung, Wong and Li^21, Reference Kupka^22]. Moreover, if the wavelength of the broadband fundamental-frequency light is changed to 800 nm and a narrow band of 1053 nm is used as the pump light, a very wide bandwidth output ( $\geqslant$ 5 nm) can be achieved near the 400 nm wavelength. This is one of the solutions developed by our laboratory to achieve a large-bandwidth output.

3.4 Engineering complexity and feasibility of engineering implementation

STPCL-Mm drivers exhibit parity in beamlet quantity with coherent laser architectures. Mature amplifier technologies, including Nd:glass systems and hybrid Nd:glass/OPA amplifier configurations, have introduced comparable engineering complexities across these platforms. Furthermore, critical beamline subsystems developed for coherent laser systems, such as synchronization mechanisms, temporal waveform modulation and auto-collimation, demonstrate direct adaptability to STPCL-Mm implementations, effectively accelerating the technological maturation trajectory.