Ultra-short pulse, ultra-intense (USUI) lasers have opened up new experimental capabilities for high-energy-density (HED) science^[1], high harmonic generation (HHG)^[2], laser-driven particle acceleration^[3], and nuclear physics^[4]. The development of chirped-pulse amplification (CPA) technology^[5] and optical parametric chirped-pulse amplification (OPCPA) technology^[6] has enabled the construction of petawatt laser systems worldwide, such as the HERCULES facility in the USA^[7], the PETAL facility in France^[8], 4 PW Laser in Republic of Korea^[9], SG-II facility^[10] and SULF-10 PW laser^[11] in China. The corresponding focused peak intensity of such femtosecond lasers has reached 1022–1023W/cm2^[12]. In the development of a USUI laser, a high-energy pump source is one of the crucial factors. To illustrate, let us consider a typical 10 PW laser pulse with an energy of 300 J and a duration of 30 fs. The energy of the laser pulse before compression should exceed 500 J, taking into account energy loss in the compressor. Consequently, the total pump energy is up to 1 kJ calculated at 50% conversion efficiency. However, it is widely known that the high energy output and high repetition rate requirements pose a contradiction, which significantly constrains the average power of solid-state lasers, particularly those utilizing rod amplifiers. Waste heat can adversely affect beam quality by causing thermal distortions, thermal lenses, and some other thermal effects. Consequently, the implementation of efficient thermal management techniques is essential for dissipating the generated heat and maintaining the desired beam quality.

Diode-pumped solid-state laser (DPSSL) systems are extensively used as pump sources for PW laser systems, primarily due to their low heat deposition and high optical-to-optical efficiency^[13-16]. However, these systems are complex and expensive, which makes manual operation challenging and necessitates a significant investment. For applications requiring lasers with high energy output but lower demands on the repetition rate, a cost-effective alternative is water-cooled rod-based lasers pumped by flashlamps^[17-19]. Among the gain media commonly used in high-energy lasers, Nd-doped gain mediums are advantageous for generating high-energy pulses due to their relatively large emission cross section, high energy storage capacity, and simple energy level structure. Commercial solutions have also been developed for pumping, such as the ATLAS 100 J/1 ppm by Thales and the 16 J/5 Hz pump laser system by Amplitude.

Based on the analysis above, Nd-doped rod amplifiers pumped by flashlamps are suitable for developing high-energy nanosecond-pumped lasers. However, rod amplifiers capable of achieving high-energy output cannot operate at high repetition rates due to their low heat dissipation ability. Thus, high energy and high repetition rates pose a constraint. In this study, taking the home-made pump units in SULF-10 PW and our previous self-built pump lasers in SEL-100 PW prototype as examples, the technical scheme of rep-rate pump lasers based on rod amplifiers is investigated. Additionally, the spatiotemporal evolution, thermal effects, and frequency-doubled progress are thoroughly studied. This study holds immense significance for the research of 10 PW USUI lasers and has crucial applications in other fields of high-energy lasers, such as advanced material processing^[20] and laser shock peening^[21].

First, the pump energy of the flashlamps can be determined based on the output energy of the power supply. Subsequently, the stored energy and thermal load can be calculated using the computed pump efficiency. Additionally, the absorption efficiency of the pump cavity and the distribution of energy storage in the gain rod can be calculated using the Monte Carlo ray tracing method. Based on the calculated energy storage above, the gain of the amplifier can be calculated using the Frantz-Nodvik equation. However, the conventional Frantz-Nodvik equation is typically used to calculate the overall gain, and adjustments are necessary to account for spatial and temporal variations of the laser pulse. Once the desired gain performance is established, thermal analysis can be conducted to determine the appropriate repetition rate of the amplifier.

Considering the cost of engineering construction, large-diameter rod gain media are typically pumped by flashlamps. The emitted flash spectrum covers the absorption peak of the gain medium, although not all spectral components emitted are absorbed. Hence, it is essential to calculate the absorption efficiency of the gain medium by analyzing its absorption spectrum and the radiation spectrum of the flashlamps. The calculations revealed that the spectrum overlap rate was around 15% based on the self-established code. The Monte Carlo ray tracing model can be employed to determine the absorption efficiency of pump light for various pump structures. In this model, the flashlamp was modeled as a Lambertian radiator. Additionally, there exists a plasma region in the center of the lamp, which absorbs most of the light as the pump light passes through this area. The reflective property was set as the scattering reflector. The absorption efficiency, as calculated by the Monte Carlo ray tracing model, was around 14%. Consequently, the overall pump efficiency was estimated to be approximately 2%, aligning with the typical expectations.

The stored energy can be calculated with pump efficiency, and the gain performance of various amplifiers can be evaluated using the well-known Frantz-Nodvik equation. Figure 1 illustrates the comparison between theoretical and experimental results in the pump unit of SULF-10 PW. The first-stage amplifier in this pump unit consists of two Nd:glass rods with diameters of 25 mm and a length of 280 mm, while the second-stage amplifier includes two Nd:glass rods with diameters of 50 mm and a length of 280 mm. All Nd:glass rods were N31 phosphate glass with a doping concentration of 0.5%. The arc length of the pump flashlamps was 250 mm. As depicted in Fig. 1, the experimental results closely match the theoretical results. Also, the output energy increases linearly with the rise of the stored energy, indicating the absence of gain saturation because the stored energy is directly proportional to the pumped energy. This suggests the potential to increase the energy of this pump unit.

However, rod amplifiers have a drawback in that the stored energy in the cross-section of the rod is rarely uniform, especially for large-diameter rods. This is attributed to the fact that the pump light is primarily absorbed at the edge of the rod, resulting in weaker pump light as it approaches the center of the rod. This non-uniformity in pumping can be improved by changing the structure of the reflective cavity, altering the doping concentration, or increasing the pump energy.

Specific experiments were conducted in the pump unit of SULF-10 PW. First, the home-made seed sources were carefully designed to produce a super-Gaussian distribution beam profile, as shown in Fig. 2(a). Subsequently, this super-Gaussian beam underwent double-pass amplification, comprising two amplification modules with 25 mm diameter Nd:glass rods. In the previous design, gold-plated involute reflectors were utilized, and the findings indicated that the stored energy in the rod’s center exceeded that at the edges due to the higher beam intensity at the center. In order to achieve a more uniform beam profile, the gold-plated involute reflectors were substituted with elliptical reflectors fabricated from a diffuse reflection material. Regrettably, as shown in Fig. 2(c), the amplified beam profile initially displayed a “hollow-shaped” intensity distribution due to the relatively high doping concentration (1.2%). Subsequently, by reducing the doping concentration to 0.5%, the beam profile achieved greater uniformity, as depicted in Fig. 2(b).

However, achieving gain uniformity becomes more challenging when using rods with larger diameters, such as 50 mm. Figure 3(a) displays the final output beam profile after amplification by a 50 mm diameter Nd:glass rod with a 0.5% doping rate. The results indicate that the gain at the edge of the Nd:glass rod is higher than that at the center. Additionally, the distribution of stored energy can be calculated using Monte Carlo ray tracing method. This calculated distribution of stored energy is depicted in Fig. 3(b) and aligns with the measured results. Furthermore, achieving a uniform beam profile can be accomplished by adjusting the injected beam profile or increasing the pump energy. The latter approach is more straightforward to implement and comprehend physically, as sufficient pump energy can maximize the excitation of particles, leading to a uniform distribution of stored energy. However, this approach may lead to inefficient use of pump energy and potential damage to subsequent optical elements due to the high energy flow density and narrowing pulse width associated with increased gain. An alternative method involves modifying the injected beam profile, which in turn places requirements on the output beam profile of the preceding stage. Specifically, in instances where there is higher stored energy at the edge, an injected Gaussian beam can be converted into a top-hat beam. Experimental validation has confirmed the effectiveness of this method.

Based on the preceding analysis and taking into account pump uniformity, it is recommended to use rod gain mediums with a diameter smaller than 50 mm. The rod gain medium with a larger diameter can achieve higher output energy but may result in an uneven beam profile. While a multi-pass amplification scheme can eliminate beam profile inhomogeneity, it also increases the risk of potential damage to the optical components due to higher output energy.

Thermal management is also a crucial aspect of the design of a high-energy pump laser. Finite element analysis (FEA) is employed to evaluate the temperature distribution of the gain rods, and from the obtained data in FEA, the calculation of thermal birefringence becomes possible. First, the temperature distribution can be determined through transient analysis or steady-state analysis, depending on the specific situation. The transient method calculated the temperature distribution incrementally combined with the Monte-Carlo ray-tracing method. Typically, the low repetition rate of the high-energy rod amplifier ensures the accuracy of this method. Eventually, the temperature reaches a quasi-steady state, allowing for the analysis of thermal effects based on the transient data when the pulse reaches. An alternative approach to deducing the thermal load is to directly analyze the steady state using the calculated stored energy and the quantum efficiency of the gain medium. Both methods are effective for analyzing thermal effects in most situations, with the transient method being more precise only in cases where the stored energy distribution in the gain medium is highly non-uniform. Taking the pump unit of SULF-10 PW as an example, Fig. 4 illustrates the comparison of these two methods based on the 50 mm Nd:glass rod in SULF-10PW, where the pump distribution is relatively uniform. The results indicate that the final calculated state was nearly identical to the maximum temperature of the gain medium. Through a basic analysis of the temperature distribution and thermal stress, the maximum repetition rate can be determined. We deduced, based on theoretical analysis and engineering experience, that the maximum safe average power for nanosecond solid-state amplifiers using flashlamp-pumped rod amplification is 200 W. In our constructed nanosecond flashlamp-pumped lasers with rod amplifiers, the high average power was measured at 60 W (1.2 J, 50 Hz) and 200 W (20 J, 10 Hz), both utilizing Nd:YAG rods. The 60 W laser employed two amplification modules with 8 mm diameter Nd:YAG rods, while the 200 W laser consisted of three amplification stages, incorporating Nd:YAG rods with diameters of 8, 12, and 21 mm. Further enlarging the diameter of the rod will decrease the heat dissipation ability. Moreover, the temperature of the laser amplifier’s shell noticeably increased when the output power reached 200 W, and this pump laser operated at less than 200 W for safety reasons. Additionally, operating at 50 Hz would evidently reduce the flashlamps’ lifetime. Therefore, based on engineering practice, 200 W appears to be the maximum average power for flashlamp-pumped high-energy rod amplifiers. Innovative thermal management approaches are necessary to assess the average power of solid-state lasers.

In addition to analyzing temperature distribution and thermal stress distribution, it is important to consider other thermal effects, such as thermal birefringence and thermal distortion. A custom code was developed to calculate the thermal birefringence and thermal distortion. The comparison between theoretical and experimental results, as shown in Fig. 5, revealed a strong agreement between the calculated and experimental outcomes.

To illustrate the validity of the calculated results, rods with diameters of 25 and 50 mm were selected. As previously mentioned, these two different types of Nd:glass rods were used in the 100 J pump unit of SULF-10 PW. The gain data were presented previously. Figure 5 demonstrates the alignment of the measured thermal depolarization patterns with the simulated results. These thermal depolarization patterns were measured through thin-film polarizers (TFPs) without any method of compensating thermal depolarization. To mitigate the adverse impact of thermal birefringence, each stage of the amplifier consisted of two amplification modules with a polarization rotator positioned between them.

Thermal distortion is a crucial factor to consider in pumping OPCPA due to the impact of wavefront distortion on the OPA process. The wavefront distortion primarily arises from three effects: thermal expansion, thermal-stress-induced index variation, and thermal-optical effects. Through FEA analysis, temperature distribution, stress distribution, and displacement data can be determined. Subsequently, the initial data were input into a self-developed code to compute the thermal distortion. As the thermal wavefront of the rod gain medium at low repetition rates is often too subtle for detection by a wavefront sensor, specific experiments were conducted at 50 Hz. Nd:YAG was selected as the gain medium due to its higher thermal transformation coefficient. In fact, for flash-pumped rod amplifiers operating at a high repetition rate, the heat generated by flashlamps was much larger than the rod gain medium itself. However, considering the cost, flashlamps are much cheaper than LD lasers. Figure 6 shows the measured and calculated wavefront distortion of the Nd:YAG rod under different pump voltages. Although the wavefront distortion distribution was ununiform, the total value and the distribution trend were consistent. The uneven distribution of the wavefront was attributed to non-uniform heat dissipation. Additionally, wavefront distortion varied during each pulse, limiting the repetition rate of the flash-pumped rod amplifier. While one solution to this issue is to use a stimulated Brillouin scattering (SBS) mirror to compensate for wavefront distortion, it may cause reflective loss. Therefore, the rod-based amplifier should operate at a low repetition rate when wavefront distortion needs to be minimized.

During the amplification process, the laser beam may experience temporal distortions due to gain saturation, leading to a narrowing of the pulse width. Reduced pulse width means increased peak intensity that can potentially harm laser components. Thus, mitigating damage caused by pulse width narrowing is crucial. The full width at half-maximum (FWHM) of the laser pulse is a widely used parameter for evaluating pulse width. By solving the modified Frantz-Nodvik equation, the evolution of the pulse width can be simulated with a specific gain factor and an appropriate size of the gain medium. Figure 7 illustrates the calculated evolution of the pulse shape following the first-stage and second-stage amplification in a 100 J pump unit in SULF-10 PW. As shown in Fig. 7, the pulse shape remains unchanged while the pulse width decreases from 17 ns in the seed laser to 15.4 ns during the amplification process, which is deemed acceptable for preventing optical damage. The measured pulse width of the final amplified pulse, as depicted in Fig. 7, is 15.6 ns, closely aligning with the theoretical results.

A further requirement of the laser pulse shape is temporal uniformity, namely, a square-wave pulse. The amplification of the leading edge of the pulse is higher than the trailing edge, which is a common phenomenon in laser amplifiers, and rod amplifiers exhibit this behavior as well. To achieve a square pulse, a triangular pulse with lower intensity at the leading edge is required. This required triangular pulse can be derived from the final output square-wave signal. Figure 8 demonstrates the evolution of the pulse shape in SEL-100 PW pump prototype. As shown in Fig. 8, the leading edge of the laser pulse has a larger gain than the trailing edge. Thus, the pulse shape gradually becomes flatter and eventually becomes a nearly square-wave signal, which has a higher doubled frequency efficiency (up to 70%). For nanosecond high-energy laser pulses, the properties of double-frequency light inherit the characteristics of fundamental frequency light such as the pulse shape and intensity distribution. Therefore, there is no need to worry about drastic changes in the temporal shape of the double-frequency light.

In conclusion, a comprehensive multiphysics analysis of liquid-cooled flashlamp-pumped Nd-doped rods in high-energy nanosecond laser amplifiers is presented. The analysis encompasses gain performance, thermal effects, and spatiotemporal evolution of laser pulses. Each aspect involves theoretical simulations and specific experiments, with results from both methods proving consistent. These analyses help define certain performance limits of flash pump rod amplifiers. From an engineering standpoint, it is advised to maintain the average power of nanosecond pump lasers with flashlamp-pumped rod amplifiers below 200 W. Furthermore, ensuring controllable wavefront distortion is crucial for pumping OPCPA, which, in turn, restricts the repetition rate of liquid-side cooling rod amplifiers. Despite inherent drawbacks associated with the rod-shaped configuration, flashlamp-pumped rod amplifiers are cost-effective, stable, and suitable for various high-energy nanosecond laser applications.