The advent of ultra-short, ultra-intense (USUI) laser systems has revolutionized extreme condition physics research, achieving unprecedentedly focused intensities exceeding 1022W/cm2^[1,2]. Critical technological breakthroughs in chirped-pulse amplification (CPA)^[3] and optical parametric chirped-pulse amplification (OPCPA)^[4] have enabled petawatt-scale implementations across major laser facilities^[5,6]. Within this technological landscape, the J-KAREN laser system^[7] represents a milestone as the first operational petawatt-class laser device, while the first operable OPCPA system was developed using the pump beam from the Luch Facility at the Institute of Applied Physics, Russian Academy of Science, Nizhny Novgorod, with a laser output of 0.2 PW in 2006^[8]. At the current stage, representative petawatt-class laser systems, including OMEGA-EP^[9], ELI-Beamlines^[10,11], and SULF-10 PW^[12], offer unprecedented opportunities in fields such as high-energy-density physics, advanced accelerator physics, novel radiation sources, laser nuclear physics, and laboratory astrophysics^[13,14].

Contemporary high-peak-power systems predominantly employ 527–532 nm pumping schemes, optimized for Ti: sapphire absorption bands and OPCPA phase-matching requirements^[15–17]. These green pump sources, typically generated via second-harmonic generation (SHG) from 1 µm fundamental waves, constitute critical subsystems in modern pump laser architecture^[6,18]. The SHG process involves two key aspects: conversion efficiency and spatial characteristics. While SHG optimization has traditionally focused on conversion efficiency maximization^[19–21], the spatial characteristics of frequency-doubled beams remain critically understudied — an oversight with significant practical consequences. Poor spatial uniformity of pump beams can negatively affect the output spot of high-energy lasers. For example, in research on the SEL-100 PW device, the near-field profile of the amplified pulse showed ring structures caused by the inhomogeneous pump beam distribution^[22]. Inhomogeneous spot intensity distribution is a major cause of optical element damage^[23,24]. Green lasers are not only employed in pumping high-peak-power laser systems but also demonstrate notable advantages in precision micromachining applications, attributed to the significantly higher absorptivity exhibited by highly reflective metals like copper and aluminum in the visible spectral range^[25]. Furthermore, green lasers serve as an ideal choice for underwater communications and laser shock peening (LSP) processes requiring penetration through aqueous media, owing to their low absorption coefficient and high transmission efficiency in water^[26–28]. Moreover, in applications related to LSP, a flat-top beam is also required^[29–31]. Due to the broad range of applications of green lasers, it is essential to ensure the spatial quality of the beam during SHG.

This work presents a comprehensive analysis of the spatial beam dynamics in type-I LBO crystals through combined theoretical modeling and experimental verification. We established a predictive spatiotemporal evolution framework validated against empirical data from our custom SHG platform [featuring 0.7% root mean square (RMS) stability]. Our results demonstrate that controlled power density escalation to 0.478GW/cm2 induces deterministic beam profile transformations, achieving 85% conversion efficiency while simultaneously improving spatial uniformity. This dual optimization enables significant mitigation of intensity hot spots, directly addressing critical challenges in high-energy laser system longevity and performance stability.

We systematically investigated the correlation between fundamental beam power density and SHG conversion efficiency through analysis of three-wave mixing dynamics. The theoretical framework originates from second-order nonlinear polarization effects, governed by the following coupled wave equations:{dE(ω1,z)dz=iω12k1c2χeff(2)E(ω3,z)E*(ω2,z)e−iΔkzdE(ω2,z)dz=iω22k2c2χeff(2)E(ω3,z)E*(ω1,z)e−iΔkzdE(ω3,z)dz=iω32k3c2χeff(2)E(ω1,z)E(ω2,z)eiΔkz.(1)

Here, ω1 and ω2 are the fundamental frequencies, while ω3 is the high-harmonic frequency, and z represents the propagation distance of the electric field in the medium. E(ω1,0) and E(ω2,0) are the initial complex amplitudes of the incident fields, while E(ω3,z) is the complex amplitude of the output field. Crucially, all three fields exhibit mutual dependence due to pump depletion effects, necessitating numerical solution of the coupled equations. The resultant electric field demonstrates periodic intensity modulation along the crystal length, with spatial periodicity governed by incident field intensity. Notably, the SHG process cannot be treated as a simple degenerate case of sum-frequency generation (ω1=ω2) due to concurrent competing nonlinear interactions. Under ideal phase-matching conditions, analytical solutions predict 100% conversion efficiency; the specialized coupled equations for SHG are therefore expressed as{dE(2ω,z)dz=2iω2k2ωc2χeff(2)E2(ω,z)dE(ω,z)dz=iω2kωc2χeff(2)E(2ω,z)E*(ω,z).(2)

However, practical limitations introduce a phase-matching factor δ that modifies the efficiency relationship: {dE(2ω,z)dz=2iω2k2ωc2χeff(2)E2(ω,z)eiδzdE(ω,z)dz=iω2kωc2χeff(2)E(2ω,z)E*(ω,z)e−iδz.(3)

This non-ideality causes periodic energy transfer between fundamental and harmonic waves, with maximum conversion occurring at characteristic crystal lengths. Building upon the preceding results, we conduct a two-dimensional analysis of the equation system. As depicted in Fig. 1, we discretize the fundamental laser energy into pixel cells and solve the coupled-wave equations for each unit along one dimension. Each pixel cell exhibits a distinct second-harmonic conversion efficiency. By integrating the fundamental laser energy with these efficiencies, the energy and spatial distribution of any beam after SHG can be theoretically determined.

The experimental setup can be classified into two principal subsystems: 1) beam shaping and amplification system; 2) SHG system. The former configuration comprises an optically addressed spatial light modulator (OASLM) and a dual-pass laser amplification stage (Fig. 2) that uses two Nd:YLF rod crystals as the gain medium, seeded by a self-built regenerative amplifier operating at 1 Hz, 1053 nm, and 5 ns pulse width with stable output energy (∼10mJ, RMS<0.8%). The amplifier comprises a 45° rotator and 0° mirrors. The seed beam (2 mm spot) is expanded to 8 mm via lenses L1 and L2, exhibiting a Gaussian profile (measured by CCD, Ophir Spiricon LT665). During dual-pass amplification, the pulse traverses the Nd:YLF modules, reflects off mirror M3, and exits. Gain saturation effects transform the injected Gaussian beam [M2<1.1; Fig. 2(a)] into a flat-top beam [Fig. 2(b)]. Post-amplification energy control is achieved via the λ/2 waveplate and polarizing beam splitter, keeping fluence below the LBO crystal damage threshold. Due to this threshold, the OASLM is placed upstream of amplification [Fig. 3(a)], with the LBO crystal at the beam’s image plane. The second-harmonic beam passes through dichroic mirrors (not shown) and a wedge prism into an image transfer system for spatial quality verification [Fig. 3(b)]. It is worth noting that LBO crystals are highly hygroscopic. To ensure stable operation, we employed a temperature-controlled crystal holder (40°C) to mitigate moisture absorption.

The nonlinear coupling between fundamental beam power density and SHG conversion efficiency was systematically investigated using the optimized beam profile from Fig. 2(b). Through phase-matched Type I interactions in a 20mm×20mm×25mm LBO crystal, experimental measurements were complemented by numerical simulations based on the derived SHG coupled equations (Fig. 4). In the calculation of SHG efficiency, the energies of the beam before and after SHG were both averaged over data collected during a period of time in the experiment to ensure the reliability of the results. Key findings reveal a characteristic efficiency-power dependence: conversion efficiency initially increases with fundamental power density before reaching saturation at ∼0.3GW/cm2 (for 25 mm crystal length), beyond which efficiency roll-off occurs due to back-conversion effects.

During the experiment, we observed a change in beam uniformity before and after the SHG process. To illustrate the relationship between SHG and beam uniformity, we shaped the seed using OASLM. Figure 5 illustrates the spatial distribution of the beam before and after shaping. In Fig. 5(a), the beam expanded by the regenerative amplifier exhibits a Gaussian distribution. By loading an appropriate grayscale pattern, we used OASLM to reduce the central intensity of the beam, as shown in Fig. 5(b). After saturation amplification by the double-pass Nd: YLF amplifier, the central intensity of the beam increased slightly, but remained significantly lower than the intensity at the edges, as shown in Fig. 5(c). This resulted in a hollow beam, with a maximum energy of 2 J, suitable for SHG. Notably, whether the OASLM was used for beam shaping, we incorporated a polarization beam splitter cube and a λ/2 waveplate combination to ensure continuous adjustment of energy of the fundamental wave. Furthermore, this adjustment method ensured that the spatial intensity distribution of the beam profile remained identical for different energies.

Guided by the experimentally measured second-harmonic efficiency curve and numerical simulations, the OASLM-structured hollow-core beam was selected as the fundamental excitation source. After passing through the LBO crystal, the transverse intensity profile of the second-harmonic beam was captured via the spot acquisition system, as shown in Fig. 6.

The spatially nonuniform SHG conversion efficiency of the incident beam induces intensity redistribution between the SHG and fundamental fields. Our analysis reveals that this intensity redistribution phenomenon manifests under two distinct regimes: 1) the power density of all (or most) regions within the beam is below the saturation point of second-harmonic efficiency; 2) the power density of all (or most) regions within the beam is near the saturation point of second-harmonic efficiency. In the non-saturation regime, regions with elevated power densities demonstrate enhanced second-harmonic conversion efficiencies, whereas areas with reduced power densities exhibit diminished efficiencies. This phenomenon amplifies the spatial heterogeneity of high-intensity regions across the beam profile. Conversely, when the power density in the high-intensity region of the beam profile decreases beyond the rollback threshold, regions with reduced power densities demonstrate enhanced second-harmonic efficiency, thereby improving the spatial uniformity of the beam profile during SHG.

In the experiment, we observed that the beam profile, characterized by a central intensity defect, becomes spatially homogenized during the SHG process. We further investigated the evolution of beam profiles exhibiting enhanced spatial uniformity. Since we used a low-repetition-rate laser source (1 Hz), the thermal effect in the LBO crystal can be considered negligible. Meanwhile, the gain saturation effect of the dual-pass laser amplifier cannot fully compensate for the spatial imperfections of the beam. Therefore, the variation in beam uniformity can be attributed to the SHG process itself. The coefficient of variation (CV) provides a statistically robust measure of dispersion for the experimentally collected beam profile dataset, owing to its invariance to measurement scale and dimensionality. Its mathematical formulation is defined as Cv=∑i=1,j=1N[E(i,j)−M]2/NM.(4)

In Eq. (4), M corresponds to the average pixel energy, E(i,j) represents the energy of an individual pixel at coordinates (i,j), and N denotes the total number of pixels. The statistical significance of CV lies in allowing comparison of the degree of dispersion between different datasets, even when their means differ. This helps eliminate the influence of individual high-intensity pixels during beam profile acquisition. Thus, the CV of pixel intensities serves as a quantitative measure for evaluating the global energy distribution uniformity of the beam profiles. Lower CV values reflect reduced regional intensity disparities, directly correlating with enhanced spatial homogeneity. To investigate this relationship, we first experimentally characterized the spatial distribution of the fundamental beam, then performed numerical simulations of the SHG process, and finally computed the post-SHG CV values across distinct pixel subsets. These results are systematically compiled in Fig. 7.

Figure 7 incorporates two key reference lines for benchmarking the results: the red line denotes the threshold for 80% SHG efficiency, while the blue line represents the CV of the fundamental beam. Together, both lines provide a dual framework for evaluating the quality of the SHG process. The red line serves as an efficiency benchmark, ensuring optimal energy conversion, whereas the blue line highlights variations in the fundamental beam’s energy distribution. By analyzing the interplay between these metrics, the figure facilitates a comprehensive assessment of the trade-offs between conversion efficiency and beam uniformity in SHG processes. This indicates that within the range shown in the figure, the CV value of the second-harmonic beam decreases with increasing fundamental power density, corresponding to the improvement of the beam uniformity in the spatial profile. Notably, the optimization of the beam uniformity aligns with the rollback region of the second-harmonic efficiency curve, concomitant with efficiency degradation. Consequently, an ideal operational region (shaded area in the figure) is identified, in which the reduction in second-harmonic conversion efficiency is confined within an acceptable threshold (>80%) without compromising beam profile optimization. In the experiment, at a fundamental power density of 0.51GW/cm2, we achieved a second-harmonic beam profile with a superior spatial distribution compared to the fundamental beam, while maintaining 85% second-harmonic efficiency. Since the improvement in beam uniformity comes at the cost of a decrease in SHG efficiency, a power density of 0.51GW/cm2 represents the balance between the two factors.

In conclusion, we demonstrated the systematic variation relationship between the SHG process in LBO and beam uniformity using the output pulse from a double-pass laser amplifier. Particularly when using a flat-top beam with a central defect as the fundamental, we observed that the defect was compensated for. This effect was further confirmed by our uniformity analysis of a perfectly flat-top beam before and after SHG, which showed that the beam’s spatial distribution was optimized when the fundamental power density reached at least 0.478GW/cm2, in agreement with our simulation results. We achieved approximately 85% SHG efficiency by injecting a high-quality flat-top beam with a power density of 0.51GW/cm2. For frequency-doubling crystals with different parameters, controlling the fundamental power density within an appropriate range allows the SHG process to optimize the beam’s spatial distribution without sacrificing much conversion efficiency. The SHG-based pump source can generate higher-quality beams by properly controlling both the fundamental power density and spatial distribution.