Objective In high-power laser facility, the ultraviolet laser damage of fused-silica optics becomes the bottleneck restricting its output capacity and is the key to affect the long-term stable operation of the laser facility. It is found that placing the fused-silica lens in front of the fourth harmonic generation (FHG) crystal can effectively reduce its damage probability. However, this scheme will cause the convergent beam to achieve harmonic conversion in the crystal and the FHG frequency decreases as a result, which limits its engineering application. It is an urgent problem to improve the FHG efficiency of convergent beam. DKDP crystal with non-critical phase matching is the best scheme to realize the high-efficiency FHG of large-aperture convergent beam. Based on the non-critical phase matching DKDP crystal, this paper firstly studies the influence of F number on the FHG efficiency of convergent beam systematically. Then the beam segmentation, temperature gradient distribution, and deuterium content gradient distribution methods are proposed to improve the FHG efficiency. This study is helpful to solve the limitation of the preposition scheme of fused-silica lens and lay a foundation for the wider application of the FHG beam in high-power laser facility.

Methods Based on the phase matching theory and the coupled wave equation, a theoretical model of FHG of large-aperture convergent beam is established. The crystal size of DKDP is set as 360 mm×360 mm×10 mm, and the crystal is not coated. The diameter double frequency (2ω) beam is 300 mm and the beam has an 11-order super-Gaussian distribution. In order to facilitate the control of crystal temperature, the non-critical phase matching temperature is set at 30 ℃, and the deuterium content of DKDP crystal is 64.5%. Based on the above theoretical model, this paper mainly studies the harmonic conversion process from 2ω to 4ω.

Results and Discussions It is found that the FHG efficiency of convergent beam decreases obviously when F≤30. The optimum matching temperature of convergent beam increases with the F number and incident 2ω intensity generally [Fig. 3(b)]. This paper theoretically simulates the change of the FHG efficiency with 2ω intensity at the optimum matching temperature for different F number systems (Fig. 4). The results show that when 2ω intensity is greater than 0.75 GW·cm ^-2the FHG efficiency of F≤10 system gradually tends to saturation, while the FHG efficiency of F=15--20 system slowly increases first and then decreases significantly with the increase of 2ω intensity. The FHG efficiency of F≥30 system continuously increases with the increase of 2ω intensity. Besides, at the same 2ω intensity the FHG efficiency rapidly increased with the F number when F≤30. As the F number reaches 30, the growth rate of FHG efficiency decreases and the FHG efficiency is more than 80%. In addition, when F≤20 the temperature and wavelength acceptance bandwidth of the system increase significantly [Fig. 3(a) and Fig. 5]. This is because the angle mismatch, temperature mismatch, and the wavelength mismatch have a superposition effect. The angle mismatch can partially compensate for the temperature mismatch and wavelength mismatch. For the FHG of convergent beams, the efficiency decrease is mainly due to the large phase mismatch at the position far away from the beam center. If the converging beam is divided into several sub-beams and each sub-beam is quadrupled individually, the FHG efficiency of the converging beam can then be effectively improved as a result (Fig. 7). Although sub-beam segmentation method can improve the FHG efficiency, it also increases the shielding area of each sub-beam. Therefore, for small F-beam systems, the beam segmentation method still makes it difficult to achieve the FHG efficiency of more than 70%. The phase mismatch is a function of angle, temperature, and deuterium content. The phase matching can be realized by adjusting the temperature or deuterium content at the non-central position point of the crystal to compensate for the deviation of incident light angle. It should be noted that this method will lead to each point temperature or deuterium content on crystal changing with crystal position, namely the temperature or deuterium content on the crystal shows a gradient distribution. Therefore, this method can make the convergent beam meet the phase matching at each point on the DKDP crystal and the FHG efficiency can reach more than 80%. The variations of temperature and deuterium content gradient distribution with convergent beam F number are given in detail (Fig. 8).

Conclusions Based on non-critical phase-matched DKDP crystal, the influence of F number on the FHG efficiency of convergent beam is systematically studied in this paper. The results show that the FHG efficiency rapidly increases with the F number when F≤30. As the F number reaches 30, the growth rate of FHG efficiency decreases and the FHG efficiency is more than 80%. In addition, when F≤20, the temperature and wavelength bandwidth of quadruple frequency are significantly increased. Therefore, the convergent beam with a small F number is more conducive to the FHG frequency of the broadband laser. For the small F-number system, this paper proposes a beam segmentation method to improve the FHG efficiency of convergent beams. However, the overall FHG efficiency is still difficult to reach more than 70% due to the beam shielding between sub-beams. In order to further improve the FHG efficiency of the convergent beam, the methods of temperature gradient distribution and deuterium content gradient distribution are proposed to make the convergent beam meet the phase matching at each point on the DKDP crystal, so that the FHG efficiency of the convergent beam can reach more than 80%. In theory, matching temperature gradient distribution and deuterium content distribution can be designed for each F number, but in practice, it is not easy to achieve the gradient distribution for large F number (F≥30) or small system (F≤5). It can be known from the above results, the beam segmentation method, the temperature gradient distribution and crystal deuterium content gradient distribution method can improve the FHG efficiency of convergent beam dramatically, but every method has its own limitations. In engineering practice, in order to reduce the implementation difficulty of every method, these methods can be combined to develop a specific scheme for the FHG of a specific F number.