The simplified regenerative amplifier concept is shown in Figure 1. The amplifier contains two cavity mirrors (CM1 and CM2), a laser head (LH), a Pockels cell (PC), a quarter-wave plate (QWP), a thin-film polarizer (P), and a lens (L). The L, CM1, and CM2 are used to determine the mode size. The PC and QWP are used to control the amplification pass. The cavity length is 6750mm, which corresponds to 45 ns round trip. Thus, a laser pulse with a maximum of 25 ns temporal width can be amplified, considering the 10 ns rising edge of the PC.

The input laser with s-polarization is injected into the amplifier cavity from P. When the input laser first passes through the PC, the PC is powered off. After a round trip, the laser turns to be p-polarization with the help of QWP. After the laser passes through the PC for the second time, the PC is powered with a quarter-wave voltage, and the function of the PC and QWP is similar to that of a half-wave plate. The laser polarization state keeps constant during a round trip, and the laser can be amplified successively until the PC is powered off. The output laser with s-polarization is outputted from P. The total amplification pass is fix(t×c/l), where t is the voltage width of PC, c is the speed of light, l is the cavity length, and the function fix(x) is obtaining the maximum integer of x.

In practical multi-beam regenerative amplifiers, due to the parameter differences of optical components (e.g., reflectivity and transmission), the total loss of the cavity is different. Moreover, in laser–matter interactions with multi-beam lasers, the moments of multi-beam lasers arriving at the target must be the same. Thus, the amplification pass (or the voltage width of PC) should be identical. In this circumstance, the output energy and energy stability exhibit obvious differences from beam to beam. Here, the typical parameters of optical components are shown in Table1. In this table, four configurations of parameters are included. The design baseline is the typical parameter of the optical component. The minimum value, maximum value, and mean value are obtained by actually measuring the minimum, maximum, and mean parameters from over 100 real components.

With these parameters in Table1, the energy curves of different configurations are calculated theoretically. In the theoretical calculations, the Avizonis–Grotbeck model is adopted [20]. Compared with the commonly used Frantz–Nodvik model [21], the Avizonis–Grotbeck model involves the loss of gain medium, which is important, especially for rod-shaped gain medium and multi-pass amplification, which makes the theoretical simulations more exact. Here, the static loss and dynamic loss of Nd:glass are 0.12% cm⁻¹ and 0.36% cm⁻¹, respectively, which are obtained from the Nd:glass factory. The static loss and dynamic loss are the loss without and with LD pumping, respectively. Combined with the input energy of ∼0.1 μJ, the theoretical energy curves are calculated, as shown in Figure 2. In the figure, the small-signal gain and the amplification pass are chosen to be 1.269 and 96-pass, respectively, so as to satisfy the output energy of 25 mJ and excellent energy stability at design baseline (black line in Figure 2). Due to the different losses, the energy and energy stability of the four configurations show undesirable disparities. For configurations of the minimum value, mean value, and maximum value, the output energy is far larger than 25 mJ before the laser is exported from the regenerative amplifier. The optical damages are probably induced and affect the safe operations. Moreover, the performances of energy stability are also different. For design baseline, the curve at 0.1 μJ is nearly flat. It means that the output energy is extremely insensitive to the fluctuations of input energy. However, for the minimum value, mean value, and maximum value, the output energy varies with the fluctuations of input energy, which will influence the power balance of multi-beam lasers.

In order to obtain uniform energy stabilities of multi-beam lasers and avoid optical damage, an active method of controlling the product of gain and loss is adopted. Here, by slightly adjusting the small-signal gain, the products of gain and loss of multi-beam regenerative amplifiers are uniform. With this method, the optimized small-signal gain of four configurations are 1.269, 1.254, 1.233, and 1.242, respectively. The optimized energy curves of four configurations are shown in Figure 3. From the figure, the output energies of four configurations are all in the safe operating range of 25–35 mJ. The energy stabilities are nearly uniform and not sensitive to the fluctuations of the input laser.

The laser head is a diode-side-pumped circular Nd:glass rod with a 1 Hz repetition rate. The profile is shown in Figure 4. The laser head consists of Nd:glass, cooling fluid, quartz tube, and pump ring module. The geometric size of the rod is Ф5 mm × 100 mm. The gain medium is N31 Nd:glass whose Nd³⁺ concentration is 2.2 wt% [22, 23]. The ends of the rod are tapered to stifle unwanted whispering gallery parasitic modes. The quartz tube is 1 mm thickness, which is used to separate the cooling fluid from the diode. The pumping ring module is a 5-side ring-pumping scheme with a pumping power of 2.5 kW, and two pumping ring modules are incorporated. The center wavelength of the diode is 802 nm.

In order to obtain excellent beam quality, the pumping is required to be uniform[24]. The simulated pumping uniformity with the aforementioned parameters is shown in Figure 5 using the ray-tracing method. The uniformity in the center 3 mm × 3 mm region is 91.8%.

The small-signal gain is estimated according to the energy-transfer mechanism [25], including the pumping transfer efficiency, absorbing efficiency, quantum efficiency, Stokes efficiency, and storage efficiency. In this case, the total energy transfer efficiency (η _tot ) is 26%. The maximum small-signal gain is estimated to be 1.95 using Eq. 1. It is slightly larger than the aforementioned required small-signal gain. G 0 = exp ( η t o t P p t p E s A ) (1) where P _p is the pumping power; t _p is the pumping width, which is 500 μs; E _s is saturated fluence (4.97 J/cm² for N31 Nd:glass); and A is the section area of the gain medium.

Due to the long cavity length, even weak focal lensing can contribute to large changes in mode radius for non-optimized cavity configurations, i.e., undesired disparities across multi-beam lasers. For multi-beam regenerative amplifiers, on the one hand, as mentioned previously, in order to obtain uniform output energy and energy stability, the small-signal gain of each beam is strongly dependent on the optical loss. This means the pumping power and induced thermal lensing are different. According to the laser head design and thermal analyses, at the small-signal gain of 1.269 (design baseline in Section 2.2), the corresponding thermal lensing is about 25 m. Furthermore, one can easily estimate the thermal lensing for the other three cavity loss configurations, which are 23.75 m (minimum value), 22.74 m (mean value), and 21.98 m (maximum value), respectively. On the other hand, the diode pumping fluctuations induced by power supply and the machining errors of optical components such as CM1, CM2, and L will also cause mode radius changing. Thus, the cavity should be stable over a large range of the thermal lens.

The cavity design is developed by utilizing an analytic ABCD description [20]. By varying the end mirror and cavity lens radius and the distance of optical components in Figure 1, the stability zones for each cavity parameter were evaluated. Only the cavity parameters meeting the following constraints were analyzed: first, the mode radius in the rod is ∼1.5 mm, so as to obtain large energy without damage risk. Second, the cavity should be stable with the thermal lensing changing from 21.98 to 25 m or more. This is necessary when changing the small-signal gain for multi-beam lasers. Third, the mode radius on the other optical components is larger than 0.5 mm to avoid optical damages. With these criteria, the optimized cavity parameters are as follows: the radii of the curvature of CM1 and CM2 are −10,000 and 10,000 mm, respectively. The focal length of L is 2,430 mm, and its position is at the center of the cavity. The distance between LH and CM2 is 2,750 mm. The corresponding stability zones are shown in Figure 6. In the figure, the x-axis is the thermal focal power, which is reciprocal to thermal lensing, and the y-axis is the mode radius. From the figure, at the thermal focal power of 4 × 10^–5 mm⁻¹ (i.e., 25 m for thermal lensing), the cavity mode size is insensitive to the thermal lensing variations. When the thermal focal power varies by ±20%, the mode radius at Nd:glass changes to smaller than 0.1%. Considering all the factors that may result in mode size changes including the diode pumping fluctuation, the thermal lensing variations, and machining errors of optical components, the total mode radius changes are small. This is negligible. With the parameters, the mode size of the regenerative cavity is calculated, as shown in Figure 7. The mode radii in the laser head (LH), lens (L), and cavity mirrors (CM1 and CM2) are 1.5, 1.75, 0.98, and 0.65 mm, respectively.