## Abstract

A numerical modeling is developed for 30-kW class liquid-convection-cooled elastically-mounted Nd:YAG multi-slab laser resonator configuration. The modeling exhibits the thermal effects and resultant wavefront aberration of the gain module under flow cooling and CW pumping at 100-kW level, the self-reproducing oscillating mode within the large-aperture cavity, as well as the beam quality enhancement by adaptive optics. The simulation results predict a CW output power of 31 kW with the optical-optical efficiency of 26.1% obtained from a modified resonator configuration with dual gain modules that have opposite flow directions, while the beam quality can be improved to *β*<2 after the correction of a deformable mirror.

© 2015 Optical Society of America

## 1. Introduction

To dissipate the heat load deposited in the gain medium, most high power solid-state lasers nowadays are conduction cooled, in which the heat is conducted to a heat sink [1, 2]. However, the conduction-cooled high power lasers still suffer from severe thermal effects, which would degrade the beam quality and limit the maximum injected pump power [3, 4].

Liquid-convection-cooled configuration [5–7], in which the circulating liquid flows over all surfaces of the gain medium to carry away the deposited heat, has gained much attention recently, mainly because it has shown advantages over traditional conduction cooling method especially in terms of producing high power lasers. In 2010, Textron Defense Inc. presented a state-of-the-art liquid-convection-cooled configuration called ThinZag that used Nd:YAG ceramic thin slabs as the gain medium while the laser beam passed through the thin slabs and flow layers in a zigzag manner. Based on ThinZag structure, 27 kW of output power was achieved from a single oscillator module, and 100 kW was obtained from the single aperture with six modules placed within the cavity [8]. General Atomics Corp. is developing the 150 kW “liquid laser” that consists of ten 15-kW liquid-cooled modules, with the total weight no more than 750 kg, which would be 10 times smaller and lighter than lasers of similar power [9–11].

Recently, our group reported a novel design of the liquid-convection-cooled Nd:YAG laser resonator [12], which employs the straight-through geometry that the oscillating laser propagates through multiple thin slabs and multiple cooling flow layers in Brewster angle. More particularly, in order to minimize the risk of thermal stress fracture under ultrahigh thermal load, the Nd:YAG thin slabs are elastically held by a flexible supporter that has several tiny grooves, which serves to greatly reduce the external constraint on the slab, permitting the deformation induced by the thermal gradient to occur freely to its fullest extent and thus dramatically alleviate the thermal stress. The experiment of Nd:YAG multi-slab resonator produced a continuous-wave (CW) output power of 3 kW with an optical-optical efficiency of 15.1% and a slope efficiency of 21.2%, while the beam quality is poor. The experiment results indicate that stable kW-class operation can be obtained from the configuration of liquid-convection-cooled multi-slab resonator, and explicitly demonstrate the feasibility and validity of the elastical mounting approach of thin slabs, of the efficient heat removal under 20-kW pumping by using the 0.5-mm-thick cooling flow layers, and of the high-intensity uniform gain profile by fan-like distribution of laser diode stacks. Furthermore, the 3 kW output curve presented by [12] has an excellent linearity with no sign of saturation, which suggests a strong power scaling capability at much higher level.

In this paper, a numerical modeling is developed for 30-kW class liquid-cooled multi-slab configuration. The modeling demonstrates the thermal effects and resultant wavefront aberration under flow cooling and pumping at 100-kW level, the self-reproducing oscillating mode within the large-aperture cavity, as well as the beam quality enhancement by adaptive optics. According to the simulation results, a CW output power of 31 kW with the optical-optical efficiency of 26.1% is expected from a modified resonator configuration with dual gain modules that have opposite flow directions, while the beam quality is predicted to be improved to *β*<2 after the correction of a deformable mirror with the interval of actuators of 4 mm.

## 2. Configuration

The configuration of liquid-convection-cooled large-aperture laser oscillator discussed in this paper (see Fig. 1) is basically same as the experiment setup reported in [12]. The resonator consists of a total-reflection mirror (HR) and an output mirror (OC), while multiple parallel-arranged elastically-mounted Nd:YAG thin slabs that are sandwiched between a pair of fused silica windows are cooled by heavy water that flows over all slab surfaces in the 0.5-mm-thick channel. Heavy water with the deuteration degree as high as 99.8% is used rather than the deionized water because of much lower absorption losses for light at 808 nm and 1064 nm. All the thin slabs and windows are uncoated, and the oscillating laser beam passes them near the incidence of Brewster angle with low reflection losses. At each side of the gain module there are 15 laser diode (LD) arrays arranged in a fan shape for achieving uniform pump profile. Each LD array that contains 20 LD bars emits a maximum output power of 2000 W at 808 nm, with the fast axis collimated by microlenses.

There are two main differences about the slab parameters between the modeling in this paper and the previous 3 kW experiment setup. One is that the slabs herein are bigger for the sake of 30 kW class power scaling, with the dimension of 172 mm × 36 mm (length × width) and the clear aperture of 165 mm × 35 mm. The other difference is that seven Nd:YAG slabs are chosen as the gain medium in the modeling, with the same doping concentration of 1 at.%. In order to balance the deposited heat power, various thicknesses from 1.1 mm to 2.2 mm are adopted individually for the seven slabs that the slab closer to the pump source is thinner, as indicated in Table 1. Although the adoption of slabs with various thicknesses for the simulation is different from the approach using slabs with various doping concentrations in [12], both methods are aimed at achieving relatively uniform heat load.

Using the software Tracepro, the simulated pump intensity at each slab surface is obtained, while 60 kW of total pump power is injected into the gain module. Figure 2 describes a highly uniform pump profile at the central region of pump surface of slab #1 (outer slab) and slab #4 (central slab), with the root meam square (RMS) value of 5.9% and 7.8% respectively. The uniform pump distribution is an important precondition for efficient power extraction. In addition, the absorbed pump intensity distribution and deposited heat distribution for all the seven slabs can be calculated from the results of ray tracing, thus providing the data for the modeling of thermal effects and optical gain in the following.

## 3. Thermal modeling

Besides the deposited heat power within the slabs that obtained in Section 2, the heat load of the cooling liquid due to the absorption of pump power and laser power is taken into account as well, considering that the absorption coefficient of heavy water with the deuteration degree of 99.8% is 0.023 cm^{−1} at 1064 nm and 0.013 cm^{−1} at 808 nm as measured in [12].

Assuming the fluid properties are constant and the flow field is three-dimensional, the Navier-Stokes equations of Newtonian fluids governing the flow field and the heat transfer process are given as

**u**is the flow velocity vector,

*u*is the velocity component in

_{i}*x*,

*y*or

*z*direction,

*p*is the pressure,

*T*is the temperature distribution,

*μ*is the dynamic viscosity coefficient of cooling liquid,

*k*is the thermal conductivity of the liquid, and

*c*is the heat capacity.

_{p}The temperature distribution of liquid layers and the liquid-cooled slabs can be obtained by the software Fluent, by combining and solving the Navier-Stokes equations and the thermal conduction equation of solid-state gain medium. Before turning to the software Fluent, we have to determine whether the flow pattern in our case is laminar or turbulent, by calculating the Reynolds number. For the flow between parallel plates, the Reynolds number is defined as

where*u*

_{0}is the average flow rate between parallel plates,

*D*= 2

_{h}*b*is the hydraulic diameter of the channel between two parallel plates while

*b*is the channel thickness, and

*ν*is the kinematic fluid viscosity coefficient. For our case, we have

*ν*= 0.9915 × 10

^{−6}m

^{2}s

^{−1}for heavy water at 25°C, while the flow layer thickness of

*b*= 0.5 mm and the cooling flow rate of

*u*

_{0}= 2 m/s are the same as that in [12], thus leading to a Reynolds number of 2017, which is smaller than the critical Reynolds number of 2300, that is, the most important factor in determining transition from laminar to turbulence. In addition, the flow is guided through a channel as narrow as the slab interval long before it arrives at the slab interface, to make sure the transition from turbulent to laminar occurs in advance. Therefore the flow pattern that interacts with the gain medium is laminar in this modeling, and the laminar model of Fluent is used for solving these thermal equations, while the analysis of wavefront aberration induced by turbulent flow field in liquid-convection-cooled lasers can be found in our previous publication [13].

The calculated three-dimensional temperature profile of the central slab is presented in Fig. 3. As the heavy water travels across the slab surface along the flow direction, it is constantly heated by the deposited heat from adjacent slabs. As the heat power accumulates, the flow has a higher temperature at the exit than at the entrance, therefore the slab temperature increases along the flow direction. Figure 3 also shows that the temperature of the central slab dramatically increases from 283 K at the flow entrance to 337 K at the flow exit, under the total pump power of 60 kW for the gain module.

## 4. Wavefront aberration

The wavefront aberration as represented by the optical path difference (OPD) can be divided into three parts: OPD due to temperature gradient (OPD_{t}), OPD due to thermal deformation of slabs (OPD_{d}), and OPD due to stress-induced change of refractive index (OPD_{s}). However, the third term has much weaker influence than the other two terms [14, 15] and thus is neglected hereinafter. It should be noted that in our modeling OPD_{t} is calculated assuming no thermal deformation is occurred, while OPD_{d} is the variation of OPD due to thickness changes of slabs and heavy water layers, since Nd:YAG slabs under large heat load expand and replace the fluid. Thus we have

*n*

_{g}(

*x*,

*y*,

*z*) and

*n*

_{f}(

*x*,

*y*,

*z*) is respectively the distribution of temperature-dependent refractive index of Nd:YAG and heavy water,

*L*

_{0}is the optical path with no deformation assumed (as shown in Fig. 4) going through seven slabs and eight liquid layers, and Δ

*L*

_{j}(

*x*,

*y*) denotes the slab deformation at each solid-liquid interface. The temperature-dependent refractive index of heavy water

*n*

_{f}is given in [16], while the refractive index of Nd:YAG can be calculated aswhere

*T*

_{0}is the original temperature, and d

*n*/d

*T*is assumed as 7.3 × 10

^{−6}/K.

In the OPD analysis, the fractional heat load of 24% and 30% are used for lasing and non-lasing conditions according to [17], while the laser beam mode size within the gain module is assumed as 90% of the pump size.

#### 4.1 Aberration due to temperature gradient

The optical path difference due to the temperature gradient, experienced by the beam going through the gain module can be obtained by integrating the temperature-dependent refractive index of thin slabs and liquid layers along the beam path at Brewster angle (see Fig. 4), while the beam path as well as relative positions of the Nd:YAG slabs, heavy water layers and fused silica windows was detailed in [12].

Figure 5(a) and 5(b) show the OPD_{t} for undeformed model of the outer slab and the central slab respectively. It is shown that the OPD_{t} is almost uniform along *y* direction except at the edges, while the OPD_{t} along *x* direction (flow direction) is much higher at the flow exit than that at the flow entrance, which coincides with the temperature distribution along *x* direction due to heating process of the heavy water. In addition, the central slab has a higher peak-to-valley (PV) value of the OPD_{t} at 0.84 μm than the outer slab since it is two times thicker despite of a slightly lower heat load.

Similarly the OPD_{t} of single liquid layer has a flat distribution along the flow direction, and the OPD_{t} of outer liquid layer has a larger PV value (1.65 μm) than that of the central layer (1.34 μm) due to stronger heat transfer, as shown in Fig. 5(c) and 5(d). However, the PV value of OPD_{t} of single liquid layer is higher at the flow entrance than at the exit since the heavy water has a negative dn/dT. Figure 6(b) describes the total thermal-gradient-induced OPD experienced by the beam passing the single gain module (including seven slabs and eight liquid layers), resulting in a PV value as high as 10.94 μm, in which the OPD_{t} of liquid layers plays the major role.

#### 4.2 Aberration due to thermal deformation

The OPD due to thermal deformations, OPD_{d}, is calculated as below, taking slab #1 in the gain module under full pump power of 60 kW as an example. The two large surfaces of slab #1 are defined as surface A (i.e. the pump surface) and surface B (the other large surface, which is farther to the pump source). The thermal deformation Δ*L*_{1}(*x*, *y*) for surface A and Δ*L*_{2}(*x*, *y*) for surface B are analyzed using the software Ansys, with the results given in Figs. 7(a) and 7(b) respectively. According to Fig. 7, most of the region at surface A expands with the deformation close to 1 μm, while in the surface B the expansion mainly occurs at two edges, with the peak value reaching 1 μm.

Thus the OPD_{d} for slab #1 is calculated as

*T*

_{1}(

*x*,

*y*) and

*T*

_{2}(

*x*,

*y*) is respectively the temperature profile at surface A and B.

Figure 7(c) describes the calculated result of OPD_{d} for slab #1, with the maximum aberration value of 0.09 μm, while the calculation of OPD_{d} for other solid-liquid interfaces can be carried out in the same way.

#### 4.3 Combined aberration

The PV values of calculated OPD_{t} and OPD_{d} for all the slabs are summarized in Table 1. The total phase aberration of the gain module combining OPD_{t} and OPD_{d} is illustrated in Fig. 8(a). It can be seen that the phase aberration is increasing monotonically along the flow direction. Since it is difficult to obtain a stable self-reproducing mode in the resonator with such an asymmetrical phase aberration, an modified resonator configuration is adopted (see Fig. 9) that two same gain modules with opposite flow directions are placed within the cavity to compensate the non-uniform distribution of refractive index mainly due to cooling flow. The combined phase aberration of the dual-module configuration is shown in Fig. 8(b), demonstrating the symmetrical aberration profiles in both *x* and *y* direction. Figure 10 compares the phase aberration along the flow direction for the cases of single module and double modules, showing that in the latter case, the maximum phase aberration drops by 63.2% from 6.65μm to 2.45μm.

## 5. Resonator output and beam quality

Based on the calculated wavefront aberration obtained in Section 4, the output mode as well as the output power of the dual-module-oscillator is studied in this section. A commercially available software GLAD that is widely applied for optical resonator analysis [18], is used to simulate the oscillating mode and thus calculate the *β* factor of the output beam quality. Detailed definition of *β* factor can be found in [19]. Furthermore, the resonator output power is predicted while considering the effect of amplified spontaneous emission (ASE).

#### 5.1 Output mode and beam quality

For the simulation of resonator mode, both the intra-cavity phase aberration as described in Fig. 10 and the optical gain are taken into account. The simulation results of GLAD indicate that there cannot exist a stable self-reproduction mode with a stable cavity configuration such as a flat-flat cavity or a flat-concave cavity. Therefore a flat-convex unstable cavity is adopted, which in the simulation can successfully realize the stable self-reproduction mode.

The case with a variable reflectivity mirror (VRM) output coupler having a 6th-order super-Gaussian reflectivity profile in radial direction is compared with the case that uses a uniform reflectivity mirror (URM) output coupler, for the flat-convex unstable cavity that has a cavity length of 3 m and the HR mirror with the radius of curvature of 50 m. The central reflectivity of the VRM output coupler is 80% while the transmission of the URM output coupler is 20%. The near-field intensity and phase distributions of the output beam for the two cases are depicted in Table 2. The *β* factor of output beam quality is *β*_{x} = 21.50, *β*_{y} = 14.07 for URM case and *β*_{x} = 6.39, *β*_{y} = 14.98 for VRM case, demonstrating the improvement of beam quality in the horizontal direction by using the VRM output coupler.

To further improve the output beam quality for VRM case, a deformable mirror outside the cavity is added in the simulation to correct the wavefront aberration of the output beam. The deformable mirror with enough clear aperture and two-dimensional array interval of actuators of 8 mm × 8 mm, 4 mm × 8 mm and 4 mm × 4 mm is adopted respectively in the simulation code, after the wavefront correction by which, the corresponding *β* factors of the output beam quality in both directions are 7.07 × 8.64, 3.36 × 1.84, and 1.29 × 1.77 individually. As shown in Table 3, the deformable mirror serves to reduce the fluctuations among near-field phase profile, thus resulting in a more concentrated far-field intensity distribution (evidently reducing the number of side lobes in both directions). In addition, as the interval of actuators reduces, the correction effect of the deformable mirror enhances significantly and the beam quality improves.

#### 5.2 Resonator output power

The oscillator output power with inclusion of ASE is studied in the following, based on the theoretical model given by G. Haag, *et al.* [20], according to which the output power with ASE can be calculated as

*R*is the reflectivity of the output coupler,

*r*is the reflectivity ASE flux at the output coupler that is assumed as 20%, and other parameters given as

*c*=

*α*/

*g*

_{0},

*α*is the loss factor,

*g*

_{0}is the small-signal gain, and

*σ*(0) can be solved from

*L*is the single-pass gain length, and

*A*

_{3}= 2[(1-

*c*) + 4

*Ac*/(1-

*c*)], in which

*A*is the coefficient that describes the source term of spontaneous emission withwhere

*τ*

_{u}is the excited-state lifetime,

*τ*

_{s}is the spontaneous lifetime, Δ

*ν*

_{s}is the bandwidth of the spontaneous fluorescence, Δ

*ν*

_{n}is the width of the narrowed ASE line, and dΩ/4π is the average solid angle contributing to ASE output withwhere

*ρ*

_{0}is the aspect ratio

*L*/

*D*

_{0}, while

*D*

_{0}is the transverse dimension of the gain medium. Apparently it can be deduced from Eq. (11) and Eq. (12) that the ASE intensity becomes stronger as the transverse size of gain medium increases. In the case of double-module resonator, we have

*D*

_{0}= (35

^{2}+ 165

^{2})

^{0.5}= 168.7 (mm) and

*ρ*

_{0}= 0.16. Thus we obtain dΩ/4π = 0.46.

The ratio of Δν_{n} to Δν_{s} in Eq. (11) can be calculated as [21]

_{n}/Δν

_{s}= 0.83, with

*g*

_{0}

*L*= 0.42,

*G*= exp(

*g*

_{0}

*L*) = 1.52, and

*g*

_{0}/

*α*= 10. Then we have

*A*= 0.34 with

*τ*

_{u}= 230 μs and

*τ*

_{s}= 260 μs. Therefore the output power can be obtained combining Eq. (8)-Eq. (10).

Figure 11 shows the oscillator output power with inclusion of ASE, varying as a function of different output coupling of the cavity, under the total pump power of 120 kW for two gain modules. The case with central transmission of VRM output coupler of *T*_{r} = 35% has the highest CW output power of 31.3 kW, corresponding to an optical-optical efficiency of 26.1% and a slope efficiency of 40.0%. Figure 12 describes the output power curve vs. pump power with *T*_{r} = 35%, comparing the cases with and without consideration of ASE effects. It is shown that with the ASE included, the predicted maximum output power reduces from 37.1 kW to 31.3 kW, while the threshold pump power increases from 31.2 kW to 41.8 kW.

## 6. Conclusion

In this letter, we present a numerical modeling for 30-kW class liquid-convection-cooled elastically-mounted Nd:YAG multi-slab laser resonator configuration. The temperature and gain distribution as well as the wavefront aberration induced by the temperature gradient and the thermal deformation is simulated for the liquid-cooled gain modules under CW pump power of 100-kW level. With the obtained gain distribution and wavefront aberration, the self-reproducing mode within the large-aperture cavity is studied. According to the simulation results, a CW output power of 31.3 kW with the optical-optical efficiency of 26.1% and the slope efficiency of 40.0% is expected to be obtained from the flat-convex resonator configuration with dual gain modules that have opposite flow directions, with the beam quality of *β*_{x} = 1.29, *β*_{y} = 1.77 after the correction of a deformable mirror with the interval of actuators of 4 mm. For practical development of a liquid-cooled dual-module slab resonator at 30-kW level, the gain condition and flow condition should be carefully maintained as equal for the two modules, as an attempt to realize the symmetrical phase aberration and thus stable large-aperture laser output.

## Acknowledgments

The research was supported in part by Tsinghua University Initiative Scientific Research Program (grant 2012THZ05108), in part by the National Natural Science Foundation of China (grant 61308043 and grant 51021064), and in part by China Postdoctoral Science Foundation funded project (2013T60108).

## References and links

**1. **A. Minassian, B. Thompson, and M. Damzen, “Ultrahigh-efficiency TEM_{00} diode-side-pumped Nd:YVO_{4} laser,” Appl. Phys. B **76**(4), 341–343 (2003). [CrossRef]

**2. **S. Tokita, J. Kawanaka, M. Fujita, T. Kawashima, and Y. Izawa, “Sapphire-conductive end-cooling of high power cryogenic Yb:YAG lasers,” Appl. Phys. B **80**(64), 634–638 (2005).

**3. **A. E. Blume and K. F. Tittel, “Thermal effects in laser amplifiers and oscillators,” Appl. Opt. **3**(4), 527–530 (1964). [CrossRef]

**4. **V. Sazegari, M. R. Milani, and A. K. Jafari, “Structural and optical behavior due to thermal effects in end-pumped Yb:YAG disk lasers,” Appl. Opt. **49**(36), 6910–6916 (2010). [CrossRef] [PubMed]

**5. **H. Okada, H. Yoshida, K. Sumimura, T. Sato, H. Fujita, and M. Nakatsuka, “Large-clear-aperture Nd:Cr:YAG split-disk laser amplifier,” in *Conference on Lasers and Electro-Optics/Pacific Rim* (2007), paper WP_015. [CrossRef]

**6. **J. Wang, J. Min, and Y. Song, “Forced convective cooling of a high-power solid-state laser slab,” Appl. Therm. Eng. **26**(5-6), 549–558 (2006). [CrossRef]

**7. **H. Okada, H. Yoshida, H. Fujita, and M. Nakatsuka, “Liquid-cooled ceramic Nd:YAG split-disk amplifier for high-average-power laser,” Opt. Commun. **266**(1), 274–279 (2006). [CrossRef]

**8. **A. Mandl and D. E. Klimek, “Textron’s J-HPSSL 100 kW ThinZag® laser program” in *Conference on Lasers and Electro-Optics* (2010), paper JThH2. [CrossRef]

**9. **V. Coffey, “High-energy lasers: new advances in defense applications,” Opt. Photonics News **25**(10), 28–35 (2014). [CrossRef]

**10. ***“*High Energy Liquid Laser Area Defense System*,”*http://en.wikipedia.org/wiki/High_Energy_Liquid_Laser_Area_Defense_System.

**11. **M. D. Perry, P. S. Banks, J. Zweiback, and R. W. Schleicher, “Laser containing a distributed gain medium,” U.S. Patent 7,366,211 (April 29, 2008).

**12. **X. Fu, P. Li, Q. Liu, and M. Gong, “3kW liquid-cooled elastically-supported Nd:YAG multi-slab CW laser resonator,” Opt. Express **22**(15), 18421–18432 (2014). [CrossRef] [PubMed]

**13. **P. Li, X. Fu, Q. Liu, and M. Gong, “Analysis of wavefront aberration induced by turbulent flow field in liquid-convection-cooled disk laser,” J. Opt. Soc. Am. B **30**(8), 2161–2167 (2013). [CrossRef]

**14. **P. Ferrara, M. Ciofini, L. Esposito, J. Hostaša, L. Labate, A. Lapucci, A. Pirri, G. Toci, M. Vannini, and L. A. Gizzi, “3-D numerical simulation of Yb:YAG active slabs with longitudinal doping gradient for thermal load effects assessment,” Opt. Express **22**(5), 5375–5386 (2014). [CrossRef] [PubMed]

**15. **S. Chénais, F. Druon, S. Forget, F. Balembois, and P. Georges, “On thermal effects in solid-state lasers: the case of ytterbium-doped materials,” Prog. Quantum Electron. **30**(4), 89–153 (2006). [CrossRef]

**16. **S. Kedenburg, M. Vieweg, T. Gissibl, and H. Giessen, “Linear refractive index and absorption measurements of nonlinear optical liquids in the visible and near-infrared spectral region,” Opt. Mater. Express **2**(11), 1588–1611 (2012). [CrossRef]

**17. **D. L. Kim and B. T. Kim, “Fracture characteristics of ceramic Nd:YAG,” Opt. Express **22**(9), 11331–11339 (2014). [CrossRef] [PubMed]

**18. **“Applied Optics Research,” www.aor.com

**19. **Y. Tan and X. Li, “Numerical analysis of beam quality factor β on coherent combination of multiple laser beams,” Proc. SPIE **8551**, 85511C (2012). [CrossRef]

**20. **G. Haag, M. Munz, and G. Marowsky, “Amplified spontaenous emission (ASE) in laser oscillators and amplifiers,” IEEE J. Quantum Electron. **19**(6), 1149–1160 (1983). [CrossRef]

**21. **A. M. Hunter II and R. O. Hunter, Jr., “Bidirectional amplification with nonsaturable absorption and amplified spontaneous emission,” IEEE J. Quantum Electron. **17**(9), 1879–1887 (1981). [CrossRef]