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Abstract
High repetition rates in high energy solid-state laser systems can yield to a rise of temperature in amplifiers despite the use of cooling systems. This effect can significantly impact the performance of amplifiers by inducing thermal stress, birefringence or thermal lensing. Here, we develop a multiphysics model to support the design, optimization and commissioning of a liquid-cooled large aperture split-slab laser glass amplifier. This multiphysics model includes optical pumping in the amplifying medium, heat loading, hydraulic effects induced by the liquid coolant, mechanical deformation and their potential coupled effects on the optical wavefront. The accuracy of each model is assessed by carrying out specific experimental measurements and characterizations. We show that this set of models allows the prediction of performance of a liquid-cooled amplifier from the flash-lamp emission to the amplified wavefront at a repetition rate of one shot per minute.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High power solid-state lasers are impacted by the generation of heat in optical amplifiers [1]. Heat created during optical pumping of the amplifier medium is usually removed by an external liquid exchanger and/or by forced gas flow. When laser systems operate at high energy and/or high repetition rate, this cooling technique cannot avoid the generation of heat that can compromise laser performance through thermal stress leading to fracture in worst cases, stress birefringence, thermal lensing among other physical mechanisms.
Liquid cooling was considered in early 1970s [2] but in practice, very few practical realizations have been referenced. Hence, liquid cooling remains very difficult to manage. The coolant surrounding the laser media has to be transparent to pumping laser wavelengths and shall preserve its transparency during long range of time, often in presence of intense ultraviolet light. Viscosity has to be low to warrant a good flow through a simple pumping system and to limit any deformation in the amplifier [3]. Moreover, the flow and the potential effect of temperature on the liquid coolant refractive index have to be managed to preserve the wavefront quality. Overall the difficulty relies on the ability to manage not only this coolant liquid flow but also its coupling with the laser amplifiers, in presence of all potential hydro-thermo-mechanical-optical effects.
More recently, liquid cooling was further investigated in the context of split thin-disk amplifiers for high average power high repetition rate laser systems. The Nd:YAG split disk geometry concept proposed by Okada delivering 17 W [4] has been improved up to the 10–100 kW range by different groups [5–8]. In this specific case, centimeter scale beam apertures with rather low beam quality is needed. The liquid flow is simulated to optimize the uniformity of the liquid coolant [6,9,10], and to estimate its effect on the wavefront aberration by integrating the thermal-induced variations of the liquid refractive index [10].
As an alternative to gas flow [11], liquid cooling is now pushed to a new extend for high power large aperture lasers in the framework of the Extreme Light Infrastructure (ELI) [12, 13] with the L4-ATON laser system. L4-ATON will be able to deliver 10 PW at 1 shot per minute at the wavelength of 1.053 µm using 300 mm aperture laser glass liquid cooled amplifiers [12].
The new challenges in high-energy solid-state lasers motivated the mounting of a 90 mm diameter aperture laser glass amplifier as a test-bed for a future large aperture liquid cooled 1 kJ, 1 shot per minute, nanosecond pulse, 1.053 µm wavelength laser system [14]. The aim of this project is to use this test-bed to improve knowledge on liquid cooled amplifier from modeling to experiments. This effort has been carried out in the framework of the LEAP project [15,16]. This amplifier consists of two 120 mm × 215 mm × 10 mm LHG-8 Nd:phosphate cladded slabs immersed in a circulating coolant (see Fig. 1). Optical pumping is performed by two sets of 10 cerium doped flash-lamps equipped with silver coated reflectors at a repetition rate of one shot per minute. Two 10 mm thick fused silica windows close to the amplifier cell allow the transmission of the laser beam at an incidence of 56.7 degrees on the entrance window. Despite a less efficient heat extraction [17,18], a laminar coolant flow is implemented to warrant an optimal control of the fluid flow in the channels to minimize its impact on the wavefront distortion [10,18].
Fig. 1. Liquid cooled amplifier. (a) Top view. (b) Side view and coolant flow. Laser beam propagates from left to right.
In this work, we present the numerical codes developed to support this effort on liquid cooling amplification. Modeling is carried out in two major steps (see Section 2). In a first step, described in Section 3, the gain and thermal load applied to the slabs are computed using the flash-lamp emission characteristics (intensity, spectrum, and duration), the properties of the ion doping (number of levels, absorption and emission spectrum,…) and the geometry of the amplifier thanks to a calculation of the ion population at the different energy levels in a non-stationary mode. In a second step, described in Section 4, thermal load is injected in a multiphysics model (COMSOL Multiphysics) [19] that takes into account mechanical, hydraulic, thermic and optical coupling effects in the amplifier. Thermo-hydraulic and gain estimated by this modeling sequence are verified with specific experiments. We conclude by showing wavefront aberrations calculated when the split slab system is in pseudo-stationary mode i.e. at one laser shot per minute repetition rate.
2. Global numerical scheme
The first step of the numerical scheme is the implementation of a model of the flash-lamps to compute the output energy and the spectrum in the slabs (Step 1). Additionally, thanks to an absorption model, the thermal load on each slab is computed, and the theoretical gain of the modeled configuration is also estimated (Step 2). These steps are subsequently detailed in Section 3. Afterward, a thermo-hydraulic transient model based on COMSOL (and benchmarked on FLUENT [20]) uses the thermal load calculated in Step 2 to compute the temperature profiles in the fluid, and the residual temperature on each slab after one minute of fluid circulation (Step 3). Then, in a COMSOL Multiphysics model, we evaluate the slab thermal distortions, and the cartographies of refractive index in the slab and in the flowing heated fluid. The deformation of the silica windows induced by the liquid is also calculated (Step 4). We finally couple the different interactions with the COMSOL Ray optic module to estimate the wavefront distortion (Step 5). It can be decomposed to obtain the effect of each contributor on the total wavefront (thermal and mechanical).
3. Gain and thermal load of the amplifying medium
A computer code has been written to give the time dependent gain of the amplifier and the amount of heat released into the amplifying medium during its pumping. The two quantities are determined through the knowledge of the population of the different atomic levels (tot) of the active species (Nd3+ in our case). Schematically the populations are calculated by the resolution of the system rate equations.
N is the population densities $\left( {\begin{array}{c} {{n_1}}\\ \vdots \\ {{n_{tot}}} \end{array}} \right)$ vector and n0 is the number density of active species: M is the transition matrix whose elements are defined by:In operating conditions, the lamps are cooled by a demineralized water flow about one centimeter thick which absorbs totally the infrared part (above 1 µm) of the spectrum radiated by the lamps. The glass of the lamps is also doped with cerium which prevents the transmission of the ultraviolet part (below 0.3 µm) of the spectrum. The different elements (silica windows, cooling liquid) being transparent between 0.3 and 1 µm, the former quantity W is the only source of thermal load of the amplifying system.
All the former equations are generic and must be particularized to the simulation we concern with. For the results presented in the following, thirteen electronic levels of Nd3+ ion are considered. Figure 2 presents these levels and the transition types between them. For the levels of higher energy than the meta (4F3/2) one, only non-radiative relaxation is considered. For the meta and lower lying levels, both radiative and non-radiative transitions are taken into account. For all radiative transitions shown in the figure, the inverse transitions are also considered in the modeling, hence ASE (Amplified Spontaneous Emission) is automatically computed.
Fig. 2. Absorption cross section of Nd3+ ion from the ground level in LH-G8 glass. The right side of the figure describes the electronic levels considered in the model, all radiative transition represented being accompanied by the inverse one.
An example of the time dependent gain calculated through this set of equations is given in Fig. 3. For the discharge current of 1300 A and 500 µs flowing through the flash-lamps, a theoretical gain of 1.32 can be obtained for cladded amplifying slabs.
To depict the kinematics of the pumping, the two quantities Qup and Qlow are introduced. Qlow represents the amount of heat released in both amplifying slabs by non-radiative relaxation of the meta and lying lower levels, Qup being the equivalent quantity due to the relaxation of the higher lying levels. If Vol is the total volume of the amplifying slab, these quantities are defined by:
Figure 4 shows the evolution of the energies involved during the optical pumping of the amplifier slabs by the 20 flash-lamps when discharge depicted in Fig. 3 is applied.
Fig. 4. Time-dependent heat of the total energy of the metastable level (4F3/2) and of heat released in the amplifying slabs by non-radiative relaxation
The 3D heat distribution of the second slab (the last slab crossed by the laser beam) is presented in Fig. 5. It can be noticed in Fig. 5(b) that the heat map distribution in the front face approximately matches the energy distribution radiated by flash-lamps. Overall the amount of heat deposited in the two slabs is close to 500 J. This corresponds to approximately 4% of the energy passing through the flash-lamps.
Fig. 5. Second slab 3D Heat source distribution: in the middle cutting plane (a), in the front face (b).
4. From thermal load to wavefront distortion
We now aim at predicting the laser wavefront deformation induced by the liquid cooling. For that purpose, we need to model the thermal load deposited by flash-lamps during optical pumping in the laser slabs (see Section 3) as well as the effect of the fluid flow and potential coupling effects between the fluid flow, mechanical deformations, thermal effects and ray propagation. We carry out a validation by part with the help of specific experiments to assess the quality of this model: hydraulic calculations and measurements on an admission device similar to that used in the split slab system (detailed in paragraph 4.1), thermo-hydraulic modeling of classical test cases using FLUENT and benchmarked with COMSOL (detailed in paragraph 4.2), and mechanical deformation of the split slab mechanical structure with COMSOL (detailed in paragraph 4.3). Finally, we model the laser optic path through the amplifier to deduce the effect of each physical phenomenon on the wavefront distortion.
4.1 Hydraulic calculation and validation
To limit optical aberration induced by the cooling fluid, the flow has to be laminar around the slabs [10,18]. A hydraulic admission was thus designed to transform a random turbulent flow into parallel path lines. Thanks to this geometry, we found that the preponderant heat transfer phenomenon is the forced convection in laminar flow. This result is important because this study frame allows us to assert that the fluid thermal distribution is controlled only by the fluid flows. This statement enables us to separate the validation of thermal quantities from velocity ones. This model is assessed by comparing modeling of both admission and flow in the split-laser slab channels with velocity field measurement on a real full-scale experiment.
The fluid domain, composed mainly of the admission and amplifier parts, is meshed with 4.8E7 elements (ANSYS MESHING). We use a laminar flow model (this assumption will be confirmed later) nearby the slab, and a k-ω SST turbulence model for the flow behind the slabs. Finally, we compute the mean velocity field obtained during five rounds of fluid elements through the domain.
A dedicated experiment using a full-scale transparent model (see Fig. 6) was built with optical polymethyl methacrylate. The flow of the fluid is monitored with a set of three Moineau cavity pumps, settled by a magnetic flowmeter. To perform particle image velocimetry measurement, the template was filled with water containing polyamide particles doped with rhodamine.
Then, a Nd:YAG laser emitting at 532nm is expanded into a collimated beam and focused into a linear beam. A CCD camera captures the illuminated particles with two frames separated by a delay Δt (from µs to s). A cross-correlation algorithm estimates the two components of the velocity field in the laser plane. Thousands of shots are done to obtain the mean velocity in each plane of measurement. The accuracy of the method depends on different parameters [25]. In our case, the measurement precision between the slabs is quite stable and the uncertainty remains between 1% and 2%.
The experimental results show that the flow stays in a laminar state in all channels from 0.5 L.s−1 to 5.0 L.s−1 and that at the lowest flow rate, the canal flow features a Poiseuille velocity profile (see Fig. 7).
Fig. 7. Comparison between a velocity distribution measured with the experimental set-up (taken at z = 120 mm, Q = 0.5 L.s−1) and an analytical Poiseuille profile.
In Fig. 7, the corresponding Poiseuille profile plot is $V(x )= {V_{\textrm{max}}}({1 - {{({{\raise0.7ex\hbox{$x$} \!\mathord{\left/ {\vphantom {x h}} \right.}\!\lower0.7ex\hbox{$h$}}} )}^2}} )$, where Vmax is the maximum velocity in the channels and h is the half of the canal width. Moreover, the velocity field around the slabs, and near their center is confirmed to be two-dimensional (velocity component along the Y axis can be neglected compared to the other components). This is an important point for the next section.
Accordingly, all the measurements validate the hydraulic entrance performance: it is confirmed that the flow is laminar around the two slabs in a large range of flow rate.
Having that solid reference, we have to validate the Computational Fluid Dynamics (CFD) model that we developed. The measured and calculated velocity fields around the two slabs are compared in Fig. 8. It turns out that the difference between measured and computed velocities ranges between 1% and 14%. The largest difference is observed at the entrance of the center channels.
Fig. 8. Longitudinal velocity field in the three channels. (top) FLUENT simulation, (bottom) Measurement.
For the sake of illustration, we show the measured and simulated velocity profiles at the center of the slabs in Fig. 9 for two flow rates. We can observe a good accuracy of the model since differences of 7% and 5% are obtained for the 0.5 L.s−1 (Fig. 9(a)) and 0.75 L.s−1 (Fig. 9(b)) flow rates respectively. This good accuracy evidences that this set-up warrants a laminar flow between the low gaps separating the slabs and validates the use of a laminar CFD model.
Fig. 9. Longitudinal velocity profile for (a) Q = 0.5 L.s−1 and (b) Q = 5.0 L.s−1 calculated and measured at z = 70 mm.
4.2 Thermo-hydraulic transient calculation and verification
We now aim at verifying the accuracy of the temperature distribution modeling in both liquid flow and laser slabs. As demonstrated in the previous section, the velocity component along the Y axis can be neglected, this allows us to perform the numerical thermo-hydraulic verification study in two-dimensions. We considered three 2-dimensions thermo-hydraulic cases representative of the cooling problem for which an analytic solution exists and can thus be compared to the corresponding numerical calculations. First, a flat plate was considered (Fig. 10(a)) to study the flow sweeping on a semi-infinite heated plan and to restitute hydraulic and thermal boundary layers with two thermal boundaries conditions: fixed temperature and fixed heat flux [26,27]. In fact, this case corresponds to the conditions observed at the very entrance of channels. Then, the Graetz-Flow was studied (Fig. 10(b)) for which the fluid flow features a parabolic Poiseuille profile, in which the thermal boundary layers are created thanks to the two heated walls with temperature or heat flux boundary conditions [28,29]. This configuration corresponds to the output region of the channels. Finally, we considered the case of the flat duct [30,31] which combines the two precedent cases (Fig. 10(c)).
Fig. 10. Verification test-cases: Flat plate (a), Graetz-Flow (b), Flat duct (c). Green lines: velocity boundary layers; red lines: thermal boundary layers. Thanks to the fluid Prandtl number, the thermal boundary layer is always thinner than the velocity boundary layer.
Then, the convergence of the model with respect to the mesh size is studied. The thermo-hydraulic solutions obtained with FLUENT are compared to the analytical results for decreasing mesh sizes. The mesh size is shorter close to the walls and channel entry and larger at the center of the channels to have a good restitution of hydraulic and thermal boundary layers, and, in consequence, of the heat transfers between the slabs and the fluid. Eventually these mesh rules are used in the FLUENT 3D thermo-hydraulic model to warrant a good restitution of the forced convection phenomenon.
We applied the mesh rules to the 3D split slab liquid cooled laser system. To model the heat generated by the pumping flash, we set a volumetric heat source on each slab. The sources are switched-on every minute to obtain a certain amount of energy per slab (established in Section 3). With the cooling system, the temperature of the slab is stabilized around a mean value, after a certain number of pump flashes as shown in Fig. 11. We consider that the system is in a pseudo-steady-state when the energy sent by the flash-lamps to the slabs is entirely dissipated in the cooling fluid.
Fig. 11. Thermal energy in one slab with respect to time for Q = 0.5L.s−1. After 300s, the system is at a pseudo stationary state.
The final objective of this work is to estimate the impact of the cooling system on the wavefront of the laser beam. This means that the thermo-hydraulic model has to be coupled with an optical model. Henceforth, we use the COMSOL software that combines multiple physic models, i.e. hydraulic, thermal mechanic and optics. Before that, we need to verify the accuracy of the thermo-hydraulic model implemented in COMSOL in comparison with FLUENT. A comparison between the two models is carried out. As both codes have different numeric methods – FLUENT uses a finite volume method and COMSOL uses a finite element method – the comparison cannot be done directly. Hence, the benchmark consists in comparing the mean flow around the slabs at the middle of the amplifier and comparing the fluid and solid temperature distribution just before a flash, and at the end of pumping when the amplifier reached the pseudo-steady state.
Figure 12 shows three velocity slices along the channels for the flow rate of 0.5 L.s−1. We cannot see any significant difference between the two software calculations whatever the flow rate from 0.5 to 5 L.s−1 and the position around the slab.
Fig. 12. Comparison between COMSOL and FLUENT velocity fields at Q = 0.5 L.s−1: at z = 1.5 mm (a) z = 20 mm (b) and z = 70 mm (c).
Then, we study the thermal quantities, when the amplifier is in the pseudo-steady state, at the end of a cooling period (t0), i.e. before pumping, and after the energy addition in the slabs (t0+Δt), via the pumping flash (Δt is the pumping time duration). In the next figure, we plot the relative difference temperature, $ {T_{rel}}$, at t = t0 (Fig. 13(a) and 13(b)) and t = t0+Δt (Fig. 13(c) and 13(d)) which is defined for any coordinate (x, y,z) inside the slabs by:
Fig. 13. Relative difference temperature field $\Delta {T_{rel}}$ between COMSOL and FLUENT: external face slab before pumping (a), internal face slab before pumping (b), external face slab after pumping (c), internal face slab after pumping (d).
Moreover, from the calculation of the slab dilatation coefficient, we can compute the slab volume increase. When the system is in the pseudo-stationary state, the volume variation is about 10−3%. This order of magnitude does not induce any significant effect on liquid path lines around and we do not consider this effect on the fluid flow. However, it is still an important contributor to the optical path.
4.3 Mechanical calculation
During the experiment, the mechanical stress of the amplifier cell does not change, and the thermal expansion of the slabs only relies on the internal temperature due to flash-lamp energy deposition. To avoid internal stress in the slabs induced by their thermal expansion, a specific mechanical holder was designed to move the slab according to two degrees of freedom (one translation and one rotation). Furthermore, the silicate windows are fixed to the inox structure with polymer glue and a system of actuators was designed to thwart the hydraulic pressure against windows.
With the slabs and the fluid domain, we model the amplifier structure, the silica windows and the interactions between each other described above. The COMSOL mechanical module is used to solve the static equation, which leads to the displacement of the silica windows induced by the hydraulic pressure and the strains of the fixture with the amplifier structure. Standard boundary conditions were used for mechanics, regarding thermal management we used either adiabatic interface (for slab edges), heat continuity (for fluid/solids interfaces, except windows) or temperature continuity (for fluid/window interfaces). During the transient thermo-hydraulic computation, strain estimation is done when the laser beam passes through the amplifier at pseudo stationary state (i.e. after 300 seconds).
This approach is supported by the fact that (i) mechanical strains remain constant (only thermo-mechanical strains evolve when flash occurs), and that (ii) the order of magnitude of the mean deformation is about one micrometer (for both strain sources) (see Fig. 14), which does not affect the channel flow. Hence, this explains why we can first perform the fluid computation and then the mechanical one.
Fig. 14. Mechanical results at pseudo stationary state at flash-lamp shot time: slab face deformation (a), window deformation (b).
4.4 Wavefront aberration calculation
At this stage, we have computed the different contributions that affect the laser beam in the pseudo-state mode, i.e. after 300 seconds (see Fig. 11). The amplifier total Optical Path Difference (OPD) at the peak of pumping is shown in Fig. 14(d); the peak-to-valley (PV) value is 4.2 µm. Thanks to the modularity of the physics modules in COMSOL, we decompose the total OPD into several OPD physical contributions. First, we compute the purely mechanical wavefront distortion, which does not depend on time. It shows the effect of the mechanical strains and the hydraulic pressure on the windows. In Fig. 15(a), the mechanical wavefront distortion has a PV value of 2.38 µm, and the surface plot almost follows the deformation of the windows. The cylinder shape observed is mostly the result of the hydrostatic pressure of the fluid inside the amplifier cell and of the windows fixtures. In Fig. 15(b), we plot the optical aberration due to refractive index modification in both fluid and slabs. The PV value is 2.07 µm. The shape observed is the result of heat dissipation mostly in the fluid during the cooling. Then, we extract the wave distortion due to the thermal expansion (Fig. 15(c)) of the two slabs during the flash pumping, and we get a PV value of 1.05µm. The slope observed is due to the fact that the distance between flash-lamps and slabs is not constant and consequently the heat deposit is not completely uniform.
Fig. 15. Optical Path Differences (OPD) for a one-pass laser beam: mechanical strains and hydraulic pressure (a), Amplifier slabs and fluid refractive index variation (b), index variation slabs thermal expansion (c) and total amplifier cell OPD (d)
It is worth underlying that the overall total wavefront aberration is mostly astigmatic (Fig. 15(d)) and can be easily corrected by a deformable mirror. Moreover, thanks to the cooling system developed in this work, millimeter to centimeter scale spatial frequencies are not created by the liquid flow which is an important result since such defects cannot be corrected.
5. Conclusion
A multiphysics model has been developed to design and to optimize a liquid cooled large aperture split-slab laser glass amplifier prototype. This model combines all physical phenomena from optical pumping, heat loading to hydraulic effects induced by the liquid coolant, mechanical deformation and their potential coupled effects on the wavefront distortion. Dedicated experiments were made to assess the accuracy of each physic model (hydraulic, heat transfer). Thanks to this coupled model, we evidence that such a liquid cooled amplifier is able to amplify laser pulses with a theoretical gain of up to 1.32. The amplified wavefront distortion is in the range of some micrometers, with low spatial frequencies easily compensated by a deformable mirror. Additionally, small and mid-spatial frequencies are suppressed from the amplified wavefront thanks to the optimization of the liquid flow in the amplifier cell.
Funding
European Commission (#3404410, ERDF #2663710); Conseil Régional Aquitaine (#DEE21-04-2019-5131820, CPER #16004205).
Acknowledgments
The authors thank L. Chattelier and G. Gommit from Université de Poitiers, Institut de recherche de Poitiers : Recherche ingénierie en matériaux, mécanique et énergétique pour les transports, l'énergie et l'environnement (Institut Pprime), groupe Hydrodynamique et Écoulements Environnementaux-(HydÉE), for hydraulic admission experiments. We also thank N. Bonod for fruitful guidance in the writing of this article.
Disclosures
The authors declare no conflicts of interest
References
1. W. Koechner, Solid-state laser engineering, Vol. 1, Springer (2013).
2. V. A. Buchenkov, B. N. Kolesnikov, V. M. Mit’kin, D. I. Perlov, and A. I. Stepanov, “Investigation of a periodically emitting neodymium glass laser under transient thermal conditions,” Sov. J. Quantum Electron. 5(4), 403–405 (1975). [CrossRef]
3. J. M. Rinefierd, S. D. Jacobs, D. C. Brown, J. A. Abate, O. Lewis, and H. Appelbaum, “Liquids for high repetition rate glass laser systems,” ‘Laser Induced Damage in Optical Materials: ASTM International, (1979).
4. 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]
5. V. Coffey, “High-energy lasers: new advances in defense applications,” Opt. Photonics News 25(10), 28–35 (2014). [CrossRef]
6. X. Fu, P. Li, Q. Liu, and M. Gong, “3 kW liquid cooled elastically-supported Nd:YAG multi-slab CW laser resonator,” Opt. Express 22(15), 18421–18432 (2014). [CrossRef]
7. K. Wang, B. Tu, C. Jia, J. Shang, X. An, Y. Liao, Z. Xu, J. Guo, J. Yi, and Y. Yu, “7 kW direct-liquid-cooled side-pumped Nd: YAG multi-disk laser resonato,” Opt. Express 24(13), 15012–15020 (2016). [CrossRef]
8. J. Yi, B. Tu, X. An, X. Ruan, J. Wu, H. Su, J. Shang, Y. Yu, Y. Liao, H. Cao, L. Cui, Q. Gao, and K. Zhang, “9 kilowatt-level direct-liquid-cooled Nd:YAG multi-module QCW laser,” Opt. Express 26(11), 13915–13926 (2018). [CrossRef]
9. Z. Ye, C. Liu, B. Tu, K. Wang, Q. Gao, C. Tang, and Z. Cai, “Kilowatt-level direct refractive index matching liquid-cooled Nd:YLF thin disk laser resonator,” Opt. Express 24(2), 1758–1772 (2016). [CrossRef]
10. X. Ruan, H. Su, B. Tu, J. Shang, J. Wu, J. Yi, H. Cao, Y. Ma, G. Wang, D. Shen, Q. Gao, K. Zhang, and C. Tang, “Analysis of output characteristics introduced by non-uniform coolant in direct-liquid-cooled multi-disk laser,” Opt. Commun. 436, 26–33 (2019). [CrossRef]
11. S. Banerjee, K. Ertel, P. Mason, P. Phillips, M. De Vido, J. Smith, T. Butcher, C. Hernandez-Gomez, R. Greenhalgh, and J. Collier, “DiPOLE: a 10 J, 10 Hz cryogenic gas cooled multi-slab nanosecond Yb:YAG laser,” Opt. Express 23(15), 19542–19551 (2015). [CrossRef]
13. E. Gaul, G. Chériaux, R. Antipenkov, F. Batysta, T. Borger, G. Friedman, J. T. Greene, D. Hammond, J. Heisler, D. Hidinger, A. Jochmann, M. Kepler, A. Kissinger, D. Kramer, J. C. Lagron, A. Meadows, B. Rus, P. Trojek, S. Vyhlídka, and T. Ditmire, “Hybrid OPCPA/Glass 10 PW laser at 1 shot a minute,” in, Optical Society of America, San Jose, California, pp. STu3M.2 (2018).
14. P. Dalbies, N. Blanchot, E. Bordenave, B. Cadilhon, S. Cavaro, R. Chonion, P. Depeyris, E. Lavastre, P. Manac’h, Y. Modin, G. Paquignon, P. Patelli, J. Sajer, D. Taroux, and J. Neauport, “Nd: Phosphate split-slab liquid cooled kJ amplifier for high power laser,” in Laser Congress 2019 (ASSL, LAC, LS&C), OSA Technical Digest (Optical Society of America, 2019), paper JTu3A.30.
15. http://www.atlas-onco.com/leap.html
16. D. Penninckx, “Advanced laser technologies for high-power energetic lasers,” Advanced Laser and Photon Sources (ALPS) conference 2020, 20-23/04/2020, Yokohama.
17. A. Bejan and A. D. Kraus, Heat Transfer Handbook (Wiley, 2003).
18. 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]
19. COMSOL Multiphysics® v. 5.4. http://www.comsol.com . COMSOL AB, Stockholm, Sweden.
20. Ansys® Academic Research Mechanical, Release 18.2, Help System, Coupled Field Analysis Guide, ANSYS, Inc.
21. D. Mihalas and B. Mihalas, “Foundations of Radiation Hydrodynamics,” Oxford University Press (1984)
22. L. H. Auer and F. Patelou, Astron. Astrophysics, 675 (1994).
23. I. Ibgui, I. Hubeny, T. Lanz, and C. Sehle, “IRIS: a generic three-dimensional radiative transfer code,” Astron. Astrophys. 549, A126 (2013). [CrossRef]
24. H. T. Powell, A. C. Erlandson, K. S. Jancaitis, and J. E. Murray, “High-Power Solid State Lasers adns Applications,” Proc. SPIE 1277, 103 (1990).
25. A. Sciacchitano, “Uncertainty quantification in particle image velocimetry,” Meas. Sci. Technol. 30(9), 092001 (2019). [CrossRef]
26. T. Cebeci and P. Bradshaw, “Momentum transfer in boundary layers,” Series in Thermal and Fluids Engineering, (1977).
27. S. W. Churchill and H. Ozoe, “Correlations for laminar forced convection over an isothermal flat plates and in developing and fully developed flow in an isothermal tube,” J. Heat Transfer 95(3), 416–419 (1973). [CrossRef]
28. R. K. Shah and A. L. London, “Laminar flow forced convection in ducts,” Academic Press, (1978).
29. A. P. Hatton and J. S. Turton, “Heat transfer in the thermal entry length with laminar flow between parallel walls at unequal temperature,” Int. J. Heat Mass Transfer 5(10), 973–980 (1962). [CrossRef]
30. H. Schlichting, “Boundary layer thoery,” McGray-Hill Book Co.,(1966)
31. L.-T. Hwang and C.-L. Fan, “Finite difference analysis of forced-convection heat transfer in entrance region of a flat rectangular duct,” Appl. Sci. Res. 13(1), 401–422 (1964). [CrossRef]
