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Abstract
Diffractive optical element (DOE) is a critical device for combining multiple laser beams into a single beam in a coherent beam combining (CBC) architecture. This study proposes a determination method for calculating the intrinsic absorption rate of the DOE, and the corresponding experimental system is established. We present a theoretical thermal deformation model of the laser-irradiated DOE based on the thermoelastic equation and thermal conduction theory. The temperature and thermal deformation of the DOE are simulated using different parameters, including the laser power density, substrate size, substrate material, laser incident time, and clamping method. The simulations indicated that the thermal deformation is directly proportional to substrate area and inversely proportional to substrate thickness. The thermal deformation of the DOE can also be decreased by using a two-surface fixing method, and the maximum decrease is 4.4%. The quantitative discussion and analysis of the DOE temperature field and thermal deformation are important for designing a DOE to increase the combining efficiency and improve the combined beam quality of a practical DOE-based CBC system.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Coherent beam combining (CBC) is a coherent combination technique that combines several lasers having identical wavelengths into one composite output beam [1–6]. In 2020, the Civan Advanced Technologies Ltd reported a 16 kW CBC based on the coherent combination of 32 parallel ytterbium doped fiber amplifiers [7]. In 2020, the Friedrich Schiller University Jena achieved an average output power of 10.4 kW with a CBC system [8]. The diffractive optical element (DOE) is characterized by high diffraction efficiency, high damage threshold, and outstanding intensity uniformity of an array of beamlets [9–11]. In recent years, one-dimensional (1D) or two-dimensional (2D) DOEs have been widely applied in high-power fiber laser CBC systems to obtain near-diffraction-limited, high-combining-efficiency, and multi-kilowatt laser outputs [12–14]. In the DOE-based CBC schemes, an array of beamlets is coherently superimposed with the DOE surface at angles analogous to the corresponding diffractive order to perform the filled aperture geometry [15]. In 2014, McNaught et al. achieved an output power of 2.4 kW using DOE-based CBC schemes [14]. In 2016, five high-power fiber lasers were combined by a DOE with a total power output of 4.9 kW [16], which is a key component for the wavefront reconstruction of beamlets. The predictable results are that with the increase of combination laser number and power, increases the temperature of the DOE, leading to the thermal deformation of the surface of the DOE. This thermal deformation of the DOE occurs in the application of CBC system, and may influence the combined beam quality and combining efficiencies. Furthermore, the thermal stress and deformation may damage the DOE as the temperature and deformation continues to increase. For DOE-based CBC system with large beamlets scale, the low thermal deformation of the DOE must be ensured to maintain the high quality of the combined beam [17]. The thermal deformation of the DOE leads to light-field intensity modulation of beamlets, which eliminates the common-aperture condition and influences the beam combination. However, there have been no reports of detailed numerical calculations and derivation of the temperature, stress, and thermal deformation of the DOE.
In this paper, a determination method is presented for obtaining the intrinsic absorption rate of the DOE, and the experiment platform is established. On this basis, we established a composite model of thermal conduction and thermal deformation of DOE. Based on thermal conduction theory and the thermoelastic equation, the temperature, stress, and thermal deformation change of the DOE is simulated. The main factors that influence the temperature field and thermal deformation of the DOE are investigated, including the laser power density, substrate size, substrate material, and clamping method. The results reveal two methods to reduce the thermal deformation of the DOE. The substrate material with a larger thermal conductivity coefficient and lower thermal expansion can effectively reduce the thermal deformation of the DOE. This work enriches the understanding of the thermal deformation change mechanism of the DOE under high-power CW laser irradiation. We believe this research can lay a foundation for exploring the dependence of DOE thermal deformation on the combined beam quality and propagation properties in DOE-based CBC systems in the future.
2. Theoretical model
2.1 Laser absorption model
When the laser is irradiated and transmitted to the DOE, the laser energy is absorbed owing to the intrinsic absorption. The Cartesian coordinate system used in the laser absorption model is shown in Fig. 1, and has its origin at the center of the back surface of the DOE.
The thickness of the surface microstructure is a few hundred nanometers [18]. Compared with the laser beam radius and substrate thickness, it is more than 1000 times smaller. Thus, the thickness of the surface microstructure can be ignored. Thus, only its intrinsic absorption is considered in our theoretical model. The thermal conduction equation of the DOE is written as follows:
where T, k, η, ρ, cp, and Pd denote the temperature of the DOE, coefficient of thermal conductivity, intrinsic absorption rate of the DOE, density of the material, specific heat of the material, and the laser power density distribution function on the surface of the DOE, respectively. Pd is expressed in Eq. (2):The boundary S includes all of the cooling boundaries of the DOE except for the laser irradiated area on the surface. Here n is the normal vector to the cooling boundaries of the DOE, h is the convection thermal transfer coefficient in air, and T0 is the environment temperature of the laboratory.
2.2 Thermoelasticity model
The temperature-field distribution of the DOE is introduced in Eq. (4), which describes the stress and strain relations [19].
The thermal deformation and strain tensor relations can be represented as Eq. (5).
where D denotes the fourth-order elasticity tensor, “:” denotes the double-dot tensor product, S denotes the stresses, S0 denotes the initial stresses, ɛth denotes the thermally induced strains, and ɛ and ɛ0 denotes the strain tensor and the initial strains, respectively. The DOE thermal deformation evolves according to the Navier-Stokes thermoelastic equation [20–21]. The equation for the simulation of the thermal deformation of the DOE is expressed by Eq. (6). where u denotes the thermal deformation distribution of the DOE, ρ is the material density, and cp is the specific heat. The symbol ν denotes the Poisson ratio, and α is the thermal expansion coefficient.3. Intrinsic absorption rate of the DOE
The intrinsic absorption rate of the DOE is the key parameter to determine the energy absorbed amount. We propose a determination method for calculating the intrinsic absorption rate of the DOE, and the corresponding experimental system is established. This method can also be used for the determination of intrinsic absorption rate in other optical devices, such as grating, lens, and endcap.
3.1 Experimental setup
The temperature of the DOE under different laser power irradiation was measured using the experimental setup shown in Fig. 2(a). A high-power fiber laser was used as the heat source for the DOE and was incident on the front surface of the DOE. The size of the fused silica substrate DOE was 50 × 50 × 1.46 mm3. The output power was in the range of 0∼1.24 kW, with a laser of wavelength 1064 nm and a radius of 1.5 mm. A power meter (PM) was used to receive the transmission laser. The transform lens was located between the laser array and PM to ensure that each individual beam spatially overlapped on the PM, and an infrared camera was employed to record the change in the front face temperature of the DOE. The temperature distribution of the DOE surface is shown in Fig. 2(b). It exhibits a near-Gaussian distribution.
Fig. 2. (a) Schematic of the configuration used for the DOE intrinsic absorption rate measurement. (b) DOE temperature distribution acquired by the infrared camera.
3.2 Experimental result
When the laser beam with high-power irradiates the DOE, the surface microstructure absorbs the energy, increasing the temperature of the DOE. The function expressing the relationship between the temperature and the intrinsic absorption rate was obtained using Eq. (1). The intrinsic absorption rate of the DOE was calculated using our model by comparing the simulation and experimental temperature fields. Figure 3(a) shows the maximum surface temperature of the DOE for different power irradiation. The temperature increases from 20.01 °C to 25.62 °C when the laser power is increased from 0 to 1.24 kW. We irradiated the DOE by a variable laser power with a period of 180 s. The blue line represents the simulation data, and the red line represents the experimental result.
Fig. 3. (a) Comparison of experimental and simulated DOE surface temperature change depends on irradiation laser power. (b) The intrinsic absorption rate of the DOE change depends on irradiation.
The temperature model was applied to calculate the intrinsic absorption rate of the DOE. Table 1 presents all of the parameters and their values used in the numerical simulations. The thermophysical parameters of the substrate materials of the DOE are shown in Table 2. The finite-element method (FEM) [22–23] was used to solve the coupling equations Eqs. (1)–(3) to obtain the temperature distribution of the DOE.
Table 1. Parameters and values for the experiment and simulation
Table 2. Thermophysical parameters of DOE (fused silica substrates)
A comparison of the experiment and simulation results indicated that the average matching error and the average relative error of this model, were 0.036 ° and 0.16% respectively. Figure 3(b) shows the intrinsic absorption rate of the DOE change dependences on irradiation laser power, then averaged the results. The average absorption rate was 20 ppm.
In order to verify our calculation results from different perspectives, we compared simulation and experimental results of variation in the DOE temperature with time, and the surface temperature distribution of the DOE. Figure 4(a) and (b) shows experiment and simulation results of the front surface maximum temperature as a function of time under different laser irradiation conditions. Figure 4(c) and (d) show a cross-sectional view of the front surface temperature in the y direction at different incidence power levels in 180 s. The simulation data is in good agreement with the experimental results.
Fig. 4. Maximum DOE front surface temperature as a function of the time and corresponding curves for different irradiation powers: (a) experiment and (b) simulation. Maximum DOE front face temperature distribution as a function of the y-axis coordinate with different irradiation powers: (c) experiment, (d) simulation.
4. Numerical simulations results and discussion
The FEM was used to solve the partial differential equations to obtain the stress field and thermal deformation distribution of the DOE. In the simulation, initial strains ɛ0 and initial stresses S0 we set to be zero, and the environment temperature T0 was set to be 20 °. All of the domains were set under the free conditions, but the boundary condition of the temperature field was as expressed by Eq. (3). Table 3 lists the preliminary condition and boundary conditions for solving the partial differential equations. The difference between single- and multiple-laser irradiation lies in the different irradiation angles, which has little influence on the overall change in both the thermal deformation and temperature field when the laser irradiation range overlaps [23]. Hence, for the simulation of the thermal behavior of the DOE, increasing the irradiation power of a single laser can replace the multiple-laser irradiation scenario for the same total incident power.
Table 3. Preliminary condition and boundary condition for solving the partial differential equations
4.1 Variation of temperature and deformation with laser power
To investigate the influence of the laser power density on the temperature and surface deformation of the DOE, we set the DOE area to 50 × 50 mm2 and a thickness of 2 mm. The power of the irradiation laser was varied from 1 kW to 10 kW, and the laser radius at the DOE was set to 2.5 mm.
Figures 5(a) and (b) show the DOE maximum surface temperature and deformation dependences on irradiation laser power), respectively. The maximum surface temperature of the DOE increases with increasing laser power. The higher the power of the irradiation beam, the higher the maximum thermal deformation of the DOE surface, as shown in Fig. 5(b). Figures 5(c) and (d) show the variation in the DOE temperature and thermal deformation with irradiation for different laser power, respectively. It is clear that a higher laser power with a fixed beam diameter will cause higher temperature and greater thermal deformation of the DOE. The temperature increases from 22.82 °C to 48.66 °C, and the maximum DOE thermal deformation increases from 3.57 nm to 35.66 nm, when the laser power is increased from 1 kW to 10 kW.
Fig. 5. Maximum surface (a) temperature and (b) thermal deformation of the DOE as a function of time for different irradiation laser powers. Maximum (c) temperature and (d) thermal deformation of the DOE surface with different powers.
4.2 Variation of temperature and deformation with DOE size
To investigate the influence of the DOE size on the temperature and surface deformation of the DOE, we set the radius of the laser to 2.5 mm and power to 10 kW. The thickness of the DOE was varied from 2 to 6 mm in 2 mm increments for DOE areas of 50×50 mm2, 40×40 mm2, and 30×30 mm2.
Figures 6(a) and (b) show that the maximum surface temperature and deformation varies over time according to the area of the DOE. For the same irradiation time, as the DOE area decreases, the maximum surface temperature increases, and thermal deformation decreases. The thermal deformation of the DOE decreased from 35.95 nm to 31.74 nm when the area of the DOE was changed from 50×50 mm2 to 30×30 mm2. The non-uniform temperature distribution in the DOE causes non-uniform thermal stress and thermal deformation. The higher the temperature of the DOE, the higher the thermally induced strains. However, as the thermal deformation increases, the strain tensor will increase, changing the stress of the DOE and resulting in the stress distributions shown in Fig. 7. This is the reason for the temperature and thermal deformation having an opposite changing direction with an increase in the area of the DOE.
Fig. 6. Maximum surface (a) temperature and (b) thermal deformation of the DOE as a function of time for different substrate areas.
Fig. 7. Stress value distributions of the DOE for substrate area of (a) 30×30 mm2, (b) 40×40 mm2 and (c) 50×50 mm2.
Figure 8(a) and (b) shows the variation in the maximum DOE surface temperature and thermal deformation with time for different DOE thicknesses. The DOE area was set to 50×50 mm2. The maximum temperature of the DOE decreased, corresponding to an increase in thickness, as shown in Fig. 8(a). When the DOE thickness was increased from 2 to 6 mm, the temperature decreased from 48.54 °C to 38.51 °C. Figure 8(b) shows the change in the thermal deformation of the DOE, which corresponds to an increase in the substrate thickness. The thermal deformation is directly proportional to DOE area and inversely proportional to DOE thickness. It is obvious that an increase in DOE thickness with total laser power held constant would decrease thermal deformation. The maximum DOE thermal deformation decreases from 35.84 nm to 34.46 nm, when the DOE thickness was increased from 2 to 6 mm. Figures 8(c) and (d) show the variation in the DOE temperature and thermal deformation with irradiation for different DOE thickness, respectively. The results show that increasing the thickness of the DOE can reduce the temperature and thermal deformation of the DOE. The maximum improvement in the temperature and deformation are 20.7% and 3.9%, corresponding to a thickness increase from 2 mm to 6 mm.
Fig. 8. Maximum (a) temperature and (b) thermal deformation of the DOE surface as a function of time for different substrate thickness. Maximum (c) temperature and (d) thermal deformation of the DOE surface with different thickness.
Cross-sectional views of the DOE temperature distributions and stress for different DOE thicknesses are shown in Fig. 9. From the DOE temperature distributions shown in Fig. 9, we conclude that most of the laser energy is concentrated within the central region of the laser irradiation area along the center line. As the thickness increases, the temperature difference between the front and back surface of the DOE gradually increases. This is because the thermal conductivity of the fused quartz is small, causing the thermal energy could not transfer rapidly in the material and stress concentration. Figure 10 presents an intuitive visualization of the front surface thermal deformation distribution for different thicknesses under the same irradiation. The thermal deformation distribution was a near Gaussian distribution.
Fig. 9. Temperature and stress value distributions of the DOE for substrate thicknesses of (a) 2 mm, (b) 4 mm and (c) 6 mm.
Fig. 10. The front surface thermal deformation distribution of the DOE for substrate thicknesses of (a) 2 mm, (b) 4 mm and (c) 6 mm.
4.3 Variation of temperature and deformation with substrate material
To investigate the effect of the substrate material on the temperature and surface deformation of the DOE, we set the size of DOE to 50 × 50 mm2 and thickness of 2 mm. The power of the irradiation laser was set to 10 kW and the radius of beam was set to 2.5 mm. The thermophysical parameters of the four substrate materials are listed in Table 4.
Table 4. Thermophysical parameters of substrates materials
Figures 11(a) and (b) show the variations of the maximum temperature and thermal deformation of the DOE front surface with time and different substrate materials, respectively. Figures 12(a) and (b) show the variation in the DOE temperature and thermal deformation with irradiation for different substrate materials, respectively. The higher the thermal conductivity of the substrate material, the lower the temperature of the DOE, as shown in Fig. 11(a) and Fig. 12(a). The YAG have a thermal conductivity that is 10 times that of the fused silica, and that of CaF2 is seven times that of the fused silica, which will quickly guide away the thermal energy, decreasing the temperature of the surface of the DOE. The thermal deformation profiles of the four substrate materials are shown in Fig. 11(b). The thermal expansion coefficient of CaF2 is about two times more than that of the YAG and BK7, and about 30 times that of fused silica. This means that even if the temperature of fused silica is several times that of other materials, the thermal deformation may remain low.
Fig. 11. Maximum surface (a) temperature and (b) thermal deformation of the DOE surface as a function of time for different substrate materials (irradiation power: 10 kW).
Fig. 12. Maximum (a) temperature and (b) thermal deformation of the DOE surface with different powers (irradiation time: 30s).
The substrate material with a relatively low thermal expansion reduces the thermally induced strains, preventing high stress and lowering the thermal deformation of the DOE. The substrate material with a large coefficient of thermal conductivity can effectively transfer the high temperature quickly, reducing the temperature and enhancing high thermal deformation. Hence, the use of the substrate material with a low thermal expansivity and large thermal conductivity coefficient is important to decrease the thermal deformation of the DOE.
4.4 Optimal clamping methods for the DOE
This section aims to investigate the influence of the clamping methods on the temperature and thermal deformation of the DOE. It has been demonstrated that clamping methods for optical elements significantly influence thermal deformation [24]. In this case, we analyzed the characteristics of several clamping methods and simulated the corresponding temperature and thermal deformation distribution of the DOE. The four clamping methods are (1) not fixed, (2) two-surface fixing method (both upper and lower walls), (3) four-point fixing method (upper and lower, right and left walls), (4) four-point fixing method (both upper and lower walls). The shading indicates the fixed area, and the clamping details are shown in Fig. 13 (a) – (b). In the fixed area, DOE is not deformed, and the boundary condition is changed to u = 0 and thermal insulation. We set the irradiation laser to a beam radius of 2.5 mm and power of 10 kW. The DOE size was set to 50 × 50 × 2 mm3, and the substrate materials of the DOE is fused silica.
Fig. 13. Comparison of maximum surface temperature and thermal deformation of different clamping methods under various irradiation times. (a), (e), and (i) Not fixed; (b), (f), and (j) Two-surface fixing method (both upper and lower walls); (c), (g), and (k) Four-point fixing method (upper and lower, right and left walls); (d), (h), and (l) Four-point fixing method (both upper and lower walls). The inset graphs: the shading indicates the fixed area.
The influence of various clamping methods was observed on the thermal deformation of the DOE in the simulation results, which is shown in Fig. 13(c)–(h). The thermal deformation occurs mainly on the DOE surface and the maximum thermal deformation is concentrated on the surface center, while a little deformation is observed at the edge of the DOE. The simulated results shown in Fig. 13(i)–(l) show that the thermal deformation of the DOE fixed in the two-surface fixing method is 34.39 nm, which has an advantage over the four-point fixing method (38.97 nm and 39.78 nm) when the irradiation time is 60 s. Compared with the not fixed situation, the maximum thermal deformation of the DOE fixed in the two-surface fixing method decreases, while in other cases increases significantly. The clamping methods for the DOE did not affect the temperature field of the DOE.
Cross-sectional views of the DOE stress value distributions for not fixed (Way 1) and the two-surface fixing method (Way 2) are shown in Fig. 14. The results show that by setting a suitable clamping method, the stress [25–26] can be distributed more evenly, decreasing the thermal deformation of the DOE.
Fig. 14. Stress value distributions of the DOE for different clamping methods: (a) Way 1; (b) Way 2.
5. Conclusion
In this paper, we proposed a temperature and thermal deformation distribution analysis model for the DOE based on the thermal conduction theory and thermoelastic equation. Factors that influence the temperature and thermal deformation of the DOE are simulated in the single-aperture coherent beam combining system, including the laser irradiation time, laser power, substrate size, substrate material, and clamping method. We calculated the intrinsic absorption rate of the DOE based on the corresponding experiments. With 1.24 kW laser irradiation, the maximum temperature was 25.62 ° for an irradiation time of 180 s, and the intrinsic absorption rate of the DOE was calculated to be 20 ppm. It is indicated from the simulations that increasing the thickness of the substrate can reduce the thermal deformation of the DOE, with an maximum decrease of 3.9%. Further, decreasing the area of the DOE can decrease the thermal deformation of the DOE effectively, with a maximum decrease of 11.7%. The use of the substrate material with a lower thermal expansion and a larger thermal conductivity coefficient is significant to decrease the thermal deformation of the DOE. The different clamping methods do not affect the temperature but influence the thermal deformation of the DOE. Under the same irradiation laser power, the thermal deformation of the DOE fixed in the two-surface fixing method is always better than other clamping methods, even the thermal deformation is reduced compared to the case of not fixed. The discussion and analysis about DOE thermal deformation will be beneficial to reduce DOE thermal deformation and improve the combined beam quality of a DOE-based CBC system.
Funding
National Key Research and Development Program of China (2018YFB0504500); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2020252); National Natural Science Foundation of China (61705243, 61735007, 61805261).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon request.
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