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Abstract
The multilayer dielectric (MLD) grating is a critical device for combining multiple laser beams into a single beam in a spectral beam combining (SBC) system. We established a theoretical thermal deformation model of the laser-irradiated MLD grating. Thermal deformation on the surface of the grating is simulated according to a series of parameters including the laser irradiation time, laser power density, and substrate size. To verify the model, we exposed a 960 l/mm, 50×50×1.5 mm3 grating to a laser power density of 3.61 kW/cm2 and observed the temperature change. We used a Twyman-Green interferometer to measure the interference fringes on the grating surface. Based on the Fourier-transform method and a Zernike polynomial fitting method, the real-time grating surface profile is reconstructed. The results show that substrate thickness increase or area decrease can reduce thermal deformation, the average decreases are 18.3% and 19.9%, respectively. The discussion and analysis of the grating thermal deformation are potentially valuable for designing grating to decrease the thermal deformation and improve the combined beam quality of a SBC system.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Spectral beam combining (SBC) is a high-power laser incoherent beam combining technique, which combines laser beams of different wavelengths into a single-aperture beam with good beam quality [1–6]. In 2016, Shanghai Institute of Optics and Mechanics reported a 10.8 kW power level SBC based on a polarization independent multilayer dielectric (MLD) grating with a combining efficiency of 94% [7]. In 2017, the Lockheed Martin Company achieved an output power of 58 kW with an SBC system [8]. The MLD grating is characterized by large wavelength bandwidth, high diffraction efficiency, high damage threshold, and polarization independence [9]. In recent years, the MLD grating has been applied to high power fiber laser SBC systems. The MLD grating is a key component for wave-front reconstruction of beamlets [10]. With the increase of irradiation laser power [11–14], the temperature of the MLD grating increases, leading to thermal deformation of the surface of the MLD grating [15–17]. The thermal deformation of the MLD grating is expected to have an effect on the beam propagation characteristics and combining efficiencies of the SBC system [18–20]. Furthermore, thermal deformation or thermal stress may lead to MLD grating damage when the combined laser power is increased. In 2018, Yunxia Jin et al. studied temperature increase of the MLD grating and derived the temperature field equation [21]. In 2016, Zhen Wu et al. investigated the effect of MLD grating thermal deformation on the combined beam properties [22]. However, detailed numerical calculations of the stress and thermal deformation distribution of the MLD grating have not been reported. Study of MLD grating thermal deformation, by further understanding its origin, can provide a method to decrease it and improve the beam quality of the combined beam. Therefore, it is important and necessary to study the influencing factors of MLD grating thermal deformation under high-power laser irradiation.
In this paper, we established a theoretical model of thermal conduction and thermal deformation of MLD gratings subjected to high-power laser irradiation, based on thermal conduction theory and the thermoelastic equation, and simulated the temperature and thermal deformation change of the grating. Factors that influence the thermal deformation of the MLD grating are investigated, including the laser irradiation time, laser power, and substrate size. And the temperature and thermal deformation of the grating under the same laser power density but different power and irradiation laser radius is simulated. To verify the accuracy of the model, the temperature and thermal deformation change of the MLD grating were measured under several different irradiation laser powers. An infrared thermal camera was employed to record the temperature distribution of the MLD grating surface. The interference fringe-pattern on the grating surface was observed by a Twyman-Green interferometer. After image processing of fringe-patterns and the Zernike polynomial fitting method, the grating surface profile was reconstructed from the acquired fringe-pattern. The results reveal two methods to reduce the thermal deformation of the MLD grating and provide a reference for understanding the thermal dynamics of the high-power CW-laser-irradiated MLD grating.
2. Theoretical consideration and modeling details
We established a model of thermal conduction and thermal deformation of MLD gratings based on thermal conduction theory and the thermoelastic equation. The thickness of the multilayer dielectric film is only a few hundred nanometers [23]; thus, the thickness of the film can be ignored and only its intrinsic absorption is considered in the numerical model. Its temperature is consistent with the surface of the substrate. When a laser beam is incident on the surface of the MLD grating, the laser energy is deposited by the intrinsic absorption of the multilayer dielectric film. The coordinate system used in the model is shown in Fig. 1, which has its origin at the center of the MLD grating back surface.
2.1 Laser absorption and thermal transfer model
According to the literature [23], the thermal conduction equation of an MLD grating can be represented as Eq. (1).
where T is the temperature distribution of the MLD grating, k is the thermal coefficient of conductivity, ρ is the material density, cp is the specific heat, and ηI is the laser energy absorbed by the MLD grating. η is the intrinsic absorption rate of the MLD grating film, and I is the laser power density distribution function in the x,y-plane, with an unit of W/cm2.2.2 Thermoelasticity model
In this study, using the thermal conduction equation Eq. (1) and boundary conditions Eq. (3), the temperature distribution of the MLD grating under high-power laser irradiation can be obtained. The nonuniform temperature change in the MLD grating causes a nonuniform thermal stress and thermal deformation. The temperature field distribution is introduced into the following equations, which describe the stress and strain relations [24].
3. Numerical simulations results and discussion
Temperature and thermal deformation distributions developed in the laser irradiated MLD grating were investigated and the effects of grating size, laser power density, and irradiation time on the thermal deformation levels were examined. In the simulation, initial strains and initial stresses we set to be zero, the environment temperature T0 was set to be 20 °C, and the convection thermal transfer coefficient h was 15 W/(m2·K). The intrinsic absorption rate of the MLD grating film was set to 1468.5 ppm, with an unit of 1/m [23]. Table 1 lists the thermophysical parameters of fused silica, which was used for the substrate. The thermophysical parameters listed in the table, including the coefficient of thermal expansion α, specific heat cp, material density ρ, thermal conductivity coefficient k, Young modulus E, and Poisson’s ratio v are regarded as temperature-independent. The difference between single-beam and the multiple-beam irradiation lies in the different incident angle, but this has little effect on the overall trend both temperature and thermal deformation. Therefore, by increasing the irradiation power of single laser beam can replace the multiple beams irradiation situation for the same irradiation power.
Table 1. Thermophysical parameters of MLD grating (fused silica substrates)
3.1 Deformation varies over time according to grating size
To explore the influence of the substrate size on the surface deformation of the MLD grating, we set the irradiation laser to a beam radius of 7.5 mm and power of 1 kW. The thickness of the substrate was varied from 6 mm to 10 mm in 2-mm increments for MLD areas of 30×30 mm2, 40×40 mm2, and 50×50 mm2.
Figures 2(a) and (b) show that the maximum surface deformation varies over time according to the substrate area and thickness of the MLD grating. With decreasing the substrate area, the maximum surface deformation decreases for the same laser irradiation time as shown in Fig. 2(a). The pink dot-dash line indicates the percentage decrease of grating thermal deformation caused by changing the area of the substrate from 50×50 mm2 to 30×30 mm2 (percentage decrease = (µ50×50 - µ30×30)/µ50×50). Figure 2(b) shows the maximum surface deformation change of the MLD grating corresponding to an increase of the substrate thickness. As the laser irradiation time increases, the thickness-dependent difference in surface deformation of the MLD grating gradually decreases. The pink dot-dash line indicates the percentage decrease of the grating thermal deformation caused by changing the thickness of the substrate from 6 mm to 10 mm (percentage decrease = (µh=6 - µh=10)/µh=6). The results show that decreasing the area of the substrate can reduce the thermal deformation of the MLD grating, with an average decrease of 18.3%. Further, for short irradiation times, increasing the thickness of the grating can decrease the thermal deformation of the MLD grating effectively, with a maximum decrease of 40.85%.
Fig. 2. Maximum surface deformation of the MLD grating as a function of (a) time for different fused silica substrate areas (substrate thickness: 10 mm), and (b) thicknesses (area: 50×50 mm2). Inset: Detail of thickness dependence for time from 50 to 60 s.
The maximum temperature change of the MLD grating for different substrate thicknesses is shown in Fig. 3(a). Figure 3(b) is a cross-sectional view of the front surface temperature distribution in the x direction at y = 0 for different substrate thicknesses and irradiation times. It is clear that the maximum temperature of the MLD grating surface increases with decreased substrate thickness. Cross-sectional views of the MLD grating stress value and temperature distributions for different substrate thicknesses and laser irradiation times are shown in Fig. 3(c).
Fig. 3. (a) Maximum temperature of the MLD grating as a function of irradiation time for different substrate thickness. (b) Maximum MLD grating front surface temperature distribution as a function of the x-axis coordinate for different substrate thicknesses and laser irradiation times. (c) and (d) stress value and temperature distributions of the MLD grating for different laser irradiation times for substrate thicknesses of 10 mm and 6 mm, respectively.
From the temperature distributions shown in Figs. 3(c) and 3(d), we infer that most of the thermal energy is concentrated within the central region of the laser irradiation area along the z-axis. This is because the thermal conductivity of the MLD grating substrate is small, causing thermal energy to be accumulated in the laser irradiation area. The higher the temperature of the grating, the higher the thermally induced strains and the greater the thermal deformation of the MLD grating, as shown by Figs. 3(c) and 3(d) (specifically, c1, c3, d1, and d3). However, as the grating thermal deformation increases, the strain tensor will increase, changing the stress of the grating and resulting in the stress distributions shown in the lower right panels of Figs. 3(c) and 3(d) (indicated as c4 and d4). This is the cause of the decrease in the maximum deformation difference of MLD gratings of different thicknesses for prolonged irradiation times.
3.2 Grating surface thermal deformation versus irradiation laser power density
To investigate the influence of the laser power density on the surface deformation of the MLD grating, we set the substrate size to a length and width of 50 mm and thickness of 10 mm. The power of the irradiation laser was varied from 1 kW to 2.5 kW with 0.5-kW increments, and the laser diameter at the MLD was set to 5 mm, 10 mm, and 15 mm.
Figures 4(a) and 4(b) show the MLD grating maximum surface deformation dependences on irradiation laser spot diameter at the MLD and laser power. The maximum surface deformation of the MLD grating increases with decreasing laser spot diameter. The higher the power of the irradiation laser, the higher the maximum surface deformation of the MLD grating surface, as shown in Fig. 4(b). It is clear that a higher laser power density will cause greater thermal deformation of the grating. Thus, emphasize the thermal analysis of the grating under the same laser power density but different power and irradiation laser spot radii, as shown in Fig. 5.
Fig. 4. (a) Maximum surface deformation of the MLD grating as a function of the time for (a) different laser spot diameters and (b) different laser powers.
Fig. 5. (a) Maximum surface thermal deformation and temperature of the MLD grating as a function of irradiation time for constant power density laser irradiation for several combinations of laser power and diameter. (b) and (c) normalized temperature and surface deformation curves.
Figure 5(a) shows how the maximum temperature and surface deformation vary over MLD grating irradiation time for different power and irradiation laser spot radii for a constant laser power density of 15.92 kW/cm2, Figs. 5(b) and 5(c) are the corresponding normalization curves. Figure 6(a) shows a cross-sectional view of the MLD grating front surface temperature distribution in the x direction for different irradiation laser powers and irradiation laser spot radii. The surface thermal deformation distribution has a near-Gaussian distribution, as shown in Figs. 6(b) and 6(c). The larger irradiation area and higher laser power result in more energy being absorbed by the grating for the same irradiation time. Compared with a lower laser power and smaller laser spot radius, the temperature of the MLD grating was higher and the degree of thermal deformation was greater than those for higher laser power and larger laser spot radius laser irradiation. The normalization curves revealed that a lower laser power and smaller laser spot radius leads to a larger maximum surface deformation increase rate and temperature increase rate of the MLD grating, as shown in Figs. 5(b) and 5(c). This is because a smaller laser irradiated area results in a smaller thermal transfer area and a lower temperature conduction speed, thus the temperature increase rate of the MLD grating will increase.
Fig. 6. (a) MLD grating front surface temperature distribution as a function of the x-axis coordinate for different irradiation laser powers and spot diameters. (b) and (c) MLD grating front surface thermal deformation distributions.
4. Experimental verification
4.1 Experimental setup
The temperature and thermal deformation of the MLD grating under different laser power irradiation were measured using the experimental setup shown in Fig. 7. A Twyman-Green interferometer was used to measure the surface thermal deformation of the MLD grating for high power laser irradiation, and an infrared thermal camera was used to measure the surface temperature distribution of the MLD grating. A He-Ne laser with a wavelength of 632 nm was used as the probe laser for the Twyman-Green interferometer. The probe laser was split into two beams using a beam splitter. One beam passed through the beam splitter, was directly incident on the MLD grating, and then reflected by the beam splitter onto a CCD. The other beam was reflected onto the CCD by reflection at the beam splitter and reference mirror. The resulting interference fringes were recorded by the CCD.
A fiber laser with a wavelength of 1064 nm was used as the heat source for the MLD grating and was incident at the MLD Littrow angle. The fiber and He-Ne lasers were adjusted to irradiate the same part of the MLD grating surface, and the radius of each laser spot was 3 mm. The size of the fused silica-substrate MLD grating was 50×50×1.5 mm3. The fiber laser power was adjustment over the range of 0 to 1.02 kW. A power meter (PM) was used to measure the -1 order diffracted laser power. Images of the interference fringes were obtained by the CCD for later analysis. The precise distribution of the grating surface deformation was reconstructed by Fourier-transform method of fringe-pattern analysis [23,27] and Zernike polynomial fitting of the fringe images [28–30].
4.2 Experimental result
The temperature distribution of the MLD grating surface was recorded by the infrared thermal camera in real-time, as shown in Fig. 8(a), and exhibited a near-Gaussian distribution. Figure 8(b) shows the interference fringes on the CCD for an irradiation laser power of 0 W, Based on Fourier-transform method of fringe-pattern analysis and image processing [23,27], the extracted interference fringe image is shown in Fig. 8(c), these fringes are solved using a Zernike polynomial fitting method to obtain the MLD grating surface wavefront information [28–30].
Fig. 8. (a) MLD grating temperature distribution acquired by the infrared thermal camera. (b) Interference fringes and (c) imaged processed fringes for 0 W laser irradiation.
Figure 9(a) and (c) shown the interference fringes on the CCD for different irradiation laser power density. Figure 9(b) and (d) shows the surface deformation distribution of the MLD grating for the irradiation power density of 2.71 kW/cm2 and 3.61 kW/cm2, the irradiation time of 60 s. The experimental results indicate that with increasing laser power, the thermal deformation of the MLD grating become more pronounced.
Fig. 9. (a)(c) Interference fringes for different irradiation laser power density. (b)(d) Surface deformation of the MLD grating for different irradiation power density, and irradiation time of 60 s.
Figure 10(a) shows the maximum surface deformation of the MLD grating for different irradiation laser powers. The maximum surface temperature of the MLD grating for different power irradiation is shown in Fig. 10(b). The temperature increases from 20.2 °C to 210 °C, and the maximum MLD grating thermal deformation increases from 0 to 210.49 nm, when the laser power is increased from 0 to 1.02 kW (The power density is increased from 0 to 3.61 kW/cm2). The blue line represents the experimental data, and the red line the simulation result. The simulation results are in good agreement with the experimental data, for both temperature and surface deformation.
Fig. 10. Comparison of experimental and simulated (a) MLD grating surface deformation and (b) temperature change dependences on irradiation laser power.
In order to verify our calculation results from different aspects, we compared by simulation and experimental result of the grating surface deformation distribution. Figures 11(b) and (d) shows a cross sectional view of the front surface thermal deformation in the y direction with different incidence power levels in 60 s. The simulation matches the experimental results very well.
Fig. 11. Simulated grating surface deformation and MLD grating front surface deformation distribution as a function of the y-axis coordinate with different irradiation powers: a) and b) 1.66 kW/cm2, c) and d) 3.61 kW/cm2.
5. Conclusion
In this paper, we have established a thermal deformation model for the laser irradiated MLD grating based on thermal conduction theory and the thermoelastic equation. The influence of the irradiation laser power and spot radius, substrate size, and irradiation time on MLD grating thermal deformation was simulated. To demonstrate the accuracy of the presented model, the thermal behavior of the MLD grating was experimentally investigated at a laser power density of 3.61 kW/cm2. A Twyman-Green interferometer was used to measure the surface thermal deformation of the MLD grating at different irradiation laser powers. The acquired fringes were image processed and a Zernike polynomial fitting method was applied to reconstruct the grating surface profile. At 3.61 kW/cm2 laser irradiation, the maximum temperature was 210 °C and the maximum thermal deformation was 210 nm for an irradiation time of 60 s. The simulation results were in good agreement with the experimental data, demonstrating the accuracy of the proposed model. Based on the simulation and experimental results, we conclude as follows: The use of a substrate with a small area and large thickness was important to reduce the maximum thermal deformation of the MLD grating. As the thickness of the substrate increases or the area of the substrate decreases, the maximum temperature and thermal deformation of the MLD grating surface gradually decrease. However, the maximum thermal deformation difference of the MLD grating for different thickness decreases gradually, corresponding to an increase of the irradiation time. With increasing irradiation laser power or decreasing laser spot radius, the maximum thermal deformation of the MLD grating increased for the same irradiation time. Compared with a low-power laser and small laser spot radius, the temperature of the MLD grating was higher and the degree of thermal deformation was greater for a larger laser spot radius at the same power density.
Funding
The Key-Area Research and Development Program of Guangdong Province (2018B090904001); Youth Innovation Promotion Association of the Chinese Academy of Sciences; Natural Science Foundation of Shanghai (19ZR1464000, 19ZR1464200); National Natural Science Foundation of China (61705243, 61735007, 61805261).
Disclosures
We declare no conflicts of interest.
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