Study of the key factors affecting temperature of spectral-beam-combination grating

Jiao Xu; Junming Chen; Peng Chen; Yonglu Wang; Yibin Zhang; Fanyu Kong; Yunxia Jin; Jianda Shao

doi:10.1364/OE.26.021675

1. Introduction

Spectral beam combination (SBC) is a promising method for high-radiance lasers, which can achieve a power of 100 kW with a good beam quality [1–3]. SBC is an incoherent combination technique, which combines the output beams of many laser arrays with different wavelengths into a single beam using an external cavity [4]. The concept of SBC was proposed for the first time in 1999 at the Massachusetts Institute of Technology (MIT) [5]. They have performed beam combination of the outputs of two fiber lasers with a grating, both pumped with approximately 1.9 W. The total power of the combined beam was 223 mW. In 2017, the Lockheed Martin Company achieved an output power of 58 kW with a polarization-dependent all-dielectric grating [6]. In 2016, the Shanghai Institute of Optics and Fine Mechanics reported an 8-element SBC of Yb-doped all-fiber super-fluorescent sources at a wavelength of approximately 1070 nm [7]. The eight output beamlets were spectrally combined using a home-made polarization-independent multilayer dielectric reflective diffraction grating; an output power of 10.8 kW was achieved with an efficiency of 94% [8]. Multilayer dielectric gratings (MDGs) are of key importance in the combination of laser arrays into a single beam in SBC external cavities [9–11]. With the increase of the combination power, the temperature of the MDGs unavoidably increases, leading to, initially, surface heat distortion of the MDG. Interference fringe curves, observed using a Michelson interferometer, indicated that MDG thermal deformation occurs under a laser irradiation of 30 W [12]. The MDG wave-front deformation due to thermal effects reaches 0.1λ at 1 μm with a laser irradiance of 1.5 kW/cm² [13]. In 2017, Linxin Li reported that the maximum MDG temperature increased to 192.1 °C, when the radiation power was increased from 0 kW/cm² to 3.6 kW/cm² [14]. The peak-to-valley value Δ of the distorted wavefront reached approximately 0.44λ at 632.8 nm. The beam quality M² in far-field degraded to 2.56 with the increase of the maximum MDG temperature to 192.1 °C. Furthermore, the surface melting or thermal stress can damage the MDG when the temperature continues to increase [15,16]. The influence of the MDG thermal deformation on an SBC system has recently been analyzed numerically; the beam quality of the combined beam degraded with the increase of the incident laser-beam power density [17]. However, detailed numerical calculations and derivation of the temperature distribution of the MDG have not been reported. In this study, the temperature field equation of the MDG is derived, and the main factors that influence the surface temperature of the MDG are investigated; the results reveal approaches to reduce the temperature. Finally, experiments are performed to confirm the calculation results.

2. Theoretical model of the MDG transient temperature

The temperature field of the MDG is obtained by solving the heat conduction equation in transient state. Owing to the sandwich structure of the MDG [8], the intrinsic absorption evenly distributes on the grating layer of the MDG. In order to simplify the calculation, the inhomogeneous impurity absorption is not considered in this study. The intrinsic absorption on the surface is mainly obtained from weak-absorption experiments on the MDG.

When the laser incidents to the MDG surface under a certain angle, the laser energy is deposited on the MDG surface owing to the intrinsic absorption, which increases the temperature of the MDG. The temperature of the MDG is continuous owing to the constant motion of the free electrons and existence of lattice vibration waves [18]. The thicknesses of the films and grating layer are generally at the nanoscale [8]. Therefore, the temperature in the films and grating layer does not considerably change in the depth direction (z-axis); it is approximately equal to the temperature of the substrate’s upper surface. Therefore, the films and grating layer can be ignored in the calculation of the temperature, and the MDG can be simplified as the substrate material with a certain intrinsic absorption rate. The substrate material is generally isotropic. The heat conduction equation of the MDG is:

c_{p} ρ \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T) + ρ f_{0}

where k is the thermal conductivity coefficient of the substrate material [W·(m·K)⁻¹], c_p is the specific heat capacity [J·(kg·K)⁻¹], and ρ is the density (kg·m⁻³). As there is no internal heat source in the MDG, f₀ is 0, hence the heat conduction equation (Eq. (1)) is simplified:

k (\frac{\partial^{2} T}{\partial x^{2}} + \frac{\partial^{2} T}{\partial y^{2}} + \frac{\partial^{2} T}{\partial z^{2}}) = ρ c_{p} \frac{\partial T}{\partial t}

As shown in Fig. 1, the heat conduction equation is solved in rectangular coordinate system. The origin of the coordinate system is located at the bottom–left corner of the MDG. The sides of the MDG are along the X, Y, and Z axes. The incidence direction of the laser is at a certain angle (θ) with respect to the Z axis in the negative direction. The MDG is cooled in the air; therefore, the boundary conditions of the MDG are:

{(k \frac{\partial T}{\partial Z} + h T)}_{z = c} = h T_{0} + η f (x, y)

{(- k \frac{\partial T}{\partial n})}_{S_{2}} = h (T - T_{0})

where c denotes the thickness of the MDG, S₂ represents all of the boundaries of the MDG except the upper surface (z = c), h is the convection heat transfer coefficient in air [W/(m²·K)], T₀ is the temperature of the laboratory, η and f (x, y) are the intrinsic absorption rate and laser power density distribution function, respectively:

f (x, y) = \frac{9 P \cos θ}{2 π r^{2}} e x p [- \frac{9 {(x - \frac{a}{2})}^{2} + 9 \cos^{2} θ {(y - \frac{a}{2})}^{2}}{2 r^{2}}]

where a denotes the length of the square MDG in Fig. 1, respectively, r and θ are the laser radius and angle of incidence, respectively, and P is the total power irradiated by the laser. The initial conditions are:

Fig. 1 MDG diagram in rectangular coordinates.

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T_{t = 0} = T_{0}

The integral-transform method [19] was used to solve the above three-dimensional heat conduction equation. The obtained expression of the transient temperature field is:

\begin{array}{l} T (x, y, z, t) = \sum_{m = 1} \sum_{n = 1} \sum_{p = 1} - \frac{X (β_{m}, x) Y (γ_{n}, y) Z (η_{p}, z)}{N (β_{m}) N (γ_{n}) N (η_{p})} \cdot \frac{{1 - e x p [- α (β_{m}^{2} + γ_{n}^{2} + η_{p}^{2}) t]}}{α (β_{m}^{2} + γ_{n}^{2} + η_{p}^{2})} \\ \cdot A (β_{m}, γ_{n}, η_{p}) + T_{0} \end{array}

In Eq. (7), the functional expressions of $X (β_{m}, x), N (β_{m}), Y (γ_{n}, y), N (γ_{n}), Z (η_{p}, z),$ and $N (η_{p})$ are:

X (β_{m}, x) = β_{m} \cos (β_{m} x) + H \sin (β_{m} x), N (β_{m}) = \frac{1}{2} [(β_{m}^{2} + H^{2}) (a + \frac{H^{2}}{β_{m}^{2} + H^{2}}) + H]

where

β_{m}

are the solutions of the equation:

\tan (a β_{m}) = \frac{2 β_{m} H}{β_{m}^{2} + H^{2}}

Y (γ_{n}, y) = γ_{n} \cos (γ_{n} y) + H \sin (γ_{n} y), N (γ_{n}) = \frac{1}{2} [(γ_{n}^{2} + H^{2}) (b + \frac{H^{2}}{γ_{n}^{2} + H^{2}}) + H]

where

γ_{n}

are the solution of the equation:

\tan (b γ_{n}) = \frac{2 γ_{n} H}{γ_{n}^{2} + H^{2}}

Z (η_{p}, z) = η_{p} \cos (η_{p} z) + H \sin (η_{p} z), N (η_{p}) = \frac{1}{2} [(η_{p}^{2} + H^{2}) (b + \frac{H^{2}}{η_{p}^{2} + H^{2}}) + H]

where

η_{p}

are the solution of the equation:

\tan (c η_{p}) = \frac{2 η_{p} H}{η_{p}^{2} + H^{2}}

The constants H and $α$ in the above formulas can be expressed as:

H = \frac{h}{k}, α = \frac{k}{ρ c_{p}}

The three-dimensional temperature field of the MDG under the laser irradiation can be obtained using the integral-transform method. Equation (7) shows that the time-dependent part is mainly related to the size and physical parameters of the substrate, such as the density, specific heat capacity, and heat conduction coefficient. With the increase of the time, the temperature of the MDG tends to be stable. In addition to the dependence on the size and physical parameters of the substrate, the steady temperature of the MDG is related to the laser power and radius. In order to investigate the relationship between the above parameters and temperature of the MDG in detail, the numerical calculation of the temperature is analyzed under different conditions.

3. Numerical simulations and analysis

The intrinsic absorption is related only to the structure of the grating layer and films, not to the substrate. In the following calculation process, the surface intrinsic absorption rate was set to 100 ppm, according to the results of MDG-surface weak-absorption experiments performed many times. The convective heat dissipation coefficient h in air was 15 W/(m²·K), while the initial temperature of the environment T₀ was 20 °C. The radius and pump power of the incident laser were variables in the calculation. The incident angle of the laser was 30°, which is close to the Littrow angle of a 960-line/mm MDG [8]. The length and thickness of the substrate were also variables. Quartz and YAG were used as the MDG substrate. The basic physical parameters of the two materials are shown in Table 1. In the following sections, we present the variations of the maximum surface temperature with time under the different irradiation and substrate parameters.

Table 1. Physical parameters of quartz and yttrium aluminum garnet (YAG).

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3.1 The maximum surface temperature under different irradiation powers

In order to investigate the influence of the irradiation power on the temperature increase of the MDG, the maximum temperature of the quartz-substrate MDG was calculated for different irradiation powers. The size of the substrate was 50 × 50 × 5 mm³. In the calculation, the radius of the laser had a constant value of 3 mm. The irradiation power was varied in the range of 1.0 kW to 1.8 kW, at an interval of 0.4 kW. We discuss below the calculation results obtained with the finite-element method.

The black curves are the variations of the maximum surface temperature with time under different irradiation powers and the blue curves are the corresponding normalization curve in Fig. 2(a). The normalization curves reveal that the MDG has the same temperature increase rate under the different irradiation powers. Figure 2(b) shows the MDG maximum temperature variations with power in the steady state. For a constant radius of the laser, the steady-state temperature of the MDG linearly increased with the irradiation power. Therefore, it can conclude that the irradiation power does not influence the temperature increase rate of MDG, but determines the steady-state temperature. When the irradiation power increases, the steady-state temperature of MDG rises, but the temperature rise rate remains unchanged.

Fig. 2 (a) Maximum MDG temperature as a function of the time and corresponding normalization curves for different irradiation powers; (b) MDG maximum temperature variations with power in the steady state.

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3.2 The maximum surface temperature under different laser radii

In order to investigate the influence of the laser radius on the temperature increase of the MDG, the maximum temperature of the quartz-substrate MDG was calculated under different laser radii. The size of the substrate was 50 × 50 × 5 mm³. In the calculation, the power of the laser had a constant value of 1.8 kW. The laser radius was varied in the range of 3 mm to 7 mm, at an interval of 2 mm. We present below the calculation results obtained with the finite-element method.

The black curves are the variations of the maximum surface temperature with time under different laser radii and the blue curves are the corresponding normalization curve in Fig. 3(a). The normalization curves reveal that a smaller laser radius leads to a larger temperature increase rate of the MDG. Figure 3(b) shows the MDG maximum temperature variations with laser radius in the steady state. For a constant power of the irradiation, the steady-state temperature of the MDG decreased as parabola type with the laser radius. Therefore, it can conclude that with the increase of the laser radius, the ability of the MDG to withstand a higher power increases; however, it reduces the temperature increase rate of the MDG, which delays the stable output of laser. When the lasers with large radii are used in SBC system, the MDG can tolerate higher laser power but the output laser takes more time to achieve steady state.

Fig. 3 (a) Variations of the maximum MDG temperature as a function of the time and corresponding normalization curves for different laser radii; (b) MDG maximum temperature variations with laser radius in the steady state.

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3.3 Maximum temperature of the MDG for different substrate sizes

In order to investigate the influence of the substrate size on the temperature increase of the MDG, the maximum temperature was calculated for different sizes of the quartz-substrate MDG. The irradiation power and radius of the laser were set to 1.8 kW and 3 mm, respectively. By separately varying the thickness and length of the substrate, the temperature increase of the MDG was studied using the finite-element method.

Figure 4(a) shows variations of the maximum temperature of the MDG with time and the corresponding normalization curves for different substrate thicknesses. The substrate length a was 50 mm. The substrate thickness c was varied in the range of 2 mm to 10 mm, at an interval of 4 mm. With the increase of the substrate thickness, the steady-state temperature of the MDG and the temperature increase rate both decrease gradually. Figure 4(b) shows MDG maximum temperature variations for different substrate thicknesses in the steady state. This reveals that with the increase of the substrate thickness, the maximum temperature of the MDG gradually decreases and becomes steady slowly. Therefore, increasing the substrate thickness within limits is beneficial to improving the ability of the MDG to withstand a higher power but the output laser takes more time to achieve steady state.

Fig. 4 (a) Maximum temperature of the MDG as a function of the time and corresponding normalization curves for different substrate thicknesses; (b) MDG maximum temperature variations for different substrate thicknesses in the steady state; (c) Maximum temperature of the MDG as a function of the time and corresponding normalization curves for different substrate lengths; (d) MDG maximum temperature variations for different substrate lengths in the steady state.

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Figure 4(c) shows the variations of the maximum temperature of the MDG with time and the corresponding normalization curves for different substrate lengths. The substrate length a was varied in the range of 20 mm to 60 mm, at an interval of 20 mm. The substrate thickness c was set to 5 mm. As shown in Fig. 4(c), the substrate length hardly influences the temperature increase rate of the MDG and the steady-state temperature, which could be attributed to the significantly larger substrate length than the diameter of the laser in the calculation. Figure 4(d) shows MDG maximum temperature variations for different substrate lengths in the steady state. This reveals that the effect of increasing the substrate lengths on reducing the temperature is too small. In practical use, the substrate length of the MDG has to be significantly larger than the laser diameter. Therefore, the change of the substrate length does not have a large contribution to the improvement of the MDG temperature. Increasing the length will not significantly reduce the temperature of MDG, but spends more time and money.

3.4 Variations of the maximum temperature of the MDG with time for different substrate materials

In order to investigate the influence of the substrate material on the temperature increase of the MDG, the maximum temperatures of the quartz- and YAG-substrate MDGs were calculated under the same irradiation conditions. The size of the substrate was 50 × 50 × 5 mm³. The radius and pump power of the incident laser were set to 3 mm and 1.8 kW, respectively. The basic physical parameters of the two materials are shown in Table 1. We present below the calculation results obtained with the finite-element method.

Figure 5 shows the variations of the maximum temperatures of the MDGs with the different substrate materials as a function of the time. The black and red curves represent the temperature variations of the quartz- and YAG-substrate MDGs, respectively. The steady-state temperatures of the MDGs with different substrate materials are significantly different. If the high thermal conductivity material is used instead of quartz as the MDG substrate, the temperature of the MDG will be significantly reduced under the same irradiation power.

Fig. 5 Variations of the maximum temperatures of the MDGs with different substrate materials as a function of the time.

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4. Experiment

4.1 Experimental conditions

In order to verify the effects of the above physical parameters on the MDG temperature increase, the experimental setup illustrated in Fig. 6(a) was employed. A pump laser was incident to the MDG surface. The maximum output power was 1.8 kW, while the wavelength of the pump laser was 1,064 nm. The incidence angle of the pump laser was near the Littrow angle; the laser radius was 7.25 mm. In the Fig. 6(a), PM denotes a power meter used to receive the −1-order diffracted laser. If a large-focal-length lens is introduced in front of the MDG, the laser radius can be changed by adjusting the distance between the lens and MDG. The infrared thermal camera is used to record the variations of the MDG surface temperature with time. Figure 6(b) shows the temperature of the MDG measured by the infrared thermal imager. The parameters of the MDGs are shown in Table 2.

Fig. 6 (a) Schematic of the MDG-temperature measurement. (b) Temperature of the MDG measured by the infrared thermal imager.

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Table 2. Parameters of the MDGs.

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4.2 Effects of the irradiation power and laser radius on the MDG temperature

Figure 7 shows variations of the MDG maximum temperature with time under different irradiation conditions. The black and red curves represent the maximum temperatures of the MDG under irradiation powers of 1 kW and 1.8 kW, respectively. The laser radius is 3.1 mm. The comparison of the two curves reveals that the temperature increase rates of the MDG are equal; however, the steady-state temperature of the MDG is higher for a larger irradiation power, which agrees with the results of the theoretical calculation.

Fig. 7 Variations of the maximum temperature of the MDG with time under different irradiation conditions.

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The blue solid curve represents the variations of the maximum temperature of the MDG with time under the large-radius irradiation. The laser radius is 7.25 mm, while the irradiation power is 1.8 kW. The comparison of the blue and red curves reveals that although the steady-state temperature of the MDG is lower under the large-radius irradiation, the temperature increase rate of the MDG decreases, which verifies the results of the theoretical calculation.

4.3 Effects of the substrate thickness and material on the MDG temperature

In order to verify the effects of the substrate thickness and material on the MDG temperature, the maximum temperatures of the S01, S02, and Y02 MDGs are measured under the same irradiation conditions: irradiation power of 1.8 kW and laser radius of 7.25 mm.

Figure 8 shows the variations of the MDG maximum temperature with time for different substrate parameters. The red and blue curves represent the maximum temperatures of the S01 and S02 MDGs, respectively. The substrate thicknesses of the S01 and S02 MDGs are 5 mm and 10 mm, respectively. Both substrate materials are quartz. The comparison of the two curves reveals that a larger thickness yields a lower steady-state temperature and slower temperature increase. The experimental results are in a good agreement with the theoretical data. The black curve in Fig. 8 represents the maximum temperature of the Y02 MDG. The substrate thickness of the Y02 MDG is 5 mm, while the substrate material is YAG. The comparison of the black and red temperature curves reveals that the MDG temperature increases for the different substrate materials are significantly different. The temperature of the YAG-substrate MDG is significantly lower than that of the quartz-substrate MDG, which agrees with the theoretical data. Therefore, the use of the substrate material with a larger thermal conductivity is of significance to reduce the temperature of the MDG.

Fig. 8 Variations of the maximum temperature of the MDG with time for different substrate parameters.

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5. Conclusion

The temperature of the MDG under continuous laser irradiation is related to various factors, such as the laser radius, irradiation power, thickness of the substrate, and substrate material. In the SBC system, the laser radius and irradiation power are mainly determined by the laser units [5] and cannot be easily changed. Therefore, the thickness of the substrate and substrate material were the only factors that could be varied to reduce the temperature of the MDG. According to the theoretical analysis and experimental verification, we can obtain the following conclusions.

1. The increase of the substrate thickness can reduce the steady-state temperature of the MDG to some degree but delays the stable output of laser in the SBC.
2. The use of a substrate material with a better thermal conductivity coefficient helps effectively reduce the temperature of the MDG.

Funding

NSAF (No. U1430121) and the National Natural Science Foundation of China (Grant No. 11604352, U1630140).

References and links

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Substrate material	Density ρ [g·cm⁻³]	Specific heat capacity c_p [J·g⁻¹·K⁻¹]	Thermal conductivity coefficient k [W·(m·K)⁻¹]
quartz	2.65	0.213	0.35
YAG	4.55	0.625	14

Substrate material	Density ρ [g·cm⁻³]	Specific heat capacity c_p [J·g⁻¹·K⁻¹]	Thermal conductivity coefficient k [W·(m·K)⁻¹]
quartz	2.65	0.213	0.35
YAG	4.55	0.625	14

Study of the key factors affecting temperature of spectral-beam-combination grating

Abstract

1. Introduction

2. Theoretical model of the MDG transient temperature

3. Numerical simulations and analysis

3.1 The maximum surface temperature under different irradiation powers

3.2 The maximum surface temperature under different laser radii

3.3 Maximum temperature of the MDG for different substrate sizes

3.4 Variations of the maximum temperature of the MDG with time for different substrate materials

4. Experiment

4.1 Experimental conditions

4.2 Effects of the irradiation power and laser radius on the MDG temperature

4.3 Effects of the substrate thickness and material on the MDG temperature

5. Conclusion

Funding

References and links

Cited By

Figures (8)

Tables (2)

Equations (12)

Optics Express

MDG number	Line density [line·mm⁻¹]	Substrate material	Substrate size [mm³]
Y02	960	YAG	50 × 50 × 5
S01	960	quartz	50 × 50 × 5
S02	960	quartz	50 × 50 × 10