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Curr. Opt. Photon. 2022; 6(6): 627-633

Published online December 25, 2022 https://doi.org/10.3807/COPP.2022.6.6.627

## 300-W-class Side-pumped Solar Laser

Hongfei Qi1, Lanling Lan1, Yan Liu1 , Pengfei Xiang1, Yulong Tang2

1Center for Astronomy and Space Science, College of Science, China Three Gorges University, Yichang, Hubei 443002, China
2Key Laboratory for Laser Plasmas (MOE), School of Physics and Astronomy, Collaborative Innovation Center of IFSA, Shanghai Jiao Tong University, Shanghai 200240, China

Corresponding author: *liuyan703@163.com, ORCID 0000-0003-2806-9264
**yulong@sjtu.edu.cn, ORCID 0000-0001-7388-2516

Received: June 30, 2022; Revised: September 19, 2022; Accepted: September 29, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

To realize uniform side pumping of solar lasers and improve their output power, a solar concentrating system based on off-axis parabolic mirrors is proposed. Four identical off-axis parabolic mirrors with focal length of 1,000 mm are toroidally arranged as the primary concentrator. Four two-dimensional compound parabolic concentrators (2D-CPCs) are designed as a secondary concentrator to further compress the focused spot induced by the parabolic mirrors, and the focused light is then homogenized by four rectangular diffusers and provides uniform pumping for a laser-crystal rod to achieve solar laser emission. Simulation results show that the solar power received by the laser rod, uniformity of the light spot, and output power of the solar laser are 7,872.7 W, 98%, and 351.8 W respectively. This uniform pumping configuration and concentrator design thus provide a new means for developing high-power side-pumped solid-state solar lasers.

Keywords: Compound parabolic concentrator, Off-axis parabolic mirror, Side pumping, Solar laser

OCIS codes: (140.3580) Lasers, solid-state; (140.5560) Pumping; (350.6050) Solar energy

A solar laser is a laser pumped by sunlight. Because sunlight is a renewable and clean energy source, solar lasers are also called “green” lasers. Solar lasers have great application prospects in various areas, including energy utilization, space debris removal, space communication, measurement, and magnesium resource recycling [1–4].

Research on solar pumped solid-state lasers began in the 1960s. In 1965, Yong et al. [5] reported a solar side-pumped Nd:YAG laser with output power of 1 W, which was the first case in which lasing was actually achieved with a solar setup. Since then, solar pumped solid-state lasers have attracted great interest [6–9]. The key element for highly efficient and high-power solar solid-state lasers is a specially designed sunlight-collection system. In 1975, Falk et al. [10] demonstrated a Cassegrain system with a conical lens as a solar collection system, and achieved 4.5 W of laser output. In 1988, Wekslar and Shwartz [11] designed a solar concentrator with a heliostat, a faceted mirror made of some 600 spherical mirrors, and a compound parabolic concentrators (CPC), and 60 W of CW laser output was obtained by side-pumping a rod-shaped Nd:YAG crystal. In 2003, Lando et al. [12] used a two-stage concentrating system composed of a primary mirror and a two-dimensional compound parabolic concentrators (2D-CPC) to side-pump a Nd:YAG rod, and obtained a laser output of 45 W. In 2007, Yabe et al. [13–15] used a Fresnel lens as the primary concentrator and a conical cavity as the secondary concentrator to side pump a Cr,Nd:YAG ceramic, but the laser output power was only 24.4 W. In 2012, they used the same concentrating system but with the method of split-cavity water cooling, and the laser output power was improved to 120 W by side-pumping a Nd:YAG rod. Liang [16–18] designed various concentrating systems such as 2D-CPC-EL, fused-silica elliptical cavity, total-internal-reflection secondary concentrator, and two-dimensional elliptical-cylindrical (2D-EL-CYL) cavity. In 2021, Liang’s group [19] used four parabolic mirrors and aspheric lenses as the solar concentrator, and obtained a laser output of 346.8 W. To our knowledge, the maximum laser output power of solar solid-state lasers has reached 500 W, for a solar concentrating system consisting of a heliostat field of 64 parabolic mirrors, a 3D CPC, and a 2D CPC [20]. Solar lasers based on other gain media were also reported. In 2013, Johnson et al. [21] demonstrated a solar pumped semiconductor laser based on multijunction photovoltaic cells and laser diodes; the laser output power was 4.31 W. In 2014, Danilov et al. [22] reported a solar pumped oxygen-iodine laser. In 2016, Payziyev et al. [23] investigated solar pumping of a neodymium-containing phosphorus oxychloride solution as a lasing medium. In 2021, Guo et al. [9] demonstrated a solar pumped Nd-doped fiber laser and obtained a laser output of 2.6 W.

Collecting more sunlight is the key to high-power solar lasers, and achieving uniform side-pumping of the laser medium can improve laser-beam quality. Compared to end pumping, side pumping features a larger light-receiving surface, and thus can collect more sunlight. What’s more, side pumping has the advantage of providing highly uniform solar pumping for a solid-state laser. Here we propose a solar concentrating system composed of off-axis parabolic mirrors, 2D-CPCs, and rectangular diffusers, and use it to collect sunlight to side-pump a Nd:YAG solid-state laser. Simulation shows that this solar collection system provides not only high concentrating power, but also excellent pumping uniformity. Total concentration of solar power as high as 7,872.7 W and side-pumping uniformity of 98% can be realized, leading to output power that reaches 351.8 W.

The primary concentrator consists of several off-axis parabolic mirrors. A fan-circular parabolic mirror with an inner radius of 1,000 mm, an outer radius of 3,000 mm, and a height of 2,000 mm can be divided into n parts, with each part being an off-axis parabolic mirror. An off-axis parabolic mirror, a 2D-CPC, and a rectangular diffuser together make up a solar concentrating subsystem, the structure of which is shown in Fig. 1.

Figure 1.Schematic of the solar concentrating subsystem.

The three-dimensional structure of the entire solar concentrator, composed of four subsystems, is shown in Fig. 2. One end of the rectangular diffuser is connected to the outlet of the 2D-CPC, and the other end is put close to the cooling-water channel. The Nd:YAG laser rod has a size of Φ6 × 30 mm, and is cooled by a circulating-water channel with outer and inner diameters of 9 mm and 6 mm respectively.

Figure 2.Schematic diagram of (a) the primary concentrating system, and (b) the laser head.

The concentrating characteristics of the primary concentrator are simulated by ray tracing with TracePro, and a circular light spot is obtained at the focal plane of each off-axis parabolic mirror. The irradiance diagram is shown in Fig. 3. According to the typical reduction of sunlight power density to 1/e of the maximum value, the focused spot has a diameter of about 20 mm. Since the symmetry axis of the off-axis parabolic mirror is consistent with that of the laser rod, and the length of the laser rod is 30 mm, only the width of the focused spot needs to be compressed. 2D-CPCs are adopted for the secondary concentrator. The outlet length of the 2D-CPC is 30 mm, which is consistent with the length of the laser rod. Since only the width of the light spot needs to be compressed, the bottom and top ends of the 2D-CPC are set to be flat. The structure of a 2D-CPC is shown in Fig. 4. The maximum acceptance half-angle of the 2D-CPC depends on the maximum divergence half-angle of sunlight passing through the off-axis parabolic mirror. The outlet width of the 2D-CPC is consistent with the chord length of the corresponding laser-rod cross section.

Figure 3.Irradiance map at the focal point of an off-axis parabolic mirror.

Figure 4.Schematic of the secondary concentrator.

### Ⅲ. OPTIMIZATION OF THE CONCENTRATING SYSTEM

To maximize the solar power converging on the side of the laser rod and improve the uniformity of the light spot, it is necessary to optimize the design of the number of parts n of the fan-circular parabolic mirror, the acceptance half angle of the 2D-CPC, and the length of the rectangular diffuser. The concentrating principle of a single off-axis parabolic mirror is shown in Fig. 5. Here CN is the center line of the off-axis parabolic mirror, OM is the rotational-symmetry axis of the parabolic mirror, and O is the focal point of the parabolic mirror. AO, CO and DO represent three converging sunlight rays. AC⊥CN, and B is the midpoint of line AC. The axis of symmetry OM is perpendicular to the plane determined by O’AC. According to geometric relations, the length of AB¯ can be expressed as

Figure 5.Schematic diagram of the condensing principle of the off-axis parabolic mirror.

a=xsinθ'

where θ′ = 360/(4n). The coordinates of A are (x, y, 0) and those of the focus O are (0, 1000, 0). Thus, the following relation can be obtained:

θ=2arcsinaL=2arcsina x2+ 1000y2

Here y can be obtained from the parabolic equation y = 14000x2. L is the distance from A to O. θ is the divergence half-angle of the convergent light rays, and x is the distance from point A to the symmetry axis OM.

The fan-shaped parabolic mirror is divided into n parts (n = 3–8). Figure 6 demonstrates the variation of the sunlight power collected by a single concentrating subsystem with the divergence half-angle of the convergent light rays, for different values of n. The sunlight reflectivity of the concentrator system is assumed to be 98%. The power density of the surface sunlight is taken as 900 W/m2 [24]. For a given n, the divergence half-angle increases from the minimum to the maximum in steps of one degree. After the optimum divergence half-angle is determined, the length of the rectangular diffuser is optimized to give the light spot the highest uniformity, and under this condition the sunlight power collected by a single concentrating subsystem is determined. In fact, the optimum divergence half-angle is the optimum acceptance half-angle of the 2D-CPC. If n is 2, the maximum divergence half-angle of the converging spot of a single off-axis parabolic mirror is 90°; under these conditions the spot cannot be compressed by the 2D-CPC, so the minimum value of n is taken to be 3.

Figure 6.Variation of the collected sunlight power of a concentrating subsystem with the divergence half-angle, for different numbers of mirror parts: (a) n = 3, (b) n = 4, (c) n = 5, (d) n = 6, (e) n = 7, (f) n = 8.

The total collected power is the sum from the n subsystems. Ray tracing with TracePro is performed to obtain the total sunlight power incident on the side of laser rod from the concentrating system, when different numbers of subsystems n are assumed. The results are shown in Fig. 7.

Figure 7.Relationship between total incident solar power and n.

When the fan-circular parabolic mirror is divided into 4 (n = 4) identical parts, the solar power received by the side of the laser rod reaches a maximum value of 7,872.7 W. In this case, the acceptance half-angle of the 2D-CPC and the length of the rectangular diffuser are 40° and 92 mm respectively. The surface of the concentrator is coated with a film with 98% reflectivity for sunlight. After the sunlight is concentrated by the off-axis parabolic mirror, 62.2% of the total sunlight power enters the 2D-CPC. After passing through the 2D-CPC, the rectangular diffuser and the cooling-water layer induce 27.5% loss, so 34.7% of the sunlight power converges on the side of the laser rod. The irradiance upon the side surface of the laser rod is shown in Fig. 8. The irradiance uniformity is defined as

Figure 8.Irradiance map of the side of the laser rod.

ΔE=1EmaxEaveEmax+Eave×100%

Here Emax and Eave are the maximum value of irradiance and the average value of the irradiance on the receiving surface respectively. When n is 4, the uniformity is 98%. We also investigated the uniformity of the light irradiance on the surface of the laser rod when n is 3, 5, 6, 7 and 8; the values are 97.4%, 98.3%, 97.4%, 98.4%, and 97.5% respectively.

### Ⅳ. ANALYSIS OF THERMAL EFFECTS AND CALCULATION OF LASER OUTPUT POWER

The heat distribution in the laser rod can be expressed as [25]

Qr=ηheatαpeαr0r

where ηheat is the fraction of the absorbed pump power that is converted to heat, α is the average absorption coefficient of the rod, p is the incident solar power per unit area, and r0 is the rod’s radius.

The temperature distribution in the laser rod is analyzed using the Comsol multiphysics software. The average absorption coefficient of the Nd:YAG laser rod across the whole solar spectrum is taken to be 0.35 cm−1 [26]. The surface heat-transfer coefficient is 1 W/(cm2·℃) and the temperature of the cooling water is 293 K. ηheat is calculated to be 0.46 [26]. After simulation, the transverse temperature distribution of the laser rod is shown in Fig. 9. The temperature is 359 K at the center of the laser rod and 325 K at the surface, with a gradient of 34 K.

Figure 9.Transverse temperature distribution in the laser rod.

Nd:YAG is a typical four-level laser medium. By solving the four-level rate equations, the laser output power can calculated with the equation [27]

Pout=21Rη1+R2α0l1nRPinIs 2α0l1nRA2η

where η = ηQηSηBηpηTηa; ηQ , ηS, and ηB are the quantum efficiency, the Stokes factor, and the beam overlap efficiency respectively; and ηp, ηT, and ηa are the laser absorption and solar-spectrum overlap, the radiation transmission efficiency, and the absorption efficiency respectively. R, a0, and l are the reflectivity of the output cavity mirror, the scattering coefficient, and the length of laser crystal respectively. Pin is the total pump power and A is the end-face area of the laser rod. IS is the saturation intensity of the laser. For a four-level system, the saturation intensity is [27]

Is=hvτfσ21

Here ν is the pump frequency and h is Planck’s constant. tf is the fluorescence lifetime and σ21 is the stimulated emission cross section. The relationship of laser output power Pout, pump threshold Pth, and slope efficiency β can be written as Pout = b (PinPth). The slope efficiency and threshold pump power are

β=21Rη1+R2α0l1nR

Pth=Is2α0l1nRA2η

When the Nd:YAG rod is pumped by sunlight, the laser absorption and solar-spectrum overlap is 0.16 [28]. The transmission efficiency is 0.85 [28], and the absorption efficiency is 0.82 in this laser system. The average weighted wavelength of sunlight is 660 nm [11], the laser output wavelength is 1,064 nm, and the Stokes factor is 0.62 [11]. The beam overlap efficiency is the ratio of the cavity mode volume to the pump volume of the laser rod; its value is assumed to be 0.91. The quantum efficiency is 0.9 [28], the fluorescence lifetime is 230 μs [27], and the stimulated emission cross section is 6.5 × 10−19 cm2 [27]. The scattering coefficient of the laser material is 0.003 cm−1 [28]. Based on Eqs. (5)–(8), the output power of this solar laser with varying reflectivity of the output cavity mirror and varying solar pump power are calculated, and the results are shown in Fig. 10. The reflectivity of the other end of the crystal is assumed to be 100%. When the reflectivity of the output cavity mirror is 83%, a maximum laser output power of 351.8 W is obtained, and the threshold pump power, the slope efficiency, and solar-to-laser conversion efficiency are 616 W, 4.85%, and 4.5% respectively.

Figure 10.Change in calculated laser output power with (a) the output cavity mirror’s reflectivity, and (b) the total input pump power.

To achieve uniform side pumping of a high-power solar laser, a solar concentrating system is proposed. The concentrator system consists of four off-axis parabolic mirrors with a focal length of 1,000 mm, and four 2D-CPCs and rectangular diffusers. To collect more solar power, the number of off-axis parabolic mirrors, the acceptance half-angle of the 2D-CPCs, and the length of the diffusers are optimized. When four off-axis parabolic mirrors are adopted, the solar power received by the side of the laser rod is the highest, being up to 7,872.7 W; the corresponding optimal acceptance half-angle of the 2D-CPCs is 40°; and the optimal length of the rectangular diffusers is 92 mm. The irradiance uniformity on the side of the laser crystal rod reaches 98%. Then the transverse temperature distribution of the laser rod is analyzed, and the center temperature is found to be 359 K, while the surface temperature is 325 K. Finally, based on the four-level rate equations, the output characteristics of the solar laser are analyzed. When the reflectivity of the output cavity mirror is 83%, the maximum laser output power, the threshold pump power, the slope efficiency, and the solar-to-laser conversion efficiency are 351.8 W, 616 W, 4.85%, and 4.5% respectively.

The authors declare no conflicts of interest.

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.

National Natural Science Foundation of China (61675129); the Natural Science Foundation of Hubei Province (2014 CFB671); Natural Science Foundation of Shanghai (19ZR1427100); Research Fund for Excellent Dissertation of China Three Gorges University (2021SSPY149).

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### Article

#### Article

Curr. Opt. Photon. 2022; 6(6): 627-633

Published online December 25, 2022 https://doi.org/10.3807/COPP.2022.6.6.627

## 300-W-class Side-pumped Solar Laser

Hongfei Qi1, Lanling Lan1, Yan Liu1 , Pengfei Xiang1, Yulong Tang2

1Center for Astronomy and Space Science, College of Science, China Three Gorges University, Yichang, Hubei 443002, China
2Key Laboratory for Laser Plasmas (MOE), School of Physics and Astronomy, Collaborative Innovation Center of IFSA, Shanghai Jiao Tong University, Shanghai 200240, China

Correspondence to:*liuyan703@163.com, ORCID 0000-0003-2806-9264
**yulong@sjtu.edu.cn, ORCID 0000-0001-7388-2516

Received: June 30, 2022; Revised: September 19, 2022; Accepted: September 29, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

To realize uniform side pumping of solar lasers and improve their output power, a solar concentrating system based on off-axis parabolic mirrors is proposed. Four identical off-axis parabolic mirrors with focal length of 1,000 mm are toroidally arranged as the primary concentrator. Four two-dimensional compound parabolic concentrators (2D-CPCs) are designed as a secondary concentrator to further compress the focused spot induced by the parabolic mirrors, and the focused light is then homogenized by four rectangular diffusers and provides uniform pumping for a laser-crystal rod to achieve solar laser emission. Simulation results show that the solar power received by the laser rod, uniformity of the light spot, and output power of the solar laser are 7,872.7 W, 98%, and 351.8 W respectively. This uniform pumping configuration and concentrator design thus provide a new means for developing high-power side-pumped solid-state solar lasers.

Keywords: Compound parabolic concentrator, Off-axis parabolic mirror, Side pumping, Solar laser

### I. INTRODUCTION

A solar laser is a laser pumped by sunlight. Because sunlight is a renewable and clean energy source, solar lasers are also called “green” lasers. Solar lasers have great application prospects in various areas, including energy utilization, space debris removal, space communication, measurement, and magnesium resource recycling [1–4].

Research on solar pumped solid-state lasers began in the 1960s. In 1965, Yong et al. [5] reported a solar side-pumped Nd:YAG laser with output power of 1 W, which was the first case in which lasing was actually achieved with a solar setup. Since then, solar pumped solid-state lasers have attracted great interest [6–9]. The key element for highly efficient and high-power solar solid-state lasers is a specially designed sunlight-collection system. In 1975, Falk et al. [10] demonstrated a Cassegrain system with a conical lens as a solar collection system, and achieved 4.5 W of laser output. In 1988, Wekslar and Shwartz [11] designed a solar concentrator with a heliostat, a faceted mirror made of some 600 spherical mirrors, and a compound parabolic concentrators (CPC), and 60 W of CW laser output was obtained by side-pumping a rod-shaped Nd:YAG crystal. In 2003, Lando et al. [12] used a two-stage concentrating system composed of a primary mirror and a two-dimensional compound parabolic concentrators (2D-CPC) to side-pump a Nd:YAG rod, and obtained a laser output of 45 W. In 2007, Yabe et al. [13–15] used a Fresnel lens as the primary concentrator and a conical cavity as the secondary concentrator to side pump a Cr,Nd:YAG ceramic, but the laser output power was only 24.4 W. In 2012, they used the same concentrating system but with the method of split-cavity water cooling, and the laser output power was improved to 120 W by side-pumping a Nd:YAG rod. Liang [16–18] designed various concentrating systems such as 2D-CPC-EL, fused-silica elliptical cavity, total-internal-reflection secondary concentrator, and two-dimensional elliptical-cylindrical (2D-EL-CYL) cavity. In 2021, Liang’s group [19] used four parabolic mirrors and aspheric lenses as the solar concentrator, and obtained a laser output of 346.8 W. To our knowledge, the maximum laser output power of solar solid-state lasers has reached 500 W, for a solar concentrating system consisting of a heliostat field of 64 parabolic mirrors, a 3D CPC, and a 2D CPC [20]. Solar lasers based on other gain media were also reported. In 2013, Johnson et al. [21] demonstrated a solar pumped semiconductor laser based on multijunction photovoltaic cells and laser diodes; the laser output power was 4.31 W. In 2014, Danilov et al. [22] reported a solar pumped oxygen-iodine laser. In 2016, Payziyev et al. [23] investigated solar pumping of a neodymium-containing phosphorus oxychloride solution as a lasing medium. In 2021, Guo et al. [9] demonstrated a solar pumped Nd-doped fiber laser and obtained a laser output of 2.6 W.

Collecting more sunlight is the key to high-power solar lasers, and achieving uniform side-pumping of the laser medium can improve laser-beam quality. Compared to end pumping, side pumping features a larger light-receiving surface, and thus can collect more sunlight. What’s more, side pumping has the advantage of providing highly uniform solar pumping for a solid-state laser. Here we propose a solar concentrating system composed of off-axis parabolic mirrors, 2D-CPCs, and rectangular diffusers, and use it to collect sunlight to side-pump a Nd:YAG solid-state laser. Simulation shows that this solar collection system provides not only high concentrating power, but also excellent pumping uniformity. Total concentration of solar power as high as 7,872.7 W and side-pumping uniformity of 98% can be realized, leading to output power that reaches 351.8 W.

### Ⅱ. STRUCTURE OF THE CONCENTRATING SYSTEM

The primary concentrator consists of several off-axis parabolic mirrors. A fan-circular parabolic mirror with an inner radius of 1,000 mm, an outer radius of 3,000 mm, and a height of 2,000 mm can be divided into n parts, with each part being an off-axis parabolic mirror. An off-axis parabolic mirror, a 2D-CPC, and a rectangular diffuser together make up a solar concentrating subsystem, the structure of which is shown in Fig. 1.

Figure 1. Schematic of the solar concentrating subsystem.

The three-dimensional structure of the entire solar concentrator, composed of four subsystems, is shown in Fig. 2. One end of the rectangular diffuser is connected to the outlet of the 2D-CPC, and the other end is put close to the cooling-water channel. The Nd:YAG laser rod has a size of Φ6 × 30 mm, and is cooled by a circulating-water channel with outer and inner diameters of 9 mm and 6 mm respectively.

Figure 2. Schematic diagram of (a) the primary concentrating system, and (b) the laser head.

The concentrating characteristics of the primary concentrator are simulated by ray tracing with TracePro, and a circular light spot is obtained at the focal plane of each off-axis parabolic mirror. The irradiance diagram is shown in Fig. 3. According to the typical reduction of sunlight power density to 1/e of the maximum value, the focused spot has a diameter of about 20 mm. Since the symmetry axis of the off-axis parabolic mirror is consistent with that of the laser rod, and the length of the laser rod is 30 mm, only the width of the focused spot needs to be compressed. 2D-CPCs are adopted for the secondary concentrator. The outlet length of the 2D-CPC is 30 mm, which is consistent with the length of the laser rod. Since only the width of the light spot needs to be compressed, the bottom and top ends of the 2D-CPC are set to be flat. The structure of a 2D-CPC is shown in Fig. 4. The maximum acceptance half-angle of the 2D-CPC depends on the maximum divergence half-angle of sunlight passing through the off-axis parabolic mirror. The outlet width of the 2D-CPC is consistent with the chord length of the corresponding laser-rod cross section.

Figure 3. Irradiance map at the focal point of an off-axis parabolic mirror.

Figure 4. Schematic of the secondary concentrator.

### Ⅲ. OPTIMIZATION OF THE CONCENTRATING SYSTEM

To maximize the solar power converging on the side of the laser rod and improve the uniformity of the light spot, it is necessary to optimize the design of the number of parts n of the fan-circular parabolic mirror, the acceptance half angle of the 2D-CPC, and the length of the rectangular diffuser. The concentrating principle of a single off-axis parabolic mirror is shown in Fig. 5. Here CN is the center line of the off-axis parabolic mirror, OM is the rotational-symmetry axis of the parabolic mirror, and O is the focal point of the parabolic mirror. AO, CO and DO represent three converging sunlight rays. AC⊥CN, and B is the midpoint of line AC. The axis of symmetry OM is perpendicular to the plane determined by O’AC. According to geometric relations, the length of $AB¯$ can be expressed as

Figure 5. Schematic diagram of the condensing principle of the off-axis parabolic mirror.

$a=xsinθ'$

where θ′ = 360/(4n). The coordinates of A are (x, y, 0) and those of the focus O are (0, 1000, 0). Thus, the following relation can be obtained:

$θ=2arcsinaL=2arcsina x2+ 1000−y2$

Here y can be obtained from the parabolic equation y = $14000x2$. L is the distance from A to O. θ is the divergence half-angle of the convergent light rays, and x is the distance from point A to the symmetry axis OM.

The fan-shaped parabolic mirror is divided into n parts (n = 3–8). Figure 6 demonstrates the variation of the sunlight power collected by a single concentrating subsystem with the divergence half-angle of the convergent light rays, for different values of n. The sunlight reflectivity of the concentrator system is assumed to be 98%. The power density of the surface sunlight is taken as 900 W/m2 [24]. For a given n, the divergence half-angle increases from the minimum to the maximum in steps of one degree. After the optimum divergence half-angle is determined, the length of the rectangular diffuser is optimized to give the light spot the highest uniformity, and under this condition the sunlight power collected by a single concentrating subsystem is determined. In fact, the optimum divergence half-angle is the optimum acceptance half-angle of the 2D-CPC. If n is 2, the maximum divergence half-angle of the converging spot of a single off-axis parabolic mirror is 90°; under these conditions the spot cannot be compressed by the 2D-CPC, so the minimum value of n is taken to be 3.

Figure 6. Variation of the collected sunlight power of a concentrating subsystem with the divergence half-angle, for different numbers of mirror parts: (a) n = 3, (b) n = 4, (c) n = 5, (d) n = 6, (e) n = 7, (f) n = 8.

The total collected power is the sum from the n subsystems. Ray tracing with TracePro is performed to obtain the total sunlight power incident on the side of laser rod from the concentrating system, when different numbers of subsystems n are assumed. The results are shown in Fig. 7.

Figure 7. Relationship between total incident solar power and n.

When the fan-circular parabolic mirror is divided into 4 (n = 4) identical parts, the solar power received by the side of the laser rod reaches a maximum value of 7,872.7 W. In this case, the acceptance half-angle of the 2D-CPC and the length of the rectangular diffuser are 40° and 92 mm respectively. The surface of the concentrator is coated with a film with 98% reflectivity for sunlight. After the sunlight is concentrated by the off-axis parabolic mirror, 62.2% of the total sunlight power enters the 2D-CPC. After passing through the 2D-CPC, the rectangular diffuser and the cooling-water layer induce 27.5% loss, so 34.7% of the sunlight power converges on the side of the laser rod. The irradiance upon the side surface of the laser rod is shown in Fig. 8. The irradiance uniformity is defined as

Figure 8. Irradiance map of the side of the laser rod.

$ΔE=1−Emax−EaveEmax+Eave×100%$

Here Emax and Eave are the maximum value of irradiance and the average value of the irradiance on the receiving surface respectively. When n is 4, the uniformity is 98%. We also investigated the uniformity of the light irradiance on the surface of the laser rod when n is 3, 5, 6, 7 and 8; the values are 97.4%, 98.3%, 97.4%, 98.4%, and 97.5% respectively.

### Ⅳ. ANALYSIS OF THERMAL EFFECTS AND CALCULATION OF LASER OUTPUT POWER

The heat distribution in the laser rod can be expressed as [25]

$Qr=ηheatαpe−αr0−r$

where ηheat is the fraction of the absorbed pump power that is converted to heat, α is the average absorption coefficient of the rod, p is the incident solar power per unit area, and r0 is the rod’s radius.

The temperature distribution in the laser rod is analyzed using the Comsol multiphysics software. The average absorption coefficient of the Nd:YAG laser rod across the whole solar spectrum is taken to be 0.35 cm−1 [26]. The surface heat-transfer coefficient is 1 W/(cm2·℃) and the temperature of the cooling water is 293 K. ηheat is calculated to be 0.46 [26]. After simulation, the transverse temperature distribution of the laser rod is shown in Fig. 9. The temperature is 359 K at the center of the laser rod and 325 K at the surface, with a gradient of 34 K.

Figure 9. Transverse temperature distribution in the laser rod.

Nd:YAG is a typical four-level laser medium. By solving the four-level rate equations, the laser output power can calculated with the equation [27]

$Pout=21−Rη1+R2α0l−1nRPin−Is 2α0l−1nRA2η$

where η = ηQηSηBηpηTηa; ηQ , ηS, and ηB are the quantum efficiency, the Stokes factor, and the beam overlap efficiency respectively; and ηp, ηT, and ηa are the laser absorption and solar-spectrum overlap, the radiation transmission efficiency, and the absorption efficiency respectively. R, a0, and l are the reflectivity of the output cavity mirror, the scattering coefficient, and the length of laser crystal respectively. Pin is the total pump power and A is the end-face area of the laser rod. IS is the saturation intensity of the laser. For a four-level system, the saturation intensity is [27]

$Is=hvτfσ21$

Here ν is the pump frequency and h is Planck’s constant. tf is the fluorescence lifetime and σ21 is the stimulated emission cross section. The relationship of laser output power Pout, pump threshold Pth, and slope efficiency β can be written as Pout = b (PinPth). The slope efficiency and threshold pump power are

$β=21−Rη1+R2α0l−1nR$

$Pth=Is2α0l−1nRA2η$

When the Nd:YAG rod is pumped by sunlight, the laser absorption and solar-spectrum overlap is 0.16 [28]. The transmission efficiency is 0.85 [28], and the absorption efficiency is 0.82 in this laser system. The average weighted wavelength of sunlight is 660 nm [11], the laser output wavelength is 1,064 nm, and the Stokes factor is 0.62 [11]. The beam overlap efficiency is the ratio of the cavity mode volume to the pump volume of the laser rod; its value is assumed to be 0.91. The quantum efficiency is 0.9 [28], the fluorescence lifetime is 230 μs [27], and the stimulated emission cross section is 6.5 × 10−19 cm2 [27]. The scattering coefficient of the laser material is 0.003 cm−1 [28]. Based on Eqs. (5)–(8), the output power of this solar laser with varying reflectivity of the output cavity mirror and varying solar pump power are calculated, and the results are shown in Fig. 10. The reflectivity of the other end of the crystal is assumed to be 100%. When the reflectivity of the output cavity mirror is 83%, a maximum laser output power of 351.8 W is obtained, and the threshold pump power, the slope efficiency, and solar-to-laser conversion efficiency are 616 W, 4.85%, and 4.5% respectively.

Figure 10. Change in calculated laser output power with (a) the output cavity mirror’s reflectivity, and (b) the total input pump power.

### Ⅴ. CONCLUSION

To achieve uniform side pumping of a high-power solar laser, a solar concentrating system is proposed. The concentrator system consists of four off-axis parabolic mirrors with a focal length of 1,000 mm, and four 2D-CPCs and rectangular diffusers. To collect more solar power, the number of off-axis parabolic mirrors, the acceptance half-angle of the 2D-CPCs, and the length of the diffusers are optimized. When four off-axis parabolic mirrors are adopted, the solar power received by the side of the laser rod is the highest, being up to 7,872.7 W; the corresponding optimal acceptance half-angle of the 2D-CPCs is 40°; and the optimal length of the rectangular diffusers is 92 mm. The irradiance uniformity on the side of the laser crystal rod reaches 98%. Then the transverse temperature distribution of the laser rod is analyzed, and the center temperature is found to be 359 K, while the surface temperature is 325 K. Finally, based on the four-level rate equations, the output characteristics of the solar laser are analyzed. When the reflectivity of the output cavity mirror is 83%, the maximum laser output power, the threshold pump power, the slope efficiency, and the solar-to-laser conversion efficiency are 351.8 W, 616 W, 4.85%, and 4.5% respectively.

### DISCLOSURES

The authors declare no conflicts of interest.

### DATA AVAILABILITY

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.

### FUNDING

National Natural Science Foundation of China (61675129); the Natural Science Foundation of Hubei Province (2014 CFB671); Natural Science Foundation of Shanghai (19ZR1427100); Research Fund for Excellent Dissertation of China Three Gorges University (2021SSPY149).

### Fig 1.

Figure 1.Schematic of the solar concentrating subsystem.
Current Optics and Photonics 2022; 6: 627-633https://doi.org/10.3807/COPP.2022.6.6.627

### Fig 2.

Figure 2.Schematic diagram of (a) the primary concentrating system, and (b) the laser head.
Current Optics and Photonics 2022; 6: 627-633https://doi.org/10.3807/COPP.2022.6.6.627

### Fig 3.

Figure 3.Irradiance map at the focal point of an off-axis parabolic mirror.
Current Optics and Photonics 2022; 6: 627-633https://doi.org/10.3807/COPP.2022.6.6.627

### Fig 4.

Figure 4.Schematic of the secondary concentrator.
Current Optics and Photonics 2022; 6: 627-633https://doi.org/10.3807/COPP.2022.6.6.627

### Fig 5.

Figure 5.Schematic diagram of the condensing principle of the off-axis parabolic mirror.
Current Optics and Photonics 2022; 6: 627-633https://doi.org/10.3807/COPP.2022.6.6.627

### Fig 6.

Figure 6.Variation of the collected sunlight power of a concentrating subsystem with the divergence half-angle, for different numbers of mirror parts: (a) n = 3, (b) n = 4, (c) n = 5, (d) n = 6, (e) n = 7, (f) n = 8.
Current Optics and Photonics 2022; 6: 627-633https://doi.org/10.3807/COPP.2022.6.6.627

### Fig 7.

Figure 7.Relationship between total incident solar power and n.
Current Optics and Photonics 2022; 6: 627-633https://doi.org/10.3807/COPP.2022.6.6.627

### Fig 8.

Figure 8.Irradiance map of the side of the laser rod.
Current Optics and Photonics 2022; 6: 627-633https://doi.org/10.3807/COPP.2022.6.6.627

### Fig 9.

Figure 9.Transverse temperature distribution in the laser rod.
Current Optics and Photonics 2022; 6: 627-633https://doi.org/10.3807/COPP.2022.6.6.627

### Fig 10.

Figure 10.Change in calculated laser output power with (a) the output cavity mirror’s reflectivity, and (b) the total input pump power.
Current Optics and Photonics 2022; 6: 627-633https://doi.org/10.3807/COPP.2022.6.6.627

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Wonshik Choi,
Editor-in-chief