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Abstract
Thermal effects under high-power pumping significantly limit the laser beam quality. To address this, we developed an M2 simulation algorithm based on ray trajectory simulation and established a corresponding experimental platform. This approach optimized the M2 factor of pulsed lasers to 2.2 and output power of 25.9 W under a repetition rate of 10 kHz. The results revealed that under specific conditions, thermal effects, typically considered detrimental to beam quality, could significantly enhance it. Compared to other methods necessitating additional optical components, our strategy offers a streamlined and straightforward solution for beam quality control under high-power pumping conditions.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The pursuit of high beam quality is the eternal goal of solid-state lasers [1–5]. However, under high-power pumping conditions, the beam quality of the laser output beam is severely impacted by both the non-uniform distribution of gain within the gain medium and the pronounced thermal effects [6,7].
To achieve optimal beam quality, there are primarily two methods in existing research. The first method is to suppress the thermal effects on beam quality as the output laser of a solid-state laser generates approximately 69% power loss, of which about 71% is derived from the thermal lens effect [8]. Under high power conditions, the thermal effects of solid-state lasers increase significantly, directly leading to the deterioration of beam quality, the increase of power loss, and the instability of system performance. Therefore, how to effectively suppress or even eliminate the influence of thermal effectss has been a crucial issue in solid-state lasers, prompting extensive research efforts by various researchers [9–12]. However, these methods have little effect under high-power pumping conditions.
The second approach entails incorporating specialized optical devices within the resonant cavity for mode selection and aberration correction of the oscillating beam [13–17].
In recent years, researchers have conducted extensive work related to improving the beam quality of high repetition rate and high power pump solid-state lasers. In 2016, M. Kaskow et al. [14] utilized a nonreciprocal, self-adaptive cavity, achieving an improvement in the beam quality factor to around 2.5 in 1 at.% doped slab Nd: YAG laser operating at a frequency of 2 Hz. In 2017, Quanwei Jin et al. [18] focused their studies on the master oscillator power amplifier (MOPA) of Nd: YAG, attaining a beam quality factor of 2.3 at 400 Hz. In 2019, Yanjie Song et al. [19] employed a multi-pass cavity (MPC) structure, realizing a beam quality factor of 1.45 with an Nd-doped concentration of 0.6 at.% at a Q-switching frequency of 10 kHz. In 2021, Wentao Zhu et al. [16] controlled the beam quality factor within 1.1 using a chirped pulse amplification (CPA) system and Yb: YAG dual-crystals, achieving an output laser pulse energy of 3.5 mJ at a repetition rate of 1 kHz. In 2022, Chao Ma et al. [20] designed a dual-pass multi-fold end-pumped slab amplifier and attained beam quality factors of 1.52 at a repetition rate of 10 kHz. Lastly, in 2023, Jun Meng et al. [21,22] optimized an Nd: YAG system, reducing the beam quality from 30.5 to within 10 through enhancements in thermal management and Gaussian mirror.
Numerous research teams have made significant efforts to enhance laser performance. However, many of these methods necessitate the integration of additional optical components, which leads to heightened complexity and reduced laser system stability. The primary challenge in current research revolves around effectively controlling solid-state laser beam quality while maintaining a compact structure.
This paper presents a different approach to delve deeper into this issue. Firstly, we propose a simulation algorithm aimed at modeling the trajectory changes of light rays within complex refractive index media. Unlike traditional single-line representations, our approach uses primary rays and accompanying rays, effectively capturing the changes in light intensity. Furthermore, by considering the interplay between thermal effects and the optical field, we have employed an iterative approach to attain a stable distribution of light rays within the gain medium. Lastly, by utilizing the stabilized oscillating light field distribution and its magnitude, we calculated the beam quality factor $M^2$ for this specific pumping condition. By thoroughly investigating the implications of thermal effects on beam quality, we have discovered that solely utilizing thermal effects for modulation can significantly enhance beam quality without altering the laser’s structure or incorporating additional beam quality control devices, and sustaining the other performance attributes of the laser. This offers a straightforward and effective approach for enhancing beam quality in solid-state lasers under high-power pumping conditions.
2. Theoretical foundations
2.1 Simulation principle of ray trajectory in variable refractive index medium
As depicted in Fig. 1, $abcd$ represents a cross-section of the medium along the optical axis. Each point within it has a distinctive refractive index $n$, and its corresponding refractive index gradient is expressed as $\overrightarrow {\nabla }n$. $\overrightarrow {L_1}$ symbolizes any ray vector propagating within the cross-section, with its starting coordinate denoted as $P_0$ and the incidence angle denoted as $\alpha _1$. The dashed line "M" represents the equiphase surface, where the optical path length or phase is equal at each point.
Fig. 1. Illustration of the deviation of the trajectory of a ray of light in a medium with variable refractive index.
To calculate the propagation trajectory of $\overrightarrow {L_1}$ within the medium, the medium can be conceptualized as a composition of numerous finite segments. Sequentially computing the endpoint coordinates and spatial angles of the ray after passing through each segment, a complete trajectory of the ray can be composed by linking these coordinates.
Assuming $\overrightarrow {L_1}$ propagates through a minute segment within the medium, it is influenced by the refractive index gradient, resulting in endpoint coordinates $P_1$ and an exit angle $\alpha _1$. To differentiate from $\overrightarrow {L_1}$, the vector is designated as $\overrightarrow {L_2}$ after undergoing these adjustments. The endpoint coordinate of $\overrightarrow {L_1}$ is employed as the starting coordinate for $\overrightarrow {L_2}$, and the exit angle is employed as its incident angle, allowing further propagation through subsequent medium segments.
The derivation of $P_1$ and $\alpha _1$ is as follows involves introducing an auxiliary vector $\overrightarrow {L_1'}$, which is parallel to $\overrightarrow {L_1}$ in the vertical direction. The starting point of $\overrightarrow {L_1'}$ is denoted as ${P_0}'$.
The refractive index affects the direction of propagation of light, mainly due to the component perpendicular to the direction of the light, which is denoted as $\overrightarrow {\nabla } { n_\bot }$, calculable using Eq. (1).
Different refractive index gradients within the gain medium result in a divergence between the propagation directions of $\overrightarrow {L_1}$, creating the optical path differences, denoted as $\Delta L_1$, as represented by Eq. (2).
Here, $|\overrightarrow {\nabla } { n _ \bot }|$ represents the magnitude of $\overrightarrow {\nabla } { n _ \bot }$, $h$ denotes the distance between the ray vectors $\overrightarrow {L_1}'$ and $\overrightarrow {L_1}$, and $L_1$ represents the length of $\overrightarrow {L_1}$.
By using the optical path difference $\Delta L_1$ and the distance $h$ between the ray vectors $\overrightarrow {L_1}$ and $\overrightarrow {L_1}'$, the deflection angle for the succeeding ray segment can be derived as in Eq. (3).
According to Eq. (3), the deflection angle of the ray is independent of the distance $h$ between any two parallel rays. Therefore, after one deflection, a new ray vector $\overrightarrow {L_2}$ is generated, with its starting coordinate ${P_1} = {P_0} + \overrightarrow {L_1}$ and an angle of incidence ${\alpha _2} = \theta - {\alpha _1}$.
In a similar way, the relative deflection direction of the ray $\overrightarrow {L_3}$, $\overrightarrow {L_4}$, $\cdots$, $\overrightarrow {L_n}$ can be computed sequentially. Linking these vectors sequentially outlines the associated trajectory of ray propagation. Thus, given known refractive index distribution, the simulation of any ray’s propagation process within the gain medium and the visual depiction of its trajectory is attainable.
2.2 Fast optical field sampling scheme
To map the propagation trajectories of the pump optical field within the gain medium, we must sample the initial pump optical field. In geometric optics, a single ray can only describe the path and propagation direction of light, without reflecting changes in optical intensity. However, in actual beam propagation, optical intensity varies. To address this shortfall, we employ a dual-ray accompanying propagation sampling method to simulate the intensity variations of the light beams within the medium. For each primary ray, we configure an accompanying ray very close to it, forming a ray pair. By tracking the variation in separation between the ray pairs, corresponding changes in optical intensity can be calculated. In the simulation calculations of the beam, we determine the sampling density of rays based on the beam intensity distribution in the cross-section.
2.2.1 Primary ray sampling
Firstly, primary rays are randomly sampled based on optical intensity, and the number of samples depends on the optical intensity. In simpler terms, locations with higher optical intensity receive a greater number of randomly sampled rays, while locations with lower optical intensity receive fewer samples. Following this approach, the weights ${w_i}$ are calculated, representing optical intensity within the sampling area, as defined by Eq. (4). Regions with greater weights have more ray samples collected.
Here, ${\sigma _l}$ denotes the entire beam area, ${\sigma _i}$ the sampled beam area, and ${m_l}$ denotes the total number of ray samples. Thus, the rays sampled within ${\sigma _i}$ are ${w_i}{m_l}$.
Primary rays also have an associated divergence angle, increasing with distance from the fiber core’s center. Direct emissions from the fiber core center have a ${0^ \circ }$ divergence angle. The actual divergence from varying positions of the fiber output is derived from experimental data.
Rays sampled from the pump optical field are represented as a set $\{ligh{{t}_{1}},ligh{{t}_{2}},\ldots ligh{{t}_{n}}\}$, where $n>0$, marking the number of primary ray samples. In theory, the larger $n$ is, the better. Each ray is described as $ligh{{t}_{i}}=(l,\vec {p},\alpha )$, where $l$ is the length of the ray vector, $\vec {p}=(x,y,z)$ denotes the starting point coordinates of the ray, and $\alpha$ represents the divergence angle of the primary ray.
2.2.2 Accompanying rays configuration
Each independent primary ray can reflect the distribution pattern of optical intensity but not the magnitude of optical intensity changes by itself. Therefore, we assign an accompanying ray to each independent primary ray to account for the variations in optical intensity during light transmission, taking advantage of the spatial distance changes between primary rays and accompanying rays. We set up an accompanying ray for each individual primary ray, denoted as $light_{i}^{\prime }=({l}',{\vec {p}}',{\alpha }')$. The newly configured accompanying ray has its starting point coordinates in the same plane as the original ray, i.e., $|{\vec {p}}'-\vec {p}|={{k}_{i}}$, where ${k_{i}}$ is the inter-ray spacing dependent on pump optical intensity, and ${{k}_{i}}>0$. The numerical value of ${{k}_{i}}$ depends on the pump optical intensity. The accompanying ray $light_{i}^{\prime }$ is parallel to the original primary ray $light_{i}$ (${\alpha }'=\alpha$), and their lengths are equal, i.e., ${l}'=l$.
As the rays propagate within the medium, changes in refractive index result in deviations between primary ray $ligh{{t}_{i}}$ and accompanying ray $ligh{{t}_{i}}^{\prime }$. Their lengths and angles are no longer equal, i.e., ${l}'\ne l$ and ${\alpha }'\ne \alpha$. Additionally, the inter-ray spacing ${{k}_{i}}$ also varies. Smaller ${{k}_{i}}$ corresponds to higher optical intensity $I$, while larger ${{k}_{i}}$ corresponds to lower optical intensity $I$.
2.2.3 Optical intensity variation calculation
Assuming an initial optical intensity distribution ${I_0}$ at $z=0$, the optical intensity becomes $I(x,y,z)$ during propagation. At the initial position, primary ray $ligh{{t}_{i}}$ and accompanying ray $ligh{{t}_{i}}^{\prime }$ are parallel, and their separation is ${k_{0}}$ related to the optical intensity. This can be expressed as given by Eq. (5).
Here, $I(x,y,z)$ smaller, ${k(x,y,z)}$ has a larger numerical value. By tracking the variation in separation between ray pairs, we can calculate the discrete field of optical intensity distribution, as shown in Eq. (6).
3. Solution design
The interaction between pump light and temperature fields within the gain medium is a crucial process, frequently unaddressed in numerous investigations. While such negligence is acceptable under low-power pump conditions but can lead to significant inaccuracies under high-power pump scenarios.
Initially, as the pump light enters the gain medium, no waste heat deposition occurs, maintaining a uniform temperature field and unaltered pump light distribution. However, with the continuation of pumping, internal thermal effects within the gain medium amplify, resulting in a non-uniform temperature distribution. This alters the distribution of the refractive index, consequently affecting the light field distribution of the pump light and further impacting the distribution of thermal power density, finally leading the temperature field within the gain medium towards stability. In the simulation, we execute a quantitative comparison of the ray tracing distribution radius $\omega _{i}^{p}(z)$ at varying iterative stages. The comparison evaluates the size deviations between consecutive iterations, denoted by $\beta = \omega _{i}^{p}(z) / \omega _{i+1}^{p}(z)$. Upon achieving a $\beta$ value exceeding 99.99%, the beam radii between the consecutive iterations are considered equivalent. This equivalence implies a stabilized ray distribution and light field, thereby obtaining the final refractive index distribution.
Upon acquiring the final refractive index distribution, we use rate equations and initial pump light parameters to solve for the beam radius and power of the fundamental mode oscillating light at the crystal end face. The oscillating light enters the final refractive index distribution formed by the pump light and the gain medium, and we use the algorithm for another iteration process. The interaction between oscillating light and thermal effects also needs to be considered, and it is also necessary to determine whether the oscillating light field is stable. Similar procedures are employed to compare the oscillating light radius $\omega _{i}^{o}(z)$ at different iteration stages, wherein the relative sizes between successive iterations are denoted by $\gamma =\omega _{i}^{o}(z)/\omega _{i+1}^{o}(z)$. Once $\gamma >99.99{\%}$ signifies a stabilized oscillating light field distribution.
Ultimately, the beam quality factor of the oscillating light field can be determined by analyzing the stabilized final oscillating light field. Therefore, the algorithm is primarily divided into two main segments.
The overall flowchart of the simulation algorithm for ray trajectories of pump light and oscillating light in a complex temperature field is shown in Fig. 2.
Fig. 2. Ray tracing simulation flowchart for pump and oscillating light fields in a solid-state laser with complex temperature distribution.
3.1 Pump light field ray trajectory simulation
The following process involves solving for the stable field distribution of the pump light in a medium with a variable refractive index.
(1) Initial the pump beam performs function fitting as $G(x, y, z)$.
(2) Employ a fast optical field sampling scheme, represent the light field distribution using ray distribution, and assign an accompanying ray for each primary ray.
(3) Combining the final stable state heat conduction equation and boundary conditions, $T(x,y,z)$ and refractive index distribution $N(x,y,z)$ of each medium cell are calculated according to $G(x,y,z)$.
(4) Calculate the ray distribution $F_{i}(x,y,z)$ and update light intensity distribution $I_{i}(x,y,z)$ and temperature distribution in gain medium $T_ {i} (x,y,z)$. Where $F_{i}(x,y,z)$ is obtained from each ray and the corresponding accompanying ray after being influenced by the refractive index distribution $N_i(x,y,z)$ of the gain medium. The ray intensity $I_{i}(x,y,z)$ at this time is inferred by the distance $k _i$ between the accompanying ray and the intensity distribution leads to a change in the temperature distribution $T_{i}(x,y,z)$ of the gain medium.
(5) Compare the pump field light distribution obtained in the previous and subsequent calculations in step (4). If the beam radius ratio at a given location is $\beta >99.99 {\%}$, the algorithm terminates, and the resulting distribution is considered the final temperature distribution, the final refractive index gradient, and the final light field distribution. Otherwise, $i=i+1$, update the temperature field distribution $T_ I (x, y, z)$ and refractive index distribution $N_ I (x, y, z)$ of the gain medium affected by the pump light distribution, then return to step (4).
(6) Obtain the final stable temperature field distribution $T_p(x, y, z)$ and refractive index distribution $N_p(x, y, z)$.
3.2 Oscillating light field ray trajectory simulation
The steps of the simulation algorithm for modeling the ray trajectory of the oscillating light in a variable refractive index medium are as follows.
(1) Input the intensity distribution of the oscillating light, $G_O^0(x, y, z)$. The power and beam radius of $G_O^0(x, y, z)$ are determined by the intensity distribution of the fundamental mode Gaussian light.
(2) Perform fast optical field sampling on $G_O^0(x, y, z)$ to obtain the initial distribution of oscillating light rays, $L_O^0(x, y, z)$, and assign an accompanying ray to each.
(3) Simulate the trajectory of oscillating light as it passes through the stable refractive index field distribution, denoted as $N_p(x, y, z)$, using the algorithm for simulating light trajectories in variable refractive index media.
(4) Simulate the ray trajectory obtained in step (3) as it propagates through the air to the output mirror’s surface. Rays reaching the output mirror are reflected, and the reflected light continues through the air to the gain medium’s rear surface. Record the ray trajectory during this process.
(5) Use the ray trajectory obtained in step (4) as input for the gain medium, injecting it from the gain medium’s rear surface into its interior. Employ the algorithm for light ray trajectories in variable refractive index media to determine the ray trajectory within the gain medium.
(6) Calculate $\gamma$ as the ratio of the beam radius at a given position to the previous computation’s beam radii. If $\gamma$ is less than 99.99%, compute the ray distribution emitted from the gain medium’s end face and return to step (3). If $\gamma$ exceeds 99.99%, conclude the computation, considering the last distribution as the final distribution of the oscillating light.
3.3 Methodologies for computing the beam quality factor $M^2$
The variation in distance between the accompanying ray and the primary ray reflects alterations in light intensity. By simulating the ray trajectory using a ray tracing algorithm, we can determine the light transmission path and intensity variation, which ultimately leads to the final distribution of the rays. Based on this, the beam radius $r$ and the corresponding intensity distribution $I(r)$ can be calculated. The beam width and spot size inside the gain medium can be determined using the method proposed by ISO 11146 [23] and other works [24–26]. The application of curve fitting enables the computation of the beam quality factor $M^2$ via Eq. (7).
In this equation, $\lambda$ represents the wavelength, the coefficients a, b, and c are fitting parameters.
4. Theoretical simulation and simulation results
4.1 Theoretical model
The resonant cavity follows a standard flat-flat configuration, as shown in Fig. 3. The gain medium employed in this study is Nd: YAG, which is a cylindrical medium with dimensions of $\phi 3\times 20$ mm. It has flat surfaces at both ends, where the front end is coated with a high-transmission film at 808 nm and a high-reflectance film at 1064 nm. The back end is coated with a high-reflectance film at 808 nm and a high-transmission film at 1064 nm.
The experimentally measured spatial distribution of the pump light intensity in the experiment closely resembles a second-order flat-top Gaussian function, as illustrated in Fig. 3(b), (c), and (d). Hence, we adopt a second-order flat-top Gaussian function to characterize the initial distribution of the pump light field.
4.2 Simulation results of the pump light field in the gain medium
The pump light rays undergo multiple iterations inside the gain medium, eventually reaching a stable distribution of ray trajectories. The following figures illustrate a cross-section of half of the gain medium, where the $x$ direction represents the diameter of the gain medium and the $z$ direction represents its length.
Using the simulation algorithm for light trajectories in variable refractive index media, we can visualize the distribution of ray trajectories within the medium. In this study, our focus is on observing variations in the propagation trajectory of the pump light resulting from changes in pump power, pump beam radius, and doping concentration.
4.2.1 Impact of pump power on the propagation trajectory of pump light rays
The initial conditions for Fig. 4(a), (b), and (c) involve a pump beam radius ($\omega$) of 0.3 mm and an $Nd^{3+}$ doping concentration of 0.3 at.%. The pump powers are set to 40 W, 50 W, and 60 W, respectively.
Fig. 4. Distribution of pump light inside Nd: YAG, with fixed $\omega$ of 0.3 mm and a doping concentration of 0.3 at.%. (a) $P=40$ W (b) $P=50$ W (c) $P=60$ W.
In general, as the pump power ($P$) increases from 40 W to 60 W, the region of light convergence shifts closer to the front-end face of the medium. This shift occurs because higher pump power leads to an increased transverse refractive index gradient within the gain medium, resulting in a larger deflection angle for the light. Consequently, the convergence region of the light moves forward.
4.2.2 Impact of pump beam radius on the propagation trajectory of pump light rays
The initial conditions for Fig. 5(a), (b), and (c) involve a pump power ($P$) of 100 W and an $Nd^{3+}$ doping concentration of 0.3 at.%. The pump beam radius ($\omega$) are set to 0.4 mm, 0.5 mm, and 0.6 mm, respectively.
Fig. 5. Distribution of pump light inside Nd: YAG, with fixed $P=100$ W, doping concentration is 0.3 at.% (a) $\omega = 0.4$ mm (b) $\omega = 0.5$ mm (c) $\omega = 0.6$ mm.
When $\omega = 0.4$ mm, a noticeable convergence point in the crystal is observed for the pump light. The convergence effect is also more pronounced when $\omega = 0.5$ mm compared to $\omega = 0.6$ mm. This suggests that a smaller pump beam radius results in a more concentrated distribution of thermal power within the gain medium. Consequently, stronger temperature gradients and refractive index gradients are established in the region where the light is distributed, leading to a more significant convergence effect of the pump light.
4.2.3 Impact of doping concentration magnitude on the propagation trajectory of pump light
The initial conditions for Fig. 6(a), (b), and (c) consist of a pump power ($P$) of 50 W and a beam radius ($\omega$) for $Nd^{3+}$ of 0.3 mm. The doping concentration of $Nd^{3+}$ is set to 0.1 at.%, 0.2 at.%, and 0.4 at.%, respectively.
Fig. 6. Distribution of pump light inside Nd: YAG, with fixed $P=50$ W, $\omega =0.3$ mm (a) doping concentration: 0.1 at.% (b) doping concentration: 0.2 at.% (c) doping concentration: 0.4 at.%.
The convergence effect becomes more prominent with a doping concentration of 0.4 at.% $\rm {Nd}^{3+}$ particles compared to 0.2 at.% and 0.1 at.%. This indicates that the doping concentration in the gain medium directly affects the absorption coefficient, with higher doping concentrations leading to larger absorption coefficients. Consequently, the distribution of pump energy near the end face becomes more concentrated, resulting in a stronger transverse temperature gradient and refractive index gradient. Consequently, the pump light exhibits a more pronounced convergence effect.
4.3 Beam quality factor of the oscillating light
Through simulation calculations, the beam quality factor of the oscillating light was studied by varying the pump power in the range of 15 to 80 W with a step size of 5 W, considering different pump beam radii. Figure 7 shows the curve of the beam quality factor of the oscillating light as a function of pump power for different beam radii.
From the graph, it can be observed that the beam quality factor of the oscillating light generally increases with increasing pump power. However, there are specific power ranges where the beam quality undergoes significant changes, resulting in improved beam quality. Specifically, for a beam radius of $\omega = 0.5$ mm, a significant improvement in beam quality is observed in the pump power range of [65, 75] W. Similarly, for a beam radius of $\omega = 0.8$ mm, a similar phenomenon is observed in the pump power range of [70, 80] W.
Under the condition of a fixed end-face beam radius, by adjusting the pump power, we can always find a specific range of pump power where the oscillating light achieves both high output power and good beam quality. Similarly, by adjusting the beam size while keeping the pump power constant, we can control the beam quality of the oscillating light.
To validate the results obtained from the simulation calculations, we will establish an experimental platform to verify the optimal pump power range for this specific beam radius.
The simulation results indicate that, with the other conditions fixed, the convergence effect of the light becomes more pronounced with increasing pump power, and the convergence point of the light approaches the end face of the gain medium where the pump light is incident.
In addition, when the beam radius of the pump light is changed at the end face of the gain medium, within a certain range, a smaller beam size leads to a more pronounced convergence effect of the pump light, with the convergence point closer to the end face of the gain medium.
Furthermore, as the beam radius decreases, the light convergence effect becomes more prominent, requiring a lower maximum pump power to achieve it. Conversely, a larger beam radius necessitates a higher maximum power for achieving the same level of light convergence.
Under identical conditions, increasing the doping concentration of the gain medium results in higher absorption efficiency of the pump light, leading to a more pronounced convergence effect to the light and a convergence point that is closer to the incident surface of the gain medium.
Analyzing the beam quality of the oscillating light reveals that, with other conditions constant, the beam quality factor increases with increasing pump power at low power densities. However, there is a notable decrease followed by an increase in the beam quality factor within a specific small power range.
Based on these findings, we can develop designs that leverage thermal effects to achieve high-power lasers with exceptional beam quality.
5. Experimental system and experimental results
5.1 Experimental system
A fiber laser (model number: DILAS Diodenlaser GmbH M1F1DS22-808-1S3.1M2) with a central wavelength of 808 nm, a numerical aperture of 0.22, and a diameter of 400 ${\mu{\rm m}}$ was chosen as the pump source for the Nd: YAG laser.
The Nd: YAG laser parameters remain the same as those described in the theoretical section, and the doping concentration was 0.3 at.%. The laser adopts a flat-flat cavity configuration with a cavity length of 12 cm and an output mirror transmittance of 70%.
The structure of the experimental Nd: YAG laser was illustrated in Fig. 8. The laser system employs an optical fiber for pump light delivery. Along the optical axis, the coupling system, gain medium, acousto-optic medium, and output mirrors were arranged in sequence. The pump light was collimated and focused by the coupling system before entering the resonant cavity comprised of the gain medium, acousto-optic medium, and output mirrors. This configuration enabled the generation of laser output.
Fig. 8. Diagram of the Nd: YAG experimental platform:① Optical fiber② Coupling system③ Laser crystal and heat sink④ Acousto-optic crystal and heat sink⑤ Output mirror.
5.2 Results of experiment
The experimental results demonstrated a gradual increase in the output power of Nd: YAG in an end-face-pumped, Q-modulated configuration as the pump power increased while keeping all other conditions constant.
By effectively managing the thermal effects in the Nd: YAG laser, we were able to achieve an output laser with a beam quality factor of 2.2, a pulse width of 4.2 ns, an average power of 25.9 W, and a peak power of approximately 617 kW. These parameters were obtained when the pump beam radius was 0.8 mm, the pump power was 72 W, and the repetition frequency was 10 kHz. Similarly, by utilizing thermal effects control of the beam quality, we obtained an output laser with a beam quality factor of 4.3, an average power of 23.7 W, a pulse width of 4.4 ns, and a peak power of 539 kW when the beam radius was 0.5 mm, the pump power was 70 W, and the repetition frequency was 10 kHz.
In Fig. 9(a), for a smaller beam radius of $\omega =0.5$ mm, the beam quality factor exhibited a rapid increase from 2.7 to 5.7 within the pump power range of 20 - 40 W. Conversely, in Fig. 9(b), for a larger beam radius of $\omega =0.8$ mm, the beam quality factor exhibited a similar rapid increase from around 2.1 to 5.0 within the pump power range of 20 - 50 W.
Fig. 9. Output power and beam quality of Nd: YAG (a) beam radius is 0.5 mm (b) beam radius is 0.8 mm.
This behavior could be attributed to the relatively low thermal effects induced by the pump power within this range, leading to a weaker convergence effect on the light and less matching between the pump light and the fundamental mode of the oscillating light. As the pump power increased, the diffraction angle of the oscillating light also increased, resulting in a gradual decrease in beam quality.
With a further increase in pump power, the thermal effects become more pronounced, leading to a stronger light convergence effect. These effects bring the oscillating light and the pumping light closer to the center axis of the gain medium, resulting in a higher degree of mode matching between the two beams. Consequently, the beam quality rapidly improved, reaching an optimal ${M^2}$ value.
With a further increase in pump power, the thermal effects became more pronounced, leading to a significant convergence effect on the pump light. This pronounced convergence effect resulted in a decrease in the alignment between the pump light and the oscillating light. It is important to note that excessive thermal effects could potentially lead to changes in the cavity shape, which might adversely affect the stability and safety of the laser system. Therefore, careful consideration and proper thermal management were crucial to ensuring optimal performance and reliable operation of the laser system.
For Fig. 9(a), the optimal range for the pump power to achieve the best beam quality factor was found to be between 65–75 W, with an optimal value of 4.3. Similarly, for Fig. 9(b), the optimal range for the pump power to achieve the best beam quality factor was determined to be between 70–75 W, with an optimal value of 2.2.
Optimizing beam quality could be achieved by identifying the optimal pumping interval and adjusting the beam radius. Experimental observations indicated that a beam radius of 0.8 mm resulted in significantly improved beam quality, as indicated by a higher beam quality factor $M^2$, compared to a beam radius of 0.5 mm.
6. Conclusion
This paper presents a simulation approach for the beam quality factor $M^2$ of solid-state lasers based on a ray trajectory simulation algorithm in a variable refractive index medium. Simulation results reveal that, as the pump power increases or the beam radius decreases, the thermal effects in the crystal tends to degrade beam quality continuously, resulting in an overall increase in the $M^2$ factor. However, there exists a narrow range of pumping conditions where the distribution of the fundamental mode oscillating light and the pump light field within the crystal are optimally matched, resulting in significantly improved beam quality. Specifically, for a given solid-state laser, when the beam radius is fixed, increasing the pump power to a certain level significantly enhances the beam quality. Similarly, with the pump power maintained at a constant level, adjusting the beam radius to a specific value also leads to a notable improvement in beam quality.
To validate the findings of this theoretical study, we conducted a corresponding experiment using a high-power end face pumped solid-state laser. We employed a 0.3 at.% Nd: YAG crystal with a $\phi 3 \times 20$ mm structure, operating at a repetition rate of 10 kHz, and the results of the experiment confirmed the theoretical conclusions. With increasing pump power, the beam quality factor $M^2$ exhibited an overall upward trend at both 0.5 mm and 0.8 mm beam radius, indicating a continuous degradation in beam quality. However, when utilizing thermal effects control with a 0.5 mm pump beam radius, an optimal pump power interval of 65 - 75 W was identified, leading to significantly improved beam quality. Similarly, at the same 0.5 mm beam radius, an optimal pump power interval of 70 - 75 W was observed using the thermal effects control, resulting in significantly enhanced beam quality. In the experiment, under the condition of a beam radius of 0.8 mm and a pump power of 72 W, resulted in the beam quality factor of the output laser was improved to 2.2, and its pulse width of 4.2 ns, the average power of 25.9 W, and the peak power of approximately 617 kW. These results demonstrate that the method proposed in the paper enhances the beam quality factor while maintaining a laser with high average power, high peak power, and narrow pulse width. This approach offers a simple structural technology for controlling beam quality in high-power solid-state lasers.
Funding
Basic Research Fund of the Ministry of Education (90501220005).
Acknowledgments
The authors thank the compact solid-state laser team of Xidian University for their help.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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