## Abstract

The fracture of laser material in a ceramic Nd:YAG laser pumped by a fiber-coupled laser diode was analyzed. The fracture of the laser material was found to occur when the critical temperature difference between the center of the material and the surface exceeded 355°C. To quantitatively analyze the material fracture, the heat-generation length and heat-generation radius of the laser material were calculated and the critical pump power per unit volume was examined. Under lasing and non-lasing conditions, the fracture of laser material occurred at 24.41 kW/cm^{3} and 19.53 kW/cm^{3}, respectively, for 2 at.% ceramic Nd:YAG and 25.57 kW/cm^{3} and 20.47 kW/cm^{3}, respectively, for 4 at.% ceramic Nd:YAG.

© 2014 Optical Society of America

## 1. Introduction

Understanding the laser output and beam characteristics is crucial in high-power solid-state laser development. To increase the laser output power, the pump light, spectrum matching of the material, and resonator need to be optimized; however, the fundamental issue is increasing the pump power. As the pump power incident to the laser material increases, the thermal effect of the material or the thermal birefringence effect induces a thermal lens effect, which diminishes the laser output characteristics [1,2]. Even if the external factors used to optimize the resonator structure or cooling system of the material are adjusted to resolve the thermal effect in the laser material, if the thermal stress of the laser material and cracking are unknown, the material is highly likely to fracture during laser operation [3]. Therefore, laser operation without fracture of the laser material requires an understanding of the critical properties of the material in terms of the pump light properties.

Ceramic Nd:YAG has begun to receive a large amount of attention for its promising potential applicability as a high-efficiency, high-power laser material [4–7]. The thermal properties of ceramic Nd:YAG, namely, its thermal conductivity and thermal expansion, are almost identical to that of the Nd:YAG crystal; however, its thermal shock parameter (1290 W/mK), flexural strength (360 MPa), Vickers hardness (16.1 GPa), and fracture toughness (8.8 MPam^{1/2}) are up to 3, 1.3, 1.1, and 5 times greater than those of the Nd:YAG crystal, respectively [8,9]. These properties imply that ceramic Nd:YAG can be applied in high-power lasers. Nevertheless, to date, no study of material fracture in ceramic Nd:YAG caused by thermal effects has been reported.

The fracture type of a material can be generally classified into ductile and brittle fractures. In the case of ductile fracture, plastic deformation, in which the external shape is deformed noticeably, occurs during the crack propagation process before fracture, and fracture requires a large amount of energy. On the other hand, brittle fracture involves little plastic deformation, and fracture results from the rapidly increasing crack propagation velocity. Ductile fracture can be observed in metallic materials, and brittle fracture can be observed in glass and ceramic materials, which are referred to as brittle materials [10]. Hence, ceramic is subject to brittle fracture, which proceeds very quickly. The defects in transparent ceramic, included porosity and any other undesired impurities, all of which would make fracture occur prematurely. In general, failure in ceramic material is a result of tensile stress. When the maximum tensile stress reaches the tensile strength of a material, the temperature changes in the center (in the case of heating) or surface (in the case of quenching) of the material, and fracture occurs on the surface [11–13]. Thus, in the case of ceramic Nd:YAG, fracture is expected to be a result of a similar cause. To use ceramic Nd:YAG in high-power applications, research on its fracturing is needed.

When analyzing laser material fracture by converting the beam diameter and the pump power into the critical pump power per unit volume, for every change in the pump beam diameter, the material’s absorption coefficient, and the laser oscillation should be considered. Thus, it is difficult to normalize the critical pump power. However, we can assume that the pump energy is absorbed by the laser material in proportion to the absorption rate as a function of material depth and that heat is produced proportionally within the laser material. Therefore, the heat generation volume of the laser material can be used to calculate the critical pump power per unit volume to normalize the critical pump power regardless of the pump beam size, allowing the quantitative analysis of material fracture.

In this study, the causes of laser material fracture during laser experiments using ceramic Nd:YAG are analyzed. The temperature distribution of ceramic Nd:YAG is determined in terms of the Nd^{3+} dopant concentration, critical temperature difference, and heat generation volume of the laser material. Based on these findings, the critical temperature difference per unit volume is calculated to assess the cause of material fracture.

## 2. Experiment setup

Figure 1 presents a schematic diagram of the laser-diode pump ceramic Nd:YAG laser used herein. The excitation source featured a fiber-coupled laser diode (Apollo Ins.: F25-808-4P) with an optical fiber core diameter of 400 μm and a numerical aperture of 0.22 operated at 15.3 W. The excitation source has a central oscillation wavelength of 808.5 nm and a full width at half maximum of 3.2 nm at an operating temperature of 27°C. The power supply (HP: 6011A) can supply a maximum of 20 V at 100 A. The pump light was focused by a focusing optical system, which focuses the pump light onto laser material with a focal length of 10.9 mm, yielding a spot diameter of 380 μm and a transmittance of 91.6%. The laser resonator consisted of a rear mirror and a concave output coupler. The rear mirror had a dichroic coating with total reflection at 1064 nm and anti-reflection at 808 nm. The output coupler (120 mm radius of curvature) had a reflectivity of 90.4% at 1064 nm. The material holder was made of copper to allow it to efficiently dissipate the heat generated by the ceramic Nd:YAG. The laser material was surrounded by indium foil, which has a high thermal conductivity, and mounted in the center of the holder. The heat generated in the laser material was recovered by a water-cooled plate attached to the bottom of the holder. In the ceramic Nd:YAG experiment, Ø5.0 × 20-mm and Ø4.52 × 20-mm laser materials (Baikowski) were used for Nd^{3+} dopant concentrations of 2 at.% and 4 at.%, respectively. The results of the experiment with a dopant concentration of 2 at.% revealed an output power of 2.21 W for a 6.54-W pump power, giving a maximum of approximately 33.3% output efficiency, 39.3% slope efficiency, and a 0.92-W lasing threshold. When the pump power exceeded 6 W, the output decreased due to the thermal lens effect, and at approximately 12 W, the laser did not oscillate [14]. During the experiment, the ceramic Nd:YAG material with 2 at.% neodymium initially did not experience any problems, but after conducting the experiment a few dozen times, the material fractured at one point. Using the 4.0 at.% ceramic Nd:YAG, the material fractured after one application of an incident pump power of approximately 15 W. Images of the fractured material can be found in Fig. 5. Based on these results, an analysis of the fracture in highly doped ceramic Nd:YAG is required.

## 3. Temperature distribution in the laser material

The most decisive parameter for fracture in laser material is the temperature difference between the surface of the material and the center of the material. This temperature difference induces thermal stress, and when the maximum tensile stress exceeds the tensile strength of the material, the material fractures. Because the end-pumping method was used in this experiment, the temperature will be the highest at the center of the material cross-section and will decrease along the cross-section radial and longitudinal direction. The material’s absorption coefficient will also influence the temperature distribution. The temperature distribution of ceramic Nd:YAG was calculated using Eq. (1) [15–18]. The steady-state temperature difference between the rod surface and the center during end-pumping by a Gaussian beam can be approximated as follows:

^{−1}and 40 cm

^{−1}are used for 2 at.% and 4 at.% Nd

^{3+}, respectively. In the above equation, $P$ is the pump power, $k$ is the thermal conductivity of the material, ${r}_{0}$ is the rod radius, ${w}_{p}$ is the pump beam radius, $L$ is the rod length, and $\eta $ is the thermal loading, for which reference values of 0.24 and 0.3 are used for lasing and non-lasing conditions, respectively [19,20]. In addition, ${E}_{i}(z)$ is the exponential integral function. The temperature at the surface of the material was assumed to be the same as the temperature of the material holder, 26.0°C. The pump power used for the calculation was the maximum value from the experiment, 15.3 W.

## 4. Thermal shock resistance

The thermal shock resistance parameter is useful in the initial stage of material selection. The following equations were used to compare the thermal shock resistance coefficients of ceramic Nd:YAG and YAG crystals [11].

The definitions of the parameters in Eqs. (2) and (3) are shown in Table 1.Equation (2) applies in the case of rapid cooling, where the maximum stress can be considered to be equal to the tensile strength of the material. However, because this extreme rapid cooling condition, which requires the value of $R$, is rare, the value of ${R}^{\prime}$ from Eq. (3), which considers thermal conductivity, is occasionally used instead.

In Table 1, the coefficients of the thermal shock resistance of ceramic Nd:YAG and Nd:YAG crystal are listed. High values of $R$ and ${R}^{\prime}$, the coefficients of thermal shock resistance, indicate that the thermal shock resistance is high. The coefficients of the thermal shock resistance of ceramic Nd:YAG, $R$ and ${R}^{\prime}$, which considers thermal conductivity, are both 1.5 times larger than that of Nd:YAG crystal. Thus, ceramic Nd:YAG is more suitable to application in high-power laser development than YAG crystal is.

## 5. Critical temperature difference

The normalized maximum stress of a material, ${\sigma}_{\mathrm{max}}$, can be found using the following equation [10]:

where $\alpha $ is the coefficient of thermal expansion; $E$ is Young’s Modulus; $\nu $ is Poisson’s ratio; $\beta $ is the Biot modulus, expressed as $\beta =h{r}_{0}/k$; ${r}_{0}$ is the radius of the material; and $k$ is the thermal conductivity. A reference value of 15 kW/m^{2}K was used for the heat transfer coefficient for the material surface covered with indium foil [21]. If the value of the Biot modulus is less than 5, the material cools slowly, if it is much greater than 5, it cools rapidly [22]. Applying the experimental conditions, the Biot modulus was found to be 3.4, corresponding to the range of slow cooling, in which the temperature of the material is lowered slowly. $f(\beta )$ is the stress reduction factor, expressed as follows [10]:

In general, the thermal stress decreases as the Biot modulus decreases, and the maximum stress within the material occurs after some time has passed. Because the material fracture due to thermal stress occurs when the maximum tensile stress and the tensile strength of the material are equal, the critical temperature difference can be expressed as the following [10]:

Using Eq. (6) and applying the experimental conditions, the critical temperature difference of the laser material in the ceramic Nd:YAG laser is calculated to be 355°C. Thus, the material fails when the temperature difference between the center of the laser material and the surface is higher than 355°C.

## 6. Difference heat generation rate in the laser material

The heat transfer rate, $\dot{Q}$, is defined as the amount of heat transferred per unit time. For one-dimensional heat transfer, Fourier expressed the heat equation as follows [23]:

where $k$ is the thermal conductivity of the material, $A$ is the heat transfer area, and $dT/dx$ is the rate of change of the temperature, $T$, with respect to the position, $x$.When the pump light is incident upon the laser material, the heat is transferred away from the point at which the beam is focused. In an end-pumped solid-state laser, heat is transferred from the center of the material cross-section along the radial direction and inside. Depending on the spot size and power of the pump light, a temperature distribution is produced, creating a temperature difference between the surface and the center of the material. The temperature difference occurs not only in the material cross-section but also on the material surface. Thus, the heat distribution within the material can be quantitatively explained by calculating the critical power per unit volume, considering the temperature difference in the radial direction based on the spot size of the pump beam and the absorbed length of the pump beam.

Because heat generation is a volumetric phenomenon that occurs within the entire material, if the difference in the heat generated in different locations is known, then the total rate of heat generation in a medium of a given volume can be calculated. In the steady state, the energy balance of the solid material can be expressed as follows [23]:

where $\dot{g}$ and $V$ are the rate of heat generation per unit volume and the volume of the medium, respectively.Assuming a cylindrical shape for the material, in the steady state, all of the heat generated from the inside of the heat generative material is transferred via the surface of material. Between the center of the cylindrical laser material and the surface, the temperature difference, $\Delta T$, can be expressed as follows, based on Eq. (1) [23]:

where ${T}_{0}$ is the temperature at the center of material, ${T}_{s}$ is the surface temperature, and ${r}_{0}$ is the radius of the material.Figure 2 presents a conceptual diagram of the heat generation of the laser material, which was calculated as follows. The heat generation length was found from absorbed length of the maximum pump power. The heat generation radius was determined as follows. From Eq. (1), the temperature difference between the material center and the surface is calculated according to the cross-sectional area of the pump beam and the maximum pump power. The calculated temperature difference is then substituted into Eq. (9) to solve for the rate of heat generation per unit volume. Finally, the calculated rate of heat generation per unit volume is substituted into Eq. (8) to obtain the heat generation radius. From the heat generation length and heat generation radius, the heat generation volume of the laser material was found and then used to calculate the critical pump power per unit volume of the laser material.

## 7. Fracture analysis results

Table 2 lists the temperature difference, heat generation radius, and heat generation length at the maximum pump power for each dopant concentration. When the pump power was 15.3 W and the beam diameter was 380 μm, the temperature differences for the 2 at.% and 4 at.% doped materials were 363.7°C and 788.2°C, respectively, and the corresponding heat generation radii were 610 μm and 662 μm, respectively. The heat generation length was calculated as the length for which the maximum pump power decreased exponentially and 99.99% of the pump power was absorbed, and the values were found to be 2.30 mm and 4.43 mm for dopant concentrations of 2 at.% and 4 at.%, respectively.

Table 3 shows the critical pump power and critical pump power per unit volume for each dopant concentration. Under the lasing condition, the material with a dopant concentration of 2 at.% fractured when the critical pump power exceeded 14.9 W, corresponding to a power per unit volume of 24.41 kW/cm^{3}. Under the non-lasing condition, the critical pump power was 11.9 W, and the corresponding critical pump power per unit volume was 19.5 kW/cm^{3}. The calculated critical pump power per unit volume is a quantitative value indicating the point at which material facture for 2 at.% ceramic Nd:YAG occurs. The critical pump powers for the material with a dopant concentration of 4 at.% under lasing and non-lasing conditions are 6.87 W and 5.50 W, respectively, and the critical pump powers per unit volume are 25.57 kW/cm^{3} and 20.47 kW/cm^{3}, respectively. Although the beam diameter of the pump changes, because the critical pump power per unit volume is constant for a given dopant concentration and lasing condition, the critical pump power can be easily predicted. Thus, for a pump beam diameter of 560 μm and a dopant concentration of 4 at.%, the heat generation radius and length are 652 μm and 2.30 mm, respectively. The critical pump powers are 7.85 W and 6.28 W for lasing and non-lasing conditions, respectively, and the critical pump powers per unit volume are 25.57 kW/cm^{3} and 20.47 kW/cm^{3}, respectively. The pump power needed to induce the critical temperature difference increased as the pump beam diameter increased because the increase in the beam diameter increases the heat generation volume, allowing much more heat to accumulate.

Figure 3 shows the material fracture characteristics for each material under lasing and non-lasing conditions. For each case depicted, the critical pump power and critical power per unit volume are marked with ●. The temperature difference as a function of pump power increased with different slopes for each case, and the material fracture seems to occur at a critical temperature greater than 355°C in all four cases. Considering a 15.3 W maximum incident pump power in the experiment, the 2 at.% ceramic Nd:YAG may have experienced thermal shock multiple times at the critical temperature difference of approximately 355°C. This occurrence is thought to have induced the gradual growth of cracks, ultimately leading to thermal fatigue failure. Even in conventional brittle materials, such as ceramic, cracks grow slowly below the critical thermal stress, depending on the environment [11]. The 4 at.% material is considered to have failed abruptly because the incident pump light exceeded the critical temperature difference. Thus, when the maximum tensile stress of the material due to the pump power exceeds the tensile strength, it can be concluded that the thermal shock enabled the growth of microcracks inside the material, reducing the strength.

Figure 4 shows the maximum pump power corresponding to the critical temperature difference as a function of material diameter. The occurrence of fracture in ceramic Nd:YAG can be reduced in two ways: slowly increasing the pump power or using a smaller sample of the material. The latter suggestion is more practical solution in terms of solid-state laser development; therefore, it was used to analyze the critical temperature difference as a function of material diameter. As the diameter of the material decreases, the pump power increases. Approximately 1.4 times more incident pump power is allowed when the diameter is 2 mm than when the diameter is 5 mm, improving the laser output power.

Figure 5 shows images of 2 at.% and 4 at.% ceramic Nd:YAG materials fractured during the experiment. As can be observed, a crack formed in the center of the material cross-section in the direction of the center of the cylinder. Looking at the cross-section, one can observe a lateral crack on the side and a radial crack in the radial direction of the center of the material, where the pump light is focused. The crack is larger for the 2 at.% material than the 4 at.% material because the thermal stress generated at a temperature higher than the critical temperature difference was applied to the 4 at.% material. This finding implies that although the pump power during laser operation did not reach the critical temperature difference of the material, cracks can occur in the laser material when the pump power changes rapidly.

## 8. Conclusion

In this study, the causes of the fracture of ceramic Nd:YAG were analyzed. Analyzing the critical temperature difference of ceramic Nd:YAG, it was found that the material fractures when the temperature difference between the center of the laser material and the surface is greater than 355°C. To normalize the pump power, which affects the critical temperature difference, the heat-generation volume per unit volume was found. The critical pump power per unit volume for the material fracture was found to be 24.41 kW/cm^{3} and 19.53 kW/cm^{3} under the lasing and non-lasing conditions, respectively, for 2 at.% ceramic Nd:YAG and 25.57 kW/cm^{3} and 20.47 kW/cm^{3} under the lasing and non-lasing conditions, respectively, for 4 at.% ceramic Nd:YAG. To reduce the occurrences of fracture in ceramic Nd:YAG, the critical temperature difference was analyzed in terms of material diameter. It was found that when the diameter decreases from 5 mm to 2 mm, 1.4 times more incident pump power is allowed. Lastly, to use ceramic Nd:YAG in a wider range of applications, more studies on the fracture characteristics of various material shapes are needed.

## Acknowledgments

This study was supported by a research grant (special project) in 2014 - 2015 from Cheongju University.

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