1. Introduction

Ceramic Nd:YAGs have the advantage of mass production with high Nd³⁺ dopant concentrations compared to YAG crystals [1–3]. However, the thermal lens and thermal birefringence of highly doped ceramic Nd:YAGs can be caused by high absorption and the thermal effect has been reported to reduce the laser power [4–7]. It has also been reported that 2 at.% and 4 at.%, ceramic Nd:YAGs are damaged at 355 °C, a critical temperature difference between the center of the material and the surface. Even below the critical thermal stress, when thermal shock generated around the critical temperature difference was applied to the medium several times, cracks slowly propagated and thermal fatigue failure eventually occurred [8]. In brittle materials such as a ceramics, micro cracks grow via a thermal shock when the maximum tensile stress reaches the tensile strength. Growing micro cracks not only reduce strength, but also rapidly increase the crack propagation velocity, causing brittle fractures [9–11]. The crack length of medium fractures varies according to the stress magnitude, and crack positions and shapes depend on the loading or unloading [12]. In general, when a load is applied to a medium using a sharp indenter, tensile stress is generated below the contact surface. Conversely, when pump power is applied to a medium, compressive stress occurs near the focal point [13–15]. When the stress distribution via pump power is analyzed in association with the result of load, the medium’s fracture process, failure origin, and crack type can be clarified.

This study describes the crack formation process of a medium based on previous research on end-pumped ceramic Nd:YAG lasers [8]. The crack type was determined by analyzing the failure surface of a damaged laser medium. The thermal stress was calculated using the finite element method, and the crack propagation was estimated by analyzing the stress field.

2. Results and discussion

Figure 1 shows the crack types of fractured 2 at.% and 4 at.% ceramic Nd:YAGs. The 2 at.% and 4 at.% media are Ø5.0 × 20 mm and Ø4.5 × 20 mm, respectively. The crack types are classified as a lateral crack generated on the rod’s sides, a radial crack generated toward the surface from the center of the rod’s cross-section, and a median crack generated toward the center inside the rod’s cross-section [11–12]. The mechanical properties of each crack are analyzed as follows.

 

Fig. 1. Crack types in fractured ceramic Nd:YAGs. (a) 2 at.% ceramic Nd:YAG and (b) 4 at.% ceramic Nd:YAG.

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The lateral crack indicates the tendency of pieces to be cut off parallel to the medium surface and is caused by erosion and abrasion. This crack is related to the residual stress that forms when the indentation load is removed. The shape of the residual stress field is generated when the tensile zone moves from the contact area toward the rod’s surface. When the pump power is focused on the center of the cross-section of the ceramic Nd:YAG, an inelastic strain occurs at the center and displacement occurs in the elastic zone near the surface. When the pump power increases, compressive stress and tensile stress occur at the rod’s center and surface, respectively, forming lateral cracks. Lateral cracks will be larger when the pump power increases instantly rather than slowly. Medium fracture many decrease if the pump power increases slowly depending on the thermal equilibrium of the ceramic YAG.

The radial crack is normal to the rod’s surface and is caused by decreased strength. This crack forms due to residual stress when the indentation load is removed. A crack is generated by self-stress that increases as the pump power increases in the ceramic Nd:YAG. The ceramic polycrystalline medium has randomly arranged single crystals. When the pump power increases, crystals with a non-cubic structure are deformed by pushing on one another due to the temperature change. This residual strain remains even if the pump power is applied constantly.

The median crack is normal to the rod’s surface and is caused by a decreased strength. The crack is generated and grows when an indentation load is applied to the medium. It occurs due to inelastic strain in the tensile zone below the contact point. When the pump power decreases to zero, the medium eventually reaches thermal equilibrium, and tensile and compressive stress occur at the rod’s center and surface, respectively, forming a median crack. When the pump power decreases rapidly, the probability of median cracks rises.

Crack length R_b and fracture stress σ_f have a branching constant relationship ${\textrm{M}_\textrm{b}} = {\mathrm{\sigma}_\textrm{f}}\sqrt {{\textrm{R}_\textrm{b}}} \; $ [12]. In Fig. 1, the lateral crack lengths in the 2 at.% and 4 at.% media are approximately 1.9 mm and 5.7 mm, respectively. Substituting the lateral crack lengths for the branching constant equation, the thermal stress applied to the 4 at.% ceramic YAG is approximately 1.7-fold larger than that applied to the 2 at.% ceramic YAG. This is mainly caused by the fact that the absorption coefficient (40 cm⁻¹) of the 4 at.% medium is approximately 2-fold larger than the absorption coefficient (20.8 cm⁻¹) of the 2 at.% medium.

The maximum tensile stress of the laser medium is expressed as Eq. (1) and the longitudinal stress as Eq. (2):

According to Eq. (1), the critical temperature differences in ceramic Nd:YAGs and crystal Nd:YAGs are approximately 355 °C and 241 °C, respectively. The smaller the stress reduction factor, the larger the critical temperature difference. Stressed rods are generally analyzed using analytical solutions for radial, tangential, and longitudinal stress [13]. The stress on the rod’s center and surface is the greatest for the longitudinal stress component represented by Eq. (2). When the temperature difference in ceramic Nd:YAGs is 355 °C, longitudinal stress at radial distances of 1, 2, and 2.5 mm from the rod’s center $0\;\textrm{mm}$ are -529, -360, 148, and 529 MPa, respectively. According to the results, the medium was fractured by stress higher than the flexural strength (360 MPa) applied to the surface. However, according to a previous study, the stress fracture safety factor of ceramic Nd:YAGs is 2.4 [4]. Hence, it is expected to fracture below the flexural strength. To complement these results, we conducted a finite element analysis according to the pump power using LASCAD 3.6 commercial software.

Figure 2 shows the temperature and stress distribution at a pump power of 14.9 W for the 2 at.% and 4 at.% ceramic Nd:YAGs. In the finite element analysis, thermal expansion, Young's Modulus, Poisson’s ratio, and the absorption coefficient are identical to the values applied to Eqs. (1) and (2). The dn/dT was 8.9 × 10⁻⁶ K⁻¹, the refractive index was 1.8169, the thermal conductivity was 10.7 W/mK in the 2 at.% medium and 9.2 W/mK in the 4 at.% medium, the super-Gaussian exponent was 2, the rod’s surface temperature was 26.0 °C, and the rod’s length was 3 mm because the temperature distribution was similar at above 3 mm. For the spot diameter, a point where the pump light intensity was 0 W at 380 µm of the spot diameter in the program was calculated, and 380 µm was implemented when the spot radius was 76.5 µm. Differences in the calculation results occurred because in this tool, only limited values can be input into the super-Gaussian exponent parameter, which determines the beam profile of pump light. To compensate for the differences in the calculation results in the analysis of the stress components, we multiplied the finite element analysis result by 1.39 by referring to a previous study [4].

 

Fig. 2. Temperature and stress distributions. (a) $2\;\textrm{at}.{\%}$ ceramic Nd:YAG and (b) $4\;\textrm{at}.{\%}$ ceramic Nd:YAG.

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In the 2 at.% ceramic Nd:YAG, the temperature difference at a pump power of 14.9 W was 344.4 °C and the xx, yy, and zz stress components were 124.3, 124.2, and 160.8 MPa, respectively. In the 4 at.% ceramic Nd:YAG, the temperature difference at a pump power of 7.5 W was 280.4 °C, and the xx, yy, and zz stress components were 85.6, 81.5, and 160.7 MPa, respectively. When the pump power reached 14.9 W, the temperature difference was 557.2 °C and the xx, yy, and zz stress components were 169.9, 161.9, and 319.1 MPa, respectively. Consequently, during lasing, the zz stress components of the 2 at.% and 4 at.% ceramic Nd:YAGs approached 160 MPa and were fractured when the pump power was 14.9 W and 7.5 W, respectively. In a previous study, 2 at.% and 4 at.% ceramic Nd:YAGs had a critical temperature of 355 °C at pump powers of 14.9 W and 6.9 W, respectively [8]. In the 2 at.% ceramic Nd:YAG, the finite element analysis of the temperature differences that caused medium fracture was similar to the results of a previous study [8]. However, in the 4 at.% ceramic Nd:YAG, the medium was fractured because the tensile stress reached 160 MPa due to a high absorption rate when the temperature difference was 280 °C, lower than the critical temperature of 355 °C.

The medium’s stress distribution σ_θθ is expressed by the Vickers hardness test as follows [9]:

The applied P value varies depending on the medium’s stress state. When a load is applied, the contact area develops tensile stress, and a residual strain appears when the load is removed. When the pump power applied to the laser medium increases, the pump light’s focal plane develops compressive stress. However, if the applied pump power decreases, tensile stress appears caused by residual stress. To express the laser medium’s stress distribution via pump power using Eq. (3), the zz stress component in the finite element analysis in Fig. 2 was substituted in the applied load by reversing the zz stress component’s sign.

Figure 3 shows the stress distributions for the 2 at.% and 4 at.% ceramic Nd:YAGs. In Fig. 3(a), the pump power applied to the 2 at.% ceramic Nd:YAG reached 14.9 W, which was the result of applying -160.8 MPa. Similar to the finite element analysis, it was in a compressive stress state below the center of the rod’s cross-section, and the surface was in a tensile stress state. According to the stress distribution analysis, the lateral, radial, and median cracks are distinguished as follows. The shape of the lateral crack is parallel to the rod’s cross-section. A 200 MPa, tensile stress forms on the surface that decreases as it becomes farther from the rod’s center. The stress increases infinitely as it moves from the rod’s surface toward the center of the cross-section, and the stress can cause inelastic strain. A radial crack is generated as the residual stress increases perpendicularly from the center of the rod’s cross-section. A median crack forms from the center of the rod’s cross-section toward the inside. The -200 MPa compressive stress decreases as it moves perpendicularly. When the pump power is reduced and removed, the area develops tensile stress, causing a median crack. Figure 3(b) shows the case when the pump power applied to the 4 at.% ceramic Nd:YAG reached 14.9 W, which was the result of applying -319.2 MPa. The stress applied to the 4 at.% ceramic Nd:YAG was approximately 2-fold higher than that of the 2 at.% ceramic Nd:YAG, similar to the branching constant analysis results. The cracks generated are shown in Fig. 1.

 

Fig. 3. Stress distributions. (a) 2 at.% ceramic Nd:YAG and (b) 4 at.% ceramic Nd:YAG.

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Figure 4 shows the crack formation process according to the pump power. In the case of the Vickers hardness test, the median crack propagates as the critical load increases, and the crack forms in the tensile zone below the contact point. The radial crack forms due to the residual strain, which increases as the load decreases, and appears perpendicular to the surface. Finally, the lateral crack forms and stops when the load is almost removed. The crack generation sequence according to the pump power differs from the process using the Vickers hardness test. If the pump power applied to the medium increases to greater than the critical power, the focal point area develops compressive stress, and the rod’s surface develops tensile stress, resulting in a lateral crack. When the pump power increases continuously, radial cracks form due to the residual strain. Lateral and radial cracks can develop almost simultaneously as the pump power increases. When the maximum pump power decreases, the focal point area develops tensile stress. When the pump power increases, the fractured surface area closes, and a median crack is generated. Its length varies depending on the maximum pump power.

 

Fig. 4. Crack formation process.

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3. Conclusion

We analyzed and described the crack formation process of ceramic Nd:YAGs. The ceramic Nd:YAGs were fractured at a tensile stress of 160 MPa or higher. According to the stress distribution analysis results, the 4 at.% ceramic Nd:YAG received approximately 2-fold higher stress than the 2 at.% medium at the same pump power. When the stress caused by the pump power increased and decreased in the ceramic YAGs above the maximum tensile stress, lateral, radial, and median cracks occurred sequentially. When the temperature increased slowly, the material was not fractured even at a temperature higher than the critical temperature. Further research should be conducted on fractures that occur in ceramic Nd:YAGs when the pump power per unit time increases or decreases.