1. INTRODUCTION

Ceramic Nd:YAG has been attracting increasing attention because it can be produced in larger sizes than YAG crystals, and the neodymium concentration can be increased by 2 at. % or more [1,2]. A higher neodymium concentration results in a higher absorption coefficient of the medium, which appears to have thermal effects at a low pump power [3,4]. An end-pumped ceramic Nd:YAG laser with a doping concentration of 2 at. % produced thermal lens effects at a pump power of 6 W or more [5], and the phase differences owing to thermal birefringence were analyzed at 38° at a pump power of 12 W [6]. The laser material was damaged when a pump power of 15 W or higher was applied to ceramic Nd:YAG [7]. The laser material was fractured when thermal stress was applied several times, even below the critical temperature differences [8]. To achieve stable laser operation, the fracture conditions of the laser material must be quantitatively analyzed in the full pump power range. The thermal effect is generated by the temperature difference between the center and surface of the laser material [9,10]. Three primary conditions exist for generating a temperature difference in the laser material during the laser operation. First, when pumping energy is applied to a laser material, a temperature difference occurs between the center and surface of the laser material, and when the pumping energy increases, the temperature difference in the laser material also increases suddenly. Second, when the pumping energy applied to a laser material no longer increases, thermal equilibrium occurs in the laser material and the temperature difference remains constant. Third, when the applied pumping energy decreases, the thermal equilibrium in the laser material changes, as does the temperature difference. Following a certain period, thermal equilibrium is reached after removing the pumping energy completely, and no temperature difference occurs. The temperature difference between the center and surface of the laser material causes thermal stress and thermal shock in the laser material.

The thermal shock resistance of ceramics can be summarized as follows [11–13]. First, if the maximum tensile stress is higher than the flexural strength of the material, the thermal shock produces microcracks in the material, which reduces its strength. The resistance of a material to a large temperature difference is known as thermal shock fracture resistance. Material damage owing to cracks, such as collapse and separation, is significant in the development of solid-state lasers. The resistance to crack growth owing to thermal shock is referred to as thermal shock damage resistance. Second, when the flexural strength of the material is higher than the maximum tensile stress, even in ceramics, which are typical brittle materials, cracks gradually grow under the critical thermal stress, depending on the environment. When the maximum tensile stress caused by thermal shock is smaller than the flexural strength of the material, if thermal shocks are applied repeatedly, cracks gradually grow and fractures will be caused by thermal fatigue. Resistance to thermal fatigue fractures is known as thermal shock fatigue resistance. The fracture strength was found to be ${341.7}{\pm}{58.3}\;{\rm MPa}$ for the noncomposite YAG ceramics and ${360.8}{\pm}{49.2}\;{\rm MPa}$ for 0.1% Nd:YAG composite ceramics [14]. Ceramic Nd:YAG with a flexural strength of 360.8 MPa was reported to fracture when the tensile stress increased above 160 MPa and then decreased [7]. If the temperature difference in the laser material is lower than the critical temperature difference, the laser material will fracture if low thermal shocks are applied several hundred or thousand times. That is, the failure of the medium can be predicted by probabilistically analyzing its fatigue life owing to stress. In general, the fatigue lifetime can be verified by repeatedly applying loads to the medium to induce damage [15,16]. However, as the laser medium is transparent, its thermal effect can be measured according to the pump power using a probe beam. The fatigue life can be obtained by correlating the thermal effect of the laser medium using a probe beam with its temperature and stress using the finite-element method (FEM). This study proposes a fatigue lifetime analysis method using a probe beam for the first time.

 

Fig. 1. Experimental setup for measuring thermal effects in laser material.

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In this study, the intensity variations of a probe beam in a ceramic Nd:YAG laser were measured using the pump power. The intensity variation in the measured probe beam was correlated with the temperature difference and stress in the laser material. The thermal shock owing to the stress generated in the ceramic Nd:YAG was calculated, and the fatigue life was analyzed to predict the damage to the medium.

2. EXPERIMENTAL SETUP

Figure 1 depicts the experimental setup used to measure the thermal effects of the ceramic Nd:YAG. An He–Ne laser was used as the probe beam to obtain the thermal effects in the laser material. The probe beam was transformed into S-polarized light after being reflected by a polarization beam splitter (PBS) with an extinction ratio of 1000:1, and it was passed through a spatial filter and a beam expander to make the beam diameter similar to that of the laser material. In general, the conoscope method is used to measure a probe beam that is depolarized by thermal birefringence [17]. In this approach, the laser material is located between a polarizer and an analyzer, the transmission axes of which are perpendicular. The disadvantage of this method is that the sensitivity of the measurement signal is low. However, when the polarization state of the probe beam incident on the medium is circularly polarized light, the measurement signal is larger than the measurement signal when using linear polarization, such as the conoscope technique. To improve the sensitivity of the measurement signals, in the experiment, the linearly polarized probe beam was changed into circularly polarized light by a quarter-wave plate. A fiber-coupled laser diode (Apollo; F25-808-4P) with a core diameter of 400 µm and a numerical aperture of 0.22 was used as the pump source. The pump source was operated using a power supply (HP; 6011A). The pump power was focused on the laser material using a focusing lens system with a spot diameter of 380 µm and an effective focal length of 10.9 mm. The resonator consisted of a dichroic-coated flat mirror with a size of ${10}\;{\rm mm} \times {20}\;{\rm mm} \times {2}\;{\rm mm}$ as a rear mirror and a concave output coupler with 90% reflectivity for the lasing wavelength. Ceramic Nd:YAG (Baikowski) with an $Nd^{3+}$ concentration of 2 at. %, a diameter of 5 mm, and a length of 10 mm was used for the laser material. The laser material was wrapped with indium foil and mounted tightly on a water-cooled copper holder, and the temperature of the material was maintained at 26°C. It has previously been reported that the thermal birefringence effect of ceramic Nd:YAG can be considered as the average of the birefringence effect in many grains of the medium [6]. The probe beam was adjusted to be parallel light by installing a convex beam with a focal length of 280 mm in front of the output coupler. The laser beam was removed by reflection by a high-reflection mirror for the lasing wavelength, and it subsequently entered into a beam dump. When the pumping light was incident on the laser material, the circularly polarized probe beam was changed to elliptically polarized light owing to phase differences. The intensity of the transmitted probe beam through the PBS used as an analyzer was changed by the angle between the inclination of the major axis of the elliptical polarization and the analyzer. A half-wave plate was used to improve the measurement sensitivity by rotating the angle of the major axis of the elliptical probe beam. A mirror that was high-reflection-coated at 808 nm and antireflection-coated at 632.8 nm was used as a filter to remove the pump light. Finally, the characteristics of the probe beam owing to the thermal effect of the laser medium were obtained using a photodiode (Thorlabs; DET 110) and a CCD camera (Sony; ST-50). The change in the average thermal birefringence value occurring in the intensity medium of the probe beam, rather than the birefringence at each position in the cross section of the probe beam, was measured using a photodetector. The transverse modes of the probe beam according to the pump power were observed using the CCD camera.

 

Fig. 2. Transverse modes and intensity profiles of probe beam. (a) Pump power of 0 W; (b) pump power of 2 W; (c) pump power of 4 W; (d) pump power of 8 W; and (e) pump power of 12 W.

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3. RESULTS AND DISCUSSION

Figure 2 presents the transverse modes and intensity profiles of the probe beam according to the pump power. The top images in Fig. 2 depict the transverse modes measured using the CCD camera, whereas the bottom images are the intensity profiles from the center to the end of the transverse modes of the probe beam. Figure 2(a) depicts the results at a pump power of 0 W. The P-polarized component of the circularly polarized probe beam was obtained using an analyzer. Figure 2(b) presents the results of the transverse modes of the probe beam measured at thermal equilibrium when there was no change in the probe beam intensity at a pump power of 2 W. A change occurred in the transverse modes when the pump power was applied to the laser material, which was thought to be caused by a change in the refractive index of the laser material owing to the pump power. Figure 2(c) shows the results at a pump power of 4 W, where an interference fringe pattern appeared in the transverse modes of the probe beam. The interference fringe pattern was caused by the change in the refractive index of the material owing to the thermal effect. The fringe results occurred a period before thermal equilibrium, in which the optical beam could have very high-intensity hot spots. This could lead to optical damage in the laser medium. The fringe appeared when the pump power was applied, and the fringe patterns changed and stabilized upon reaching thermal equilibrium. Figure 2(d) presents the results at a pump power of 8 W. The number of interference fringes in the transverse modes of the probe beam was higher than that of the 4 W pump power owing to the thermal effects. Figure 2(e) shows the results at a pump power of 12 W. The number of interference fringes increased with the pump power and a clear fringe pattern appeared. The interference fringe pattern indicates that the intensity of the laser beam is focused on a specific part of the medium. If the beam intensity is locally focused inside the medium, the thermal shock is also locally focused, which increases the probability of damage to the medium.

Figure 3 presents the variations in the probe beam intensity at different pump powers. Figure 3(a) shows the measurement results at a pump power of 2 to 12 W. The entire experiment was conducted for 1800 s. The pump power was incident on the laser material approximately 60 s after the start of the experiment, and it was removed at approximately 1200 s. The circularly polarized probe beam was transformed into an elliptically polarized probe beam by the thermal birefringence generated in the medium by the pump power. As the pump power in the laser medium increased, the thermal birefringence and thermal lens effects increased simultaneously. The phase difference and major axis of the elliptically polarized light increased with the pump power. The angle of the major axis of the elliptically polarized probe beam was controlled using the half-wave plate to obtain the maximum intensity with a photodiode. The S-polarized component of the elliptically polarized light was obtained using an analyzer. The variations in the intensities of the six different configurations of the S-polarized light of the probe beam were observed using an analyzer and a photodiode. The intensity of the measurement signal was normalized to 1.0 at a pump power of 0 W. In general, the thermal equilibrium time constant in a medium varies depending on the pump power. The intensity measurement results of the probe beam at different pump powers demonstrated that the laser material reached thermal equilibrium, and the intensity of the probe beam was almost constant after 1000 s. In general, the thermal equilibrium time according to the pump power is considered when analyzing the thermal time constant of the medium. This study focused on the change in the intensity of the probe beam according to the pump power. The variations in the probe beam intensity increased with the pump power, and the variations in the probe beam intensity suddenly increased when the pump power was applied and removed. Figure 3(b) depicts an enlarged image of when the pumping started, as shown in Fig. 3(a). When the pump power increased suddenly, the temperature difference between the center and surface of the laser material increased rapidly, which significantly changed the refractive index of the laser material. The change in the refractive index affected the polarization component of the probe beam. As a result, when the pump power was applied and removed, the probe beam intensity change was greater than that in the thermal equilibrium state. For a pump power of 4 to 12 W, approximately 4 s were required until the peak value of the probe beam intensity, which increased suddenly, decreased to the level of thermal equilibrium. Figure 3(c) presents an enlarged image of when the pumping stopped, as shown in Fig. 3(a). The intensity of the probe beam increased suddenly when the pump power decreased rapidly because the refractive index of the laser material changed rapidly as the pump power decreased. When the pumping was stopped, the peak intensity of the probe beam was lower than that of the probe beam when the pumping started. This result indicates that a large thermal shock was applied to the laser material when the pumping started. Approximately 20 s after the pump power was stopped, the probe beam intensity stabilized to that of the initial state, which was related to the thermal time constant of the laser material. According to the results of the probe beam intensity as a function of the pump power, when the pump power suddenly increased or decreased, a large thermal shock was suddenly applied to the laser material.

 

Fig. 3. Variations in probe beam intensity. (a) Normalized intensity as function of time; (b) values when pumping started; and (c) values when pumping stopped.

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Figure 4 presents the variation in the probe beam intensity according to the pump power. In Fig. 4, the red dot indicates the probe beam intensity when the laser material was in thermal equilibrium, as illustrated in Fig. 3(a), whereas the blue dashed line indicates the fitted value. The equation for the fitting curve of the probe beam intensity variation as a function of the pump power is expressed as $y = 1.09891 + 0.13864x + 0.01689{x^2}$. The green triangle indicates the maximum value of the probe beam intensity when the pumping started, as shown in Fig. 3(b), and the black square indicates the maximum value of the probe beam intensity when the pumping stopped, as illustrated in Fig. 3(c). As the pump power increased, the intensity of the probe beam increased nonlinearly. At a pump power of 12 W, the intensity of the probe beam increased 5.7 times compared to a pump power of 0 W when the pumping started, 5.2 times when the pumping stopped, and 5.2 times during pumping. The average variation in the probe beam intensity was 1.2 times higher than that during pumping when the pumping started and 1.1 times higher when the pumping stopped. The intensity of the probe beam was expected to increase when the pump power increased by more than 12 W. The variation in the probe beam intensity, as shown in Fig. 4, was correlated with the calculated values for the temperature difference and stress of the laser material, as shown in Fig. 5.

 

Fig. 4. Variation in probe beam intensity according to pump power.

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Fig. 5. Results of FEM simulation. (a) Temperature distributions and (b) stress distributions.

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The mechanical properties of the ceramic Nd:YAG were analyzed using FEM with reference to the results of previous studies [7]. Figure 5 depicts the FEM simulation results for the 2 at. % ceramic Nd:YAG. Figures 5(a) and 5(b) present the temperature distribution and zz stress component of the laser material at a pump power of 14.9 W. The input parameters for the FEM simulation were as follows: thermal expansion, ${8.0} \times {{10}^{- 6}}\;{{\rm K}^{- 1}}$; Young’s modulus, 280 MPa; Poisson’s ratio, 0.25; dn/dT, ${8.9} \times {{10}^{- 6}}\;{{\rm K}^{- 1}}$; refractive index, 1.8169; thermal conductivity, 10.7 W/mK; laser material surface temperature, 26.0°C; and pumping beam spot diameter, 153 µm [7,8]. Moreover, the length of the laser material was set to 3 mm because the temperature distributions were similar at 3 mm or longer. The analysis tool could not accurately represent the pumping beam profile because the input value of the super-Gaussian exponent, which determines the beam profile of the pump beam, was limited to a minimum value of 2.0, which resulted in differences in the calculation results. To compensate for the pump beam, the calculation results were multiplied by 1.39. The fracture conditions of the 2 at. % ceramic Nd:YAG have been reported as follows: a temperature difference of 344.4°C between the center and surface of the laser material at a pump power of 14.9 W and xx, yy, and zz stress components of 124.3, 124.2, and 160.8 MPa, respectively [7,8]. The zz stress component was mainly analyzed because it had the greatest effect among the stress components. The FEM simulation results demonstrated that, at pump powers of 2, 4, 6, 8, 10, and 12 W, the temperature differences between the center and surface of the laser material were 45.6, 91.8, 138.0, 184.2, 230.3, and 276.5°C, respectively, whereas the maximum zz stress components were 21.3, 42.9, 64.6, 86.2, 107.9, and 129.5 MPa, respectively. The temperature and stress in the ceramic Nd:YAG, as shown in Fig. 6, were analyzed by correlating the analysis of the thermal effect that occurred in the material with the variation in the probe beam intensity.

 

Fig. 6. Temperature differences and zz stress components according to pump power.

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Figure 6 depicts the temperature differences and zz stress components in the laser material according to the pump power. The temperature differences and zz stress components according to the intensity variations in the steady state are provided in the small image in the figure. The data in Fig. 6 were calculated using the following procedure: The probe beam intensity change data during the pumping condition shown in Fig. 4 and the temperature difference data shown in Fig. 5 were matched according to the pump power. The red dots could be derived as a result. The experimental and calculation results could be matched because they were based on thermal equilibrium conditions. The start pumping and stop pumping data were obtained by calculating the experimental results obtained in Fig. 4 based on the thermal equilibrium state results. The solid blue line indicates the temperature difference and zz stress components in the medium according to the pump power calculated using the FEM, as shown in Fig. 5. The temperature differences and zz stress components increased linearly with the pump power. The temperature difference in the laser material at a pump power of 6 W was 193.7°C when the pumping started, 139.1°C during pumping, and 160.0°C when the pumping stopped. The maximum value of the zz stress component was 90.6 MPa when the pumping started, 65.1 MPa during pumping, and 74.8 MPa when the pumping stopped. At a pump power of 6 W, the value when the pumping started was 1.39 times higher than the value during pumping. At a pump power of 12 W, the temperature difference was 289.9°C when the pumping started, 275.1°C during pumping, and 282.4°C when the pumping stopped. The maximum value of the zz stress component was 135.6 MPa when the pumping started, 128.7 MPa during pumping, and 132.1 MPa when the pumping stopped. At a pump power of 12 W, the values when the pumping started were 1.05 times higher than those during pumping. The thermal shock to the laser material caused by the momentary temperature difference was the highest when the pumping started and the pump power was initially applied. It decreased slightly at thermal equilibrium during pumping. The thermal shock occurred owing to the change in the temperature difference when the pumping stopped. The greatest difference between the steady state and the peak value at a pump power of 6 W could be considered as a general experimental error. The data in Fig. 6 were derived by correlating the experimental results using the probe beam shown in Fig. 4 and the FEM results presented in Fig. 5. The thermal shock illustrated in Fig. 7 was calculated using the temperature difference and zz stress component, whereas the fatigue life, as shown in Fig. 8, was analyzed based on the thermal shock.

 

Fig. 7. Thermal shock according to pump power.

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Fig. 8. Fatigue lifetime according to pump power.

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Figure 7 presents the thermal shock according to the pump power. The red dots in the figure indicate the values obtained using the zz stress component when the pumping started, during pumping, and when the pumping stopped, as shown in Fig. 6. The blue dashed lines indicate the fitting values. The equation for the fitting curve of the thermal shock as a function of the pump power x is expressed as $y = - 0.02750 + 0.05713x - 0.00128{x^2}$. The thermal shock resistance parameter was calculated using $R^\prime = \sigma k({1 - \upsilon})/\alpha {\rm E}$ [12], where $\sigma $ is the flexural strength, $k $ is the thermal conductivity, $\upsilon $ is the Poisson’s ratio, $\alpha $ is the thermal expansion, and E is the Young’s modulus. The zz stress component was used as the input to obtain the flexural strength. The thermal conductivity, Poisson’s ratio, thermal expansion, and Young’s modulus were the same as the input parameters in the FEM simulation depicted in Fig. 5. The thermal shock was gradually saturated as it increased to 0.09, 0.22, 0.32, 0.36, 0.43, and 0.49 kW/m at a pump power of 2, 4, 6, 8, 10, and 12 W, respectively. The zz stress component at which the 2 at. % ceramic Nd:YAG fractured, as indicated in Fig. 5, was 160.8 MPa at a pump power of 14.9 W. When the zz stress component of 160.8 MPa increased in flexural strength, the thermal shock resistance parameter was 0.57 kW/m and the pump power of the thermal shock resistance using the fitting value was 17.2 W. In the literature, the flexural strength of ceramic Nd:YAG is 360.8 MPa [14], and in this study, the thermal shock resistance parameter was 1.3 kW/m. According to the analytical results of the laser material using the thermal shock, the experimental value was approximately half of that reported in the literature. Based on this result, two or more safety factors should be considered for thermal shock when developing ceramic Nd:YAG lasers.

A previous study on a 2 at. % ceramic Nd:YAG laser reported that the pump power increased up to 15.3 W [5], and the laser material broke approximately 20 months after the experiment. The results of the current study demonstrate that the thermal shock that occurred around the thermal shock resistance parameter of 0.57 kW/m was applied to the ceramic Nd:YAG several hundred times, and cracks gradually grew inside the medium before the laser material eventually fractured. The fractures in the laser material caused by the increasing pump power were analyzed using a stress–life (S–N) diagram, which indicated the relationship between the thermal stress and fatigue lifetime. In general, the S–N diagram can be obtained not only by testing, but also using the following equation [15,16]:

Figure 8 depicts the fatigue lifetime according to the pump power. In the figure, the red dots indicate the values when the pumping started. The blue solid line indicates the values obtained by entering the results shown in Fig. 6 into Eqs. (1) and (2). In Eq. (1), the value of 21.3 MPa, which was calculated at a pump power of 2 W, was substituted into ${\sigma _{{\min}}}$, the zz stress component of the pump power in ${\sigma _{{\max}}}$, the maximum value of the zz stress component, 160.8 MPa, was substituted into ${{S}_u}$, and ${{S}_a}$ of Eq. (2) was substituted into ${{S}_e}$. The pump powers of 5, 10, 15, and 20 W corresponded to 4.4, 8.8, 13.2, and ${17.6}\;{{\rm kW/cm}^2}$. The fatigue lifetime decreased as the pump power increased, and it reached ${4.5} \times {{10}^4}$ cycles at a pump power of 12 W. The fractured laser material during the ceramic Nd:YAG laser experiment was analyzed as follows: In Fig. 8, the green square indicates the results at a pump power of 15.0 W. The maximum stress was 162.0 MPa and the fatigue lifetime was 238 cycles. Consequently, it was determined that 240 or more applications of a pump power of 15.0 W, which corresponded to a stress of 162.0 MPa, applied to the 2 at. % ceramic Nd:YAG would damage the laser material. When a pump power of 17.2 W was applied to the thermal shock resistance parameter of 0.57 kW/m, which is the analytical result presented in Fig. 7, the fatigue lifetime was four cycles. That is, when the pump power was 17.2 W, the material had a very high fracture probability when the pumping started. When developing 2 at.% ceramic Nd:YAG lasers, considering the safety factors of the fatigue lifetime, it would be appropriate to set the maximum pump power to lower than 12.0 W, which is approximately 70% of 17.2 W.

4. CONCLUSIONS

This study analyzed the thermal shock applied to a laser material in a ceramic Nd:YAG laser using a probe beam and considered the fatigue lifetime of the damage to the laser material. Moreover, the transverse modes and intensity variations of the probe beam according to the pump power were measured. As the pump power increased, the number of interference fringes in the transverse modes increased, and the fringe clearly appeared. The fringe results occurred a period before thermal equilibrium, at which the optical beam could have very high-intensity hot spots. These phenomena can cause optical damage. The variation in the probe beam intensity when the pump power was applied to or removed from the laser material was larger than that in the thermal equilibrium state. A large thermal shock was suddenly applied to the laser material when the pump power suddenly increased or decreased. The temperature differences and stress in the laser material were calculated using the FEM, and the results were correlated with the probe beam intensity. The temperature difference in the laser material when the pumping started was 289.9°C and the zz stress component was 135.6 MPa at a pump power of 12 W. Based on the experimental results, the thermal shock resistance parameter was 0.57 kW/m at a pump power of 14.9 W and a zz stress component of 160.8 MPa. The fatigue lifetime results demonstrated that the laser material would fracture when a stress of 160.8 MPa was applied to the 2 at. % ceramic Nd:YAG more than 240 times. In the development of ceramic Nd:YAG lasers, the safety factors can be predicted by analyzing the thermal shock and fatigue lifetime. Further research on the theoretical analysis of damage to the medium by changing the interference fringe of the probe beam according to the pump power will be necessary in the future.