## 1.

## Introduction

High-power all-solid-state 2-*μ*m laser systems have played an increasingly important role in the technological landscape during the past few years. Applications have ranged from an abundance of medical uses, such as laser surgery and therapy, to more technical applications, such as laser ranging and remote sensing.^{1}2.3.^{–}^{4} It is important to develop a laser system with high efficiency, long lifetime, and high stability; however, the laser output window may compromise the system’s performance because of the thermal and mechanical effects caused by the window absorption in the process of high-power laser-beam outputting, which not only limits the service life, but also affects the power scaling of the system.^{5} The thermal-mechanical effects and the damage mechanism of the laser output window have to be studied for further improving the power levels of the system before the onset of window damage.^{6} These issues have received a great deal of attention, and various models have been proposed including the heating conduction model, the multiphoton ionization model, and cumulative ionization breakdown. Studies have shown^{7}^{,}^{8} that the multiphoton ionization model and cumulative ionization breakdown dominate only in the case of high-power pulsed output, such situations have difficulties in quantitative analysis, and the heating conduction model is frequently used in the interpretation of continuous wave (cw) laser damage of output windows.^{9}

In this article, we establish a three-dimensional (3-D) thermal model of the laser output window. Expressions for the transient thermal and mechanical distributions are deduced in the case of continuous laser output using the integral-transform method. Then, taking the infrared quartz window material as an example, the temperature and mechanical field distributions of a high-power all-solid-state 2-*μ*m laser system window are simulated. Finally, the laser-induced damage mechanism is analyzed.

## 2.

## Model Building

## 2.1.

### 2-*μ*m Laser System

The 2-*μ*m Tm:YAG laser system contains a two-rod scheme, in which either rod employs a laser diode (LD) side-pumping laser module. The module contains five diode arrays, and each array consists of 12 diodes operating in cw-mode with a central wavelength of 785 nm. The rod is 4 mm in diameter and 69 mm in length with a ${\mathrm{Tm}}^{3+}$-ion doping concentration of 3.5%. The Tm:YAG rod is cooled to 8°C by deionized water. M1 is the flat rear mirror coated to obtain a reflectivity more than 99.5% at 2 *μ*m, and M2 is the output coupler with a transmission of 5% around 2 *μ*m; in this instance, the wavelength of the output laser is 2.07 *μ*m, and the maximum cw output power is 115 W. The schematic of the laser system is shown in Fig. 1 (Ref. 10).

The selection of the laser window is extremely important during the design of lasers. Materials such as ${\mathrm{CaF}}_{2}$, ${\mathrm{Al}}_{2}{\mathrm{O}}_{3}$, and ${\mathrm{BaF}}_{2}$ are commonly used for optical windows,^{11} and quartz glass, specifically, exhibits outstanding thermodynamic properties, which makes it an attractive selection as the window material of a high-power 2-*μ*m laser system.

## 2.2.

### Thermal Model Description

Heat deposition arises due to the absorption of the laser-beam energy when a high-energy laser beam passes through the output window, resulting in temperature rise as well as thermal stress. The thermal model of the output window is defined and illustrated in Fig. 2. A laser beam with a radius ${\omega}_{0}$ is incident along the positive $z$-direction for the window M2, which has a radius ${r}_{0}$ and a thickness $L$.

The 3-D heat transfer equation of the window can be expressed as

## (1)

$$\frac{{\partial}^{2}T}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial T}{\partial r}+\frac{{\partial}^{2}T}{\partial {z}^{2}}+\frac{q(r,z,t)}{k}=\frac{\rho C}{k}\frac{\partial T}{\partial t},$$## (2)

$$q(r,z,t)=\{\begin{array}{ll}{I}_{0}{e}^{-\alpha z}& 0\le r\le {\omega}_{0},\phantom{\rule[-0.0ex]{1em}{0.0ex}}t\ge 0\\ 0& \text{else}\end{array},$$## (4)

$$\{\begin{array}{ll}T={T}_{\infty}& t=0\\ \frac{\begin{array}{c}\partial T\end{array}}{\partial r}=0& r={r}_{0}\\ \frac{\begin{array}{c}\partial T\end{array}}{\partial z}=0& z=0,l\end{array}$$## 3.

## Theoretical Analysis

There are many analytical methods for solving Eq. (1) based on the initial and boundary conditions in Eq. (4).^{12} Numerical methods, including the finite-element method^{13}^{,}^{14} and the finite-difference method,^{15} do not involve derivation of formulas or transformations, and so the solution procedure is convenient, especially in simulating multidimensional systems.^{16} However, numerical solutions cannot provide an intuitive understanding of the relationships among the various physical data, whereas analytical solutions can clearly depict how the temperature-stress field distribution evolves with the parameters of the laser and window material.^{17}^{,}^{18} Therefore, we employ the approach of using the integral-transform method to obtain such analytical solutions in this article.

Based on the model established above, we obtain the following formulas through utilizing the integral-transform method to solve Eqs. (1)–(4) and performing the positive and inverse integral transforms for variables $r$ and $z$:^{19}

## (5)

$$T(r,z,t)=\sum _{m=1}^{\infty}\frac{{R}_{0}({\beta}_{m},r)}{N({\beta}_{m})}\overline{T}({\beta}_{m},z,t)$$## (6)

$$\overline{T}({\beta}_{m},z,t)={\int}_{r\prime =0}^{{r}_{0}}{r}^{\prime}{R}_{0}({\beta}_{m},{r}^{\prime})T({r}^{\prime},z,t)\mathrm{d}r.$$Here, $T(r,z,t)$ is the temperature distribution of the window, and Eq. (6) is the inverse transform of Eq. (5). The eigenfunctions ${R}_{0}({\beta}_{m},r)$ and the norm $N({\beta}_{m})$ are obtained as

## (8)

$$\frac{1}{N({\beta}_{m})}=\{\begin{array}{ll}\frac{2}{{r}_{0}^{2}{J}_{0}^{2}({\beta}_{m}{r}_{0})}& m\ne 0\\ \frac{2}{{r}_{0}^{2}}& m=0\end{array},$$The eigenfunctions $Z({\eta}_{p},z)$ and the norm $N({\eta}_{p})$ are obtained as

## (9)

$$Z({\eta}_{p},z)=\{\begin{array}{ll}\mathrm{cos}\text{\hspace{0.17em}}{\eta}_{p}z& p\ne 0\\ 1& p=0\end{array}$$## (10)

$$N({\eta}_{p})=\{\begin{array}{ll}\frac{2}{l}& p\ne 0\\ \frac{\begin{array}{c}2\end{array}}{l}& p=0\end{array},$$The 3-D temperature distribution of the output window is given by Eq. (11) according to the inferential method from Eqs. (5)–(10):

## (11)

$$T(r,z,t)=\frac{4{P}_{0}(1-\tau )}{{\mathrm{kr}}_{0}^{2}l\pi {\omega}_{0}}\left\{\begin{array}{l}\begin{array}{c}\{\sum _{m=1}^{\infty}\sum _{p=1}^{\infty}\frac{\alpha {J}_{1}({\beta}_{m}{\omega}_{0})[1-\mathrm{exp}(-\alpha l){(-1)}^{p}]}{{\beta}_{m}{J}_{0}^{2}({\beta}_{m}{r}_{0})({\alpha}^{2}+{\eta}_{p}^{2})({\beta}_{m}^{2}+{\eta}_{p}^{2})}{J}_{0}({\beta}_{m}r)\mathrm{cos}({\eta}_{p}z)\times \{1-\mathrm{exp}[-\frac{k}{\rho C}({\beta}_{m}^{2}+{\eta}_{p}^{2})t]\}\}\end{array}\\ +\{\sum _{m=1}^{\infty}\frac{{J}_{1}({\beta}_{m}r)[1-\mathrm{exp}(-\alpha l)]}{2\alpha {J}_{0}^{2}({\beta}_{m}{r}_{0}){\beta}_{m}^{4}}{J}_{0}({\beta}_{m}r)[1-\mathrm{exp}(-\frac{k}{\rho C}{\beta}_{m}^{2}t)]\}\\ +\{\sum _{m=1}^{\infty}\frac{\alpha {\omega}_{0}{J}_{1}({\beta}_{m}r)[1-\mathrm{exp}(-\alpha l){(-1)}^{p}]}{2{\eta}_{p}^{2}({\alpha}^{2}+{\eta}_{p}^{2})}\mathrm{cos}({\eta}_{p}z)[1-\mathrm{exp}(-\frac{k}{\rho C}{\beta}_{m}^{2}t)]\}\end{array}\right\}\phantom{\rule{0ex}{0ex}}+\frac{{P}_{0}(1-\tau )[1-\mathrm{exp}(-\alpha l)]}{\pi \alpha \rho C{r}_{0}^{2}l}t+{T}_{\infty}.$$The nonuniform temperature distribution experienced by the output window causes mechanical stresses, which can be obtained from the stress-strain equation and Hooke equation^{20}

## (12)

$${\sigma}_{r}=\frac{\eta E}{1-\nu}(\frac{1}{{r}_{0}^{2}}{\int}_{0}^{{r}_{0}}Tr\mathrm{d}r-\frac{1}{{r}^{2}}{\int}_{0}^{r}Tr\mathrm{d}r),$$## (13)

$${\sigma}_{\phi}=\frac{\eta E}{1-\nu}(\frac{1}{{r}_{0}^{2}}{\int}_{0}^{{r}_{0}}Tr\mathrm{d}r+\frac{1}{{r}^{2}}{\int}_{0}^{r}Tr\mathrm{d}r-T),$$## (14)

$${\sigma}_{z}=\frac{\eta E}{1-\nu}(\frac{2}{{r}_{0}^{2}}{\int}_{0}^{{r}_{0}}Tr\mathrm{d}r-T),$$## (15)

$${\sigma}_{r}=-\frac{4\eta E{P}_{0}(1-\tau )}{{\mathrm{kr}}_{0}^{2}l\pi {\omega}_{0}(1-\nu )}\left\{\begin{array}{l}\{\sum _{m=1}^{\infty}\sum _{p=1}^{\infty}\frac{\alpha {J}_{1}({\beta}_{m}{\omega}_{0})[1-\mathrm{exp}(-\alpha l){(-1)}^{p}]}{{\beta}_{m}^{2}{J}_{0}^{2}({\beta}_{m}{r}_{0})({\alpha}^{2}+{\eta}_{p}^{2})({\beta}_{m}^{2}+{\eta}_{p}^{2})}\frac{{J}_{1}({\beta}_{m}r)}{r}\mathrm{cos}({\eta}_{p}z)\times \{1-\mathrm{exp}[-\frac{k}{\rho C}({\beta}_{m}^{2}+{\eta}_{p}^{2})t]\left\}\right\}\\ +\sum _{m=1}^{\infty}\frac{{J}_{1}({\beta}_{m}r)[1-\mathrm{exp}(-\alpha l)]}{2\alpha {J}_{0}^{2}({\beta}_{m}{r}_{0}){\beta}_{m}^{4}}\frac{{J}_{1}({\beta}_{m}r)}{r}[1-\mathrm{exp}(-\frac{k}{\rho C}{\beta}_{m}^{2}t\left)\right]\end{array}\right\}$$## (16)

$${\sigma}_{\phi}=\frac{4\eta E{P}_{0}(1-\tau )}{{\mathrm{kr}}_{0}^{2}l\pi {\omega}_{0}(1-\nu )}\left\{\begin{array}{l}\left\{\sum _{m=1}^{\infty}\sum _{p=1}^{\infty}\frac{\alpha {J}_{1}({\beta}_{m}{\omega}_{0})[1-\mathrm{exp}(-\alpha l){(-1)}^{p}]}{{\beta}_{m}{J}_{0}^{2}({\beta}_{m}{r}_{0})({\alpha}^{2}+{\eta}_{p}^{2})({\beta}_{m}^{2}+{\eta}_{p}^{2})}\right[\frac{{J}_{1}({\beta}_{m}r)}{{\beta}_{m}r}-{J}_{0}({\beta}_{m}r)]\frac{{J}_{1}({\beta}_{m}r)}{r}\mathrm{cos}({\eta}_{p}z)\times \{1-\mathrm{exp}[-\frac{k}{\rho C}({\beta}_{m}^{2}+{\eta}_{p}^{2})t]\left\}\right\}\\ +\sum _{m=1}^{\infty}\frac{{J}_{1}({\beta}_{m}r)[1-\mathrm{exp}(-\alpha l)]}{2\alpha {J}_{0}^{2}({\beta}_{m}{r}_{0}){\beta}_{m}^{3}}[\frac{{J}_{1}({\beta}_{m}r)}{{\beta}_{m}r}-{J}_{0}({\beta}_{m}r)][1-\mathrm{exp}(-\frac{k}{\rho C}{\beta}_{m}^{2}t\left)\right]\end{array}\right\}$$## (17)

$${\sigma}_{z}=-\frac{4\eta E{P}_{0}(1-\tau )}{{\mathrm{kr}}_{0}^{2}l\pi {\omega}_{0}(1-\nu )}\left\{\begin{array}{l}\{\sum _{m=1}^{\infty}\sum _{p=1}^{\infty}\frac{\alpha {J}_{1}({\beta}_{m}{\omega}_{0})[1-\mathrm{exp}(-\alpha l){(-1)}^{p}]}{{\beta}_{m}^{2}{J}_{0}^{2}({\beta}_{m}{r}_{0})({\alpha}^{2}+{\eta}_{p}^{2})({\beta}_{m}^{2}+{\eta}_{p}^{2})}{J}_{0}({\beta}_{m}r)\mathrm{cos}({\eta}_{p}z)\times \{1-\mathrm{exp}[-\frac{k}{\rho C}({\beta}_{m}^{2}+{\eta}_{p}^{2})t]\left\}\right\}\\ +\sum _{m=1}^{\infty}\frac{{J}_{1}({\beta}_{m}r)[1-\mathrm{exp}(-\alpha l)]}{2\alpha {J}_{0}^{2}({\beta}_{m}{r}_{0}){\beta}_{m}^{4}}{J}_{0}({\beta}_{m}r)[1-\mathrm{exp}(-\frac{k}{\rho C}{\beta}_{m}^{2}t\left)\right]\end{array}\right\}\mathrm{.}$$## 4.

## Numerical Computation and Discussions

In this section, some numerical calculations concerning the thermal–mechanical effects are performed, and the damage mechanism of the laser output window is investigated in the case of taking infrared quartz glass as the output window. Infrared quartz glass is used as the 2-*μ*m laser optical window material owing to its fine performance, whose application wavelength is in the range of 260 to 3500 *μ*m and the infrared transmittance $\tau $ is 85%. Some parameters of the glass in the calculation are listed in Table 1. We assume that the diameter $D$ and the thickness $L$ of the window are, respectively, 25 and 6 mm, the laser beam radius ${\omega}_{0}$ is 2 mm, and the ambient temperature ${T}_{\infty}$ is 20°C.

## Table 1

Physical parameters of infrared quartz glass.

Density, ρ (g/cm3) | 2.2 |

Heat capacity, C (J/g °C) | 0.67 |

Thermal conductivity, k (W/cm °C) | 0.014 |

Absorption coefficient, αv (cm−1) | 1×10−4 |

Poisson’s ratio, v | 0.16 |

Young’s modulus, E (MPa) | 72,000 |

Thermal coefficient of expansion η (°C−1) | 5.5×10−7 |

## 4.1.

### Analysis of Temperature and Stress Fields

Figure 3 illustrates the 3-D temperature distribution of the window in the case of a laser system producing continuous output for 8 s when the laser output power density is $1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{cm}}^{2}$. Figure 4 displays the temperature distributions of the window surface ($L=0$) for different laser-beam radii and different durations when the laser operates at a maximum output power of 115 W.

As can be seen in Fig. 3, in the radius direction, the nearer the sample is to the center of the window, the higher the temperature is, whereas in the axial direction, the temperature does not change significantly. At the edge of the laser beam, the temperature gradient increases sharply, reflecting the thermal expansion of the actively heated area against the passively heated area of the window. Figure 4 shows that the temperature rises higher with longer run times and larger laser-beam radii. It can also be seen that the maximum temperature rise is 289°C when the laser operates at maximum output power.

Figures 5(a), 5(b), and 5(c) depict the stress distributions of the window in the radial, hoop, and axial directions, respectively. Figures 6(a), 6(b), and 6(c) illustrate the situation for different run times and different laser-beam radii, all of which depict how the radial, hoop, and axial stresses evolve at the window surface ($L=0$) when the laser operates at the maximum output power.

From Fig. 5, it can be seen that all stresses are compressive on the centerline of the window, where the maximum heat deposition occurs. Radial stress is always compressive; the hoop and axial stresses, however, turn tensile at the outer rim of the window. Figure 6 illustrates that the stress gradients become greater with longer run times or larger laser-beam radii.

## 4.2.

### Damage Mechanism Analysis

The infrared quartz window may be damaged because of either the temperature reaching the melting point (1780°C) or the tensile (compressive) stress exceeding the tensile (compressive) strength of 48 MPa (1100 MPa). From the analysis mentioned above, it can be seen that the temperature of window would exceed the melting point while the thermal stresses would not reach the limits, providing that the laser power density reaches $1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kW}/{\mathrm{cm}}^{2}$ and the continuous laser output lasts for 8 s. The thermal stress on the quartz material caused by the temperature gradient are relatively small due to the good thermal stability of the quartz; thus the window is primarily damaged by melting in the area where the temperature exceeds the melting point of the material, so that melting, ablation, or even perforation arises as a result.

## 5.

## Laser Output

The wavelength of the 2-*μ*m laser system can switch depending on the transmission of the output window. When the coupler M2 has a transmission of 10%, the wavelength of the output laser is 2.02 *μ*m, and the maximum cw output power is 77.1 W. When the coupler has a transmission of 5%, the wavelength of the laser output is shifted to 2.07 *μ*m, and the maximum cw output power is 115 W. The laser spectra at the maximum output power were recorded by an optical spectrometer (NIRQuest256-2.5), as shown in Fig. 7.

Damage to the output window can be avoided by controlling the output power of the laser system. The dependence of the laser output power on the LD pump power is obtained with a power meter (Ophir F300A-SH), as illustrated in Fig. 8. With a coupler transmission of 5%, the maximum output power is 115 W under an LD pump power of 1188 W, while the maximum power of 77.1 W is achieved with a coupler transmission of 10% under an LD pump power of 1188 W.

## 6.

## Conclusions

In summary, we have defined a thermal model of a laser window and deduced expressions for the transient thermal and mechanical distributions. Taking the infrared quartz glass as an example, the temperature and mechanical field distributions of a high-power all-solid-state 2-*μ*m laser system window were simulated, and the laser-induced damage mechanism was analyzed. The results showed that the window damage mechanism is mainly due to the melting in the areas where the temperature exceeds the melting point of the material, when a 2-*μ*m high-power laser continuously outputs the optical power. We believe that the research methods and results presented in this article will be helpful for the selection of laser windows.

## Acknowledgments

This work was supported by the National Natural Science Foundation under Grant No. 61074158 (China).

## References

## Biography

**Wenwen Liu** received her BS degree in optical engineering from Beihang University, Beijing, China, in 2010. She is currently pursuing her MS degree in optical engineering at Beihang University. Her research is focused on laser technology.