# Online Measurement of Internal Parameters in a Quasi-CW Diode-Pumped Nd:YAG Laser
## 2 Materials and Methods
From Equation (7), one has  
\[t_{2}=T(\theta)\big{|}_{r_{1}=r_{2}}+\delta_{i}=T(\theta_{r})+\delta_{i}. \tag{14}\]  
The behavior of variable polarization reflective loss generated by the rotation of the analyzer actually functions as another variable cavity mirror. A rotating angle of \(\theta_{r}\), which meets \(r_{1}(\theta)=r_{2}\), is obtained from  
\[E_{O}\big{|}_{\theta_{r}}=E_{MO}\big{|}_{\theta_{r}}. \tag{15}\]  
The transmittance \(t_{2}\) of the coupling-cavity mirror M2 in a laser system is measured by Equations (14) and (15). The threshold condition of a laser oscillator with a measurement cell is written as  
\[2g_{0}l=\Gamma(\theta_{m})+\delta_{T}, \tag{16}\]  
where \(\Gamma(\theta_{m})\) is the losses produced by the measurement cell to make the laser just cease to oscillate and \(\theta_{m}\) is the corresponding rotating angle. \(\Gamma(\theta_{m})\) is determined by  
\[\Gamma\,\theta_{m}]=-ln[1-T\,\theta_{m}]-\delta_{i}]. \tag{17}\]  
The internal optical loss of \(\delta_{T}\) is the sum of the coupling-out loss of mirror M2 and the round-trip net optical loss \(\delta\) of a Nd:YAG crystal, written as  
\[\delta_{T}=-ln(r_{2})+\delta. \tag{18}\]With the help of Equation (16), the small-signal gain coefficient of \(g_{0}\) is measured. Applying the measured parameters of \(\delta\), \(g_{0}\), and \(t_{2}\) to the expression of \(E_{O}\) in Equation (10), the saturation fluence parameter of \(I_{S}\) is further obtained and the saturation parameter of \(I_{S}\) is determined by Equation (11). By virtue of the obtained results of \(g_{0}\), \(\delta\), and \(I_{S}\), the single-pulse output energy \(E_{Out}\) of the Nd:YAG laser product without the measurement cell is described as  
\[E_{Out}=\frac{J_{S}t_{2}}{\delta+t_{2}}\Big{[}g_{0}l+ln\sqrt{r_{2}}\Big{]}. \tag{19}\]  
With the aid of Equation (19), we can optimize the output coupler and maintain a laser product. The details of the online measurement process are described as follows. The first step is to measure the transmittance \(t_{2}\) of cavity mirror M2. Rotate the PL2 of the measurement cell to find the angle of \(\theta_{r}\) at which the condition of Equation (15) is satisfied and then record the values of \(\theta_{r}\). The transmittance \(t_{2}\) of the output mirror is measured with the help of Equations (1) and (14). The second step is the measurement of the round-trip net optical loss \(\delta\) and the small-signal gain coefficient of \(g_{0}\). Rotate PL2 to make the laser just cease to oscillate and then record \(\theta_{m}\). Next, rotate PL2 to the direction of diminishing \(\theta\) and find the angles of \(\theta_{st}\) at which the outputs of \(E_{O}\) and \(E_{MO}\) are both large enough and stable enough to reduce the measurement error and then record the values of \(E_{O}\) and \(E_{MO}\). The round-trip net optical loss \(\delta\) is determined by Equations (1) and (13). Further, obtain the absorption loss coefficient \(\alpha\) of the Nd:YAG crystal by Equation (8). Then, apply the parameters determined of \(\delta\), \(\theta_{m}\), and \(t_{2}\) to Equation (16) and measure the small-signal gain coefficient of \(g_{0}\). The final step is the measurement of saturation parameter \(I_{S}\). Apply the measurement results of \(\delta\), \(g_{0}\), and \(t_{2}\) to the expression of \(E_{O}\) in Equation (10), then calculate the saturation fluence parameter of \(I_{S}\) by the recorded values of \(E_{O}\). Finally, determine the saturation parameter of \(I_{S}\) by Equation (11).