# Online Measurement of Internal Parameters in a Quasi-CW Diode-Pumped Nd:YAG Laser
## 2 Materials and Methods
The mirror M1 is a total reflector with a high reflectivity and the mirror M2 is an output coupler with a partial transmission in a laser oscillator. We denote the power densities of \(I_{+,-}(Z)\) in the \(+Z\) and \(-Z\) directions of laser propagation by the wave traveling to the right from \(I_{+}(0)\) to \(I_{+}(l)\) and the wave traveling to the left from \(I_{-}(l)\) to \(I_{-}(0)\), respectively. The two-side laser output losses of the Nd:YAG active medium have the \(E_{MO}\) from the measurement cell and \(E_{O}\) from mirror M2. In Figure 2, the equations governing the power density of the wave traveling to the right \(I_{+}(Z)\) and the power density of the wave traveling to the left \(I_{-}(Z)\) in the Nd:YAG active medium [19] are written together as  
\[\frac{dI_{+,-}(Z)}{d(Z)}=\frac{\pm g_{H}^{0}(\
u)I_{+,-}(Z)}{1+\left[I_{+}(Z)+ I_{-}(Z)\right]/I_{sv}}\mp\alpha I_{+,-}(Z), \tag{6}\]  
where \(\alpha\) is the absorption loss coefficient of the Nd:YAG crystal. The gain equations of Equation (6) are transcendental equations because of the absorption loss coefficient \(\alpha\) in the gain medium. The solution to Equation (6) is obtained by [PERSON]'s approach that the losses of the resonator are assumed to be concentrated near the mirrors [20]. The round-trip net optical loss of the Nd:YAG crystal is assumed to be uniformly distributed at mirrors M1 and M2. The location of the measurement cell is between mirror M1 and the Nd:YAG crystal, and therefore the polarization reflective transmittance loss and round-trip insertion loss of the measurement cell, as well as the round-trip net optical loss distributed at mirror M1, are concentrated as an effective reflectance loss \(r_{1}(\theta)\) of mirror M1 at location Z of 0. An effective reflectance loss \(r_{2}\) of mirror M2 at location Z of \(l\) includes the transmittance of mirror M2 and the round-trip net optical loss distributed at mirror M2. The effective reflectance losses of \(r_{1}(\theta)\) and \(r_{2}\) in the two sides of the Nd:YAG crystal at locations Z of 0 and \(l\), respectively, are expressed as  
\[\left\{\begin{array}{ll}r_{1}[\theta=1-\{T[\theta+\delta_{i}+\delta/2&\quad \text{ \Using [PERSON]'s approach [20], the single-pulse output energy \(E_{o}\) from the mirror M2 and the output loss \(E_{Mo}\) from the measurement cell in a quasi-CW diode-pumped Nd:YAG laser are solved in the form  
\[\left\{ \begin{array}{l}E_{O}=\frac{J_{S}t_{2}\sqrt{r_{1}(\theta)}}{\big{[} \sqrt{r_{1}(\theta)+\sqrt{r_{2}}}\big{]}[1-\sqrt{r_{1}(\theta)r_{2}}]}\Big{[}g _{0}l+ln\sqrt{r_{1}(\theta)r_{2}}\Big{]}\\ E_{MO}=\frac{J_{S}[T(\theta+\delta_{1}\sqrt{r_{2}}}{\big{[}\sqrt{r_{1}(\theta)+ \sqrt{r_{2}}}\big{]}[1-\sqrt{r_{1}(\theta)r_{2}}]}\Big{[}g_{0}l+ln\sqrt{r_{1}( \theta)r_{2}}\Big{]}\end{array} \right., \tag{10}\]  
where \(g_{0}\) is the integrated small-signal gain coefficient in multimode oscillators and the saturation fluence parameter of \(J_{S}\) is given by  
\[J_{S}=\tau_{f}AI_{S}, \tag{11}\]  
where \(\tau_{f}\) is the fluorescence lifetime, \(A\) is the beam cross-section in the Nd:YAG crystal, and \(I_{S}\) is the integrated saturation power density parameter in multimode oscillators. From Equation (10), the output ratio is derived as  
\[\frac{E_{MO}}{E_{O}}=\frac{[T[\theta+\delta_{i}]\sqrt{r_{2}}}{t_{2}\sqrt{r_{1} (\theta)}}. \tag{12}\]  
Applying Equation (7) into Equation (12), the round-trip net optical loss \(\delta\) is derived as  
\[\delta=2[y^{2}1-T\theta)-\delta_{i})-1-t_{2})]/(y^{2}-1), \tag{13}\]  
where \(y=E_{MO}t_{2}/[E_{O}T\theta]+\delta_{i}]]\). If we obtain the transmittance \(t_{2}\) of mirror M2, the round-trip net optical loss \(\delta\) of the laser medium is determined by Equation (13). The absorption loss coefficient \(\alpha\) of the Nd:YAG crystal is further obtained by Equation (8). The two sides of the Nd:YAG crystal have the same loss in a symmetrical resonator defined by \(r_{1}(\theta)=r_{2}\).