# Online Measurement of Internal Parameters in a Quasi-CW Diode-Pumped Nd:YAG Laser
## 2 Materials and Methods  
The measurement cell illustrated in Figure 1a consists of a linear polarizer (PL1) and a linear polarization analyzer (PL2), in which PL2 rotates about the laser propagation axis and thus polarization reflective loss \(E_{MO}\) is generated. The linear polarization direction of PL1 is parallel to the X axis as shown in Figure 1a and the rotating angle \(\theta\) of the analyzer PL2 is defined as an angle between the linear polarization directions of PL2 and PL1. Therefore, PL2 has the vertical axis (the X axis) at the rotating angle of 0 degrees and the horizontal axis (the Y axis) at 90 degrees. The polarization reflective loss \(E_{MO}\) varies with the rotating angle \(\theta\) of the analyzer PL2 while the measurement cell is placed in a laser resonator. As a result, the measurement cell actually serves as another variable output coupler in a laser system. According to the Malus Law, an equivalent transmittance \(T(\theta)\) of the measurement cell after a laser beam makes a round trip is  
\[T(\theta)=\sin^{2}(\theta)\Big{[}1+\cos^{2}(\theta)\Big{]}. \tag{1}\]  
The equivalent transmittance \(T(\theta)\) of the measurement cell as a function of the rotating angle \(\theta\) of the PL2 analyzer is shown in Figure 1b. The curve of \(T(\theta)\) is mirror symmetrical in four quadrants, and a wide range of equivalent transmittance from 0 to 100% is provided within the limits of the rotating angle of 90 degrees in a quadrant. So, we can conveniently choose any one of the four quadrants to complete the measurements of the internal parameters in a laser. For a four-level solid-state laser employing the Nd:YAG crystal active medium shown in Figure 2, the power density saturation is predominantly due to a homogeneous broadening mechanism [13]. The homogeneous broadening mechanisms of Nd:YAG have a frequency dependence on the Lorentzian lineshape for a atomic response. So, a saturated gain [19] is reduced according to  
\[g_{H}(\
u,I)=g_{H}^{0}(\
u_{0})\,\frac{\left(\Delta\
u_{H}/2\right)^{2}}{\
u- \
u_{0})^{2}+\,\Delta\
u_{H}/2)^{2}(1+I/I_{\
u_{0}})}, \tag{2}\]  
where \(g_{H}(\
u,I)\) is the saturated gain coefficient depending on the frequency \(\
u\) and the power density \(I\), and \(g_{H}^{0}(\
u_{0})\) is the unsaturated gain coefficient at the center frequency of the atomic lineshape. \(\Delta\
u_{H}\) is the full width at half-maximum of the unsaturated Lorentzian, and \(I_{\
u_{0}}\) is the saturation parameter. With some algebra, the frequency term was incorporated into the gain and saturation parameters. Equation (2) can be rewritten as  
\[g_{H}(\
u,I)=\frac{g_{H}^{0}(\
u)}{1+I/I_{\
u}}, \tag{3}\]  
where \(g_{H}^{0}(\
u)\) is the unsaturated gain coefficient at the frequency of \(\
u\), given by  
\[g_{H}^{0}(\
u)=g_{H}^{0}(\
u_{0})\Bigg{[}\frac{\left[\frac{\Delta\
u_{H}}{2} \right]^{2}}{\left[\
u-\
u_{0}\right]^{2}+\left[\frac{\Delta\
u_{H}}{2} \right]^{2}}\Bigg{]}, \tag{4}\]  
Figure 1: Schematic of the measurement cell and the equivalent transmittance \(T(\theta)\) curve. (**a**) Schematic of the measurement cell. (**b**) The equivalent transmittance \(T(\theta)\) of the measurement cell as a function of the rotating angle \(\theta\). Figure 2: Schematic of measurement by introducing polarization reflective loss. where the expression in the square bracket is the Lorentzian frequency factor. The parameter \(I_{Sv}\) defines a saturation power density in the active material at which the unsaturated gain coefficient \(g_{H}^{0}(\
u)\) is reduced by one-half [13], given by  
\[I_{sv}=I_{sv_{0}}/\left[\frac{\left[\frac{\Delta\
u_{0}}{2}\right]^{2}}{\left[ \
u-\
u_{0}\right]^{2}+\left[\frac{\Delta\
u_{0}}{2}\right]^{2}}\right]. \tag{5}\]  
From Equation (3), we see that the transition will saturate uniformly under the influence of a sufficiently strong signal applied anywhere within the atomic linewidth of the Lorentzian line shape. Figure 2 illustrates a laser power circulating in a laser oscillator.