## 2 Experimental Setup  
Figure 1 demonstrates two types of experimental setups for generating the 559 nm and 589 nm beams. One type shown in Figure 1a illustrated that we used a size of 3 \(\times\) 3 \(\times\) 20 mm\({}^{3}\) a-cut 0.3-at.% Nd:YVO\({}_{4}\) crystal as the gain medium and the Raman active medium at the same time. The facet of the Nd:YVO\({}_{4}\) crystal toward the pump was coated high reflectance (R \(>\) 99.9%) at 1030-1200 nm and high transmission at 808 nm (T \(>\)95%). The second type was that an a-cut 0.35-at.% Nd:YVO\({}_{4}\) crystal with the dimension of 3 \(\times\) 3 \(\times\) 10 mm\({}^{3}\) was utilized to generate the fundamental wave. Moreover, a size of 3 \(\times\) 3 \(\times\) 20 mm\({}^{3}\)\(N_{\rm p}\)-cut KGW was placed behind Nd:YVO\({}_{4}\) crystal to generate the first Stokes field as shown in Figure 1b. Two LBO crystals, with different cutting angle \(\theta\) = 90\({}^{\circ}\) and \(\varphi\) = 3.9\({}^{\circ}\); \(\theta\) = 90\({}^{\circ}\) and \(\varphi\) = 8\({}^{\circ}\), were exploited to achieve the optimal phase-matching in nonlinear process of SHG for the first Stokes field and SFG for the first Stokes field and the fundamental wave, respectively. The dimensions for both LBO crystals were 3 \(\times\) 3 \(\times\) 8 mm\({}^{3}\) and the surface coatings for both were an antireflection coating at 1064 nm, 1176 nm, and 530-600 nm (reflectance \(<\) 0.2%). All laser crystals were wrapped with an indium foil and housed at an upper copper heat sink. The upper copper heat sink was mounted at the lower copper heat sink which had the water cooling at the temperature of 20 \({}^{\circ}\)C. For the LBO crystals, the thermoelectric cooler (TEC) was placed between the upper and lower copper heat sink. Therefore, we can precisely control the temperature of LBO crystals to obtain the optimal phase-matching. The output coupler was a concave mirror with 75 mm radius of curvature. Both facets of the output coupler were coated at high reflectance at 1050-1180 nm (R \(>\) 99%) and high transmission at 540-600 nm (T \(>\) 98%), which decreased the loss of the Stokes field and the lasing threshold. We utilized a 40 W fiber-coupled laser diode (BWT K808 DNINR-40.00W) as a pump source, which was made up of a core diameter 200 \(\upmu\)m, a numerical aperture 0.22, and emitting central wavelength at 808 nm. The laser diode was operated at quasi-CW mode with the repetition rate 50 Hz and the duty cycle 50%. A telescope arrangement was exploited to collimate and focus the pump beam into the Nd:YVO\({}_{4}\) with a diameter of approximately 500 \(\upmu\)m. The entire resonator length of self-Raman Nd:YVO\({}_{4}\) laser and Nd:YVO\({}_{4}\)/KGW Raman laser were 60 mm and 65 mm, respectively.  
The designed coating at the facet of KGW toward the Nd:YVO\({}_{4}\) was highly reflective at 1150-1180 nm (R \(>\) 98%) and highly transmissive at 1064 nm (T \(>\) 95%) as shown in Figure 2a. By applying this coating on the surface of KGW crystal, the resonators of Stokes field and that of fundamental wave were totally separated. In order to prevent the visible lights to propagate into the gain medium or Raman medium which caused the reabsorption and decreased the output powers, the highly reflective at 530-590 nm (R \(>\) 99.5%) was coated on the other facet of 20 mm Nd:YVO\({}_{4}\) crystal and KGW crystal as shown in Figure 2b.