Analyzing the Intracavity Frequency Doubling of the Simultaneous Q-Switched Mode-Locked Diode-Pumped Pulsed $Nd^{3+}$ Laser with $Cr^{4+}$:YAG as a Polarized Saturable Absorber

The intracavity frequency doubling of a simultaneous passively Q-switched mode-locked diode-pumped $Nd^{3+}$ laser is studied with a polarized isotropic $Cr^{4+}$:YAG saturable absorber. A general recurrence formula for the mode-locked pulses under the Q-switched envelope at a fundamental wavelength has been reconstructed in order to analyze the temporal shape behaviour of a single Q-switched envelope with mode-locking pulse trains. This formula has been derived taking into account the impact of the intracavity frequency doubling and polarized $Cr^{4+}:YAG$ saturable absorber (the Fresnel losses). The presented mathematical model essentially describes the self-induced anisotropy appearing in the polarized $Cr^{4+}:YAG$ in the nonlinear stage of the giant pulse formation. For the anisotropic $Nd^{3+}:YVO_4$ active medium, the generated polarized waves are fixed through the lasing cycle. Second harmonic peak power, pulse width, pulse duration, shift pulse position of central mode and rotational angle as a function of the absorber initial transmission are estimated. The calculated numerical results are in good qualitative and quantitative agreement with the available experimental data reported in references.