Mode-Locking Polarization State Dynamics of an Erbium-Doped Fiber Laser Based on 3D Parametric Mapping

For the first time, we report the 3D parametric region of mode-locking (ML) generation based on complete angular space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\left\langle \varphi _{1}, \varphi _{2}, \varphi _{3} \right\rangle$</tex-math></inline-formula> of a motorized polarization controller (PC) inside an erbium-doped fiber laser cavity. This parametric region revealed the predominant role of the bound state soliton as the key to ML self-starting, with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varphi _{3} = 45^{\circ }$</tex-math></inline-formula> plane exhibiting the best stability and reproducibility of the pulsed regime. The contour maps of the mean power showed a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$1.5~{\mathrm{dB}}$</tex-math></inline-formula> increase in unpolarized light (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P_{U}$</tex-math></inline-formula>) relative to the simultaneous increase in polarized light (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P_{Pol}$</tex-math></inline-formula>), with a ratio of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P_{Pol}/P_{U} = 14.1 \pm 0.7$</tex-math></inline-formula> in the ML regimes. These regimes represented 0.6% and 2.0% of the possible output polarization states achievable with a PC, corresponding to the complete angular space and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varphi _{3}$</tex-math></inline-formula>, respectively. This enabled the quantification of the effective ML window of the hybrid saturable absorption system, NPR and monolayer graphene, in terms of bounded Stokes parameters, translating to constraints for the polarization ellipse within the ranges: <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\psi = (147.8 \pm 1.6)^{\circ }$</tex-math></inline-formula>, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\chi = (-11.1 \pm 3.1)^{\circ }$</tex-math></inline-formula>, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\delta = (205.5 \pm 7.3)^{\circ }$</tex-math></inline-formula> for azimuth, ellipticity, and phase shift, respectively. These restrictions imposed on the resulting vector solitons defined the 'Stokes fingerprint' of the laser cavity, characterized by ratios between the Stokes parameters such as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$|s_{2}| \approx 2|s_{1}|$</tex-math></inline-formula>, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$|s_{1}| \approx |s_{3}|$</tex-math></inline-formula>, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$s_{3} &lt; 0$</tex-math></inline-formula>, with the latter indicating left-handed polarization for all the ML regimes obtained, resulting in a proportion <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$E_{0x}/E_{0y} \approx 1.5$</tex-math></inline-formula> between the electric field components. This motorized parametric characterization of the cavity facilitated both the automatic generation of ML and the explanation of anomalous cavity behaviors under conditions of pulsed regime loss, attributed to induced chirality in regions of maximum <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$P_{U}$</tex-math></inline-formula> emission. Additionally, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\psi$</tex-math></inline-formula> was proposed as a key parameter in recovering the pulsed regime after external perturbation. Finally, it was found that the Stokes parameters <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$s_{2}$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$s_{3}$</tex-math></inline-formula> are strongly correlated, which proved useful for dimensionality reduction of the described system, achieving an accumulated variance of 82% and 93% for three and four principal components, respectively.