Optical beam propagation inside a graded-index fiber with saturable nonlinearity

TL;DR

The paper analyzes two-dimensional optical beam propagation in a graded-index (GRIN) fiber with saturable nonlinearity. Using a variational approach and a Gaussian ansatz, it derives an amplitude–width relation that reveals bistable spatial solitons whose stability is enabled by saturation, and it confirms these findings with full numerical solutions. A linear stability analysis shows neutral stability (purely imaginary eigenvalues) on the bistable branches, while a more general variational treatment yields spatial similaritons with periodic width–amplitude oscillations whose period increases with saturation. The study provides insight into stable self-guiding in GRIN fibers and characterizes the dynamic regimes (solitons vs. similaritons) as a function of the saturation parameter, with potential applications in beam shaping and nonlinear fiber optics.

Abstract

We study theoretically the spatial evolution of optical beams inside a graded-index fiber exhibiting saturable nonlinearity. Utilizing an approach based on the variational principle, we identify the existence of bistable spatial solitons inside such a nonlinear medium, whose stability, analyzed through a linear stability analysis, is due to the saturating nature of the nonlinearity. Spatial solitons adhere to a specific amplitude-width relationship. Any deviation from this relationship leads to oscillating-type solutions with a period that increases with saturation level of the nonlinearity. Theoretically calculated values of this period agree well with numerical findings.

Figures (5)

• Figure 1: ($a$) Schematic of a GRIN fiber with ($b$) a parabolic index profile and ($c$) a saturable nonlinearity.
• Figure 2: ($a$) Comparison of analytical (solid lines) and numerical (dashed lines) curves showing the $\mathcal{A}$-$\mathcal{R}$ relationship for three values of $s$. ($b$)-($d$) Soliton-like propagation of in three cases indicated by circles. ($e$)-($g$) Amplitude variations in the same three cases.
• Figure 3: Stability features of bistable solitons for four different combinations of $\mathcal{A}$ and $s$. Top two rows correspond to the upper bistable branch, and the bottom two rows to the lower bistable branch. In each case, the first column shows the real and imaginary parts of the eigenvalues and the second column shows the evolution of a perturbed soliton. The third and fourth columns show spatial profiles of the soliton at the input and output ends of the GRIN fiber.
• Figure 4: Bistable $\mathcal{A}$-$\mathcal{R}$ curve obtained analytically (solid line) and numerically (dotted line) using $s=0.8$. Points and mark two amplitudes allowed for a specific width of the soliton. Two insets shows the potential that governs beam-width oscillations of similaritons at points and . ($b$)-($e$) Evolution of Gaussian-beams at these four points. ($f$)-($k$) Periodic variations of beam's amplitudes and widths in the four cases. Solid lines correspond to variational results and dotted lines show numerical data.
• Figure 5: Changes in $Z_p$ as a function of $s$ (solid line) predicted by the variational result in Eq.\ref{['Eq18']}. The corresponding numerical results are shown by a dotted line for $\mathcal{A}_0=8$ and $\mathcal{R}_0=1$. ($b$)-($d$) Evolution of a Gaussian beam corresponding to points-. Periodic changes in the beam's width ($\mathcal{R}$) and amplitude ($\mathcal{A}$) are also shown as a function of $\xi$. In all cases, the solid and dotted lines correspond to numerical and variational results, respectively.