Polarization and tranverse mode nonlinear dynamics in a multimode VCSEL

Abstract

We theoretically analyze the nonlinear dynamics and routes to chaos in a multimode vertical cavity surface-emitting laser (MM-VCSEL) in free-running operation. Including higher order transverse modes (TMs) results in additional bifurcations at higher currents not found for single-mode VCSELs (SM-VCSELs). The resulting dynamics involve competition between modes with different transverse profiles and polarization and show good qualitative agreement with recent experiments.

Figures (4)

• Figure 1: (a) LI-curves for the LP$_{01,x}$ (red), LP$_{01,y}$ (black), LP$_{11,x}$ (green) and LP$_{11,y}$ (blue) modes for $f_{p,0}$ = 9.7 GHz and $\gamma_{a}$ = 1 ns$^{-1}$. Inset: 2D intensity profiles of the modes LP$_{01}$ and LP$_{11}$ (superposition of both orientations). (b) Bifurcation diagram and (c) RF-spectrum of the LP$_{01,y}$ mode. The green dashed line is the ROF. The three points A, B and C in (b) correspond to Fig. \ref{['fig:trace_spectre_quasi_per_chaos']}(A), Fig. \ref{['fig:trace_spectre_quasi_per_chaos']}(B) and Fig. \ref{['fig:polarization_transverse_modes_switching']}, respectively.
• Figure 2: (A.1) Temporal trace and (A.2) RF-spectrum for $\mu = 3.94$ and (B) for $\mu = 6.20$ with $f_{p,0} = 9.7$ GHz and $\gamma_a = 1~\mathrm{ns}^{-1}$. Mode LP$_{01,x}$ (LP$_{01,y}$) is shown in red (black). The birefringence splitting $f_{p,0}$ and its harmonics are indicated in blue.
• Figure 3: (a) Temporal traces of the LP$_{01,x}$, (b) the LP$_{01,y}$ and (c) the LP$_{11,y}$ modes for $\mu = 11$, filtered by a Butterworth low-pass filter of order 4 with cutoff frequency $f_{p,0} = 9.7$ GHz. Other parameters as in Fig. \ref{['fig:diag_bifur_spectre_RF']}.
• Figure 4: Dynamics maps for different values of the injection current $\mu$ and the birefringence splitting $f_{p,0}$. The dynamics of the multimode model are displayed for (a) $\gamma_{a}$ = -1 ns$^{-1}$, (b) $\gamma_{a}$ = 0 ns$^{-1}$ and (c) $\gamma_{a}$ = 1 ns$^{-1}$. (d) The dynamics of the single-mode model is displayed for $\gamma_{a}$ = 1 ns$^{-1}$. The same color code as in Fig. \ref{['fig:diag_bifur_spectre_RF']} is used. The periodic states are indicated in light blue, where the gradient of blue quantifies the value of the dominant frequency. The green dashed lines indicate $f_{RO}/2$, $f_{RO}$ and $3 f_{RO}/2$. The white and magenta curves respectively represent the stability boundaries of the LP$_{01,x}$ and the LP$_{01,y}$ modes as computed from the Hopf bifurcation expressions $\mu_{x,H}$ and $\mu_{y,H}$ detailed in the text.