Figure 1

Schematic diagram of the cross-coupled laser experiment with delayed negative optoelectronic feedback: ${\mathrm{L}\mathrm{D}}_1$ and ${\mathrm{L}\mathrm{D}}_2$, laser diodes; ${\mathrm{P}\mathrm{D}}_1$ and ${\mathrm{P}\mathrm{D}}_2$, photodiodes; $L_1$ and $L_2$, optical fibers; OSC, oscilloscope; $V_1$ and $V_2$, photodiode output voltages; $A_1$ and $A_2$, electronic amplifiers; ${\mathrm{A}\mathrm{t}\mathrm{t}\mathrm{n}}_1$ and ${\mathrm{A}\mathrm{t}\mathrm{t}\mathrm{n}}_2$, variable electronic attenuators. The thin lines indicate the optical signal path through the fiber and the thick lines indicate the electronic signal path through transmission cables.Reuse & Permissions

Figure 2

Experimental observations and analysis. (a) Plot of intensity oscillation amplitude measured by ${\mathrm{P}\mathrm{D}}_2$, ${\mathrm{V}}_2$, versus coupling strength $d_2$ for different coupling strengths $d_1$. Plus signs show the amplitude for $d_1=0.71$, squares for $d_1=0.80$, circles for $d_1=0.89$, triangles for $d_1=1.00$, diamonds for $d_1=1.12$, crosses for $d_1=1.26$, and stars for $d_1=1.42$. (b) Plot of ${log}_{10}(\mathrm{\text{coupling strength}}{d}_2)$ vs ${log}_{10}(\mathrm{\text{coupling strength}}{d}_1)$ at the emergence of oscillations. The line shows the best-fit linear model. (c) Rescaled intensity amplitudes, $V_2/\sqrt{d_1}$, versus the product of coupling strengths ($d_1{d}_2$), showing data collapse. (d) Plot of the ratio of the rescaled amplitudes $(V_1/\sqrt{d_2})/(V_2/\sqrt{d_1})$ vs ($d_1{d}_2$).Reuse & Permissions

Figure 3

Numerical computations. (a) Plot of amplitude of the intensity fluctuations normalized to steady state level $y_2$ versus coupling constant ${\delta }_2$ for different coupling constants ${\delta }_1$. Plus signs show the amplitude for ${\delta }_1=1.90$, squares for ${\delta }_1=1.92$, circles for ${\delta }_1=1.94$, triangles for ${\delta }_1=1.96$, diamonds for ${\delta }_1=1.98$, crosses for ${\delta }_1=2.00$, stars for ${\delta }_1=2.02$, asterisks for ${\delta }_1=2.04$, and points for ${\delta }_1=2.06$. (b) Plot of ${log}_{10}({\delta }_2)$ versus ${log}_{10}({\delta }_1)$ at the emergence of oscillations. The line shows the best-fit linear model. (c) Plot of rescaled variable $y_2/\sqrt{{\delta }_1}$ versus the product of the coupling constants ${\delta }_1{\delta }_2$, showing the data collapse. The values of the parameters used are $a=2$, $b=1$, $ϵ=\sqrt{0.001}$, and $\tau =30$ in dimensionless units. (d) The same as (c) with $\tau =150$. The same marker types are used for same ${\delta }_1$ values in (a), (c), and (d).Reuse & Permissions