Fig. 1. Schematic of the proposed PT-symmetric fiber-ring laser, composed of two coupled fiber-ring cavities with gain and losses. The coupling between cavities is controlled by means of phase shifts φ1 and φ2. Arrows indicate the direction of propagation.

Fig. 2. (a) Ratio of two linear mode eigenvalues, |μ+/μ−| and (b) the relative phase of the eigenmode amplitudes arg(u1/u2)μ+=Re(ν+) versus the difference of phases and gain/losses in two fiber cavities. White dotted lines indicate the PT-breaking boundary. (c), (d) The absolute eigenvalues shown with solid (|μ+|) and dashed (|μ−|) lines versus the gain coefficient for fixed losses (g2=−0.7) and different phases (c) Δφ=1.5 and (d) Δφ=1. Horizontal dotted line marks the level |μ|=1 corresponding to stationary modes with balanced gain and losses.

Fig. 3. (a) Stationary regimes of laser operation with nonlinear gain saturation: no lasing (white background), pair of PT-symmetric laser modes (grey shading), or one mode in PT-broken regime (yellow shading). (b), (c) Characteristic mode amplification versus power for points A and B marked in (a) corresponding to different lasing regimes. Solid circles mark stable and the open circle marks unstable regimes with balanced gain and loss (zero mode amplification). Background shading marks PT-symmetric and broken regimes. Saturable gain parameter gh=0.23 (1 dB).

Fig. 4. Power dependencies in stationary lasing regimes. (a) A ratio of power generated in passive and active cavities, P2/P1, is unity in PT-symmetrical region and less than unity in PT-broken area. (b), (c) Dependence of the lasing power in two cavities on the gain in PT-symmetric and PT-broken regimes corresponding to different phase shift Δφ=1.5 and 0.8, respectively, both shown with dashed lines in plot (a).

Fig. 5. (a) Dependence of generated power P1 on phase shift Δφ and loss g2 at fixed gain g1=1.0 has non-trivial form resulting from PT transition. In the PT-symmetric area, the higher are the losses |g2|, the lower is the lasing power as it should be in a conventional laser, whereas in the case of PT-broken regime, the generation power increases with the increase of losses. Panels (b)–(d) are cross sections of a 3D surface indicated on panel (a) over dotted lines.

Fig. 6. Dynamical properties of a PT-symmetric fiber laser. (a) Two trapping regions shown with shading according to Eq. (18), shown in the plane of relative phases and powers in two fiber cavities. Laser dynamics is confined to one region according to the initial conditions. Solid and dashed lines indicate possible stationary lasing states: PT-symmetric—vertical lines at P2/P1=1, and PT-broken—horizontal lines at relative phases {0,±π}. (b), (c) Dynamical evolution demonstrating bi-stability on the PT-symmetric regime. Shown are relative (b) phases and (c) powers, which converge to one of two stationary states marked with solid circles in (a). Parameters are g1=2.3, g2=−0.7, and Δφ=2.95.