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Figure 1 shows a schematic diagram of the proposed polarimetric PT-symmetric photonic system consisting of two polarimetric loops. The two polarimetric loops, with independently adjustable eigenfrequencies and round-trip gain coefficients, are implemented based on polarimetric diversity in a single physical loop. Specifically, the birefringent path in the fiber ring laser loop creates two polarimetric loops, and the coupled path allows coupling between the two polarimetric loops. Two polarization controllers (PCs), in conjunction with two polarizers (Pol. 1 and Pol. 2), are used to tune the gain and loss coefficients of the eigenvalues in the two polarimetric loops to achieve PT symmetry. As the gain and loss coefficients are increased to values greater than the coupling coefficient, PT symmetry breaking occurs. The two polarimetric loops form a PT-symmetric fiber ring laser with one polarimetric loop supporting the gain modes and the other supporting the loss modes. The polarimetric PT symmetry then strongly enhances the gain difference between the dominant mode and the sidemodes, making single-mode lasing possible without using an ultranarrow bandpass filter.
Fig. 1 Schematic diagram of the proposed polarimetric PT-symmetric photonic system.
The system consists of a single spatial loop, in which two equivalent polarimetric loops are formed by recirculating light waves of orthogonal polarization states in the loop. To achieve PT symmetry, the phase retardance, power ratio, and coupling coefficient between the orthogonally polarized light waves are tuned by controlling PC1 in the birefringent path, and the lasing threshold is tuned by controlling PC2 in the coupled path. PC: polarization controller; Pol.: polarizer; EDFA: erbium-doped fiber amplifier; OC: optical coupler; TOF: tunable optical filterAs shown in Fig. 1, the photonic system has a single unidirectional physical fiber loop that supports two polarimetric loops. An erbium-doped fiber amplifier (EDFA) is incorporated to provide an optical gain. The polarimetric diversity is implemented by controlling the polarization states of light in the fiber loop. The dominant mode can be coarsely chosen with a tunable optical filter (TOF). The output of the system is derived using a 3-dB optical coupler (OC), which is sent to an optical spectrum analyzer (OSA) for spectrum analysis, an optical homodyne system for mode analysis, and an optical self-heterodyne system for linewidth measurement.
A three-paddle fiber-optic PC has a sandwiched structure with a half-wave plate located between two quarter-wave plates. The transfer function of a three-paddle PC is given by28
$$ F_{PC} = \left[ {\begin{array}{*{20}{c}} {\exp \left( {i\varphi _2} \right)} & 0 \\ 0 & 1 \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} {\cos \theta } & { - \sin \theta } \\ {\sin \theta } & {\cos \theta } \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} {\exp \left( {i\varphi _1} \right)} & 0 \\ 0 & 1 \end{array}} \right] $$ (1) where φ1 and φ2 are the phase retardances introduced to the two orthogonally polarized light waves by the two quarter-wave plates and θ is the rotation angle of the polarization direction introduced by the half-wave plate. As can be seen, a PC can introduce independent polarization direction rotation and polarization phase retardance to an incident light. In our system, polarimetric PT symmetry is achieved by controlling three critical angles in PC1 and PC2, which are listed in Table 1.
Angle Definition Functionality Tuning method θr Rotation angle of quarter-wave plate to tune polarization phase retardance PT symmetry of the real part of the eigenfrequency of the polarimetric loops Quarter-wave plate of PC1 θi Optical axis rotation angle from Pol. 1 to Pol. 2 PT symmetry of the imaginary part of the eigenfrequency of the polarimetric loops Half-wave plate of PC1 θt Optical axis rotation angle from Pol. 2 to Pol. 1 Lasing threshold Half-wave plate of PC2 Table 1. The critical angles that are tuned to achieve polarimetric PT symmetry
Specifically, the bending-induced birefringence of PC1 leads to a polarization phase retardance. Assuming that the fast and slow polarization components are Ex and Ey, respectively, the phase retardance can be tuned by changing θr using the equivalent quarter-wave plates in PC1, which aligns the eigenmodes of the two polarimetric loops. The m-th order eigenmode of a loop resonator is given by
$$ \omega _m = \frac{{\left( {2m\pi + \varphi } \right)c}}{{n_{eff}L}} $$ (2) where φ is the phase shift within the laser ring cavity, c is the light velocity in a vacuum, and neff and L are the effective refractive index and the length of the optical fiber within the laser ring cavity, respectively. Although the physical lengths of the two polarimetric loops are the same, the eigenfrequencies of the cavities with the respective polarizations may be perturbed due to the residual birefringence originating from the fiber bending and the polarization mode dispersion (PMD) of the optical components, resulting in a mismatch of the eigenmodes between the two polarimetric cavities, as shown in Fig. 2a. Assuming that the birefringent phases of the two polarizations are φx and φy, the separation between the localized eigenfrequencies is given by
Fig. 2 Polarimetric PT symmetry achieved by controlling the polarizations by tuning PC1 and PC2 in the photonic system.
a Compensation for eigenfrequency separation between the polarimetric loops when the phase retardances between Ex and Ey are tuned by the quarter-wave plate of PC1; b round-trip gain and loss coefficient variations of the polarimetric loops when the polarization directions of Ex and Ey are tuned by the half-waveplate of PC1; and c illustration of the gain, loss and coupling of the polarization components as light at PC1 propagates through Pol. 1, PC2, and Pol. 2 and then returns to PC1. Part of the polarization component Ex is coupled to the other polarization component Ey, and vice versa$$ \delta \omega _m = \frac{{( {\varphi _x - \varphi _y})c}}{{n_{eff}L}} $$ (3) To achieve PT symmetry, it is required that the eigenfrequency separation is zero, which can be achieved by tuning the equivalent quarter-wave plate in PC1, i.e., θr, to match the values of φx and φy, thus aligning the eigenfrequencies of the polarimetric cavities for the implementation of PT symmetry between the real parts of the eigenfrequencies29.
On the other hand, the tuning of θi using the half-wave plate in PC1 can result in a rotation of the optical axis in the optical path from Pol. 1 to Pol. 2, which, in conjunction with the polarizers, adjusts the round-trip gain, loss, and coupling coefficients of the two polarimetric loops. PT symmetry between the imaginary parts of the eigenfrequencies of the two polarimetric loops can be achieved. We first assume that the polarization direction of the light is preserved from Pol. 2 to Pol. 1, i.e., θt = 0, and that the directions of Pol. 1 and Pol. 2 are aligned such that they are the same. The joint operation between Pol. 1 and Pol. 2 is then equivalent to a single polarizer. The relations between the electric fields at PC1 before and after one round trip in the cavity, denoted as E(0) and E(1), are given by
$$ \left[ {\begin{array}{*{20}{c}} {E_x^{\left( 1 \right)}} \\ {E_y^{\left( 1 \right)}} \end{array}} \right] = \gamma _0\left[ {\begin{array}{*{20}{c}} {{\cos}^2\theta _i} & {\sin \theta _i\cos \theta _i} \\ {\sin \theta _i\cos \theta _i} & {{\sin}^2\theta _i} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {E_x^{\left( 0 \right)}} \\ {E_y^{\left( 0 \right)}} \end{array}} \right] $$ (4) where γ0 is the round-trip gain of the electric field of a polarization component when its polarization direction is perfectly aligned with that of the polarizer.
As the polarization components Ex and Ey recirculate in the fiber loop, their incidental angles respective to the principal axes of Pols. 1 and 2 result in round-trip gains, given by
$$ \gamma _x = \gamma _0{\cos}^2\theta _i $$ (5) $$ \gamma _y = \gamma _0{\sin}^2\theta _i $$ (6) and a round-trip coupling strength given by
$$ \kappa = \gamma _0\sin \theta _i\cos \theta _i $$ (7) The round-trip gains γx and γy can be tuned continuously from 0 to a maximum value of γ0. At PT symmetry, gain and loss balance should be achieved between the two polarimetric loops, i.e., γxγy = 1. Based on Eq. (5), we find that the angle of polarization rotation for the implementation of PT symmetry for the imaginary part of the eigenfrequency is
$$ \theta _\gamma = \pm \frac{1}{2}\arcsin \left( {\frac{2}{{\gamma _0}}} \right) $$ (8) Last, the optical path from Pol. 2 to Pol. 1 is initially configured to be nonbirefringent by tuning PC2 to fully compensate for the polarization mode dispersion contributed by all optical components. The tuning of the equivalent half-wave plate in PC2 then introduces a propagation loss for any incident light to Pol. 1, as the polarization direction of light from Pol. 1 is no longer perfectly aligned with that of Pol. 2. The maximum net round-trip gain becomes γ0cosθt instead of γ0, where θt is the polarization rotation angle. Tuning θt is different from the case where θi is tuned. The tuning of θi introduces a variation in the gain difference between the polarimetric loops and can be used to achieve gain and loss balance. By contrast, the tuning of θt introduces a universal gain variation of cosθt for both polarimetric loops and thus affects only the lasing threshold of the PT-symmetric laser. Tuning θt is an alternative way to adjust the lasing threshold, which is more precise and convenient than changing the pump current to the EDFA.
We then convert the round-trip gain, loss and coupling to the corresponding per-time-unit coefficients (see Supplementary Information 1), and the optical coupling between the two polarimetric loops can be written as (see Supplementary Information 2)
$$ \frac{{dE_x}}{{dt}} = - i\omega _mE_x + i\kappa E_y + \gamma _xE_x $$ (9) $$ \frac{{dE_y}}{{dt}} = - i\omega _mE_y + i\kappa E_x + \gamma _yE_y $$ (10) where κ is a real number30. The solution to the coupling equations shows that a PT-symmetric system has complex eigenfrequencies when the gain or loss coefficients have a higher magnitude than the coupling coefficient, in which the imaginary part represents the gain or loss coefficient of the mode14. When operating in PT symmetry, the system can provide a gain difference enhancement between the longitudinal mode with the highest round-trip gain (dominant mode, γ0) and that with the second highest gain (secondary mode, γ1). The enhancement factor is given by
$$ F = \frac{{{\Delta} g_{PTS}}}{{{\Delta} g_{Hermitian}}} = \frac{{\sqrt {\gamma _0^2 - \gamma _1^2} }}{{\gamma _0 - \gamma _1}} $$ (11) where ΔgPTS and ΔgHermitian are the gain differences between the dominant and secondary modes in a PT-symmetric system and a single-loop Hermitian system, respectively. The enhancement of the gain difference with a factor of F would significantly reduce the difficulty in achieving stable single-mode lasing.