Tunable degree of polarization in a Figure-8 fiber laser

We experimentally study a fiber loop laser in the Figure-8 configuration, and explore the dependency of the degree of polarization on controlled parameters. To account for the experimental observations, a mapping is derived to evaluate the polarization time evolution. Nonlinearity induced by Kerr effect and gain saturation gives rise to rich dynamics. We find that degree of polarization can be increased by tuning the system into a region where the mapping has a locally stable fixed point.


I. INTRODUCTION
Scrambling is commonly employed to lower the degree of polarization (DOP) of optical sources, when effects such as polarization hole-burning and polarization mode dispersion are unwanted [1][2][3].On the other hand, high DOP is desired for some other applications, including the detection of magneto-optic [4,5] and optomechanical effects [6].
Polarization evolution in optical fibers with spatially varying birefringence has been explored in [29].Polarization evolution of solitary waves in fiber lasers is complex as optical components e.g.polarization controllers (PC), laser gain medium, optical couplers, wavelength filters as well as multiple cavity round trips, contribute to polarization evolution apart from birefringence and dispersion of optical fibers.Ref. [24] has shown that the soliton polarization rotation is periodic where the period can be multiple of cavity round trip time in a fiber ring laser.It has been shown that soliton dynamics in a mode-locked fiber laser can produce either multiple fixed polarization states occurring periodically or no fixed polarization state at a fixed location of laser cavity [30].This can lead to different DOPs depending on the soliton dynamics.F8L is promising as this fiber laser configuration can give rise to sub-picosecond laser pulses in ML region [31,32].However, polarization instability is a challenging issue in mode-locked F8L not based on polarization-maintaining fibers.
In our experiment we employ the rotating quarter-wave plate method [33] to measure the DOP in both continu-ous wave (CW) and ML regions.We find that for both cases the DOP can be tuned.

II. EXPERIMENTAL SETUP
A sketch of the experimental setup is shown in Fig. 1.The F8L consists of two fiber optical loops, which are connected through a 50 : 50 optical coupler (OC).PCs are integrated into both loops.
The right loop serves as a fiber optical loop mirror (FOLM) [34][35][36].An erbium doped fiber (EDF) of length 3.6 m is integrated into the FOLM to provide gain in the telecom band.The EDF is pumped using a 980 nm diode laser and a wavelength division multiplexer (WDM).The EDF is connected to a long nonlinear single mode fiber (NSMF) of length 50 m.Band selection is performed using a tunable wavelength filter (WLF) of 0.42 nm width and −3 dB insertion loss.A 99 : 1 OC in the right loop is connected to a photodetector (PD), which is probed by a radio frequency spectrum analyzer (RFSA).
For clockwise unidirectional propagation of light in the left loop, two optical isolators (OI) are employed (labeled by arrows in Fig. 1).A 90:10 OC in the left loop is connected to an optical spectrum analyzer (OSA), a PD, an oscilloscope (OS), and a rotating quarter-wave plate polarimeter (PM)(Thorlabs PA430).

III. MEASUREMENTS
The temporal F8L output signal measured by the OS in the region of ML is shown in Fig. 2(a).Figure 2(b) shows the corresponding autocorrelation (AC) signal as a function of a delay time.Figure 2(c) shows the AC signal calculated using Eq.(A4) of appendix A. The calculated AC signal approximately imitates the coherence artifact (spike) and broad pedestal (wings) of the measured AC.The pulse width is extracted by fitting the data with Eq. (A4) (see caption of Fig. 2).
Plots of the optical spectrum measured by the OSA, the RF spectrum measured by the RFSA, the averaged optical power measured by the PD in the right loop, and the DOP measured by the PM, are shown in Fig. 3 as a function of the diode current I D .For the measurements shown in Fig. 3(a1), (b1), (c1) and (d1), both PCs are tuned to maximize the DOP in the ML regime, whereas the DOP in the ML regime is minimized for the measurements shown in Fig. 3(a2), (b2), (c2) and (d2).For the first (second) case the CW threshold occurs at diode current of I D = 0.1 A (I D = 0.15 A), and the ML threshold at I D = 0.26 A (I D = 0.37 A).As can be seen by comparing Fig. 3(d1) and (d2), DOP in the ML regime can be tuned.The DOP plot in Fig. 3(d1) reveals a sharp change in the state of polarization (SOP) at the transition from CW to ML regime occurring at diode current of I D = 0.26 A. For this case, the DOP increases with diode current I D in both CW and ML regions.A SOP change occurring at the transition from CW to ML regions is also seen in Fig. 3(d2).However, for this case, the DOP remains very low above the transition.

IV. FOLM
To account for the experimental observations, scattering and birefringence in the FOLM are theoretically studied.The 50:50 OC is characterized by forward (backward) transmission t (t ′ ) and reflection r (r ′ ) amplitudes.Time reversal and mirror symmetries together with unitarity imply that t ′ = t, r ′ = r = it |r/t|, and   a1) is given by [37] , where the total gain g is real, the is the Pauli's z matrix, and J + (J − ) is the Jones matrix corresponding to circulating the FOLM loop in the clockwise (counter clockwise) direction.The assumption that both J + and J − are unitary, i.e.J † + J + = J † − J − = 1, together with the relations and where yield where the operator S is given by (note that σ 2 z = 1) These relations imply the unitarity condition A derivation (see appendix B) similar to the one that yields the Heisenberg uncertainty principle is employed to derive a lower bound upon the FOLM reflectivity The relation A † T A T +A † R A R = 1 together with inequality (8) yields an upper bound upon the FOLM transmissivity P T = p| A † T A T |p / p |p .However, both P R and P T become unbounded when J † + σ z , J − σ z − = 0.In particular, no bound is imposed by the inequality (8) when J − = σ z J + σ z .As will be shown below, the condition J − = σ z J + σ z is satisfied when the SOP evolution along the FOLM depends only on the geometry of the fiber spacial curve.

V. SOP EVOLUTION
The Jones matrices J + and J − can be calculated by integrating the equation of motion for the SOP along the FOLM.Consider an optical fiber winded in some spacial curve in space.Let r (s) be an arc-length parametrization of this curve, i.e. the tangent ŝ =dr/ds is a unit vector.The normal and binormal Serret -Frenet unit vectors are denoted by ν and b, respectively.The curve torsion τ is defined as d b/ds = −τ ν [38].A unit vector parallel to the electric field is denoted by ê0 .In the Serret -Frenet frame the unit vector ê0 , which is expressed as ê0 = e ν ν + e b b, evolves according to the parallel transport equation of geometrical optics [39][40][41] d ds where K = K g + K f .The geometrical birefringence K g is given by [40,42] whereas K f is the fiber birefringence induced by elastooptic, electro-optic, or magneto-optic effects.For a lossless fiber the matrix K is Hermitian.Consider the case where K f vanishes.For this case the solution to Eq. ( 9) is given by [38] where the rotation angle θ is given by the integrated torsion along the fiber curve For the case where the curve end points are parallel, i.e. ŝ (s) = ŝ (0), the following holds θ = Ω, where Ω is the solid angle subtends by the closed curve ŝ (s ′ ) at the origin.This geometrical rotation of polarization, which is closely related to the Berry's phase [43], has been experimentally measured in [42,44].
For the case K f = 0 the following holds J − = σ z J + σ z , and consequently . Hence, for this case both reflectivity P R and transmissivity P T becomes independent on the input SOP.For a 3 dB OC, i.e. |t| 2 = |r| 2 = 1/2, and in the linear limit, i.e. for Θ = 0, the FOLM for the same case where K f = 0 becomes perfectly reflecting, i,e, P T = 0 and P R = 1.
For treating the general case (where K f cannot be disregarded, and consequently FOLM perfect reflectivity can not be guaranteed) it is convenient to express the matrix J − as J − = J m σ z J + σ z , where J m is unitary, i.e.J † m J m = 1.Using this notation one finds that where and where V m = σ z (J m − 1) σ z .Note that J m = 1 and V m = 0 for the case K f = 0.

VI. MAPPING
In this section a cycle to cycle mapping is derived to analyze the SOP evolution.Stability analysis of the mapping allows the DOP evaluation in steady state.The input SOP |p i after n cycles is denoted by |p n .The mapping between |p n and |p n+1 is given by where the Jones matrix of the clockwise unidirectional left loop, which is denoted by J L , is assumed to be unitary, i.e.J † L J L = 1.The mapping ( 16) is nonlinear, since A T depends on the nonlinear phase Θ, which, in turn, depends on the intensity I n = p n |p n .In addition, the gain g may vary due to saturation.In general, the nonlinear phase Θ may also depend on the SOP, however, this dependency is expected to be relatively weak since our setup is based on regular (rather than polarization maintaining) single mode fibers.The amplification factor I n+1 /I n corresponding to the SOP |p n is denoted by F (|p n ), where F (|p ) = g 2 p| A † T A T |p / p |p .The blue lines in Fig. 4 represent the values of |λ 1 | and |λ 2 | as a function of Θ for four different cases, where λ 1,2 are the eigenvalues of J L A T .The unitary matrices J + , J m and J L are expressed as and where all angles θ, ϕ and φ are real.The matrix U (θ, ϕ, φ) represents SOP rotation around the unit vector (sin θ cos ϕ, sin θ sin ϕ, cos θ) by an angle φ.The red lines in Fig. 4 represent the values of Λ  maximized (for a given Θ).If, in addition, |p F is an eigenvector of the mapping operator gJ L A T , then the DOP is expected to be relatively high in steady state, provided that |p F is a locally stable fixed point of the mapping (16).For the case shown in Fig. 4(a), for which the operator A T is given byA T = i sin (Θ/2) σ z , the SOP |p F is an eigenvector of gJ L A T for all Θ [note that |λ 1,2 | = Λ  3(d1) and (d2)] occurring in the transition from CW to ML is attributed to the change in peak power, which in turn, affects SOP evolution due to nonlinearity of the cycle to cycle mapping 16 (induced by both Kerr effect and gain saturation).The drop in PDO represents the change in the averaged optical loss per cycle, which depends on the FOLM reflectivity P R .A bifurcation diagram example of the mapping ( 16) is shown in Fig. 5.The parameters that are used for the calculation are listed in the figure caption.To account for saturation, the gain g is assumed to be given by where g 0 is the small signal gain, and I s is the saturation intensity.The Kerr effect induced nonlinear phase Θ is assumed to be given by Θ = ζI, where ζ is a positive constant, and where I = p |p is the intensity.For the example shown in Fig. 5, a period doubling bifurcation occurs at g 0 = 4.18.Relatively low DOP is expected in the region 4.28 ≤ g 0 ≤ 4.51, where the period time of the mapping ( 16) becomes much longer than the loop fundamental period.Note that the mapping 16 strongly depends on the unitary matrices J + , J m and J L .In the experiment, this dependency can be explored by tuning both PCs.The dependency shown in Fig. 3, of both PDO and DOP on diode current I D , can be significantly modified by retuning both PCs.Similarly, the bifurcation diagram shown in Fig. 5 can be significantly modified by changing the assumed unitary matrices.In a a fiber-based setup, it is very difficult to independently measure each of these unitary matrices, for any given paddle configuration of both PCs.However, when these unitary matrices are treated as fitting parameters, good agreement between theory and experiment can be obtained.

VII. SUMMARY
In summary, DOP tunability is experimentally demonstrated in a F8L, in both CW and ML regions.A mapping ( 16) is employed to derive the SOP time evolution.High DOP can be obtained in the regions where the mapping has a locally stable fixed point.Nonlinearity of the cycle to cycle SOP mapping 16 gives rise to complex dynamics.The current experimental setup, which is based on the rotating quarter-wave plate method, does not allow monitoring the relatively fast SOP cycle to cycle evolution, since the fiber ring period time is much shorter than the plate rotation period time.Future experiments, which will employ a faster polarimeter, will be devoted to the cycle to cycle complex dynamics of this system.

FIG. 2 :
FIG. 2: Mode locking.(a) OS signal as a function of time at diode current ID = 1 A. (b) Measured AC as a function of delay time at diode current ID = 1 A. (c) Calculated AC signal using Eq.(A4) with pulse width of σc = 1.8 ps and varying splitting time, which follows a normal distribution having a standard deviation of σs = 12 ps.

FIG. 3 :
FIG. 3: Dependency on diode current ID.(a) Optical spectrum (in dBm units) at wavelength λ, (b) RF spectrum (in dBm units) at frequency f, (c) photodetector output power (PDO) and (d) DOP.Left (right) panels, which are labeled by '1' ('2'), corresponds to maximized (minimized) DOP in ML regime.Note that the gain of the PD employed to measure the RF spectrum is set at 10 6 with bandwidth of 3 MHz.The PD limited bandwidth gives rise to low pass filtering artifact in the data presented in (b1) and (b2).
are the non-negative eigenvalues of the Hermitian and positive-definite operator A † T A T .Let |p F be a SOP, for which the amplification factor F (|p ) is