Polarization pulling in Raman assisted fiber optical parametric amplifiers

S. H. Wang; Xinchuan Xu; P. K. A. Wai

doi:10.1364/OE.24.006884

1. Introduction

Nonlinear polarization pulling is a promising mechanism to control all-optically the states of polarization (SOPs) of the polarization-scramble signals in optical fiber transmission links [1–16]. Recently, a number of such polarization control techniques were proposed by using different nonlinear effects in optical fibers, such as third-order parametric process [5–9], stimulated Raman scattering [11–15,17–20], and stimulated Brillouin scattering [16] in which the pump beam and the input signal are either counter-propagating [13–19] or co-propagating [5–9,11,12]. In these proposed schemes, the pump (or control) beam provides not only gain but also polarization control to the signal through cross-polarization modulation (XPolM).

If the pump and signal beams are counter-propagating, polarization pulling will pull the output SOP of the signal towards the input SOP of the pump. If the input SOP of the pump is fixed, the output signal will be fully repolarized and stabilized. This effect is known as polarization attraction [1,13–15,17–20]. However, counter-propagation requires a higher gain [9] and a longer response time to achieve signal repolarization than co-propagation. Co-propagating schemes, on the other hand, can only achieve partial repolarization even though instantaneous polarization control can be achieved with a small gain [1,9]. This is mainly due to the randomly varying fiber birefringence making the output SOP of the co-propagating pump unpredictable. The polarization pulling effect in bi-directionally pumped schemes however has not been investigated.

In this paper, we will investigate the polarization pulling effect in Raman assisted fiber optical parametric amplifiers (RA-FOPAs) which use bi-directionally propagating pumps [21–26]. In RA-FOPAs, a counter-propagating Raman pump is added to the co-pumped FOPA to enhance the parametric gain and extend the gain bandwidth [21–26]. For FOPAs, broadband polarization pulling had been experimentally demonstrated [7] and theoretically analyzed [7–9]. Compared to the fiber Raman amplifiers (FRAs) and FOPAs, the polarization pulling effect in the RA-FOPAs is rather complex because of the energy transfer from the counter-propagating Raman pump to the co-propagating parametric pump through the Raman gain. Both the Raman and the parametric pump provide gain and polarization pulling to the signal and the idler. Thus the variations of the powers and the SOPs of the Raman and parametric pumps along the fiber will determine the overall polarization pulling effect to the signal and the idler in RA-FOPAs.

In this paper, we report a comprehensive theoretical model to investigate polarization pulling in bi-directionally pumped degenerate RA-FOPAs using randomly birefringent fibers. In this model, the contributions of chromatic dispersion, polarization mode dispersion (PMD), Raman gain, and nonlinear effects to the phase matching in RA-FOPAs are investigated. We will also study polarization pulling in different states of RA-FOPAs to determine how to achieve optimum polarization attraction.

2. Theoretical model

In RA-FOPAs, the co-propagating parametric pump will be amplified by the counter-propagating Raman pump which in turn amplifies the signal and the idler through parametric gain [23]. In parametric amplification, the phase matching condition has to be satisfied or nearly satisfied to provide high parametric gain [5,8,9]. However, the phase matching condition can be easily broken by fiber chromatic dispersion, PMD, Raman process, and other nonlinear effects [9,23]. Thus, investigating the phase matching condition is crucial to the characterization of the polarization pulling effect in the RA-FOPAs.

The phase matching in RA-FOPAs is determined by the contributions from all the combinations of both of the components along the two principal axes of the electric fields of the pump, the signal, and the idler [9]. Similar to FOPAs, one particular combination out of the 16 combinations of the optical beams is sufficient to derive the full phase term in RA-FOPAs, which is given by Θ(z) = Θ_xxxx = Δkz + θ_sx(z) + θ_ix(z) − 2θ_px(z), where Δk ≈β₂(ω_s – ω_p)² is the phase mismatch caused by chromatic dispersion and β₂ is the group-velocity dispersion (GVD) coefficient [9]. The parameters ω_R, ω_p, ω_s, and ω_i are the angular frequencies of the Raman pump, parametric pump, signal, and idler, respectively. Thus, the evolution of the Raman pump $\bar{R} = {(\begin{matrix} R_{1}, & R_{2}, & R_{3} \end{matrix})}^{T},$ parametric pump $\bar{P} = {(\begin{matrix} P_{1}, & P_{2}, & P_{3} \end{matrix})}^{T},$ signal $\bar{S} = {(\begin{matrix} S_{1}, & S_{2}, & S_{3} \end{matrix})}^{T},$ idler $\bar{D} = {(\begin{matrix} D_{1}, & D_{2}, & D_{3} \end{matrix})}^{T},$ and phase combination Θ in Stokes space can be described by the following coupled equations [9,22,23]

\frac{d \bar{R}}{d z} = α_{R} \bar{R} + \frac{g_{R}^{Ω}}{2} \frac{ω_{R}}{ω_{p}} [P_{0} \bar{R} + R_{0} \bar{P} + μ (3 P_{0} \bar{R} + R_{0} \bar{P} - 2 R_{0} {\bar{P}}_{3})] - (\underline{M} ω_{R} \bar{b} + {\bar{W}}_{R}) \times \bar{R},

\frac{d \bar{P}}{d z} = - α_{p} \bar{P} + \frac{g_{R}^{Ω}}{2} [R_{0} \bar{P} + P_{0} \bar{P} + μ (3 R_{0} \bar{P} + P_{0} \bar{R} - 2 P_{0} \bar{R_{3}})] + (ω_{p} \bar{b} + \bar{W_{p}}) \times \bar{P} + \bar{K},

\frac{d \bar{S}}{d z} = - α_{p} \bar{S} + \frac{g_{R}^{Ω - Λ}}{2} [R_{0} \bar{S} + S_{0} \bar{R} + μ (3 R_{0} \bar{S} + S_{0} \bar{R} - 2 S_{0} {\bar{R}}_{3})] + (ω_{s} \bar{b} + {\bar{W}}_{s}) \times \bar{S} + \bar{M},

\frac{d \bar{D}}{d z} = - α_{p} \bar{D} + \frac{g_{R}^{Ω + Λ}}{2} [R_{0} \bar{D} + D_{0} \bar{R} + μ (3 R_{0} \bar{D} + D_{0} \bar{R} - 2 D_{0} {\bar{R}}_{3})] + (ω_{i} \bar{b} + {\bar{W}}_{i}) \times \bar{D} + \bar{N},

\frac{d Θ}{d z} = Δ k + \frac{b_{2}}{4} (2 \frac{ω_{p} P_{2}}{I_{p x}} - \frac{ω_{s} S_{2}}{I_{s x}} - \frac{ω_{i} D_{2}}{I_{i x}}) + Q + U + C_{x x x x} + C_{x x y y} + C_{y y x x} + C_{x y x y} + C_{y x x y},

where α_R and α_p ≈ α_s ≈ α_i are fiber losses at the Raman pump wavelength and the parametric pump wavelength respectively. The parameter

g_{R}^{ω}

is the parallel Raman gain coefficient at a frequency shift of ω. The Raman shift between the Raman and parametric pump is Ω = ω_R − ω_p and the frequency difference between the parametric pump and the signal is Λ = ω_s − ω_p. The parameter μ = 0.18 is the ratio of the orthogonal to parallel Raman gain coefficients [13].

The vector $\overset{⇀}{b} = {(\begin{matrix} b_{1}, & b_{2}, & b_{3} \end{matrix})}^{T} = b {(\begin{matrix} \cos φ (z), & \sin φ (z), & 0 \end{matrix})}^{T}$ describes the linear random-ly varying fiber birefringence at the pump wavelength, where b is the strength of fiber birefringence and φ is the orientation angle of the birefringent axes. Here, for RA-FOPAs with less than 3 THz gain bandwidth [23,24], we can assume that the birefringence axes at the wavelengths of the pump, the signal and the idler are correlated. For the FRAs [13,14], both theoretical and experimental results indicate that the counter-propagating Raman pump provides an average gain which is insensitive to the decorrelation of the birefringence axes. Thus we assume that the birefringence axes at the Raman pump wavelength and the Stokes wavelengths are correlated. The matrix $\underline{M} = diag (\begin{matrix} 1, & 1, & - 1 \end{matrix})$ represents the backward propagation of the Raman pump.

The vectors ${\overset{⇀}{W}}_{R} = (2 γ_{R} / 3) [{\overset{⇀}{R}}_{3} + 2 \sum_{X}^{P, S, D} ({\overset{⇀}{X}}_{3} - \overset{⇀}{X})],$ ${\overset{⇀}{W}}_{p} = (2 γ_{p} / 3) [{\overset{⇀}{P}}_{3} + 2 \sum_{X}^{R, S, D} ({\overset{⇀}{X}}_{3} - \overset{⇀}{X})],$ ${\overset{⇀}{W}}_{s} = (2 γ_{s} / 3) [{\overset{⇀}{S}}_{3} + 2 \sum_{X}^{R, P, D} ({\overset{⇀}{X}}_{3} - \overset{⇀}{X})],$ ${\overset{⇀}{W}}_{i} = (2 γ_{i} / 3) [{\overset{⇀}{D}}_{3} + 2 \sum_{X}^{R, P, S} ({\overset{⇀}{X}}_{3} - \overset{⇀}{X})],$ where ${\overset{⇀}{P}}_{3} = P_{3} {\hat{e}}_{3}$ , ${\overset{⇀}{S}}_{3} = S_{3} {\hat{e}}_{3}$ , ${\overset{⇀}{D}}_{3} = D_{3} {\hat{e}}_{3}$ , govern the effects of self-phase modulation (SPM), cross-phase modulation (XPM), nonlinear polarization rotation (NPR) effects, and XPolM in the Raman pump, parametric pump, signal, and idler, respectively. The four wave mixing (FWM) terms of the pump, the signal and the idler are the same as that of the FOPAs shown in Eqs. (5)-(7) in [9].

To simplify the model of RA-FOPAs, we assume a uniform Raman gain profile, i.e. $g_{R}^{Ω} \approx g_{R}^{Ω + Λ} \approx g_{R}^{Ω - Λ}$ , and assume the nonlinear coefficient γ = γ_p ≈ γ_s ≈ γ_i ≈ γ_R as in [23]. In Eq. (5), the evolution of Θ is determined by the phase mismatches caused by the following:

(1) The chromatic dispersion, and
(2) The PMD effect described by the second term on the right hand side of Eq. (5), where I_ab, a = p, s, i, and b = x, y are the intensities of the x- and y-component of the pump, signal and idler, respectively,
(3) The term Q give the effects of SPM, XPM, NPR, XPolM as $\begin{array}{l} Q = \frac{γ}{3} [2 (2 P_{0} - S_{0} - D_{0}) + (2 P_{1} - S_{1} - D_{1}) \\ - (R_{2} + P_{2} + S_{2} + D_{2}) (\frac{2 P_{2}^{}}{I_{p x}} - \frac{S_{2}^{}}{I_{s x}} - \frac{D_{2}^{}}{I_{i x}}) + (\frac{P_{2}^{2}}{I_{p x}} - \frac{S_{2}^{2}}{2 I_{s x}} - \frac{D_{2}^{2}}{2 I_{i x}})] . \end{array}$

Although the nonlinear contribution

(2 γ / 3) (2 P_{0} + P_{1})

can compensate the chromatic dispersion induced phase mismatch, the other nonlinear effects can still cause significant phase mismatch when the combined contribution of the rest of the terms in the right hand side of Eq. (6) is not close to zero,

(4) The Raman gain induces phase mismatch given by

U = \frac{g_{R}^{Ω}}{4} [(1 + μ) R_{2} (\frac{S_{3}}{I_{s x}} + \frac{D_{3}}{I_{i x}} - 2 \frac{P_{3}}{I_{p x}}) + (1 - μ) R_{3} (2 \frac{P_{2}}{I_{p x}} - \frac{S_{2}}{I_{s x}} - \frac{D_{2}}{I_{i x}})],

We note that when the SOP of the Raman pump is (1, 0, 0), Eq. (7) will reduce to zero. In this case, the contribution of the Raman gain to the phase combination Θ is minimized.

(5) The last five terms represent the FWM effects among the parallel components and that among the orthogonal components which are the same as those in [9]. When Θ is close to (n + 1)π/2, the term C_xxxx is minimized and the signal and idler will achieve a maximum gain from the parametric process.

Equations (1) to (5) completely characterize the effects listed above on the phase matching of RA-FOPAs using randomly birefringent fibers. As discussed in [9], the evolution of Eqs. (1)-(5) is determined by the corresponding twelve phase combinations, all of which can be derived from $\overset{⇀}{R}$ , $\overset{⇀}{P}$ , $\overset{⇀}{S}$ , $\overset{⇀}{D}$ , and Θ.

When the powers of the signal and the idler are much smaller than those of the Raman and parametric pumps, Eq. (2) can be further simplified by assuming undepleted pump pairs which is commonly used in the modeling of small signal FRAs and FOPAs [27]. We have

\frac{d \overset{⇀}{P}}{d z} = - α_{p} \overset{⇀}{P} + \frac{g_{R}^{Ω}}{2} [R_{0} \overset{⇀}{P} + P_{0} \overset{⇀}{R} + μ (3 R_{0} \overset{⇀}{P} + P_{0} \overset{⇀}{R} - 2 P_{0} {\overset{⇀}{R}}_{3})] + (ω_{p} \overset{⇀}{b} + {\overset{⇀}{W}}_{p}) \times \overset{⇀}{P} .

If we further neglect the XPM and XPolM effects caused by the signal and the idler, we can simplify the vectors as

{\overset{⇀}{W}}_{R} = (2 γ / 3) [{\overset{⇀}{R}}_{3} + 2 ({\overset{⇀}{P}}_{3} - \overset{⇀}{P})]

and

{\overset{⇀}{W}}_{p} = (2 γ / 3) [{\overset{⇀}{P}}_{3} + 2 ({\overset{⇀}{R}}_{3} - \overset{⇀}{R})] .

Note that in the model described by Eqs. (1) and (8), the energy coupling between the Raman pump and parametric pump cannot be neglected because their powers are comparable.

3. Results and discussion

3.1 Numerical method

In RA-FOPAs, the variations of the power and SOP of both pumps along the fiber determine the overall polarization pulling on the signal and the idler. It is rather complex to figure out numerically the variations in RA-FOPAs with bi-directionally propagating pumps. This is because there is not only energy exchange, but also strong mutual polarization pulling between the Raman and parametric pumps. Besides, the SOPs of both pumps vary randomly along the fiber because of the randomly varying fiber birefringence. Thus, initial alignment of the input signal SOP with one of the principal axes at the input end of the fiber cannot guarantee the maximum gain in RA-FOPAs.

In this paper, we adopt the same approach as in [9] to define a “principal” SOP referring to the state with optimum phase matching in RA-FOPAs. In the simulation, since the input SOPs of the Raman pump, the parametric pump, and the signal have to be randomized, it is more computational intensive to determine the “principal” SOP with optimum phase matching in RA-FOPAs than the FOPAs in [9].

To simplify the problem, we firstly fixed the input SOP of one of the pump (Pump1) to be linearly polarized, which is also aligned with the birefringence axis at one of the fiber ends. Then, we change the input SOPs of the other pump (Pump2), the signal, and the realizations of the randomly varying fiber birefringence which were generated by a random modulus model [28], to determine the maximum gain. Since we assumed a linearly polarized Pump1, in general global optimum phase matching cannot be achieved. In the simulation, we follow the following four steps for each fiber realization.

Step I: First generate 100 Pump2s and 100 signals the input SOPs of which are random and uniformly distributed on the Poincaré sphere. This step simulates the experimental procedure in which one adjusts the polarization controllers of the input Pump2 and signal to search for the optimal combination of input SOPs for maximize gain.

Step II: For each of the 100 Pump2s, the profiles of the Pump2 and corresponding Pump1 along the fibers are firstly calculated using shooting method [13]. These pump profiles are then used to simulate the RA-FOPA gains of the 100 input signals generated in Step I and the maximum gain of this choice of Pump2 will be recorded. In this step, the input signal wavelength has to locate around the gain peak of the RA-FOPA.

Step III: Compare the maximum gains of the 100 Pump2s and determine the values of Pump2s with the optimal input SOPs of the overall maximize gain. We assume the combination of this Pump2 and the corresponding Pump1 can provide the optimum phase matching in the RA-FOPAs using this fiber realization. The profiles the Stokes vectors of this pump combination along the fiber will be recorded for the next step.

Step IV: the wavelengths of the 100 input signals generated in Step 1 are varied to cover the gain bandwidth of the RA-FOPAs. For all the wavelengths chosen, the profiles of the Raman pump and the parametric pump in Stokes space recorded in Step III are used to obtain the simulation results.

In the results, the DOPs of the signal and the idler are defined as $D O P_{G} =$ ${\sqrt{\sum_{i = 1}^{3} {〈 S_{i} (z) 〉}_{SOP}^{2}} / 〈 S_{0} (z) 〉}_{SOP}$ and $D O P_{C E} = {\sqrt{\sum_{i = 1}^{3} {〈 D_{i} (z) 〉}_{SOP}^{2}} / 〈 D_{0} (z) 〉}_{SOP}$ respectively, where ${〈 \cdot 〉}_{SOP}$ is the average taken over 100 input signals which are generated in Step I. The fourth order Runge-Kutta method is used in the numerical integration. The maximum step size chosen is 10 mm which is sufficient to follow the local birefringence variations. Finally, the mean DOP is determined as ${〈 D O P 〉}_{fiber},$ where ${〈 \cdot 〉}_{fiber}$ is the average taken over an ensemble of 100 fiber realizations [9,13].

3.2 Comparison with experimental results

To verify our model, we compared the simulation results with the experimental results in [23,24]. As shown in Table 1, both the experiments used 1 km long HNLFs from the same manufacturer with the same chromatic dispersion D (equivalently β₂ = –0.154 ps²/km). Table 1 shows the fiber parameters of HNLF1 and HNLF2 which are used in the RA-FOPA experiments in [23] and [24], respectively. In the simulations, 9/8 times the measured γ [9,28] and double of the measured $g_{R}^{Ω}$ [10] of HNLFs are used. The nominal PMD coefficients D_PMD of HNLF1 and HNLF2 are also shown in Table 1. The power of the input signal is fixed at 10 µW and the fiber correlation length L_C is fixed at 10 m [9].

Table 1. Parameters of the Fibers Used in the RA-FOPA Experiments [23,24]

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Here, we note that unpolarized counter-propagating Raman pumps were used in [23] and [24]. However, since the RA-FOPAs operate in the deep saturation state (as discussed in Section 3.4), the co-propagating parametric pump provides the dominant gain and much stronger polarization pulling to the signal than the polarized Raman pump. The weak polarization pulling from the polarized Raman pump can therefore be neglected. In addition, for HNLF1 and HNLF2, the polarized counter-propagating Raman pump provides an averaged gain which is polarization insensitive and similar to that using an unpolarized Raman pump [10]. Therefore, we can assume a polarized Raman pump in our model and use the simulation results to approximate the experimental overall maximum gains in [23] and [24] which are determined by optimal combination of the pump SOP and signal SOP.

Figure 1(a) show the simulated maximum gain profiles of the FOPAs versus the signal detuning Δυ = (ω_s − ω_p)/2π using HNLF1 but with different D_PMD, where the pump wavelength is fixed at 1555 nm and the input powers of the Raman pump and the parametric pump are 1.17 W and 128 mW, respectively. In Fig. 1(a), the experimental data of [23] are also shown in solid circles for comparison. We note that from Fig. 1(a) the numerical results obtained with D_PMD = 0.02 ps/km^1/2 agree well with the experimental results from [23] up to detuning of ~0.9 THz, but the nominal D_PMD was four times smaller than that provided by the manufacturer, which are consistent with the observation in [9]. In Fig. 1(a), we also note that when the detuning is larger than 0.9 THz, the simulated results overestimate the RA-FOPA gain because of the assumptions of uniform Raman gain profile and correlated birefringence axes.

Fig. 1 The maximum gains with different D_PMD versus the signal detuning for the RA-FOPAs using (a) HNLF1 with the Raman pump at 1.17 W and parametric pump at 128 mW, and (b) HNLF2 with the Raman pump 1.15 W and the parametric pump at 148 mW. The corresponding experimental data (solid circles) of [23] and [24] are also shown for comparison.

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Figure 1(b) show the experimental measurement results in RA-FOPAs using HNLF2 [24] and the corresponding simulated results, where the input powers of the Raman pump and the parametric pump are 1.148 W and 148 mW, respectively. Similar to Fig. 1(a), the numerical results obtained with a smaller D_PMD = 0.04 ps/km^1/2 agree well with the experimental results from [23]. If the nominal values of D_PMD of HNLF1 (0.08 ps/km^1/2) and HNLF2 (0.06 ps/km^1/2) are accurate, then the simulated results underestimate the measured maximum gains (31.8 dB and 26.8 dB) of the RA-FOPAs by 6.6 dB and 2.7 dB, respectively. However, the numerical simulation can still describe the general behavior of the maximum gain and the gain profile of the RA-FOPAs.

We note that in both RA-FOPAs in [23] and [24], the Raman gain is highly depleted by the relatively large parametric pump which will result in the differences between the simulation and experimental results. We also found that the maximum net simulated Raman gain of the output parametric pump, defined as the ratio of output to input parametric pump power, is 8.01~8.05 dB for Fig. 1(b), is close to the 8.9 dB measured on-off gain (i.e. 8.11 dB net gain if α_R = 0.79 dB/km) as in [24]. The result indicates that the assumptions of using 9/8 times of measured γ and twice the measured $g_{R}^{Ω}$ of HNLFs is reasonable.

In Figs. 1(a) and 1(b), we fixed the input SOPs of the Raman pump to (1, 0, 0). That is, we set the Raman pump as Pump1 (Section 3.1), and generated 100 parametric pumps (Pump2s) with random input SOPs. In Figs. 2(a)-2(c), we compare the simulation results of the global maximum gains, mean gains, and the gain-DOPs relationship of RA-FOPAs using two different initial conditions of the pump SOP. Here, the term of global maximum gain refers to the maximum gain that can be achieved in the100 fiber realizations. Condition I is the same as that in Fig. 1(b) when the input SOPs of the Raman pump are fixed. In Condition II, the input SOPs of the parametric pump is fixed to (0, 1, 0), and 100 random input SOPs of the Raman pump were generated in the simulation.

Fig. 2 Comparison of the (a) global maximum gains and (b) mean gains as a function of the signal detuning and (c) the relationship between the mean gain and output DOPs of RA-FOPAs using pumps with different initial SOPs conditions. The parameters are the same as that used in Fig. 1(b) for D_PMD = 0.04 ps/km^1/2.

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As shown in Figs. 2(a) and 2(b), the case with fixed input Raman pump can provide larger gain than those with fixed input parametric pumps. This is because, as discussed in Section 2, the Raman gain induced phase mismatch is minimized when the SOP of the Raman pump is (1, 0, 0). However, in Fig. 2(c), when compared with Condition I, we observe that higher output DOP can be achieved using smaller gain when the input SOPs of parametric pump is fixed. The result indicates that in bi-directionally pumped RA-FOPAs, a linearly polarized counter-propagating Raman pump with fixed input SOPs can achieve higher gain but weaker polarization pulling than those fixed the input SOPs of parametric pumps when the Raman gain is deeply saturated. In the following discussion, we will set the Raman pump as Pump1 and fix its input SOP to (1, 0, 0).

3.3 Mean gains and conversion efficiencies

Figures 3(a) and 3(b) show the simulated mean gain and mean conversion efficiencies of the FOPAs using HNLF2 with different D_PMD versus the signal detuning. The parameters are the same as those in Fig. 1(b). Similar to the FOPAs using HNLF1 [9], when the D_PMD of fiber decreases, strong polarization pulling occurs and the output SOPs of the signal become more predictable. Figures 3(c) and 3(d) show the corresponding mean DOP_G of the signals of Fig. 3(a) and the mean DOP_CE of the idlers of Fig. 3(b), respectively. Since we assume uniform Raman gain profile and neglect the decorrelations between the birefringence axes at the wavelengths of the pump, signal and idler, we note that for FOPAs with >3 THz bandwidth, the gain bandwidth reduction will not occur as shown in Fig. 1 of [3] and Figs. 3(a)-3(b) in [23].

Fig. 3 The (a) average gains, (b) average conversion efficiencies, (c) mean signal output DOP_G, and (d) mean idler output DOP_CE of the RA-FOPAs using HNLF2 with D_PMD = 0.02 (triangles), 0.04 (circles), and 0.06 ps/km^1/2 (squares) versus signal detuning. The corresponding standard deviations of the (e) average gains, (f) mean signal output DOP_G, (g) average conversion efficiencies, and (h) mean idler output DOP_CE are also shown, respectively.

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Figures 3(e) and 3(f) show the corresponding standard deviations of the mean gains and mean conversion efficiencies of Figs. 3(a) and 3(b), respectively. As shown in Fig. 3(e), when the mean gains increase, their standard deviations will also increase. For the mean conversion efficiencies, similar to FOPAs [9], when the conversion efficiencies increase, the corresponding standard deviations will first increase and then decrease as shown in Fig. 3(f).

Figures 3(g) and 3(h) show the standard deviations of Figs. 3(c) and 3(d), respectively. In RA-FOPAs, even at large parametric gain the output DOPs can still be close to unity with small variances. However, when compared with FOPAs [9], the bi-directionally pumped setup need higher gain to achieve that. We also found that, as shown in Figs. 3(g) and 3(h), when the mean gains and mean conversion efficiencies increase, the corresponding standard deviations of their output DOPs at the gain peak will first increase and then decrease.

3.4 Characterizing the polarization pulling in bi-directionally pumped schemes

In both co-propagating [9,12] and counter-propagating nonlinear fiber amplifiers [13], the mean DOPs are related to the mean gain which can be written as

D O P_{G} = 1 - \exp [- G_{on-off} (dB) / Γ]

where G_on-off is the on-off gain of the signal. In this section, we investigate whether bi-directionally pumped RA-FOPAs obey a similar relationship.

Figures 4(a)-4(c) show the simulation results of the output mean DOP_G as a function of the mean gain over the gain band of RA-FOPAs using HNLF1 when the Raman pump powers are 1.17, 0.8, and 0.6 W, respectively. The symbols show the simulation results when the parametric pump powers P_P vary from 2 to 200 mW. For example, gray squares refer to the mean DOP_G versus the mean gain over the parametric gain band when P_P = 200 mW. We also note that the data point at the far right is the peak of the mean gains of RA-FOPAs. Here, the average is taken over an ensemble of 100 fiber realizations. In Figs. 4(a)-4(c), the parameter D_PMD equals to 0.02 ps·km^−1/2. The interpolation functions of Eq. (9) for co-propagating FOPAs (Γ = 6.2 [9]), counter-propagating FRAs (Γ = 10.2 [13]) are also shown. We also found that, for bi-directionally pumped RA-FOPAs without deeply saturated Raman gain, in the optimum cases (when the parametric pump powers are 63 and 80 mW in Fig. 4(b)) the mean DOP_G and mean gain are related by Eq. (9) with Γ = 8.5 which was also shown in Figs. 4(a)-4(c).

Fig. 4 The mean DOP_G as a function of the mean gain over the gain band of the RA-FOPAs using HNLF1 when the Raman pump powers are (a) 1.17, (b) 0.8, and (c) 0.6 W, respectively. The symbols represent the simulation results when parametric pump powers vary from 2 to 200 mW. Here, the PMD coefficient D_PMD equals to 0.02 ps/km^1/2. The interpolation functions for co-propagating FOPAs (Γ = 6.2 [9]), counter-propagating FRAs (Γ = 10.2 [13]), and optimum bi-directionally pumped RA-FOPAs (Γ = 8.5) are also shown.

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Figure 5(a) shows the mean gains (open symbols), the global maximum gains (solid symbols), and the standard deviations of the gains (gray symbols) at the gain peaks of the RA-FOPAs as a function of the input parametric pump powers in the same RA-FOPAs as in Figs. 4(a)-4(c). As shown in Fig. 5(a), when the parametric pump power increases, both the mean gains and the global maximum gains first decrease because of the Raman gain saturation, but then increase because of the domination of parametric gain. We note that, since the co-propagating parametric amplification is polarization sensitive, the variances of the gain increase when the parametric pump power increases. Thus the rate of increase of the maximum gains is higher than that of the mean gains when the parametric pump power is larger than 17 dBm.

Fig. 5 (a) The mean gains (open symbols), the global maximum gains (solid symbols), and the standard deviations of the gains (gray symbols) at the gain peak of the RA-FOPAs as a function of the input parametric pump power for the RA-FOPAs in Figs. 4(a)-(c) when the Raman pump powers are 1.17, 0.8, and 0.6 W, respectively. (b) The mean DOPs (gray symbols) and corresponding standard deviations of DOPs (open symbols) of Fig. 5(a).

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Figure 5(b) shows the mean DOPs (gray symbols) and corresponding standard deviations of DOPs (open symbols) of Fig. 5(a). In Fig. 5(b), the corresponding output DOPs first decrease slightly, then increase when the parametric pump powers are larger than ~20 dBm at which the parametric gain saturates the Raman gain and provides the dominant gain. We also note that when the parametric pump power is ~19 dBm, the mean DOPs of the RA-FOPAs increase as shown in Fig. 5(b) which indicates that strong polarization pulling occurs in RA-FOPAs before Raman gain saturation. We also note that, as shown in Fig. 5(b), the variances of the output DOPs are <0.15 which indicates the differences among the output DOPs of different fiber realizations are small (more details are shown below).

Figure 6(a) shows the mean output DOP_P of the parametric pump with 100 different input SOPs (as mentioned in Step I, Sec. 3.1) over the 100 fiber realizations as a function of input parametric pump powers when the Raman pump powers are 1.17, 0.8, and 0.6 W, respectively. As mentioned in Step III, Sec. 3.1, the profiles of the one with overall maximum gain in this 100 random input SOPs along the fiber were used for the simulation of Figs. 4(a)-4(c). Figure 6(b) shows the corresponding average output powers as a function of the input parametric pump powers. Figure 6(c) shows the corresponding mean output DOP_P as a function of averaged Raman gain to the parametric pumps.

Fig. 6 (a) The mean output DOP_P of the parametric pump with 100 different input SOPs (as mentioned in Step I, Sec. 3.1) over the 100 fiber realizations as a function of input parametric pump powers when the Raman pump powers are 1.17, 0.8, and 0.6 W, respectively. (b) The corresponding average output powers as a function of input parametric pump powers. (c) The corresponding mean output DOP_P of the parametric pump as a function of average Raman gain to the parametric pumps.

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In Fig. 6(c), we observed that when the parametric pump power is small, the relationship between the DOP_P and Raman gain are close to the interpolation function of Eq. (9) with Γ = 10.2, which behaves as counter-propagating FRAs. When we increase the input parametric pump power, as shown in Figs. 6(a) and 6(c), the mean output DOP_P decreases along with the reducing Raman gain. When the input parametric pump is smaller than 10% of input Raman pump power, despite the Raman gain depletion, the corresponding data points can still be above the curve of Eq. (6) with Γ = 10.2 except when the Raman pump power is 1.17 W. We observed that higher DOP_P can be achieved in Fig. 6(c) than that of the small signal FRAs. It is because of the strong polarization attraction between the Raman pump and parametric pump. As shown in Figs. 6(b) and 6(c), when the input powers of parametric pump are around one-sixth of the input Raman pump powers, the Raman pump will be deeply saturated and the corresponding DOP_P will be smaller than 0.35. It indicates that in this case the competition between these two pumps will cause unpredictable mutual polarization pulling to each other. If we keep increasing the parametric pump power, when the output powers of the parametric pump are comparable with the input Raman pump powers, the corresponding DOP_P will begin to increase. In this case, the relatively large parametric pump overcomes the polarization pulling from the counter propagating Raman pump and behaves like an FOPA.

Figure 7(a) shows the relationships between the mean DOPs and the peak of the mean gains over the bandwidth of RA-FOPAs as shown in Figs. 4(a)-4(c). From Fig. 7(a), the relationships of the mean DOPs and the peak gains are similar for different RA-FOPAs. When the input parametric pump powers increase, initially the mean DOPs slightly increases but the peak gains decrease. Then, both the mean DOPs and the peak gains decrease and then increase. Figure 7(b) shows the details of the dynamics when P_R = 0.8 W. The numbers in the parentheses refer to parametric pump powers as shown in the legend of Fig. 4.

Fig. 7 (a) The relationships of the mean DOP and the mean gain at the gain peak of the RA-FOPAs when the Raman pump powers are 1.17 (squares), 0.8 (circles), and 0.6 W (triangles), respectively. (b) The four different operating states, FRAs, RA-FOPAs, deep saturation of the FRAs, and FOPAs of the RA-FOPAs when P_R = 0.8 W. The numbers in the parentheses refer to different parametric pump powers as shown in the legend of Fig. 4. Selected data points with different pump combinations are also shown for Fig. 8.

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Fig. 8 Simulated output SOPs of the signal (top row) and the idler waves (bottom row) in the realizations with the global maximum gain for different pump combinations, (a) P_R = 0.6 W, P_P = 2 mW, (b) P_R = 0.8 W, P_P = 80 mW, (c) P_R = 0.8 W, P_P = 128 mW, (d) P_R = 0.6 W, P_P = 160 mW, as shown in Fig. 6(a). The SOPs are normalized by the maximum S₀(L) and D₀(L). Most of the data points thus are inside instead of on the surface of the Poincare sphere. The output SOPs of the parametric pump are also shown in black solid diamonds. The input SOP of the Raman pump is fixed at (1, 0, 0). The details of the corresponding results are shown in Table 2.

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Figure 9 The output DOP_G as a function of the mean gains of the RA-FOPAs using 100 different fiber realizations (open circles) for the four pump combinations shown in Table 2. The data points of the corresponding realizations shown in Figs. 8(a)-8(d) are also shown in solid diamonds.

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According to the discussion on the mutual polarization pulling between the bi-directionally propagating pumps in Figs. 6(a)-6(c), we categorize the general behaviors of the polarization pulling in RA-FOPAs into four operating states as follows.

(1) FRA state. As shown in Figs. 4(a)-4(c), when the parametric pump is small, for each pump combinations, the data points cluster together. In this case, the RA-FOPAs operate as counter-propagating FRAs. The mean DOP_G as well as mean gain are related by Eq. (9) as that of the counter-propagating FRAs with Γ = 10.2 [13]. The result is consistent with the uniform Raman gain profile approximation used which indicates that the Raman gain dominates the operation of the RA-FOPAs. Figure 8 shows the corresponding simulated output SOPs of the signals and the idlers in the realizations with the global maximum gain of four typical pump combinations in each of these four operating states. Here, the SOPs of the signal and the idler are normalized by the maximum S₀(L) and D₀(L) of the 100 generated input signals described in Section 3.1, respectively. Thus most of the data points of the SOPs are inside instead of on the surface of the Poincaré sphere. The output SOPs of the parametric pump are also shown in black solid diamonds. The details of the corresponding results of Fig. 8 are shown in Table 2
Table 2. Corresponding Simulation Results of Fig. 8
View Table | View all tables in this article
. Fig. 8(a) shows the output SOPs of the signals (top) and the idlers (bottom) of a typical pump combination (Combination 1) operating in the FRA state. From the bottom of Fig. 8(a), since the parametric process merely generates the idlers, the output DOPs of the idlers are pulled by the Raman pump and thus cannot be as high as that in the FOPAs [9].
(2) RA-FOPA state. When parametric pump power is smaller than 10% of input Raman pump power, the amplified parametric pump will provide significant gain but not large enough to saturate the Raman gain. As discussed in Fig. 6(c), in this state the strong polarization attraction between the pumps results in a higher DOP_P of the parametric pump than that of small signal FRAs. For the output signal, as shown in Fig. 4, most of the corresponding data points lie between the interpolation functions with Γ = 8.5 and 10.2. Depending on whether the parametric gain dominates the overall gain, the RA-FOPAs states can be further separated into two sub-categories.
- (a) Enhanced FRA. In this sub-category, higher parametric gain will induce stronger polarization pulling in RA-FOPAs than that of counter-propagating FRAs. The mean output powers of the idlers are 3~10 dB smaller than those of the output signals. In the realizations with the global maximum gain, both of the DOPS of the output signal and the idler can be >0.95
- (b) Optimum RA-FOPA. In this sub-category, a) Enhanced FRA. In this sub-category, higher parametric gain will induce stronger polarization pulling in RA-FOPAs than that of counter-propagating FRAs. The mean output powers of the idlers are 3~10 dB smaller than those of the output signals. In the realizations with the global maximum gainthe parametric gain is large and it dominates. As a result, the data points scatter because of the gain differences across the parametric gain bandwidth as shown in Figs. 4(b)-4(c). As shown in Fig. 6(c) and Fig. 8(b), the slightly depleted Raman pump can still attract the output SOP of the parametric pump and pull it close to the input SOP of Raman pump. These two pumps in turn attract the SOPs of signal and the idler together. Thus, different from the Enhanced FRA state, as shown in Figs. 4(b)-4(c), the data points in this state are approximated well by Eq. (9) with Γ = 8.5 across the gain bandwidth of RA-FOPAs. In the realization with the global maximum gain, as shown in Fig. 8(b) and Table 2, the signals and idlers at similar power level and larger than 0.97 output DOPs can be achieved. We note that the output DOP of the idler is 0.999443 which indicates that the optimum polarization attraction was achieved in this state. From Figs. 4(b)-4(c), the mean gain difference over the parametric gain bandwidth is less than 2 dB indicating that the polarization attraction can be achieved over the parametric gain bandwidth.
  (3) Deep saturation state. As shown in Figs. 6(b) and 6(c), when the input parametric pump is larger than 10% of and smaller than one-sixth of input Raman pump power, the output powers of the parametric pump become comparable with the input Raman pump powers. In this case, the competition between these two pumps will cause unpredictable mutual polarization pulling leading to significant phase mismatch and thus reduces both the overall gains and the output DOPs of the RA-FOPAs. We observed that the DOP_P of the amplified parametric pump decreases to < 0.35 with the Raman gain shown in Fig. 6(c). As shown in Fig. 7(b) and Figs. 4(a)-4(c), most of the data points in this state are distributed below the curve of Eq. (9) with Γ = 10.2 which indicates that the RA-FOPAs operating in the deep saturation state requires higher gain than that of counter-propagating FRAs to achieve the same output DOPs. Figure 9 shows the output DOP_G as a function of the mean gains of RA-FOPAs using 100 different fiber realizations (open circles) for the four pump combinations shown in Table 2. The data points of the corresponding realizations shown in Figs. 8(a)-8(d) are shown as solid diamonds. In this case, as shown in Figs. 9(b)-9(c) and Table 2, although high maximum gain and high DOP_G can still be achieved, the variance of the mean gain and mean conversion efficiency will be 3-5 dB larger than that in the optimum RA-FOPAs state.
  (4) FOPA state. As discussed in Fig. 6(c), when the input powers of parametric pump are larger than one-sixth of the input Raman pump powers, the RA-FOPAs behave as FOPAs. From Fig. 7(b), we observed that in this state, the data points corresponding to the parametric gain peak are distributed above the interpolation functions of Eq. (9) with Γ = 8.5, which indicates higher DOPs can be achieved in this state at the parametric gain peak. We also note that, in Fig. 7(a), both of the values of the mean gains and DOPs of Combination 4 are similar to that of Combination 2 (in optimum RA-FOPAs state). Figs. 9(b) and 9(d) show the details of the distributions of the output in 100 different fiber realizations with the same pump combinations. Although a higher maximum gain and conversion efficiency can be achieved using Combination 4 (in FOPA state), as shown in Fig. 8(d) and Table 2, the corresponding DOPs of gain and conversion efficiency are smaller than that of Combination 2 in the realizations with the global maximum gains. This is because the mutual polarization pulling between the counter-propagating pumps prevents the polarization attraction in RA-FOPAs operating in the FOPAs state. In Figs. 5(a)-5(b) and Figs. 9(b) and 9(d), the variances of both the mean gains and mean DOP_G of the Combination 4 are larger than those of Combination 2 indicating that the output SOPs of the signals are more unpredictable in the FOPAs state. Moreover, Fig. 9(d) also shows that the overall maximum gain and maximum mean gain cannot be achieved in the same realization. Finally, as shown in Figs. 4(a)-4(c), the >10 dB mean gain difference over the parametric gain bandwidth indicates that in the FOPAs state such strong polarization pulling can only be achieved within the parametric gain peak.

From Fig. 7, the optimum RA-FOPAs state is difficult to achieve in RA-FOPAs because the Raman gain can be deeply saturated easily. Although higher Raman pump can provide higher gain to the parametric pump, it will make the RA-FOPAs directly go from the enhanced FRAs state into the deep saturation state but no optimum RA-FOPA state, as shown in Fig. 4(a). In this case, the amplified parametric pump then not only saturates the Raman gain but causes unpredictable mutual polarization pulling which makes its DOP_P be smaller than that of small signal FRAs as shown in Fig. 6 (c). Thus the input powers of the Raman and parametric pumps have to be carefully chosen to avoid that.

3.5 Discussion on the simulation results

Similar to the FOPAs discussed in [9], initial alignment of the input Raman pump SOP with one of the fiber principal axes cannot guarantee the global maximum gain. In other words, the optimal input SOP of the Raman pump is neither linearly polarized nor aligned with the fiber principal axes. In experiments, the optimum input SOP of the Raman pump is found by adjusting the polarization controllers [14]. However, the results in Fig. 1 suggested that the proposed model and numerical method can be used to find the state with optimum phase matching in RA-FOPAs. The corresponding statistical results characterize the general behavior of the polarization pulling in RA-FOPAs. We note that the large relative intensity noise (RIN) induced by input signal SOP fluctuations can be suppressed when the Raman gain is depleted [29]. Thus RA-FOPAs is a promising candidate to suppress RIN for practical applications of polarization pulling.

4. Conclusion

In conclusion, we propose a comprehensive theoretical model to investigate polarization pulling in bi-directionally pumped Raman assisted degenerate fiber optical parametric amplifiers using randomly birefringent fibers. In this model, for the first time the contributions of chromatic dispersion, PMD, Raman gain, and various nonlinear effects to the phase matching in RA-FOPAs are investigated. We found that the RA-FOPAs using a linearly polarized counter-propagating Raman pump with fixed input SOPs can achieve higher gain than those with fixed input SOPs of parametric pumps. We characterize four different states of polarization pulling effects in RA-FOPAs. We also found that polarization attraction effect can be achieved in the optimum phase-matching state of the bi-directionally pumped RA-FOPAs if the parametric pump power is chosen carefully to avoid deep saturation of Raman gain.

Acknowledgments

This work was supported by National Natural Science Foundation of China (NSFC) under Project 61205049, and Natural Science Foundation of Fujian Province, under Project 2013J05089.

References and links

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14. F. Chiarello, L. Palmieri, M. Santagiustina, R. Gamatham, and A. Galtarossa, “Experimental characterization of the counter-propagating Raman polarization attraction,” Opt. Express 20(23), 26050–26055 (2012). [CrossRef] [PubMed]

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Parameters	α (dB/km)	L (m)	γ (W⁻¹·km⁻¹)	$g_{R}^{Ω}$ (W⁻¹·km⁻¹)	D (ps/nm/km)	D_PMD (ps/km^1/2)
HNLF1 [23]	0.91@1550 nm*	1000*	13.6^†	4.8^†	0.12@1555 nm*	0.08*
HNLF2 [24]	0.79@1550 nm*	1000*	11.2^#	3.8^†	0.12@1555 nm*	0.06*

Parameters	Combination 1	Combination 2	Combination 3	Combination 4
State	1)FRAs	2-b) Optimum RA-FOPAs	3) Deep Saturation	4) FOPAs
R₀(L) (W)	0.6	0.8	0.8	0.6
P₀(0) (mW)	2	80	128	160
<G> (dB)	13.9743	17.6942	20.2996	15.0783
*<σ_G> (dB)*	10.0802	6.5054	9.39294	6.04545
G_max (dB)	16.3836	22.7045	26.4358	24.7394
DOP_G	0.883823	0.969985	0.982487	0.67361
<CE> (dB)	−28.7463	17.4182	20.2077	14.3939
*<σ_CE> (dB)*	0.000774	8.06294	12.4766	10.7237
CE_max (dB)	−26.4454	22.5225	26.4087	24.7522
DOP_CE	0.617027	0.999443	0.999102	0.944372
P₀(L) (W)	0.0383317	0.555814	0.536373	0.534083

Parameters	α (dB/km)	L (m)	γ (W⁻¹·km⁻¹)	$g_{R}^{Ω}$ (W⁻¹·km⁻¹)	D (ps/nm/km)	D_PMD (ps/km^1/2)
HNLF1 [23]	0.91@1550 nm*	1000*	13.6^†	4.8^†	0.12@1555 nm*	0.08*
HNLF2 [24]	0.79@1550 nm*	1000*	11.2^#	3.8^†	0.12@1555 nm*	0.06*

Parameters	Combination 1	Combination 2	Combination 3	Combination 4
State	1)FRAs	2-b) Optimum RA-FOPAs	3) Deep Saturation	4) FOPAs
R₀(L) (W)	0.6	0.8	0.8	0.6
P₀(0) (mW)	2	80	128	160
<G> (dB)	13.9743	17.6942	20.2996	15.0783
*<σ_G> (dB)*	10.0802	6.5054	9.39294	6.04545
G_max (dB)	16.3836	22.7045	26.4358	24.7394
DOP_G	0.883823	0.969985	0.982487	0.67361
<CE> (dB)	−28.7463	17.4182	20.2077	14.3939
*<σ_CE> (dB)*	0.000774	8.06294	12.4766	10.7237
CE_max (dB)	−26.4454	22.5225	26.4087	24.7522
DOP_CE	0.617027	0.999443	0.999102	0.944372
P₀(L) (W)	0.0383317	0.555814	0.536373	0.534083

Polarization pulling in Raman assisted fiber optical parametric amplifiers

Abstract

1. Introduction

2. Theoretical model

3. Results and discussion

3.1 Numerical method

3.2 Comparison with experimental results

3.3 Mean gains and conversion efficiencies

3.4 Characterizing the polarization pulling in bi-directionally pumped schemes

3.5 Discussion on the simulation results

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Tables (2)

Equations (9)

Optics Express