## Abstract

In this paper, Raman induced polarization dependent gain (PDG) in orthogonally pumped optical parametric amplifiers is investigated. Based on the Manakov Eqs., complete coupled Eqs. are derived and numerically solved. Analytical approximate solutions are derived. The simulation results show that in orthogonally pumped optical parametric amplifiers, the Raman effect between the pump and the signal contributes more prominently to the PDG than that induced by asymmetrical pump depletion.

© 2006 Optical Society of America

## 1. Introduction

Optical parametric amplifier (OPA) has the potential to provide much wider amplification bandwidth with relatively low Raleigh scattering noise and high gain than other optical amplifiers [1]. The main withdraw is that its gain highly depends on the polarization state of the pumps so that high polarization dependent gain (PDG) may prevent it from practical applications for long haul DWDM transmission systems. Though various studies have been carried out and methods have been proposed to reduce PDG by, e.g., employing two orthogonally polarized pumps [2–5], experiment shows that PDG in such OPAs is not negligible [5–7]. Theoretical analysis has been carried out on the OPA with two orthogonal pumps [8–9]; however, there has been no satisfactory explanation on such experimental results of remaining PDG. One possible factor accounting for the discrepancy between theoretical and experimental results is the inaccuracy arising from the assumption of no pump depletion. In some applications, high pump efficiency is usually desired and power depletion as high as 92% has been reported [10]. Pump depletion in two pump OPA has also been investigated in Refs [11–12], and PDG due to asymmetrical pump depletion in the orthogonally pumped OPA is analyzed in Ref [13]. However, the theoretical result is much less significant than the mentioned experimental result. Gain obtained via stimulated Raman scattering in the OPA will be the main resource of large PDG. As is known, high nonlinear fiber has relatively larger Raman gain in comparison with conventional single-mode optical fiber. Therefore, Raman amplification should not be neglected [14–16] so that there will be asymmetry gain spectrum and enlarged PDG in OPAs. Related studies include Refs [17–18], which investigated Raman effect and FWM simultaneously, and Refs [19–20], which analyzed Raman induced noise in OPA.

In this paper, we present a relatively complete model for orthogonally pumped OPA that includes the Raman effect and the asymmetrical pump depletion. Analytical proximate solution is provided and compared with the numerical results. Simulation results with different OPA parameters are given and explanations are presented.

## 2. Theoretical model

In orthogonally pumped OPAs, the center wavelength of the two pumps is usually placed near the zero dispersion wavelength to produce non-degenerated four-wave-mixing for signal amplification. The wavelength separation should, however, be large enough to avoid the degenerated four-wave-mixing, which may introduce severe inter-channel interactions [2] [4], [21–24]. Since the pump wavelength is far from the zero dispersion wavelength, the two idlers, with frequencies at 2*ω*
_{1} - *ω*
_{3} and 2*ω*
_{2} - *ω*
_{4}, can be neglected. They may alter the gain slightly when *ω*
_{3} is close to *ω*
_{1}. When *ω*
_{3} is far from *ω*
_{1}, the effect is negligible.

Henceforth, some degenerated terms in the mathematical model proposed in Ref [13] can be removed. Raman interactions take place among the pumps, the signals and the idlers. To have better performance, fiber with low birefringence is required. While fabricating the low birefringent fiber, it is difficult to manufacture almost perfect circular cross-sections reproducibly; therefore, the birefringence axes and birefringence value vary randomly with distance [9]. Therefore we assume that the optical fiber used in this paper is randomly birefringent and the pulse propagation in such fiber can be characterized by Manakov Eqs. [9, 25–28]. In the analysis of Ref [5], Manakov-Eq. model was not adopted. However, in the subsequent paper, the authors modify their model similar to Manakov-Eq. to explain their experiment with better accuracy [29]. Henceforth, taking the Raman scattering into account, we derive our Eq. based on Manakov Eq. model:

$$\frac{\partial {A}_{\mathrm{iy}}}{\partial z}=\left(\mathrm{jR}\left(\mid {A}_{\mathrm{iy}}^{2}\mid +2\sum _{j=1,j\ne 1}^{4}\mid {A}_{\mathrm{jy}}^{2}\mid +\sum _{j=1}^{4}\mid {A}_{\mathrm{jx}}^{2}\mid \right)+\frac{1}{2}\sum _{j=1,j\ne 1}^{4}\left({g}_{\parallel}\left({v}_{j},{v}_{i}\right)\mid {A}_{\mathrm{jy}}^{2}\mid +{g}_{\perp}\left({v}_{j},{v}_{i}\right)\mid {A}_{\mathrm{jx}}^{2}\mid \right)\right){A}_{\mathrm{iy}}+\mathrm{jR}\left({A}_{3x}{A}_{4y}{A}_{3-i,x}^{*}+{A}_{4x}{A}_{3y}{A}_{3-i,x}\right)\mathrm{exp}\left(j\Delta \mathrm{\beta z}\right)$$

$$\frac{\partial {A}_{\mathrm{iy}}}{\partial z}=\left(\mathrm{jR}\left(\mid {A}_{\mathrm{iy}}^{2}\mid +2\sum _{j=1,j\ne 1}^{4}\mid {A}_{\mathrm{jy}}^{2}\mid +\sum _{j=1}^{4}\mid {A}_{\mathrm{jx}}^{2}\mid \right)+\frac{1}{2}\sum _{j=1,j\ne 1}^{4}\left({g}_{\parallel}\left({v}_{j},{v}_{i}\right)\mid {A}_{\mathrm{jy}}^{2}\mid +{g}_{\perp}\left({v}_{j},{v}_{i}\right)\mid {A}_{\mathrm{jx}}^{2}\mid \right)\right){A}_{\mathrm{iy}}+\mathrm{jR}\left({A}_{1x}{A}_{2y}{A}_{7-i,x}^{*}+{A}_{2x}{A}_{1y}{A}_{7-i,x}\right)\mathrm{exp}(-j\Delta \mathrm{\beta z})$$

where *γ* is the nonlinear coefficient and $\u0331R=\frac{8}{9}\gamma $. Δ*β* is the phase mismatch. *A*
_{1x} and *A*
_{2y} are the amplitudes of the two orthogonal pumps. We add *A*
_{2x} and *A*
_{1y} into Eq. (1) for the completeness of the model. Although they are equal to zero at the input, the decay of signal and idler photons may produce pump photons.̱*A*
_{3x} and *A*
_{3y} are the signal amplitudes of the two orthogonal states. *A*
_{4x} and *A*
_{4y} are the idler amplitudes respectively. Figure 1 shows the relationship between them. It is worth noting that x and y stand for two orthogonal polarization components which vary randomly together with the birefringent axes during the wave propagation.*g*
_{∥} (*V _{i}*,

*V*) and

_{j}*g*

_{⊥}(

*V*,

_{i}*V*) are the co-polarized and orthogonal Raman gain coefficient between frequency

_{j}*V*and

_{i}*V*, respectively:

_{j}where *A _{eff}* is the effective area of the fiber.

*g*

_{∥/⊥i}(

*V*-

_{i}*V*) is the co-polarized/orthogonal Raman gain spectrum. Since the orthogonal Raman coefficient is an order-of-magnitude smaller than the co-polarized coefficient, it can be neglected in applications [30–31].

_{j}The above set of Eqs. is highly coupled and nonlinear, and is hard to obtain analytical solutions. If no special technique reusing the pump power is taken, the pump depletion due to parametric amplification is relatively small in comparison with the Raman depletion, as will be shown in the next section. Furthermore, the amplitude of signal and idler is relatively small in calculating the phase shift. Also the contribution of *A*
_{2x} and *A*
_{1y} is small and can be neglected for simplification. Based on the above consideration, the nonlinear Eq. can be decoupled and simplified to the following form:

$$\frac{\partial {A}_{2y}}{\partial z}=\mathrm{jR}\left(\mid {A}_{2y}^{2}\mid +\mid {A}_{1x}^{2}\mid \right){A}_{2y}\phantom{\rule{4em}{0ex}}i=\mathrm{3,4}$$

$$\frac{\partial {A}_{\mathrm{ix}}}{\partial z}=\mathrm{jR}\left(2\mid {A}_{1x}^{2}\mid +\mid {A}_{2y}^{2}\mid \right){A}_{\mathrm{ix}}+\frac{1}{2}[{g}_{\parallel}({\omega}_{1},{\omega}_{3})\mid {A}_{1x}^{2}\mid +{g}_{\perp}({\omega}_{2},{\omega}_{3})\mid {A}_{2y}^{2}\mid ]{A}_{\mathrm{ix}}+\mathrm{jR}\left({A}_{1x}{A}_{2y}{A}_{7-i,y}^{*}\right)\mathrm{exp}\left(-j\Delta \mathrm{\beta z}\right)$$

$$\frac{\partial {A}_{\mathrm{iy}}}{\partial z}=\mathrm{j\gamma}\left(2\mid {A}_{2y}^{2}\mid +\mid {A}_{1x}^{2}\mid \right){A}_{\mathrm{iy}}+\frac{1}{2}[{g}_{\parallel}({\omega}_{2},{\omega}_{3})\mid {A}_{2y}^{2}\mid +{g}_{\perp}({\omega}_{1},{\omega}_{3})\mid {A}_{1x}^{2}\mid ]{A}_{\mathrm{iy}}+\mathrm{jR}\left({A}_{1x}{A}_{2y}{A}_{7-i,x}^{*}\right)\mathrm{exp}\left(-j\Delta \mathrm{\beta z}\right)$$

We see that *A*
_{3x} and *A*
_{4y} as well as *A*
_{4x} and *A*
_{3y} form two sets of de-coupled Eqs. and can be solved analytically. If there is no idler power at the input, we can obtain the analytical solutions by the procedure similar to Refs [17] and [18]:

$${A}_{4y}\left(L\right)=\mathrm{exp}\left({M}_{4y}L\right)\mathrm{exp}\left(-j\frac{{\kappa}_{x}}{2}L\right)\frac{\mathrm{j\gamma}\sqrt{{P}_{1x}{P}_{2y}}}{{g}_{x}^{*}}\mathrm{sinh}{\left({g}_{x}L\right)}^{*}{A}_{3x}\left(0\right)$$

where:

$${M}_{4y}=\mathrm{jR}\left(2{P}_{2y}+{P}_{1x}\right)+\frac{1}{2}{g}_{\parallel}({\omega}_{2},{\omega}_{4}){P}_{2y}+\frac{1}{2}{g}_{\perp}({\omega}_{1},{\omega}_{4}){P}_{1x}$$

$${\kappa}_{x}=\Delta \beta +R\left({P}_{1x}+{P}_{2y}\right)+j[\frac{1}{2}{g}_{\parallel}({\omega}_{2},{\omega}_{3}){P}_{2y}+\frac{1}{2}{g}_{\perp}({\omega}_{1},{\omega}_{3}){P}_{1x}-\frac{1}{2}{g}_{\parallel}({\omega}_{1},{\omega}_{3}){P}_{1x}-\frac{1}{2}{g}_{\perp}({\omega}_{2},{\omega}_{3}){P}_{2y}]$$

$${\kappa}_{\mathrm{}y}=\Delta \beta +R\left({P}_{1x}+{P}_{2y}\right)+j[\frac{1}{2}{g}_{\parallel}({\omega}_{2},{\omega}_{4}){P}_{2y}+\frac{1}{2}{g}_{\perp}({\omega}_{1},{\omega}_{4}){P}_{1x}-\frac{1}{2}{g}_{\parallel}({\omega}_{1},{\omega}_{4}){P}_{1x}-\frac{1}{2}{g}_{\perp}({\omega}_{2},{\omega}_{4}){P}_{2y}]$$

$${g}_{x}^{2}={R}^{2}{P}_{1x}{P}_{2y}-{\left(\frac{{\kappa}_{x}}{2}\right)}^{2}$$

$${g}_{y}^{2}={R}^{2}{P}_{1x}{P}_{2y}-{\left(\frac{{\kappa}_{y}}{2}\right)}^{2}$$

*A*
_{4x} and *A*
_{3y} can be obtained similarly. Raman induced PDG can also be calculated analytically.

## 3. Numerical results

Reference [14] gives measured Raman gain coefficient spectrum of the high nonlinear fiber. The other parameters used are: nonlinear coefficient *γ* is 18 *W*
^{-1}/*km*, the dispersion slope is 0.031 *ps*/*nm*
^{2} ● *km*, and the zero dispersion wavelength is at 1540.2 nm. The parameter of the fiber used in the simulation is exactly the same as the one in Ref [14] with the length of 1km. The pumps with power of 22.5 dBm are at 1525.1 nm and 1555.4 nm, respectively. The separation of the pump wavelength is about 30nm similar to Ref [5] and large enough to avoid the degenerated four-wave-mixing. The signal wavelengths are in the 1550 nm band and the power is set at 0 dBm per channel.

Figure 2 shows the gain spectra with and without the Raman effect. The input signals are supposed to be linearly polarized with an angle of 45 degrees to the x-axis. The numerical results obtained directly from Eq. (1) (triangle) and the analytical results given by Eq. (3) (circle) are in good agreement. The gain without the effect of stimulated Raman scattering but taking the pump depletion into consideration is calculated numerically and plotted as the square line shows. It can be clearly seen that the Raman effect changed gain profile in an asymmetrical way. Such result has been observed experimentally in Ref [14]. Therefore in orthogonally pumped OPAs, the Raman effect should not be neglected.

The PDGs with and without the Raman effect, calculated both numerically and analytically at one of the signal channels at wavelength of 1529 nm, are presented in Fig. 3. The PDG due to stimulated Raman scattering is calculated numerically and analytically (the triangle and circle lines). The remaining PDG due to the pump asymmetrical depletion, without considering the Raman effect, is plotted by the square line. From the simulation we see that the Raman effect adds about 1.6 dB additional PDG to the asymmetrical pump depletion induced PDG (about 0.1 dB). The reason why Raman effect induces so large PDG is as follows. In OPA the signal wavelengths are between the two pump wavelengths as illustrated in Fig. 4.

The signal obtains the maximum Raman gain from the shorter wavelength pump and has minimum depletion on the longer wavelength pump if its polarization state is parallel to that of the former. On the other hand, if the signal has the polarization state paralleled to that of longer wavelength pump, it has the minimum Raman gain and maximum depletion.

Then we investigated the dependence of PDG on parameters such as the nonlinear coefficient and the dispersion slope. The dispersion slope is evaluated at the zero dispersion wavelength. The signal wavelength is 1529nm and the input power is 0dBm. The results are shown in Fig. 5 and Fig. 6.

From Fig. 5 we can see that the PDG increases with the nonlinear coefficient. This can be explained as larger nonlinear coefficient usually brings about greater Raman gain coefficient. Though stimulated Raman scattering is independent on the dispersion, the parametric gain depends on the even-order dispersion coefficients at the center frequency of the pumps for it affects the phase mismatch in FWM. And the second-order dispersion coefficient at the center frequency of the pumps is in proportion to the dispersion slope at the zero dispersion wavelength. The fourth order dispersion coefficient is also associated with it [14]. This explains why PDG has dependence dispersion slope as Fig. 6 shows. The results tell us that for the dispersion slope of interests, the PDG is usually small. Extreme value of the dispersion slope may leads to large PDG.

## 4. Conclusion

In this paper, we analyzed factors that leads to large polarization dependent gain in orthogonally pumped optical parameter amplifiers and presented a relatively complete model that includes the effects of asymmetrical pump depletion and stimulated Raman scattering. Theoretical analysis and numerical simulation were performed. The analytical solution, which is useful in practical applications such as system optimization, is also given and the results agree well with the numerical ones. Simulation shows that in OPAs, PDG is dominated by Raman effect. Over 1.5 dB Raman induced PDG has been observed. It may give rise to power penalty and should be taken into consideration in system design and analysis.

## Acknowledgments

This work is partially supported by NSFC (ID: 60377013, 90204006, 60507013), Ministry of Education, China (ID:20030248035) and STCSM(ID: 036105009)

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