Optimization of Raman-Assisted Fiber Optical Parametric Amplifier Gain

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JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 29, NO. 8, APRIL 15, 2011

Optimization of Raman-Assisted Fiber Optical Parametric Amplifier Gain Shao Hao Wang, Member, IEEE, Lixin Xu, P. K. A. Wai, Senior Member, IEEE, and Hwa Yaw Tam, Senior Member, IEEE

Abstract—We studied the gain of Raman-assisted fiber optical parametric amplifiers (RA-FOPAs) both theoretically and experimentally. We investigated the relationship between the overall gain and different combinations of Raman and parametric pump powers using contour maps. We derived a normalized phase-matched model to determine the general behavior the peak gains of RA-FOPAs operating in the small signal region. The contour maps of the combined gain enhancement can be used to optimize both indirect and direct Raman gain of the signal/idler in RA-FOPAs. We showed that, in a given fiber, it is possible to optimize the Raman and parametric pump powers in RA-FOPAs for high gain and high efficiency. Index Terms—Nonlinear optics, optical fiber amplifiers, optical parametric amplifiers, Raman scattering.

I. INTRODUCTION

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PTICAL amplifiers are essential components in fiberoptic communication systems [1]–[6]. In optical transmission links, erbium-doped fiber amplifiers (EDFAs) boosts the capacity of the fiber-optic communication systems because its gain bandwidth coincides with the third transmission window of silica-based optical fibers. In recent years, Internet bandwidth usage keeps increasing. The annual growth was more than 60% globally in 2009 [7]. Future fiber-optic communication systems must continue to increase their capacities to meet the increasing demand. One way to increase system capacity is to increase the spectral efficiency using advanced modulation formats. An alternate way to increase system capacity is to increase the transmission bandwidth by using novel fibers. For example, AllWave fiber can provide a low loss window from 1.3 to 1.65 m which is over 40 THz transmission bandwidth [8], [9]. However, EDFAs become a bottleneck because they can only provide about 40 nm gain bandwidth covering the C- or L-band [10]. New optical amplifiers are required for future fiber-optic communication systems. Next generation optical amplifiers should Manuscript received November 11, 2010; revised January 29, 2011; accepted January 31, 2011. Date of publication February 10, 2011; date of current version April 01, 2011. This work was supported by the Research Grant Council of the Hong Kong Special Administrative Region, China, under Project PolyU 5326/ 09E. S. H. Wang is with the Department of Microelectronic, Fuzhou University, Qi Shan Campus, Fuzhou 350108, China (e-mail: [email protected]). L. Xu is with the Department of Optics and Optical Engineering, University of Science and Technology of China, Hefei 230026, China. P. K. A. Wai is with the Photonic Research Center, Department of Electronic and Information Engineering, the Hong Kong Polytechnic University, Hung Hom, Hong Kong. H. Y. Tam is with the Photonic Research Center, Department of Electrical Engineering, the Hong Kong Polytechnic University, Hung Hom, Hong Kong. Digital Object Identifier 10.1109/JLT.2011.2112636

have broad gain bandwidth, wide tuning range, low noise figure, polarization-independence, in addition to large gain. Semiconductor optical amplifiers (SOAs) are attractive as semiconductor devices are compact and can be integrated with other photonic components. However, relatively high insertion loss and optical signal-to-noise ratio (OSNR) degradation hinder commercialization of SOAs [2]. For future fiber optical amplifiers, the leading candidates are fiber Raman amplifiers (FRAs), fiber optical parametric amplifiers (FOPAs), and hybrid fiber amplifiers (HFAs) [3]–[6], [11]–[20]. A single pump fiber Raman amplifiers (FRAs) can provide about 3 THz gain band at arbitrary wavelength. Compared to EDFAs, a multi-pump FRAs can provide over 100 nm flat gain spectrum using a single piece of fiber [3]. However, relatively low Raman gain coefficient prevents the commercial use of Raman-only amplifiers in long-haul transmission. Moreover, although distributed FRAs can provide negative effective noise figure, double Rayleigh scattering limits the Raman gain to 10 to 15 dB [5]. FOPAs have higher gain efficiency than FRAs because the nonlinear coefficient in different types of optical fibers is at least twice larger than the Raman coefficient [5], [6]. Single pump FOPA can provide over 100 nm continuous gain bandwidth with dB gain [11]. A single stage one-pump FOPA can also provide 70-dB peak gain, which is larger than that of single stage FRAs and EDFAs [12]. Moreover, phase-insensitive one-pump FOPA has a 3-dB noise figure limit, which is similar to the EDFA and better than discrete FRAs [13]–[15]. Despite these advantages over EDFAs and FRAs, in conventional FOPAs the selection of parametric pump wavelength is limited by the gain bandwidth of booster EDFAs typically used in the FOPAs. Hybrid fiber amplifiers (HFAs) combined different types of fiber optical amplifiers, such as EDFAs and FRAs [16]. HFAs are attractive because of their abilities to tailor the gain profile, compensate fiber dispersion and loss, and suppress the spontaneous noise [16]–[18]. Raman-assisted fiber optical parametric amplifiers (RA-FOPAs) are one kind of HFAs, which combine different third-order nonlinear effects in optical fibers, namely stimulated Raman scattering (SRS) and four-wave mixing (FWM) processes. As both FRAs and FOPAs can provide gain at arbitrary transparency wavelength of optical fibers, it is advantageous to combine FRAs with FOPAs in order to amplify optical signals over a broad wavelength range [19]–[21]. Moreover, we have demonstrated that close to 3-dB noise figure can be achieved in RA-FOPAs by suppressing the external noise sources and carefully choosing pump powers [22]. Typical HFAs cascade different amplifiers. Thus, the gains of the HFAs are the sum of the gains of the individual amplifiers. In RA-FOPAs,

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WANG et al.: OPTIMIZATION OF RAMAN-ASSISTED FIBER OPTICAL PARAMETRIC AMPLIFIER GAIN

Fig. 1. Energy flow in RA-FOPA.

both the FRA and the FOPA use the same section of fiber, thus the optical signal are amplified by the SRS and FWM processes simultaneously. In other words, the two amplifiers are nonlinearly combined together. The gain behavior of RA-FOPAs are rather complex. The overall gain can be larger than the sum of the individual gain of Raman and parametric amplifiers [19], [20], [23]. The gain properties of RA-FOPAs thus need to be carefully characterized for optimization. In this paper, we use contour maps to describe the relationship between the overall gain and different combination of Raman and parametric pump powers. A normalized phase-matched model was derived to predict the contour maps in different fibers. We proposed to use contour maps of the combined gain enhancement to optimize both indirect and direct Raman gain to the signal/idler in RA-FOPAs. The remainder of this paper is organized as follows. In Section II, we derive the normalized phase-matched model for RA-FOPAs. In Section III, we plot the contour maps according to both experimental results and numerical simulation results using the normalized phase-matched model. We will optimize RA-FOPAs using the model developed in Section IV and discuss the phase matched model in Section V. Section VI concludes the paper. II. THEORY In RA-FOPA systems, the Raman pumps are the main source of energy. Fig. 1 shows the two paths of energy flow from the Raman pump to the signal/idler; direct Raman amplification and indirect Raman amplification. In direct Raman amplification, the Raman pump couples energy directly to the signal/ idler through the Raman process. In indirect Raman amplification, the Raman pump amplifies the parametric pump, and the enhanced parametric pump in turn amplifies the signal/idler through the parametric process. As the coupled Raman gain is proportional to the power of the Stokes wave, most energy from the Raman pump will flow into the parametric pump and only a small portion of the Raman energy will directly amplify the input signal and the idler. On the other hand, the parametric process is more efficient than the Raman process because in most optical fibers the nonlinear coefficient is larger than the Raman coefficient. The ratios between nonlinear coefficients and Raman coefficients are typically between 2.2 and 3.4 [6], [25]–[27]. Because of the

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higher efficiency in the parametric process, the input signals and the idlers can get more energy through parametric amplification than through Raman amplification, sometimes even when the parametric pump power are smaller than the Raman pump power. The gain efficiency of RA-FOPAs can thus be larger than FRAs. The RA-FOPAs have a wider gain bandwidth than that of FRAs because the signal and the idler can still get amplification through the parametric process even when their wavelengths are outside the Raman gain bandwidth. Finally, because of the smaller contribution of direct Raman amplification, the gain variance caused by the variation of Raman gain profile can also be reduced in RA-FOPAs. It is well-know that both Raman and parametric processes are polarization dependent and optical fibers are birefringent. Fiber birefringence will eventually destroy any initial polarization alignment of the pump and signal. We also note that if the fiber is sufficient long or the polarization mode dispersion (PMD) coefficient is large, such as the 1-km HNLF used in our experiment, the birefringence orientation will be fully randomized. In this case, the effective Raman coefficient is halved and the effective nonlinear coefficient is 8/9 of the effective value [28], [29]. According to [30], spectrally randomly varying birefringence sets a major limit on the gain bandwidth of FOPA, which is determined by the diffusion length. If the diffusion lengths within the parametric gain bandwidth are much longer than the fiber length, the states of polarization the parametric pump, the signal and the idler wavelengths, are correlated. In our experiment, we used one piece 1-km long HNLF with a PMD coefficient of 0.08 ps/km . The diffusion length of pump and signal separated by 12 nm is about 5.3 km. We can therefore assume that the parametric components, i.e., the parametric pump, the signal and the idler, are correlated. For RA-FOPAs, the polarizations of the Raman pump and the parametric components are decorrelated because of they are counter-propagating and the large detune ( nm) between them. Thus, the parametric components will experience the effective Raman gain coefficient due to the averaging effects. In the following we assumed that the Raman pump, the parametric pump, the signal and the idler are collinearly polarized monochromatic waves and used the measured effective coefficients for the fiber and the effective Raman gain. As is well known, in RA-FOPA, the parametric gain depends on the phase matching condition. In the Appendix, we derive a set of coupled mode equations [cf (A3)–(A6)] to describe the evolution of the amplitude of the lightwaves along the fiber in backward-pumped RA-FOPAs. Equations (A3)–(A6) can be simplified using the relative phase to describe the evolution of phase matching condition. Neglecting pump depletion caused by stimulated Brillouin scattering (SBS) and Rayleigh scattering, we can rewrite (A3)–(A6) as (1)

(2)

(3)

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(4)

(5) where and , and , as the intensities of Raman pump, parametric pump, signal and idler, respectively. Equations (1)–(4) govern the evolutions of the Raman pump, the parametric pump, the signal and the idler. The evolution of phase difference is described in (5). In (5), the first two terms on the right-hand side corresponds to chromatic dispersion and Raman pump induced phase mismatches, respectively. Phase mismatch due to chromatic dispersion can , which is the difference be written as between the wave numbers of the parametric pump, the signal and the idler. This phase mismatch can be approximately written as , where and are the group-velocity dispersion (GVD) coefficient and the fourth order dispersion coefficient at the parametric pump frequency, respectively. The rest of the terms on the right-hand side of (5) are phase-mismatch caused by nonlinear effects. Equations (1)–(5) can be further simplified by neglecting intra-Raman effects of parametric pump, signal and idler, and assuming that Raman gain profile is flat, i.e., and . This assumption is justified for RA-FOPAs with less than several tens of nanometers gain bandwidth or when the indirect Raman gain dominant. As the intensities of the signal and the idler are small when compared with parametric pump power, we can rewrite (1)–(5) as (6) (7) (8) (9)

parametric pumps of RA-FOPAs. Note that in (6)–(10), we have assumed uniform Raman gain for all the lightwaves involving the parametric process. As shown in Fig. 1, when the indirect Raman gain through the parametric pump dominates, the enhanced parametric pump will become the major energy source of the signal and the idler. By using this assumption, we can derive a normalized model from (6)–(10). We use the products of Raman coefficient and input Raman pump and fiber length to describe the Raman gain assistance in different fibers. The normalized model can be written as (12) (13) (14) (15)

(16) where , and . For small signal, the Raman and parametric pumps together serve as one virtual pump that provides gain to the signal and the idler. We can then assume that the virtual pump is undeplete, which is used in the modeling of FRAs and OPAs [5], [24]. If the Raman gain is uniform, for small signal RA-FOPAs with the same Raman and parametric pump powers, the peak gain values remain the same even when the pump wavelengths change. For FOPAs whose pumps operate in anoumlous dispersion region, although the gain profiles vary with the dispersion profiles, the peak gains only depend on the pump powers, fiber lengths, and nonlinearities [24]. We observed that the same is also true for RA-FOPAs. We also note that the phase-matching conditions at the parametric gain peak wavelength are satisfied or nearly satisfied along propagation distance. Since we focus on the peak gain value of RA-FOPAs only, the normalized model can be further simplified by assuming that the phase-matching condition is always satisfied in the peak region. Thus, (12)–(16) can be written as (17)

(10) (18) Here, we define the nonlinear coefficient as and assume

(19) (11)

where and are the frequencies of parametric pump and signal, respectively. Equations (6)–(10) can be reduced to the model in [30] by turning off the Raman pump and neglecting fiber loss at the parametric pump frequency. In the following, we will use (6)–(10) to characterize the relationship between the overall gains and different combinations of the Raman and

(20) where , are intensities. Note that we assume the losses at the wavelengths of parametric pump, the signal and the idler are nearly equal, , which is true for the new dry fiber having a low-loss window and the HNLF in our experiment [23], [25]. In this model, the number of independent


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Fig. 2. Experimental setup of RA-FOPA. DFB: distributed feedback laser, TLS: tunable laser source, PC: polarization controller, PM: phase modulator, BPF: bandpass filter, CIR: circulator.

parameters is reduced to four, which are , and . The parameter is the product of Raman coefficient, Raman pump power and fiber length and represents the Raman gain; is the ratio between nonlinear coefficient and Raman coefficient; the parameter represents the input parametric pump power and represents the fiber loss and fiber length. By using this normalized phase-matched model, we can characterize the gain of the RA-FOPAs for different fibers and pumps. We also note that both of the models given by (7)–(11) and (12)–(16) neglect higher order idlers. When the gain of RA-FOPAs is saturated, the power in higher order idlers will become comparable to that of the output signal and fundamental idler. The system power will spread out and our model will become invalid. A number of nonlinear effects will occur during saturation, such as backward SBS, double Rayleigh scattering, and SRS among the FWM components, which will manifest as noise sources and further deplete the Raman and parametric pumps. In our experiment, we also observed a SBS Stokes of 16.0 dBm when a 2.5 GHz phase modulated signal at 23.1 dBm is amplified by a 1.0 W Raman pump. Hence, these extra nonlinear effects should be avoided. III. CHARACTERIZATION OF RA-FOPA GAIN A. Experimental Setup and Results Fig. 2 shows a typical experimental setup of an RA-FOPA. The parametric pump was modulated by a phase modulator (PM) using pseudo-random binary sequence (PRBS) at 2.5 GHz in order to suppress SBS. We used a 1-km long dispersionflattened HNLF with a chromatic dispersion of 0.12 ps/nm km at 1555 nm, a dispersion slope of 0.01 ps/nm km, a nonlinear km , a Raman gain coefficient of coefficient of 13.6 4.8 W km , and an attenuation of 0.91 dB/km at 1550 nm. A 1455-nm CW Raman laser serving as the Raman pump was launched into the HNLF in the counter-propagation direction. The total insertion loss for the input signal was measured to be 12.2 dB, which is mainly due to the 90:10 coupler. The net wavelength conversion efficiencies were measured in the optical spectrum analyzer by comparing the signal input to the corresponding idler output. Fig. 3 show the (a) on-off gain and (b) wavelength conversion efficiency spectra for wavelength 1554.9 nm. Squares, diamonds and circles represent the measured results of the Raman-assisted FOPA, FOPA without Raman pump, and Raman amplifier only, respectively. The parametric pump power after CIR1 was 128 mW and the Raman pump power

Fig. 3. (a) On-off gain and (b) wavelength conversion efficiency spectra for wavelength 1554.9 nm. Squares, diamonds and circles represent the measured results of the Raman-assisted FOPA, FOPA without Raman pump, and Raman amplifier only, respectively. The solid line and the dashed line were simulation results of the FOPA with and without Raman assistance, respectively. The parametric pump power after CIR1 was 128 mW and the Raman pump power after CIR2 was 1.17 W. The input signal power was fixed at 14 dBm.

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after CIR2 was 1.17 W. The input signal power was fixed at dBm. The solid line and the dashed line were simulation results of the FOPA with and without Raman assistance, respectively, using (6)–(10). In the simulation, we first used a shooting method to solve for the Raman power profile along the fiber, which is a two-point boundary value problem. Then the evolutions of parametric pump, the signal and the idler are calculated using the predetermined Raman power profiles [31]. Improved Euler method was used, which is second-order accurate. From Fig. 3(a), the on–off gain of the parametric pump without Raman assistance has a peak value of 7.5 dB at 1560.5 nm. The Raman pump can offer a maximum on–off gain of 20.8 dB at 1555 nm and the 3-dB bandwidth is 23.5 nm. When Raman amplifier is combined with the parametric amplifier, the maximum gain of this RA-FOPA is 33.6 dB and the 3-dB bandwidth for both gain bands is nm. The wavelengths of both gain peaks were shifted 2.6 nm away from the parametric pump wavelength of 1554.9 nm when compared with that of the FOPA without Raman assistance. Thus, there is a 27.2 dB enhancement in the FOPA gain in the presence of the Raman pump. We note that the RA-FOPA provides an additional 6.4 dB gain enhancement when compared to the sum of individual peak gains of the Raman and parametric amplifiers, which is larger than the gain enhancement of 3.8 dB

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Fig. 4. Measured (symbols) and simulated (dashed curves) contour maps of the RA-FOPA gains. Gray solid circles and solid lines represent the measured and simulated results when the gain of the RA-FOPA equals to the gain with the Raman amplifier, respectively. The input signal power was fixed at 14 dBm.

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in [17] and 3.0 dB in [18]. In Fig. 3(a), on both the lower-left and lower-right corners, the overall gain of RA-FOPA can be smaller than the Raman gain. This is due to the phase mismatch in the parametric process. The signal and idler can only get direct Raman gain, which is smaller than that of Raman-only case because of the Raman pump depletion. This result supports the discussion in Section II that the indirect Raman gain of the signal and the idler can dominate over direct Raman gain. In Fig. 3(b), the profiles of the net conversion efficiency spectra are similar to that of the hybrid on-off gain. The peak conversion efficiencies were 20.7 dB and dB with and without Raman assistance, respectively. Fig. 3(a) and (b) also show the corresponding simulation results using the model described in (6)–(10). The simulation results assuming a uniform Raman gain agree very well with the measurement results. We also note that our simulation results show less than 10% direct Raman gain contribution at the output end of fiber. B. Contour Gain Map of RA-FOPAs Fig. 4 shows the measured peak gain values of the RA-FOPAs for different parametric and Raman pump powers when the input signal power after circulator CIR1 is fixed at dBm. From Fig. 4, for a given fiber, different choices of Raman and parametric pump powers can give the same overall gain. The overall gain can be increased by increasing either the Raman or parametric pump power. The corresponding simulation results are also shown in Fig. 4 using the model described in (6)–(10) and the parameters used in experiment. The simulation results match the measurement results very well in the small gain region ( dB). In the high pump powers region, the simulation results overestimate the RA-FOPA gain because higher order parametric idlers were not included in the model. In Fig. 4, the gray circles and solid lines respectively show the experimental and simulation results when the overall gain of the RA-FOPAs approximately equals to the gain with the Raman amplifier alone. For parametric pump power below this limit, the input signal can get less gain from the RA-FOPA than that from using Raman-only amplifiers. This is because

in RA-FOPA the parametric pump couples most of the energy of the Raman pump, the input signal therefore gets less energy from the Raman pump due to pump depletion. As a result, the overall gain will be reduced when the parametric pump cannot offer sufficient gain to compensate for the reduced Raman gain. Although not suitable to be used as amplifiers, RA-FOPAs operating in this region can be used for signal processing applications [34]. Note that in RA-FOPAs, a flat gain for wide gain bandwidth can be achieved if the Raman gain peak covers the zero dispersion wavelength (ZDW) region [24], [29]. In this case, the parametric pump, which should be located at the ZDW region to satisfy the phase matching condition, can receive the maximum Raman amplification and thus produce the broadest gain bandwidth. In our experiments, the gain peak of Raman pump is about 20 nm longer than the ZDW of the fiber. Since the purpose of this work is to characterize RA-FOPA gain, we chose to inject the parametric pump at the wavelength close to the Raman gain peak to maximize the Raman assistance effect. The parameter pump is therefore not at the ZDW region. Because of the large chromatic dispersion at the parametric pump wavelength, the gain bandwidths of RA-FOPAs were limited to less than 5 dB for each side band. In the simulation, we neglect the effect of fourth order dispersion because of the large chromatic dispersion. Fig. 4 shows the simulation results, which agree very well with the experimental results. Such agreement might appear to be somewhat surprising because both Raman and the parametric processes are polarization dependent but we have used acollinear polarized model with measured effective fiber coefficients. The agreement validates the assumption that if the fiber is sufficient long or the PMD coefficient is large such that the polarizations of the Raman pump, the parametric components are decorrelated, collinear polarized model using the effective Raman coefficient and the effective nonlinear coefficient can be used to describe the gain behavior of RA-FOPAs. C. Combined Gain Enhancement For a hybrid optical amplifier, when compared to the sum of the gains of component amplifiers, the gain enhancement of the hybrid amplifier can be written as (21) where and are the peak overall gain of th component amthe hybrid amplifier and peak gains of the plifier, respectively. A positive represents additional gain enhancement while a negative represents a penalty when combining different kinds of amplifiers together. As discussed in Section II, when indirect Raman gain dominates, the overall gain of RA-FOPAs can be larger than the sum of the gains of individual Raman and parametric amplifiers. When the direct Raman gain is not saturated, a maximum gain enhancement can thus be achieved in RA-FOPAs. Fig. 5(a) shows the contour map of the combined gain enhancement of the RA-FOPAs, which is obtained by a second order least square fitting of the experimental data shown in Fig. 4. In Fig. 5(a), the maximum gain enhancement is obtained at the center portion of the contour map. If we increase


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TABLE I COMPARISON OF THE APPROXIMATE VALUES OF E IN DIFFERENT TYPES OF TELECOMMUNICATION FIBER

Fig. 5. Measured (a) and simulated (b) contour maps of the combined gain enhancements in the RA-FOPA.

the parametric pump power, the combined gain enhancement will first increase, then decrease and eventually becomes negative. Fig. 5(b) shows the corresponding simulation results of Fig. 5(a), which match well with the measurement results when mW). The the parametric pump power is small (say, simulation results overestimated the overall gain of RA-FOPAs when the parametric pump is large because other nonlinear effects such as SBS were not included in the model. In the experiment, we observed an SBS Stokes of 16.0 dBm when a 23.1 dBm induced parametric pump was amplified by a 1 W Raman pump. Therefore, the parametric pump is also reduced owing to power depletion caused by the backward SBS [33]. IV. OPTIMIZING THE RA-FOPAS As mentioned in Section II, the number of independent parameters in the phase-matched model is reduced to four. They are , and . In the normalized model of (12)–(16), the Raman gain and the parametric gain are decided by and , respectively. The parameter is the ratio between nonlinear coefficient and Raman coefficient and it will affect the proportion of direct and indirect Raman contributions in RA-FOPAs. Besides , the parameter is responsible for the fiber loss and its length. From (17)–(20), and represent the relationship between Raman gain and fiber loss in RA-FOPAs. In this section, we will discuss how to optimize the powers of Raman and parametric pumps in RA-FOPAs for different parameters and . A. Effects of In Table I, we show the typical values of in different types of optical fiber [6]. Silica fiber has an close to 3. In special fibers, such as dispersion flattened highly nonlinear fibers (DFHNLFs), dispersion shifted fibers (DSFs), and Raman fibers, the values of cannot deviate far beyond the range of values shown in Table I because of fiber dispersion. In Fig. 6, we show the contour gain maps and their relative combined gain enhancements in fibers with different values of . In this simulations, we set dB and fixed the input signal at and idler power (noise level) at , which is much smaller than input signal.

Fig. 6. Effects of different values of " on the overall gain and combined gain enhancement of RA-FOPAs. (a), (b), and (c) show the simulated contour maps of the overall gains for " values of 4, 3, and 2 respectively. (d), (e), and (f) show the combined gain enhancement for " values of 4, 3, and 2 respectively. The contour curves are labeled in decibel.

Fig. 6 plots the effects of different values of on the overall gain and combined gain enhancement of RA-FOPAs. Figs. 6(a), (b), and (c) show the simulated contour maps of the overall gains for values of 4, 3, and 2 respectively. Figs. 6(d), (e), and (f) show the combined gain enhancement for values of 4, 3, and 2 respectively. Comparing Figs. 6(a) and (d) with (b) and (e), and (c) and (f), we observed that a larger will result in a larger combined gain enhancement. Since the nonlinear coefficient is much larger than the Raman coefficient, the parametric pump serves as an effective medium to transfer energy from the Raman pump to the signal and the idler. Fig. 6(d) predicts over 12-dB combined gain enhancement at the upper part of the contour map for an overall gain of 60 dB. In Fig. 6(b) and (e), the value of used is close to the real value of in the HNLF under test. Our model predicts that over 70 dB overall gain can be achieved in RA-FOPAs using higher nonlinearity. Note that our model is only valid

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in small signal region. In our experiment, the linewidth of the parametric pump was broadened through phase-modulation dBm up to to increase the threshold of SBS effect from 20 dBm. However, when the Raman gain increases, the parametric pump power will increase and eventually surpasses the SBS threshold. SBS effect will deplete the forward propagate parametric pump. We also note that, since a short HNLF is used in the RA-FOPA, in our experiment the threshold of the double Rayleigh scattering effect is lower than that of SBS effect, which is different from that for FRAs using long transmission fiber as gain medium. Fig. 6(e) shows that the maximum combined gain enhancement can be obtained at the left part of the contour map, which agrees with the experimental results shown in Fig. 5. From Fig. 6(b) and (e), high combined gain enhancement can be achieved by optimizing the powers of the pump pair and keeping them out of the saturation region and parametric gain limit. If we keep increasing the pump powers to get higher overall gain, the combined gain enhancement will dramatically decrease and eventually become negative because of pump depletion. Fig. 6(c) and (f) show the case when equals to 2. We observed that in this case when the Raman pump power increases, the overall gains of RA-FOPAs do not increase as much as the cases with larger values. This is because the direct contributions from the Raman pump to the signal and idler are comparable the indirect one. Therefore, positive combined gain enhancement is possible only when the Raman and parametric pumps are small. In this case, the efficiency of RA-FOPA is low. The results indicate that fibers with small value of , such as Raman fiber, are not suitable for RA-FOPAs because of the relatively low efficiency.


Fig. 7. Effects of different values of on the overall gain and combined gain enhancement of RA-FOPAs. Simulated contour maps of (a)–(c) the overall gains and (d)–(f) the combined gain enhancement for values of 0.5, 1, and 3, respectively. The contour curves are labeled in decibel.

lose more energy in fiber with larger , which prevents the optimization of the RA-FOPA at small pump powers. B. Effect of V. DISCUSSIONS Note that and represent the relationship between Raman gain and fiber loss in RA-FOPAs, respectively. For example, when the Raman pump power is 1 W, for a 10 km SMF, is 3.4 and equals to 2.5 dB; for a 1 km HLNF, is 4.8 and equals to 1.0 dB. Fig. 7 shows the effects of different values of on the overall gain and combined gain enhancement of RA-FOPAs for , and 3. Fig. 7(a)–(c) show the simulated contour maps of the overall gains for values of 0.5, 1, and 3 respectively. Fig. 7(d)–(f) show the combined gain enhancement for values of 0.5, 1, and 3 respectively. From Fig. 7, we note the overall gain and the maximum combined gain enhancement are nearly the same when is small. For fibers with small , pump depletion is small and so the pumps can provide high gain to the signal and the idler. For fibers with large , which means the fiber is long or lossy, higher pump powers are required in RA-FOPAs to obtain high overall gain and combined gain enhancement as shown in Fig. 7(c) and (f). From Fig. 7(d)–(f), the maximum combined gain enhancement is similar for fibers with different values of , which indicates that the maximum values of combined gain enhancement of RA- FOPAs are not significantly affected by the fiber length or fiber loss. Although the same maximum combined gain enhancement can be obtained, Raman and parametric pumps will

A. Phase-Matched Model To determine the peak gains of RA-FOPAs using the model of (12)–(16), it is necessary to search the whole gain band of the RA-FOPAs which is a time consuming process. We therefore simplify the calculation by assuming that the phase-matching condition is always satisfied at the gain peak region as in (17)–(20). By using this phase-matched model, we can directly determine the peak gain values of RA-FOPAs without searching gain band. It is important to check the validity of the phase-matched model. In Fig. 8, we compared the peak gain values determined using the phase matched model of (17)–(20) and those using the model of (12)–(16). We use the same set of parameters as in Fig. 7(b) and (e). From Fig. 8, the predictions from the phase-matched model agrees quite well with that from the full model of (12)–(16) especially when the input Raman pump is relatively small. Thus, the phase-match model can be used to characterize the general behaviors of the gain of RA-FOPAs. When the Raman amplification is large, the phase matched model overestimates the peak gain because the Raman pump amplifies the parametric pump, which in turn changes the phase matching condition and make the phase matched point move outward from the parametric pump wavelength.


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APPENDIX Assuming that the fiber is linearly birefringent and the induced optical field is collinear polarized, the general form of the third-order nonlinear polarization in silica fiber is given by [35] (A1) where, governs the delay Raman response and is responsible for the instantaneous electronic response of the nonlinear medium. In RA-FOPAs, if the transverse distribution of optical field does not change along -direction, the total optical field can be written as Fig. 8. Difference between the simulation results in decibel using model of (12)–(16) and the phase matched model. We use the same set of parameters as in Fig. 7(b) and (e).

The wavelength satisfying the phase matching condition is thus not fixed along the propagation distance. Signals at these wavelengths however are still nearly phase-matched and can still get amplified through parametric process. B. Optimization of the Noise Figure In the design of RA-FOPAs, besides the gain, the noise figure also needs to be optimized. In RA-FOPAs, Raman and parametric pump combinations with relative large parametric pumps can obtain better noise figure through suppressing the spontaneous Raman emission at the input end [22]. This indicates that pump combinations with better noise figure can be obtained at the lower-right corner of the contour maps. Thus, in order to design RA-FOPAs with high combined gain enhancement and low noise figure, the parameters must be carefully chosen to move the maximum gain enhancement region to the lower-right corner of the contour maps.

(A2) where and , and , are the angular frequencies and wave numbers, respectively. The indexes , and represent Raman pump, parametric pump, signal and idler, re) is the transverse distribution of optical field spectively. and , and , are the slowly varying amplitudes of optical fields at frequencies of Raman pump, parametric pump, the signal and the idler, respectively. Substituting (A2) into (A1), there will be 48 frequency combinations, three of which are degenerated in FWM process. We neglect high frequency components, high-order FWM idlers, anti-Stokes terms, and FWM terms generated by Raman pump, and consider lightwaves at the frequencies of Raman pump, parametric pump, signal and idler only. Thus, assuming collinear monochromatic waves, we obtain the coupled mode equation as

VI. CONCLUSION In conclusion, we used contour maps to describe the relationship between the overall gain and different combination of the powers of Raman and parametric pumps in Raman-assisted fiber optical parametric amplifiers. We derived a normalized phase-matched model to determine the general behaviors of the peak gains of RA-FOPAs operating in the small signal region. The simulation results show good agreement with measured data. Our simulation results indicate that higher small signal gain can be obtained in RA-FOPAs by preventing pump depletion caused by other nonlinear effects, such as backward SBS. We demonstrated that the models can be used to design and optimize the Raman and parametric pump pair of RA-FOPAs for high gain and high efficiency in different fibers. Our results also show that fibers with small ratio of nonlinear coefficient to Raman coefficient ratio, such as Raman fiber, are not suitable for use in RA-FOPAs. In fiber with low loss, high Raman coefficient and nonlinear coefficient, and high ratio of nonlinear to Raman coefficient, the optimum combined gain enhancements can be achieved by using small Raman and parametric pump powers.

(A3)

(A4)

(A5)

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(A6) is the fiber loss, is the Raman shift where and is the detune between signal and parametric pump. We introduce the nonlinear coefficients (A7) where is the effective core area, and are reflective index of the fiber and speed of light in vacuum. We also define the frequency dependent nonlinear coefficient differences, which can be written as (A8) where is the Fourier transform of coefficients, which are

, and Raman gain (A9)

If we neglect the Raman effects among parametric pump, the signal and the idler and frequency dependent nonlinear coefficient differences, (A3)–(A6) will reduced to the model described in [21], which focused on the amplitudes of input optical field. REFERENCES [1] E. Desurvire, Erbium Doped Fiber Amplifiers, Principle and Application. New York: Wiley, 1994. [2] M. J. Connelly, Semiconductor Optical Amplifiers. Boston, MA: Kluwer, 2002. [3] H. Kidorf, K. Rottwitt, M. Nissov, M. Ma, and E. Rabarijaona, “Pump interactions in a 100-nm bandwidth Raman amplifier,” IEEE Photon. Technol. Lett., vol. 11, pp. 530–532, 1999. [4] C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, “Parametric amplifiers driven by two pump waves,” IEEE J. Sel. Topics Quantum Electron., vol. 8, pp. 538–547, 2002. [5] M. N. Islam, Raman Amplifiers for Telecommunications 1: Physical Principles. Berlin, Germany: Springer, 2004. [6] C. Headley and G. P. Agrawal, Raman Amplification in Fiber Optical Communication Systems. Amsterdam, The Netherlands: Elsevier, 2005. [7] Global Bandwidth Research Service Exclusive Summery p. 2, 2010. [Online]. Available: http://www.telegeo graphy.com/product-info/gb/, Telegeography research, PriMetrica, Inc. [8] R. C. Alferness, P. A. Bonenfant, C. J. Newton, K. A. Sparks, and E. L. Varma, “A practical vision for optical transport networking,” Bell Labs Tech. J., pp. 3–18, 1999. [9] D. Mazzarese, “AllWave FLEX ZWP fiber white paper,” in OFS Technical Resource, 2006 [Online]. Available: http://www.ofsoptics.com/ resources/AllWave FLEXFiberWP-web.pdf, Avalible: [10] C. R. Davidson, C. J. Chen, M. Nissov, A. Pilipetskii, N. Ramanujam, H. D. Kidorf, B. Pedersen, M. A. Mills, C. Lin, M. I. Hayee, J. X. Cai, A. B. Puc, P. C. Corbett, R. Menges, H. Li, and A. Elyamani, “1800 Gb/stransmission of one hundred and eighty 10 Gb/sWDM channels over 7000 km using the full EDFA C-band,” in Proc. Optical Fiber Commun. Conf., 2000, paper PD25. [11] T. Torounidis and P. A. Andrekson, “Broadband single-pumped fiberoptic parametric amplifiers,” IEEE Photon. Technol. Lett., vol. 19, pp. 650–652, 2007. [12] T. Torounidis, P. A. Andrekson, and B. Olsson, “Fiber-optical parametric amplifier with 70-dB gain,” IEEE Photon. Technol. Lett., vol. 18, pp. 1194–1196, 2006. [13] J. L. Blows and S. E. French, “Low-noise-figure optical parametric amplifier with a continuous-wave frequency-modulated pump,” Opt. Lett., vol. 27, pp. 491–493, 2002.

[14] K. K. Y. Wong, K. Shimizu, M. E. Marhic, K. Uesaka, G. Kalogerakis, and L. G. Kazovsky, “Continuous-wave fiber optical parametric wavelength converter with 40-dB conversion efficiency and a 3.8-dB noise figure,” Opt. Lett., vol. 28, pp. 692–694, 2003. [15] P. Kylemark, P. O. Hedekvist, H. Sunnerud, M. Karlsson, and P. A. Andrekson, “Correction to ‘Noise characteristics of fiber optical parametric amplifiers’,” J. Lightw. Technol., vol. 23, p. 2192, 2005. [16] H. Masuda, S. Kawai, and K. I. Suzuki, “Optical SNR enhance amplification in long-distance re-circulating loop WDM transmission experiment using 1580 nm band hybrid amplifier,” Electron. Lett., vol. 35, pp. 411–412, 1999. [17] H. S. Seo, W. J. Chung, and J. T. Ahn, “A novel hybrid silica wideband amplifier covering S+C +L bands with 105-nm bandwidth,” IEEE Photon. Technol. Lett., vol. 17, pp. 1830–1832, 2005. [18] H. S. Seo, Y. G. Choi, B. J. Park, D. H. Cho, and K. H. Kim, “Simultaneous amplification by Er ions and SRS in an Er-doped germane—silica fiber,” IEEE Photon. Technol., vol. 15, pp. 1198–1200, 2003. [19] D. A. Chestnut, C. J. S. de Matos, and J. R. Taylor, “Raman-assisted fiber optical parametric amplifier and wavelength converter in highly nonlinear fiber,” J. Opt. Soc. Amer. B, vol. 19, pp. 1901–1904, 2002. [20] C. J. S. de Matos, D. A. Chestnut, P. C. Reeves-Hall, and J. R. Taylor, “Continuous-wave-pumped Raman-assisted fiber optical parametric amplifier and wavelength converter in conventional dispersion-shifted fiber,” Opt. Lett., vol. 26, pp. 1583–1585, 2001. [21] J. F. L. Freitas, N. B. Costa e Silva, S. R. Lüthi, and A. S. L. Gomes, “Raman enhanced parametric amplifier based S-C band wavelength converter: Experiment and simulations,” Opt. Commun., vol. 255, pp. 314–318, 2005. [22] S. H. Wang, L. Xu, and P. K. A. Wai, “Noise characterization of Raman-assisted fiber optical parametric amplifiers,” in Proc. 14th OptoElectron. Commun. Conf., 2009, paper TuG5. [23] S. H. Wang, L. Xu, P. K. A. Wai, and H. Y. Tam, “6.4-dB enhancement of the gain of a Raman-assisted fiber optical parametric amplifier over the sum of the gains of individual amplifiers,” in Proc. Opt. Fiber Commun. Conf., San Diego, CA, 2008, paper JThA13. [24] J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Topics Quantum Electron., vol. 8, pp. 506–520, 2002. [25] TrueWave® RS Fiber LWP Specification Sheet. Denmark, 2009, OFS Fitel LLC.. [26] Highly Non-Linear Fiber Specification Sheet. Denmark, 2009, Ver. 070401 LGN, OFS Fitel LLC.. [27] Raman Fiber Specification Sheet. Denmark, 2007, Ver. 041307, OFS Fitel LLC.. [28] P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence,” J. Lightw. Technol., vol. 14, pp. 148–157, 1996. [29] Q. Lin and G. P. Agrawal, “Vector theory of stimulated Raman scattering and its application to fiber-based Raman amplifiers,” J. Opt. Soc. Amer. B., vol. 20, pp. 1616–1631, 2003. [30] Q. Lin and G. P. Agrawal, “Effects of polarization-mode dispersion on fiber based parametric amplification and wavelength conversion,” Opt. Lett., vol. 29, pp. 1114–1116, 2004. [31] B. Min, W. J. Lee, and N. Park, “Efficient formulation of Raman amplifier propagation equations with average power analysis,” IEEE Photon. Technol. Lett., vol. 12, pp. 1486–1488, 2000. [32] S. H. Wang, L. Xu, P. K. A. Wai, and H. Y. Tam, “All-optical wavelength conversion using multi-pump Raman-assisted four-wave mixing,” in Proc. Opt. Fiber Commun. Conf., 2007, paper OWQ1. [33] A. Koybyakov, M. Mehendale, M. Vasilyev, S. Tsuda, and A. F. Evans, “Stimulated brillouin scattering in the Raman-pumped fibers: A theoretical approach,” J. Lightw. Technol., vol. 20, pp. 1635–1643, 2002. [34] G. Cappellini and S. Trillo, “Third-order three-wave mixing in singlemode fibers: Exact solutions and spatial instability effects,” J. Soc. Amer. B, vol. 8, no. 4, pp. 824–838, 1991. [35] R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantam Electron., vol. 5, pp. 1–68, 1977.

Shao Hao Wang (M’09), biography not available at the time of publication.

Lixin Xu, biography not available at the time of publication.


P. K. A. Wai (SM’96) received the B.S. (Hons.) degree from the University of Hong Kong in 1981, and the M.S. and Ph.D. degrees from the University of Maryland, College Park, in 1985 and 1988, respectively. In 1988, he joined Science Applications International Corporation, McLean, VA, where he was a Research Scientist involved with the Tethered Satellite System project. In 1990, he became a Research Associate with the Department of Physics, University of Maryland, College Park, and the Department of Electrical Engineering, University of Maryland, Baltimore County. In 1996, he joined the Department of Electronic and Information Engineering, The Hong Kong Polytechnic University. He became Chair Professor of Optical Communications in 2005. Currently he is the Vice President (Research Development). His research interests include soliton, fiber lasers, modeling and simulations of optical devices, long-haul optical fiber communications, all-optical packet switching, and network theories. He is an active contributor to the field of photonics and optical communications, having authored or coauthored over 300 international refereed publications. Currently he is an editor of Optics Express. Prof. Wai is a Fellow of the Optical Society of America. He serves as a guest editor of the IEEE JOURNAL OF SELECTED AREAS OF QUANTUM ELECTRONICS.

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Hwa Yaw Tam (SM’09) received the B.Sc. degree and Ph.D. degree in electrical and electronic engineering from The University of Manchester, Manchester, U.K., in 1985 and 1990, respectively. From 1989 to 1993, he was with Hirst Research Center, GEC-Marconi Ltd. in the U.K., first as a Research Scientist, then as a Senior Research Scientist working on WDM systems and fiber amplifiers. He also worked briefly for Marconi-Italiana of Italy as a consultant before joining the Department of Electrical Engineering of the Hong Kong Polytechnic in January 1993. Currently, he is a Chair Professor of Photonics with the Department of Electrical Engineering, The Hong Kong Polytechnic University, where he is also the Director of the Photonics Research Centre, Faculty of Engineering. His current research interests are silica and polymer fibre fabrication, FBGs and FBG-based transducers, fiber amplifiers, optical communication systems, and all-optical signal processing. He has secured over HK$100 million research funds since 1995. He has authored or coauthored about 400 technical papers and holds eight patents in the areas of fiber optics.