1. Introduction

Stimulated Brillouin scattering (SBS) is a fundamental nonlinear optical process in optical-fiber communication systems [1]. Classically, in SBS a portion of laser power incident on an optical medium is converted to a second wave, slightly shifted in frequency. This wave is known to propagate in the backward direction to the incoming photons, and is called back-scattered Stokes wave. The process of frequency shift, which is also called the Brillouin shift, comes from the Doppler effect. The incident coherent pump light generates an acoustic wave in the optical fiber through the process of electrostriction, which causes an index grating to form in the fiber due to the longitudinal nature of the acoustic wave. This induced grating scatters the pump wave by Bragg reflection via the elasto-optic phenomenon, generating the Stokes wave and producing the attenuation of the pump and the amplification of the Stokes wave [2, 3]. SBS has been subject of intensive theoretical as well as experimental investigations over the years. In this regard, various techniques have been developed to suppress SBS [4–7]. Various analyses of SBS have been considered in the past by different researchers. However, recent research has proved that SBS can be utilized for highly desirable applications [8–13]. The start of SBS is dictated by an input power called SBS threshold [14]. Achieving a low threshold value is of utmost importance in most applications. SBS threshold reduction, through bidirectional pumping methods, in order to quicken the SBS establishment time have also been considered by a number of researchers [15–17]. Modeling of SBS has been considered [18] in order to predict the strength of the phenomenon in optical fibers.

In this paper, we report the development, for the first time to the best of our knowledge, a theoretical model that can be used to predict SBS threshold in an optical fiber where Brillouin pump signal is recycled back into the Brillouin gain medium by the use of an optical mirror, thereby acting as a bidirectional pump. We propose a new expression of stimulated Brillouin scattering threshold that depends on fiber length and the degree of reflectivity of an optical mirror. Results obtained from solving our model equations show that SBS threshold is reduced by ≈50% in a 5 km long single mode fiber, thereby validating experimental results obtained in [16].

2. Theoretical model

We start the modeling by referring to the experimental setup given in Fig. 1 [16], and by finding the evolution of the pump power $P_p(z)$with distance as reported in [1]

 

Fig. 1 Schematic diagram of Brillouin pump power recycling technique [16].

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Let $P_{pf}(z)$ represents the Brillouin pump (BP) power coupled into the fiber front end and $P_{pb}(z)$ represents the recycled BP power coupled into the other fiber end, to account for pumping from the two ends of the fiber. In this case, the pump absorption characteristics from these two directions are assumed to be independent. In the absence of pump power depletion at the threshold, we neglect the first term at the right hand side of Eq. (1) as proposed in [1] and obtain the pump power along the length of the fiber from the two sides,

Equations (2a) and (2b) describe the evolution of the initial BP and recycled BP respectively as a function of fiber length. Integrating both sides of Eq. (2a) from the near end of the optical fiber to z, and Eq. (2b) from the far end of the optical fiber to z, we get

The change in the power of the first-order backward scattered Stokes waves at point 2 in Fig. 1, is given as reported in [1,19],

Substituting Eq. (3a) into Eq. (5) and integrating it in order to find an equation for the first-order Stokes wave power as a function of fiber length

We need to consider the contribution of the recycled BP power in generating a Stokes wave. Therefore we introduce $P_{sf}(z)$as the Stokes wave generated by $P_{pb}(z)$, which propagates in the same direction as $P_{pf}(z)$, is derived as

In this case, $P_{sf}(z)$propagates in the same direction of $P_{pf}(z)$ and then it is reflected back from the far end of optical fiber by the optical mirror. This reflected $P_{sf}(z)$ becomes the additional Stokes wave propagating again to the near end of the fiber (z = 0) in the same direction of $P_{sb}(z)$, and hence we have

The most important parameter for characterizing and determining the SBS threshold is the Stokes wave gain parameter, $G_{SBS}=P_s(0)/P_s(L)$. The use of Eq. (11) requires$P_{pf}(0)$, $P_{sf}(0)$ and$P_{sb}(L)$at the near end and the far end of the optical fiber. If $P_{sf}(0)$and $P_{sb}(L)$are equal to zero, the optical power of the Stokes waves, $P_{sf}(z)$ and $P_{sb}(z)$ are also equal to zero based on Eq. (6) and 10 respectively. This reflects to the fact that these two Stokes waves must start from a seed power. In practice, $P_{sf}(0)$ and $P_{sb}(L)$are generated by a fictitious Stokes photon from noise or spontaneous Brillouin scattering occurring throughout the fiber length [2]. In the SBS process, this is done by summing all the contributions of each frequency component of energy (hv) and multiplying by$G_{SBS}$. This is equivalent to injecting a single Stokes photon per longitudinal mode falling within the Brillouin bandwidth at the point where the gain equals the loss of the fiber [2],

There are various definitions of the SBS threshold in optical fibers. One is based on the condition when the backward scattered Stokes optical power at (z = 0) is equal to the input power. However, SBS threshold power can also be defined as the input power where the backward scattered Stokes optical power is equal to some fraction, of the pump power. One might also identify the SBS Brillouin threshold based on the input pump power at which the backward scattered Stokes optical power becomes equal to the pump power at the fiber length (l) [20]. Then,

3. Experimental validation

In this paper, both experimental and simulation works using single mode fiber (SMF) as the Brillouin gain media are performed. The experimental work is very critical to validate our mathematical models. There are two configurations studied in this research work; conventional and BP recycling techniques. The conventional technique refers to the configuration which does not have a mirror to reflect the residual BP back into the fiber. Parameters employed in the simulation by using MATLAB software are given in Table 1 . To perform simulation for the conventional technique, we set R and $P_{sf}(0)$ equal to zero.

Table 1. Parameters used for the simulation work

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Figure 2 shows the Stokes wave power dependence of the BP power in the SMF. In Fig. 2, the solid line and squares show the theoretical and experimental results, respectively. For these findings, the BP wavelength is located near 1550 nm.

 

Fig. 2 Stokes wave power with variation in BP power for 5 km long SMF, (a) conventional technique, and (b) recycling pump technique.

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It could be noted that by increasing the pump power from 0 to 30 mW for SMF, the backward scattered Stokes wave power increases. We consider that the SBS threshold power P_th is the condition where the back-scattered Stokes wave power begins to rise above the spontaneous back-scattered power (when the exponential growth of the Stokes wave due to electrostriction is larger than the linear growth of the Stokes wave due to the thermal acoustic waves). At a low pump power the SBS shows no real gain and is very small and the transmitted power experiences only linear loss due to the loss of the fiber. When this threshold power is reached, the SBS power grows exponentially and the pump power starts to show depletion due to its conversion to the backward scattered Stokes wave. The pump depletion causes the gain of the process, g_BP_pf, to decrease and so the rate of increase of the backward scattered Stokes wave decreases as the pump power is further increased. Therefore, the value of $P_{th}\cong$16 mW by using the conventional technique when the residual BP power is not recycled back into the fiber. Thus, the reduction coefficient in the SBS threshold is nearly ≈50%, by using the pump recycling technique to $P_{th}\cong$8 mW. This calculation result of the SBS threshold is evaluated by using Eq. (14). The additional energies provided by the recycling transmitted BP power to the Brillouin gain medium increases the effective fiber length, thus initiating SBS emissions in the forward direction ($P_{sf}(z)$). The forward Stokes wave is then reflected by the optical mirror ($P_{sfb}(z)$) and it becomes a probe signal to the initial BP ($P_{pf}(z)$). Consquently, the SBS process increases which translates to the additional gain of the scattered Stokes wave power ($P_{sb}(z)$). The reduction coefficient pattern is similar to our experimental coefficient. Based on the findings, the agreement is very good between theoretical and experimental results.

The simulation results of the backward scattered Stokes wave for different lengths of SMF for conventional and recycling pump techniques have each been compared with the plots obtained from our experimental results, which are shown in Fig. 3 . For the simulation results, the Stokes wave power is calculated in the range of 4.0 and 11.5 mW for conventional and pump power recycling techniques respectively. The increment of Stokes wave power for the proposed technique is due to the additional gain provided by the recycled BP. Physically, this represents a complete conversion of the pump power above the SBS threshold into backward scattered Stokes wave power and represents the saturation of the gain. At high power it can be seen that the transmitted pump power saturates and shows no significant increase with the increment of pump power. Therefore, the maximum power that can be transmitted by an optical fiber is limited by the process of SBS.

 

Fig. 3 Output Stokes wave power distribution at different SMF lengths for 30 mW BP power, (a) conventional technique, and (b) recycling pump technique.

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Figure 4 shows the characteristics of SBS threshold power with respect to the fiber length. In this case, the theoretical SBS threshold power is calculated using Eq. (14). The experimental and theoretical results show small discrepancy which indicates a good agreement between those values. For the conventional technique, the SBS threshold power decreases from 15 to 7 mW with the increment of fiber length from 5 to 17 km respectively. On the other hand, the SBS threshold power reduces from 7.5 to 4.5 mW with respect to the variation of fiber length from 5 to 17 km respectively.

 

Fig. 4 Brillouin threshold power at different SMF lengths, (a) conventional technique, and (b) recycling pump technique.

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Based on these findings, the BP power recycling technique lowers the SBS threshold as the fiber length increases because the interaction length for Brillouin Stokes wave generation is increased. With attenuation loss and reflectivity of the optical mirror taken into account, the backward scattered Stokes intensity distribution in the fiber has a strong dependence on the pump power coupled into the fiber. At high pump power, the Stokes wave grows very quickly near the front of the fiber resulting in a short effective length. At low pump power, the Stokes wave has a slow growth in a backward direction, resulting in a relatively long effective length. We found that with longer fiber lengths in the BP power recycling method, the Brillouin threshold saturates at the input signal power of 4.5 mW. With the use of high reflectivity mirror, the SBS establishment time is significantly shorter, which means less pump power is required. Adding reflectivity reduces the SBS threshold, increases the SBS conversion efficiency, reduces the saturation level for the transmitted pump power and has the same effect as increasing the Brillouin gain.

4. Conclusion

A new theoretical model of Brillouin threshold reduction through BP recycling has been presented. The SBS threshold of different fibers can be estimated by using the derived equations. The modeling results are compared and concur with our experimental results. Typical threshold power for SBS is about 16 mW in a simulation when the BP power is not recycled while for the BP recycling technique, it is found to be 8 mW, at 5 km long SMF. The reduction of SBS threshold for the BP recycling technique is due to the increment of interaction length for Brillouin Stokes wave generation. In a different perspective, the additional energy provided by the reflected Stokes wave from the recycled BP becomes a probe signal that increasingly stimulates the process of SBS. Thus, the SBS threshold is reduced as the result of output Stokes wave power increment.