1. Introduction

Stimulated Brillouin scattering (SBS) slow light in an optical fiber has attracted a wide attention since the first demonstration [1, 2]. It has several advantages such as the simplicity, the low threshold of pump power, the compatibility to fiber-optic communication systems and the flexibility of the configuration. It shows many applications such as data buffering, data synchronization, optical switch, optical memory and optical signal processing [3–5].

The basic principle of SBS slow light in an optical fiber can be depicted as follows: Two light beams are simultaneously injected into a single mode fiber (SMF) from its two ends. One beam is a strong continuous-wave (cw) laser acting as the pump light and the other one is a weak pulse laser acting as the signal pulse. Because of the electrostriction effect the two beams give rise to a density fluctuation in the fiber, generating an index grating or an acoustic wave which travels in the same direction with the pump beam. The frequency of the acoustic wave field is equal to the frequency difference between the pump beam and the signal beam. The pump beam produces strong backward scattering because of the moving index grating, which down shifts the pump frequency to the same level of the signal and simultaneously produces phonons. As a result both the signal pulse and the acoustic field are constructively enhanced. The strongest SBS occurs when the frequency shift equals to the inherent Brillion frequency shift Ω_B (decided by the electrostriction response characteristic of the fiber material). Around this resonance and in a normal dispersion fiber the index of refraction changes rapidly with frequency, inevitably leading to the increase of the group index and thus the decrease of the group velocity of the signal pulse. This is the so called SBS slow light.

However, the inherent SBS bandwidth of SMF is only tens of MHz (defined by the inverse phonon lifetime in silica fiber which is in the 10-ns range). This narrow gain bandwidth causes the amplification of only a small portion of the pulse spectrum near the Brillouin resonance and inevitably yields pulse broadening while propagating along the fiber [6]. But in practical applications, it is more important to achieve slow light for short pulses of sub-nanosecond durations corresponding to Gbits/sec data streams with little broadening. To overcome this obstacle, many ways of broadening the gain spectrum have been proposed. This includes overlapping the multi-line frequencies of a single pump laser [7, 8] and modulating the pump laser by frequency- [6, 9], amplitude- [10–12] and phase-modulation [13, 14]. All of these methods have only solved the problem to some extent. Even using the broadband pump scheme [15, 16], large pulse broadening still exists.

In this paper we propose a new method of constructing and optimizing the SBS gain spectrum by use of two Gaussian-shaped broadband pump beams. Our idea is that the top of the gain spectrum of one pump can be partially counteracted by the loss spectrum of the other. As a result we can construct a broadband gain spectrum with a special profile, which provides us a possibility to completely eliminate the SBS-related pulse broadening under a specific condition. Although the idea of broadening SBS bandwidth through the compensation between the gain spectrum of one pump laser and the loss spectrum of the other has ever been reported in Ref [17] and Ref [18], the dependence of pulse broadening on the SBS gain profile has not been reported so far. For the first time to our knowledge we theoretically investigated and numerically analyzed the dependence of pulse broadening on the gain profile constructed.

The spectral configuration of our double broadband pump scheme is shown in Fig. 1. The frequency separation between the central frequency of pump1 (ω_p01) and that of pump2 (ω_p02) is 2Ω_B. Therefore, the Stocks gain spectrum of pump1 has the same central frequency with the anti-Stocks loss spectrum of pump2, denoted by ω₀=ω_p01-Ω_B=ω_p02+Ω_B. We can see that a wide gain spectrum with a flat-top is produced by overlapping gain1 and loss2. Suppose a signal pulse with a carrier frequency at ω₀ oppositely launches the fiber, it will experience the slow-light propagation. In what follows we will theoretically analyze the characteristics of the time delay and the broadening of the signal pulse using such a pump scheme we proposed.

 

Fig. 1. Spectral configuration of double broadband pump beams and the corresponding Brillouin gain as well as loss spectra of them. The total gain profile is constructed by overlapping the Stokes gain spectrum (gain1) of pump1 and the anti-Stokes loss one (loss2) of pump2, where the gain1 and the power spectrum of pump1 are normalized to unity, respectively. Here Δω_p2=Δω_p1/(2)^1/3, I_p1/I_p2=2 and Δω_p1=0.5Ω_B.

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2. Theory

The SBS gain is given by the convolution of the intrinsic SBS gain spectrum g₀(ω) with the power spectrum of the tailored pump beam I_p(ω), as given by [15]

where

and

where g^B is the intrinsic Brillouin gain coefficient, γ is the half width at half maximum of inherent Brillouin gain spectrum, +/- correspond to the Stokes gain and the anti-Stokes loss, respectively; Δω_p1 (Δω_p2) and I_p1 (I_p2) are the 1/e half width and the intensity of pump1 (pump2), respectively.

Inserting Eqs. (2) and (3) into Eq. (1) and assuming the conditions that γ≪Δω_p1 and γ≪Δω_p2 (they are satisfied in our case), one can approximate g(ω) to

where ξ₁=(ω-ω₀)/Δω_p1 and ξ₂=(ω-ω₀)/Δω_p2, erfc is the complementary error function. The real part of g(ω) leads to an amplification of the counter propagating signal pulse, whereas the imaginary part results in an accompanied phase shift of it. They can be expressed from Eq. (4) as follows:

Moreove, the strong frequency dependence of the propagation constant change of the signal pulse leads to a change in the group velocity which in turn changes the arrival time of the pulse at the fiber output. This SBS-related time delay ΔT_d(ω) (being defined as a difference between the transit times of the pulse with and without pump beams) of the signal pulse at ω is given by [15]

where z is the fiber length.

As can be seen from Eq. (5) a flat-top gain can be obtained when d²Re[g(ω)]/d²ω=0 at the central frequency ω₀. This condition is equivalent to I_p1/I_p2=Δω_p1 ³/Δω_p2 ³. Let us denote it as I_p1/I_p2=Δω_p1 ³/Δω_p2 ³=b for convenience. With this condition, at the resonance frequency ω₀ thus ξ₁=ξ₂=0, Eq. (7) can be reduced to the following simple form:

Equation (8) indicates that slow light (ΔT_d(ω₀)>0) can only be produced when b>1. Equivalently, the intensity and the spectral width of pump1 should be larger than that of pump2, i.e., I_p1>I_p2 and Δω_p1>Δω_p2.

Moreover, as shown in Fig. 1 the net gain spectrum is the superposition of a positive Gaussian-shaped gain spectrum (gain1) with a negative Gaussian-shaped loss one (loss2). Mathematically, the segment between the two peaks of the first derivative of a Gaussian function can be approximated to a straight line. And taking into account the condition of Δω_p1>Δω_p2 we select the frequency separation between two peaks of the first derivative of loss2 as the flat-top width of the net gain spectrum, denoted by W_flat-top and it is easy to get:

In this paper we assume a Gaussian-shaped signal pulse having the carrier frequency of ω₀ experiences slow-light propagation in the fiber. And we select the 1/e half bandwidth of the pulse (denoted by 1/δ_in, where δ_in being the 1/e half duration) to be the half of W_flat-top, ensuring that the entire spectrum of the signal pulse may fit within the frequency range between ω₀-W_flat-top/2 and ω₀+W_flat-top/2. In this case, we have

Note from Eq. (10) that W_flat-top determines the maximum spectral width (or the minimum temporal width δ_in) of the signal pulse. It is easy to see from Eq. (8) that the time delay ΔT_d(ω₀) increases with b, while from Eq. (9) that the width of flat-top W_flat-top decreases with b. These relationships are shown in Fig. 2. With a trade-off between ΔT_d(ω₀) and W_flat-top, throughout the paper we select b=2 as an example for simulations. Such a selection does not lose the generality of the method we proposed here. In Sec. 3 we will investigate the dependences of the time delay and the temporal broadening of signal pulse on SBS gain profile.

 

Fig. 2. Dependences of the time delay and the width of the flat-top on parameter b. Here Δω_p1=0.5ω_B, P_p1=0.2 W and I_p1/I_p2=Δω_p1 ³/Δω_p2 ³=b.

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3. Results and discussion

Let us introduce the parameters we used for our simulations. These parameters are selected either corresponding to the properties of the common single-mode fiber SMF-28 at 1550 nm or according to the possible experimental conditions in practice: γ/(2π)=20 MHz, g_B=5×10^-11 m/W, Ω_B/(2π)=10.8 GHz and the effective mode area A_eff=50 µm²; We set Δω_p1 to its maximum, i.e., Δω_p1=0.5Ω_B, for obtaining a complete separation between gain1 and loss1 (as shown in Fig. 1); We set the fiber length at z=3.5 km and the laser power of pump1 (P_p1) at 0.2 W, note that I_p1 depends on P_p1 and A_eff through I_p1=P_p1/A_eff. Taking into account of Δω_p1=0.5Ω_B and b=2 we get δ_in=52 ps from Eq. (10). In what follows we will use these parameters unless otherwise is stated.

To construct different gain profiles, in this paper we fix I_p1 (by fixing P_p1 at 0.2 W) and change I_p2 and let I_p1/I_p2=a for the convenience. Note that from now on b=2 is always kept but it just means Δω_p2=Δω_p1/(2)^1/3. The dependences of Brillouin gain and time delay on the parameter a, as respectively described by Eqs. (5) and (7), are firstly investigated. Figure 3(a) shows the spectral dependence of Brillouin gain exponent for different a. Three typical gain profiles are obtained. We can see that a flat-top is indeed produced, as analyzed in Sec. 2, when a=2; a dip around the central frequency ω₀ is produced when a<2; and the normal case where gain is maximum at ω₀ is yielded when a>2. Figure 3(b) shows the spectral dependence of time delay for different a. We can clearly see that at the flat-top gain situation (a=2) the time delay does not show a flat-top; while a flat-top delay is obtained around ω₀ when a=2.5. At the wings, the time delay becomes negative which means the signal would experience superluminal process.

 

Fig. 3. Spectral dependences of (a) the gain exponent and (b) the time delay for different values of a. Here Δω_p1=0.5Ω_B, P_p1=0.2 W and Δω_p2=Δω_p1/(2)^1/3.

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Before we discuss the relationship of the pulse broadening with gain profiles, let us quantitatively analyze the pulse broadening related to the SBS process. Suppose that a signal pulse at the carrier frequency ω_c has a Gaussian-shaped intensity profile such that I_in(t)=I₀exp[-(t/δ_in)²], with δ_in being the 1/e pulse width in the temporal domain. Then in the spectral domain it can be expressed as I_in(ω)=I₀δ_in ²exp[-δ_in ²(ω-ω_c)²], with 1/δ_in being the 1/e bandwidth. Specifically, the pulse emerging from the fiber is amplified through the SBS process and it still has a Gaussian-shaped intensity profile, being determined by the relation of I_out(ω)=I_in(ω)exp[Re[g(ω)z]] [2]. If we assume that the carrier frequency of signal pulse is set at the SBS bandwidth center, i.e., ω_c=ω₀, we obtain the output pulse duration δ_gain given by 1/Δω_out, with Δω_out being the 1/e half-width of I_out(ω), namely I_out(ω₀±Δω_out)=I_out(ω₀)/e. Then, a gain-dependent broadening factor B_gain may be given by δ_gain/δ_in. As is clearly seen from Eq. (7), the time delay of signal pulse is accompanied with a broadening related to the group velocity dispersion (GVD). This GVD-dependent broadening may be approximated to a difference between time delays at ω₀ and ω₀±Δ_out. This yields a GVD-dependent broadening factor B_GVD given by [ΔT_d(ω₀)-ΔT_d(ω₀±Δ_out)]/δ_in. In the practical optical communication systems it is required that the maximum pulse broadening be no more than a factor of 2, i.e., B=B_gain+B_GVD≤2 [2, 19].

According to the above parameters, we simulate the dependence of pulse broadening on the parameter a, as shown in Fig. 4. It is seen from Fig. 4 that although both B_gain and B_GVD increase with the increase of a, they imply different meanings. B_gain is always positive, meaning that the front edge of the signal pulse always keeps in the front during the process of gain-dependent broadening. Note that B_gain<1 means the pulse is compressed and B_gain>1 means the pulse is broadened; while B_GVD could be negative, this means the front edge of the signal pulse could fall behind of the back edge of it. Examining the total broadening curve we find that there exists a special point at a=2.073 where we get B=1. At this special point, a little positive B_gain (thus a slight pulse broadening) is compensated by the low negative B_GVD (thus a slight pulse compression), leading to zero-broadening of the signal pulse.

 

Fig. 4. Dependences of B_gain, B_GVD and B on parameter a. Here Δω_p1=0.5ω_B, P_p1=0.2 W and Δω_p2=Δω_p1/(2)^1/3.

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To see the significance of B=1 in more detail, the spectral distributions of the Brillouin gain and the intensities of the input and output signal pulses are plotted in Fig. 5 where a=2.073. Figure 5(a) shows the real and imaginary parts of gain g(ω)z. It can be seen that, owing to the special value of a, the gain spectrum shows a wide flat-top in its real part and a good linearity in its imaginary part. Such a gain profile ensures low pulse distortions of SBS-based slow light. Figure 5(b) shows the intensities of the input and output pulses as well as the exponential gain. It should be pointed out that in Fig. 5(b) we only consider the gaindependent broadening effect. From Fig. 4 we know that B_gain is a little larger than 1 when a=2.073, i.e., the output pulse is just a little broader than the input one in the time domain. This is the reason why the bandwidth of the output pulse is a little narrower than that of the input one as shown in Fig. 5(b). However if we simultaneously consider the SBS-related GVD effect the bandwidths of them would be the same, as one can see from Fig. 4 that B_GVD is a little negative when a=2.073. Figure 5 means that such a gain profile can effectively restrain pulse broadening. Moreover, although we omitted the effect of the inherent fiber’s GVD in our calculations (i.e., a zero-GVD dispersion fiber is implicitly assumed), it can also be seen from Fig. 4 that the pulse could be compressed by setting an appropriate value of a, indicating that the total pulse broadening could be adjusted to be zero even though the inherent GVD is taken into account.

Furthermore, Eqs. (5) through (7) intuitively tell us that the pulse broadening may be dependent on Δω_p1 and I_p1, i.e., the spectral width and the intensity of pump1. Therefore, it is significant for us to investigate the dependences of a (for B=1) on Δω_p1 and I_p1 (or the laser power of pump1, i.e., P_p1). Figure 6(a) shows the dependence of a on Δω_p1 where P_p1 is fixed at 0.2 W. We can see that although a decreases with Δω_p1, the variation of a is very inappreciable even in a large range of Δω_p1. The insert of Fig. 6(a) shows the allowable signal pulse-width δ_in under different Δω_p1 as determined by Eq. (10) where b=2. Given a certain signal pulse to experience slow propagation we can decide Δω_p1 according to the pulse’s width δ_in, then from Fig. 6(a) there exists an optimal a for obtaining B=1. Figure 6(b) indicates the dependence of a on P_p1 where Δω_p1 is fixed at 0.5Ω_B. It is seen that a increases slightly with P_p1. Both Figs. 6(a) and 6(b) denote that a can be regarded as nearly independent of the spectral width and the power of pump1 (Δω_p1 and P_p1). This insensitive characteristic of a (nearly constant) for B=1 is owing to the specific gain profile we constructed. A signal pulse will experience group delay without broadening (related to SBS process) regardless of the pump power and the propagation distance.

 

Fig. 5. Spectral dependences of Brillouin gain and signal pulse intensity when a=2.073 for B=1. (a) Real and imaginary parts of gain; (b) Intensities of input and output pulses as well as the exponential gain. The 1/e full bandwidths for both the input and output pulses are shown in Fig. 5(b). Here Δω_p1=0.5Ω_B, P_p1=0.2 W and Δω_p2=Δω_p1/(2)^1/3.

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Fig. 6. Dependences of a for B=1 on the spectral width (Δω_p1) and the laser power (P_p1) of pump1 in our double pump scheme. (a) on Δω_p1, where P_p1 is fixed at 0.2 W; and (b) on P_p1, where Δω_p1 is fixed at 0.5ω_B. The insert of Fig. 6(a) shows the allowable signal pulse-width δ_in calculated by Eq. (10) for different Δω_p1. Here Δω_p2=Δω_p1/(2)^1/3.

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Finally, we will study the dependences of time delay and pulse broadening of signal pulse on the laser power of pump1, P_p1, in the double broadband pump scheme (pump1 and pump2). Here we select a=2.085 as an example, which is a mediate value of a when P_p1 is between 0 and 1 W (as shown in Fig. 6(b)). Figure 7 shows these relationships under the condition of B≤2. The results of using a single broadband pump beam (only pump1) are also shown for comparison. We can see from Fig. 7(a) that for both cases the time delays increase linearly with the increase of P_p1, indicating the tunable time delay of slow light, but the time delay for double pump case is smaller as compared with that for single pump case under the same pump power. Meanwhile, this reduction of time delay can be compensated by increasing the pump power, the fiber length and/or using highly nonlinear optical fibers. Figure 7(b) shows the variations of pulse broadening with P_p1. It is seen that B_gain, B_GVD and B monotonously increase with the increase of P_p1 for the single pump case, leading to a large broadening of signal pulse. On the contrary, for the double pump case the broadening factor B is nearly independent of the pump power, keeping almost unity in the process of Brillouin scattering. This is owing to the compensation between the positive gain-dependent broadening B_gain and the negative GVD-dependent broadening B_GVD. Essentially, this property is attributed to the specific gain profile we designedly constructed.

 

Fig. 7. Dependences of the time delay and the pulse broadening on the laser power of pump1 under both the single pump scheme and the double pump scheme we proposed. (a) for time delay and (b) for pulse broadening factors B_gain, B_GVD and B. Here δ_in=52 ps, a=2.085, Δω_p1=0.5Ω_B and Δω_p2=Δω_p1/(2)^1/3.

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4. Conclusion

In conclusion we have proposed a new SBS gain spectrum tailoring method for slow light propagation in a single mode optical fiber using two broadband pump beams with different powers and spectral widths. Theoretical analyses are given to discuss the characteristics of Brillouin gain and pulse broadening by use of this method. The dependence of pulse broadening on the SBS gain profile is investigated for the first time to our knowledge. It is shown that varying gain profiles can be obtained, including the narrow arched, the flat-top, the slight center-dipped and the bimodal ones, provided that the intensity and spectral width proportions of the two pump beams are properly chosen. Among them there exists a specific gain profile, i.e., the narrow arched gain profile, which will result in an approximate zero-broadening of signal pulse. This is owing to the compensation between the positive gain-dependent broadening and the negative GVD-dependent broadening. Before ending this paper, we also point out that the law we got in this paper is also applicable for the linear propagation of arbitrarily shaped light pulses. This is because that through a resonant amplifying medium the pulse can acquire a nearly Gaussian shape, regardless of its initial temporal shape and spectral profile. And the incident pulses having different shapes but the same area and the same width can be reshaped in the medium to give approximately the same Gaussian-shaped pulse, as reported in Ref. [20]. This zero-broadening property we presented in this paper is very significant for slow light propagation, especially in the high-speed short-pulse systems. It indicates that we can control the pulse broadening of SBS slow light not only by just increasing the bandwidth of SBS gain spectrum, but also by artificially constructing and optimizing the profile of it.