We present a theoretical analysis of the slow-light tunable optical buffers based on fiber Brillouin amplifiers (FBA). Its storage capacity was discussed for return-to-zero (RZ) and non-return-to-zero (NRZ) bit streams. Gain saturation and pulse broadening are two key factors which limit the buffer capacity. Gain saturation is an inherent characteristic of the FBA. Broadening of the amplified pulse always accumulates with increasing gain owing to the dispersion during stimulated Brillouin scattering (SBS) process. It is shown that the maximum buffer capacity varies with data bit rate and for continuous wave (CW) or quasi-CW pump it is 0.53 and 1.04 bit for RZ and NRZ respectively. Also, the optimum data bit rate to achieve the best storage capacity is obtained.
© 2007 Optical Society of America
Tunable optical buffers (TOB) are key components in future all-optical routers (AOR). The lack of feasible AOR is a well-known bottleneck of all-optical communication network. To realize the TOB, one needs to control the group velocity of light. In the last two years, slowlight using the dispersion associated with a laser-induced amplifying resonance, such as SBS in optical fiber, has been widely studied both experimentally and theoretically. Compared with former schemes, SBS seems to be the most promising one for its large fractional delay, roomtemperature operation, low threshold, possibility of inducing delays at any wavelength, and compatibility with existing optical communication systems.
In early SBS slow-light experiments [1, 2], the delay is relatively small and the applicable bandwidth is limited by tens of MHz due to the narrow gain bandwidth. Further experiments extend the delay to several pulse widths by employing cascaded scheme  and increase the bandwidth to hundreds of MHz , and 12GHz  by using a modulated pump, even exceed 25GHz by employing a double pump method . However, pulse distortion is an inevitable phenomenon in all SBS slow-light experiments, although it can be reduced, at the cost of more pump consuming, by using a broadband pump which can be achieved by using a variety of modulating schemes [4, 5, 7–9] or various gain profiles, such as gain doublet , or multi-line Brillouin gain spectrum [7, 11, 12], instead of a single Lorentzian gain line.
Generally, one bit comprises at lease one pulse period (at least two pulse widths) for the real signal bit streams. However, in all SBS slow-light experiments up to date, no public reports have evidently shown the delay exceeded one bit, although it exceeded one or two pulse widths in rare previous experiments by employing short pulses  or cascaded scheme . Reference  analyzed the performance limits of the delay lines based on optical amplifiers where a Gaussian gain profile was employed. Commonly the Brillouin gain spectrum should be Lorentzian shape. The physics origin of this limitation for SBS has not been clarified as clearly as possible. In addition, most of the SBS slow-light experiments and theoretical analyses [14, 15] concentrated on single pulse case (pulse period ≫ pulse width), in which pulse distortion is not a serious problem. However, for the bit stream of real signals, signal distortion has a fatal affection on the buffer capability and system performance.
E. Shumakher et al.  investigated the relationship between delay, bandwidth and signal distortion in SBS slow-light systems where the signal bandwidth is of the order of the entire gain spectrum, which are prone to distortions. However, a detailed theoretical model of SBS slow-light for bit streams has not been proposed yet and the problem of the real storage capacity of the TOB based on FBA remains unsolved. In this paper, we highlight SBS slowlight systems where the signal bandwidth is smaller than the Brillouin gain bandwidth and propose a detailed analysis on the optimum buffer ability of SBS slow-light, in which the pulse broadening has been considered. Our analysis shows that the maximum of storage bits is dominated by the available net gain, which is limited by both the gain saturation and pulse broadening. Detailed analysis shows the storage capacity of SBS slow-light buffer varies with data bit rate and best storage capacity is around one bit.
2. SBS-based slow-light buffer
SBS can be viewed as a nonlinear interaction, in which the pump wave interacts with Stokes fields through an acoustic wave and produces narrowband gain or loss .
Figure 1 shows the characteristics of the SBS resonance. The probe pulse is being amplified (attenuated) due to the spectral resonance, which has a complex response function. Thus, according to the Kramers-Kronig (KK) relations, the amplification (attenuation) will result in a sharp transition in the refractive index of the material. Consequently, the group index will experience a strong transition, which is responsible for the pulse delay or advancement. The delay time is tunable via simply changing the pump power. For a CW or quasi-CW pump, the Brillouin gain has a Lorentzian spectrum of the form ,
where, g p is the peak value of the Brillouin-gain coefficient, I p is the pump intensity, Γ B is the Brillouin gain linewidth, and ω0 is the frequency in the gain line center. KK relations can be written as
where, P before the integral means to take the Cauchy principal value, nr is the real part of the phase refractive index. Substituting (1) into (2), by solving the integral we obtain
Consequently, with the group velocity vg=c(nr+ωdnr/dω)-1 and the group delay Td=L(c/vg-nr)/c, pulse delay can be approximated as
Note that Eqs. (3) and (4) can also be obtained by employing the method in ref. . Nevertheless it is more convenient to derive them from KK relations. When [2(ω-ω0)/ΓB]2≪1, Eq. (4) can be approximated by Td≈GB/ΓB, where GB=gpIp L is described as the Brillouin gain parameter. We first consider the RZ data with 50% dutycycle, pulse period τ, bit interval bit T and corresponding bit rate Bbit. The temporal duration (FWHM) of Gaussian pulses can be expressed as τFWHM ≈τ/2≈Tbit/4=1/4Bbit. For the case of NRZ
data, τFWHM≈τ/2≈Tbit/2=1/2Bbit. The storage bits of the data in the buffer Nbit (namely delay-bandwidth product), which represents the storage capacity , can be approximated as
We can see from Eq. (5), Nbit is proportional to GB for a given τFWHM, and inversely proportional to τFWHM for a given GB, as has been experimentally verified [1, 2]. However, deviations occur, for instance, due to pulse distortion when the bandwidth of input pulses is larger than SBS gain bandwidth and due to gain saturation when the input intensity of the probe pulses is too high.
3. Numerical investigation and discussions
In fact, delay is always accompanied by pulse distortion during the SBS process. In our case (signal pulses longer than tens of nanoseconds and CW or quasi-CW pump), the dispersion caused pulse broadening dominates the changing of the pulse shape. For an input Gaussian pulse with width τin (FWHM), we can obtain a longer output pulse width τout (FWHM). The output pulse remains Gaussian shape and the pulse broadening factor B is given by 
For a given tolerable B, the gain parameter is limited by
Figure 2(a) shows the maximum of broadening-constrained gain parameter GB,max as a function of input pulse width. For practical TOB, high data fidelity is desired. For any datacoding format the broadening factor B should be bounded to the condition B≤τ/τFWHM, where τ is a full pulse period, otherwise the pulses cannot be identified in the receiver. For Gaussian pulses, the limit condition is B≤2 for real signal bit streams.
On the basis of the analyses above, the gain parameter GB in Eq. (5) is dominated by two key factors. One is gain saturation induced by the high intensity of the probe pulses. The other is the pulse broadening that simultaneously occurs with the delay due to the dispersion in SBS process. In fact, GB is also limited by the process of SBS seeded by spontaneous Brillouin scattering . When GB reaches the Brillouin threshold, around 21  (It depends on the fiber parameters), spontaneous Brillouin amplification will deplete the pump power in the absence of any input probe field. Z.M. Zhu et al.  has shown that the delay increases with increasing gain in small signal regime and decreases with increasing gain in the gain saturation regime . Therefore, to achieve the maximum storage bits, GB≤Gsat needs to be satisfied, where Gsat is the saturation gain parameter which is associated with the intensity of input pulse Is(in) for a given pump power. Is(in) can be written as Is(in)≈PpeakτinBbit/Aeff, where, Ppeak, Aeff and Bbit are the peak power of the input signal pulse, the mode area of the fiber and the signal bit rate respectively. We consider here no pump depletion and ignore the fiber loss. Then we can use the method in Ref.  to calculate the saturation gain
Where ξ=Is(out)/Ip(in), and Ip(in), Is(out) are the input intensity of the pump and output intensity of the signal pulses respectively, GA=gBIp(in)L is the non-saturation gain parameter. In our numerical simulations, we use the following values for standard single mode fiber at 1550nm, Aeff=50µm2, ΓB/2π=35 MHz, g0=5×10-11 m/w, effective fiber length Leff=1km, Ppump=21mw, corresponding GA=21. We first investigated the relations between Gsat and Ppeak for RZ and NRZ bit streams by numerically solving Eq. (8). It can be seen from Fig. 2(b) that the saturation gain goes up sharply as signal peak power decreases. We then compared Gsat and GB,max as a function of input NRZ pulse width for different input pulse peak powers (see Fig. 2(c)). The maximum available gain parameter can be obtained from Fig. 2(c). Further, according to Eq. (5), we can plot Fig. 3, which denotes the maximum of storage bits Nmax varying with different bit rates for variable peak powers or tolerable broadening factors.
Figures 3(a) and 3(c) show that the maximum of storage bits Nmax first goes up linearly as bit rate increases for a given peak power. However, as the bit rate increases, the data pulse become shorter, so the broadening limit is reached quickly as the gain increases and consequently the available net gain gets smaller [see Fig. 2(c)]. As a result, Nmax then goes down as bit rate continuously increases. Thus Nmax is limited owing to the limited net gain and we can obtain an optimal bit rate to achieve the maximum Nmax. For instance, under the condition of 1 peak P=nw, B=2, the maximum Nmax is 0.53 for RZ data (bit rate 7MHz) and 1.04 for NRZ data (bit rate 14.2MHz). The difference between RZ data and NRZ data is result from the nature of the two coding methods, as can be seen from Eq. (5).
To apply the FBA-based buffer for high bit rate data streams, a broad gain bandwidth is necessary, which can be achieved by utilizing a broadband-modulated pump. However, the pump modulation modifies the shape of SBS gain spectrum beyond broadening . As a result, the signal pulse may experience unexpected distortion after being amplified by a modulated pump. More severely, for the same amount of signal gain an increase of the gain
bandwidth comes at the expense of a reduction of the delay of the same order of magnitude . Since the achievable net gain is limited, for modulated pump, the buffer ability will decrease in comparison with CW or quasi-CW pump. In fact, several other factors also play important roles in the SBS slow-light buffer. The pulse distortion can be impressively large under conditions of high bit rate (under this condition serious deviation from Eq. 6 will occur due to the signal bandwidth exceeds gain bandwidth. E.g. in Ref. , when τin=15ns the theoretical value of B is 1.95, while the experiment value is 1.4). Besides, the data pattern and signal detuning from the gain line centre also affect the system performance .
We have studied theoretically the SBS slow-light tunable optical buffer, and have highlighted its storage abilities. Our results show that the maximum storage capacity is around 1 bit. It is limited by not only gain saturation but also pulse broadening. This is maybe one possible reason why no delay over one bit was reported. The former limitation can be overcome by employing attenuators in cascaded schemes. However the latter, up to now, remains unsolved. To solve this problem, new methods need to be developed. In spite of small buffer capacity, the tunable SBS slow-light buffer can play important roles in some circumstances such as accurate data synchronization and data bit equalization. In addition, we believe similar method can be used to analyze the resonance induced slow-light buffers including stimulated Raman scattering, optical parametric amplification, electromagnetically induced transparency etc.
The authors acknowledge the support from National Natural Science Foundation of China under the grants 60577048, the Science and Technology Committee of Shanghai Municipal under the contracts 04DZ14001/05ZR14078, and the Program for New Century Excellent Talents in University of China.
References and links
1. K. Y. Song, M. G. Herraez, and L. Thevenaz, “Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering,” Opt. Express 13, 82–88 (2005). [CrossRef] [PubMed]
2. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Phys. Rev. Lett. 94, 153902 (2005). [CrossRef] [PubMed]
4. M. G. Herraez, K. Y. Song, and L. Thevenaz, “Arbitrary-bandwidth Brillouin slow light in optical fibers,” Opt. Express 14, 1395–1400 (2006). [CrossRef]
5. Z. M. Zhu, M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “12-GHz-Bandwidth SBS slow light in optical fibers,” in proceedings of OFC 2006, paper PDP1.
8. A. Zadok, A. Eyal, and M. Tur, “Extended delay of broadband signals in stimulated Brillouin scattering slow light using synthesized pump chirp,” Opt. Express 14, 8498–8505 (2006). [CrossRef] [PubMed]
9. L. L. Yi, L. Zhan, W. S. Hu, and Y. X. Xia, “Delay of broadband signals using slow light in stimulated Brillouin scattering With phase-modulated pump,” IEEE Photon. Technol. Lett. 19, 619–621 (2007). [CrossRef]
11. T. Schneider, M. Junker, K. U. Lauterbach, and R. Henker, “Distortion reduction in cascaded slow light delays,” Electron. Lett. 42, 1110–1112 (2006). [CrossRef]
14. Z. M. Zhu and D. J. Gauthier, “Numerical study of all-optical slow-light delays via stimulated Brillouin scattering in an optical fiber,” J. Opt. Soc. Am. B 22, 2378–2384 (2005). [CrossRef]
16. E. Shumakher, N. Orbach, A. Nevet, D. Dahan, and G. Eisenstein, “On the balance between delay, bandwidth and signal distortion in slow light systems based on stimulated Brillouin scattering in optical fibers,” Opt. Express 14, 5877–5884 (2006). [CrossRef] [PubMed]
17. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2005), Chap. 9.
18. R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: Capabilities and fundamental limitations,” J. Lightw. Technol. 23, 4046–4066 (2005). [CrossRef]
19. L. Zhang, T. Luo, W. Zhang, C. Yu, Y. Wang, and A. E. Willner, “Optimizing operating conditions to reduce data pattern dependence induced by slow light elements,” in proceedings of OFC 2006, paper OFP7.