Broadband SBS Slow Light in an Optical Fiber

We investigate slow-light via stimulated Brillouin scattering in a room temperature optical fiber that is pumped by a spectrally broadened laser. Broadening the spectrum of the pump field increases the linewidth $\Delta\omega_p$ of the Stokes amplifying resonance, thereby increasing the slow-light bandwidth. One physical bandwidth limitation occurs when the linewidth becomes several times larger than the Brillouin frequency shift $\Omega_B$ so that the anti-Stokes absorbing resonance cancels out substantially the Stokes amplifying resonance and hence the slow-light effect. We find that partial overlap of the Stokes and anti-Stokes resonances can actually lead to an enhancement of the slow-light delay - bandwidth product when $\Delta\omega_p \simeq 1.3 \Omega_B$. Using this general approach, we increase the Brillouin slow-light bandwidth to over 12 GHz from its nominal linewidth of $\sim$30 MHz obtained for monochromatic pumping. We controllably delay 75-ps-long pulses by up to 47 ps and study the data pattern dependence of the broadband SBS slow-light system.


I. INTRODUCTION
T HERE has been great interest in slowing the propagation speed of optical pulses (so-called slow light) using coherent optical methods [1]. Slow-light techniques have many applications for future optical communication systems, including optical buffering, data synchronization, optical memories, and signal processing [2], [3]. It is usually achieved with resonant effects that cause large normal dispersion in a narrow spectral region (approximately equal to the resonance width), which increases the group index and thus reduces the group velocity of optical pulses. Optical resonances associated with stimulated Brillouin scattering (SBS) [5]- [9], stimulated Raman scattering [10] and parametric amplification [11] in optical fibers have been used recently to achieve slow light.
The width of the resonance enabling the slow-light effect limits the minimum duration of the optical pulse that can be effectively delayed without much distortion, and therefore limits the maximum data rate of the optical system [12]. In this regard, fiber-based SBS slow light is limited to data rates less than a few tens of Mb/s due to the narrow Brillouin resonance width (∼30 MHz in standard single-mode optical fibers). Recently, Herráez et al. [13] increased the SBS slowlight bandwidth to about 325 MHz by broadening the spectrum of the SBS pump field. Here, we investigate the fundamental limitations of this method and extend their work to achieve a SBS slow-light bandwidth as large as 12.6 GHz, thereby supporting data rates of over 10 Gb/s [14]. With our setup, we delay 75-ps pulses by up to 47 ps and study the data pulse quality degradation in the broadband slow-light system. This paper is organized as follows. The next section describes the broadband-pump method for increasing the SBS slow-light bandwidth and discuss its limitations. Section III presents the experimental results of broadband SBS slow light, where we investigate the delay of single and multiple pulses passing through the system. From the multiple-pulse data, we estimate the degradation of the eye-diagram as a function of delay, a first step toward understanding performance penalties incurred by this slow-light method. Section IV concludes the paper.

II. SBS SLOW LIGHT
In a SBS slow-light system, a continuous-wave (CW) laser beam (angular frequency ω p ) propagates through an optical fiber, which we take as the −z-direction, giving rise to amplifying and absorbing resonances due to the process of electrostriction. A counterpropagating beam (along the +zdirection) experiences amplification in the vicinity of the Stokes frequency ω s = ω p − Ω B , where Ω B is the Brillouin frequency shift, and absorption in the vicinity of the anti-Stokes frequency ω as = ω p + Ω B .
A pulse (denoted interchangeably by the "probe" or "data" pulse) launched along the +z-direction experiences slow (fast) light propagation when its carrier frequency ω is set to the amplifying (absorbing) resonance [5]- [9]. In the small-signal regime, the output pulse spectrum is related to the input spectrum through the relation E(z = L, ω) = E(z = 0, ω) exp[g(ω)L/2], where L is the fiber length and g(ω) is the complex SBS gain function. The complex gain function is the convolution of the intrinsic SBS gain spectrumg 0 (ω) and the power spectrum of the pump field I p (ω p ) and is given by where g 0 is linecenter SBS gain coefficient for a monochromatic pump field, and Γ B is the intrinsic SBS resonance linewidth (FWHM in radians/s). The real (imaginary) part of g(ω) is related to the gain (refractive index) profile arising from the SBS resonance.
Equation (1) shows that the width of the SBS amplifying resonance can be increased by using a broadband pump. Regardless of the shape of the pump power spectrum, the resultant SBS spectrum is approximately equal to the pump spectrum when the pump bandwidth is much larger than the intrinsic SBS linewidth. This increased bandwidth comes at some expense: the SBS gain coefficient scales inversely with the bandwidth, which must be compensated using a higher pump intensity or using a fiber with larger g 0 .
To develop a quantitative model of the broadband SBS slowlight, we consider a pump source with a Gaussian power spectrum, as realized in our experiment. To simplify the analysis, we first consider the case when the width of the pump-spectrum broadened Stokes and anti-Stokes resonances is small in comparison to Ω B , which is the condition of the experiment of Ref. [13]. Later, we will relax this assumption and consider the case when ∆ω p ∼ Ω B where the two resonances begin to overlap, which is the case of our experiment.
In our analysis, we take the pump power spectrum as Inserting this expression into Eq. (1) and evaluating the integral results in a complex SBS gain function given by where w(ξ + iη) is the complex error function [15], ξ = (ω + Ω B − ω p0 )/∆ω p , and η = Γ B /(2∆ω p ). When η ≪ 1 (the condition of our experiment), the gain function is given approximately by where erfc is the complementary error function. The width (FWHM, rad/s) of the gain profile is given by Γ = 2 √ ln 2∆ω p , which should be compared to the unbroadened resonance width Γ B . The line-center gain of the broadened resonance is given by G = √ πηG 0 . The SBS slow-light delay at line center for the broadened resonance is given by A Gaussian pulse of initial pulse width T 0 (1/e intensity halfwidth) exits the medium with a broader pulse width T out determined through the relation Assuming that a slow-light application can tolerate no more than a factor of two increase in the input pulse width (T out = 2T 0 ), the maximum attainable delay is given by which is somewhat greater than that found for a Lorentzian line [16]. From Eq. (7), it is seen that large absolute delays for fixed ∆ω p can be obtained by taking T 0 large. We now turn to the case when the pump spectral bandwidth ∆ω p is comparable with the Brillouin shift Ω B . In this situation, the gain feature at the Stokes frequency ω p0 − Ω B overlaps with the absorption feature at the anti-Stokes frequency ω p0 + Ω B . The combination of both features results in a complex gain function given by where ξ ± = (ω ± Ω B − ω p0 )/∆ω p . As shown in Fig. 1, the anti-Stokes absorption shifts the effective peak of the SBS gain to lower frequencies when ∆ω p is large, and reduces the slope of the linear phase-shift region and hence the slowlight delay. For intermediate values of ∆ω p , slow-light delay arising from the wings of the anti-Stokes resonances enhances the delay at the center of the Stokes resonance. Therefore, there is an optimum value of the resonance linewidth that maximizes the delay. Figure 2 shows the relative delay as a function of the resonance bandwidth, where it is seen that the optimum value occurs at ∆ω p ∼ 1.
3 Ω B and that the delay falls off only slowly for large resonance bandwidths. This result demonstrates that it is possible to obtain practical slow-light bandwidths that can somewhat exceed a few times Ω B .

III. EXPERIMENTS AND RESULTS
As discussed above, the SBS slow-light pulse delay T del is proportional to G/Γ. The decrease in G that accompanies the increase in ∆ω p needs to be compensated by increasing the fiber length, pump power, and/or using highly nonlinear optical fibers (HNLF). In our experiment, we use a 2-km-long HNLF (OFS, Denmark) that has a smaller effective modal area and therefore a larger SBS gain coefficient g 0 when compared with a standard single-mode optical fiber. We also use a high-power Erbium-doped fiber amplifier (EDFA, IPG Model EAD-1K-C) to provide enough pump power to achieve appreciable gain.
To achieve a broadband pump source, we directly modulate the injection current of a distributed feedback (DFB) singlemode semiconductor laser. The change in injection current changes the refractive index of the laser gain medium and thus the laser frequency, which is proportional to the currentmodulation amplitude. We use an arbitrary waveform generator (TEK, AWG2040) to create a Gaussian noise source at a 400-MHz clock frequency, which is amplified and summed with the DC injection current of a 1550-nm DFB laser diode (Sumitomo Electric, STL4416) via a bias-T with an input impedance of 50 Ohms. The resultant laser power spectrum is approximately Gaussian. The pump power spectral bandwidth is adjusted by changing the peak-peak voltage of the noise source.
The experiment setup is shown schematically in Fig. 3. Broadband laser light from the noise-current-modulated DFB laser diode is amplified by the EDFA and enters the HNLF via a circulator. The Brillouin frequency shift of the HNLF is measured to be Ω B /2π = 9.6 GHz. CW light from another tunable laser is amplitude-modulated to form data pulses that counter-propagate in the HNLF with respect to the pump wave. Two fiber polarization controllers (FPC) are used to maximize the transmission through the intensity modulator and the SBS gain in the slow-light medium. The amplified and delayed data pulses are routed out of the system via a circulator and detected by a fast photoreceiver (12-GHz bandwidth, New Focus Model 1544B) and displayed on a 50-GHz-bandwidth sampling oscilloscope (Agilent 86100A). The pulse delay is determined from the waveform traces displayed on the oscilloscope.
To quantify the effect of the bandwidth-broadened pump laser on the SBS process, we measured the broadened SBS gain spectra by scanning the wavelength of a CW laser beam and measuring the resultant transmission. Figure 4(a) shows an example of the spectra. It is seen that the features overlap and that Eq. (4) does an excellent job in predicting our observations, where we adjusted Γ to obtain the best fit. We find Γ/2π = 12.6 GHz (∆ω p /Ω B ∼ 0.8), which is somewhat smaller than the optimum value. We did not attempt to investigate higher bandwidths to avoid overdriving the laser with the broadband signal. This non-ideality could be avoided by using a laser with a greater tuning sensitivity. Based on the measured SBS bandwidth, we chose a pulsewidth (FWHM) of ∼75 ps (T 0 ∼ 45 ps) produced by a 14 Gb/s electrical pulse generator. Figures 4(b)-(d) show the experimental results for such input pulses. Figure 4(b) shows the pulse delay as a function of the gain experienced by the pulse, which is determined by measuring the change in the pulse height. A 47-ps SBS slow-light delay is achieved at a pump power of ∼580 mW that is coupled into the HNLF, which gives a gain of about 14 dB. It is seen that the pulse delay scales linearly with the gain, demonstrating the ability to control all-optically the slow-light delay. The dashed line in Fig. 4(b) is obtained with Eq. (5), which tends to underestimate the time delay that is enhanced by the contribution from the anti-Stokes line (see Fig. 2). Figure 4(c) shows the width of the delayed pulse as a function of gain. The data pulse is seen to be broadened as it is delayed, where it is broadened by about 40% at a delay of about 47 ps. The dashed curve in Fig. 4(c) is obtained with Eq. (6). Figure 4(d) shows the waveforms of the undelayed and delayed pulses at a gain of 14 dB. We observe pulse delays that are due to fiber lengthening under strong pump conditions due to fiber heating. These thermallyinduced delays are not included in Fig. 4(b). To investigate how the pulse broadening seen in Fig. 4(c) might impact a communication system, we examine the pattern dependence of the pulse distortion. For example, in NRZ data format, a single '1' pulse has a different gain than consecutive '1' pulses [17]. The pattern-dependent gain could induce a different '1' level in the whole data stream, while patterndependent delay can lead to a large timing jitter. Figures 5(a)-(c) show the delayed pulse waveforms of three simple NRZ data patterns with a bit-rate of 14 Gb/s. It is clear that the pulses overlap when they are closer to each other, which degrades the system performance. To quantify the signal quality degradation, we use Q-factor (signal quality factor) of input and output pulses, which is defined as (m 1 − m 0 )/(σ 1 + σ 0 ), where m 1 , m 0 , σ 1 , σ 0 are the mean and standard deviation of the signal samples when a '1' or '0' is received. We examine the Q-penalty (decrease in Qfactor) produced by the broadband SBS slow-light system by numerical simulations. Figure 5(d) shows the Q-penalty as a function of time delay for 10 Gb/s and 13.3 Gb/s bit-rate data streams, respectively. In the simulations, the '1' pulse is assumed to be Gaussian-shaped with a pulsewidth (FWHM) of the bit time (100 ps for 10 Gb/s, 75 ps for 13.3 Gb/s). The slow-light delay is normalized by the bit time so that Q-penalties in different bit-rate systems can be compared. It is seen that the Q-penalty increases approximately linearly with the normalized delay, and that the 13.3 Gb/s data rate incurs a higher penalty than the 10 Gb/s data rate. The penalty is higher at the higher data rate because the higher-speed signal is more vulnerable to the pattern dependence, especially when the slow-light bandwidth is comparable to the signal bandwidth. Error-free transmission (BER < 10 −9 ) is found at a normalized delay of 0.25 or less. In an optimized system, it is expected that the pattern dependence can be decreased using a spectrum-efficient signal modulation format or the signal carrier frequency detuning technique [17], for example.

IV. CONCLUSION
In summary, we have increased the bandwidth of SBS slow light in an optical fiber to over 12 GHz by spectrally broadening the pump laser, thus demonstrating that it can be integrated into existing data systems operating over 10 Gb/s. We observed a pattern dependence whose power penalty increases with increasing slow-light delay; research is underway to decrease this dependence and improve the performance of the highbandwidth SBS slow-light system.