We describe systematic measurements of the gain and delay spectra in a slow light system based on stimulated Brillouin scattering in optical fibers. The measurements yield the system complex transfer function with which delays and signal distortion can be calculated for any input signal. The theoretical predictions are confirmed experimentally for single pulses as well as 50 Mb/s data streams in a system which employs pump modulation to modify the gain and delay spectra of the SBS process.
©2006 Optical Society of America
Tunable optical delay lines based on the ability to vary the group velocity in the spectral vicinity of absorption or gain resonances have been studied extensively in the last few years. Several mechanisms taking place in different nonlinear media have been employed in what has become to be known as slow and fast light propagation [1–7]. Optical fibers have several advantages as slow and fast light media since their various nonlinearities provide narrow band gain. Indeed, most common fiber nonlinearities; stimulated Brillouin scattering (SBS) [8–10], Raman scattering , Raman assisted parametric amplification  and a hybrid combination of four wave mixing and linear dispersion  have been successfully demonstrated.
The most widely used fiber nonlinearity for slow and fast light experiments is SBS. Even though its naturally narrow bandwidth (tens of MHz) is incompatible with digital signals in the Gb/s range, its robustness and ease of control make it an ideal vehicle to demonstrate basic principles of slow and fast light. Moreover, recent experiments have demonstrated that the SBS bandwidth can be widened by pump modulation [14, 15] at the expense of the maximum obtainable delay so the relatively large delay–bandwidth product of SBS can be tailored for specific applications. This approach is highlighted by the recent demonstration of pump modulation by a noise source  which pushed the SBS bandwidth to its limit (~12 GHz) which is essentially equal to the fiber Brillouin shift.
Most slow and fast light experiments report on the achievable delay and bandwidth and do not quantify the fidelity of the delayed signal. Distortion of the delayed signal is however a key parameter which determines the ability to employ slow and fast light systems in applications. Some papers [14, 16–19] address pulse broadening in slow and fast light systems in the context of delay-bandwidth product and storage density of optical buffers. Some qualitative observation of distortion obtained by delaying two pulses  has been reported and modifications to data streams which were delayed by a narrow bandwidth SBS system have been experimentally measured . Predictions of distortions in delayed pulses based on the flatness of gain and delay spectra were presented in  for Gaussian pulses. However, the effect of non ideal pulse shapes and more important, of random data streams (whose fidelity depends on pattern effects) have not been considered.
This paper describes a systematic experimental investigation which quantifies the balance between delay, bandwidth and distortion. The technique is general and can be applied to any system and any type of input signal. We highlight systems where the signal bandwidth is of the order of the entire gain spectrum (as in ), which are prone to distortions. We measured small signal gain and delay spectra in an SBS slow light system, similarly to  and used the resulting complex transfer function to calculate the delay and temporal changes of single pulses and pseudo random bit streams (PRBS). We employ moderate SBS bandwidth broadenings (obtained by pump modulation) and a 50 Mb/s PRBS whose bandwidth occupies essentially the entire SBS gain spectrum. The results we present are applicable to any system where the bandwidth is fully utilized by the delayed pulses and data streams.
We also address the manner in which the delay is determined. In all limited bandwidth systems, some distortions occur and the delay manifests itself not only as a true temporal shift but also as a lengthening of the delayed signal transition times. This causes pattern effects so that determination of the delay based on a temporal shift in the pulse leading edge (for example at its half power) may be ambiguous. Instead we determine the delay of a PRBS by the optimum sampling time  which yields the largest eye opening. We show that under various operating conditions, the delay, so defined, differs from the one set by the shift in the pulse leading edge.
2. Small signal measurements
The experimental system used to characterize the SBS small signal response is described in Fig. 1.
A tunable laser emitting at a carrier frequency ν0 is modulated by MZ1 (in some cases) and amplified to a level of up to +14 dBm to serve as an SBS pump for a 10 km long dispersion shifted fiber. The pump power is measured at the output of the EDFA and includes therefore a significant amount of broadband noise which does not contribute to the SBS gain. The stated power values so defined are used in all measurements and hence serve to properly compare between different cases. The pump generates a narrow Brillouin gain region whose center frequency is down shifted from the carrier by 10.55 GHz. A second output of the laser is linearly modulated using MZ2 by a signal from a vector network analyzer (VNA) scanning the frequency range of f M=10.3-10.75 GHz. Following amplification, a narrow fiber Bragg grating (FBG) diminishes the spectral line at ν0+f M. The carrier and the side band at ν0-f M counter propagate in the fiber with the pump. The side band at ν0-f M scans the SBS gain spectrum as the VNA frequency is swept while the carrier remains constant. Following detection, the VNA yields the small signal gain (defined as 10·log10 of the gain experienced by the optical field) and phase spectra.
Figure 2 shows small signal characteristics for an unmodulated pump. Gain, phase and delay spectra are shown for different pump levels. The measurements reveal the usual dependencies on pump level with a gain bandwidth of slightly more than 20 MHz and peak delay values which are comparable to previously reported results [8, 9].
Broader bandwidths can be obtained by pump modulation as reported in Refs. [14–16]. The simplest scheme uses sinusoidal pump modulation. Each side band acts as a separate pump and the overall gain spectrum effectively convolves the individual gain regions resulting in a gain bandwidth broadening. Sinusoidal pump modulation was examined with modulation frequencies of 10 MHz to 30 MHz with the pump modulator (MZ1) being biased at Vπ thereby suppressing the carrier and generating two pump spectral lines. The gain phase and delay spectra for a pump power of +13 dBm are shown in Fig. 3.
Figure 3 demonstrates that indeed, the overall spectrum comprises two contributions which completely separate for high modulation frequencies. The overall gain spectrum is wider than for the unmodulated pump but for some frequencies it has a rather irregular shape which naturally affects pulse distortion. The spectral irregularities of sinusoidal pump modulation can be corrected by using a broad band pump modulation, for example, a pseudo random drive signal  or a noise source . This was indeed found to be the case as seen in Fig. 4 which describes gain, phase and delay spectra for a pseudo random pump modulation at different rates between 75 Mb/s and 500 Mb/s.
3. Large signal measurements
Modulator MZ2 is biased now at Vπ and driven by a microwave synthesizer at the Brillouin frequency shift f M=10.55 GHz. Following amplification, a narrow FBG selects a single spectral line at ν 0-f M which serves as the probe. The probe is modulated by MZ3 which is driven from a programmable pulse generator or a PRBS source. The delayed probe is detected at the output and fed into a sampling oscilloscope.
3.1. Single pulses
In the first series of experiments we examined the response of a single pulse under several conditions. Figure 6 compares the output signal with no pump (blue curve) to that obtained with different pump configurations.
Figure 6(a) shows the case of an unmodulated +10 dBm pump (solid green curve) where the pulse broadens significantly, the rise and fall times lengthen and no true delay is observed. The small signal response (Fig. 2) was used to calculate the response of the 20 nsec pulse and the results is shown in Fig. 6(a) as a broken green line. The theoretical prediction is based on a frequency domain convolution among the complex transfer function (comprising the measured gain and phase spectra) and the signal spectrum, obtained by Fourier transforming the measured temporal shape. The predicted response fits the measurement very well.
Figure 6(b) shows the responses of the 20 nsec pulse with a sinusoid ally modulated pump at 10 MHz (light blue) and 20 MHz (purple). Once more the responses were calculated using the corresponding small signal data of Fig. 3 and are shown as broken lines. With the pump modulated at 10 MHz, the output pulse shows a true delay although the rise and fall times as well as the pulse width increase. The calculation and experiment fit well in this case. For a 20 MHz modulation of the pump we observe a true delay and only a small change in pulse width but the rise and fall times still increase. However, there appears also a significant trailing pulse which is a direct result of the split gain and delay spectra shown in Fig. 3. The trailing pulse is predicted by the calculation although the calculated amplitude is lower than the measured one. A different manifestation of the distortions under sinusoidal pump modulation at high frequencies is described in Fig. 6(c). Here we examine the response to an input step (implemented in the form of a very wide, 120 nsec, pulse with a fast rise time). The case of no pump is shown in the blue curves while the other colors correspond to different pump modulation frequencies. As the modulation frequency increases, the rise time shortens and a clear overshoot appears. The overshoot becomes severe for frequencies that split the small signal spectra and results in intolerable distortions. These overshoots are predicted very well by the calculation as shown in the corresponding broken lines.
Measured and calculated responses of the single 20 nsec pulse under PRBS pump modulation are shown in Fig. 6(d). Once more we show measurements as solid lines and calculations, based on the small signal spectra of Fig. 4, as broken lines. The figure compares the output with no pump (blue curve) with the delayed response under pump modulation at 75 Mb/s and 100 Mb/s. Both cases yield broadened gain and delay spectra with regular, smooth shapes (see Fig. 4) and consequently the output shows a true delay with some distortions (somewhat increased rise and fall times and a degree of pulse broadening). These small distortions can be reduced with further increase of the pump modulation rate.
3.2. Pseudo random data stream
The various distortions observed for single pulses are emphasized significantly when pseudo random data streams are delayed by the SBS medium. This was demonstrated using a 50 Mb/s signal and moderate SBS bandwidth enhancements. Figure 7 describes measured (green trace) and calculated (red curve) responses. The case for no pump is shown in all cases in blue.
Figure 7(a) shows the unmodulated pump case where the large distortions cause a completely closed eye pattern. Figures 7(b) and 7(c) show the responses for sinusoidal pump modulation at 10 MHz and 20 MHz, respectively. The responses in Figs. 7(b) and 7(c) show some eye opening but the distortion is large, consistent with the corresponding single pulse responses and the small signal spectra. For 10 MHz, the bandwidth enhancement is insufficient while for the 20 MHz case, the SBS spectra split resulting in significant distortions. In all cases, the experiments and calculations fit well.
Better results were obtained for broad band pump modulation as described in Fig. 8. The figure shows measured (green trace) and calculated (red curves) responses for three rates of the pump modulation, 75 Mb/s, 125 Mb/s and 250 Mb/s. Also shown in each case is the response with the pump off (shown in blue).
The broad band pump modulation yields the low distortion responses shown in Fig. 8 with the calculated responses fitting the measurements very well. The response shown in Fig. 8(c) exhibits the best result but it is clearly somewhat distorted. In particular, the rise time is long as expected when the bandwidth is limited. Indeed, Fig. 4 shows that for a pump modulation rate of 250 Mb/s, the gain and delay spectra are not much wider than the spectral width of a 50 Mb/s signal.
The distortion experienced by the signal raises the issue of how to properly determine the delay. In most reported cases, the delay is determined by the temporal shift of the half power point in the leading edge. This approach is accurate when the bandwidth is large compared to the pulse spectrum and the pulse experiences a true delay. However, it becomes ambiguous in limited bandwidth cases where some of the delay manifests itself as an increase of the pulse rise time. We propose here a different way to determine the delay of a data stream that experiences some distortions. The delay is defined by the sampling temporal position which yields the largest eye opening  and is found by an optimization procedure. The delay, so defined, and the normalized eye opening as a function of the pump modulation rate are shown in Fig. 9
The measured (cyan) and calculated (purple) eye opening curves are normalized to the opening of the signal with no pump. They fit very well with the eye opening stabilizing at a value of 0.8. The figure also shows three delay curves. The red trace represents the delay measured at half the leading edge amplitude. As stated before, this can be a misleading measure in bandwidth limited cases. The delay values based on the optimum sampling point are shown in the blue (calculated) and green (measured) curves fit well to each other and are always larger than the value in the red curve. For small bandwidth enhancements (low pump modulation bit rate), the differences are small but they increase significantly as the SBS bandwidth widens.
To conclude, we have presented a systematic characterization of the delay and the signal fidelity in a slow light system where the signal occupies essentially the entire gain and delay bandwidth. We use an SBS fiber system whose bandwidth is widened by sinusoidal or broad band pump modulation and a 50 Mb/s pseudo random data sequence.
Using small signal measurements of the gain and delay spectra, we demonstrate that the temporal position and shape of any input signal are completely predictable. The predictions are confirmed in a series of experiments. Since the system is bandwidth limited, all delayed signals experience some degree of distortion and this requires using care when determining the delay value. We propose to use the temporal position of the sampling point which yields the largest eye opening as a measure for the delay. In cases where the bandwidth is relatively wide, this leads to delay values which exceed those determined by the commonly used shift of the pulse leading edge. This difference diminishes when the gain and delay spectra are much wider than the signal spectral width; a condition not examined here.
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