1. Introduction

Beside a fundamental physical point of view, slow and fast light offers a way to many interesting applications in different fields. Among them are nonlinear optics, optical signal processing, time resolved spectroscopy and optical telecommunications. For the control of the group velocity of an optical pulse a strong dispersion is required. Therefore, many different physical mechanisms and material systems can be used for this purpose. But, due to its inherent advantages, the effect of stimulated Brillouin scattering (SBS) is of special interest [1, 2]. With SBS very simple experimental set ups are possible which delay pulses up to four times their initial pulse width [3] and for high data rates the SBS bandwidth can be enhanced by a simple modulation of the pump wave [4, 5]. Recently, it was for instance shown that with SBS based slow light 10 and 2.5Gbit/s differential phase-shift keying signals can simultaneously be demodulated and slowed down [6].

However, a severe disadvantage is that every delay comes at the expense of pulse distortions. The tolerable amount of these distortions for the given application limits the maximum time delay. A broadening and tailoring of the pump profile can reduce the pulse distortions [7, 8]. But, for the same amount of gain a broadening of its bandwidth reduces the time delay of the same order of magnitude [4].

Here we will show how a drastic reduction of the pulse distortions at high delays can be achieved by a superposition of a broadened gain with narrow losses at its wings. If the pulses are delayed by 1Bit, in a natural Brillouin gain bandwidth, the output pulse width is 1.72 times the input pulse width. For a superposition of gain and losses the fractional pulse width was only 1.32 at the same delay. Therefore, the pulse broadening was reduced by around 23%.

2. Theory

A narrow band pump wave propagating in one direction of an optical fiber can produce a gain and a loss for counter propagating pulses. The pulses will be amplified if their carrier frequency is downshifted by the Brillouin shift f _B and their bandwidth fits in the Brillouin bandwidth γ. If their carrier frequency is upshifted they will be attenuated. The generated SBS gain and loss have both a Lorentzian shape. Due to a dispersion, accompanied with gain and loss, counter propagating pulses within the gain bandwidth will be delayed and pulses within the loss bandwidth will be accelerated. In the center of the Lorentzian distribution the gain is g=g _P P _P L_eff/A_eff, where g _P is the peak value of the SBS-gain coefficient, P _P is the input pump power, L_eff and A_eff are the effective length and effective area of the fiber. In the line center the pulse delay can be calculated as Δt=g/γ [9].

Every delay is accompanied by a broadening of the delayed pulses. The broadening factor B=τ _out/τ _in is the relation between the temporal output and input pulse width, for a Lorentzain gain and Gaussian pulses it can be written as [9]:

With T=Δt/τ _in as the fractional delay. As can be seen from Eq. (1), the pulse broadening decreases with γ^0.5. So, the pulse broadening can be reduced if the Brillouin bandwidth is increased. This can easily be done by a direct modulation of the pump laser with a noise signal [4]. According to Eq. (1), for a given tolerable broadening factor B the maximum gain is limited by [10]:

But, if this gain can really be reached depends on the physical properties of the delay line. A SBS based delay line can be seen as an amplifier. As in every amplifier, the maximum gain is restricted by a saturation of the amplification process. This saturation depends on the power of the input pulses [10]. The highest gain can be achieved for very low input pulse powers. If the input pulse power is not much higher than the noise floor in the fiber, the maximum gain is restricted by the threshold of SBS. For a low loss uniform fiber this gain is g _TH=19 [11]. More realistically, the input pulse is for instance 1nW (-60dBm). In this case the maximum available gain reduces to g _-60=14.02 [12]. Since the delay increases with increasing gain in the small signal regime and it decreases with increasing gain in the saturation regime [9], for high fractional delays the gain has to be smaller than this maximum value.

Figure 1 (a) shows the maximum gain versus the Brillouin bandwidth for two different broadening factors (solid lines) and the corresponding fractional delay for a 30ns input pulse (dashed dotted lines). For a broadening factor of B=2, the maximum fractional delay is T=1.94 if the Brillouin bandwidth is 38MHz. For lower broadening factors the bandwidth has to be increased. But, at the same time the fractional delay decreases. For B=1.32 the maximum fractional delay is only 0.97. This means that if only a Brillouin gain is available, the reduction in time delay due to the bandwidth broadening cannot be compensated by higher pump powers.

 

Fig. 1. (a). Maximum gain versus bandwidth for two different broadening factors (solid lines) and corresponding fractional delay (dashed dotted lines) for 30ns pulses. (b) Normalized time delay versus k and n for m=1.

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On the other hand, two frequency separated losses can lead to a time delay as well [13]. If a Lorentzian gain is superimposed with two Lorentzian losses at its wings [3], the time delay in the line center can be written as:

with the normalized gain m=g ₂/g ₁, the normalized frequency shift n=δ/γ₁, and the normalized bandwidth k=γ₂/γ₁. Here g _1,2 and γ _1,2 are the gain coefficient and the line width of the Brillouin gain and loss and δ is the frequency shift of the losses in respect to the line center of the gain. As can be seen, for n>k the time delay in the line center is increased by the losses at its wings. The normalized time delay as a function of k and n is shown in Fig. 1(b) for equal amounts of gain and loss (g ₁=g ₂). Especially if the gain is much broader than the loss (k≪1) - which is for instance the case in high data rate applications - the enhancement of time delay by the loss can be very high. Furthermore, the losses can be arranged in a manner that the delay bandwidth is very flat which reduces higher order dispersion terms [3]. But, even for k=1 the loss power can compensate the time delay reduction.

There are two reasons for the temporal pulse broadening during the delay process; one is the spectral narrowing which is caused by the fact that frequencies away from the line center of the pulse are subjected to lower amplification, the other one is group velocity dispersion [14]. Both are reduced by this method, as can be seen from Fig.2. Figure 2(a) shows the normalized gain versus normalized frequency for a single Brillouin gain with a bandwidth γ (dashed line) and a broadened Brillouin gain superimposed with two losses (γ₁=2γ, m=3, k=0.25 and n=1). As can be seen from the inset (vertically shifted solid line), the bandwidth of both gains is almost equal. But for the superimposed case, the spectrum is more flat. At the same time, the superimposed case requires much less gain for the same delay. In Fig. 2(b) to (d) the group index change and its first and second derivation is shown. As can be seen, due to the flat group velocity spectrum the higher order dispersions are almost zero in the bandwidth region.

 

Fig. 2. Normalized gain (a), normalized group index change (b), derivation of the group index change (c) and two times the derivation of the group index change (d) versus normalized frequency for a single Brillouin gain (dashed lines) and a broadened Brillouin gain superimposed with two losses (solid lines). For the case of superposition the parameters are; γ₁=2γ, m=3, k=0.25, n=1. Please note that the abscissa scale of (a) and (b) is four times the scale of (c) and (d). The inset shows a zoom into the gain spectra for a vertically shifted solid line.

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In practice a broadening of the Brillouin bandwidth leads to a Gaussian rather than a Lorentzian shape of the gain spectrum. However, for a Gaussian gain spectrum which is superimposed with two losses similar results as presented here can be seen [3].

3. Experiment

To verify our theoretical predictions we delayed pulses in a standard single mode fiber (SSMF) with the parameters α=0.209dB/km, A_eff=86µm², g _P≈2×10^-11m/W γ/2π≈30MHz and L=50.45km. We used such a very long fiber in order to reduce the pump power requirements.

The schematic experimental set up is shown in Fig. 3(a). Pulses with a temporal width of 30ns were generated by an electrical pulse generator (Pulse). A Mach Zehnder modulator (MZM2) and a fiber laser (Fiber) with a carrier wavelength of 1550nm were used to transform these pulses into the optical domain. The pulses were coupled into the SSMF via a polarizer (not shown). From the other side we coupled three pump waves, which produced the gain and losses, into the same fiber via an optical circulator (C). The gain (inset (a)) was produced by a distributed feedback laser (DFB) diode (Gain Pump) and amplified by an Erbium doped fiber amplifier (EDFA1). In order to broaden the gain spectrum, “Gain Pump” was directly modulated with an electrical noise signal (Noise). The two loss spectra were produced by another DFB laser diode (Loss Pump) which was externally modulated by a MZM1 with a sinusoidal signal (Sinus). The MZM was driven in a suppressed carrier regime so that only two sidebands are present at its output. Two times the frequency of the sinusoid determines the frequency separation between the losses. In order to superpose the gain with the two losses, “Loss Pump” was downshifted in frequency in respect to “Gain Pump” by twice the Brillouin shift in the fiber (f _B≈22GHz, inset (c)). The superposition of gain and loss is shown in inset (b). For an independent control of the loss power, the two sidebands were amplified by another EDFA2. The delayed pulses were coupled out by an optical circulator (C), transformed into the electrical domain by a photodiode (PD) and analyzed with an oscilloscope (Osci).

 

Fig. 3. (a). Schematic experimental set up. The insets show the allocation of the pump lights and of the corresponding gains and losses. (b) Pulse broadening factor vs. fractional delay for a natural Brillouin gain (squares), a broadened gain (stars) and a broadened gain superimposed with two losses (δ=30MHz) at its wings (dots). The lines show a square root and a linear fitting of the corresponding data.

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The pulse broadening factor B as a function of the fractional delay T can be seen in Fig. 3(b). The squares show the pulse broadening for the natural Brillouin gain without additional losses. For this measurement we varied the pump power (Gain Pump) between -3 and 7dBm. As can be seen from the fitting curve, for low fractional delays, the measured values deviate from the square root dependence of B vs. T. Since, for our very long fiber, low delays are accompanied with low pulse powers, we address this deviation to uncertainties in the determination of the pulse width. For a fractional delay of 1 Bit (30ns) the broadening factor was 1.72.

If the bandwidth of the Brillouin gain is doubled by a direct modulation of the pump laser with a noise signal, the broadening factor can be drastically reduced. But, as expected from the theory and shown with the stars in Fig. 3(b), the maximum fractional delay is reduced as well. For a pump power of around 14dBm it is only around 0.5 Bit. If we increase the pump power further, the delay line goes into the saturated regime which can be seen by a reduction of the fractional delay.

Higher fractional delays can only be achieved by the incorporation of additional losses, as shown with the dots in Fig. 3(b). For this measurement the frequency difference between the two losses was 60MHz. If we leave the optical power of the gain laser constant and increase the power of the loss laser to 12dBm we achieve again a fractional delay of 1Bit. But now with a much smaller broadening factor of only 1.32. This corresponds to a distortion reduction of around ¼ for the same fractional delay. If a gain is superimposed with additional losses Eq. (1) no longer holds. Hence, as can be seen from the fitting curve, for this arrangement we have a rather linear dependence between B and T.

The normalized pulse power versus time delay for the natural Brillouin bandwidth (a) and the doubled bandwidth with two additional losses (b) is shown in Fig. 4. As can be seen, for the same fractional delay, the pulse distortions are in fact much smaller if additional loss spectra are incorporated in the set up.

 

Fig. 4. (a) Normalized pulse power vs. time delay for a natural Brillouin gain and a doubled Brillouin gain superimposed with two losses at its wings (b), δ=30MHz.

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4. Conclusion

We have shown that by the superposition of different gain and loss spectra the broadening of pulses in SBS based slow light systems can be drastically reduced. For the superposition of a 60MHz broad gain with two narrowband losses at its wings, we achieved a pulse width reduction of around ¼ for a fractional delay of 1Bit.

In the presented method the time delay reduction due to the gain broadening was compensated by the power of additional losses at the wings of the gain. Since the loss laser produces simultaneously two gains, the maximum loss power is restricted by the threshold of Brillouin scattering in the fiber [12]. Therefore, higher fractional delays at small distortions can be achieved if these gains are compensated by additional losses produced by a third laser which is downshifted in respect to the first by 4 times the Brillouin shift. Another possibility of enhancing the fractional delay is the incorporation of cascaded delays [15, 16].

A further reduction of the distortions, especially for data signals, can be reached by an adaptation of the Brillouin bandwidth to the given data spectrum [7]. Here the additional losses can as well be used to compensate for the resulting reduction in fractional delay.

Although we have only shown here the distortion reduction for pulses with a rather low bandwidth, we cannot see any restriction for high data rate systems.