1. Introduction

Slow and fast light offers a way to many interesting applications in the fields of telecommunications, optical signal processing, time resolved spectroscopy, nonlinear optics and many more [1, 2]. Different mechanisms can be exploited for the control of the group velocity in a medium. But in principle, all of them are based on a strong dispersion. A method to produce an artificial dispersion in a material system is the effect of stimulated Brillouin scattering (SBS) in optical fibers [3, 4]. SBS based slow light offers several advantages over other mechanisms and the group velocity can be controlled in a very large range. Disadvantages of the SBS, like the small bandwidth, can be circumvented by a simple modulation of the pump wave [5] and by a superposition of gain and loss spectra the achievable time delays can be drastically enhanced [6, 7].

On the other hand, every delay in a SBS based delay line is accompanied by a broadening of the pulse. In a data stream with an equal width of logical “zeros” and “ones” the maximum broadening has to be smaller than two times the initial pulse width. Otherwise the “zeros” will be detected as “ones”. This broadening limits the maximum time delay. A reduction of the pulse broadening can be achieved by a tailoring of the pump profile and by an enhancement of the gain bandwidth [8, 9]. In a flat broad gain, for instance generated by a frequency comb, very low distortions can be achieved [10]. But, a broadened gain reduces the time delay of the same order of magnitude [5]. The time delay reduction can be compensated by a cascading of several delay lines [10, 11, 12], but this drastically increases the complexity of the system. On the other hand, two frequency separated losses can lead to an enhancement of the time delay in the center as well [13]. Hence, if a broad gain is superimposed with two losses; low distortions at high delays are possible [14]. Recently it was shown in theory that by the superposition of a broad gain with a narrower loss in its center, SBS based delay lines with zero-broadening are possible [15]. Since no accumulated broadening takes place, every time delay could be produced by cascading several such delay lines. Unfortunately, the broad gain reduces the achievable time delay and the loss leads to a further reduction. Hence, with this method a lot of stages would be required for fractional delays of several Bits.

Here we will show for the first time, to the best of our knowledge, experimentally that zero-broadening SBS based slow light is in fact possible. Contrary to Ref. [15] we used a superposition of a broad gain with two losses at its wings. With this arrangement much higher time delays can be achieved and the pulses can even be compressed.

2. Theory

A narrow band pump wave propagating in one direction of a fiber can produce a gain and a loss for counter propagating pulses. Gain and loss spectrum have a Lorentzian shape. The pulses experience amplification if their carrier frequency is downshifted by the Brillouin shift f _B and their bandwidths fit in the Brillouin bandwidth γ ₀. They will be attenuated if their carrier bandwidth is upshifted by f _B. According to the Kramers-Kronig relations, the pulses will be slowed down in the gain and accelerated in the loss spectrum. If a Lorentzian gain is superimposed with two losses at its wings the over all gain can be written as [6]:

the corresponding time delay is:

Here all frequencies and bandwidths were normalized to the gain bandwidth and the induced loss was normalized to the induced gain, so that Ω=ω-ω₀)/γ₀, d=δ/γ ₀, k=γ ₁/γ ₀ and m=g ₁/g ₀ with g _0,1=g _P P _P,L L _eff/A _eff as the induced gain and loss in the line center, P _P,L is the pump and loss power at the fiber input, respectively, g _P is the peak value of the SBS gain coefficient, L _eff and A _eff are the effective length and area of the fiber, γ_0,1 is the half FWHM Brillouin gain and loss bandwidth ω=2πf is the optical frequency, ω₀ the center frequency of the gain and δ is the frequency shift of the losses in respect to ω₀ (see the inset in Fig. 1(a)).

A normalized Gaussian shaped input pulse power spectrum can be written as P _in(Ω)=exp[-ln(2)(Ω/W)²], where W=Δω/γ₀ is the relation between the half-width of the Gaussian pulse and the half-width of the Lorentzian gain. The pulse will be amplified in the gain spectrum. Hence, the amplified output pulse is P _out(Ω)=P _in(Ω)exp(G _n). Since the Brillouin gain spectrum leads to a spectral narrowing of the pulse, (i.e. frequencies away from the pulse line center are subjected to lower amplification than the frequencies around the center) the pulse will be broadened. The gain dependent time broadening can be seen as the relation between the spectral width of the input and output pulse [15] B _gain=Δω_in/Δω_out. But, spectral narrowing is not the only reason for pulse broadening the other one is the group velocity dispersion (GVD). This GVD-dependent broadening can be approximated to a difference between time delays at the center frequency ΔT _d(ω₀) and at the FWHM bandwidth ΔT _d(±Δω), hence [15]: B _GVD=Δω_in[ΔT _d(ω₀) - ΔT _d(ω₀±Δω)]/ln(2). The pulse broadening is than B=B _gain+ B _GVD.

 

Fig. 1. Normalized gain (a) and time delay (b) versus normalized frequency Ω for a single gain (dashed lines) and a gain superimposed with two losses at its wings (solid lines). The inset shows the spectral arrangement of gain (dotted line), losses (dashed line) and the superposition (solid line) for an enhanced frequency scale. In the Fig. (c) and (d) the broadening due to the gain bandwidth (dashed dotted lines), due to the GVD (dashed lines) and the overall broadening (solid lines) versus the pulse width parameter W is shown for a single gain (c) and a gain superimposed with two losses (d) (g₀/g₀=1, γ₀Ω=1, m=5, d=1.1, k=0.1).

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The dashed curves in Fig. 1(a) and (b) show the gain and the normalized time delay for a single Brillouin gain. The corresponding pulse broadening versus the pulse width parameter W is shown in Fig. 1(c). As can be seen, B _gain and B _GVD are always positive and they increase with W. Therefore, the output pulse width is always higher or equal than the width of the input pulse B≥1. A very low or nearly zero broadening B≈1 can only be achieved for small values of W. This means that in a single gain delay line the broadening is negligible small if the SBS bandwidth γ₀ is much higher than the spectral pulse width Δω. But since ΔT _n=g₀/γ₀, only little fractional delays can be achieved for large Brillouin bandwidths [14]. If a broadened gain is superimposed with a loss at the same central frequency like in Ref. [15], the time delay in the line center (Ω=0) is [6]: ΔT _n=g₀/γ₀(1-m/k). Hence, since m and k are always positive, the time delay will be further reduced by the loss.

The solid curves in Fig. 1(a) and (b) show the normalized gain and time delay for a gain superimposed with two losses at its wings (d=1.1). The gain is 10 times broader and 5 times lower than the loss (k=0.1, m=5). The corresponding broadening is shown in Fig. 1(d). As can be seen, the losses reduce the gain bandwidth. At the same time the gain and therefore the amplification of the spectral components is reduced. But contrary to [15], the time delay will be increased by the losses, it is [14]: ΔT _n=g₀/γ₀(1-2mk(k ² - d ²)/(k ²+d ²)²)≈1.8g₀/γ₀. If the spectral components of the pulse are within the flat time delay region (W<0.5), the broadening due to GVD is about zero and the whole broadening comes mainly from a spectral narrowing of the pulse. Although B _gain will be increased by the losses, B is smaller than in the single gain case. At the same time higher time delays can be achieved [14]. However, if the pulse width is increased, the spectral components at the wings experience a higher time delay than in the pulse center. The result is a negative B _GVD which is able to compensate the positive B _gain. Therefore, due to the losses, the pulse can achieve high time delays without a temporal broadening. If the spectral pulse width is increased further, the GVD overcompensates the broadening by spectral narrowing. Hence, the pulse will be compressed by the delay line.

 

Fig. 2. Normalized time delay (a) and normalized broadening (b) as a function of m versus d for g₀/g₀=1, γ₀Ω=1, W=1.0, k=0.1.

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Figure 2 shows the normalized time delay (a) and broadening (b) as a function of m versus d. As can be seen from Fig. 2(b), there is a wide range in which B<2. Especially for 0.4≤d≤0.5 and m>3 very high time delays with minor or zero broadening are possible. However, in the theory presented above we neglected higher order effects than the GVD. Especially a curve like the solid one in Fig. 1(b) leads to higher order distortions of the pulse shape for W>0.5. The effects of these higher order dispersions on real data streams and possible ways to compensate them require further investigations. Furthermore, in practice a broadening of the Brillouin bandwidth leads to a Gaussian rather than a Lorentzian shape of the gain spectrum. This makes the theory more complicated. However, for a Gaussian gain spectrum which is superimposed with two losses, similar results can be seen but the parameters vary [6].

3. Experiment

For the experimental verification of our theoretical predictions we used a similar setup as already described in [14]. Pulses with a width of 1.5ns were delayed in a 25km standard single mode fiber (α=0.21dB/km, A _eff=86µm², g_P≈2×10^-11m/W, γ₀/2π≈15MHz). The Pulses were generated by a pulse pattern generator. In order to achieve a Gaussian pulse shape we enhanced the electrical pulses by an amplifier with a low pass characteristic. The gain was produced by a distributed feedback laser diode (DFB). In order to broaden the gain spectrum, the DFB was directly modulated with an electrical noise signal. The two loss spectra were produced by another DFB laser diode which was externally modulated by a Mach Zehnder Modulator (MZM) with a sinusoidal signal, driven in a suppressed carrier regime. In order to superpose the gain with the two losses, the loss laser was downshifted in frequency in respect to the gain laser by twice the Brillouin shift in the fiber (f _B≈22GHz).

 

Fig. 3. Normalized pulse power vs. time delay for different frequency shifts between the loss and the gain. The shift was increased in 10MHz steps from 100 (right) to 400MHz (left). The four bold curves correspond to a frequency shift of 140MHz, 220MHz, 280MHz and 400MHz (right to left). Please compare to the four insets in Fig. 4. (Media 1).

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The delayed pulses together with the reference pulse (zero time delay) can be seen in Fig. 3. Due to the fiber dispersion, the non delayed output pulses had a temporal width of 1.9ns. For this measurement the gain (γ₀/2π≈165MHz, 12.35dBm) and loss power (15MHz, 15.73dBm) were set to be constant and we varied the frequency shift of the two losses in respect to the gain between 100 and 400MHz. The fractional pulse width – as a relation between the pulse widths of the delayed pulse and the reference (τ/τ_ref) – versus the frequency shift of the losses can be seen in Fig. 4(a). Figure 4 (b) shows the fractional delay (relation between the time delay and the reference pulse width ΔT/τ_ref) versus the frequency shift of the losses. The insets in Fig. 4 show the four bold pulses from Fig. 3 together with the reference. As can be seen, if the losses are far away from the gain center (δ=400MHz for inset II in Fig. 4(a)), the losses have only little influence on the pulse delay. The fractional delay is rather small and the pulses will be broadened by the spectral narrowing and the GVD. The fractional delay and broadening is nearly the same as if only the gain would be present. If the pulses are closer together (δ=280MHz for inset IV in Fig. 4(b)), the fractional delay will be increased by the losses. But, at the same time the pulse will be broadened due to the smaller gain bandwidth and the higher GVD. If the separation between the pulses is decreased further (δ=220MHz for inset III in Fig. 4(b)), the fractional delay will be increased. At the same time the GVD becomes negative and will be able to compensate the high broadening by spectral narrowing. In consequence, the fractional pulse width is decreased. For a further decrease of the frequency separation (δ=140MHz for inset I in Fig. 4(a)) the pulse achieves a high fractional delay with around 1.5. At the same time the fractional pulse width is only 0.8. This means the delayed pulse has a smaller pulse width than the reference. Therefore, due to the GVD, the pulse will be compressed by the delay line. The dashed line in Fig. 4(a) shows the relation between the input pulse width and the reference. As can be seen, the fractional pulse width of the delayed pulse comes with 0.8 very close to that for the input pulse (0.79). Hence, the delay line compensates the fiber dispersion. A control of the pulse delay for constant broadening can be achieved by a variation of the gain. For a constant m, the loss has to be changed as well.

 

Fig. 4. Fractional pulse width (a) and fractional delay (b) versus frequency shift of the losses. The insets show selected delayed pulses together with the reference (please compare to the four bold curves in Fig. 3). [Media 1]

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On the other hand, the compression comes at the expense of a pulse shape distortion. As can be seen from Fig. 3, for higher fractional delays a part separates from the main pulse and forms an additional pulse with a higher group velocity than the reference. We address this behavior to higher order dispersion effects. However, in optical communication systems a logical “one” is detected at 50% of the maximum pulse power. Since the additional pulse peak is only around 20% of the main pulse, it increases the noise. But in Eq. (1) and (2) are many free parameters, so we believe that an optimization of the slow light spectrum which includes higher order dispersion effects could minimize or even cancel these distortions.

4. Conclusion

We have shown that by the superposition of two loss spectra at the wings of a broadened gain, zero-broadening slow light at high pulse delays can be achieved. Since no broadening accumulation takes place, every pulse delay could be possible by a cascading of several such delay lines. Due to the GVD produced by the two losses even a compensation of the fiber dispersion was observed. On the other hand, the compression is accompanied by a pulse shape distortion. If it will be possible to minimize or even cancel these distortions by an optimization requires further investigations.