1. Introduction

The concepts of phase, group, information, and signal velocity have been the subject of numerous recent studies [1,2]. These ideas have been explored in great detail over many decades [3], but recent interest in this area stems from dramatic experimental demonstrations of fast and slow light [4–8] and from the likelihood that these effects can be used in novel laboratory and commercial devices. Some of the proposed uses for slow light include optical buffers, continuously variable optical delays, low-power optical switching, and quantum memory. The motivation behind our research is to identify the ideal, fiber-based technique(s) for generating slow light for a variety of applications based on ease of use, wavelength agility, allowable pulse durations, conceptual simplicity and maximum total and/or relative delay [9–11].

In this paper, we report the first demonstration of slow-light using a Raman fiber amplifier. This system has the advantage of being able to accommodate the bandwidth of and to controllably delay pulses less than 1-ps duration. Specifically, we show for 430-fs input pulses that a controllable change in delay of 85% of a pulse width is possible. The system has many of the same advantages as demonstrated using stimulated Brillouin scattering [9, 12–14], that is, the slow-light resonance can be positioned at the desired wavelength by tuning the wavelength of the pump, the use of a highly nonlinear optical fiber permits the use of relatively low pump powers, operation is at room temperature, and it is compatible with telecommunication technology. We believe this demonstration represents the first step in the application of Raman amplifiers as controllable delay lines for ultrafast systems and in optical communication devices.

We find that in this system several mechanisms which impact the group velocity of an amplified pulse must be considered, such as gain-dependent self-phase modulation and cross-phase modulation, and wavelength-dependent gain combined with nonzero group-velocity dispersion. Indeed, an understanding of pulse propagation in Raman amplifiers is critical for developing a complete understanding of the impact of Raman in-line amplifiers in fiber communication systems [15, 16]. A relevant study of the optically-induced pulse delay from a solid-state Raman amplifier based on a Ba(NO₃)₂ crystal is reported in Ref. [17]. In that work, 90-ps pulses of wavelength 1.2 μm were delayed by as much as 105 ps.

The fundamental concepts of slow light based on stimulated scattering [9, 13] have been described elsewhere [2, 18]. The central idea is that for optical fields whose frequency falls near an optical resonance, there exists a variation with optical frequency in both the imaginary and real components of the refractive index. The variation in the imaginary component leads to absorption or gain, while the variation in the real component leads to a change in the group velocity. The real and imaginary components of the refractive index are Hilbert transform pairs, often referred to as Kramers-Kronig transformations. For the case of Raman amplification in optical fibers, these quantities are plotted in Refs. [15, 19]. One of the advantages of using stimulated scattering to modify the group velocity of light, rather than a permanent absorption feature or a population-related gain feature, is that it allows for the position and the magnitude of the resonance to be controlled completely by the wavelength and the power of the pump field, respectively.

 

Fig. 1. The experimental setup used to demonstrate slow light in a Raman fiber amplifier (det, detector; FPC, fiber-polarization conroller; other abbreviations are described in the text). The reference pulse is used for the FTSI measurement of the signal delay.

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As described in our previous work [9, 12], when the frequency of a pulse is tuned exactly to the peak of the gain feature, the change in group delay resulting from the resonance is given approximately by

where G = g ₀ P_pL/A_eff is the gain parameter, Γ is the linewidth of the resonance in units of angular frequency, g ₀ is the gain coefficient at the peak of the Raman resonance, P_p is the peak pump power, and A_eff and L are the effective area and length, respectively, of the fiber. The power gain is related to G by P_sig-out/P_sig-in = e^G, where P_sig-out and P_sig-in are the output and input signal powers, respectively. The assumptions implicit in Eq. (1) are that the input pulse spectrum is smaller than the bandwidth of the Lorentzian resonance, that higher-order dispersion is negligible, and that the input signal does not saturate the Raman gain.

The experimental setup is shown in Fig. 1. Signal pulses are derived from a Ti:Sapphire-pumped optical parametric oscillator (OPO). The wavelength of the OPO is continuously tunable over the entire range of wavelengths used in this experiment (1590–1643 nm), and it emits nearly transform-limited pulses. For pulses centered at a wavelength of 1640 nm, the spectral width is 9.1 nm-FWHM corresponding to a transform-limited temporal width of 430 fs. Prior to entering the Raman amplifier, the signal propagates through several optics and 8 m of standard optical fiber (SMF-28e).

In our setup, we have chosen to use Fourier-transform spectral interferometry (FTSI) to measure the delays of the signal pulses since it provides the ability to accurately measure delays as small as tens of fs between pulses of very low peak power. By using pulses whose peak power is kept <1 mW, self-phase modulation of the signal pulses is negligible. Temporally-separated reference pulses, which are used to measure the signal delay, are generated by passing the signal pulse train through an asymmetric Michelson interferometer. The resulting temporal separation is approximately 1.7 ns. Synchronous pump pulses are created by detecting the signal pulse train on a photodiode and using that electronic signal to amplitude modulate the output of a CW tunable diode laser. The resulting 500 ps-long, 1535 nm pump pulses are optically amplified and combined synchronously with the signal via a wavelength-division multiplexer (WDM).

 

Fig. 2. (a) Plots of gain versus pump peak power for several different signal wavelengths. (b) Gain slope versus signal-pump frequency detuning. The measured slopes plotted on the left axis are compared with the ASE spectrum which is plotted on the right axis.

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In the Raman amplifier portion of the system, signal, reference and pump pulses co-propagate through a 1-km span of highly-nonlinear dispersion-shifted fiber (HNLF). The zero-dispersion wavelength of the HNLF was measured to be 1551±2nm, with a slope of 0.04 ps/(nm²km), and a nonlinear coefficient of 11 (W-km)^-1. As shown in Fig. 1, the signal and pump pulses temporally overlap resulting in amplification and induced delay of the signal pulse. The reference pulse propagates without interacting with the pump, and so it remains unamplified. The pump pulses are then spectrally filtered from the signal and reference pulses using a second WDM.

The gain characteristics of the Raman amplifier are shown in Fig. 2. Using the OSA, we measure the gain as a function of pump peak power for a fixed pump wavelength (1535 nm) and for different settings of the signal wavelength. Here, the value for the gain is determined by the increase in signal power when the pump is turned on as compared with that when the pump is off. From Fig. 2(a) we see that the relationship between gain and pump power is linear and that the gain slope increases as the signal is tuned closer to the peak of the Raman gain. Plotted in Fig. 2(b) on the left axis is the value for the gain slope as a function of signal-pump frequency detuning. The spectral dependence of the gain slope follows exactly that of the Raman amplified spontaneous emission (ASE) measured for this system, which is plotted on the right axis of Fig. 2(b). The maximum Raman gain achieved using this system (not shown) is 35 dB for a peak pump power of 2.6 W. The maximum power is an experimental limitation imposed by the power handling capability of the WDM used for combining and filtering the pump and the signal. Over this range of pump powers, we did not observe saturation of the gain.

After propagating through the Raman amplifier, the signal and reference pulses are passed through an asymmetric Mach-Zehnder interferometer (MZI) having a path difference that compensates for the temporal separation between the reference and signal pulses. Temporally, the output of the MZI consists of the following four components which traverse different arms of the interferometer: reference pulses that traverse the short arm, reference pulses that traverse the long arm, signal pulses that traverse the short arm, and signal pulses that traverse the long arm. The system is configured so that the reference pulses traversing the long arm and signal pulses from the short arm are nearly temporally coincident. When the output is measured by an optical spectrum analyzer (OSA), a spectral interferogram similar to that shown in Fig. 3(a) is observed. The OSA output can be expressed as [20]

where E ₀ and E_a are the reference and Raman-amplified electric fields, respectively, ω is the frequency of the field, and ϕ_a and ϕ ₀ are the phases of the Raman-amplified and reference fields, respectively. The final term in Eq. (2) is critical because it contains the spectral modulation due to the interference between the temporally separated Raman-amplified and reference fields. The phase difference in Eq. (2) must account for gain-induced spectral shift of the amplified field as well as for cross-phase modulation between the pump and the signal. Pulse delays resulting from gain-induced spectral shifts can be included by adding a temporal component according to ∆T_SS = D∆λℓ/c, where D is the group-velocity dispersion coefficient given in units of [ps/(nm km)], ∆λ is the gain-induced spectral shift, and ℓ is the fiber length. In practice, most of the contribution to ∆T_SS occurs in the standard fiber in between the Raman amplifier and the OSA. The phase shift due to cross-phase modulation between the quasi-CW pump and signal arises within the Raman amplifier and is given by ϕ_XPM = 2γP_pL, where L is the length of the HNLF.

The amplitude of the Fourier transform of the spectral interferogram [see Fig. 3(b)] shows a peak at (∆ϕ) whose width is related to the bandwidth-implied, transform-limited temporal width of the pulse. Note that Fig. 3(b) is a time-domain representation where the x-axis has been expressed in fs. Calibration of the x-axis is achieved by translating one of the arms in the Michelson interferometer and recording interferograms for several positions.

Measurements of slow light in this Raman amplifier are shown in Fig. 4 where the signal wavelength is 1637 nm, which is very close to the peak of the Raman gain profile. The gain parameter (left axis) and signal time delay (right axis) are plotted as functions of pump peak power. We see for moderate and high pump powers that both the gain and delay vary linearly with respect to pump power. The slight nonlinearity in the curves at low pump powers is due to the difficulty in measuring the gain at these small values. The slope of the delay curve with respect to G is 15 fs which, according to the approximate relation given in Eq. (1), corresponds to a gain bandwidth of 8 THz. This derived value for the gain bandwidth is larger than the 4-THz width of the linear portion of the real part of the nonlinear coefficient as plotted by Voss, et. al. [15] and by Stolen, et. al [19]. The origin of this discrepancy is likely the result of the fact that data in Fig. 4 was recorded as a part of the sequence of measurements for several different wavelengths (see Fig. 2) and that the Raman gain spectrum is not a Lorentzian as assumed in deriving Eq. (1). Optimization of the gain and delay were not performed prior to recording data for each signal wavelength. If the polarization overlap of the pump and signal is not optimal, the output signal pulse will be broadened temporally due to gain-induced polarization mode dispersion, and the measured delay for a given gain will be reduced. The maximum delay achieved for the data depicted in Fig. 4 is 140 fs, which is 40% of the transform-limited pulse width.

Figure 5 shows three delayed signal pulses in the Fourier-transformed, time-domain representation where the system was adjusted to obtain as large a delay as possible prior to each measurement. For these pulses, the relative signal delay was varied from 0 to 370±30 fs, which at the maximum is 85% of a transform-limited pulse duration. The measured gain for the case of maximum delay was G = 7, which implies a Raman gain bandwidth of 3 THz extrapolated from Eq. (1). Note that the peak positions read from the plot do not correspond to the actual Raman group delay. To obtain the delay, corrections due to cross-phase modulation (50 fs less delay for the 370 fs case) and wavelength shift (82 fs greater delay for the 370 fs case) must be applied. As inferred from Fig. 5, we observe that the spectral width of the pulses does not change from the zero-delay case as the delay is varied.

 

Fig. 3. (a) Spectral interferograms along with the (b) Fourier-transformed pulse position.

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Fig. 4. Plots of gain and delay vs. pump power.

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Fig. 5. Amplitude of the transformed interferograms for pulse delay changes of 0 fs, 135 fs and 370 fs. The delay of 370 fs is the maximum delay achieved with our setup.

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In summary, we have demonstrated an ultrafast all-optical controllable delay in an optical fiber that utilizes the real part of the nonlinear coefficient that accompanies stimulated Raman scattering. Using this device, we have delayed a 430-fs pulse by 85% of its pulse width. This approach shows that short pulses of light with wide spectral bandwidths can be delayed using slow-light processes similar to those used in past demonstrations with relatively long pulses of light. The technique has the additional advantage of being sufficiently fast that it can be used to generate an independent delay for each consecutive pulse in a pulse train, owing to the ultrafast response of the Raman effect in silica glass.

The authors would like to thank Pablo Londero, Dan Gauthier, Zhaoming Zhu, Paul Voss, Bob Boyd, Alan Willner and Yan Wang for useful discussions. Thanks also to Chris Xu for providing HNLF and numerous instruments. The authors would also like to acknowledge financial support from the DARPA DSO Slow-Light Program and from the Center for Nanoscale Systems, supported by the NSF under grant No. EEC-0117770.