## Abstract

We experimentally demonstrate an all-optical tunable delay line based on slow and fast light using soliton collision in an optical fiber. By varying the amplitude, wavelength, and number of the control soliton pulses, we accomplish both distortion-less slow and fast light generation and achieve the consecutive temporal shift up to 7.3ps for a 4.1ps-width pulse, which corresponds to the maximum delay-to-pulse-width ratio of 1.8.

©2006 Optical Society of America

## 1. Introduction

Optical buffers will be key components for the realization of routers for all-optical packet switching networks. Recently, optical flip-flops, optical fiber delay lines (FDLs), and slow light schemes have been extensively studied in hopes of realizing a practical optical buffer. The deterioration of the signal transmission quality and system integration challenges are two of the drawbacks that have been pointed out for these schemes.

Slow light in optical fiber is based on the change of the group velocity of the light via stimulated Brillouin scattering [1] and stimulated Raman scattering [2], or on the change of the wavelength via parametric process [3]. These conventional slow lights also have some inherent limitations, i.e., the narrow bandwidth, the pulse distortion, and the small amount of temporal shift. A hybrid system consisting of a wavelength conversion and a light propagation in a dispersive fiber [4] may be able to overcome such limitations, while the system becomes complicated. By the way, it is well known that optical solitons at different frequencies can pass through each other without variations of their waveforms, frequencies, and energies except their temporal positions and phases in loss-less optical fiber [5, 6, 7, 8]. This peculiar feature of optical soliton can be applied to distortion-less tunable delay line. By varying wavelength of control soliton pulses, both slow and fast light can be easily generated.

In this paper, we propose a novel principle of a tunable delay line (TDL) using soliton collision in an optical fiber and analyze the amount of the temporal shift using the numerical simulations and the inverse scattering transformation (IST) [8, 9]. We also conduct proof-of-principle experiments to confirm the feasibility of the proposed TDL.

## 2. Theory

Figure 1 shows the schematic diagram of the proposed all-optical TDL. A signal soliton pulse with a central wavelength of *λ* and control soliton pulses with *λ*+Δ*λ* are coupled by an optical coupler (OC) and launched into an anomalous dispersion fiber (ADF). In the ADF, the control soliton pulses pass through the signal soliton pulse, then the signal pulse is temporally shifted due to cross phase modulation (XPM). The sign of the temporal shift, Δ*t*, depends on the sign of Δ*λ*. At the output of the ADF, the signal pulse is selectively filtered out by an optical band pass filter (OBPF) with a central wavelength of *λ*. Consequently, we obtain the distortion-less signal pulse with a time delay of Δ*t* and a central wavelength of *λ*.

We consider that an optical pulse with the width of *t _{s}* [ps] and the central wavelength of

*λ*[

*µ*m] is launched into an ADF with a dispersion parameter

*D*[ps/(nm ·km)], a nonlinear coefficient

*N*

_{2}/

*A*

_{eff}[×10

^{-9}/W] and a loss coefficient α [dB/km]. The pulse dynamics in the ADF is described by the nonlinear Schrödinger equation (NLSE) [5],

We here summarize the units of the quantities appeared in Eq. (1).

z [m] | : | propagation distance, |

t [s] | : | time moving at the group velocity, |

E(z, t) (|E|^{2}[W]) | : | complex envelope of electric field, |

${\beta}_{2}\left[\frac{{S}^{2}}{m}\right]=-\frac{{\left(\lambda \left[\mu m\right]\right)}^{2}D\left[\frac{\mathrm{ps}}{(\mathrm{nm}\xb7\mathrm{km})}\right]}{2\pi c\left[\frac{m}{s}\right]}\times {10}^{18}$ | : | group velocity dispersion, |

$\gamma \left[\frac{1}{(m\xb7W)}\right]=\frac{2\pi}{\lambda \left[\mu m\right]}\frac{{N}_{2}}{{A}_{\mathrm{eff}}}[\times \frac{{10}^{-9}}{W}]\times {10}^{3}$ | : | nonlinearity, |

$g\left[\frac{1}{m}\right]=\frac{\mathrm{ln}(\alpha \left[\frac{\mathrm{dB}}{\mathrm{km}}\right]\times {10}^{-2})}{20}$ | : | fiber loss, |

where *c* is the speed of light in vacuum. In order to normalize Eq. (1), we introduce the nondimensional quantities as,

and we obtain the perturbed NLSE,

where

For loss-less case (Γ=0), solving Eq. (3) by using the IST [9], we obtain the fundamental soliton solution as

Here, *η*, *κ*, *T*
_{0}, and *θ*
_{0} are constants which represent the soliton’s amplitude, frequency, initial central position, and initial phase, respectively. In Fig. 1, the signal pulse (*q _{s}*) and the control pulses (

*q*) launched into the ADF are set as

_{c}Here, Δ*η* and Δ*κ* represent the amplitude and frequency difference between the signal and control pulses, respectively. Δ*T*
_{0} is the initial temporal spacing between the signal pulse and the control pulse for *n*=1 and Δ*T _{i}* is the intervals of control pulses. Δ

*θ*represents the phase difference between signal pulse and the

_{n}*n*-th control pulse. Let us consider the following initial value problem of Eq. (3) with Γ=0,

Since we can neglect the interactions between control pulses for large Δ*T _{i}*, the normalized temporal shift Δ

*T*of the signal pulse induced by the XPM after completely passed through

*N*control pulses can be calculated as

by using the IST [8, 9]. Partially differentiating Eq. (8) by Δ*η* and setting the derivative to be zero, we obtain the optimum Δ*η* at which the maximum temporal shift is obtained for a fixed Δ*κ* as

The maximum temporal shift for a fixed Δ*κ* is then given by

Figure 2 shows the temporal shifts as functions of the frequency difference Δ*κ* for *N*=1 and Δ*T*
_{0}=10. Solid and dotted lines show the theoretical prediction of Eq. (8) with Δ*η*=0 and Eq. (10), respectively. Open and closed circles show the temporal shifts obtained by numerically solving Eq. (5) with Γ=0 for Δ*η*=0 and Δ*η*
_{opt}, respectively. The temporal shifts are observed at *Z*=2Δ*T*
_{0}/Δ*κ*. The control pulse initially located at *T*=-Δ*T*
_{0} completely passes through the signal pulse located around *T*=0 at *Z*=2Δ*T*
_{0}/Δ*κ* for Δ*T*
_{0}≫1 because Δ*κ* represents the speed of the control pulse. Note that the temporal shift increases with decreasing the frequency difference.

Figure 3 shows the temporal shift as a function of amplitude difference Δ*η* for *N*=1, Δ*T*
_{0}=10, and Δ*κ*=3.0. Solid line and open circles show the theoretical prediction of Eq. (8) and the results of numerical simulation observed at *Z*=2Δ*T*
_{0}/Δ*κ*, respectively. The maximum temporal shift of 0.607 is obtained at Δ*η*
_{opt}=2.16 which can be predicted by Eq. (9). We should note here that, however, in actual case the maximum temporal shift can not be demonstrated because an adequate wavelength difference is required to remove control pulses by an OBPF at the output of an ADF. The solid line in Fig. 4 shows the maximum temporal shift in an actual system when the limitation of Δ*κ*≥(*δκ _{s}*+

*δκ*), i.e. Δ

_{c}*κ*Δη≥1.12×(2+Δ

*η*) is imposed. Here,

*δκ*and

_{s}*δκ*are signal and control pulse’s spectral width, respectively. The dashed lines show the temporal shift given by Eq. (8) for various Δ

_{c}*κ*. We obtain the maximum practical temporal shift of 0.584 for Δ

*η*=0.0042.

Next, we change the number of control pulses to increase the temporal shift. Figure 5 shows temporally shifted waveforms obtained at *Z*=*N*×2Δ*T*
_{0}/κ by numerical simulation for Γ=0 and various number of control pulses of *N*. We obtain the temporal shifts of 0.607, 1.21, 1.82, 2.43 and 3.04 for *N*=1, 2, 3, 4 and 5, respectively. As we can see from these results, the temporal shift is almost proportional to the number of control pulses for loss-less case.

## 3. Experimental demonstration

#### 3.1. Slow light

Figure 6 shows the experimental setup for demonstrating the slow light generation. The wavelength of the signal pulse is longer than that of the control pulse. The optical pulse sequence generated by a fiber ring laser (FRL) with the repetition rate of 10GHz and the center wavelength of 1560nm is amplified by an erbium doped fiber amplifier (EDFA)1. An OBPF1 is inserted to mitigate the amplifier noise. The pulse sequence is split by a 3dB optical coupler (OC)1. While the signal pulses are generated at the lower branch (A), the control pulse is generated at the upper branch (B). At the branch (A), the center wavelength of the signal pulse is the same as that of the pulse source, i.e., 1560nm. An optical time division multiplexer (OTDM)1 is used to generate a pair of signal pulses with an interval of 25ps and an EDFA3 is used to adjust the optical power to satisfy the condition for forming soliton pulses. At the branch (B), by passing through a normal dispersion flattened fiber (DFF) with the dispersion of -1.93ps/(nm·km), nonlinearity of 3.91/(W·km) and length of 1.5km, the spectrum of the control pulse is broadened due to the self-phase modulation. The broadened spectrum is filtered out by an OBPF2 with a bandwidth of 1nm at a desired wavelength. The time position of the control pulse is set to the center of a 25ps separated pair of the signal pulses by adjusting a variable delay line (VDL). An EDFA2 is also used to adjust the optical power to satisfy the condition for forming a soliton pulse. The signal pulses with the width of 6.1ps and control pulse of 4.1ps are coupled by a 3dB-OC2 and launched into a non-zero dispersion shifted fiber (NZ-DSF) with the total length of 6km which is long enough for signal and control pulses to pass through each other. The parameters of the NZ-DSF are summarized in Table 1. A polarization controller (PC) is used to coincide the polarization states of signal and control pulses to maximize the efficiency of XPM. At the output of the NZ-DSF, the signal pulses are filtered out by an OBPF5 with a bandwidth of 0.5nm at 1560nm to remove the control pulse. The autocorrelation trace of a pair of signal pulses is observed by an autocorrelator (AC).

Figure 7 shows the observed autocorrelation traces without and with collision represented by dashed and solid lines, respectively. Figure 8 shows the autocorrelation traces enclosed by a square in Fig. 7 after the fitting. The temporal shift is 1.0ps for the wavelength difference Δ*λ* of 2.0nm. Furthermore, the pulse distortion due to the soliton collision is small, because the estimated pulse width is 5.6ps with the assumption of the hyperbolic secant pulse shape. Figure 9 shows the temporal shifts as a function of the wavelength differnce. Open circles, solid line, and dashed line show the results obtained by the experiments, the numerical simulation, and the theoretical prediction of Eq. (8), respectively. One can see that the pulse is delayed with decreasing the wavelength difference. Furthermore, the experimental results are in good agreement with the theoretical prediction and the numerical simulation.

Next, we increase the number of control pulses to observe larger amount of temporal shift. The experimental setup is shown in Fig. 10 and is the mostly same as one shown in Fig. 6. At the branch (B), each control pulse is divided by OTDM2 into the four-bit pulse train. The parameters of NZ-DSF1 and 2 are summarized in Table 2. The waveform of the signal pulse is detected by a photo detector (PD) with a bandwidth of 50GHz and observed by a sampling oscilloscope (OSC). The pulse widths of signal and control pulse are 4.1ps and 3.1ps, respectively.

Figures 11 show the experimentally observed waveforms without control pulse and with control pulses for the wavelength difference of 1.8nm and 1.3nm. Note that the waveforms observed by the OSC are broadened due to the bandwidth limit of the PD used in the experiment. The temporal shift increases with decreasing the wavelength difference as the same as we have seen in Fig. 9. Figure 12 summarizes the temporal shift from the time position of the signal pulse without collision as a function of the wavelength difference. Open circles and solid line are the results obtained by the experiment and the numerical simulation, respectively. The maximum temporal shift in the experiment is 7.3ps at the wavelength difference of 1.2nm, which corresponds to a delay-to-pulse-width ratio of 1.8. Furthermore, the experimental results are in good agreement with the result obtained by the numerical simulation. In Fig. 13, the experimentally observed pulse width and time-bandwidth-product (TBP) at the output are plotted with open circles and triangles, respectively. Here, the observed pulses are assumed to have a hyperbolic secant shape. The pulse width is broadened to around 13ps from the original pulse width of 4.1ps because the signal pulse is filtered out by the OBPF5 with the bandwidth of 5nm. The corresponding pulse width of 13.7ps is obtained by the numerical calculation with considering the fiber loss and the OBPF. The TBP ranges from 0.35 to 0.45, which shows that the pulse distortion is negligible. The deviation of the TBP from 0.315 corresponding to that of a chirp free hyperbolic-secant shape pulse may be due to the accumulated frequency chirp during propagation through the NZ-DSFs. To improve the TBP near 0.315, it may be required that the bandwidth of an OBPF5 is carefully selected and the frequency chirp is appropriately compensated for.

#### 3.2. Fast light

Figure 14 shows the experimental setup for demonstrating the fast light generation. The setup is the mostly same as one shown in Fig. 6 and the parameters of the NZ-DSF are summarized in Table 1. The wavelength of the signal pulses is shorter than that of the control pulse. While the signal pulses are generated at the upper branch (A), the control pulse is generated at the lower branch (B1). At the branch (B1), an optical attenuator (ATT) is used to adjust the optical power to satisfy the condition for forming a soliton pulse and the center wavelength of the control pulse is fixed at 1560nm. The pulse widths of the signal and control pulse are 5.0ps and 3.1ps, respectively. To change the control’s peak power associated with the pulse width, the branch (B1) is replaced by (B2) shown in Figure 15. At the branch (B2), the pulse width of control pulse is broadened to 3.9ps by the OBPF5 with the bandwidth of 1nm.

Figure 16 summarizes the temporal shift observed in cases 1 and 2 when the wavelength of signal pulses is tuned from 1558nm to 1559nm. Circles and lines show the experimental results and the theoretical prediction of Eq. (8), respectively. The experimental results are in good agreement with the theoretical prediction.

Next, we increase the number of control pulses to observe the larger amount of temporal shift. The experimental setup is shown in Fig. 17 and is the mostly same as one shown in Fig. 10. At the branch (B), each control pulse is divided by OTDM into the four-bit pulse train. The parameters of NZ-DSF1 and 2 are summarized in Table 2. The pulse widths of the signal and control pulses are 4.3ps and 5.2ps, respectively.

Figures 18 show the experimentally observed waveforms without control pulse and with control pulses for the wavelength difference of 1.8nm and 1.3nm. The waveforms observed by the OSC are broadened due to the bandwidth limit of the PD used in the experiment as discussed before. The temporal shift increases with decreasing the wavelength difference as the same as Figs. 11. Figure 19 summarizes the temporal shift from the time position of the signal pulse without collision as a function of the wavelength difference. Open circles and solid line are results obtained by the experiments and the numerical simulation, respectively. The maximum temporal shift in the experiment is 5.2ps at the wavelength difference of 1.3nm, which corresponds to a delay-to-pulse-width ratio of 1.2. Furthermore, the experimental results are in good agreement with the results obtained by the numerical simulation.

In Fig. 20, the experimentally observed pulse width and TBP at the output are plotted with open circles and triangles, respectively. Here, the observed pulse is assumed to have a hyperbolic-secant shape. The pulse width is broadened to around 16ps from the original pulse width of 4.3ps because signal pulses are filtered out by the OBPF4 with the bandwidth of 0.5nm. The corresponding pulse width of 14ps is obtained by the numerical calculation with considering the fiber loss and the OBPF. The agreement between them is pretty good. The TBP ranges from 0.50 to 0.66. The deviation of the TBP from 0.315 corresponding to that of a chirp free hyperbolic-secant shape pulse may be due to the accumulated frequency chirp during propagation through the NZ-DSFs. To improve the TBP, it may be required that the bandwidth of an OBPF4 is carefully selected and the frequency chirp is appropriately compensated for.

## 4. Conclusion

We have proposed and experimentally demonstrated a novel all-optical TDL based on distortion-less slow and fast light using soliton collision in optical fiber. Since soliton’s pe-culiar feature guarantees the distortion-less property, the TDL would be useful and practical. We have achieved a continuous temporal shift up to 7.3ps for a 4.1ps-width pulse, which corresponds to the maximum delay-to-pulse-width ratio of 1.8 for slow light, and a continuous shift up to 5.2ps for a 4.3ps-width pulse, which corresponds to the maximum delay-to-pulsewidth ratio of 1.2 for fast light.With using more highly nonlinear fiber as a recently developing nonlinearity-enhanced photonic crystal fiber and adopting time gating scheme instead of OBPF for removing control pulses, we may improve the performance of the TDL.

## References and links

**1. **K. Y. Song, M. G. Herraez, and L. Thevenaz, “Observation of pulse delaying and advancement in optical fibers using stimulated Brillouin scattering,” Opt. Express **13**, 82–88 (2005). [CrossRef] [PubMed]

**2. **J. E. Sharping, Y. Okawachi, and A. L. Gaeta, “Wide bandwidth slow light using a Raman fiber amplifier,” Opt. Express **13**, 6092–6098 (2005). [CrossRef] [PubMed]

**3. **D. Dahan and G. Eisenstein, “Tunable all optical delay via slow and fast light propagation in a Raman assisted fiber optical parametric amplifier : a route to all optical buffering,” Opt. Express **13**, 6234–6249 (2005). [CrossRef] [PubMed]

**4. **J. E. Sharping, Y. Okawachi, J. van Howe, C. Xu, Y. Wang, A. E. Willner, and A. L. Gaeta, “All-optical, wavelength and bandwidth preserving, pulse delay based on parametric wavelength conversion and dispersion,” Opt. Express **13**, 7872–7877 (2005). [CrossRef] [PubMed]

**5. **A. Hasegawa and Y. Kodama, *Solitons in optical communications*, Oxford University Press, Oxford (1995).

**6. **L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. **9**, 362–367 (1991). [CrossRef]

**7. **S. R. Friberg, “Demonstration of colliding-soliton all-optical switching,” Appl. Phys. Lett. **63**, 429–431 (1993). [CrossRef]

**8. **T. Okamawari, A. Hasegawa, and Y. Kodama, “Analyses of soliton interactions by means of a perturbed inverse-scattering transform,” Phys. Rev. E **51**, 3203–3220 (1995).

**9. **V. E. Zakharov and A. E. Shabat, “Exact theory of two-dimensional self focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP **34**,. 62–69 (1972).

**10. **Y. Kodama and A. Hasegawa, “Effect of initial overlap on the propagation of optical solitons at different wavelengths,” Opt. Lett. **16**, 208–210 (1991). [CrossRef] [PubMed]