## Abstract

Chirped pulse scheme is shown to be highly effective to attain large tunable time shifts via slow and fast light for an ultra-short pulse through a semiconductor optical amplifier (SOA). We show for the first time that advance can be turned into delay by simply reversing the sign of the chirp. A large continuously tunable advance-bandwidth product (ABP) of 4.7 and delay-bandwidth product (DBP) of 4.0 are achieved for a negatively and positively chirped pulse in the same device, respectively. We show that the tunable time shift is a direct result of self-phase modulation (SPM). Theoretical simulation agrees well with experimental results. Further, our simulation results show that by proper optimization of the SOA and chirper design, a large continuously tunable DBP of 55 can be achieved.

©2009 Optical Society of America

## Introduction

Control of velocity of light, usually referred to as slow or fast light, has attracted significant attention recently due to exciting applications in nonlinear science, beam steering for radars, and all-optical networks [1, 2]. The term slow light refers to a significant decrease in the group velocity of light, while fast light refers to an increase. Tunable delay of light pulses is desirable in many applications, including contention resolution and clock synchronization for photonic switching. In addition, the phase shift of an RF modulated optical wave can be used to steer the direction of an RF beam in phased array antennas and to realize fast tunable RF filters [3]. For these applications, it is important to achieve group velocity change at large bandwidths (> 10 GHz). An additional important metric is the delay/advance bandwidth product (A/DBP), which is approximately equivalent to time shift measured in terms of number of pulse-widths. Various techniques and media have been proposed to attain a large tunable time delay [4–10]. The use of a semiconductor optical amplifier (SOA) offers significant advantages because of high bandwidth and facilitation of electrical tunability. Recently, using ultra-fast nonlinear processes in SOAs, we demonstrated an ABP of 2.5 at THz bandwidth in a 1 mm compact device [11]. Furthermore, by imposing a chirp on an ultra-short pulse, we reported a novel chirped pulse scheme to extend the ABP to 3.5 [12].

In this paper, we show experimental results of a large tunable advance of ultra-short pulses using a novel chirped-pulse amplifier, and demonstrate that the advance can be turned into a delay by reversing the sign of the chirp. We achieved a record tunable ABP and DBP of 4.7 and 4.0 via a simple reversal of the chirp while keeping the entire optical system under the same conditions. We show that this interesting but not intuitively obvious result is due to the spectral shift from self-phase modulation (SPM) in an SOA. We developed a theoretical model and simulated the propagation of an ultra-short pulse through an SOA using the formulism developed in [13]. Simulation results for a 370 fs pulse agree very well with our experimental results. By proper choice of SOA parameters and optimization of chirper design, we predict a large DBP of 55.

## 1. Physical principle

An ultra-short pulse propagating through an SOA biased in a gain region experiences an advance due to ultra-fast nonlinear processes such as spectral-hole burning and carrier-heating [11]. Since the efficiency of these nonlinear processes depends on SOA gain, the advance can be controlled electrically by changing the SOA bias. This picture is further complicated when one considers the linewidth enhancement factor in SOAs. The linewidth enhancement factor, also known as the α-factor, is a phenomenological parameter quantifying the change in refractive index with respect to carrier density due to the asymmetry of the gain spectrum. The gain spectrum is determined by the density of states of the semiconductor medium; thus the α-factor is dependent on material, wavelength and design, with typical values ranging from 3 to 10. As the pulse propagates in an SOA, it causes a reduction in carrier density proportional to the pulse intensity. The large α-factor causes a change in carrier density which leads to a large phase change for the pulse, an effect commonly known as self-phase modulation (SPM). This dynamic phase change during the propagation results in a change in carrier frequency or, equivalently, a wavelength shift for the pulse. Since the pulse intensity is proportional to the gain in the device, the wavelength shift due to SPM increases with increasing SOA gain. By adding a dispersive element (which introduces a wavelength dependent group delay for the pulse), we can obtain a large time shift for the pulse with increasing SOA bias. Since a transform limited pulse propagating through this element obtains a chirp, we refer to the dispersive element by the term “chirper”. As explained below, a unique advantage of this scheme is the ability to switch from advance to delay by simply reversing the sign of the chirp while keeping the optical system unchanged.

In our experiments we use grating-based chirpers as dispersive elements to demonstrate the concept. However, more compact elements such as dispersive fibers or Bragg gratings can also be used. As explained earlier, tunable advance in this scheme is proportional to the SPM-induced wavelength shift or chirp. It is well known that a positive α-factor causes a negative chirp (red-shift of wavelength) for a pulse propagating through the SOA biased in the gain regime [13]. Hence, we can obtain an advance by adding a chirper with a negative dispersion coefficient or obtain a delay by adding a chirper with positive dispersion coefficient. Both advance and delay through an SOA-chirper system can be increased by increasing the magnitude of the chirper dispersion coefficient β_{2}. However, increasing β_{2} beyond a certain value causes significant pulse broadening because different frequency components of the pulse experience different group delays through the chirper. In this paper we demonstrate both theoretically and experimentally that the addition of an input chirper before the SOA with a dispersion coefficient opposite to that of the chirper after the SOA (output chirper), we can achieve a large tunable advance and delay while minimizing the pulse broadening at the output. From here on, we refer to the input chirper as the “chirper” and the output chirper as the “compensator” to avoid any confusion.

The schematic of the chirped-pulse scheme is shown in Fig. 1. There are three basic stages: the chirper, the SOA, and the compensator. A sub-picosecond pulse first passes through a grating-based chirper (stage 1) to introduce a group delay that decreases (negative chirp) or increases (positive chirp) linearly with frequency. In the time domain this difference in propagation time experienced by different frequency components stretches the pulse duration and adds an approximately linear shift in optical frequency with time. Despite the change in pulse duration, the power spectrum and bandwidth remain unchanged and the pulse is no longer transform limited. The magnitude and sign of the chirp can be tuned by adjusting the grating separation.

The chirped pulse then enters the SOA (stage 2) and experiences an advance with increasing current due to nonlinear gain dynamics. In addition to this advance, the pulse also acquires additional chirp due to self-phase modulation in the device. As depicted in the figure, SPM induced chirp is temporally nonlinear due to its dependence on pulse intensity. The contribution from SPM is small relative to the linear chirp from the input chirper. This is due to the energy-conserving reduction in peak pulse intensity as the input chirper stretches the pulse duration from 370 fs to 16 ps. This decrease in intensity reduces SPM induced chirp to the point where it may seem to be a second-order correction to the linear chirp. However, the compensator after the SOA (stage 3) removes the linear chirp while the chirp due to the SPM remains.

The compensator is identical to the chirper but with a chirp of opposite sign. Because the compensator removes chirp via a frequency-dependent delay, the residual SPM induced chirp translates into additional time shift as the pulse passes through the compensator. If the chirper and compensator are set for large chirp values, the contribution to time shift due to SPM is much larger than that due to non-linear processes such as gain saturation, spectral hole burning and carrier heating. If the input pulse is negatively chirped, then the compensator is set to induce a positive chirp and thus the SPM induced negative chirp results in an advance as the SOA current is increased. If the input pulse is positively chirped, then the situation is reversed and the negative SPM induced chirp results in a delay with increasing current.

## 2. Experimental results

The experimental setup is shown in Fig. 2. A mode-locked fiber laser operating at 1550 nm produces 370 fs pulses with a 20 MHz repetition rate. The output of the laser is split into two branches. One branch acts as a reference for cross correlation measurements. The other branch goes through a linear chirper which stretches the pulse to 16 ps. Next a high NA lens is used to couple the pulse to the SOA with approximately 3 dB of coupling loss. The SOA used in the experiments is a quantum well device operating at room temperature. An electrical bias current is used to control the carrier density within the device. The pulse propagation time through the SOA is tuned by adjusting this current. From the SOA the pulse then proceeds through the compensator, which removes the original chirp. An EDFA is used afterwards to compensate for the coupling losses in the compensator. The output of the EDFA is combined with the reference beam to measure temporal shifts via optical cross-correlation measurements.

Figure 3(a) shows the cross-correlation traces for a negatively chirped pulse with increasing SOA bias. The pulse energy at the input of the SOA is close to 6 pJ corresponding to a peak power of 0.4 W. The pulse advance is measured as the SOA linear gain is varied from 0 dB to 30 dB. It is important to note that cross correlation traces appear broader than the actual pulse due to the finite width of the reference. To obtain the actual pulse width, the cross-correlation width is divided by a correlation factor (1.54 for sech-pulses). As can be seen from the time traces, advance increases with increasing SOA current. A maximum advance of 1.7 ps is observed as the SOA current is increased from 50 mA (near transparency) to 300 mA (maximum gain). This corresponds to a large ABP of 4.7.

Figure 3(b) shows the ABP and broadening as a function of linear gain in the SOA. Pulse advance increases almost linearly with increasing gain. The maximum broadening across the entire tuning range is less than 75%. Advance with increasing gain can be understood by examining the self-phase modulation induced by the SOA. As the pulse propagates in an SOA, it causes a reduction in carrier density via stimulated emission. A reduction in carrier density causes a negative chirp (decrease in center frequency) for the pulse. For the case of a negatively chirped pulse entering the device (low frequency components enter the SOA last), the compensator after the SOA is adjusted to advance the low frequency components in order to compress the pulse back to its original width. Hence a negative chirp induced by the SOA translates into a pulse advance. Since SOA-induced chirp is proportional to gain, pulse advance increases with increasing SOA bias. Time traces show a small pedestal for the pulse. This is because the linear chirper employed in this experiment cannot exactly compensate for the nonlinear chirp induced by the SOA. Tailored chirpers can be employed in the future to obtain a better pulse shape.

Figure 4(a) shows the cross-correlation time traces for a positively chirped pulse. A total delay of 1.5 ps corresponding to a DBP of 4.0 is observed as the SOA current is increased from 45 mA (transparency) to 300 mA (maximum gain). Delay is observed with increasing current because the sign of the compensator is in opposite direction compared to the earlier case. Figure 4(b) shows the delay linearly increasing with SOA gain and maximum broadening across the entire tuning range is less than 80%. The results presented here show that the maximum pulse broadening is less than 100% for both the cases as the SOA current is varied. This indicates that the residual chirp of the output pulse is indeed very small. This can be understood by noting that the residual chirp at the output is due to the self-phase modulation in the SOA (as depicted in Fig. 1). Since the SPM induced chirp is proportional to the peak intensity, the stretching of the input pulse to a large value (from 370 fs to 16 ps) before it enters the SOA results in a small SPM induced chirp.

We propose a novel scheme to combine the advance and delay results. This scheme consists of two crossbar switches as shown in Fig. 5. In the bar configuration, the pulse first passes through a negative chirper, then the SOA, and finally a positive chirper. In this configuration, the pulse advances as the current is increased from transparency to maximum current. In the cross configuration, the pulse first passes through a positive chirper, the SOA, and finally a negative chirper. This configuration yields a delay with increasing current. Since the pulse doesn’t experience a time shift at transparency, a continuous tunable DBP of 8.7 can be achieved by switching between bar and cross configurations. The novelty of this scheme is the ability to switch from advance to delay by reversing the sign of input chirp while keeping the remainder of the system in the same condition.

## 3. Theory and discussion

In this section, we describe the methodology to simulate the pulse propagation through various optical components. Pulse propagation through the grating based chirpers can be modeled by adding a quadratic phase to each of the frequency components [14]. The chirpers used in the experiments induce a linear chirp on the pulse, i.e. the center frequency changes linearly from the leading edge to the trailing edge. In this study, the pulses are stretched to 16 ps before entering the SOA. Hence, we can neglect the contributions due to ultra-fast nonlinear processes including spectral-hole burning and carrier heating which become prominent for shorter pulse-widths (< 5 ps). With this simplification, propagation of the pulse can be modeled using an SPM-based formalism. The equations governing the pulse amplitude and phase of the pulse are given by [13]

where *P* is the pulse power, *g* is the gain, *α _{int}* is the internal loss, φ is the phase and a is the linewidth enhancement factor which characterizes the index/phase change corresponding to a change in carrier density. As mentioned earlier, a pulse propagating in an SOA experiences a phase change due to self-phase modulation. The large linewidth enhancement factor in semiconductors contributes to large SPM. Using Eqs. (1) and (2) with the gain dynamics Eq. (3), the pulse amplitude and phase at the SOA output can be solved

where *τ* is the retarded time, *g _{o}* is the small signal gain,

*τ*is the carrier life-time and

_{c}*E*is saturation energy. We use finite difference scheme to solve the equations numerically. Simulation results for a negatively chirped pulse after propagation through the chirpers and the SOA are shown in Fig. 6(a). To elucidate the importance of SPM, we simulate pulse propagation when SPM is absent by setting

_{sat}*α*= 0. From the time traces we can clearly see that the pulse doesn’t experience any advance as the gain of the SOA is increased. However for

*α*= 3, SPM leads to large advance as the linear gain is increased from transparency (0 dB) to maximum gain (30 dB). This clearly shows that advance is entirely due to self-phase modulation when the pulse is chirped to large values before entering the SOA. We also observe the appearance of a pedestal at high gain values. As mentioned earlier, SOA-induced chirp due to SPM is nonlinear because it depends on pulse amplitude. Linear chirpers employed in this study cannot exactly compensate this nonlinear chirp. However, we can reduce the pedestal by filtering out the spectral components that contribute to this pedestal. For a negatively chirped pulse, filtering out the red frequency components helps improve the pulse shape significantly as can be seen from Fig. 6(b). In experiments, we implement this spectral filtering technique to obtain a better pulse shape. It should be noted that this selective frequency filtering does not significantly broaden the pulse because SPM also leads to spectral broadening [13]. Frequency filtering in this context can be understood as removing the unnecessary frequency components that contribute to the pulse pedestal.

Next, we simulate the propagation of a 370 fs pulse stretched to 16 ps in our device. The SOA used in this study is a quantum well device operating at 1550 nm with a maximum small signal gain of 30 dB. The length of the device is 1 mm and the cross-section is 1.3^{*}0.11 µm^{2}. Using a standard pump-probe technique [11], we measured a carrier lifetime of 40 ps in our device at an SOA bias of 300 mA. The internal loss is estimated to be close to 9 dB. For the numerical simulations, we assumed a confinement factor (Γ) of 0.1, linewidth enhancement factor (α) of 4 and a differential gain (dg/dN) of 4^{*}10^{-16} cm^{2}. These values of material parameters are typical for semiconductor based devices [15].

Figure 7 shows the simulated and experimental results for a chirped pulse. Advance is achieved for a negatively chirped pulse and a delay for a positively chirped pulse. We obtain excellent match with our experimental results for both cases.

Next, we address the potential to extend the time shifts using this scheme by employing SOAs with a higher linewidth enhancement factor (α). As can be seen from Eq. 2, SPM is greatly enhanced for a device with high α factor. In this simulation, we increased *α* from 4 to 7 while all other parameters remain fixed. Large linewidth enhancement factor can be realized by operating closer to the band edge of a semiconductor [16]. The results of simulation are shown in Fig. 8. A large ABP of 10.3 is observed for a negatively chirped pulse as the linear gain (*g _{o}L*) is increased from 0.5 to 9. For a positively chirped pulse, a large DBP of 10.2 is observed. By using crossbar switches as mentioned earlier, a combined DBP of 20.5 can be obtained for a 370 fs pulse. Pulses for both cases appear broader compared to earlier results because the output chirper cannot exactly compensate for the SOA chirp at all gain values. Tailored nonlinear chirpers can be employed to obtain better pulse shape at the output as we will show later in this paper.

Time shifts using this scheme can also be enhanced by the design of SOAs with low saturation energy *E _{sat}*. Self-phase modulation is greatly enhanced in devices with low saturation energy. Saturation energy can be decreased by increasing the confinement factor or by increasing the differential gain. Large confinement factors can be achieved by proper design of SOAs [17]. Fig. 9(a) shows the results of the simulation for a negatively chirped pulse as the confinement factor is increased from 0.1 to 0.7 while other material parameters are kept constant. For this case, an ABP of 10.3 is observed. Differential gain in SOAs can be increased by optimizing the strain in quantum wells [18]. Fig. 9(b) shows the simulation results as the differential gain is increased from 4.10

^{-16}cm

^{2}to 2.10

^{-15}cm

^{2}. In this case, we obtain a large ABP of 10.6. By using positively chirped pulses, a large DBP greater than 10 can be achieved for both cases.

Finally, we increased the linewidth enhancement factor and the saturation energy to achieve large advance and delay. Figure 10 shows the results when *α* is increased to 10 and the confinement factor to 0.8. The SOA linear gain is increased from transparency to a maximum gain of 26 dB. For a negatively chirped pulse a large ABP of 26.5 is observed while for a positively chirped pulse a large DBP of 28.8.

As mentioned earlier, SOA induced chirp increases with increasing gain. Hence, a linear output chirper cannot exactly compensate for the SOA induced chirp for all gain values which results in pulse broadening. Better pulse shape at the output can be obtained by using a quadratic chirp instead of linear chirp. Using quadratic chirp, we obtained large advance and delay with maximum broadening less than 160% for both the cases. Further we assumed a high internal loss in the system (17 dB) to reduce the pulse pedestal at the output. By using crossbar switches, advance and delay results can be combined to achieve a continuously DBP of 55. Further theoretical investigation is under progress to increase the DBP by optimization of various parameters and by cascading multiple SOAs.

## 4. Comparison of chirped pulse scheme with unchirped case

In this section, we provide a direct comparison between the unchirped case (without input chirper but using a compensator after the SOA) and the chirped pulse scheme (with input chirper and compensator) to emphasize the importance of input chirp on the pulse. An input pulse width of 200 fs is chosen for the simulations. In the first scenario, a transform limited pulse is enters the SOA and experiences an advance due to ultra-fast non-linear processes described in [11]. These processes also induce a nonlinear chirp on the pulse. A compensator after the SOA is used to leverage this nonlinear chirp in order to achieve large advance. Using the density matrix formulism, the pulse propagation through the SOA-compensator system is simulated as the SOA gain is varied [19]. The results of the simulation are shown in Fig. 11. In this case, we observe an ABP of 4.7 as the SOA current is increased to 300 mA corresponding to a linear gain of 30 dB. In the second scenario, the same input pulse (200 fs) is chirped out to 20 ps before entering the SOA. A compensator is added after the SOA to compress the pulse back to its original width. In this case, the SPM induced chirp results in a large ABP of 9.6 compared to the unchirped case (ABP of 4.7). Further, pulse broadening at the output in this case is only 40% compared to a broadening of 100% for unchirped case. This clearly shows that by chirping out the pulse to large values before it enters the SOA, we can obtain a large advance while reducing the pulse broadening at the output.

## 5. Summary

In conclusion, we have shown theoretically and experimentally that self-phase modulation in semiconductor optical amplifiers can be used to achieve large tunable advance and delay at THz bandwidth. We experimentally demonstrated an ABP of 4.7 for a negatively chirped pulse and a DBP of 4.0 for a positively chirped pulse using the same device. Using a novel scheme based on crossbar switches, we showed for the first time that advance can be changed to delay simply by reversing the sign of chirp while the remainder of the system is kept under the same condition. Using this scheme, advance and delay results can be combined to achieve a continuously tunable DBP of 8.7. Electrical tuning of advance and delay makes this scheme extremely suitable for various slow light applications. Our simulation results show that by proper optimization of the SOA and chirper designs, DBP can be extended to 55. Further, we clearly demonstrated that by using the chirped pulse scheme, self-phase modulation in SOAs can be effectively leveraged to achieve large advance while reducing the pulse broadening at the output.

## Acknowledgments

The authors would like to thank the support of DARPA grants F30602-02-2-0096 and Airforce contract FA 9550-04-1-0196.

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