A numerical study on Mamyshev signal regeneration realized on silicon photonic wires is reported. Unlike fiber-optics Mamyshev regenerators employing cross-phase modulation, silicon photonic wires have to include two-photon absorption and the two-photon-absorption-induced free-carrier effect. By well adjusting time delay between the co-propagating signal and clock pulses, both cross-phase modulation and free-carrier dispersion could induce nonlinear wavelength shift, which is essential for signal recovery in the Mamyshev regeneration scheme. A simulation result shows the quality factor of signal eye diagram improved by more than 4 dB for Return-to-Zero signals with pulse width 10 ps, peak power 6.5 W, and operation speed 10 Gbit/s through a 1-cm silicon photonic wire.
©2010 Optical Society of America
In optical communication networks, signal quality is easily degraded by chromatic dispersion, fiber nonlinearities, and noise from optical amplifiers or cross connects before the signals reach the destination. Bit logic ‘0’ and ‘1’ of transmitted binary signals become difficult to be identified by the receivers, possibly resulting in transmission errors. To reduce the bit-error rate, distorted signals need to be regenerated to recover the original signal quality. Generally speaking, the ways of signal regeneration can be classified into three categories, which are reamplification, reshaping and retiming (3R) . Several approaches have been reported for optical 3R regeneration such as using semiconductor optical amplifiers (SOA) through cross-gain- or cross-phase modulation  or electro-absorption modulators (EAM) via cross-absorption modulation .
Nonlinear fiber optics can be exploited for signal regeneration. In 1998, Mamyshev  utilized self-phase modulation (SPM) in highly nonlinear dispersion shifted fibers (HNL-DSF) to offset the carrier wavelength and used optical filters to recover signals. Although signal reamplification and reshaping (2R) were achieved, timing jitter was still a problem. In 2005, Suzuki et al. proposed a modified Mamyshev regenerator , using cross-phase modulation (XPM) in HNL-DSF to effectively eliminate timing jitter through a synchronized clock pulse train. Other nonlinear optics schemes include optical parametric amplification (OPA) [6,7] or four-wave mixing (FWM)  using a clock or continuous-wave pump in HNL fibers. Although high-speed and high-quality signal regeneration is accomplished, those approaches usually require very long HNL-DSF. The cost is high, which may impede practical applications.
Nonlinear integrated optics is a candidate for a miniaturized signal regenerator since nonlinear waveguides as well as passive photonic devices such as optical filters can be monolithically integrated on the same chip. Ta’eed et al. demonstrated all-optical 2R regenerators based on Kerr optical nonlinearities in chalcogenide glass waveguides with integrated Bragg grating filters . Alternatives are silicon photonic wires. The optical Kerr effect in silicon is 100 times larger than silica and the fabrications of silicon waveguides are consistent with state-of-the-art complementary metal-oxide-semiconductor (CMOS) technologies. Due to high index of refraction (3.45) in silicon, the core dimension of silicon photonic wires is typically sub-micrometer. Thus, the power density can be magnified by several orders of magnitude if tapered couplers are incorporated for effectively coupling light from a laser beam or a fiber to the silicon wire. Optical 2R or 3R regeneration by silicon waveguides was successfully demonstrated through SPM  or FWM . However, other accompanying nonlinear processes including two-photon absorption (TPA) and TPA-induced free carriers have to be considered, and they could impact device performance. In this paper, we investigate the TPA-induced free-carrier effects on a Mamyshev regenerator implemented on silicon photonic wires. With a proper operation condition, the TPA-induced free carries could aid optical 3R regeneration.
2. Numerical model
Figure 1 is a conceptual diagram of illustrating the setup of Mamyshev 3R regeneration based on a silicon photonic wire. An input Return-to-Zero (RZ) signal with carrier wavelength λ1 is prior magnified by an erbium-doped fiber amplifier (EDFA) and then goes through an optical filter to eliminate amplified-spontaneous-emission (ASE) noise. Meanwhile, a pristine and synchronized clock pulse train with carrier wavelength λ2 is added on the input signals after passing through a tunable delay line. The time delay of clock pulses is adjusted so each clock pulse is exactly located on the trailing edge of the signal pulse. The combined signal and clock are launched into the silicon photonic wire simultaneously.
As the signal is at the state of bit logic ‘1’, blue-shift of clock wavelength occurs owing to XPM from the signal pulse. For silicon photonic wires, however, TPA and TPA-induced free carriers should be considered as well. These TPA-induced free carriers could contribute extra loss and chromatic dispersion, which further increases the wavelength shift of clock pulses if the time delay is properly tuned. In addition to free-carrier dispersion (FCD), both the signal and clock pulses are attenuated by TPA and free-carrier absorption (FCA). At the end of the silicon wire, another optical band-pass filter with central wavelength λ2 is installed to reject the signal and the blue-shifted clock pluses. Therefore, the original bit logic ‘1’ is converted to ‘0’. On the other hand, no wavelength shift of clock pulses occurs as the signal is at the state of bit logic ‘0’. Consequently, the clean clock pulses can be filtered out, resulting in bit logic ‘1’. The regenerated signal with inverted logic has a better quality since it doesn’t suffer from pulse distortion by other high-order nonlinear or dispersive effects.
The subscript s and c represent the signal and clock pulses respectively. A is the amplitude of electric field, β1s,c and β2s,c the first- and second-order dispersion coefficients, γss,sc,cs,cc the nonlinear coefficients defined by 2πn2/(λifij) where i, j = s or c, n2 is the nonlinear refractive index and fij are given by
In Eq. (3), F(x,y) represents the signal or clock optical mode distribution transverse to the propagation direction. Strictly speaking, fss, fcc, fsc and fcs differ from each other, but the difference is usually small . Therefore, these four parameters are approximately equal and expressed by Aeff, the effective area of silicon photonic wires. Moreover, α represents the linear loss, βT the TPA coefficient, αf the FCA coefficient and Δn the index variation caused by FCD. Δn and αf in Eq. (1) and (2) are strongly correlated with ΔNh and ΔNe, the free-carrier densities of excess holes and electrons induced by TPA and cross-TPA (X-TPA) in silicon. These time-dependent excess hole and electron densities can be described by the rate equation:16]:
From Eq. (1) to (5), a time-dependent free-carrier density along with a pulse can be analyzed numerically. Since signal pulses are the major pump in the operation scheme of the 3R regeneration, the excess free-carrier density is dominated by the signal power and is also determined by FCT in silicon wires. The FCT in silicon wires typically ranges from tens of picoseconds to nanoseconds, depending on the device dimension, structural profile and defect concentration. Figure 2(a) shows the simulated free-carrier density varying across a single pulse with different FCT. The pulse width is assumed to be 10 ps and the peak power is 6 W. Other parameters of silicon wires used through this paper are shown in Table 1 , according to the data presented in . One thing to note is that the group velocities of signal and clock pulses β1s, β1c are very close if the deviation of two central wavelengths is small. In this study, the walk-off length is 30 cm, which is much loner than the waveguide length which is only about 1 cm. For long FCT, the free-carrier density always increases within a pulse period, but it could rapidly decrease at the trailing edge of the pulse if the FCT is shorter than the pulse width. Moreover, free-carrier absorption would attenuate the signal power. Figure 2(b) shows the peak FCA coefficient of the 6W signal pulse as a function of FCT. Although the free-carrier density monotonically increases with FCT, the peak absorption coefficient saturates because of the short pulse duration.
3. Nonlinear processes in silicon photonic wires for Mamyshev signal regeneration
As the signal and clock pulses co-propagate along the silicon photonic wire, the spectrum of the clock pulse would be either red- or blue-shifted due to XPM and FCD. In the following analysis, 10-ps Gaussian pulses with a peak intensity of 3.5 GW/cm2 are used for the signal and 3-ps pulses with a peak intensity of 0.25 GW/cm2 (peak power of 0.5W) are used for the clock. The silicon photonic wire is assumed to be 1 cm long. Figure 3(a) displays three cases of the clock synchronizing with the signal in time domain. The clock pulse could be either localized at the (i) leading edge, (ii) central peak, or (iii) trailing edge of the signal pulse by tuning the time delay. In Fig. 3(b), during propagation, the signal pulse experiences asymmetrical spectral broadening due to SPM, TPA, FCA and FCD, which is insensitive of the location of clock pulses. Similar spectral broadening for a single pump pulse was already reported by experiments  and theoretical analysis . On the other hand, the spectral response for clock pulses is strongly dependent on the time delay. Figure 4 shows simulated clock pulses in time and frequency domains after propagation in silicon wires. In case (i), the spectrum of a clock pulse is slightly extended to long wavelength, while a spectral offset is manifest in case (iii). Additionally, in case (ii), the intensity of a clock pulse is attenuated most due to strong cross two-photon absorption (XTPA) at the central peak of a signal pulse.
The spectral response of a clock pulse can be explained by XPM and TPA-induced free-carrier density varying across a signal pulse, as shown in 5(a). At the leading edge (Zone I), since the free carriers begins to accumulate, the frequency chirp by FCD (blue shift) is against XPM (red shift). At the trailing edge, however, XPM induces a blue shift but FCD causes the clock spectrum either shifted to short wavelength (Zone II) or long wavelength (Zone III), depending on whether the free-carrier density increases or decreases. In Zone II, both XPM and FCD induce a blue shift, resulting in a large wavelength offset. The time-dependent free-carrier density is mainly determined by FCT. If the FCT is longer than the pulse width, the free carriers continue increasing even at the trailing edge. Thus, by well adjusting the time delay of clock pulses to be within Zone II, the maximum wavelength shift is achieved. Figure 5(b) shows the optimal time delay to get the maximal wavelength shift as a function of free-carrier lifetime. In our simulation, the optimal time delay is around 7.5 ps if the FCT is longer than the pulse width. The maximum wavelength shift versus free-carrier lifetime is also plotted in Fig. 5(c), which shows the wavelength shift actually decreasing first and then gradually increasing. It is explained by the fact that for short FCT, the free-carrier density decreases immediately at the trailing edge, resulting in FCD against XPM in frequency domain.
To realize a Mamyshev 3R regenerator, synchronizing the clock at the trailing edge of signal pulses with a positive time delay is desirable to get a large wavelength shift. Figure 6(a) displays the signal and clock pulses in time domain as the arrival signal logic is ‘1’ or ‘0’. After propagation through the silicon wire, the spectrum of a clock pulse is blue-shifted in the presence of a signal pulse, as illustrated in Fig. 6(b). A Gaussian bandpass filter is used to reject the blue-shifted clock pulses and only keep the pristine one. The filter bandwidth is designed so the passing-through clock pulses are reshaped to have the same pulse width as the signal. Figure 6(c) shows the clock transmittance versus the peak power of signal pulses. For long FCT, large contrast of the nonlinear clock transmittance is observed.
4. Q-factor of regenerated signals through silicon photonic wires
To examine the system performance of Mamyshev 3R regeneration on silicon photonic wires, a numerical code was developed to simulate the signal and clock pulses in a system depicted in Fig. 1. 500 random bits were used in this simulation. The pulse widths of the signal and clock are 10 ps and 3.5 ps, respectively. Before going into the silicon wire, the signal passes through an EDFA as well as an optical filter with a central wavelength at 1550 nm and a bandwidth of 1 nm. After amplification, white noise is introduced into the signal and the quality factor is assumed to be 10 dB with an extinction ratio of 20 dB. The peak power of signal pulses is boosted to be 6.5 W. The clock wavelength is 1555 nm with a peak power of 0.5 W. Before combing with the signal, the clock pulses go through a tunable delay line with a time delay of 3 ps. At the output of the silicon wire, another optical filter with a narrow bandwidth 0.35 nm and a central wavelength 1555 nm is connected to allow un-shifted clock pulses and reshape the pulse width to be equal to 10 ps. A logic-inverted pulse train is thus created on the clock. Figure 7(a) shows the eye diagrams of the input signal and output clock at two operation speeds, 4 Gbit/s and 20 Gbit/s respectively. The signal quality of output eye diagrams is generally improved. However, the output eye diagram at 4 Gbit/s is much open, compared with the one at 20 Gbit/s. It is owing to the long lifetime of free carriers. According to Fig. 6(c), although long FCT is advantageous to enhance wavelength shift, it may restrict the operation speed because the amount of free carriers yielded by the previous signal pulses could affect the next one. Therefore, the upper edge of the output eye diagram is thicker and the quality factor decreases at high data rate. In this simulation, the FCT of silicon photonic wires is assumed to be 30 ps, which is attainable by ion implantation . Figure 7(b) shows the quality (Q) factors of the input signal and output clock at different data rates.
Figure 8 shows the Q-factors of input and output eye diagrams as functions of optical signal-to-noise ratio (OSNR) for FCT equal to 1, 30, 60 and 90 ps. The transmitted data rate is fixed at 10 Gbit/s. The Q-factor is not improved in the case of 90-ps FCT, where the free-carrier lifetime is comparable to the data bit interval. This long FCT could result in inter-digit interference through FCA. On the other hand, the Q-factor is slightly improved if the FCT is 1 ps. This improved Q-factor (2 dB) is mainly contributed by the XPM from the signal. In the case of 10 Gbit/s RZ signal with pulse width of 10 ps, the optimal FCT is 30 ps, corresponding to a value slightly larger than the pulse width but shorter than the data bit interval (100 ps). Under such a condition, both the XPM and FCD introduce a strong wavelength shift required for the Mamyshev regeneration scheme. The Q-factor is improved by more than 4 dB. A general consideration of the free-carrier lifetime relative to the pulse width and the data bit interval on the quality factor is shown in Fig. 9 . The data bit interval and the free-carrier lifetime are normalized to the pulse width. In general, the Q-factor improvement decreases as the duty cycle increases, resulting from a strong inter-digit interference. The optimal free-carrier lifetime reduces with the duty cycle too. Despite this dependence on duty cycle, with free carriers, the quality factor of eye diagram is enhanced if the FCT is shorter than the data bit interval.
In conclusion, we gave a numerical study on the Mamyshev 3R regeneration implemented on silicon photonics wires. Due to TPA and TPA-induced free carriers, the wavelength shift of clock pulses in silicon wires is very different than that exhibited in HNL-DSF alone, which is predominantly determined by XPM. The TPA-induced free carriers not only incur propagation loss but induce extra wavelength shift that could enhance or suppress the XPM effect. If the free-carrier lifetime is longer than the signal pulse while shorter than the data bit interval, the wavelength offset of synchronized clock at signal bit logic ‘1’ is maximized as the clock pulses are placed at the trailing edge of the signal pulses via tuning the time delay. In our simulation, for a 1-cm long silicon wire, the Q-factor of signal can be improved by 4 dB in the case of signal pulse width 10 ps, FCT 30 ps and peak power 6.5 W, operating at speed of 10 Gbit/s.
The authors acknowledge the support of the National Science Council (NSC96-2628-E-007-143-MY2, NSC98-2622-E-007-002-CC1) and Nanodevice Laboratory (NDL97-C02M3C-046) in Taiwan for this analysis work.
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