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We investigate the use of all-optical regenerators to correct pulse distortions introduced by group delay ripple. Group delay ripple creates unwanted satellite pulses and intensity fluctuations. By placing an all-optical regenerator after a device that introduces group delay ripple, we show that the signal distortions can be effectively reduced. This has the benefit of opening the signal eye at the receiver. The performances of both self-phase modulation and four-wave mixing based regenerators in reducing ripple induced system penalties are examined. We find that the regenerator based on four-wave mixing achieves better suppression of group delay ripple distortions than the self-phase modulation based alternative. The eye closure penalty introduced by group delay ripple is reduced by the four-wave mixing based regenerator by 1dB.
©2004 Optical Society of America
1. Introduction
All-optical regenerators will be an economical means to combat signal degradation as transmission systems migrate towards higher bit-rates [1]. One cause of degradation is group delay ripple. Group delay ripple originates from imperfections in various photonic components, for example, grating-based dispersion compensators. Group delay ripple accumulates along the fibre-optic link and cannot be easily compensated with a linear device such as a dispersion compensator. Fig. 1 illustrates the propagation of a dispersed signal, from point A, through an in-line dispersion compensator exhibiting group delay ripple, to point B. The dispersed pulses are imperfectly recompressed after the dispersion compensator because of group delay ripple, which creates satellite pulses and causes intensity fluctuation. This is a form of inter-symbol interference, which is the redistribution of pulse energy among neighbouring bits, and which increases system penalties [2]. Without regeneration, the signal distortion remains and accumulates along the link. Therefore the number of in-line photonic components and the transmission distance is limited by the adverse effects of group delay ripple [3].
Fig. 1. Using all-optical regeneration to correct group delay ripple induced pulse degradation. Point A: dispersed pulses, Point B: dispersion compensated but group delay ripple distorted pulses and Point C: regenerated pulses.
All-optical regenerators have been studied in the context of noise and level fluctuation suppression. Fibre-based regenerators are particularly attractive because they can be easily incorporated into existing systems. 2R regenerators perform signal re-amplification and reshaping. Pulse re-shaping is accomplished by a nonlinear power transfer function as shown by the “Pin -Pout ” curve in Fig. 1 to reduce low-level noise and equalise peak pulse power.
In this paper we investigate the use of all-optical regenerators to minimise the detrimental effects of group delay ripple and thus relax the group delay ripple requirements of photonic devices deployed in optical systems. Eye-closure penalty (ECP) is used as a measure of signal quality degradation. The change in eye-closure penalty after two different regenerators is calculated by solving the nonlinear Schrödinger equation (NLSE) using the split-step Fourier pulse propagation method [4]. The first one is based on self-phase modulation (SPM), and the second is based on four-wave-mixing (FWM). In Section 2 the effects of group delay ripple on optical pulses are reviewed. Section 3 discusses the properties and highlights the differences of the two regenerators. In Section 4 the optical filtering requirements are discussed. In section 5 we deduce the regenerative properties of both regenerators. In section 6 the ECP attributed to group delay ripple and the ECP improvement after each regenerator is calculated.
2. Effect of group delay ripple on optical pulses
The group delay response τg (ν) of a device can be written as a sum two components, that is, τg (ν) = (ν) + Δτg (ν) where ν is the optical frequency, is the ideal response and Δτg is a perturbation to the ideal response called the group delay ripple. In the case of a dispersion compensator, is a linear function of frequency. The effect of group delay ripple Δτg can be understood using harmonic ripple analysis [5,6], in which Δτg has fixed amplitude and varies sinusoidally with frequency:
where ar is the peak-to-peak ripple amplitude, pr is the ripple period and ϕr is the phase angle between the pulse spectral peak and ripple peak.
The effect of group delay ripple on an isolated pulse is summarised in Fig. 2. The group delay ripple does not change the optical power spectrum, however it does change the temporal pulse shape. The main pulse sheds energy to create trailing and leading satellite pulses. This causes a reduction in the power of the main pulse. Satellite pulses, which are separated by the reciprocal of the ripple period 1/pr , may interfere with neighbouring pulses (1’s) or remain as noise pulses in the spaces (0’s). Group delay ripple thus manifests as intensity fluctuations of the marks and spaces and pulse-width fluctuations of the marks.
Fig. 2. Effect of harmonic group delay ripple (dotted line) on an isolated pulse, whose power spectrum (solid line) is shown on the left. The resulting temporal pulse shape is shown on the right. pr and 1/pr are the ripple period and frequency, respectively.
3. SPM- and FWM-based regenerators
We have investigated two regeneration schemes, SPM-based and FWM-based. Both schemes use a nonlinear medium in conjunction with an optical filter.
The SPM-based regenerator [7] consists of an optical amplifier, dispersion-shifted highly-nonlinear fibre (DS-HNLF) and a filter offset from the input wavelength (Fig. 3 (top left)). A pulse entering the regenerator undergoes SPM-induced spectral broadening within the fibre, where the spectral width increases with the peak power. Pulses with high peak power experience sufficient spectral broadening to be passed, in part, by the filter. Noise and low-intensity pulses remain spectrally narrow and are thus rejected by the filter (Fig. 3). A small amount of normal dispersion aids the minimisation of ripple on the SPM spectrum [7], which in the time domain also changes the input pulse into a square-like pulse with a linear chirp [8].
The power transfer function and intensity-dependent timing jitter of the SPM-based regenerator are shown in Fig. 3 (bottom left). In this scheme, although pulse-to-pulse intensity fluctuations are reduced, it is inherently translated into timing jitter. The curves for the SPM-based regenerator are calculated by launching pulses of increasing intensity into the regenerator and measuring the integrated power of the output pulses. We use integrated power instead of peak power because it is a better indicator of the eye-opening. For example, using peak power would underestimate the eye-opening of a double-peaked pulse. Also, using cw power would not hold because in this case there would be no spectral broadening. The pulse centre is defined using the mean offset of the pulse relative to the clock. We do not use the point of peak intensity in order to accommodate the possibility of double-peaked pulses.
Fig. 3. SPM-based (left) and FWM-based (right) optical regenerators. Their power transfer functions are shown as solid lines. Powers are normalised to the nominal integrated input power. Timing jitter shown as dashed lines is normalised to the 25 ps bit period. The input pulse full-width at half maximum is 8.25 ps. The top diagrams show the spectral content of the signal after the DS-HNLF in both regenerators.
The shape of the power transfer function is dependent on the ratio LD /LNL , where LD = /∣β 2∣ is the dispersion length and LNL = (γP0 )-1 is the nonlinear length. T0 is the 1/e-intensity pulse half width, β 2 is the group velocity dispersion, γ is the fibre nonlinearity and P0 is the pulse peak power [7]. A larger ratio gives a power transfer function with ripple on the upper plateau, whereas a smaller ratio makes the upper plateau slope up instead of being level. A compromise between the two is when this ratio equals 0.01 – 0.02. The choice of this ratio determines the required fibre dispersion for a given fibre nonlinearity, input power and pulse width. In particular, the required dispersion increases quadratically with pulse width and increases linearly with fibre nonlinearity and input power. We emphasise that the power transfer function is dependent on the details of the input pulse. The parameters are listed in Table 1.
A FWM-based regenerator [9] is shown in Fig. 3 (top right). A signal pulse enters the DS-HNLF together with an intense continuous-wave (CW) pump. Through multiple FWM processes new frequency sidebands are generated [9–11]. At low input signal power the first cascaded sideband power scales quadratically with signal input power. At high input signal powers the cascaded sideband power reaches a plateau. This leads to a step-like nonlinear power transfer function. A calculated power transfer function and intensity-dependent timing jitter are shown in Fig. 3. Parameters of this regenerator are listed in Table 1. In practice the use of a high power cw pump requires spectrally broadening the pump by, for example, phase modulation to suppress stimulated Brillouin scattering. This was not considered in the simulations presented here. It has been shown in Ref. [12] however that pump broadening results in broadening of the regenerated signal, resulting in random signal distortions and an increase in signal to noise ratio. The authors proposed a regenerator scheme employing a dual-pump configuration to minimise signal broadening while suppressing Brillouin effects.
Table 1. Parameters of the SPM-based and FWM-based optical regenerators
The two regenerators have two major performance differences. Firstly, although the two power transfer functions are qualitatively the same, there is an inherent gain of about 4.8 dB (excluding coupler loss) when regenerated via FWM, compared to -10 dB when regenerated via SPM. Gain in the FWM regenerator originates from parametric gain, while the loss in the SPM regenerator is a result of spectral filtering. The second difference is that timing jitter in the FWM-based regenerator is approximately fifty times less than that in the SPM-based regenerator, 0.1% as opposed to 5% of one bit-period. The FWM-based regenerator also has a flatter upper plateau.
4. Filter requirements
The optical filter plays a critical role in determining the quality of the regenerated signal. The filters within the two regenerators are quite different. In the FWM-based regenerator, the filter only acts as a bandpass filter to suppress widely separated FWM products and the pump. The in-band spectral content should be left unchanged to preserve the signal content. We have chosen a 4th-order Gaussian filter to introduce minimal spectral distortion.
In case of regeneration via SPM the filter spectrally truncates a broadband spectrum generated by high-intensity pulses and rejects low-intensity fluctuations. The output pulse shape thus depends on the filter’s transmission profile. We have chosen a first-order Gaussian filter to give Gaussian-like output pulses. Filter bandwidth is chosen to produce output pulse width the same as input pulse width. Filter offset has also been shown to be a key parameter for optimization [13]. We found that a filter offset at 0.75 nm gives the minimum eye-closure penalty when regenerating undistorted pulses, where eye-closure penalty is the difference in eye-closure before and after the regenerator. Filter offsets away from this optimal value are found to increase the eye-closure penalty even for undistorted input pulses.
5. Regenerating pulses with intensity and pulse-width fluctuations
To better understand the regenerative capabilities of the two regenerators on group delay ripple distortion we simulate the regeneration of pulses with either peak-intensity or pulse-width fluctuations. Simulations were performed by numerically solving the nonlinear Schrödinger equation using the split-step Fourier method [4]. Effects included in the simulations are attenuation, group-velocity dispersion, cubic dispersion and Kerr nonlinearity. Pulses with intensity fluctuation or pulse-width fluctuation are launched into each regenerator. Fig. 4 (top row) shows the distorted eye diagrams associated with these fluctuations before regeneration. The regenerated outputs from the SPM- and FWM-based regenerators are shown in Fig. 4 in the middle and bottom rows, respectively. Although both regenerators are tolerant to intensity fluctuation in both the 1’s and 0’s, the FWM-based regenerators are superior in two ways. Firstly, it introduces negligible timing jitter associated with the intensity fluctuation. Pulses regenerated via SPM clearly have timing jitter. This is illustrated by the left-hand column in Fig. 4 and is consistent with the transfer functions plotted in Section III. Secondly, the FWM-based regenerator does not correct pulse-width fluctuation. The right-hand column in Fig. 4 shows that the SPM-based regenerator translates pulse-width fluctuation of the input into amplitude fluctuation at the output.
Fig. 4. Regenerator’s tolerance to intensity (left-hand column) and pulse-width fluctuations (right-hand column). Top: Before regeneration, middle: regenerated via SPM, bottom: via FWM.
The origin of timing jitter in the SPM-based regenerator is illustrated schematically in Fig. 5. Pulses with different intensities acquire different amounts of SPM-induced frequency chirp at the end of the fibre. The use of a fixed filter only selects the part of a pulse with a specific instantaneous frequency. Depending on the pulse chirp rate, this frequency can occur at a time that varies from pulse to pulse, resulting in timing jitter. Yet with an appropriate retiming technique such as synchronous modulation 1,000,000 km error-free transmission incorporating SPM-based regenerators has been successfully demonstrated, compared to only 5,000 km without synchronous modulation [14]. Since we have not included a re-timing element, timing jitter results in horizontal eye-closure. When regenerated via FWM timing jitter is small because of the quasi-instantaneous response of the FWM process.
Fig. 5. Intensity-dependent timing jitter in the SPM-based regenerator..Pulse shapes at various stages of the SPM-based regenerator are shown as solid lines, frequency chirp as dotted lines, and the filter passband as grey band. Point A: input pulses with intensity fluctuation, point B: pulses under the effects of SPM and normal dispersion, and Point C: output pulses with timing jitter.
There is a different response from the two regenerator types to pulse width fluctuations. In the SPM-based regenerator the pulse spectral broadening is sensitive to the input pulse width. Because the optical power spectrum within the passband of the Gaussian filter changes with the input pulse width the transfer function also changes with pulse width. Pulse-width fluctuation becomes amplitude jitter (see Fig. 4 middle-right). In particular, the upper plateau of the transfer function fluctuates up and down as the pulse width changes. In the FWM-based regenerator pulse width fluctuation is translated into spectral width fluctuations of the cascade. The broad and flat 4th-order Gaussian filter ensures no filtering of this cascade. In the FWM-based regenerator the output filter does spectrally clip which makes it more robust to pulse-width fluctuations than SPM-based regeneration.
The output pulses from the FWM-based regenerator are double-peaked, as seen in Fig. 4 (bottom left). This is the result of pump and signal depletion at the higher-intensity part of the pulse and not spectral filtering. Pulse asymmetry is the consequence of walk-off between the signal pulse and the regenerated pulse along the length of the fibre. The signal pulse has a greater group velocity than the regenerated pulse. The leading edge of the regenerated pulse thus interacts the most with the signal pulse and acquires more parametric gain.
6. Simulations: harmonic and realistic group delay ripple
In this section, we present results from pulse propagation simulations to evaluate the impact of group delay ripple and the subsequent regeneration of the distorted pulse sequence. We will first consider harmonic ripple as described by Eq. (1), and then consider realistic ripple from a real and practical device. In both cases, we examine the eye-diagrams and calculate the eye-closure (Ceye ) both before and after the regenerators to find the eye-closure penalties (ECP). ECP is defined as the increase in eye-closure relative to the input Ceye . We adopt the following definition of eye-closure [15], Ceye =- 10log10(Pr /Pave ), where Pr is the height of the highest rectangle whose width equals 20% of a bit-period, and Pave is the average signal power. A 40 Gbits/s 128 bit return-to-zero pseudo-random bit sequence consisting of unchirped Gaussian pulses of 8.25 ps (full-width at half maximum) is used as the input signal.
6.1 Harmonic ripple analysis
This analysis allows us to identify the range of ripple periods and amplitudes that can be effectively corrected by the regeneration schemes. In this simulation, group delay ripple described by Eq. (1) is added to the input pulse sequence. As the two parameters ar and pr are varied, the ECPs before and after the regenerators are calculated and plotted as contour diagrams in Fig. 6.
Fig. 6. Harmonic ripple analysis: Eye-closure penalty (dB) before and after each regenerator. Darker regions represent higher penalty. Top and bottom rows illustrate the cases for ϕr = 0 and ϕr = π/2, respectively. Contour lines are spaced by 0.5 dB. Ripple amplitude and period are normalised to the bit-period (T) and the bit-rate (1/T), respectively.
Figure 6 shows the eye-closure penalty (in dB) contour diagrams for ϕr = 0 (top left) and ϕr = π/2 (bottom left) before regeneration. Ripple amplitude is normalised to the bit-period T, and ripple period is normalised to the bit-rate 1/T. Note that when pr > 2/T, ripple of this period is so large that Δτg varies nearly monotonically with frequency across the pulse bandwidth. In this case Δτg (ν) is seen as group delay dispersion rather than ripple. Hence, we only consider ripple period up to this value. Consistent with numerical results obtained by Eggleton et al. [16], where the authors evaluated ripple-induced power penalty, ripple-distortion causes greatest penalty when pr is close to the system bit-rate (pr ≈ 1/T). At this ripple period, satellite pulses of a particular bit overlap exactly with the bits just before and after itself. This corresponds to the worst-case inter-symbol interference for a given ripple amplitude.
Also, the ECP for the case when ϕr = 0 is generally greater than when ϕr = π/2. When ϕr = 0, the leading and trailing satellite pulses have opposite phase, leading to asymmetric inter-symbol interference. Some bits interfere with satellite pulses constructively while others interfere destructively. This worsens the intensity fluctuation and increases eye-closure. When ϕr = π/2, all satellite pulses have the same phase, and interfere with other bits in the same way.
The middle and right-hand columns in Fig. 6 show a comparison between the two regenerators. When regenerated through FWM (middle column), we see that the regenerator is effective in suppressing ripples for both phase angles for all values of ar and pr shown. For example, at ar = T and pr = 1/T, ECP is reduced by 1.7 dB and 2.2 dB for ϕr = 0 and ϕr = π/2, respectively. The SPM-based regenerator however (right column) is less effective in suppressing ripple distortions and even increases the penalty at certain ripple periods in the case of ϕr = π/2, for example, at pr ≈ 1.5/T. This harmonic analysis suggests that regeneration via FWM is capable of reducing ripple-induced penalty for a range of ripple periods, while regeneration via SPM provides only marginal improvement.
6.2 Realistic ripple analysis
Here we evaluate how optical regeneration can suppress group delay ripple degradation introduced by a device measured in our laboratory. The device is a grating-based tunable dispersion compensator designed to operate up to 40 Gbit/s [16]. It has a centre wavelength at 1555 nm and a 3-dB bandwidth of 1.15 nm and its dispersion is tunable between ±400 ps/nm. By setting the dispersion to zero, the group delay ripple is measured using the modulation phase-shift method [17]. The measured group delay ripple is shown in Fig. 6 together with the spectrum of the pulse used in the simulation. The measurement system has temporal and spectral resolutions of 0.01 ps and 5 pm, respectively. Because multiple dispersion compensators can be cascaded in a fibre-optic link the accumulated group delay ripple is larger than for a single device. Thus, the ripple data used in this simulation is three times the measured ripple as shown in Fig. 7. This approach is equivalent to placing the device as a dispersion compensator in a circulating loop experiment with three circuits [3].
Fig. 7. Measured group delay ripple. Dotted line shows the spectrum of a 8.25 ps unchirped Gaussian pulse.
The measured ripple is added to the simulated ideal input pulse sequence, which is then regenerated in one of the regenerators. Figure 8 shows a portion of the pulse sequence and the optical eye diagrams at various stages of the simulation. The first row shows the original and the second row shows the ripple-distorted pulse sequence. Satellite pulses and intensity fluctuation can be seen in the ripple-distorted pulse sequence. Close examination of the corresponding eye diagram (second row right) also reveals pulse-width fluctuation because of ripple distortion. The ECP introduced by three cascaded dispersion compensators is 1.8 dB.
Fig. 8. Realistic ripple analysis: (Partial) pulse sequences, optical eye diagrams and eye-closure penalty before and after each regenerator. Shaded rectangles representing the eye-closures are used in the calculation of Ceye .
The third and fourth rows in Fig. 8 show the pulse sequence after the SPM- and the FWM-based regenerators, respectively. When the pulses are regenerated via SPM, there are significant residual peak-intensity fluctuations. The fluctuations are re-introduced by the SPM-based regenerator as a result of its signal pulse-width dependence of output intensity. Horizontal eye-closure is exacerbated because intensity fluctuations result in timing jitter in the SPM-based regenerator. These two effects combine to degrade the ECP by a further 0.2 dB.
Intensity fluctuations are reduced significantly in the FWM-based regenerator for both the 0’s and 1’s, thereby increasing the vertical eye-opening. The intensity fluctuation of 54% on the distorted eye has been reduced to less than 17% after regeneration. Since pulse timing is not altered by the FWM-based regenerator the horizontal eye-opening remains approximately the same. Consequently, the ECP is improved from 1.8 dB to 0.6 dB.
6.3 Eye-closure penalty as a function of wavelength drift
To allow for the possibility of source wavelength drift because of, for example, aging and temperature changes, we continue the realistic ripple analysis and evaluate the ECP improvement at a narrow wavelength range near the centre of the device passband. The input wavelength is tuned ± 0.08 nm from 1555 nm. Here we focus on the FWM-based regenerator only. The ECP before and after the FWM-based regenerator is shown in Fig. 9.
Fig. 9. Eye-closure penalty before (filled circles) and after (open circles) the FWM-based regenerator for a range of input wavelength drifts.
The ECP before the regenerator fluctuates between 1.8 ± 0.4 dB in the range shown. However regardless of the wavelength shift and ECP introduced by the device, the regenerator reduces the ECP by 1.0 ± 0.2 dB across the range shown. This result further suggests that this FWM-based regenerator has good tolerance on the wavelength drift.
7. Conclusions
We investigated the viability of two types of all-optical regenerators, FWM-based and SPM-based, in suppressing group delay ripple effects. Two major pulse distortions caused by group delay ripple are intensity fluctuation and pulse width fluctuation. The SPM-based regenerator introduces extra timing jitter as a result of signal amplitude jitter: fluctuations in the signal intensity translate into pulse timing changes at the output. The regenerator is also intolerant to pulse-width fluctuations, which are translated by the regenerator into intensity jitter. These two effects lead to a poor suppression of group delay ripple distortions by the SPM-based regenerator. Harmonic group delay ripple analysis reveals that the SPM-based regenerator is not effective in reducing the eye-closure penalty caused by group delay ripple. Simulation of signal quality degradation by three cascaded tunable dispersion compensators confirms a lack of improvement in the eye-closure penalty. The power transfer function alone does not convey the full properties of the regenerator.
The quasi-instantaneous response of the FWM-based regenerator is well characterised by its power transfer function. Unlike the SPM-based regenerator, the FWM-based regenerator introduces negligible timing jitter to the output pulses. Harmonic ripple analysis confirms that the FWM-based regenerator is effective in reducing the eye-closure penalty caused by a wide range of group delay ripple amplitudes and frequencies. Furthermore, since the output filter only serves as a bandpass filter, and not to pulse-shapes as in the case in the SPM-based regenerator, the regenerator is more robust to pulse width fluctuation caused by group delay ripple. These attributes make FWM-based regenerator a better candidate than its SPM-based counterpart in suppressing group delay ripple induced system penalties. The FWM-based regenerator achieves signal reshaping and reduces the eye-closure penalty caused by group delay ripple. The eye-closure penalty introduced by three cascaded tunable dispersion compensators is shown to be reduced by 1.0 ± 0.2 dB when the source wavelength is allowed to vary between ± 0.08 nm. When placed before a receiver, a FWM-based regenerator increases the eye-opening of a group delay ripple distorted signal, relaxing the stringent specifications of device group delay ripple.
Acknowledgments
This work was produced with the assistance of the Australian Research Council under the ARC Centres of Excellence program.
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