## Abstract

Temporal self-imaging effects (TSIs) are observed when a *periodic* pulse train propagates through a first-order dispersive medium. Under specific dispersion conditions, either an exact, rate multiplied or rate divided image of the input signal is reproduced at the output. TSI possesses an interesting self-restoration capability even when acting over an *aperiodic* train of pulses. In this work, we investigate and demonstrate, for the first time to our knowledge, the capability of TSI to produce periodic sub-harmonic (rate-divided) pulse trains from aperiodic sequences. We use this inherent property of the TSI to implement a novel, simple and reconfigurable sub-harmonic optical clock recovery technique from RZ-OOK data signals. The proposed technique features a very simple realization, involving only temporal phase modulation and first-order dispersion and it allows one to set the repetition rate of the reconstructed clock signal in integer fractions (sub-harmonics) of the input bit rate. Proof-of-concept experiments are reported to validate the proposed technique and guidelines for optimization of the clock-recovery process are also outlined.

© 2015 Optical Society of America

## 1. Introduction

The temporal self-imaging effect (TSI), also referred to as temporal Talbot effect, is the time-domain counterpart of the well-known spatial self-imaging phenomenon [1,2]. This space-time duality is related to the mathematical isomorphism existing between the equations describing the paraxial diffraction of beams in space and the first-order temporal dispersion of optical pulses. The TSI occurs when a periodic train of optical pulses propagates through a dispersive medium in a first-order approximation; such a dispersive medium is characterized by a linear all-pass amplitude response and a quadratic phase variation in frequency. An appropriate amount of dispersion, given by the so-called self-imaging condition [1,2], leads to either an exact reproduction of the original pulse train (integer TSI) or to repetition-rate multiplication by an integer factor (fractional TSI).

Besides the intrinsic physical interest of the TSI phenomenon, it has also been put into practical use in several areas. In particular, TSI has been extensively studied as a lossless mean to multiply the repetition rate of a periodic pulse train, through fractional TSI [1–7]. This phenomenon also has been investigated theoretically to be an accurate technique for measuring the first-order dispersion coefficient of dispersive media [2]. Moreover, Pudo et al. [8,9] have shown the inherent buffering ability of TSI to generate a periodic output, even from an aperiodic input pulse train. They have used this property to perform *base-rate* clock recovery (BRCR) of a return-to-zero on–off keying (RZ-OOK) optical data signal by utilizing single-mode fibers (SMFs) or linearly chirped fiber Bragg gratings (LCFBGs) as dispersive media.

Very recently a new feature of TSI, referred here to as inverse temporal self-imaging (I-TSI), has been reported to demonstrate passive amplification of repetitive optical signals [10]. I-TSI involves repetition-rate division of an incoming repetitive pulse train by an integer factor (*m* = 2, 3, …) through an ideally lossless redistribution of the input signal energy, effectively implementing a process equivalent to coherent addition of each *m* individual consecutive pulses. Similar to the TSI, the I-TSI is based on the distributed interference of dispersed pulses, i.e., there is no direct one-to-one correspondence between the input and output pulses. As a consequence, we anticipate that this effect should also possess the inherent property to generate a periodic output (with sub-harmonic repetition rate), yet from an aperiodic input pulse train. This feature, therefore, could be appropriately used to recover the *m*-sub-harmonic clock signal of a RZ-OOK data signal. In optical communications, optical sub-harmonic clock recovery (SHCR) is a fundamental functionality for de-multiplexing optical time-division multiplexed (OTDM) systems and subsequent processing operations [11–14].

In this paper, we propose and experimentally demonstrate the realization of a reconfigurable SHCR technique based on I-TSI. The proposed concept can be implemented using a very simple setup, involving a suitable combination of temporal phase modulation and dispersive spectral phase filtering of the original aperiodic (data) pulse train. This setup can be designed to reconstruct a periodic optical pulse train (clock) in which the pulses are spaced by *m* times the bit period of the input data signal, i.e., the output is a sub-harmonic clock signal with a bit rate division factor of *m*. As detailed below, quality of the recovered clock signal is strongly related to the duty-cycle of the input pulses and the amount of dispersion used to implement the required spectral phase filter.

The proposed technique offers similar advantages to those of the TSI-based BRCR method [8,9], namely simplicity, the potential for timing-jitter mitigation and high energy efficiency. The latest is associated with the fact that fundamentally, no energy is lost in the recovery process, since this involves only manipulations of the signal’s phase information, in the time and frequency domains. Moreover, the SHRC scheme can be easily reconfigured to achieve clock recovery at the input signal’s bit rate (BRCR, *m* = 1) by simply bypassing the temporal phase modulation process.

## 2. Concept and operation principle

Figure 1 illustrates the principle of our proposed concept. Figure 1(a) represents the well-known standard temporal self-imaging (Talbot) effect in which a flat-phase repetitive input train of pulses (signal at *z* = 0, where *z* represents the axial propagation coordinate) is self-imaged, after dispersive propagation through an integer multiple of the Talbot length, *z _{T}* (integer Talbot images). In this representation, first-order dispersive propagation, involving a linear group delay profile, is assumed. Additionally, rate-multiplied self-images of the original input waveform train are obtained at fractional values of the fundamental Talbot distance,

*z*, as defined by the “Talbot Carpet” [1]; see examples in Fig. 1(a) at the fractional Talbot distances

_{T}*z*/2, 2

_{T}*z*/3 and 3

_{T}*z*/4. Dispersive propagation speeds up and slows down the different frequency-components, originally in-phase, of the pulse train, thus redistributing the energy of the signal into the mentioned different temporal intensity patterns. An integer self-image exhibits the same repetition rate as the input signal, whereas in the multiplied self-images, the repetition rate is increased. The repetition rate-multiplication factors for the multiplied self-images shown in Fig. 1(a) at

_{T}*z*/2, 2

_{T}*z*/3 and 3

_{T}*z*/4 are 2, 3 and 4, respectively.

_{T}In an integer self-image, the uniform temporal phase profile of the input is restored. However, the multiplied self-images, such as those observed at distances *z _{T}*/2, 2

*z*/3 and 3

_{T}*z*/4 are affected by a deterministic pulse-to-pulse residual temporal phase profile (dashed black). This residual temporal phase represents instances where the pulse field-amplitude has been advanced or delayed in relation to the envelope center [1,10]. The residual phase for a particular fractional image can be mathematically calculated and pre-introduced to the input signal by means of a phase-modulation mechanism. This emulates the effect of a previous propagation through dispersion of the input signal. Further dispersive propagation of such signals to the distance

_{T}*z*produces an output with a reduced repetition rate. The repetition rate of a pulse train starting at the fractional distance

_{T}*z*/2 will be reduced by a factor of

_{T}*m*= 2 at the output as depicted in Fig. 1(b). Likewise, the repetition rate of a pulse train starting at the fractional distance 2

*z*/3 will be reduced by a factor of

_{T}*m*= 3.

Similarly, we anticipate that an aperiodic input pulse train (e.g. a RZ-OOK data signal) undergoing similar processing steps will also produce a periodic pulse train output, with a repetition rate that is reduced by an integer factor of *m* with respect to the input bit rate, as shown in Fig. 1(c). In this paper, we demonstrate and study this interesting property of the temporal Talbot effect, for the first time to our knowledge. We also show how this property can be applied towards a simple, passive optical sub-harmonic clock recovery scheme for OOK-RZ transmission systems. Notice that the required processes involve only a suitable manipulation of the input signal temporal and spectral phase profiles, ensuring that the signal energy is ideally preserved. The complete Talbot carpet provides an infinite amount of fractional self-image conditions and corresponding phase profiles, so that any desired repetition rate division factor can be obtained, only limited in practice by the degree of control that one can achieve on the temporal phase modulation mechanism and dispersion-induced spectral phase filtering.

The required temporal phase modulation profile can be deduced from the self-imaging theory [1,10] so that in order to recover the sub-harmonic clock signal of a RZ-OOK input signal with clock division factor of *m* ( = 2, 3, 4, …), a quadratic temporal phase variation must be applied, i.e.

*n*-th bit (

*n*= 0,1,2,…,

*m*-1) of the input data signal. If these phase shifts are reduced to a 2π range, a periodic sequence of phase steps with a fundamental period equal to the clock division factor

*m*, (i.e.${\phi}_{n}={\phi}_{n+m}$) is obtained. These phase shifts induce a spectral self-imaging effect on the modulated data signal [15,16]. In particular, the temporal phase modulation process reduces the spectral comb spacing of the clock component (i.e. the spectral separation between two adjacent lines) of the input data signal by an integer factor of

*m*.

As per the TSI theory [1,10], the subsequent dispersive medium should introduce a total dispersion value given by Eq. (2):

where*s*= 0,1,2,…,

*T*is the bit period of the input data signal (inverse of the input bit rate), and ${\varphi}_{T}^{(2)}$is the slope of the medium’s group delay vs. radial frequency. Notice that

*s*= 0 ensures the minimum dispersion value required to obtain the target sub-harmonic clock signal. It should be mentioned that in the absence of temporal phase-modulation, the dispersion ${\varphi}_{T}^{(2)}$ will produce an integer Talbot self-image of the original pulse train at the output of the dispersive medium. This fact allows one to re-program the scheme to achieve also clock recovery at the basic input bit rate (BRCR) by simply bypassing the temporal phase-modulation mechanism.

In order to present a more intuitive insight into the operation principle of the proposed sub-harmonic clock recovery technique, we use joint time-frequency (TF) representations of the involved signals, first for a periodic input pulse train as shown in Fig. 2, and then for an aperiodic input pulse train (e.g. a RZ-OOK data signal), as shown in Fig. 3. For each TF representation, the bottom plot represents the temporal variation of the pulse train and the plot at the left represents the corresponding spectrum, with the 2D energy distribution shown in the larger central plot.

Figure 2 shows the case when the input signal is a periodic pulse train and we target a repetition rate division factor of two (*m* = 2), for example. In the absence of temporal phase-modulation, each repeating pulse of the input pulse train along the temporal domain is composed by all the discrete frequency components of the original pulse train, as shown in Fig. 2(a), left. The dispersive medium, with dispersion coefficient of ${\varphi}_{T}^{(2)}=2{T}^{2}\text{/2}\pi $[from Eq. (2), with *s* = 0], introduces a group delay difference of 2*T* between two adjacent discrete frequency components with frequency spacing of *F* = 1/*T*. Spectral superimposition of the resultant dispersed pulse train produces an exact (integer) Talbot self-image of the original pulse train at the fiber output, as illustrated in Fig. 2(a), right. Application of the prescribed temporal phase-modulation profile for the case *m* = 2, ${\phi}_{n}$ = {0, π/2, 0, π/2,…}, derived from Eq. (1), to the input periodic pulse train, produces new frequency components, reducing the comb frequency spacing of the input signal by a factor of 2 (i.e. to *F*/2), as illustrated in Fig. 2(b), left. Moreover, the induced pulse-to-pulse phase variation translates into instantaneous frequency fluctuations among the different optical pulses in such a way that alternating optical pulses exhibit a different frequency-shifted spectral content. Subsequent propagation of the phase-conditioned pulse train through the same amount of dispersion ${\varphi}_{T}^{(2)}$now induces a dispersive delay of *T* between two adjacent discrete frequency components (with frequency spacing of *F*/2). After the temporal redistribution induced by the frequency-dependent group delay, the individual frequency components of the pulse train will add coherently to build a new pulse train with twice the temporal period of the original signal. In other words, this delay, indeed, coherently shifts energy such that the output repetition rate is reduced by a factor of two as shown in Fig. 2(b), right.

Next, we assume an aperiodic pulse train such as a RZ-OOK data signal as the input signal, which comprises some missing pulses along time domain in comparison with a purely periodic optical pulse train. Figure 3(a), left shows the evaluation of such data signal along the temporal and spectral domains before dispersion in the absence of the prescribed temporal phase modulation. Notice that the spectrum of an arbitrary RZ-OOK data, indeed, consists of strong line spectral components (clock components, spaced in frequency by the input bit rate) and a ‘weaker’ noise-like background that is distributed continuously along the signal spectrum between the frequency clock lines. Therefore, each individual pulse of the temporal signal actually consists of a continuous band of frequency components. However, herein, for illustrative purposes, to facilitate the interpretation of the resulting plots and without loss of generality, we just consider the signals’ clock components, which play the main role in the clock recovery process. The influence of the weak noise-like spectral background will be briefly discussed later in this section.

As compared to Fig. 2, the same dispersive medium (${\varphi}_{T}^{(2)}=2{T}^{2}\text{/2}\pi $) introduces a group delay of 2*T* between two adjacent discrete clock frequency components with frequency spacing of *F* = 1/*T*. The spectral superimposition of the dispersed pulses on the temporal axis recovers the missing pulses of the input RZ signal and thereby produces a periodic pulse train with a repetition period equal to the input bit period (i.e. a base-rate clock signal with repetition rate of 1/*T*). Non-uniform distribution of the instantaneous frequencies among the different optical pulses, however, causes pulse-to-pulse amplitude variations in the recovered clock signal; such amplitude variations are more pronounced in the case of an input data signal with a longer series of zero bits. Nevertheless, higher-quality clock pulses with small pulse-to-pulse amplitude variations can be recovered when the input pulses are sufficiently short in comparison with the bit period, *T*, (i.e. low duty cycle). In this case a larger number of spectral clock components will contribute to the pulse reconstruction process so that the dispersed pulses can interfere with temporally apart pulses as well as adjacent ones.

Figure 3(b) shows how if we condition the input data signal by a proper temporal phase modulation derived from Eq. (1) ({0, π/2, 0, π/2,…}) when we target to perform sub-harmonic clock recovery with clock division factors of *m* = 2. This leads to the anticipated spectral self-imaging and reduces the comb frequency spacing of the input signal by a factor of *m* = 2. Subsequent propagation through the dispersion ${\varphi}_{T}^{(2)}$ induces a dispersive delay of *T* between two adjacent discrete clock frequency components with frequency spacing of *F*/2. This frequency-dependent delay will rearrange the instantaneous frequencies along the time domain and their coherent superimposition will produce a periodic pulse train with a repetition period twice that of the input signal (i.e. with the target sub-harmonic repetition rate). Similar to the preceding case, the recovered sub-harmonic clock pulses, however, suffer from pulse-to-pulse amplitude variations, which are particularly significant for input signals with a longer series of consecutive zero bits. Amplitude variations in the recovered clock envelope are less pronounced for lower duty-cycle input pulses. Still, if needed, these fluctuations could be further minimized by use of a higher amount of dispersion [i.e., higher value of the parameter *s* in Eq. (2)], or by incorporating an additional power equalizer, e.g. a semiconductor-optical-amplifier (SOA)-based fiber ring laser (SOA-FRL) [17]. A detailed numerical analysis and further discussions on the issue of amplitude fluctuations in the recovered clock signals are provided in the following Section 3.

Finally, we should mention that the continuous, noise-like spectral background component of the input signal adversely affects the clock recovery process and degrades the clock signal quality. After temporal dispersion, such a component will be spread randomly all over the time domain, inducing a certain pedestal on the output generated pulses, as illustrated in Fig. 1(c).

## 3. Experimental demonstration and discussion

We next report on a proof-of-concept experiment to validate the proposed concept by performing SHCR with clock division factors of *m* = 2, *m* = 3 and *m* = 4, as well as BRCR from a 9.7Gbit/s data signal. Notice that, any desired repetition rate division factor can be essentially obtained in practice by appropriate control of the temporal phase modulation and dispersion-induced spectral phase filtering. Nevertheless, to obtain higher SHCR orders, higher dispersion values are required to realize the desired spectral phase filter. In this experiment, the highest repetition rate division factor (*m* = 4) was practically limited by the available dispersive medium in our laboratories.

The experimental setup used in our demonstration is sketched in Fig. 4. The RZ-OOK data signal is produced by intensity modulation of ~4 ps (intensity FWHM) optical pulses generated by a mode-locked laser with a 9.7GHz repetition rate at a central wavelength of 1550nm. The modulator driver is a 2^{7}-1 pseudo-random bit sequence (PRBS), thus generating a RZ-OOK modulated signal. The generated data signal is then delivered to the clock recovery circuit, consisting of a commercial fiber-integrated electro-optic phase modulator driven by an arbitrary waveform generator (AWG), followed by dispersion-compensating fibers (DCFs).

Figure 5 shows the prescribed electro-optic phase modulation profiles in our reported experiments for the targeted SHCR factors of 2, 3 and 4, respectively. The periodic temporal phase profiles derived from Eq. (1) are {0, π/2, 0, π/2, …} for *m* = 2, {0, 2π/3, 2π/3, 0, …} for *m* = 3, and {0, 3π/4, π, 3π/4, 0, …} for *m* = 4. The dashed red lines show the ideal temporal phase profiles, and the solid blue lines show the actual phase drives delivered by the AWG. Figure 6 presents the optical spectra of the input data signal before (solid blue) and after (dashed red) temporal phase modulation, recorded with a standard optical spectrum analyzer. These measurements show the predicted spectral self-imaging effect, leading to the expected decrease in the spectral comb spacing by factors of 2, 3 and 4, respectively.

The required amount of first-order dispersion can be calculated using Eq. (2). In this set of experiments, we have considered the minimum required dispersion and set *s* = 0 in Eq. (2). The corresponding dispersion values would be 3376 ps^{2}/rad (≈2647 ps/nm), 5065 ps^{2}/rad (≈3972 ps/nm) and 6752 ps^{2}/rad (≈5294 ps/nm) for SHCR orders of *m* = 2, 3 and 4, respectively. In practice, the required dispersion was provided by *m* dispersion-compensating fiber modules (Corning PureForm DCM-D-080-04, each having a nominal group-velocity dispersion of ≈1324 ps/nm) for the respective SHCR order. In this case (i.e. when *s* = 0), the dispersion values increases with the SHCR factor *m*. As mentioned above, when the temporal phase modulation is off, the defined dispersion directly implements BRCR from the input RZ-OOK signal.

Figure 7 shows the temporal waveform, eye diagram and RF spectrum of the input data signal used in the experiments. The RF spectrum shows a clear bit rate frequency component at 9.7GHz and the PRBS data components around the bit rate frequency.

The experimental results of the demonstrated SHCR with clock division factors of 2, 3 and 4 are presented in Fig. 8. Figures 8(a)–8(c) display the measured temporal waveforms and eye diagrams of the 4.85GHz (*m* = 2), 3.23GHz (*m* = 3), and 2.43GHz (*m* = 4) sub-harmonic clock signals, respectively, measured with a 40-GHz photodetector attached to an electrical sampling oscilloscope. The eye diagrams are depicted together with their amplitude histograms at the center of a pulse timeslot. The results clearly indicate the presence of a periodic pulse train at the output, highlighted by the absence of a baseline in the eye diagrams. Amplitude variations in the output pulses are evident, though, through the smearing in “one” level of the output eye diagram; nevertheless, the pulse’s peaks are always well above the “zero” level.

Figures 8(d)–8(f) show the measured RF spectra corresponding to the temporal waveforms shown in Figs. 8(a)–8(c), respectively. Clearly visible peaks appear at 4.85GHz [Fig. 8(d)], 3.23GHz [Fig. 8(e)] and 2.42GHz [Fig. 8(f)], confirming the target SHCR processes for *m* = 2, 3 and 4, respectively. Comparison of the measured RF spectra in Fig. 8 and Fig. 7 reveals that the frequency components of pseudo-randomly intensity-modulated signals [visible in Fig. 7(b)] have been greatly suppressed through the clock recovery process. This proves that TSI operates as a periodic RF bandpass filter with central frequencies located at ${(mT)}^{-1}$ GHz and its harmonics. The pass bandwidth of each harmonic is inversely proportional to *m* (assuming *s* = 0 for all cases). As can be seen, the pass bandwidth becomes narrower with increasing *m*, and the RF spectral intensity of data components (restricted to the vicinity of clock frequency) is increasingly suppressed. Notice that a narrower pass bandwidth and a higher suppression rate of the data components in the RF spectra are indications of a higher-quality output clock signal, e.g., with lower pulse-to-pulse amplitude variations, also associated with an improved eye opening for the output pulse train. Therefore, provided that the minimum required dispersion is employed in each case [*s* = 0 in Eq. (2)], a higher quality optical clock is recovered as the SHCR order is increased; this can be attributed to the fact that a higher order *m* requires a higher dispersion in the recovery system and increased dispersion induces a broader spreading of each input individual pulse, facilitating the desired inter-pulse interactions at the system output.

In line with our discussions in Section 2, the ability of the temporal self-imaging effect to produce uniform amplitude optical pulses from the input RZ-OOK data signal is strongly dependent on: 1- the number of spectral clock components in the input signal, namely the input pulse duty-cycle, 2- the amount of required dispersion, depending on both *s* and *m*, and 3- the PRBS pattern length. In order to quantitatively evaluate the magnitude of pulse-to-pulse amplitude variations in the clock signal in regard to these parameters, we define the “relative amplitude variation factor (RAVF)” as the ratio of the difference between the highest pulse’s peak power (*P _{max}*) and the lowest pulse’s peak power (

*P*) to the highest pulse’s peak power:

_{min}Figure 9(a) plots the output pulses’ amplitude variation (i.e. RAVF) as a function of the input pulse duty-cycle for different values of *m* and *s*. In this simulation, we used a RZ-OOK data with a 2^{7}-1 PRBS. One may notice that for a given *m* and *s*, the amplitude variations of recovered clock pulses are alleviated as the input pulse duty-cycle decreases. As mentioned earlier, lower input pulse duty-cycle incorporate a larger number of spectral clock components in the pulse reconstruction process. Therefore, pulses are temporally spread out more significantly by the same amount of dispersion and this facilitates their interference with temporally apart pulses as well as adjacent ones, so that in average the input pulses’ energy is more uniformly redistributed over all the newly generated pulses.

In addition, for a given input pulse duty-cycle and SHCR order, the amplitude modulation of the recovered clock signal can be noticeably reduced when a higher value of dispersion is employed [i.e. higher *s* in Eq. (2)]. This can be attributed to the fact that a higher dispersion induces a broader temporal spreading of each input individual pulse, leading to more pronounced inter-pulse interactions. For example, assume the input pulse duty-cycle of 33% and SHCR order of *m* = 6, RAVF of the clock signal is reduced from 0.87 to 0.5 when *s* increases from 0 to 2.

Figure 9(b) shows that the amplitude variations at the output pulse train increases as the PRBS pattern length (2^{k}-1) of the input data increases. For example, for *m* = 2 and *s* = 0, RAVF reaches almost 1 for PRBS patterns with lengths more than 2^{23}-1. Note that a PRBS with a length of 2^{k}-1 may have a sequence of up to “k” (or “k-1”) consecutive zeros in its pattern. In this case, the temporally apart pulses in the two sides of the long sequence of zeros (i.e. long zero gap), indeed, do not get to interfere with each other and, therefore, the clock pulses along this sequence is not recovered. However, for SHCR order of *m* = 2 with *s* = 2, for example, the required dispersion is five times that for *m* = 2 and *s* = 0 and then, pulses are temporally dispersed more strongly, so that they get to pass the long zero gap and interfere with the other side pulses, creating the desired new pulses in the long zero gap zone. The pulse time-width in the input data pulse train is fixed to 4 ps in this simulation.

Finally, Fig. 10 shows the outcome of the same clock recovery circuit, in the absence of temporal phase modulation. In this case, a 9.7GHz base-rate clock signal can be obtained at the output of the circuit, no matter which SHCR order (*m*) is used, as confirmed by the corresponding RF spectra in Figs. 10(d)–10(f). Similarly to the SHCR case, the optical clock quality increases as the factor *m* is increased.

It is important to mention that the bit rate in our proof-of-concept experiment was limited by available equipment. It is well known that TSI is more easily observable as the pulse bit rate is increased since a lower dispersion amount is then required. Moreover, alternatively, the quadratic time phase modulation can be applied using non-linear optical processes, e.g. cross-phase-modulation with a parabolic pump pulse train, overcoming the intrinsic speed limitations of electro-optic phase modulation [18].

## 4. Conclusions

In summary, the effect of inverse temporal self-imaging (I-TSI) on aperiodic pulse trains have been investigated and studied. Results on this study suggest a novel scheme for reconfigurable sub-harmonic optical clock recovery based on I-TSI, involving phase-only temporal modulation and dispersive spectral filtering of the input data pulse train. The scheme can be easily reconfigured to achieve base-rate clock recovery by simply bypassing the temporal phase-modulation step. The proposed scheme has been validated through experimental demonstrations, and in particular, we have reported successful extraction of 4.85-GHz and 3.23-GHz sub-harmonic clock signals as well as the 9.7-GHz base-rate clock signal from a 9.7-Gbit/s RZ-OOK data signal under a 2^{7}-1 PRBS pattern. The quality of the recovered optical clock signal is improved for a higher sub-harmonic rate factor, a higher dispersion amount for temporal self-imaging [higher ‘s’ in Eq. (2)], or lower duty-cycle input pulses.

## Acknowledgments

This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canada Research Chair in “Ultrafast Photonic Signal Processing”. R.M. acknowledges financial support from the Ministère de l'Éducation, du Loisir et du Sport (MELS) du Québec through the Merit Scholarship Program for Foreign Students.

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