## Abstract

We analyze simultaneous amplitude fluctuation and timing jitter performance of a set of commonly believed equivalent spectrally periodic phase-only filters for implementing pulse repetition rate multiplication. Whereas amplitude noise and time jitter mitigation is observed in all cases, our analysis reveals different noise performance to that obtained with the classical Talbot filter implementation based on a single dispersive medium. Moreover, different noise improvements are achieved depending on the filter’s spectral period and a mutual interaction between amplitude noise and timing jitter is also observed.

© 2011 OSA

## 1. Introduction

The generation of ultrashort (femtosecond/picosecond) optical pulses with high or ultra-high repetition rates has attracted considerable attention in recent years since this plays a role of increasing importance in a number of applications such as ultrahigh-bit-rate optical communications, optical sampling, frequency metrology, optical clock schemes [1] and microwave photonics systems [2]. Particularly concerning this last mentioned topic, millimeter-wave (MMW) carrier generations using optical pulse trains with 100 GHz repetition-rate or higher have been recently investigated for gigabits wireless access applications [2] and distribution of synchronized photonic MMW waveforms through radio-over-fiber technology [3–5]. The ability to generate *stable* pulse sources is extremely important in all these applications, particularly considering that pulse trains produced through conventional techniques, such as actively, passively or hybrid mode-locking, typically suffer from different noise contributions. To be more concrete, pulse-to-pulse amplitude fluctuations and timing jitter are two key factors that limit the performance of high-repetition-rate optical pulse trains in any application relying on the precise regularity and periodicity of these pulse trains.

Despite the great deal of effort in finding solutions for directly generating ultrahigh repetition rate pulse trains from semiconductor or fiber lasers, still an attractive and easy way to reach pulse rates beyond those achievable by conventional methods is to multiply the repetition rate of a lower rate source outside the laser cavity [6,7]. Besides a method based on the nonlinear effect named modulation instability [8], used in the past to experimentally demonstrate pulse rates in the terahertz range, two linear solutions for pulse repetition rate multiplication (PRRM) have been demonstrated and deeply investigated in the recent past, namely the so-called fractional temporal Talbot effect [6,7, 9–12], and another solution based on spectrally periodic quadratic-phase-only filtering [13,14]. It has been recently demonstrated that a single first-order dispersive medium, generally used to implement temporal Talbot effect, is only a particular case belonging to the complete set of spectrally periodic filters presented in reference [7]. Indeed, the solutions experimentally demonstrated in [13] and [14] also belong to this general class of periodic linear filters implementing the fractional temporal Talbot effect. On the other hand, it has been previously shown that the temporal Talbot effect has an intrinsic property of mitigating the standard deviation of both *amplitude fluctuations* and *timing jitter* of an optical pulse train [15–18], either when used for multiplying the initial repetition rate (fractional Talbot effect) or when the repetition rate is kept unchanged (integer Talbot effect). In particular, in reference [18] the only experimental demonstration of timing jitter reduction through integer temporal Talbot effect was reported. Therefore, being all the periodic phase filters presented in reference [7] equivalent from the PRRM point of view, it has been generally accepted that they provide the same performance in terms of noise reduction. In this paper we demonstrate, for the first time to our knowledge, that each exemplar belonging to a family of equivalent spectrally periodic quadratic-phase filters, as described in reference [7], performs very differently from both amplitude noise and timing jitter viewpoints, in particular strongly depending on the spectral period of the PRRM filter itself. Furthermore, complementing the previous studies on temporal Talbot effect [6,7,9], we analyze here the case of *simultaneous* amplitude fluctuation and timing jitter to recreate more realistic, thus more interesting, cases. Broadly speaking, we show that all the Talbot filtering configurations for PRRM can provide a mitigation of both the amplitude fluctuations and the timing jitter of the input pulse train; however, different filtering configurations provide a different level of mitigation and general noise performance. At last, performance of the PRRM filters concerning the pedestal of the output pulse train is also evaluated here.

Notice that the optical filters described in reference [7] can be implemented using simple and compact linearly-chirped fiber Bragg gratings (LC-FBGs) or superimposed LC-FBGs [13,14]. Indeed, in reference [18] it is pointed out how, in order to get an actual timing jitter reduction, a compact solution employing a LC-FBG is practically much more feasible than making use of a spool of fiber, which comes out to be too sensitive to environmental disturbance occurred in the fiber length. For this reason, we believe that the analysis reported here should prove important to design and realize PRRM filters with a customized performance in terms of amplitude fluctuations and timing jitter.

## 2. Filters design

Based on the general theory presented in reference [7], the same Talbot-based PRRM process can be implemented through different linear optical filter designs, which are formally equivalent. In particular, assuming an input optical pulse train with a repetition rate *f _{in}* and supposing a desired PRRM process by a factor

*m*, this PRRM operation can be achieved by filtering the original pulse train through either a single dispersive medium (e.g. a spool of fiber or a LC-FBG) or a spectrally periodic phase-only filtering device with a spectral period fixed by the output repetition rate, i.e.

*f*, where

_{out}=p(mf_{in})*p*can be any positive integer. Considering the first case, i.e. linear propagation of an input pulse train through a single first-order dispersive medium, fractional temporal Talbot effect is implemented when the medium’s dispersion coefficient satisfies the following condition:

*T*is the input pulse period,

_{in}= 1/f_{in}*q = 1, 2, 3,...*and

*m = 1, 2, 3,...*such that

*(q/m)*is a non-integer and irreducible rational number [6]. The corresponding base-band frequency transfer function of the dispersive medium is

*H(f) ∝ e*with:where the dispersion coefficient ${\ddot{\Phi}}_{0}$ is in [ps

^{jΦ(f)}^{2}/rad].

Assuming an ideal input pulse train, not affected by any kind of noise or jitter, the repetition rate of this pulse train would be multiplied by *m* following linear propagation through a dispersion medium with the first-order chromatic dispersion in Eq. (1). Figure 1
reports the phase function calculated from Eq. (2) with a dashed red line, considering the case of a 10-to-40 GHz PRRM process, with *q* = 1 and *m* = 4,

On the other hand, as previously mentioned and as reported in reference [7], an identical PRRM process can be implemented by use of a spectrally periodic phase-only optical filter providing a general spectral phase transfer function:

*pm·f*and

_{in}*p = 1, 2, 3,...*. Equation (3) basically implies that in each period ${\Phi}_{p}(f)=\Phi (f-kpm{f}_{in})$ as defined by the Talbot condition in Eqs. (1)-(2), for

*f*∈ (-kpmf

_{in}/2; kpmf

_{in}/2) and

*k = 0, ±1, ±2, ±3,...*. Thus, for a given input pulse rate

*f*and a given multiplication factor

_{in}*m*, we may identify a family made up of an infinite number of phase-only filters, just by changing

*p*(i.e. changing the spectral period of the filter), for each valid value of

*q*. In Fig. 1 several examples of ${\Phi}_{p}(f)$ for different values of

*p*, i.e. for different spectral periodicity, are reported, considering the same 10-to-40GHz PRRM process (multiplication factor

*m*= 4) and the two cases

*q*= 1,

*q*= 3. All these filters formally implement the same PRRM process. Please note that a single dispersive medium (i.e. single Talbot filter, dashed and dotted red lines in Fig. 1) is only a particular case of the general family defined by Eq. (3) for the case

*p→∞*. We recall that in practice, the spectrally-periodic quadratic phase filters defined by Eq. (3) can be implemented using superimposed LC-FBGs. A higher value of either the parameter

*p*(spectral period) or the parameter

*q*translates into a larger group-delay excursion in the filter’s transfer function, thus requiring the use of a longer device (the single Talbot filter, based on a single dispersive medium, LC-FBG, is the longest device among all the equivalent filtering configurations).

The main purpose of this paper is to analyze the evolution of simultaneous amplitude fluctuations and timing jitter from an input pulse train to the output one of a PRRM designed to achieve an *m*-fold PRRM process. Different filtering configurations, i.e. characterized by different spectral periods (*p* parameters) and different *q* parameters, as defined in the general Eq. (3), will be evaluated and compared. Amplitude noise and timing jitter smoothing by temporal Talbot effect in a single dispersive medium has been previously studied [15–17]; however, to our knowledge, no studies on the noise performance of the more general Talbot filters defined by Eq. (3), e.g. as the spectral period of the filters is varied, has been reported to date. Noise mitigation of optical pulse trains through integer or fractional temporal Talbot effect is a very well known property of dispersive media, but no comparison between the noise performance of periodic phase-only filters and that of single dispersive media, to achieve the same Talbot-based PRRM process, have been carried out so far. Furthermore, whereas in reference [15] a detailed amplitude noise and timing jitter performance analysis of Talbot-based PRRMs has been carried out (for the single dispersive medium case), authors did not consider the situation of simultaneous amplitude fluctuations and timing jitter; our numerical analysis reported here considers this more practical situation, showing how these two kinds of noise may in a certain way affect each other.

## 3. Simulation results and discussion

Let us first define the input signal employed for the numerical analysis presented in this paper. In order to test the performance of the above-mentioned spectral phase-only filters, an input train of *N _{in}* = 1000 Gaussian pulses with a period

*T*= 100 ps (

_{in}*f*= 10GHz) and an individual pulse width (FWHM)

_{in}*τ*=5 ps is considered. To emulate a realistic case, we assume that each of the input pulses is affected by both amplitude and timing jitter. Thus, the input pulse train may be expressed by:

_{FWHM}*p(t)*represents the individual short Gaussian pulse with a normalized peak amplitude,

*A*is the

_{n}*n*-th extraction of a normal random variable with mean 1 and standard deviation

*σ*= 0.06 (6% of ideal amplitude), governing the random independent amplitude fluctuation, and

_{A}*τ*is the

_{n}*n*-th extraction of a normal random variable with mean 0 and standard deviation

*σ*= 0.3 ps (6% of the FWHM), governing the random independent timing jitter.

_{τ}Considering the generic filter transfer function *H _{p}(f) = e^{jΦp(f)}* where ${\Phi}_{p}(f)$ is described by Eq. (3), through numerical simulations, we calculated the output signal as:

*S(f)*is the Fourier transform of

*s(t)*and

*F*stands for the inverse Fourier transform.

^{−1}Considering again the example of a 10-to-40 GHz PRRM process (with *q* = 1), Fig. 2(a)
shows the overlapped temporal amplitude traces of the *N _{in}* input periods and Fig. 2(b)-(d) show the output (temporal amplitude traces) of the periodic filter for the two cases p = 3 and

*p*= 15 and the output of the single Talbot filter (single dispersive medium). A few comments are pretty evident from the figure. Consistently with previous works [15–17], there is a visible mitigation of both amplitude fluctuation and timing jitter from input to output in all the reported cases, as it can be inferred from the reduction of the thickness of the overlapped pulses’ trace on both the edges and the top of the trace. On the other hand a deterioration on the pedestal of each output pulse train is also evident (no pedestal at the input signal is assumed, as it can be deduced from Fig. 2(a)). An improvement in terms of both amplitude fluctuation and timing jitter performance can be noticed as the spectral period of the Talbot filter is increased, e.g. from

*p*= 3 to

*p*= 15. At last, we can also anticipate a practically identical performance for the single Talbot filter and the periodic one with

*p*= 15 (Fig. 2(c) and 2(d) look essentially the same). This last statement can be easily understood by considering the behavior of the periodic parabolic phase variation as the parameter

*p*is increased in Fig. 1: for

*p→∞*the periodic filter tends to be identical to the single Talbot filter along the frequency range corresponding to the entire support of the input pulse spectrum.

In order to evaluate both amplitude fluctuations and timing jitter of the output pulse train, the temporal position and amplitude of the peak at each output period (with temporal duration *T _{in}/m*) was collected, extracting for each set the standard deviation

*σ*and

_{τ}^{out}*σ*, respectively. Then, for each kind of filter, a timing jitter ratio

_{A}^{out}*τ*and an amplitude fluctuation ratio

_{ratio}= σ_{τ}^{out}/σ_{τ}*A*have been defined so as to estimate the improvement in terms of noise performance for each filter. The behaviors of timing jitter and amplitude fluctuation ratios are reported in Fig. 3 -4 , for the case

_{ratio}= σ_{A}^{out}/σ_{A}*m*= 4,

*T*= 100ps, with

_{in}*q*= 1 and

*q*= 3, respectively. In both figures,

*τ*and

_{ratio}*A*are plotted for different spectral periods (i.e. versus

_{ratio}*p*) and compared with the single Talbot filter case (dashed lines). Moreover, for each value of

*q*, we reported three different cases when only timing jitter (a), only amplitude fluctuation (b) and both of them (c) are applied. When

*q*= 1 timing jitter and amplitude fluctuation seem to be essentially independent (

*τ*in Fig. 3(a), (c) are identical,

_{ratio}*A*in Fig. 3(b), (c) are identical), whereas for

_{ratio}*q*= 3 amplitude fluctuation performance reveals a worsening when timing jitter is present as well (

*A*in Fig. 4(b), (c) are different).

_{ratio}The general fundamental remark here is that even though mitigation of the amplitude fluctuation and timing jitter is always observed, each periodic filter (i.e. each value of *p*) presents a different, specific noise performance and is not equivalent to the single Talbot filter case, unless the factor *p* is sufficiently high. In the case *q* = 1, the amplitude fluctuation and time jitter improvements are both more significant as the spectral period (*p* parameter) is increased, regardless of whether the two noise contributions are considered separately or simultaneously; as a result, the best performance is always achieved for the single Talbot filter case. Focusing on the case where both noises are present (Fig. 3(c) and Fig. 4(c)), another interesting observation is the highly different behavior of the amplitude fluctuation between the two cases *q* = 1 and *q* = 3, revealing a sort of mutual interaction between the two kinds of noise for *q* = 3. For higher values of *q* (*q* = 5, 7, 9...) we have observed that the amplitude fluctuation exhibits a similar trend to that reported in Fig. 4(c) for *q* = 3. To be more concrete, in the case *q* = 3 (simultaneous presence of amplitude and time jitters, Fig. 4(c)), for several specific values of *p*, the amplitude fluctuation improvement is more pronounced than that of the single Talbot filter and in particular, for *p* = 4, 5 this improvement is even slightly better than that of the single Talbot filter for *q* = 1 (Fig. 3(c)).

In general, the noise ratios (both *τ _{ratio}* and

*A*) thresholds for the single Talbot filter decrease as the input pulse duty cycle (or the multiplication factor

_{ratio}*m*[15]) is decreased, see results in Fig. 5(a) . Furthermore, for a given duty cycle, by increasing the value of

*q*from 1 to 3, the timing jitter ratio is improved whereas the amplitude fluctuation ratio gets worse (Fig. 5(a)); then, for higher values of

*q*, i.e. 5, 7, 9..., the results in terms of amplitude fluctuation and time jitter performances are essentially the same as those obtained for

*q*= 3.

To make more clear why the trend of plots in Fig. 4(c) (*q* = 3) looks so different from Fig. 3(c) (*q* = 1), we believe that the key point is that amplitude fluctuation and timing jitter are not completely independent with respect to each other for the cases where q>1. From additional simulations (not included in the present manuscript), it comes out that the filtering process itself leading to a mitigation of either timing jitter or amplitude fluctuation (through any of the alternative Talbot filtering configurations) induces additional amplitude fluctuation or timing jitter component (which increases with *p*) at the filter output, respectively. Furthermore, the amount of such induced amplitude fluctuation/timing jitter increases by increasing the input timing jitter/amplitude fluctuation to be mitigated. Thus, the induced amplitude fluctuation/timing jitter can degrade the filter performance, especially for the single Talbot filter or for the periodic phase filters with high values of *p*. As an example, we show in Fig. 5 that considering a timing jitter with *σ _{t}* = 2% and an amplitude jitter with

*σ*= 12%, a result with opposite behavior to the one reported in Fig. 4(c) has been obtained.

_{A}Another important figure of merit in a PRRM process is the extinction ratio (*ER*) of the output pulse train, generally defined as the ratio between the amplitude peak of each pulse and the pedestal level between pulses. Since in our simulations no pedestal is assumed at the input pulse train, the input ER is consequently assumed to approach infinity. Considering a signal affected by stochastic noises, as in our analysis, we can define here an average output *ER* as the ratio between the average of the pulse peak amplitudes of all the *m*·*N _{in}* output pulses and the average of the output pedestal level over the entire pulse train. In particular, we define the pedestal of the

*n*-th output period to be the signal ranging from the “time position of the

*n*-th peak” + 2

*τ*to the “time position of the

_{FWHM}*(n+1)*-th peak” - 2

*τ*. The average

_{FWHM}*ER*for the single Talbot filter decreases as the input duty cycle is increased, as reported in Fig. 6(b) . Moreover, this result is almost independent on the value of

*q*. Finally, when analyzing the output average

*ER*behavior for different spectral periods (Fig. 6(c)), we observe that the

*ER*fluctuates over a range of less than 1 dB and there is not a particularly relevant dependence with respect to the parameter

*q*; this observation applies to all cases except for

*p*= 1, for which the

*ER*reaches ~60 dB for any value of

*q*.

## 4. Conclusion

We have presented a detailed numerical analysis on the simultaneous timing jitter and amplitude fluctuation performance of the complete set of formally equivalent spectrally periodic parabolic phase-only filters aimed at implementing Talbot-based pulse rate multiplication of optical pulse trains. In particular we have demonstrated different noise performance depending on the filters’ spectral periodicity. In general, the filters help mitigating the input amplitude and time-jitter noise characteristics while providing the desired rate multiplication operation. The improvements in the amplitude fluctuation and time jitter are generally more pronounced when using a filter with an increased spectral period, the best improvement being then achieved for the single Talbot filter (single dispersive medium). However, we have identified a number of cases (for higher values of the parameter *q*) for which the improvements in the amplitude fluctuation are better than in the equivalent single Talbot case. Furthermore we analyzed noise performance for different input pulse train duty cycles and at last, we also evaluated the effect of these filters on the pedestal level between output pulses.

The analysis reported here should prove very useful to realize, e.g. through superimposed linearly-chirped FBGs, pulse rate multipliers with optimized amplitude fluctuation and timing jitter performances.

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