## Abstract

We present a simple way to analytically predict the effect of the temporal Talbot self-imaging process on random amplitude noise and timing jitter in periodic optical pulse trains. The analysis is general and can be applied to any pulse shape; simulation results are in excellent agreement with the predicted values. In addition, the results clearly show that the temporal Talbot effect has an inherent property of mitigating the standard deviation of both pulse amplitude noise and timing jitter.

© 2007 Optical Society of America

## 1. Introduction

Amplitude fluctuations and timing jitter are two major issues limiting the performance of any application relying on the precise regularity and periodicity of optical pulse trains. All-optical sampling, frequency metrology, and optical clock schemes [1] are especially sensitive to any variations in the pulse amplitude and timing.

The temporal Talbot effect [2] is a promising technique for pulse repetition rate multiplication [3] and all-optical clock recovery [4]. In short, the temporal self-imaging process occurs in a first order dispersive medium provided that the dispersion *Φ* (ps^{2}) and the pulse period *T* (ps) satisfy the following condition: *T*
^{2} = *m*/*s*⋅*2π*⋅*Φ* where *m* and *s* are integers such that *s*/*m* is an irreducible rational number. Recently, the Talbot effect has been studied
by Fernández-Pousa *et al*. [5, 6] from the perspective of timing jitter reduction. In particular, they demonstrated that a fluctuating timing jitter within a single input pulse period, where for example, the pulse’s edges suffer from larger variations than their peak, is smoothened at the output. In the temporal Talbot effect, each output pulse results from cumulative interference of dispersed input pulses. Thus, we expect intuitively that amplitude noise or timing jitter “averaging” processes should occur on a larger scale, thereby reducing intra-pulse amplitude and timing variations in the incident pulse train.

In this paper, we develop simple expressions to predict the standard deviations of the amplitude fluctuations and timing jitter of an input pulse train subject to the temporal Talbot effect. The significance and simplicity of our expressions lie in the fact that only an *a priori* knowledge of the incident pulse shape, repetition rate, and the Talbot parameters *m*, *s* is required. We compare the results from numerical simulations with those predicted from the simple expressions and find that they are in very good agreement. Moreover, the results show that the temporal Talbot effect reduces significantly the standard deviation of both amplitude fluctuations and timing jitter in the output train, i.e. it is an efficient means for suppressing irregularities in the amplitude and timing of pulses in the input pulse train.

## 2. Temporal Talbot effect: principle

In the temporal Talbot effect, the dispersion broadens individual pulses to the point where they overlap in such a way that their mutual interference results in the self-imaging process. Each individual output pulse thus results from the cumulative contribution of a sequence of input pulses, as illustrated in Fig. 1. The number of pulses which contribute effectively to each output pulse is determined by the medium dispersion, pulse train period, and pulse width.

We define the input pulse train *a _{in}(t)* as

where *a _{0}(t)* represents an individual sufficiently narrow pulse (to satisfy the Talbot self-imaging conditions) centered at

*t*= 0 with a peak amplitude of 1. After propagating through a lossless linearly dispersive medium, the output

*o(t)*will simply consist of a superposition of dispersed versions

*d(t-pT)*of consecutive input pulses

*a*:

_{o}(t-pT)For particular values of dispersion that satisfy the Talbot condition, the self-imaging effect manifests through the following relation between the amplitude envelope of the output signal |*o(t)*| and the input pulses:

The output is a periodic pulse train with a repetition rate *m* times higher than the input train. For (*m*⋅*s*) odd, the output pulse train also undergoes a temporal shift equal to half of its period. In addition, the pulse shape remains unchanged – only its amplitude is scaled down by √*m* as required by energy conservation. Assuming (*m*⋅*s*) even for simplicity, we can relate Eq. (3) to Eq (2) as follows:

The above establishes a relationship between the sum of the overlapping, dispersed pulses, and the output pulse train magnitude.

## 3. Effect on amplitude noise

Without loss of generality, we focus our analysis on a specific output pulse peak, at *t* = 0:

Eq. (5) shows that it is sufficient to sum the samples at *t* = *pT* of one dispersed pulse *d(t)* to obtain the output pulse train peak amplitude (equal to 1/√*m*). Now, let each input pulse be subject of a random independent fluctuation in its amplitude (relative to the peak pulse amplitude), governed by a normal random variable *N _{P}* with mean 1 and a standard deviation

*σ*. The expression for the input signal can then be written as

_{noise-in}As above, we consider the cumulative interference of dispersed pulses at *t* = 0 and obtain, as in Eq. (5), the following:

Since the above equation comprises a sum where each random variable *N _{p}* is governed by fixed coefficients corresponding to the values of the dispersed pulse at different time samples, we can easily determine the standard deviation

*σ*of the output amplitude random variable at

_{noise-out}*t*= 0. Invoking the well-known formula to determine the standard deviation of a linear combination of independent variables, we obtain:

Here, *m* was introduced to scale the output standard deviation as we are interested in the relative amplitude variation with respect to the pulse peak amplitude. The key feature of the above result is that we can predict the standard deviation of the noise in the output based solely upon values of one dispersed pulse, taken at successive intervals separated by *T*. A specific example can be given using Gaussian pulses where an analytic expression exists for the dispersed pulses. In particular, an initially transform-limited Gaussian pulse with a *1/e* width of *T _{0}* which has subsequently propagated through a lossless, first order dispersive medium with dispersion

*Φ*can be expressed as:

In the right hand side of the above equation we used the Talbot equality *Φ* =*T _{2}*⋅

*s*⋅

*m*⋅

^{-1}*(2π)*to substitute for the dispersion term

^{-1}*Φ*. This, in turn, allows us to expand the general expression in (8) for the output noise standard deviation to be:

It is straightforward to see the strength of Eq. (10) insofar as it allows us to foresee the relationship between the amplitude noise of the output and input pulse trains. Moreover, we only need to know the input pulse width *T _{0}*, pulse train period

*T*, and the Talbot parameters

*m*and

*s*. Finally, we can easily re-write the above expression in terms of the filter dispersion

*Φ*by relating it to

*T*,

*m*, and

*s*through the Talbot condition

*T*

_{2}=

*m*/

*s*⋅2

*π*⋅

*Φ*.

## 4. Effect on timing jitter

A similar reasoning can be used to determine a relationship between the timing jitter of the input and output pulse trains. We start by noting that it is possible to approximate the sum of *n* pulses *p(t)*, having different amplitudes *A _{1}*…

*A*and small temporal offsets

_{n}*δ*…

_{1}*δ*by a similar, scaled pulse whose central location is determined by weighting the coefficients of the contributing pulses:

_{n}$${\phantom{\rule{.2em}{0ex}}\mathrm{where}\phantom{\rule{.2em}{0ex}}\delta}_{\mathit{out}}=\frac{{A}_{1}{\delta}_{1}+{A}_{2}{\delta}_{2}+\dots +{A}_{n}{\delta}_{n}}{{A}_{1}+{A}_{2}+\dots +{A}_{n}}$$

The above approximation neglects the reduction in the peak output amplitude due to the offset of the peaks of the contributing pulses. In addition, it is only valid under the assumption of a small temporal offset with respect to the individual pulse width; otherwise the output pulse shape becomes significantly different from the input shape. Nonetheless, under the aforementioned conditions, such reasoning allows us to establish a linear dependence between the temporal offset of the interfering input pulses *δ _{1}* …

*δ*, and the resulting offset of the resulting output pulse, namely

_{n}*δ*.

_{out}We then consider the same input pulse train as in Section 2, this time with each pulse being offset from its ideal position by a normal random variable *D _{p}* with mean 0 and standard deviation

*σ*:

_{jitter-in}Note that here, the standard deviation of the temporal jitter is defined in absolute values, and not with respect to the pulse train period. Since propagation in a first order dispersive medium does not affect the temporal location of a pulse (other than the propagation time which is the same for all pulses), the output will consist simply of equally shifted, dispersed versions of the incident pulses:

In order to determine the behaviour at the output, let us define *D _{out}* as the random variable corresponding to the offset of one output pulse, ideally centered at

*t*= 0. Following on our assumption that the input time jitter is small with respect to the pulse width (which will be addressed in the next section), we can examine the output at

*t*= 0:

which, through the relation established in Eq. (11), leads us to the expression of *D _{out}* , representing the random variable governing the temporal offset of the output pulse:

Invoking once again the variance of a linear combination of independent variables, and noting that the absolute value of the denominator is equal to 1/√*m* as per Eq. (5) we determine the standard deviation of *D _{out}*, the timing jitter of the output pulse train, to be:

We can see immediately that the resulting expression is identical to that obtained in Eq. (8) for amplitude noise. The analytical expression given in Eq. (10) is therefore also valid for timing jitter.

## 5. Results and discussion

In order to verify our approach, we first simulate the propagation of an ideal train of 20,000 Gaussian pulses through a first order dispersive medium so as to eliminate the possible influence of non-idealities caused by overlapping pulses [7], filter group delay ripple [8], or finite-duration pulse bursts [9]. The default simulation assumes 10 ps full-width half-maximum pulses at a repetition rate of 10 GHz and a dispersive medium where the dispersion is set according to the Talbot condition and (*m* = 1, *s* = 1). The amplitude for each input pulse was varied according to a normal distribution with a mean of 1 and a standard deviation of *σ _{noise-in}* = 0.1. We then compared the ratio of

*R*=

_{noise}*σ*/

_{noiSe-out}*σ*from the simulations to the value predicted using Eq. (10). Figure 2 shows the ratio of

_{noise-in}*R*as a function of four variables: (a)

_{noise}*s*, (b)

*m*, (c) the input repetition rate, and (d) the pulse width. Note that in Figs. 2(b)–2(d), both curves are essentially superimposed.

Next, we consider an input pulse train in which the position of each pulse is governed by a normal random variable with *σ _{jitter-in}* = 1 ps, so as to satisfy the requirement of a small timing jitter (compared to the pulse width). Figure 3 summarizes the results comparing the predicted and simulated values of

*R*=

_{jitter}*σ*/

_{jitter-out}*σ*.

_{jitter-in}From Fig. 2 and Fig. 3, we observe an excellent agreement between the simulated and predicted values. The discrepancy can be attributed to the simulation accuracy insofar as they consider a limited train of pulses, and henceforth a limited set of random numbers used to generate noise and jitter values. The results show that for the cases considered here, both *R _{noise}* and

*R*are always smaller than 1 and can be as low as 0.2, indicating a 5-fold reduction in the amplitude noise or timing jitter in the pulse train. In addition, both ratios are relatively constant for a given Talbot multiplication factor

_{jitter}*m*, regardless of

*s*. Although a larger s is equivalent to a larger filter dispersion, the cumulative interference still results in the same amplitude profile (except for a

*T*/2 shift for

*m⋅s*odd), and thereby the effective contribution from the dispersed pulses remains the same. On the other hand, increasing the pulse width, the repetition rate, or

*m*decreases the amount of suppression in the relative noise or jitter. In the first case, longer pulses result in a smaller spectral content, and therefore less inter-pulse interference. Increasing the repetition rate on the other hand reduces the required dispersion quadratically, also decreasing the effective amount of pulse overlap.

While both the amplitude noise and timing jitter approaches are inherently equivalent, one important difference has to be highlighted. Indeed, in the case of timing jitter, we need to make the assumption of a small timing jitter with respect to the pulse width since the analysis is performed at ideal sampling points. On the other hand, as amplitude fluctuations do not affect the location of each pulse, such a restriction is absent in the amplitude noise derivations. In order to define better the boundaries of the aforementioned assumption, we compare the % error between simulated and predicted values of the standard deviation of the output timing jitter for *σ _{jitter-in}*.in increasing from 1 ps to 5 ps, while all the remaining parameters remain the same as above (input pulse width of 10 ps, 10 GHz repetition rate,

*m*=

*1*, and

*s*=

*1*.). The results are shown in Fig. 4(a). Figure 4(b) shows the same analysis for

*σ*increasing from 10% to 50% of the input pulse amplitude. As expected, for the case of timing jitter, the % error increases with increasing

_{noise-in}*σ*; on the other hand, it is relatively constant for the case of amplitude variations. This analysis allows us to state that, for example, requiring a prediction error of 10% with a 10 GHz input pulse train of 10 ps pulses undergoing a Talbot self-imaging process with (

_{jitter-in}*m*= 1,

*s*= 1), the timing jitter must not exceed 3 ps.

Recently, Duchesne *et al*. expanded the Talbot condition for temporal self-imaging to include higher order dispersion terms [10]. Provided that the higher order dispersion terms satisfy the general Talbot conditions, then insofar as the principle of the self-imaging phenomenon remains unchanged, so does our approach towards determining the resulting amplitude noise and timing jitter mitigation. In particular, we simply need to redefine *d(t)* as being the result of the original pulse having propagated through a phase-only filter comprising many dispersion orders. As a result, we can conjecture that amplitude noise and timing jitter mitigation will still occur, although the improvement can only be established by determining the dispersed pulse shape *d(t)*.

## 6. Conclusion

We have presented a very simple and straightforward method to estimate the amplitude noise and timing jitter of an output pulse train generated using the temporal Talbot effect. Our approach, confirmed by a good agreement with numerical simulations, is valid for any amplitude fluctuation values, and for relatively small timing jitter. We observe that the temporal Talbot effect reduces significantly both considered pulse train impairments. Such behaviour is intuitively consistent with the self-imaging process, as each output pulse is due to the contribution of neighboring, dispersed input pulses. Our analysis constitutes a natural extension of the timing jitter fluctuations attenuation within a single pulse period, as reported by Fernández-Pousa *et al*. The predictions are confirmed by simulation results, and lead us to conclude that the Talbot effect can be easily applied to increase the precision of optical clock signals, as well as to reduce inter-pulse variations for applications such as all-optical sampling, metrology, or timing.

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