## Simple estimation of pulse amplitude noise and timing jitter evolution through the temporal Talbot effect

Optics Express, Vol. 15, Issue 10, pp. 6351-6357 (2007)

http://dx.doi.org/10.1364/OE.15.006351

Acrobat PDF (300 KB)

### 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

2. J. Azaóa and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,“ IEEE J. Sel. Top. Quantum. Electron. **7**, 728–744 (2001). [CrossRef]

3. D. Pudo and L R. Chen, “Tunable passive all-optical pulse repetition rate multiplier using fiber Bragg gratings,“ J. Lightwave Technol. **23**, 1729–1733 (2005). [CrossRef]

*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

5. C. R. Fernández-Pousa, F. Mateos, L Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, and C. Gómez-Reino, “Timing jitter smoothing by Talbot effect. I. Variance,“ J. Opt. Soc. Am. B **21**, 1170–1177 (2004). [CrossRef]

6. C. R. Fernández-Pousa, F. Mateos, L Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, and C. Gómez-Reino, “Timing jitter smoothing by Talbot effect. II. Intensity spectrum,“ J. Opt. Soc. Am. B **22**, 753–763 (2005). [CrossRef]

*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

*a*as

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

_{0}(t)*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)*o(t)*| and the input pulses:

*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:

## 3. Effect on amplitude noise

*t*= 0:

*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*with mean 1 and a standard deviation

_{P}*σ*. The expression for the input signal can then be written as

_{noise-in}*t*= 0 and obtain, as in Eq. (5), the following:

*N*is governed by fixed coefficients corresponding to the values of the dispersed pulse at different time samples, we can easily determine the standard deviation

_{p}*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:

*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*which has subsequently propagated through a lossless, first order dispersive medium with dispersion

_{0}*Φ*can be expressed as:

*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:

*T*, pulse train period

_{0}*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

*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}*δ*…

_{1}*δ*, and the resulting offset of the resulting output pulse, namely

_{n}*δ*.

_{out}*D*with mean 0 and standard deviation

_{p}*σ*:

_{jitter-in}*D*as the random variable corresponding to the offset of one output pulse, ideally centered at

_{out}*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:

*D*, representing the random variable governing the temporal offset of the output pulse:

_{out}*m*as per Eq. (5) we determine the standard deviation of

*D*, the timing jitter of the output pulse train, to be:

_{out}## 5. Results and discussion

7. D. Pudo and L R. Chen, “Estimating intensity fluctuations in high repetition rate pulse trains generated using the temporal Talbot effect,“ IEEE Photon. Technol. Lett. **18**, 658–660 (2006). [CrossRef]

8. J. T. Mok and B. J. Eggleton, “Impact of group delay ripple on repetition-rate multiplication through Talbot self-imaging effect,“ Opt. Commun. **232**, 167–178 (2004). [CrossRef]

9. J. Azaña, “Temporal self-imaging effects for periodic optical pulse sequences of finite duration,“ J. Opt. Soc. Am. B **20**, 83–90 (2003). [CrossRef]

*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*= 0.1. We then compared the ratio of

_{noise-in}*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.

*σ*= 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

_{jitter-in}*R*=

_{jitter}*σ*/

_{jitter-out}*σ*.

_{jitter-in}*R*and

_{noise}*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.

*σ*.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,

_{jitter-in}*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.

*et al*. expanded the Talbot condition for temporal self-imaging to include higher order dispersion terms [10

10. D. Duchesne, R. Morandotti, and J. Azaña, “Temporal Talbot phenomena in high-order dispersive media,“ J. Opt. Soc. Am. B **24**, 113–125 (2007). [CrossRef]

*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

*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.

## References and links

1. | P. J. Delfyett, S. Gee, S. Ozharar, F. Quinlan, K. Kim, S. Lee, and W. Lee, “Ultrafast modelocked semiconductor laser - techniques and applications in networking, instrumentation and signal processing,“ in |

2. | J. Azaóa and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,“ IEEE J. Sel. Top. Quantum. Electron. |

3. | D. Pudo and L R. Chen, “Tunable passive all-optical pulse repetition rate multiplier using fiber Bragg gratings,“ J. Lightwave Technol. |

4. | D. Pudo, M. Depa, and L R. Chen, “All-optical clock recovery using the temporal Talbot effect,“ in |

5. | C. R. Fernández-Pousa, F. Mateos, L Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, and C. Gómez-Reino, “Timing jitter smoothing by Talbot effect. I. Variance,“ J. Opt. Soc. Am. B |

6. | C. R. Fernández-Pousa, F. Mateos, L Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, and C. Gómez-Reino, “Timing jitter smoothing by Talbot effect. II. Intensity spectrum,“ J. Opt. Soc. Am. B |

7. | D. Pudo and L R. Chen, “Estimating intensity fluctuations in high repetition rate pulse trains generated using the temporal Talbot effect,“ IEEE Photon. Technol. Lett. |

8. | J. T. Mok and B. J. Eggleton, “Impact of group delay ripple on repetition-rate multiplication through Talbot self-imaging effect,“ Opt. Commun. |

9. | J. Azaña, “Temporal self-imaging effects for periodic optical pulse sequences of finite duration,“ J. Opt. Soc. Am. B |

10. | D. Duchesne, R. Morandotti, and J. Azaña, “Temporal Talbot phenomena in high-order dispersive media,“ J. Opt. Soc. Am. B |

**OCIS Codes**

(060.2330) Fiber optics and optical communications : Fiber optics communications

(320.5550) Ultrafast optics : Pulses

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: February 23, 2007

Revised Manuscript: May 6, 2007

Manuscript Accepted: May 6, 2007

Published: May 8, 2007

**Citation**

Dominik Pudo and Lawrence Chen, "Simple estimation of pulse amplitude noise and timing jitter evolution through the temporal Talbot effect," Opt. Express **15**, 6351-6357 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6351

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### References

- P. J. Delfyett, S. Gee, S. Ozharar, F. Quinlan, K. Kim, S. Lee, and W. Lee, "Ultrafast modelocked semiconductor laser - techniques and applications in networking, instrumentation and signal processing," in Proceedings of the 18th Lasers and Electro-Optics Society Annual Meeting (2005).
- J. Azaña and M. A. Muriel, "Temporal self-imaging effects: theory and application for multiplying pulse repetition rates," IEEE J. Sel. Top. Quantum. Electron. 7, 728-744 (2001). [CrossRef]
- D. Pudo and L. R. Chen, "Tunable passive all-optical pulse repetition rate multiplier using fiber Bragg gratings," J. Lightwave Technol. 23, 1729-1733 (2005). [CrossRef]
- D. Pudo, M. Depa, and L. R. Chen, "All-optical clock recovery using the temporal Talbot effect," in Proceedings of the Optical Fiber Communications Conference (2007).
- C. R. Fernández-Pousa, F. Mateos, L. Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, and C. Gómez-Reino, "Timing jitter smoothing by Talbot effect. I. Variance," J. Opt. Soc. Am. B 21, 1170-1177 (2004). [CrossRef]
- C. R. Fernández-Pousa, F. Mateos, L. Chantada, M. T. Flores-Arias, C. Bao, M. V. Pérez, and C. Gómez-Reino, "Timing jitter smoothing by Talbot effect. II. Intensity spectrum," J. Opt. Soc. Am. B 22, 753-763 (2005). [CrossRef]
- D. Pudo and L. R. Chen, "Estimating intensity fluctuations in high repetition rate pulse trains generated using the temporal Talbot effect," IEEE Photon. Technol. Lett. 18, 658-660 (2006). [CrossRef]
- J. T. Mok and B. J. Eggleton, "Impact of group delay ripple on repetition-rate multiplication through Talbot self-imaging effect," Opt. Commun. 232, 167-178 (2004). [CrossRef]
- J. Azaña, "Temporal self-imaging effects for periodic optical pulse sequences of finite duration," J. Opt. Soc. Am. B 20, 83-90 (2003). [CrossRef]
- D. Duchesne, R. Morandotti, and J. Azaña, "Temporal Talbot phenomena in high-order dispersive media," J. Opt. Soc. Am. B 24, 113-125 (2007). [CrossRef]

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