## Spectral self-imaging of time-periodic coherent frequency combs by parabolic cross-phase modulation |

Optics Express, Vol. 21, Issue 23, pp. 28824-28835 (2013)

http://dx.doi.org/10.1364/OE.21.028824

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### Abstract

Integer and fractional spectral self-imaging effects are induced on infinite-duration periodic frequency combs (probe signal) using cross-phase modulation (XPM) with a parabolic pulse train as pump signal. Free-spectral-range tuning (fractional effects) or wavelength-shifting (integer effects) of the frequency comb can be achieved by changing the parabolic pulse peak power or/and repetition rate without affecting the spectral envelope shape and bandwidth of the original comb. For design purposes, we derive the *complete family* of different pump signals that allow implementing a desired spectral self-imaging process. Numerical simulation results validate our theoretical analysis. We also investigate the detrimental influence of group-delay walk-off and deviations in the nominal temporal shape or power of the pump pulses on the generated output frequency combs.

© 2013 Optical Society of America

## 1. Introduction

1. 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**(4), 728–744 (2001). [CrossRef]

2. J. Azaña, “Spectral Talbot phenomena of frequency combs induced by cross-phase modulation in optical fibers,” Opt. Lett. **30**(3), 227–229 (2005). [CrossRef] [PubMed]

3. J. Caraquitena, M. Beltrán, R. Llorente, J. Martí, and M. A. Muriel, “Spectral self-imaging effect by time-domain multilevel phase modulation of a periodic pulse train,” Opt. Lett. **36**(6), 858–860 (2011). [CrossRef] [PubMed]

4. A. Malacarne and J. Azaña, “Discretely tunable comb spacing of a frequency comb by multilevel phase modulation of a periodic pulse train,” Opt. Express **21**(4), 4139–4144 (2013). [CrossRef] [PubMed]

*m*) while the comb spectral envelope remains unchanged (fractional SSI). Undistorted wavelength-shifting by half the FSR can also be induced on the original frequency comb (integer SSI). Besides the intrinsic physical interest of SSI phenomena, the capability to controlling key features (e.g. FSR or spectral line location) of periodic optical frequency combs can be potentially interesting for a wide range of applications, such as precision spectroscopy [5

5. L. Consolino, G. Giusfredi, P. De Natale, M. Inguscio, and P. Cancio, “Optical frequency comb assisted laser system for multiplex precision spectroscopy,” Opt. Express **19**(4), 3155–3162 (2011). [CrossRef] [PubMed]

7. P. Del'Haye, O. Arcizet, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Frequency comb assisted diode laser spectroscopy for measurement of microcavity dispersion,” Nat. Photonics **3**(9), 529–533 (2009). [CrossRef]

8. Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature **416**(6877), 233–237 (2002). [CrossRef] [PubMed]

9. S. C. Chan, G. Q. Xia, and J. M. Liu, “Optical generation of a precise microwave frequency comb by harmonic frequency locking,” Opt. Lett. **32**(13), 1917–1919 (2007). [CrossRef] [PubMed]

10. S. Tainta, W. Amaya, M. J. Erro, M. J. Garde, S. Sales, and M. A. Muriel, “WDM compatible and electrically tunable SPE-OCDMA system based on the temporal self-imaging effect,” Opt. Lett. **36**(3), 400–402 (2011). [CrossRef] [PubMed]

11. S. Fukushima, C. F. C. Silva, Y. Muramoto, and A. J. Seeds, “10 to 110 GHz tunable opto-electronic frequency synthesis using optical frequency comb generator and uni-travelling-carrier photodiode,” Electron. Lett. **37**(12), 780–781 (2001). [CrossRef]

12. T. Healy, F. C. Garcia Gunning, A. D. Ellis, and J. D. Bull, “Multi-wavelength source using low drive-voltage amplitude modulators for optical communications,” Opt. Express **15**(6), 2981–2986 (2007). [CrossRef] [PubMed]

13. A. Alatawi, R. P. Gollapalli, and L. Duan, “Radio-frequency clock delivery via free-space frequency comb transmission,” Opt. Lett. **34**(21), 3346–3348 (2009). [CrossRef] [PubMed]

2. J. Azaña, “Spectral Talbot phenomena of frequency combs induced by cross-phase modulation in optical fibers,” Opt. Lett. **30**(3), 227–229 (2005). [CrossRef] [PubMed]

*with limited time duration only*, greatly limiting the practical use of the method. Recently, the SSI effect has been implemented by EOM using discrete (multi-level) temporal phase modulation of the original pulse train in a periodic fashion [3

3. J. Caraquitena, M. Beltrán, R. Llorente, J. Martí, and M. A. Muriel, “Spectral self-imaging effect by time-domain multilevel phase modulation of a periodic pulse train,” Opt. Lett. **36**(6), 858–860 (2011). [CrossRef] [PubMed]

4. A. Malacarne and J. Azaña, “Discretely tunable comb spacing of a frequency comb by multilevel phase modulation of a periodic pulse train,” Opt. Express **21**(4), 4139–4144 (2013). [CrossRef] [PubMed]

*infinite-duration*coherent optical frequency comb using nonlinear XPM with a

*periodic parabolic pulse train*, overcoming the speed limitations of EOM-based methods. Our proposal relies on the fact that the needed quadratic phase-modulation profile at the discrete input pulse locations is a temporally periodic function and as such, a cumulative phase chirp is not necessary for implementation of the effect [14

14. J. Azaña and S. Gupta, “Complete family of periodic Talbot filters for pulse repetition rate multiplication,” Opt. Express **14**(10), 4270–4279 (2006). [CrossRef] [PubMed]

15. C. Finot, J. M. Dudley, B. Kibler, D. J. Richardson, and G. Millot, “Optical parabolic pulse generation and applications,” IEEE J. Sel. Top. Quantum Electron. **45**(11), 1482–1489 (2009). [CrossRef]

21. T. Hirooka and M. Nakazawa, “All-optical 40-GHz time-domain Fourier transformation using XPM with a dark parabolic pulse,” IEEE Photon. Technol. Lett. **20**(22), 1869–1871 (2008). [CrossRef]

19. T. T. Ng, F. Parmigiani, M. Ibsen, Z. Zhang, P. Petropoulos, and D. J. Richardson, “Compensation of linear distortions by using XPM with parabolic pulses as a time lens,” IEEE Photon. Technol. Lett. **20**(13), 1097–1099 (2008). [CrossRef]

21. T. Hirooka and M. Nakazawa, “All-optical 40-GHz time-domain Fourier transformation using XPM with a dark parabolic pulse,” IEEE Photon. Technol. Lett. **20**(22), 1869–1871 (2008). [CrossRef]

22. Y. Ozeki, Y. Takushima, K. Aiso, and K. Kikuchi, “High repetition-rate similariton generation in normal dispersion erbium-doped fiber amplifiers and its application to multi-wavelength light sources,” IEICE Trans. Electron. **88**(5), 904–911 (2005). [CrossRef]

24. T. Hirooka and M. Nakazawa, “Parabolic pulse generation by use of a dispersion-decreasing fiber with normal group-velocity dispersion,” Opt. Lett. **29**(5), 498–500 (2004). [CrossRef] [PubMed]

25. F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, “Pulse retiming based on XPM using parabolic pulses formed in a fiber bragg grating,” IEEE Photon. Technol. Lett. **18**(7), 829–831 (2006). [CrossRef]

27. D. Krcmarík, R. Slavík, Y. Park, and J. Azaña, “Nonlinear pulse compression of picosecond parabolic-like pulses synthesized with a long period fiber grating filter,” Opt. Express **17**(9), 7074–7087 (2009). [CrossRef] [PubMed]

25. F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, “Pulse retiming based on XPM using parabolic pulses formed in a fiber bragg grating,” IEEE Photon. Technol. Lett. **18**(7), 829–831 (2006). [CrossRef]

26. T. Hirooka, M. Nakazawa, and K. Okamoto, “Bright and dark 40 GHz parabolic pulse generation using a picosecond optical pulse train and an arrayed waveguide grating,” Opt. Lett. **33**(10), 1102–1104 (2008). [CrossRef] [PubMed]

27. D. Krcmarík, R. Slavík, Y. Park, and J. Azaña, “Nonlinear pulse compression of picosecond parabolic-like pulses synthesized with a long period fiber grating filter,” Opt. Express **17**(9), 7074–7087 (2009). [CrossRef] [PubMed]

28. J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. **15**(4), 581–583 (2003). [CrossRef]

## 2. Operation principle

1. 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**(4), 728–744 (2001). [CrossRef]

*T*( = 1/

*f*) propagates through a dispersive medium in a first-order approximation; such, which dispersive medium exhibits a linear all-pass response that is characterized by a quadratic phase variation in frequency, namely

_{0}1. 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**(4), 728–744 (2001). [CrossRef]

29. J. Azaña, N. K. Berger, B. Levit, and B. Fischer, “Spectral Fraunhofer regime: time-to-frequency conversion by the action of a single time lens on an optical pulse,” Appl. Opt. **43**(2), 483–490 (2004). [CrossRef] [PubMed]

30. E. R. Andresen, C. Finot, D. Oron, and H. Rigneault, “Spectral analog of the Gouy phase shift,” Phys. Rev. Lett. **110**(14), 143902 (2013). [CrossRef]

2. J. Azaña, “Spectral Talbot phenomena of frequency combs induced by cross-phase modulation in optical fibers,” Opt. Lett. **30**(3), 227–229 (2005). [CrossRef] [PubMed]

**7**(4), 728–744 (2001). [CrossRef]

*T*is the fundamental period of the input optical pulse train (i.e.

*f*= 1/

_{0}*T*is the FSR of the input frequency comb), the positive integer

*m*is the FSR division factor induced by the SSI effect under consideration (

*m*= 1 for integer effects and

*m*= 2, 3, … for fractional effects),

*s*is an arbitrary positive integer such than

*s*and

*m*are co-prime. We recall that when Eq. (1) is satisfied, the comb spectral envelope is unchanged after temporal phase modulation (time-lens) either (i) keeping the same FSR as the input (integer effects,

*m*= 1) or (ii) with a reduced FSR by a factor

*m*(fractional effects,

*m*= 2, 3, 4, …). Furthermore, similarly to its temporal counterpart [1

**7**(4), 728–744 (2001). [CrossRef]

*s*×

*m*) is an even number (direct effects); however, the lines are additionally frequency shifted by half the output FSR when the product (

*s*×

*m*) is an odd number (reversed effects).

*mT*[3

3. J. Caraquitena, M. Beltrán, R. Llorente, J. Martí, and M. A. Muriel, “Spectral self-imaging effect by time-domain multilevel phase modulation of a periodic pulse train,” Opt. Lett. **36**(6), 858–860 (2011). [CrossRef] [PubMed]

14. J. Azaña and S. Gupta, “Complete family of periodic Talbot filters for pulse repetition rate multiplication,” Opt. Express **14**(10), 4270–4279 (2006). [CrossRef] [PubMed]

*periodic*parabolic optical pulse train (pump). Figure 2 explains schematically the principle operation of SSI by XPM of an infinite-duration periodic pulse sequence (input periodic frequency comb, probe signal in

*m*= 2 and 4, which are respectively induced by two parabolic pump signals with identical repetition periods and different peak powers. The probe and the pump signals are assumed to be centered at different wavelengths such that their respective spectra do not overlap. The pump signal is a periodic parabolic pulse train which exhibits a repetition period of

*L*of a highly nonlinear optical fiber (HNLF). Under ideal conditions (ignoring the dispersive and group-delay walk-off effects at first), the pump signal will induce a nonlinear phase shift of

20. T. Hirooka and M. Nakazawa, “Optical adaptive equalization of high-speed signals using time-domain optical Fourier transformation,” J. Lightwave Technol. **24**(7), 2530–2540 (2006). [CrossRef]

21. T. Hirooka and M. Nakazawa, “All-optical 40-GHz time-domain Fourier transformation using XPM with a dark parabolic pulse,” IEEE Photon. Technol. Lett. **20**(22), 1869–1871 (2008). [CrossRef]

*A*is the effective fiber core area,

_{Eff}*n*is nonlinear refractive index and

_{2}*m*( = 1, 2, 3, …) can be derived to satisfy the following condition:

**30**(3), 227–229 (2005). [CrossRef] [PubMed]

*mT*[3

**36**(6), 858–860 (2011). [CrossRef] [PubMed]

14. J. Azaña and S. Gupta, “Complete family of periodic Talbot filters for pulse repetition rate multiplication,” Opt. Express **14**(10), 4270–4279 (2006). [CrossRef] [PubMed]

**36**(6), 858–860 (2011). [CrossRef] [PubMed]

*fundamental period*given by

*any*integer multiple of the fundamental period, i.e. with a temporal period given bywhere

*p*= 1, 2, 3, …. Additionally, as discussed above, the individual pulses in the optical pump train must exhibit a parabolic temporal intensity shape, with duration equal to

*T*, Eq. (4), and with a peak power satisfying Eq. (3). This set of design specifications defines a

_{pu}*complete family*of different pump signals that allow implementing a desired SSI process by simply suitably changing the peak power or/and the repetition rate of the pump pulse train.

## 3. Simulation results

*T*= 25 ps), and numerically simulate a periodic train of 1,500 Gaussian pulses, each with a 1.6-ps FWHM, corresponding to a Gaussian spectral envelope with a 275.8-GHz FWHM. In all simulated cases, we assume a typical HNLF with

*L*= 1km,

^{−1}km

^{−1}, zero-dispersion wavelength at λ

_{0}= 1551nm and a flat dispersion profile with a slope of 0.017 ps/nm

^{2}·km [2

**30**(3), 227–229 (2005). [CrossRef] [PubMed]

**20**(22), 1869–1871 (2008). [CrossRef]

32. M. Galili, L. K. Oxenløwe, H. C. H. Mulvad, A. T. Clausen, and P. Jeppesen, “Optical wavelength conversion by cross-phase modulation of data signals up to 640 Gb/s,” IEEE J. Sel. Top. Quantum Electron. **14**(3), 573–579 (2008). [CrossRef]

_{0}to avoid walk-off between the two pulses and second, we place them close enough to λ

_{0}to reduce dispersion-induced pulse broadening of the propagating pulses, provided also that the respective spectra do not overlap (λ

*= 1560nm and λ*

_{Probe}*= 1542nm). Thus, in our simulations, the probe pulses are not affected by dispersion during propagation through the HNLF considering the GVD specifications of the fiber and bandwidth of the considered probe signal. For simplicity, the polarization effects are ignored assuming that the pump and probe signals propagate in the fiber with the same polarization state [31]. Moreover, the probe peak power should be low enough to neglect the corresponding self-phase modulation (SPM) contribution. The probe peak power of*

_{Pump}*m*= 2 and

*p*= 1, as illustrated in Fig. 4(a) , top. The pump peak power is then fixed to satisfy the SSI condition in Eq. (3) with

*s*= 1 and

*m*= 2, i.e.,

*m*= 2. For comparison, the spectral Gaussian envelope (GE) of the input frequency comb is also illustrated (dotted dark green lines) in all the spectral simulation results to show the fidelity of the Gaussian spectral envelope shape of the output signals with respect to the input one.

*s*= 1 and

*m*= 3, such that

*m*ideally yields the same order of reduction in the power spectral density (PSD) amplitude.

*only*, different SSI effects can be induced, namely integer (

*m*= 1) and fractional (

*m*= 2, 4) effects, as shown in Figs. 5(a)–5(e). In particular, in Fig. 5(c) the pump peak power is fixed to satisfy the SSI condition in Eq. (3), with

*s*= 1 and

*m*= 1, such that

*reversed integer*SSI condition).

*m*= 2 and

*m*= 4 are also achieved, as shown in Figs. 5(d) and 5(e), respectively. One can readily notice that as anticipated, the same SSI process for

*m*= 2 has been obtained using different pump signals: one with period of

*p*= 1) and peak power of

*p*= 2) and peak power of

**20**(22), 1869–1871 (2008). [CrossRef]

26. T. Hirooka, M. Nakazawa, and K. Okamoto, “Bright and dark 40 GHz parabolic pulse generation using a picosecond optical pulse train and an arrayed waveguide grating,” Opt. Lett. **33**(10), 1102–1104 (2008). [CrossRef] [PubMed]

*m*= 1, 2 and 4. Figures 5 (f)–5(j) show the induced SSI results which are nearly identical to the corresponding results for bright parabolic pump pulses, depicted in Figs. 5(a)–5(e). It should be, however, noted that a dark pulse requires half of the average power of a bright pulse to produce the same magnitude of chirp, since the average power of bright and dark pulses is given by

26. T. Hirooka, M. Nakazawa, and K. Okamoto, “Bright and dark 40 GHz parabolic pulse generation using a picosecond optical pulse train and an arrayed waveguide grating,” Opt. Lett. **33**(10), 1102–1104 (2008). [CrossRef] [PubMed]

**7**(4), 728–744 (2001). [CrossRef]

*m =*2, 3, 4 and 5. For comparison, the parabolic pulse train is also depicted (dotted red lines) along the time domain to show the difference between the ideal parabolic pulse train and the simulated sinusoidal pump signal. First, we assume a sinusoidal pump signal with a period set to

*m*is increased. In the case of

*m*= 3, Fig. 7(b), only one out of each three consecutive probe pulses receives the correct phase shift and two of them receive a phase shift that deviates ~0.33 rad from the nominal one, significantly affecting the fidelity of the Gaussian envelope shape in the output frequency comb. Figure 7(c) shows that for an SLD factor of

*m*= 4, half of each four consecutive pulses are phase modulated correctly whereas phase deviations of ~0.78 rad are induced on the rest of the pulses. distortions on the output frequency comb envelope are more significant as the FSR division factor

*m*is increased, as observed for instance for the case of

*m*= 5 in Fig. 7(d).

**20**(22), 1869–1871 (2008). [CrossRef]

32. M. Galili, L. K. Oxenløwe, H. C. H. Mulvad, A. T. Clausen, and P. Jeppesen, “Optical wavelength conversion by cross-phase modulation of data signals up to 640 Gb/s,” IEEE J. Sel. Top. Quantum Electron. **14**(3), 573–579 (2008). [CrossRef]

## 4. Conclusions

## Acknowledgments

## References and links

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

2. | J. Azaña, “Spectral Talbot phenomena of frequency combs induced by cross-phase modulation in optical fibers,” Opt. Lett. |

3. | J. Caraquitena, M. Beltrán, R. Llorente, J. Martí, and M. A. Muriel, “Spectral self-imaging effect by time-domain multilevel phase modulation of a periodic pulse train,” Opt. Lett. |

4. | A. Malacarne and J. Azaña, “Discretely tunable comb spacing of a frequency comb by multilevel phase modulation of a periodic pulse train,” Opt. Express |

5. | L. Consolino, G. Giusfredi, P. De Natale, M. Inguscio, and P. Cancio, “Optical frequency comb assisted laser system for multiplex precision spectroscopy,” Opt. Express |

6. | F. Adler, M. J. Thorpe, K. C. Cossel, and J. Ye, “Cavity-enhanced direct frequency comb spectroscopy: Technology and applications,” Annu. Rev. Anal. Chem. |

7. | P. Del'Haye, O. Arcizet, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Frequency comb assisted diode laser spectroscopy for measurement of microcavity dispersion,” Nat. Photonics |

8. | Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature |

9. | S. C. Chan, G. Q. Xia, and J. M. Liu, “Optical generation of a precise microwave frequency comb by harmonic frequency locking,” Opt. Lett. |

10. | S. Tainta, W. Amaya, M. J. Erro, M. J. Garde, S. Sales, and M. A. Muriel, “WDM compatible and electrically tunable SPE-OCDMA system based on the temporal self-imaging effect,” Opt. Lett. |

11. | S. Fukushima, C. F. C. Silva, Y. Muramoto, and A. J. Seeds, “10 to 110 GHz tunable opto-electronic frequency synthesis using optical frequency comb generator and uni-travelling-carrier photodiode,” Electron. Lett. |

12. | T. Healy, F. C. Garcia Gunning, A. D. Ellis, and J. D. Bull, “Multi-wavelength source using low drive-voltage amplitude modulators for optical communications,” Opt. Express |

13. | A. Alatawi, R. P. Gollapalli, and L. Duan, “Radio-frequency clock delivery via free-space frequency comb transmission,” Opt. Lett. |

14. | J. Azaña and S. Gupta, “Complete family of periodic Talbot filters for pulse repetition rate multiplication,” Opt. Express |

15. | C. Finot, J. M. Dudley, B. Kibler, D. J. Richardson, and G. Millot, “Optical parabolic pulse generation and applications,” IEEE J. Sel. Top. Quantum Electron. |

16. | J. P. Limpert, T. Schreiber, T. Clausnitzer, K. Zöllner, H. J. Fuchs, E. B. Kley, H. Zellmer, and A. Tünnermann, “High-power femtosecond Yb-doped fiber amplifier,” Opt. Express |

17. | P. Dupriez, C. Finot, A. Malinowski, J. K. Sahu, J. Nilsson, D. J. Richardson, K. G. Wilcox, H. D. Foreman, and A. C. Tropper, “High-power, high repetition rate picosecond and femtosecond sources based on Yb-doped fiber amplification of VECSELs,” Opt. Express |

18. | F. Parmigiani, C. Finot, K. Mukasa, M. Ibsen, M. A. F. Roelens, P. Petropoulos, and D. J. Richardson, “Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating,” Opt. Express |

19. | T. T. Ng, F. Parmigiani, M. Ibsen, Z. Zhang, P. Petropoulos, and D. J. Richardson, “Compensation of linear distortions by using XPM with parabolic pulses as a time lens,” IEEE Photon. Technol. Lett. |

20. | T. Hirooka and M. Nakazawa, “Optical adaptive equalization of high-speed signals using time-domain optical Fourier transformation,” J. Lightwave Technol. |

21. | T. Hirooka and M. Nakazawa, “All-optical 40-GHz time-domain Fourier transformation using XPM with a dark parabolic pulse,” IEEE Photon. Technol. Lett. |

22. | Y. Ozeki, Y. Takushima, K. Aiso, and K. Kikuchi, “High repetition-rate similariton generation in normal dispersion erbium-doped fiber amplifiers and its application to multi-wavelength light sources,” IEICE Trans. Electron. |

23. | S. Pitois, C. Finot, J. Fatome, B. Sinardet, and G. Millot, “Generation of 20-Ghz picosecond pulse trains in the normal and anomalous dispersion regimes of optical fibers,” Opt. Commun. |

24. | T. Hirooka and M. Nakazawa, “Parabolic pulse generation by use of a dispersion-decreasing fiber with normal group-velocity dispersion,” Opt. Lett. |

25. | F. Parmigiani, P. Petropoulos, M. Ibsen, and D. J. Richardson, “Pulse retiming based on XPM using parabolic pulses formed in a fiber bragg grating,” IEEE Photon. Technol. Lett. |

26. | T. Hirooka, M. Nakazawa, and K. Okamoto, “Bright and dark 40 GHz parabolic pulse generation using a picosecond optical pulse train and an arrayed waveguide grating,” Opt. Lett. |

27. | D. Krcmarík, R. Slavík, Y. Park, and J. Azaña, “Nonlinear pulse compression of picosecond parabolic-like pulses synthesized with a long period fiber grating filter,” Opt. Express |

28. | J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. |

29. | J. Azaña, N. K. Berger, B. Levit, and B. Fischer, “Spectral Fraunhofer regime: time-to-frequency conversion by the action of a single time lens on an optical pulse,” Appl. Opt. |

30. | E. R. Andresen, C. Finot, D. Oron, and H. Rigneault, “Spectral analog of the Gouy phase shift,” Phys. Rev. Lett. |

31. | G. P. Agrawal, |

32. | M. Galili, L. K. Oxenløwe, H. C. H. Mulvad, A. T. Clausen, and P. Jeppesen, “Optical wavelength conversion by cross-phase modulation of data signals up to 640 Gb/s,” IEEE J. Sel. Top. Quantum Electron. |

**OCIS Codes**

(060.5060) Fiber optics and optical communications : Phase modulation

(070.6760) Fourier optics and signal processing : Talbot and self-imaging effects

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 17, 2013

Revised Manuscript: October 29, 2013

Manuscript Accepted: October 30, 2013

Published: November 15, 2013

**Virtual Issues**

Nonlinear Optics (2013) *Optics Express*

**Citation**

Reza Maram and José Azaña, "Spectral self-imaging of time-periodic coherent frequency combs by parabolic cross-phase modulation," Opt. Express **21**, 28824-28835 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-28824

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### References

- 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(4), 728–744 (2001). [CrossRef]
- J. Azaña, “Spectral Talbot phenomena of frequency combs induced by cross-phase modulation in optical fibers,” Opt. Lett.30(3), 227–229 (2005). [CrossRef] [PubMed]
- J. Caraquitena, M. Beltrán, R. Llorente, J. Martí, and M. A. Muriel, “Spectral self-imaging effect by time-domain multilevel phase modulation of a periodic pulse train,” Opt. Lett.36(6), 858–860 (2011). [CrossRef] [PubMed]
- A. Malacarne and J. Azaña, “Discretely tunable comb spacing of a frequency comb by multilevel phase modulation of a periodic pulse train,” Opt. Express21(4), 4139–4144 (2013). [CrossRef] [PubMed]
- L. Consolino, G. Giusfredi, P. De Natale, M. Inguscio, and P. Cancio, “Optical frequency comb assisted laser system for multiplex precision spectroscopy,” Opt. Express19(4), 3155–3162 (2011). [CrossRef] [PubMed]
- F. Adler, M. J. Thorpe, K. C. Cossel, and J. Ye, “Cavity-enhanced direct frequency comb spectroscopy: Technology and applications,” Annu. Rev. Anal. Chem.3(1), 175–205 (2010). [CrossRef] [PubMed]
- P. Del'Haye, O. Arcizet, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Frequency comb assisted diode laser spectroscopy for measurement of microcavity dispersion,” Nat. Photonics3(9), 529–533 (2009). [CrossRef]
- Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature416(6877), 233–237 (2002). [CrossRef] [PubMed]
- S. C. Chan, G. Q. Xia, and J. M. Liu, “Optical generation of a precise microwave frequency comb by harmonic frequency locking,” Opt. Lett.32(13), 1917–1919 (2007). [CrossRef] [PubMed]
- S. Tainta, W. Amaya, M. J. Erro, M. J. Garde, S. Sales, and M. A. Muriel, “WDM compatible and electrically tunable SPE-OCDMA system based on the temporal self-imaging effect,” Opt. Lett.36(3), 400–402 (2011). [CrossRef] [PubMed]
- S. Fukushima, C. F. C. Silva, Y. Muramoto, and A. J. Seeds, “10 to 110 GHz tunable opto-electronic frequency synthesis using optical frequency comb generator and uni-travelling-carrier photodiode,” Electron. Lett.37(12), 780–781 (2001). [CrossRef]
- T. Healy, F. C. Garcia Gunning, A. D. Ellis, and J. D. Bull, “Multi-wavelength source using low drive-voltage amplitude modulators for optical communications,” Opt. Express15(6), 2981–2986 (2007). [CrossRef] [PubMed]
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