## Unified approach to describe optical pulse generation by propagation of periodically phase-modulated CW laser light

Optics Express, Vol. 14, Issue 8, pp. 3171-3180 (2006)

http://dx.doi.org/10.1364/OE.14.003171

Acrobat PDF (340 KB)

### Abstract

The analysis of optical pulse generation by phase modulation of narrowband continuous-wave light, and subsequent propagation through a group-delay-dispersion circuit, is usually performed in terms of the so-called bunching parameter. This heuristic approach does not provide theoretical support for the electrooptic flat-top-pulse generation reported recently. Here, we perform a waveform synthesis in terms of the Fresnel images of the periodically phase-modulated input light. In particular, we demonstrate flat-top-pulse generation with a duty ratio of 50% at a quarter of the Talbot condition for the sinusoidal phase modulation. Finally, we propose a binary modulation format to generate a well-defined square-wave-type optical bit pattern.

© 2006 Optical Society of America

## 1. Introduction

1. W. H. Knox, “Ultrafast technology in telecommunications,” IEEE J. Sel. Top. Quantum Electron. **6**, 1273–1278 (2000). [CrossRef]

2. M. Suzuki, H. Tanaka, and Y. Matsushima, “InGaAsP electroabsorption modulator for high-bit-rate EDFA systems,” IEEE Photon. Technol. Lett. **4**, 586–588 (1992). [CrossRef]

5. J. E. Bjorkholm, E. H. Turner, and D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor,” Appl. Phys. Lett. **26**, 564–566 (1975). [CrossRef]

9. H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, “Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators,” IEEE J. Sel. Top. Quantum Electron. **6**, 1325–1331 (2000). [CrossRef]

10. K. Sato, “Optical pulse generation using Fabry-Pérot lasers under continuous-wave operation,” IEEE J. Sel. Top. Quantum Electron. **9**, 1288–1293 (2003). [CrossRef]

11. S. E. Harris and O. P. McDuff, “Theory of FM laser oscillation,” IEEE J. Quantum Electron. **QE-1**, 245–262 (1965). [CrossRef]

15. T. Khayim, M. Yamamuchi, D. Kim, and T. Kobayashi, “Femtosecond optical pulse generation from a CW laser using an electrooptic phase modulator featuring lens modulation,” IEEE J. Quantum Electron. **35**, 1412–1418 (1999). [CrossRef]

*ps*temporal duration with a duty ratio of 11% by means of a LCFG and an electrooptic phase modulator (EOPM) [16

16. T. Komukai, T. Yamamoto, and S. Kawanishi, “Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings,” IEEE Photon. Technol. Lett. **17**, 1746–1748 (2005). [CrossRef]

*B*, defined essentially as the product between the frequency chirping rate and the GDD coefficient, provides a rough estimation for the optimum bunching of the frequency components. The case of

*B*= 1 gives the condition under which the CW light is optimally compressed. This method shows a low pulse extinction ratio. In fact, the optical frequency of the sinusoidally phase-modulated light is assumed to be linearly chirped within half a period. Nonlinear chirped frequency components yield other substructures or broad wings, so that a considerable part of the energy lies outside the main pulse. On the other hand, note that blue-chirping and red-chirping regions are repeated in every modulation period. As a result, both the normal GDD and the anomalous one are effective for this method. The normal dispersion corresponds to compression of red-chirped portions of the input field, whereas blue-chirped portions are compressed by an anomalous dispersion circuit. Therefore, approximately half of the energy in the input field does not contribute to the bunching and generates an undesirable dc floor level. Some attempts have been done in the past few years for highly extinctive electrooptic pulse pattern generation [17

17. T. Otsuji, M. Yaita, T. Nagatsuma, and E. Sano, “10-80-Gb/s highly extinctive electrooptic pulse pattern generation,” IEEE J. Sel. Top. Quantum Electron. **2**, 643–649 (1996). [CrossRef]

16. T. Komukai, T. Yamamoto, and S. Kawanishi, “Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings,” IEEE Photon. Technol. Lett. **17**, 1746–1748 (2005). [CrossRef]

18. N. K. Berger, B. Levit, A. Bekker, and B. Fischer, “Compression of periodic optical pulses using temporal fractional Talbot effect,” IEEE Photon. Technol. Lett. **16**, 1855–1857 (2004). [CrossRef]

*f*But the output intensity is also periodic with the GDD coefficient Φ

_{2}. The period is given by the so-called temporal Talbot dispersion relationship, Φ

_{2T}= 1/

*πf*

^{2}[20–22

20. J. Azaña and M. A. Muriel, “Technique for multiplying the repetition rates of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings,” Opt. Lett. **24**, 1672–1674 (1999). [CrossRef]

23. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. **30**, 1951–1963 (1994). [CrossRef]

24. J. M. Cowley and A. F. Moodie, “Fourier images. IV: the phase grating,” Proc. Phys. Soc. London Sec. B **76**, 378–384 (1960). [CrossRef]

*π*/4 providing the sought theoretical support of the experimental results reported in [16

16. T. Komukai, T. Yamamoto, and S. Kawanishi, “Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings,” IEEE Photon. Technol. Lett. **17**, 1746–1748 (2005). [CrossRef]

18. N. K. Berger, B. Levit, A. Bekker, and B. Fischer, “Compression of periodic optical pulses using temporal fractional Talbot effect,” IEEE Photon. Technol. Lett. **16**, 1855–1857 (2004). [CrossRef]

## 2. Theoretical analysis

*E*is the constant amplitude,

_{o}*ω*denotes the carrier optical frequency, and

_{o}*V*(

*t*) is the phase modulation function. For our purposes, we assume that

*V*(

*t*) is a periodic function with period

*T*. Note that the perfect sinusoid is enclosed as a particular case. As a result of the periodicity of the phase

*V*(

*t*), we can rewrite Eq. (1) in terms of a Fourier series expansion, namely,

*I*(

_{m}*t*)= ∣

*E*∣

_{o}^{2}. This implies that

*C*= δ

_{N}_{N,0}, where δ

_{N,0}denotes the Kronecker delta function.

_{1}and Φ

_{2}denoting the group delay and the GDD coefficient, respectively. Note that we assume no losses neither in the coupling of the input into the dispersive circuit or in the propagation. If we consider that the GDD circuit is implemented using a SMF, Φ

_{1}=

*β*

_{1}

*z*and Φ

_{2}=

*β*

_{2}

*z*, with

*z*the propagation distance. The parameters

*β*

_{1}and

*β*

_{2}are the inverse of the group velocity and the group velocity dispersion (GVD) parameter of the fiber, respectively.

*β*

_{2}/

*β*

_{3}∣ , with

*β*

_{3}the TOD parameter of the fiber, and the power carried by individual pulses is not enough to excite nonlinear mechanisms in the fiber [27

27. G. P. Agrawal, *Fiber-Optic Communication Systems*, 3rd edition, Wiley Interscience, New York2002. [CrossRef]

*τ*=

*t*-Φ

_{1}. From Eq. (6), the output intensity can be written as

*I*(

_{out}*τ*,Φ

_{2}) is a periodic function of

*τ*. Its period is, in principle, equal to the modulation period

*T*. Second, from Eqs. (7) and (8) it is clear that the output optical intensity changes periodically with the dispersion coefficient Φ

_{2}. The period is just the Talbot dispersion, Φ

_{2T}= 1/

*πf*

^{2}. From Eq. (8) we note that

*C'*(Φ

_{N}_{2}+ Φ

_{2T}/2) = exp(

*jπN*

^{2})

*C'*(Φ

_{N}_{2}). In this way, we obtain

*I*(τ,Φ

_{out}_{2}) =

*I*(τ +

_{out}*T*/2,Φ

_{2}+Φ

_{2T}/2). This means that a change in the dispersion by Φ

_{2T}/2 is equivalent to a temporal shift of half a period at the output intensity. We explore further implications of the above facts.

*V*(

*t*) = Δ

*θ*sin(2

*πf*

*t*). Here, Δ

*θ*is the modulation index in radians. Of course

*f*= 1/

*T*. For this case, the Fourier coefficients are expressed by the Bessel functions of the first kind,

*c*=

_{n}*J*(Δ

_{n}*θ*). Therefore,

_{2}and

*f*are set to Φ

_{2}=Φ

_{2T}/16 and

*40*GHz, respectively. Different new pulse waveforms not yet reported are obtained by changing Δ

*θ*. In particular, we mention short pulse generation for Δ

*θ*=

*π*/4 . Here, a duty cycle (DC) of 33% is achieved. Note that in this case the signal is free of annoying wings and tails but a high dc-floor level is present, as shown in Fig. 2(a). In Fig. 2(b) the numerical simulation shows a short pulse with a DC of approximately 18%. Although part of the energy lies outside the main pulse, the remaining dc-floor level is low. Note that, aside for a temporal shift of half a period, the same profiles are achieved when the dispersion is set to Φ

_{2}=Φ

_{2T}(8

*q*+ 1)16 where

*q*is an arbitrary integer. We also claim that the above shapes can be achieved with normal GDD as well as with anomalous one.

## 3. Flat-top-pulse generation

_{2}=Φ

_{2T}/4 we obtain (see Appendix)

25. V. Arrizón and J. Ojeda-Castañeda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A **9**, 1801–1806 (1992). [CrossRef]

26. J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta **18**, 677–682 (1971). [CrossRef]

*V*(

*τ*). If we consider the sinusoidal modulation

*V*(

*τ*) = Δ

*θ*sin(2/

*πf*

*τ*), then

*θ*=

*π*/4, the argument within the exterior trigonometric function ranges from -

*π*/2 to

*π*/2 for τ ∈ [-

*T*/2,

*T*/2]. The analytical curve shown in Eq. (11) is plotted in Fig. 3 for Δ

*θ*=

*π*/4 and an input frequency of

*40*GHz. The temporal width of the individual pulses is

*12.5*ps. In this way, a nearly flat-top-pulse with a DC of 50% is achieved. Equation (11

11. S. E. Harris and O. P. McDuff, “Theory of FM laser oscillation,” IEEE J. Quantum Electron. **QE-1**, 245–262 (1965). [CrossRef]

15. T. Khayim, M. Yamamuchi, D. Kim, and T. Kobayashi, “Femtosecond optical pulse generation from a CW laser using an electrooptic phase modulator featuring lens modulation,” IEEE J. Quantum Electron. **35**, 1412–1418 (1999). [CrossRef]

17. T. Otsuji, M. Yaita, T. Nagatsuma, and E. Sano, “10-80-Gb/s highly extinctive electrooptic pulse pattern generation,” IEEE J. Sel. Top. Quantum Electron. **2**, 643–649 (1996). [CrossRef]

_{2}= Φ

_{2T}(2

*q*+1)/4, with

*q*an arbitrary integer. The existence of multiple GDD amounts was pointed out in [16

**17**, 1746–1748 (2005). [CrossRef]

*T*being the period. The modulation

*V*(τ) is plotted in Fig. 4(a). For this case, the argument inside the trigonometric function in Eq. (10) has two values, -

*π*/2 and

*π*/2, respectively. Consequently, the output intensity shows a binary shape at Φ

_{2}= Φ

_{2T}/4 , namely

*I*(τ,Φ

_{out}_{2}) for the phase modulation in Eq. (12) and Φ

_{2}ranging the whole first Talbot period. As expected, for dispersions Φ

_{2}= 0, Φ

_{2}= Φ

_{2T}/2, and Φ

_{2}= Φ

_{2T}, the irradiance presents a constant value. Whereas for Φ

_{2}=Φ

_{2T}/4, according to Eq. (13) an ultra-flat-top optical pulse train is obtained, see Fig. 4(c). This kind of pulse could be employed for RZ modulation formats in optical signal transmission and, in particular, for differential phase-shift-keyed transmission.

## 4. GDD circuit analysis

### 4.1 Standard SMF

*ω*, should be limited to Δ

*ω*< 3∣

*β*

_{2}/

*β*

_{3}∣. For the case of perfect sinusoidal modulation, we have

*c*=

_{n}*J*(Δ

_{n}*θ*) . To obtain a rough estimation for the optical bandwidth of the phase-modulated signal, we plot in Fig. 5

*J*(Δ

_{n}*θ*) versus the modulation index Δ

*θ*. Four different values of the order

*n*have been considered. The modulation index ranges within the interval

*0*< Δ

*θ*< 10 . From this plot we can assume that the main contribution to the output intensity comes from the Bessel functions with an order lower than 10. Thus, the condition for the validity of the parabolic approximation reads 20

*f*< 3∣

*β*

_{2}/

*β*

_{3}∣. If we have an optical source peaked at the 1.55

*μm*window, for a standard SMF we obtain

*β*

_{2}=-2.168×10

^{-2}

*ps*

^{2}/

*m*and

*β*

_{3}= 1.2661×10

^{-4}

*ps*

^{3}/

*m*. In this way, the value of the term (3/20)∣

*β*

_{2}/

*β*

_{3}∣ is approximately 25

*THz*. So, the above inequality is widely satisfied even for the fastest commercially available electrooptical modulator, which works in the

*GHz*range.

### 4.2 LCFG

## 5. Conclusions

## Acknowledgments

## References and links

1. | W. H. Knox, “Ultrafast technology in telecommunications,” IEEE J. Sel. Top. Quantum Electron. |

2. | M. Suzuki, H. Tanaka, and Y. Matsushima, “InGaAsP electroabsorption modulator for high-bit-rate EDFA systems,” IEEE Photon. Technol. Lett. |

3. | K. Wakita, K. Sato, I. Kotaka, M. Yamamoto, and M. Asobe, “Transform-limited 7-ps optical pulse generation using a sinusoidally driven InGaAsP/InGaAsP strained multiple-quantum-well DFB laser/modulator monolithically integrated light source,” IEEE Photon. Technol. Lett. |

4. | V. Kaman, S. Z. Zhang, A. J. Keating, and J. E. Bowers, “High-speed operation of travelling-wave electroabsorption modulator,” Electron. Lett. |

5. | J. E. Bjorkholm, E. H. Turner, and D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor,” Appl. Phys. Lett. |

6. | T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, “Optical pulse compression using high-frequency electrooptic phase modulation,” IEEE J. Quantum Electron. |

7. | E. A. Golovchenko, C. R. Menyuk, G. M. Carter, and P. V. Mamyshev, “Analysis of optical pulse train generation through filtering of an externally phase-modulated signal from a CW laser,” Electron. Lett. |

8. | D. Kim, M. Arisawa, A. Morimoto, and T. Kobayashi, “Femtosecond optical pulse generation using quasi-velocity-matched electrooptic phase modulator,” IEEE J. Sel. Top. Quantum Electron. |

9. | H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, “Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators,” IEEE J. Sel. Top. Quantum Electron. |

10. | K. Sato, “Optical pulse generation using Fabry-Pérot lasers under continuous-wave operation,” IEEE J. Sel. Top. Quantum Electron. |

11. | S. E. Harris and O. P. McDuff, “Theory of FM laser oscillation,” IEEE J. Quantum Electron. |

12. | L. F. Tiemeijer, P. I. Kuindersma, P. J. A. Thijs, and G. L. J. Rikken, “Passive FM locking in InGaAsP semiconductor lasers,” IEEE J. Quantum Electron. |

13. | K. A. Shore and W. M. Yee, “Theory of self-locking FM operation in semiconductor lasers,” IEE Proceedings-J. |

14. | W. M. Yee and K. A. Shore, “Multimode analysis of self locked FM operation in laser diodes,” IEE Proceedings-J. |

15. | T. Khayim, M. Yamamuchi, D. Kim, and T. Kobayashi, “Femtosecond optical pulse generation from a CW laser using an electrooptic phase modulator featuring lens modulation,” IEEE J. Quantum Electron. |

16. | T. Komukai, T. Yamamoto, and S. Kawanishi, “Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings,” IEEE Photon. Technol. Lett. |

17. | T. Otsuji, M. Yaita, T. Nagatsuma, and E. Sano, “10-80-Gb/s highly extinctive electrooptic pulse pattern generation,” IEEE J. Sel. Top. Quantum Electron. |

18. | N. K. Berger, B. Levit, A. Bekker, and B. Fischer, “Compression of periodic optical pulses using temporal fractional Talbot effect,” IEEE Photon. Technol. Lett. |

19. | A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, and C. Dorrer, “Spectrally efficient (0.8 b/s/Hz) 1-Tb/s (25x42.7 Gb/s) RZ-DQPSK transmission over 28 10-km spans with 7 optical add/drops,” in ECOC 2004 Proc., 2004, Postdeadline paper Th4.4.1, pp. 40–41. |

20. | J. Azaña and M. A. Muriel, “Technique for multiplying the repetition rates of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings,” Opt. Lett. |

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

22. | J. Lancis, J. Caraquitena, P. Andrés, and M. A. Muriel, “Temporal self-imaging effect for chirped laser pulse sequences: repetition rate and duty cycle tunability,” Opt. Commun. |

23. | B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. |

24. | J. M. Cowley and A. F. Moodie, “Fourier images. IV: the phase grating,” Proc. Phys. Soc. London Sec. B |

25. | V. Arrizón and J. Ojeda-Castañeda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A |

26. | J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta |

27. | G. P. Agrawal, |

28. | F. Ouellette, J. F. Cliche, and S. Gagnon, “All-fiber devices for chromatic dispersion compensation based on chirped distributed resonant coupling,” J. Lightwave Technol. |

**OCIS Codes**

(060.5060) Fiber optics and optical communications : Phase modulation

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

(320.1590) Ultrafast optics : Chirping

(320.5390) Ultrafast optics : Picosecond phenomena

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: February 14, 2006

Revised Manuscript: April 11, 2006

Manuscript Accepted: April 11, 2006

Published: April 17, 2006

**Citation**

Víctor Torres-Company, Jesús Lancis, and Pedro Andrés, "Unified approach to describe optical pulse generation by propagation of periodically phase-modulated CW laser light," Opt. Express **14**, 3171-3180 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3171

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### References

- W. H. Knox, "Ultrafast technology in telecommunications," IEEE J. Sel. Top. Quantum Electron. 6, 1273-1278 (2000). [CrossRef]
- M. Suzuki, H. Tanaka, and Y. Matsushima, "InGaAsP electroabsorption modulator for high-bit-rate EDFA systems," IEEE Photon. Technol. Lett. 4, 586-588 (1992). [CrossRef]
- K. Wakita, K. Sato, I. Kotaka. M. Yamamoto, and M. Asobe, "Transform-limited 7-ps optical pulse generation using a sinusoidally driven InGaAsP/InGaAsP strained multiple-quantum-well DFB laser/modulator monolithically integrated light source," IEEE Photon. Technol. Lett. 5, 899-901 (1993). [CrossRef]
- V. Kaman, S. Z. Zhang, A. J., Keating, and J. E. Bowers, "High-speed operation of travelling-wave electroabsorption modulator," Electron. Lett. 35, 993-995 (1999). [CrossRef]
- J. E. Bjorkholm, E. H. Turner, and D. B. Pearson, "Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor," Appl. Phys. Lett. 26, 564-566 (1975). [CrossRef]
- T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, "Optical pulse compression using high-frequency electrooptic phase modulation," IEEE J. Quantum Electron. 24, 382-387 (1988). [CrossRef]
- E. A. Golovchenko, C. R. Menyuk, G. M. Carter, and P. V. Mamyshev, "Analysis of optical pulse train generation through filtering of an externally phase-modulated signal from a CW laser," Electron. Lett. 31, 2198-2199 (1995). [CrossRef]
- D. Kim, M. Arisawa, A. Morimoto, and T. Kobayashi, "Femtosecond optical pulse generation using quasi-velocity-matched electrooptic phase modulator," IEEE J. Sel. Top. Quantum Electron. 2, 493-499 (1996). [CrossRef]
- H. Murata, A. Morimoto, T. Kobayashi, and S. Yamamoto, "Optical pulse generation by electrooptic-modulation method and its application to integrated ultrashort pulse generators," IEEE J. Sel. Top. Quantum Electron. 6, 1325-1331 (2000). [CrossRef]
- K. Sato, "Optical pulse generation using Fabry-Pérot lasers under continuous-wave operation," IEEE J. Sel. Top. Quantum Electron. 9, 1288-1293 (2003). [CrossRef]
- S. E. Harris and O. P. McDuff, "Theory of FM laser oscillation," IEEE J. Quantum Electron. QE-1, 245-262 (1965). [CrossRef]
- L. F. Tiemeijer, P. I. Kuindersma, P. J. A. Thijs, and G. L. J. Rikken, "Passive FM locking in InGaAsP semiconductor lasers," IEEE J. Quantum Electron. 25, 1385-1392 (1989). [CrossRef]
- K. A. Shore and W. M. Yee, "Theory of self-locking FM operation in semiconductor lasers," IEE Proceedings-J. 138, 91-96 (1991).
- W. M. Yee and K. A. Shore, "Multimode analysis of self locked FM operation in laser diodes," IEE Proceedings-J. 140, 21-25 (1993).
- T. Khayim, M. Yamamuchi, D. Kim, T. Kobayashi, "Femtosecond optical pulse generation from a CW laser using an electrooptic phase modulator featuring lens modulation," IEEE J. Quantum Electron. 35, 1412-1418 (1999). [CrossRef]
- T. Komukai, T. Yamamoto, and S. Kawanishi, "Optical pulse generator using phase modulator and linearly chirped fiber Bragg gratings," IEEE Photon. Technol. Lett. 17, 1746-1748 (2005). [CrossRef]
- T. Otsuji, M. Yaita, T. Nagatsuma, and E. Sano, "10-80-Gb/s highly extinctive electrooptic pulse pattern generation," IEEE J. Sel. Top. Quantum Electron. 2, 643-649 (1996). [CrossRef]
- N. K. Berger, B. Levit, A. Bekker, and B. Fischer, "Compression of periodic optical pulses using temporal fractional Talbot effect," IEEE Photon. Technol. Lett. 16, 1855-1857 (2004). [CrossRef]
- A. H. Gnauck, P. J. Winzer, S. Chandrasekhar, and C. Dorrer, "Spectrally efficient (0.8 b/s/Hz) 1-Tb/s (25x42.7 Gb/s) RZ-DQPSK transmission over 28 10-km spans with 7 optical add/drops," in ECOC 2004 Proc., 2004, Postdeadline paper Th4.4.1, pp. 40-41.
- J. Azaña and M. A. Muriel, "Technique for multiplying the repetition rates of periodic trains of pulses by means of a temporal self-imaging effect in chirped fiber gratings," Opt. Lett. 24, 1672-1674 (1999). [CrossRef]
- 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]
- J. Lancis, J. Caraquitena, P. Andrés, and M. A. Muriel, "Temporal self-imaging effect for chirped laser pulse sequences: repetition rate and duty cycle tunability," Opt. Commun. 253, 156-163 (2005). [CrossRef]
- B. H. Kolner, "Space-time duality and the theory of temporal imaging," IEEE J. Quantum Electron. 30, 1951-1963 (1994). [CrossRef]
- J. M. Cowley and A. F. Moodie, "Fourier images. IV: the phase grating," Proc. Phys. Soc. London Sec. B 76, 378-384 (1960). [CrossRef]
- V. Arrizón and J. Ojeda-Castañeda, "Irradiance at Fresnel planes of a phase grating," J. Opt. Soc. Am. A 9, 1801-1806 (1992). [CrossRef]
- J. P. Guigay, "On Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects," Opt. Acta 18, 677-682 (1971). [CrossRef]
- G. P. Agrawal, Fiber-Optic Communication Systems, 3rd edition, Wiley Interscience, New York 2002. [CrossRef]
- F. Ouellette, J. F. Cliche, and S. Gagnon, "All-fiber devices for chromatic dispersion compensation based on chirped distributed resonant coupling," J. Lightwave Technol. 12, 1728-1738 (1994). [CrossRef]

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