## Effect of dispersion fluctuations on widely tunable optical parametric amplification in photonic crystal fibers

Optics Express, Vol. 14, Issue 20, pp. 9491-9501 (2006)

http://dx.doi.org/10.1364/OE.14.009491

Acrobat PDF (133 KB)

### Abstract

The effect of dispersion fluctuations on the conversion efficiency of large frequency shift parametric sidebands is studied by numerical simulation and experiment. Numerical results based on periodic and random dispersion models are used to fit the experimental results. The fitting parameters provide a measure of the uniformity of the photonic crystal fiber used in the experiment. This allows us to place limits on the required uniformity of a photonic crystal fiber for strong frequency conversion.

© 2006 Optical Society of America

## 1. Introduction

1. C. Lin, W. A. Reed, A. D. Pearson, and H. T. Sbang, “Phase matching in the minimum-chromatic-dispersion region of single-mode fibers for stimulated four-photon mixing,” Opt. Lett. **6**, 493–5 (1981). [CrossRef] [PubMed]

2. J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. **28**, 2225–7 (2003). [CrossRef] [PubMed]

3. A. Y. H. Chen, G. K. L. Wong, S. G. Murdoch, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Widely tunable optical parametric generation in a photonic crystal fiber,” Opt. Lett. **30**, 762–764 (2005). [CrossRef] [PubMed]

4. S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. **226**, 415–22 (2003). [CrossRef]

5. M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Wideband tuning of the gain spectra of one-pump fiber optical parametric amplifiers,” J. Sel. Top. Quantum. Electron. **10**, 1133–1141 (2004). [CrossRef]

6. G. K. L. Wong, A. Y. H. Chen, S. G. Murdoch, R. Leonhardt, J. D. Harvey, N. Y. Joly, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Continuous-wave tunable optical parametric generation in a photonic-crystal fiber,” J. Opt. Soc. Am. B **22**, 2505–2511 (2005). [CrossRef]

7. N. J. Smith and N. J. Doran, “Modulational instabilities in fibers with periodic dispersion management,” Opt. Lett. **21**, 570–2 (1996). [CrossRef] [PubMed]

15. B. Kibler, C. Billet, J. M. Dudley, R. S. Windeler, and G. Millot, “Effects of structural irregularities on modulational instability phase matching in photonic crystal fibers,” Opt. Lett. **29**, 1903–1905 (2004). [CrossRef] [PubMed]

7. N. J. Smith and N. J. Doran, “Modulational instabilities in fibers with periodic dispersion management,” Opt. Lett. **21**, 570–2 (1996). [CrossRef] [PubMed]

12. F. K. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A **220**, 213–218 (1996). [CrossRef]

12. F. K. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A **220**, 213–218 (1996). [CrossRef]

15. B. Kibler, C. Billet, J. M. Dudley, R. S. Windeler, and G. Millot, “Effects of structural irregularities on modulational instability phase matching in photonic crystal fibers,” Opt. Lett. **29**, 1903–1905 (2004). [CrossRef] [PubMed]

## 2. Parametric gain in fibers with fluctuating dispersion

*n*~10

^{-4}). As a result when light is

*A*

*) and the two sidebands (amplitudes*

_{p}*A*

*and*

_{a}*A*

*) symmetrically detuned by a frequency Ω from the pump as [16*

_{s}16. G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B **8**, 824–838 (1991). [CrossRef]

*γ*is the nonlinear interaction coefficient and α is the fiber attenuation coefficient. For simplicity we assume the same attenuation coefficient for each of the three waves.

*ϕ*is the phase of the parametric process given by

*ϕ*

_{a,s,p}are the phases of the anti-Stokes, Stokes and pump waves, respectively. Δ

*β*

*is the linear wavevector mismatch and varies along the length of the fiber as the dispersion varies,*

_{L}*β*, around the pump frequency allows us to rewrite Eq. (6) as a sum of the even powers of dispersion,

*β*

_{2}and

*β*

_{4}) are sufficient to accurately predict the sideband’s frequency shift [4

4. S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. **226**, 415–22 (2003). [CrossRef]

5. M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Wideband tuning of the gain spectra of one-pump fiber optical parametric amplifiers,” J. Sel. Top. Quantum. Electron. **10**, 1133–1141 (2004). [CrossRef]

*β*directly from a model (discussed in the next section) and then calculating the linear wavevector mismatch using Eq. (6).

*A*is the sideband amplitude and

*P*is the pump power at the beginning of the fiber. Equation (8) can be directly integrated to obtain the sideband’s small signal intensity gain per unit length for a fiber of length

*L*,

*ϕ*which for an arbitrary dispersion has no analytic solution. Nonetheless Eqs. (9) and (10) give us a simple method to numerically calculate the parametric gain for an optical fiber with an arbitrary dispersion fluctuation. From Eq. (10) it is then simple to calculate the relative small signal conversion efficiency of a sideband with frequency shift Ω compared to a zero frequency shift sideband as

*β*(Ω,

_{L}*z*)| is much larger than

*γP*. Under these assumptions the evolution of the parametric phase becomes linear,

*β*=0, with an initial phase of

_{L}*φ*(0)=

*π*/2. Under these conditions, the phase does not evolve and the parametric sidebands experience a constant small signal gain of 2

*γP*. For a fiber with fluctuating dispersion, the maximum sideband gain occurs at the frequency shift where the integral of Δ

*β*is zero [9

_{L}9. S. G. Murdoch, R. Leonhardt, J. D. Harvey, and T. A. B. Kennedy, “Quasi-phase matching in an optical fiber with periodic birefringence,” J. Opt. Soc. Am. B **14**, 1816–1822 (1997). [CrossRef]

*β*(Ω,

_{L}*z*)|, and hence a decrease in parametric gain. This can be seen by considering the first term of the expansion in Eq. (7). Equations (12) and (13) also allow us to identify a figure of merit for the parametric gain for a fiber with a periodic dispersion fluctuation, namely the product of the fluctuation amplitude and the fluctuation length. This confirms the previously identified effect that parametric gain is more sensitive to dispersion fluctuations with long rather than short fluctuation periods [14

14. M. Farahmand and M. de Sterke, “Parametric amplification in presence of dispersion fluctuations,” Opt. Express **12**, 136–142 (2004). [CrossRef] [PubMed]

## 3. Results

17. K. L. Reichenbach and C. Xu, “The effects of randomly occurring nonuniformities on propagation in photonic crystal fibers,” Opt. Express **13**, 2799–2807 (2005). [CrossRef] [PubMed]

19. A. Mussot, E. Lantz, A. Durecu-Legrand, C. Simonneau, D. Bayart, T. Sylvestre, and H. Maillotte, “Zero-dispersion wavelength mapping in short single-mode optical fibers using parametric amplification,” IEEE Photon. Technol. Lett. **18**, 22–4 (2006). [CrossRef]

20. G. K. L. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Y. Joly, “Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability,” Opt. Express **13**, 8662–8670 (2005). [CrossRef] [PubMed]

20. G. K. L. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Y. Joly, “Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability,” Opt. Express **13**, 8662–8670 (2005). [CrossRef] [PubMed]

^{-1}km

^{-1}at 650 nm. The dispersion of the PCF’s high group-index mode has been calculated from an analysis of the sideband frequency shifts and is shown in the inset to Fig. 1 [20

20. G. K. L. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Y. Joly, “Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability,” Opt. Express **13**, 8662–8670 (2005). [CrossRef] [PubMed]

*µ*m, and the large index step (this PCF has a cladding air filling fraction of 88%) produce a large waveguide dispersion which shifts the ZDW to 671 nm. We find that this dispersion is well modeled by that of a step-index fiber with a core diameter set to the effective core diameter of the PCF, and a cladding index set to the mean refractive index of the PCF’s cladding [20

**13**, 8662–8670 (2005). [CrossRef] [PubMed]

6. G. K. L. Wong, A. Y. H. Chen, S. G. Murdoch, R. Leonhardt, J. D. Harvey, N. Y. Joly, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Continuous-wave tunable optical parametric generation in a photonic-crystal fiber,” J. Opt. Soc. Am. B **22**, 2505–2511 (2005). [CrossRef]

6. G. K. L. Wong, A. Y. H. Chen, S. G. Murdoch, R. Leonhardt, J. D. Harvey, N. Y. Joly, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Continuous-wave tunable optical parametric generation in a photonic-crystal fiber,” J. Opt. Soc. Am. B **22**, 2505–2511 (2005). [CrossRef]

## 3.1. Periodic dispersion fluctuations

*dL*and percentage amplitude fluctuation±

*da*. For the PCF discussed above, a ±1% fluctuation in the core diameter corresponds to a ±0.7 ps

^{2}/km fluctuation in the dispersion (±2 nm in the zero dispersion wavelength). In Fig. 3 we plot the experimentally observed relative conversion efficiency of the anti-Stokes (frequency up-shifted) and Stokes (frequency down-shifted) sidebands as a function of frequency shift. The data obtained with the pump polarized parallel to the high group-index mode, and those obtained with the pump polarized parallel to the low group-index mode, show the same characteristic fall-off in conversion efficiency as the sideband frequency shift increases. This suggests that the core diameter fluctuations of the two axes are of the same order of magnitude, as might be expected. Fig. 3 also shows the effect of stimulated Raman scattering (SRS) on the sidebands. Below 20 THz SRS is responsible for an amplification of the Stokes sideband and an attenuation of the anti-Stokes sideband. Above 20 THz the Raman gain drops rapidly and the effect of SRS is negligible. This means our simple theory presented in Section 2 should be valid for sideband frequency shifts above 20 THz.

*β*(Ω

_{L}_{pm},

*z*)|, is indeed much larger than the nonlinear term,

*γP*, so the effect of the periodic fluctuations is well described by the proposed figure of merit, i.e. the product of the fluctuation magnitude and the fluctuation period. The solid line superimposed on Fig. 3 is the calculated relative conversion efficiency for a stepwise periodic fluctuation with

*da*·

*dL*=±0.022%·m. The agreement between the modeled and measured conversion efficiencies is very good, except at low frequencies where the effects of Raman gain are visible on the experimentally measured points. For the product

*da*·

*dL*=±0.022%·m, we find the numerically calculated relative conversion efficiency changes negligibly as the fluctuation period

*dL*is varied from 1 cm to 5 m. This confirms the validity of our figure of merit. For periods longer than 5 m, the partial periods in the 43 m fiber start to have a noticeable effect on the calculated conversion efficiency. For periods shorter than 1 cm, the magnitude of the core diameter fluctuation becomes so large that the relationship between the core diameter fluctuation and the resultant dispersion fluctuation is no longer linear. To demonstrate the sensitivity of the conversion efficiency to the product

*da*·

*dL*, we superimpose on Fig. 3 the calculated relative conversion efficiencies obtained with

*da*·

*dL*=±0.011%·m and

*da*·

*dL*=±0.044%·m. These superimposed traces show that the fall off in parametric gain as a function of frequency shift is a strong function of the product

*da*·

*dL*.

_{3dB}, the sideband frequency shift at which the relative conversion efficiency falls 3 dB below that of the zero frequency shift sideband, as a function of

*da*·

*dL*. For the PCF used in this paper the fitted fluctuation product is

*da*·

*dL*=±0.022%·mwhich gives an Ω

_{3dB}bandwidth of approximately 47 THz. Figure 4 shows the Ω

_{3dB}bandwidth decreases as the magnitude of the dispersion fluctuations increases, and allows us to place an upper bound on the magnitude of the dispersion fluctuations that can be tolerated for a desired Ω

_{3dB}bandwidth.

## 3.2. Random dispersion fluctuations

14. M. Farahmand and M. de Sterke, “Parametric amplification in presence of dispersion fluctuations,” Opt. Express **12**, 136–142 (2004). [CrossRef] [PubMed]

*α*

_{0}, standard deviation σ

*, and a length given by*

_{a}*U*is a uniformly distributed random variable with a range from 0 to 1, and

*L*

*is the correlation length, the typical length scale of the fluctuations. This gives the autocorrelation function of the core diameter fluctuations the physically reasonable form of*

_{c}*L*

*=1 m.*

_{c}*L*or σ

_{c}_{a}results in decreased conversion efficiency for the large frequency shift sidebands. While Fig. 6 shows the best fit for a correlation length of 1 m, it is possible to adjust σ

*to obtain an equally good fit to the data for other correlation lengths. In Fig. 7 we plot this ‘best fit’ curve in the (*

_{a}*L*,σ

_{c}*) parameter space. It shows that there is a clear relationship between the correlation length and the standard deviation of the core diameter fluctuation, with an increase in*

_{a}*L*requiring a corresponding decrease in σ

_{c}*to maintain a good fit to the experimentally measured conversion efficiency versus frequency shift curve.*

_{a}## 4. Conclusion

*da*·

*dL*=±0.022%·mfor the periodic model, and σ

*~0.001-0.005% of the mean core diameter for the random model. As the largest fluctuation period of the random model is in the order of the fiber length, one can see that the fluctuation amplitudes predicted by the two models are of the same order of magnitude, and conclude that the models presented are indeed consistent. This work suggests that efficient large frequency shift (~100 THz) parametric amplification in long (>50 m) fibers would require a fiber uniformity two to three times higher than the above values. Further progress in fabrication technology will be needed to achieve this goal.*

_{a}## Acknowledgements

## References and links

1. | C. Lin, W. A. Reed, A. D. Pearson, and H. T. Sbang, “Phase matching in the minimum-chromatic-dispersion region of single-mode fibers for stimulated four-photon mixing,” Opt. Lett. |

2. | J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett. |

3. | A. Y. H. Chen, G. K. L. Wong, S. G. Murdoch, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Widely tunable optical parametric generation in a photonic crystal fiber,” Opt. Lett. |

4. | S. Pitois and G. Millot, “Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber,” Opt. Commun. |

5. | M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, “Wideband tuning of the gain spectra of one-pump fiber optical parametric amplifiers,” J. Sel. Top. Quantum. Electron. |

6. | G. K. L. Wong, A. Y. H. Chen, S. G. Murdoch, R. Leonhardt, J. D. Harvey, N. Y. Joly, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Continuous-wave tunable optical parametric generation in a photonic-crystal fiber,” J. Opt. Soc. Am. B |

7. | N. J. Smith and N. J. Doran, “Modulational instabilities in fibers with periodic dispersion management,” Opt. Lett. |

8. | J. C. Bronski and J. N. Kutz, “Modulational stability of plane waves in nonreturn-to-zero communications systems with dispersion management,” Opt. Lett. |

9. | S. G. Murdoch, R. Leonhardt, J. D. Harvey, and T. A. B. Kennedy, “Quasi-phase matching in an optical fiber with periodic birefringence,” J. Opt. Soc. Am. B |

10. | E. Ciaramella and M. Tamburrini, “Modulation instability in long amplified links with strong dispersion compensation,” IEEE Photon. Technol. Lett. |

11. | A. Kumar, A. Labruyere, and P. Tchofo Dinda, “Modulational instability in fiber systems with periodic loss compensation and dispersion management,” Opt. Commun. |

12. | F. K. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, “Modulational instability in optical fibers with variable dispersion,” Phys. Lett. A |

13. | M. Karlsson, “Four-wave mixing in fibers with randomly varying zero-dispersion wavelength,” J. Opt. Soc. Am. B |

14. | M. Farahmand and M. de Sterke, “Parametric amplification in presence of dispersion fluctuations,” Opt. Express |

15. | B. Kibler, C. Billet, J. M. Dudley, R. S. Windeler, and G. Millot, “Effects of structural irregularities on modulational instability phase matching in photonic crystal fibers,” Opt. Lett. |

16. | G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects,” J. Opt. Soc. Am. B |

17. | K. L. Reichenbach and C. Xu, “The effects of randomly occurring nonuniformities on propagation in photonic crystal fibers,” Opt. Express |

18. | D. Derickson, |

19. | A. Mussot, E. Lantz, A. Durecu-Legrand, C. Simonneau, D. Bayart, T. Sylvestre, and H. Maillotte, “Zero-dispersion wavelength mapping in short single-mode optical fibers using parametric amplification,” IEEE Photon. Technol. Lett. |

20. | G. K. L. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Y. Joly, “Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability,” Opt. Express |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(060.2320) Fiber optics and optical communications : Fiber optics amplifiers and oscillators

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.4970) Nonlinear optics : Parametric oscillators and amplifiers

**ToC Category:**

Photonic Crystal Fibers

**History**

Original Manuscript: July 5, 2006

Revised Manuscript: August 24, 2006

Manuscript Accepted: September 1, 2006

Published: October 2, 2006

**Citation**

J. S. Y. Chen, S. G. Murdoch, R. Leonhardt, and J. D. Harvey, "Effect of dispersion fluctuations on widely tunable optical parametric amplification in photonic crystal fibers," Opt. Express **14**, 9491-9501 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9491

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### References

- C. Lin, W. A. Reed, A. D. Pearson, and H. T. Sbang, "Phase matching in the minimum-chromatic-dispersion region of single-mode fibers for stimulated four-photon mixing," Opt. Lett. 6, 493-5 (1981). [CrossRef] [PubMed]
- J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, "Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber," Opt. Lett. 28, 2225-7 (2003). [CrossRef] [PubMed]
- A. Y. H. Chen, G. K. L. Wong, S. G. Murdoch, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, "Widely tunable optical parametric generation in a photonic crystal fiber," Opt. Lett. 30, 762-764 (2005). [CrossRef] [PubMed]
- S. Pitois and G. Millot, "Experimental observation of a new modulational instability spectral window induced by fourth-order dispersion in a normally dispersive single-mode optical fiber," Opt. Commun. 226, 415-22 (2003). [CrossRef]
- M. E. Marhic, K. K. Y. Wong, and L. G. Kazovsky, "Wideband tuning of the gain spectra of one-pump fiber optical parametric amplifiers," J. Sel. Top. Quantum. Electron. 10, 1133-1141 (2004). [CrossRef]
- G. K. L. Wong, A. Y. H. Chen, S. G. Murdoch, R. Leonhardt, J. D. Harvey, N. Y. Joly, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, "Continuous-wave tunable optical parametric generation in a photonic-crystal fiber," J. Opt. Soc. Am. B 22, 2505-2511 (2005). [CrossRef]
- N. J. Smith and N. J. Doran, "Modulational instabilities in fibers with periodic dispersion management," Opt. Lett. 21, 570-2 (1996). [CrossRef] [PubMed]
- J. C. Bronski and J. N. Kutz, "Modulational stability of plane waves in nonreturn-to-zero communications systems with dispersion management," Opt. Lett. 21, 937-9 (1996). [CrossRef] [PubMed]
- S. G. Murdoch, R. Leonhardt, J. D. Harvey, and T. A. B. Kennedy, "Quasi-phase matching in an optical fiber with periodic birefringence," J. Opt. Soc. Am. B 14, 1816-1822 (1997). [CrossRef]
- E. Ciaramella and M. Tamburrini, "Modulation instability in long amplified links with strong dispersion compensation," IEEE Photon. Technol. Lett. 11, 1608-1610 (1999). [CrossRef]
- A. Kumar, A. Labruyere, and P. Tchofo Dinda, "Modulational instability in fiber systems with periodic loss compensation and dispersion management," Opt. Commun. 219, 221-239 (2003). [CrossRef]
- F. K. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, "Modulational instability in optical fibers with variable dispersion," Phys. Lett. A 220, 213-218 (1996). [CrossRef]
- M. Karlsson, "Four-wave mixing in fibers with randomly varying zero-dispersion wavelength," J. Opt. Soc. Am. B 15, 2269-2275 (1998). [CrossRef]
- M. Farahmand and M. de Sterke, "Parametric amplification in presence of dispersion fluctuations," Opt. Express 12, 136-142 (2004). [CrossRef] [PubMed]
- B. Kibler, C. Billet, J. M. Dudley, R. S. Windeler, and G. Millot, "Effects of structural irregularities on modulational instability phase matching in photonic crystal fibers," Opt. Lett. 29, 1903-1905 (2004). [CrossRef] [PubMed]
- G. Cappellini and S. Trillo, "Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects," J. Opt. Soc. Am. B 8, 824-838 (1991). [CrossRef]
- K. L. Reichenbach and C. Xu, "The effects of randomly occurring nonuniformities on propagation in photonic crystal fibers," Opt. Express 13, 2799-2807 (2005). [CrossRef] [PubMed]
- D. Derickson, Fiber Optic Test and Measurement, 1st ed. (Prentice Hall, Upper Saddle River, NJ, 1998).
- A. Mussot, E. Lantz, A. Durecu-Legrand, C. Simonneau, D. Bayart, T. Sylvestre, and H. Maillotte, "Zerodispersion wavelength mapping in short single-mode optical fibers using parametric amplification," IEEE Photon. Technol. Lett. 18, 22-4 (2006). [CrossRef]
- G. K. L. Wong, A. Y. H. Chen, S. W. Ha, R. J. Kruhlak, S. G. Murdoch, R. Leonhardt, J. D. Harvey, and N. Y. Joly, "Characterization of chromatic dispersion in photonic crystal fibers using scalar modulation instability," Opt. Express 13, 8662-8670 (2005). [CrossRef] [PubMed]

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