## Mode profile dispersion in the generalized nonlinear Schrödinger equation

Optics Express, Vol. 15, Issue 24, pp. 16110-16123 (2007)

http://dx.doi.org/10.1364/OE.15.016110

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### Abstract

The formulation of Schrödinger-like equations for nonlinear pulse propagation in a single-mode microstructured optical fiber with a strongly frequency-dependent guided-mode profile is investigated. A correct account of mode profile dispersion in general necessiates a generalization of the effective area concept commonly used in the generalized nonlinear Schrödinger equation (GNLSE). A numerical scheme to this end is developed, and applied to a solid-core photonic bandgap fiber as a test case. It is further shown, that a simple reformulation of the GNLSE, expressed only in terms of the traditional frequency-dependent effective area, yields a good agreement with the more complete theory.

© 2007 Optical Society of America

## 1. Introduction

2. A. Fuerbach, P. Steinvurzel, J. Bolger, A. Nulsen, and B. Eggleton, “Nonlinear propagation effects in antiresonant high-index inclusion photonic crystal fibers,” Opt. Lett. **30(8)**, 830–832 (2005). [CrossRef]

3. A. Fuerbach, P. Steinvurzel, J. Bolger, and B. Eggleton, “Nonlinear pulse propagation at zero dispersion wavelength in anti-resonant photonic crystal fibers,” Opt. Express **13(8)**, 2977–2987 (2005). [CrossRef]

4. D. Ouzounov, F. Ahmad, D. Muller, N. Venkataraman, M. Gallagher, M. Thomas, J. Silcox, K. Koch, and A. Gaeta, “Generation of megawatt optical solitons in hollow-core photonic band-gap fibers,” Science **301(5640)**, 1702–1704 (2003). [CrossRef]

5. D. G. Ouzounov, C. J. Hensley, A. L. Gaeta, N. Venkateraman, M. T. Gallagher, and K. W. Koch, “Soliton pulse compression in photonic band-gap fibers,” Opt. Express **13(16)**, 6153–6159 (2005). [CrossRef]

6. C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, “Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers,” Opt. Express **15(6)**, 3507–3512 (2007). [CrossRef]

7. F. Gerome, K. Cook, A. George, W. Wadsworth, and J. Knight, “Delivery of sub-100fs pulses through 8m of hollow-core fiber using soliton compression,” Opt. Express **15(12)**, 7126–7131 (2007). [CrossRef]

8. J. Lægsgaard, N. A. Mortensen, J. Riishede, and A. Bjarklev, “Material effects in airguiding photonic bandgap fibers,” J. Opt. Soc. Am. B **20**, 2046–51 (2003). [CrossRef]

6. C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, “Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers,” Opt. Express **15(6)**, 3507–3512 (2007). [CrossRef]

*effective area*parameter,

*A*. The effective area contributes to the nonlinear coefficient

_{eff}*γ*, which is the prefactor of the nonlinear term in the GNLSE, as follows:

## 2. Derivation of a 1+1D propagation equation

*ε*(

**r**⊥) is the transverse distribution of the dielectric constant defining the waveguide. The fields are partitioned as:

**e**

*m*and

**h**

*are assumed to be eigenstates of the linear eigenvalue problem for a*

_{m}*z*-propagating mode, with propagation constants

*β*(

_{m}*ω*). In Ref. [13

13. M. Kolesik, E. Wright, and J. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers.” Appl. Phys. B: Lasers & Optics **79(3)**, 293–300 (2004). [CrossRef]

*ansatz*was shown to lead to the 1+1D propagation equation:

*N*(

_{m}*ω*) is a normalization parameter given by:

*R(t)*is the Raman response function and

*χ*is the full third-order susceptibility tensor. Both

^{(3)}*R*and

*χ*

^{(3)}will in this work be assumed independent of position. It is important to note that, at this stage, the field

**E**(

**r**,

*t*) is purely real, which means that the frequency integrals in Eqs. (3), (4) extend over both positive and negative frequencies, with

*β*(-

_{m}*ω*)=-

*β*(ω),

_{m}**e**

*(r⊥,-*

_{m}*ω*)=

**e***

*(*

_{m}**r**⊥,

**ω**) etc. Inserting the frequency-domain expansion of

**E**(

**r**,

*t*), Eq. (3), one obtains:

*n*=

_{eff}*cβ/ω*,

**x**̂, ŷ are unit vectors, and

*F*is a real function. In the example PBG fiber, full-vectorial control calculations showed that the fraction of field energy carried by the neglected vector components was at most on the order of 10

^{-3}. For the following discussion it is useful to introduce the normalizations:

*E*is the total pulse energy, and

_{p}*n*is some representative refractive index, here chosen to be 1.45.

_{0}*ω*

_{0}/3, where ω0 is some suitably chosen base frequency [16

16. K. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. **25(12)**, 2665–2673 (1989). [CrossRef]

*t*′=

*t*-

*β1z*, is done, where

*β*

_{1}=

*dβ/dω*, evaluated at

*ω*

_{0}. This is equivalent to transforming

*β*(

*ω*) into

*β(ω)*-

*β*

_{1}(

*ω*)×(

_{0}*ω-ω*). In addition,

_{0}*ω*is shifted by the base frequency

*ω*

_{0}and

*β*(

*ω*) is correspondingly measured relative to

*β*. With the normalization conventions in Eq. (14), and the above changes, Eq. (11) becomes:

_{0}=β(ω_{0})*n*

_{2}differs from the conventional definition of

*n*

_{2}=

*3χ*/4

^{(3)}_{xxxx}*n*, which follows from approximating

^{2}_{0}ε_{0}c*ω*/(

*β*(

*ω*)in the prefactor on the RHS of Eq. (16) by 1/

*n*

_{0}, a convention not adopted in this work. It is from now on implicit that frequency integrations are done over positive frequencies only. Apart from some differences with respect to normalizations these results correspond to the earlier findings in Refs. [12

12. P. Mamyshev and S. Chernikov, “Ultrashort-pulse propagation in optical fibers,” Opt. Lett. **15(19)**, 1076–1078 (1990). [CrossRef]

*z*, the soliton experiences a changing effective area parameter, even if its spectral width at any particular value of

*z*is small enough to merit a neglect of the frequency variation of

*F*. Another case where the use of Eq. (22) may be adequate is that in which the different spectral components of the propagating light are well separated in time (e.g. in a strongly chirped pulse). Since the Kerr interaction is strictly local in time, and the Raman response function has a width of ~100 fs, the different frequency components in such pulses will mainly experience nonlinear interactions with themselves. The difference between Eq. (16) and (22) becomes apparent when frequency components with a significant difference in the mode profile have a temporal overlap. This observation is quite interesting, since in PBG fibers a strong mode profile dispersion is often associated with a strong group velocity dispersion, meaning that different frequency components of a short pulse will quickly disperse away from each other.

## 3. Approximate forms of the propagation equation

*K*in the double frequency integral in Eq. (16) implies that this integral cannot be evaluated as a series of convolutions, as is the case for the integrals in Eq. (20), and (22). Since convolutions can be done by the fast Fourier transform (FFT) with a numerical effort on the order of

*Nln(N)*where

*N*is the number of points on the frequency mesh, whereas a full 2D integral, evaluated at all values of

*ω*, would require

*N*

^{3}operations, this is a very significant drawback, severely limiting the practical usefulness of Eq. (16). Therefore, it is important to derive approximate approaches to this equation which are numerically feasible. Kolesik

*et al*. [13

13. M. Kolesik, E. Wright, and J. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers.” Appl. Phys. B: Lasers & Optics **79(3)**, 293–300 (2004). [CrossRef]

12. P. Mamyshev and S. Chernikov, “Ultrashort-pulse propagation in optical fibers,” Opt. Lett. **15(19)**, 1076–1078 (1990). [CrossRef]

### 3.1. Basis set expansion

*K*into one-dimensional basis functions:

*O*[

*Nln*(

*N*)] operations using fast Fourier transforms. The price is, that the number of such integrals to be evaluated now increases with the number of terms in the expansion of

*K*.

*F*(

**r**⊥,

*ω*), into a set of frequency-independent basis functions:

*F*could in principle be determined as the frequency derivatives of

_{µ}*F*at

*ω*, but a more robust procedure is to use a polynomial fitting scheme as detailed below.

_{c}*F*(

**r**⊥,

*ω*) of an all-silica PBG fiber. The structure chosen for study consists of a triangular array of Ge-doped rods in a background of pure silica, with a missing rod comprising the low-index core defect. The structure is depicted by the black circles in Fig. 1. Only a quarter of the structure is shown, since the finite-difference calculation was restricted to modes that are even under reflection in the

*x*- and

*y*-axes. Losses are not accounted for in the calculation, so periodic boundary conditions have been assumed for simplicity. Such fibers have been extensively studied in the last few years [18

18. J. Riishede, J. Lægsgaard, J. Broeng, and A. Bjarklev, “All-silica photonic bandgap fibre with zero dispersion and large mode area at 730 nm,” J. Opt. A: Pure and Applied Optics **6**, 667–70 (2004). [CrossRef]

19. A. Argyros, T. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. S. J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express **13**, 309–14 (2005). [CrossRef] [PubMed]

20. G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (<20 dB/km) around 1550 nm,” Opt. Express **13**, 8452–9 (2005). [CrossRef] [PubMed]

21. A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express **10**, 1320–33 (2002). [PubMed]

22. N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Resonances in microstructured optical waveguides,” Opt. Express **11**, 1243–51 (2003). [CrossRef] [PubMed]

23. J. Lægsgaard, “Gap formation and guided modes in photonic bandgap fibres with high-index rods,” J. Opt. A: Pure and Applied Optics **6**, 798–804 (2004). [CrossRef]

23. J. Lægsgaard, “Gap formation and guided modes in photonic bandgap fibres with high-index rods,” J. Opt. A: Pure and Applied Optics **6**, 798–804 (2004). [CrossRef]

18. J. Riishede, J. Lægsgaard, J. Broeng, and A. Bjarklev, “All-silica photonic bandgap fibre with zero dispersion and large mode area at 730 nm,” J. Opt. A: Pure and Applied Optics **6**, 667–70 (2004). [CrossRef]

*V*=3.95 to

*V*=5.0 is considered here, where

*V*is the classical

*V*-parameter of the high-index rods:

*V*-range is well within the bandgap-guiding region, but still a significant mode profile dispersion occurs at the edges, as shown below.

*V*-parameter range from 3.95 to 5.0, with Λ=12

*µ*m. The fields in the center and at the edges of this range are shown in Fig. 1. Subsequently, at each point on the finite-difference grid, the 43 field values were fitted to a

*M*th-order polynomium in the normalized and centered frequency variable (

*ω-ω*)/

_{c}*ω*where

_{c}*ω*was taken in the center of the frequency range. The set of

_{c}*M*+1 polynomial coefficients from each gridpoint then constitute the

*M*+1 basis functions in Eq. (28). Examples of these basis functions are shown in Fig. 2, for

*M*=2.

*F*(

**r**⊥,

*ω*), or the approximate mode profiles given by the polynomial expansion in Eq. (28). In Fig. 3, the ‘exact’ effective-area curve is compared to curves calculated with expansions of

*M*=2 (i.e. 3 basis functions in total), and

*M*=3. It is seen, that already the

*M*=2 expansion provides a useful approximation to the effective-area curve, and that the

*M*=3 expansion gives a very satisfactory agreement. The

*M*=2 expansion implies 81 terms in the sum occurring in Eq. (24), whereas

*M*=3 implies 256 terms. The sums can be somewhat reduced by using the symmetry of

*K*under permutation of the frequency arguments. In the implementation used here, a single evaluation of the nonlinear term using the M=3 expansion requires 164 complex and 320 real FFT operations, compared to just two complex and two real FFT’s when using Eq. (22). In the latter case, computational overheads arising from the evaluation of exponential functions and the use of a fairly advanced (Bulirsch-Stoer) stepper algorithm are not completely insignificant, so the total difference in runtime between the two approaches was about a factor of 50. While the use of Eqs. (16), (24) thus lead to a significantly increased computational load, they are no longer unfeasible, and can therefore be used to check the accuracy of simpler formulations. Still further reductions in the computational overhead may be achieved by casting away small terms in

*K*, and/or do linear transformations within the determined basis set to minimize some of these coefficients, but such schemes were not studied in this work.

_{µνγδ}### 3.2. Modified GNLSE

*K*

_{0000}term in Eq. (24) is much more important than the other terms. If only the

*K*

_{0000}term is retained, this equation becomes:

12. P. Mamyshev and S. Chernikov, “Ultrashort-pulse propagation in optical fibers,” Opt. Lett. **15(19)**, 1076–1078 (1990). [CrossRef]

*µ*>0 in Eq. (28) have effective areas much larger than

*F*

_{0}. In the PBG fiber example discussed here, the

*F*

_{1}state, shown in the middle panel of Fig. 2, has an effective area ~2.5 times larger than that of

*F*

_{0}(left panel), and the area of the

*F*

_{2}state (right panel) is about 6 times larger than the

*F*

_{0}area. Thus, the validity of Eq. (33) is not obvious in this case. There are, however, two other arguments of a more general nature which favour the use of Eq. (33) over Eq. (22). The first is that both Eq. (16) and Eq. (33) conserve the classical photon number, which with the normalizations used here is proportional to:

16. K. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. **25(12)**, 2665–2673 (1989). [CrossRef]

*ω*-

*ω*

_{c}as will now be demonstrated. If one assumes that the relation

*K*can be discarded, the expression for

*K*becomes:

## 4. Numerical results

*M*=3 as described in the previous section. For a fair comparison, the GNLSE and M-GNLSE are solved using the effective area calculated with the

*M*=3 expansion, e.g. the green curve in Fig. 3. Material dispersion was included in the finite-difference calculations, assuming that the index difference between the high-index rods and the pure-silica background remained constant. The dispersion curve of the fiber with Λ=12

*µ*m is shown in Fig. (4). The zero-dispersion point is at 1.025

*µ*m. The calculations were done on a frequency grid with 2

^{15}points, extending from

*λ*

_{0}=0.91

*µ*m to λ0=1.15

*µ*m. The value for

*n*

_{2}, as defined by Eq. (19), was put to 3.857·10

^{-8}W/

*µ*m

^{2}, and the Raman response function was parametrized according to Ref. [17].

*T*

_{0}matched to the peak power

*P*

_{0}, the effective area and the dispersion coefficient of the fiber according to the well-known relation [17]:

*β*

_{2}is the second derivative of

*β*with respect to

*ω*, and

*γ*is defined in Eq. (1). The dispersion coefficient of the fiber at 1100 nm is 146 ps/nm/km, and a

*P*

_{0}of 100 kW corresponds to a

*T*

_{0}of about 32 fs. Due to the Raman effect, the soliton will shift towards longer wavelengths during propagation [24

24. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. **11(10)**, 662–664 (1986). [CrossRef]

*P*

_{0}of 100 kW, as found with the three calculational approaches discussed above. The calculation using the

*M*=3 expansion took about 28 hours on a 3GHz PC, whereas solution of the M-GNLSE required about 35 minutes. The M-GNLSE is seen to provide a near-perfect match to the results of the

*M*=3 expansion, whereas the wavelength shift found with the GNLSE deviates by about 8.4% after 10 meters of propagation. The redshifting of the wavelength happens quickly at the launch end of the fiber, but gradually slows down during propagation. This is understandable, since the soliton adjusts its width to the increasing group velocity dispersion at longer wavelengths, thereby increasing

*T*

_{0}according to Eq. (42). The rate of redshifting decreases with increasing

*T*

_{0}, and is roughly proportional to

*T*

^{-4}

_{0}for long pulses [24

24. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. **11(10)**, 662–664 (1986). [CrossRef]

*P*

_{0}, calculated with either the GNLSE or M-GNLSE. At the lowest peak power of 10 kW, the difference between the predicted redshifts is about 2.5% of the total shift, indicating that the deviation between the GNLSE and M-GNLSE follows the spectral width of the soliton. It should also be noted that the relative deviations found are of the same order of magnitude as the relative variation in

*A*over the width of the soliton.

_{e f f}*M*=3 expansion is reflected in the calculated spectra after 10 meters of propagation, as shown in Fig. 6 for

*P*

_{0}=100 kW. Apart from being shifted in frequency, the GNLSE spectrum is wider than those predicted by the other methods, because it resides at wavelengths with a lower group velocity dispersion.

*M*=3 expansion.

## 5. Conclusion

## Acknowledgements

## References and links

1. | J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78(4) (2006). |

2. | A. Fuerbach, P. Steinvurzel, J. Bolger, A. Nulsen, and B. Eggleton, “Nonlinear propagation effects in antiresonant high-index inclusion photonic crystal fibers,” Opt. Lett. |

3. | A. Fuerbach, P. Steinvurzel, J. Bolger, and B. Eggleton, “Nonlinear pulse propagation at zero dispersion wavelength in anti-resonant photonic crystal fibers,” Opt. Express |

4. | D. Ouzounov, F. Ahmad, D. Muller, N. Venkataraman, M. Gallagher, M. Thomas, J. Silcox, K. Koch, and A. Gaeta, “Generation of megawatt optical solitons in hollow-core photonic band-gap fibers,” Science |

5. | D. G. Ouzounov, C. J. Hensley, A. L. Gaeta, N. Venkateraman, M. T. Gallagher, and K. W. Koch, “Soliton pulse compression in photonic band-gap fibers,” Opt. Express |

6. | C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, “Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers,” Opt. Express |

7. | F. Gerome, K. Cook, A. George, W. Wadsworth, and J. Knight, “Delivery of sub-100fs pulses through 8m of hollow-core fiber using soliton compression,” Opt. Express |

8. | J. Lægsgaard, N. A. Mortensen, J. Riishede, and A. Bjarklev, “Material effects in airguiding photonic bandgap fibers,” J. Opt. Soc. Am. B |

9. | N. Karasawa, S. Nakamura, and N. Nakagawa, “Comparison Between Theory and Experiment of Nonlinear Propagation for A-Few-Cycle and....” IEEE J. Quantum Electron. |

10. | G. Chang, T. B. Norris, and H. G. Winful, “Optimization of supercontinuum generation in photonic crystal fibers for pulse compression,” Opt. Lett. |

11. | B. Kibler, J. M. Dudley, and S. Coen, “Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area,” Appl. Phys. B |

12. | P. Mamyshev and S. Chernikov, “Ultrashort-pulse propagation in optical fibers,” Opt. Lett. |

13. | M. Kolesik, E. Wright, and J. Moloney, “Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers.” Appl. Phys. B: Lasers & Optics |

14. | A. Ferrando, M. Zacares, P. de Cordoba, D. Binosi, and A. Montero, “Forward-backward equations for nonlinear propagation in axially invariant optical systems,” Phys. Rev. E (Statistical, Nonlinear, and Soft Matter Physics) |

15. | Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, “Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond compression and carrier envelope phase effect,” Phys. Rev. A (Atomic, Molecular, and Optical Physics) |

16. | K. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quantum Electron. |

17. | G. P. Agrawal, |

18. | J. Riishede, J. Lægsgaard, J. Broeng, and A. Bjarklev, “All-silica photonic bandgap fibre with zero dispersion and large mode area at 730 nm,” J. Opt. A: Pure and Applied Optics |

19. | A. Argyros, T. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. S. J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express |

20. | G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (<20 dB/km) around 1550 nm,” Opt. Express |

21. | A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express |

22. | N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Resonances in microstructured optical waveguides,” Opt. Express |

23. | J. Lægsgaard, “Gap formation and guided modes in photonic bandgap fibres with high-index rods,” J. Opt. A: Pure and Applied Optics |

24. | J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. |

**OCIS Codes**

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(060.4005) Fiber optics and optical communications : Microstructured fibers

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 18, 2007

Revised Manuscript: November 13, 2007

Manuscript Accepted: November 16, 2007

Published: November 20, 2007

**Citation**

Jesper Laegsgaard, "Mode profile dispersion in the generalised nonlinear Schrödinger equation," Opt. Express **15**, 16110-16123 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-16110

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### References

- J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135 (2006).
- A. Fuerbach, P. Steinvurzel, J. Bolger, A. Nulsen, and B. Eggleton, "Nonlinear propagation effects in antiresonant high-index inclusion photonic crystal fibers," Opt. Lett. 30, 830-832 (2005). [CrossRef]
- A. Fuerbach, P. Steinvurzel, J. Bolger, and B. Eggleton, "Nonlinear pulse propagation at zero dispersion wavelength in anti-resonant photonic crystal fibers," Opt. Express 13, 2977-2987 (2005). [CrossRef]
- D. Ouzounov, F. Ahmad, D. Muller, N. Venkataraman, M. Gallagher, M. Thomas, J. Silcox, K. Koch, and A. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003). [CrossRef]
- D. G. Ouzounov, C. J. Hensley, A. L. Gaeta, N. Venkateraman, M. T. Gallagher, and K. W. Koch, "Soliton pulse compression in photonic band-gap fibers," Opt. Express 13, 6153-6159 (2005). [CrossRef]
- C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, "Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers," Opt. Express 15, 3507-3512 (2007). [CrossRef]
- F. Gerome, K. Cook, A. George, W. Wadsworth, and J. Knight, "Delivery of sub-100fs pulses through 8m of hollow-core fiber using soliton compression," Opt. Express 15, 7126-7131 (2007). [CrossRef]
- J. Lægsgaard, N. A. Mortensen, J. Riishede, and A. Bjarklev, "Material effects in airguiding photonic bandgap fibers," J. Opt. Soc. Am. B 20, 2046-51 (2003). [CrossRef]
- N. Karasawa, S. Nakamura, and N. Nakagawa, "Comparison between theory and experiment of nonlinear propagation for a-few-cycle and ultrabroadband," IEEE J. Quantum Electron. 37, 398-404 (2001). [CrossRef]
- G. Chang, T. B. Norris, and H. G. Winful, "Optimization of supercontinuum generation in photonic crystal fibers for pulse compression," Opt. Lett. 28, 546-548 (2003). [CrossRef]
- B. Kibler, J. M. Dudley, and S. Coen, "Supercontinuum generation and nonlinear pulse propagation in photonic crystal fiber: influence of the frequency-dependent effective mode area," Appl. Phys. B 81, 337-342 (2005). [CrossRef]
- P. Mamyshev and S. Chernikov, "Ultrashort-pulse propagation in optical fibers," Opt. Lett. 15, 1076-1078 (1990). [CrossRef]
- M. Kolesik, E. Wright, and J. Moloney, "Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers," Appl. Phys. B: Lasers Opt. 79, 293-300 (2004). [CrossRef]
- A. Ferrando, M. Zacares, P. de Cordoba, D. Binosi, and A. Montero, "Forward-backward equations for nonlinear propagation in axially invariant optical systems," Phys. Rev. E 71, 16,601 (2005).
- Y. Mizuta, M. Nagasawa, M. Ohtani, and M. Yamashita, "Nonlinear propagation analysis of few-optical-cycle pulses for subfemtosecond compression and carrier envelope phase effect," Phys. Rev. A 72, 63,802 (2005).
- K. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989). [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).
- J. Riishede, J. Lægsgaard, J. Broeng, and A. Bjarklev, "All-silica photonic bandgap fibre with zero dispersion and large mode area at 730 nm," J. Opt. A: Pure and Applied Optics 6, 667-70 (2004). [CrossRef]
- A. Argyros, T. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. S. J. Russell, "Photonic bandgap with an index step of one percent," Opt. Express 13, 309-314 (2005). [CrossRef] [PubMed]
- G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, "Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm," Opt. Express 13, 8452-8459 (2005). [CrossRef] [PubMed]
- A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, "Analysis of spectral characteristics of photonic bandgap waveguides," Opt. Express 10, 1320-1333 (2002). [PubMed]
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