## Coupled core-surface solitons in photonic crystal fibers

Optics Express, Vol. 12, Issue 20, pp. 4841-4846 (2004)

http://dx.doi.org/10.1364/OPEX.12.004841

Acrobat PDF (789 KB)

### Abstract

We predict existence and study properties of the coupled core-surface solitons in hollow-core photonic crystal fibers. These solitons exist in the spectral proximity of the avoided crossings of the propagation constants of the modes guided in the air core and at the interface between the core and photonic crystal cladding.

© 2004 Optical Society of America

## 1. Introduction

1. P.St.J. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

2. C.M. Smith, N. Venkataraman, M.T. Gallagher, D. Müller, J.A. West, N.F. Borrelli, D.C. Alan, and K.W. Koch, “Low-loss hollow-core silica/air photonic bandgap fiber,” Nature **424**, 657–659 (2003). [CrossRef] [PubMed]

1. P.St.J. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

3. F. Benabid, J.C. Knight, G. Antonopoulos G, and P.S.J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science **298**, 399–402 (2002). [CrossRef] [PubMed]

4. S.O. Konorov, A.B. Fedotov, and A.M. Zheltikov, “Enhanced four-wave mixing in a hollow-core photonic-crystal fiber,” Opt. Lett. **28**, 1448–1450 (2003). [CrossRef] [PubMed]

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

6. F. Luan, J.C. Knight, P.S.J. Russell, S. Campbell, D. Xiao, D.T. Reid, B.J. Mangan, D.P. Williams, and P.J. Roberts, “Femtosecond soliton pulse delivery at 800nm wavelength in hollow-core photonic bandgap fibers,” Opt. Express **12**, 835–840 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-835 [CrossRef] [PubMed]

2. C.M. Smith, N. Venkataraman, M.T. Gallagher, D. Müller, J.A. West, N.F. Borrelli, D.C. Alan, and K.W. Koch, “Low-loss hollow-core silica/air photonic bandgap fiber,” Nature **424**, 657–659 (2003). [CrossRef] [PubMed]

7. D.C. Alan, N.F. Borrelli, M.T. Gallagher, D. Müller, C.M. Smith, N. Venkataraman, J.A. West, P. Zhang, and K.W. Koch, “Surface modes and loss in air-core photonic band-gap fibers,” Proc. of SPIE **5000**, 161–174 (2003). [CrossRef]

8. K. Saitoh, N.A. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express **12**, 394–400 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394 [CrossRef] [PubMed]

9. J.A. West, C.M. Smith, N.F. Borrelli, D.C. Alan, and K.W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express **12**, 1485–1496 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485 [CrossRef] [PubMed]

10. G. Humbert, J.C. Knight, G. Bouwmans, P.St.J. Russell, D.P. Williams, P.J. Roberts, and B.J. Mangan, “Hollow core photonic crystal fibers for beam delivery,” **12**1477–1484, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1477

7. D.C. Alan, N.F. Borrelli, M.T. Gallagher, D. Müller, C.M. Smith, N. Venkataraman, J.A. West, P. Zhang, and K.W. Koch, “Surface modes and loss in air-core photonic band-gap fibers,” Proc. of SPIE **5000**, 161–174 (2003). [CrossRef]

9. J.A. West, C.M. Smith, N.F. Borrelli, D.C. Alan, and K.W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express **12**, 1485–1496 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485 [CrossRef] [PubMed]

2. C.M. Smith, N. Venkataraman, M.T. Gallagher, D. Müller, J.A. West, N.F. Borrelli, D.C. Alan, and K.W. Koch, “Low-loss hollow-core silica/air photonic bandgap fiber,” Nature **424**, 657–659 (2003). [CrossRef] [PubMed]

7. D.C. Alan, N.F. Borrelli, M.T. Gallagher, D. Müller, C.M. Smith, N. Venkataraman, J.A. West, P. Zhang, and K.W. Koch, “Surface modes and loss in air-core photonic band-gap fibers,” Proc. of SPIE **5000**, 161–174 (2003). [CrossRef]

11. D.L. Miles, *Nonlinear Optics* (Springer, Berlin, 1998). [CrossRef]

13. V.M. Agranovich, D.M. Basko, A.D. Boardman, A.M. Kamchatnov, and T.A. Leskova, “Surface solitons due to second order cascaded nonlinearity,” Opt. Commun. **160**, 114–118 (1999). [CrossRef]

10. G. Humbert, J.C. Knight, G. Bouwmans, P.St.J. Russell, D.P. Williams, P.J. Roberts, and B.J. Mangan, “Hollow core photonic crystal fibers for beam delivery,” **12**1477–1484, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1477

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

6. F. Luan, J.C. Knight, P.S.J. Russell, S. Campbell, D. Xiao, D.T. Reid, B.J. Mangan, D.P. Williams, and P.J. Roberts, “Femtosecond soliton pulse delivery at 800nm wavelength in hollow-core photonic bandgap fibers,” Opt. Express **12**, 835–840 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-835 [CrossRef] [PubMed]

## 2. Model

**5000**, 161–174 (2003). [CrossRef]

9. J.A. West, C.M. Smith, N.F. Borrelli, D.C. Alan, and K.W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express **12**, 1485–1496 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485 [CrossRef] [PubMed]

*A*

_{c}and

*A*

_{s}are the slowly varying envelopes of the core and surface states. The reference frequency

*ω*

_{ref}is assumed to be the frequency at the center of the avoided crossing. Note, that the true eigenmodes of the fiber, which we term below as supermodes, are of course not coupled linearly. However, here, as in many other physical contexts [14

14. C.M. de Sterke and J.E. Sipe, “Coupled modes and the nonlinear Schrodinger-equation,” Phys. Rev. A **42**, 550–555 (1990). [CrossRef]

*Z*is the coordinate along the fiber and

*T*is time.

*κ*is the coupling between the core and surface states. Γ is the loss arising from the coupling of the surface state with cladding modes.

*α*

_{c,s}are the slopes of the graphs of the propagation constants of the core and surface states as functions of frequency.

*𝓝*

_{c,s}are the nonlinear responses of the surface and core states. Operators

*D*

_{c,s}(

*i∂*

_{T}) describe dispersions of the second order and higher. For large detunings from

*ω*

_{ref}each of the supermodes asymptotically tends either to the pure surface or to the pure core mode, see Fig. 1(a).

**424**, 657–659 (2003). [CrossRef] [PubMed]

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

**5000**, 161–174 (2003). [CrossRef]

8. K. Saitoh, N.A. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express **12**, 394–400 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394 [CrossRef] [PubMed]

**12**, 1485–1496 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485 [CrossRef] [PubMed]

*λ*

_{ref}= 2

*πc*/

*ω*

_{ref}= 1580nm,

*α*

_{c}= 1.01/

*c*,

*α*

_{s}= 1.4/

*c*and

*κ*= 103m

^{-1}, where

*c*= 3 × 10

^{8}m/s. Effective refractive indices for the two supermodes and corresponding group velocity dispersions (GVDs) derived from Eqs. (1,2) for

*D*

_{c,s}= 0 are shown in Fig. 1. According to [5

**301**, 1702–1704 (2003). [CrossRef] [PubMed]

8. K. Saitoh, N.A. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express **12**, 394–400 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394 [CrossRef] [PubMed]

*λ*

_{0}≃ 1425nm, which can be matched by taking

*D*

_{c}(

*i∂*

_{T}) = -

*i*β ˜

_{2}

β ˜

_{3}

β ˜

_{2}≃ - 0.036ps

^{2}/m and

_{3}= 0.0001ps

^{3}/m. One can show that the

*D*

_{c}term becomes important only for detunings from

*ω*

_{ref}approaching 2

*π*× 15THz. In this frequency range the single mode generalized nonlinear Schrödinger equation for the amplitude of the core mode can be used to model propagation of femto-second solitons at mega-watt powers [5

**301**, 1702–1704 (2003). [CrossRef] [PubMed]

*D*

_{c,s}terms can be safely neglected. This is because values of GVD created by the coupling of the modes in this spectral region are ~ 100ps

^{2}/m, see Fig. 1(b), which is four orders of magnitude larger than the correction

β ˜

_{2}entering into

*D*

_{c,s}terms.

*n*

_{2c}≃ 3 · 10

^{-23}m

^{2}/W, which is 3 orders of magnitude lower than

*n*

_{2}for silica,

*n*

_{2s}≃ 2.4 · 10

^{-20}m

^{2}/W. Assuming the effective area of the core mode is

*S*

_{c}≃ 60

*μ*m

^{2}, we find that the nonlinear parameter

*γ*[15] of the core mode is

**301**, 1702–1704 (2003). [CrossRef] [PubMed]

6. F. Luan, J.C. Knight, P.S.J. Russell, S. Campbell, D. Xiao, D.T. Reid, B.J. Mangan, D.P. Williams, and P.J. Roberts, “Femtosecond soliton pulse delivery at 800nm wavelength in hollow-core photonic bandgap fibers,” Opt. Express **12**, 835–840 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-835 [CrossRef] [PubMed]

*S*

_{s}≃ 1

*μ*m

^{2}. Thus, the nonlinear parameter for the surface mode is

*γ*

_{cs}≃

*ε*

_{c}

*γ*

_{0}, where

*ε*

_{c}is the phenomenological coefficient characterizing the ratio of the intensity of the core mode at the interface to the intensity maximum of the core mode. The estimate for the nonlinear cross-coupling of the surface mode to the core mode is

*γ*

_{sc}≃

*ε*

_{s}

*γ*

_{0}, where

*ε*

_{s}characterizes the ratio of the intensity of the core mode at the interface to the intensity maximum of the surface mode. Values of

*ε*

_{c,s}depend on the fiber design [10

10. G. Humbert, J.C. Knight, G. Bouwmans, P.St.J. Russell, D.P. Williams, P.J. Roberts, and B.J. Mangan, “Hollow core photonic crystal fibers for beam delivery,” **12**1477–1484, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1477

*ε*

_{c,s}are order or less than 0.1, i.e.,

*γ*

_{cs,sc}~ 10

^{-4}W

^{-1}m

^{-1}. Thus,

*γ*

_{c,cs,sc}/

*γ*

_{s}≪ 1 and therefore we can safely assume in our calculations that

*𝓝*

_{s}=

*γ*

_{s}|

*A*

_{s}|

^{2}

*A*

_{s}and

*𝓝*

_{c}= 0. Note, however, that we have verified robustness of the numerical results presented below by introducing slightly exaggerated, upto 0.05

*γ*

_{s}, coefficients of the nonlinear cross-coupling between the core and surface modes.

*v*= 2/[

*α*

_{s}-

*α*

_{c}],

*t*= [

*T*-

*αZ*]/

*τ*,

*τ*= 1/[|

*v*|

*κ*],

*α*= [

*α*

_{c}+

*α*

_{s}]/2,

*z*=

*κZ*,

*P*=

*κ*/

*γ*

_{s}and Γ = Γ/

*κ*. For the parameters chosen above

*sgn*(

*v*) = 1,

*τ*= 0.6ps and

*P*= 10kW. Γ can be estimated at 4m

^{-1}[9

**12**, 1485–1496 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485 [CrossRef] [PubMed]

^{-3}.

## 3. Core-surface solitons

14. C.M. de Sterke and J.E. Sipe, “Coupled modes and the nonlinear Schrodinger-equation,” Phys. Rev. A **42**, 550–555 (1990). [CrossRef]

16. S. Wabnitz, “Forward mode-coupling in periodic nonlinear-optical fibers - Modal dispersion cancellation and resonance solitons,” Opt. Lett. **14**1071–1073 (1989). [CrossRef] [PubMed]

17. G. Van Simaes, S. Coen, M. Haelterman, and S. Trillo, “Observation of resonance soliton trapping due to a photoinduced gap in wave number,” Phys. Rev. Lett. **92**, 223902 (2004). [CrossRef]

11. D.L. Miles, *Nonlinear Optics* (Springer, Berlin, 1998). [CrossRef]

14. C.M. de Sterke and J.E. Sipe, “Coupled modes and the nonlinear Schrodinger-equation,” Phys. Rev. A **42**, 550–555 (1990). [CrossRef]

**42**, 550–555 (1990). [CrossRef]

16. S. Wabnitz, “Forward mode-coupling in periodic nonlinear-optical fibers - Modal dispersion cancellation and resonance solitons,” Opt. Lett. **14**1071–1073 (1989). [CrossRef] [PubMed]

17. G. Van Simaes, S. Coen, M. Haelterman, and S. Trillo, “Observation of resonance soliton trapping due to a photoinduced gap in wave number,” Phys. Rev. Lett. **92**, 223902 (2004). [CrossRef]

*F*

_{c,s}(

*z,t*) =

*f*

_{c,s}(

*ξ*)

*e*

^{iqz}, where

*ξ*=

*t*-

*wz*and

*q*measures the detuning of the wavenumber from the gap center. Solving the system of ordinary differential equations

*ξ*| → ∞, which implies

*q*

^{2}+

*w*

^{2}< 1. We have confirmed numerically that the latter inequality gives the existence boundary for the soliton solutions. For

*w*controls the relative power of the two components, it is instructive to consider the soliton as superposition of its Fourier harmonics. Propagation constant of the harmonic with frequency

*ω*is given by

*β*

_{sol}(

*ω*) =

*β*

_{0}+

*qκ*+ {

*ω*-

*ω*

_{ref}}{

*wκ*- [

*α*

_{c}+

*α*

_{s}]/2}, where

*β*

_{0}is the propagation constant at the central point of the gap. Then by plotting the effective refractive index of the soliton,

*n*

_{eff/sol}=

*β*

_{sol}

*c*/

*ω*, as function of

*λ*, we find that

*n*

_{eff/sol}tends to approach the effective index of the surface mode if

*w*> 0 and of the core mode if

*w*< 0, see Fig. 1(a). Therefore, if

*w*> 0, then the larger portion of energy is concentrated inside the glass. This reduces the peak powers required to support solitons down to 10kW and below, see Fig. 2(b).

18. F. Biancalana, D.V. Skryabin, and A.V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E **70**, 016615 (2004). [CrossRef]

**12**, 394–400 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394 [CrossRef] [PubMed]

**5000**, 161–174 (2003). [CrossRef]

## 4. Excitation of solitons

*R*(

*t*) = [1 -

*θ*]Δ(

*t*) +

*θα*Θ(

*t*) exp(-

*t*/

*τ*

_{2}) sin(

*t*/

*τ*

_{1}). Here

*α*= [

*τ*

_{1}

*t*) and Θ(

*t*) are, respectively, delta and Heaviside functions,

*θ*= 0.18,

*τ*

_{1}= 12.2

*fs*/

*τ*,

*τ*

_{2}= 32

*fs*/

*τ*[15]. For

*θ*= 0, i.e. without the Raman effect, Eq. (6) transforms into Eq. (4). To solve Eqs. (3), (6) we used split-step method. At the first step we solved decoupled linear equations

*∂*

_{z}

*F*

_{c,s}±

*∂*

_{t}

*F*

_{c,s}= 0. At the second step, we took account of the linear coupling and at the third step, we solved the nonlinear part of the equation for the surface mode. The integral in Eq. (6) has been calculated in the Fourier domain using the convolution theorem. We have also used absorbing boundary conditions in order to minimize reflection of the radiation from the boundaries.

*θ*= 0 we have checked that the solitons found as stationary solutions of Eqs. (5) show stable propagation in

*z*for various values of

*q*and

*w*. This has also given us confidence in reliability of our numerical approach. Then, to check possibility of the experimental excitation of the core-surface solitons we have taken simplest and probably most practical initial conditions, when the pump pulse couples only to the core state, i.e.

*F*

_{s}= 0 for

*z*= 0. In Fig. 3 we present results of the numerical modelling with the 1ps pump pulse having 100kW peak power. One can see, that the initial pulse quickly couples to the surface mode. The latter acquires well pronounced localized component and some dispersive radiation. It is clear that already after about 1cm of propagation the coupled core-surface soliton is formed and propagates further without significant distortion of its shape. For longer pulses we have observed excitation of two or more solitons. An example of propagation with 5ps pump pulse is shown in Fig. 4. We have checked that all the excited solitons retain the pump frequency, which indicates that the characteristic length at which solitons are formed is much shorter than the length at which Raman effect becomes noticeable.

## References and links

1. | P.St.J. Russell, “Photonic crystal fibers,” Science |

2. | C.M. Smith, N. Venkataraman, M.T. Gallagher, D. Müller, J.A. West, N.F. Borrelli, D.C. Alan, and K.W. Koch, “Low-loss hollow-core silica/air photonic bandgap fiber,” Nature |

3. | F. Benabid, J.C. Knight, G. Antonopoulos G, and P.S.J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science |

4. | S.O. Konorov, A.B. Fedotov, and A.M. Zheltikov, “Enhanced four-wave mixing in a hollow-core photonic-crystal fiber,” Opt. Lett. |

5. | D.G. Ouzounov, F.R. Ahmad, D. Müller, N. Venkataraman, M.T. Gallagher, M.G. Thomas, J. Silcox, K.W. Koch, and A.L. Gaeta, “Generation of megawatt optical solitons in hollow-core photonic band-gap fibers,” Science |

6. | F. Luan, J.C. Knight, P.S.J. Russell, S. Campbell, D. Xiao, D.T. Reid, B.J. Mangan, D.P. Williams, and P.J. Roberts, “Femtosecond soliton pulse delivery at 800nm wavelength in hollow-core photonic bandgap fibers,” Opt. Express |

7. | D.C. Alan, N.F. Borrelli, M.T. Gallagher, D. Müller, C.M. Smith, N. Venkataraman, J.A. West, P. Zhang, and K.W. Koch, “Surface modes and loss in air-core photonic band-gap fibers,” Proc. of SPIE |

8. | K. Saitoh, N.A. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express |

9. | J.A. West, C.M. Smith, N.F. Borrelli, D.C. Alan, and K.W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express |

10. | G. Humbert, J.C. Knight, G. Bouwmans, P.St.J. Russell, D.P. Williams, P.J. Roberts, and B.J. Mangan, “Hollow core photonic crystal fibers for beam delivery,” |

11. | D.L. Miles, |

12. | V.M. Agranovich, V.I. Rupasov, and V.Y. Chernyak, “Self-induced transparency of surface-polaritons,” JETP Lett. |

13. | V.M. Agranovich, D.M. Basko, A.D. Boardman, A.M. Kamchatnov, and T.A. Leskova, “Surface solitons due to second order cascaded nonlinearity,” Opt. Commun. |

14. | C.M. de Sterke and J.E. Sipe, “Coupled modes and the nonlinear Schrodinger-equation,” Phys. Rev. A |

15. | G.P. Agrawal, |

16. | S. Wabnitz, “Forward mode-coupling in periodic nonlinear-optical fibers - Modal dispersion cancellation and resonance solitons,” Opt. Lett. |

17. | G. Van Simaes, S. Coen, M. Haelterman, and S. Trillo, “Observation of resonance soliton trapping due to a photoinduced gap in wave number,” Phys. Rev. Lett. |

18. | F. Biancalana, D.V. Skryabin, and A.V. Yulin, “Theory of the soliton self-frequency shift compensation by the resonant radiation in photonic crystal fibers,” Phys. Rev. E |

**OCIS Codes**

(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons

(240.4350) Optics at surfaces : Nonlinear optics at surfaces

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 31, 2004

Revised Manuscript: September 22, 2004

Published: October 4, 2004

**Citation**

Dmitry Skryabin, "Coupled core-surface solitons in photonic crystal fibers," Opt. Express **12**, 4841-4846 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4841

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### References

- P. St.J. Russell, "Photonic crystal fibers,�?? Science 299, 358-362 (2003). [CrossRef] [PubMed]
- C.M. Smith, N. Venkataraman, M.T. Gallagher, D. Müller, J.A. West, N.F. Borrelli, D.C. Alan, and K.W. Koch, "Low-loss hollow-core silica/air photonic bandgap fiber,�?? Nature 424, 657-659 (2003). [CrossRef] [PubMed]
- F. Benabid, J.C. Knight, G. Antonopoulos G, P.S.J. Russell, "Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,�?? Science 298, 399-402 (2002). [CrossRef] [PubMed]
- S.O. Konorov, A.B. Fedotov, and A.M. Zheltikov, "Enhanced four-wave mixing in a hollow-core photonic-crystal fiber,�?? Opt. Lett. 28, 1448-1450 (2003). [CrossRef] [PubMed]
- D.G. Ouzounov, F.R. Ahmad, D. Müller, N. Venkataraman, M.T. Gallagher, M.G. Thomas, J. Silcox, K.W. Koch, A.L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers,�?? Science 301, 1702-1704 (2003). [CrossRef] [PubMed]
- F. Luan, J.C. Knight, P.S.J. Russell, S. Campbell, D. Xiao, D.T. Reid, B.J. Mangan, D.P. Williams, and P.J. Roberts, "Femtosecond soliton pulse delivery at 800nm wavelength in hollow-core photonic bandgap fibers,�?? Opt. Express 12, 835-840 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-835">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-835</a>. [CrossRef] [PubMed]
- D.C. Alan, N.F. Borrelli, M.T. Gallagher, D. Müller, C.M. Smith, N. Venkataraman, J.A. West, P. Zhang, and K.W. Koch, "Surface modes and loss in air-core photonic band-gap fibers,�?? Proc. of SPIE 5000, 161-174 (2003). [CrossRef]
- K. Saitoh, N.A. Mortensen, M. Koshiba, "Air-core photonic band-gap fibers: the impact of surface modes,�?? Opt. Express 12, 394-400 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394</a>. [CrossRef] [PubMed]
- J.A. West, C.M. Smith, N.F. Borrelli, D.C. Alan, and K.W. Koch, "Surface modes in air-core photonic band-gap fibers,�?? Opt. Express 12, 1485-1496 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485</a>. [CrossRef] [PubMed]
- G. Humbert, J.C. Knight, G. Bouwmans, P.St.J. Russell, D.P. Williams, P.J. Roberts, and B.J. Mangan, �??Hollow core photonic crystal fibers for beam delivery,�?? 12 1477-1484, (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1477">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1477</a>.
- D.L. Miles, Nonlinear Optics (Springer, Berlin, 1998). [CrossRef]
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- V.M. Agranovich, D.M. Basko, A.D. Boardman, A.M. Kamchatnov, T.A. Leskova, �??Surface solitons due to second order cascaded nonlinearity,�?? Opt. Commun. 160, 114-118 (1999). [CrossRef]
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