OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 20 — Oct. 4, 2004
  • pp: 4841–4846
« Show journal navigation

Coupled core-surface solitons in photonic crystal fibers

Dmitry V. Skryabin  »View Author Affiliations


Optics Express, Vol. 12, Issue 20, pp. 4841-4846 (2004)
http://dx.doi.org/10.1364/OPEX.12.004841


View Full Text Article

Acrobat PDF (789 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

Optical fibers with hollow cores and photonic crystal cladding [1

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

] are fabricated now with the level of losses ~ 10dB/km [2

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]

] and below. These fibers are already used in various nonlinear optical applications and have potential to find their niche in communication technologies [1

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

]. For example, hollow-core photonic crystal fibers (PCFs) filled with gases allow enhancement of the efficiency of the stimulated Raman scattering and four-wave mixing by many orders of magnitude [3

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

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]

]. Another interesting nonlinear application of such fibers is delivery of megawatt optical solitons over long distances [5

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

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]

].

An important feature of the guided modes in the hollow-core PCFs is existence of the avoided crossings of the propagation constants of the modes guided in the fiber core with the surface modes guided at the interface between the core and photonic crystal cladding [2

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

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

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

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

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,” 121477–1484, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1477

]. An elegant coupled mode theory describing linear coupling between the modes localized inside the core and at the core walls has been reported in [7

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

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]

]. This theory explains experimentally measured variations of the PCF losses with wavelength [2

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

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]

]. Note, that the surface modes are the guided modes, not the leaky ones. Excitation of the surface modes increases the loss level because their overlap with the leaky cladding modes is greater than the overlap between the core and cladding modes.

Of course, optical effects at interfaces have been an active research area well before advent of PCFs [11

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

]. One of the interesting objects, which can exist at nonlinear interfaces is the optical surface soliton, see e.g., [12

12. V.M. Agranovich, V.I. Rupasov, and V.Y. Chernyak, “Self-induced transparency of surface-polaritons,” JETP Lett. 33, 185–188 (1981).

, 13

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]

]. Unfortunately, existence of these structures still lucks its clear experimental verification. Main and obvious practical difficulties preventing their observation are the low levels of nonlinearities and small life-times of the surface waves. It seems, however, that these problems can be overcome in PCFs. Indeed, surfaces modes at the interface between air and photonic crystal have been observed in hollow-core PCFs [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,” 121477–1484, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1477

] and, as it is demonstrated below, nonlinear coefficient of the surface mode is large enough to support solitons at the pump powers reachable with available laser sources.

The aim of this paper is to study nonlinear regimes of pulse propagation and solitons in hollow-core PCFs pumped in the proximity of the avoided crossing. It’ll be demonstrated that combined action of the linear coupling between the core and surface modes on one side and Kerr nonlinearity of the interface between the core and cladding on the other can support coupled core-surface solitons. Peak powers required for observation of these structures can be one to two orders of magnitude less than the mega-watt powers required for excitation of the core solitons with the central frequency detuned far from the avoided crossing [5

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

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

Model we use for theoretical and numerical analysis is the extension of the existing linear theory [7

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

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]

]:

ZAc+αcTAcAs=iDc(iT)Ac+i𝓝c,
(1)
ZAs+αsTAsAc=iDs(iT)As+i𝓝sΓAs.
(2)

Here Ac and As are the slowly varying envelopes of the core and surface states. The reference frequency ωref 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]

], introduction of the linearly coupled basis states helps to simplify theoretical treatment and provides more intuitive way for understanding of the problem. 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. αc,s are the slopes of the graphs of the propagation constants of the core and surface states as functions of frequency. 𝓝c,s are the nonlinear responses of the surface and core states. Operators Dc,s (i∂T ) describe dispersions of the second order and higher. For large detunings from ωref each of the supermodes asymptotically tends either to the pure surface or to the pure core mode, see Fig. 1(a).

Equations (1–2) are quite general and can be substantially simplified after the balance of different terms is taken into consideration for the realistic values of the PCF parameters. To estimate the values of parameters in Eqs. (1,2) we use data from Refs. [2

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]

, 5

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]

, 7

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

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

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]

], which all study very similar PCFs. We take λref = 2πc/ωref = 1580nm, αc = 1.01/c, αs = 1.4/c and κ = 103m-1, where c = 3 × 108m/s. Effective refractive indices for the two supermodes and corresponding group velocity dispersions (GVDs) derived from Eqs. (1,2) for Dc,s = 0 are shown in Fig. 1. According to [5

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]

, 8

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]

] there is a zero GVD point of the core mode at λ 0 ≃ 1425nm, which can be matched by taking Dc (i∂T ) = -iβ˜ 2 T2 + β˜ 3 T3 with β˜ 2 ≃ - 0.036ps2/m and β˜ 3 = 0.0001ps3/m. One can show that the Dc term becomes important only for detunings from ωref approaching 2π × 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

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]

]. The focus of this paper, however, is the spectral proximity of the avoided crossing, where the Dc,s terms can be safely neglected. This is because values of GVD created by the coupling of the modes in this spectral region are ~ 100ps2/m, see Fig. 1(b), which is four orders of magnitude larger than the correction β˜ 2 entering into Dc,s terms.

Fig. 1. (a) Black lines (full and dashed) show the effective refractive indices of the two supermodes undergoing avoided crossing at 1580nm. Straight red, blue and green lines show effective refractive indices of the Fourier components of the coupled core-surface soliton for q = 0 and different values of w. (b) Group velocity dispersion parameter β 2 as function of the wavelength for the same supermodes.

Now we turn our attention to the nonlinear properties of PCFs in the proximity of the avoided crossing. Typical Kerr nonlinearity of gases inside the fiber core, e.g., air is n 2c ≃ 3 · 10-23m2/W, which is 3 orders of magnitude lower than n 2 for silica, n 2s ≃ 2.4 · 10-20m2/W. Assuming the effective area of the core mode is Sc ≃ 60μm2, we find that the nonlinear parameter γ [15

15. G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).

] of the core mode is γc2πλrefn2cSc~106W1m1, which matches the value reported in [5

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

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]

]. The realistic estimate for the area of the silica interface between the core and cladding is Ss ≃ 1μm2. Thus, the nonlinear parameter for the surface mode is γs2πλrefn2sSs~101W1m1. After some algebra one can estimate the nonlinear cross-coupling of the core mode to the surface mode as γcsεc γ 0, where γ02πλrefn2cSc+n2sSsScSs~103W1m1. Here εc 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 γscεs γ 0, where εs characterizes the ratio of the intensity of the core mode at the interface to the intensity maximum of the surface mode. Values of εc,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,” 121477–1484, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1477

] and should be evaluated on the case by case basis. Here we consider the typical situation, when the spread of the core mode to the silica is reasonably small and εc,s are order or less than 0.1, i.e., γcs,sc ~ 10-4W-1m-1. Thus, γc,cs,sc /γs ≪ 1 and therefore we can safely assume in our calculations that 𝓝s = γs |As |2 As and 𝓝c = 0. Note, however, that we have verified robustness of the numerical results presented below by introducing slightly exaggerated, upto 0.05γs , coefficients of the nonlinear cross-coupling between the core and surface modes.

After rescalling to dimensionless units we transform Eqs. (1,2) to

zFcsgn(v)tFciFs=0,
(3)
zFs+sgn(v)tFsiFc=iFs2FsΓ˜Fs.
(4)

Here v = 2/[αs - αc ],t = [T - αZ]/τ, τ = 1/[|v|κ], α = [αc + αs ]/2, z = κZ, Fc,s=Ac,sP, P = κ/γs 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

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]

], which gives Γ˜ = 4 × 10-3.

Fig. 2. (a) Temporal profiles of the amplitudes of the core (full line) and surface (dashed line) components of the coupled core-surface soliton for q = -0.7 and w = -0.5. (b) Dependencies of the peak power vs FWHM for the core surface solitons calculated for q = 0 and different values of w. Full green lines correspond to the core and dashed red lines to the surface components, respectively. For q = w = 0 the amplitude profiles of the core and surface components are identical, which explains overlap of the two lines.

3. Core-surface solitons

Equations (1,2) and (3,4) describing evolution of the co-propagating core and surface modes belong to the general class of models exhibiting the wavenumber band-gap [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]

, 16

16. S. Wabnitz, “Forward mode-coupling in periodic nonlinear-optical fibers - Modal dispersion cancellation and resonance solitons,” Opt. Lett. 141071–1073 (1989). [CrossRef] [PubMed]

, 17

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]

], as opposite to the frequency band-gap for the coupled counter-propagating waves [11

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

, 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]

]. Note, that the nonlinear parts of Eqs. (3,4) are substantially different from the symmetric with respect to the permutation of the two fields nonlinear response occurring in the conventional fiber systems [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]

, 16

16. S. Wabnitz, “Forward mode-coupling in periodic nonlinear-optical fibers - Modal dispersion cancellation and resonance solitons,” Opt. Lett. 141071–1073 (1989). [CrossRef] [PubMed]

, 17

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]

].

We seek localized solutions of Eqs. (3,4) in the form Fc,s (z,t) = fc,s (ξ)eiqz , where ξ = t - wz and q measures the detuning of the wavenumber from the gap center. Solving the system of ordinary differential equations

i[w+1]ξfc=fsqfc,i[w1]ξfs=fcqfs+fs2fs.
(5)

numerically we have found a family of solutions representing the coupled core-surface solitons. Typical temporal profiles of these solitons are shown in Fig. 2(a) and dependencies of the peak powers vs full width at half maximum (FWHM) of the amplitude are shown in Fig. 2(b). Tails of the bright solitons are naturally expected to decay exponentially to 0 for |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 1q2<w<0 the surface component of the soliton contains less power than the core component and it is vise versa for 0<w<1q2, see Fig. 2. In order to understand why parameter 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 βsol (ω) = β 0 + + {ω - ωref }{ - [αc + αs ]/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, neff/sol = βsolc/ω, as function of λ, we find that neff/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).

Note, that an important condition for the soliton solutions reported above to exist in their ideal form is that neither of the two supermodes in question should have the second avoided crossing in the spectral proximity of the first one. More precisely the frequency detuning between the two avoided crossings of the same supermode should be at least greater than characteristic spectral width of the soliton. Presence of the second crossing will imply existence of the resonance between the linear dispersive wave and soliton. These kind of resonances do not usually destroy solitons, but lead to the emission of radiation [18

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]

]. Detail study of this effect, however, goes beyond our present scope. Typical spectral width of the solitons reported above is less or order of 10nm and therefore PCFs with dispersion characteristics shown, e.g., in Fig. 2 in [8

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]

] and in Fig. 7 in [7

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]

] are suitable for observation of the core-surface solitons.

Fig. 3. Results of the numerical modelling of Eqs. (3,6) showing z evolution of the squared amplitudes of the core (a) and surface (b) modes resulting in formation of the coupled core-surface soliton. Only the core mode is excited initially. Initial conditions are shown by the red lines in (a) and (b). Peak pump power is 100kW, pump wavelength is 1580nm and pulse duration is 1ps.

4. Excitation of solitons

After existence of the coupled core-surface solitons has been established, the natural problem to study is whether they can be excited by sech-like laser pulses. In order to check this we have carried out series of numerical experiments. To be closer to reality we have also included Raman nonlinearity of the glass, i.e., we replaced Eqs. (4) with

zFs+sgn(v)tFsiFc=iFs+R(t)Fs(tt,z)2dtΓ˜Fs,
(6)
Fig. 4. The same as Fig. 3, but with 5ps pump pulse.

where R(t) is the response function: R(t) = [1 - θ]Δ(t) + θαΘ(t) exp(- t/τ 2) sin(t/τ 1). Here α = [τ12 + τ22]/[τ 1 τ22], Δ(t) and Θ(t) are, respectively, delta and Heaviside functions, θ = 0.18, τ 1 = 12.2fs/τ, τ 2 = 32fs/τ [15

15. G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).

]. 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 zFc,s ± tFc,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.

First, by taking 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. Fs = 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.

In summary: We predicted existence and demonstrated feasibility of experimental observation of the coupled core-surface solitons in hollow-core photonic crystal fibers. Author acknowledges discussions with D. Bird and J. Knight.

References and links

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]

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]

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,” 121477–1484, (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1477

11.

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

12.

V.M. Agranovich, V.I. Rupasov, and V.Y. Chernyak, “Self-induced transparency of surface-polaritons,” JETP Lett. 33, 185–188 (1981).

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]

14.

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

15.

G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).

16.

S. Wabnitz, “Forward mode-coupling in periodic nonlinear-optical fibers - Modal dispersion cancellation and resonance solitons,” Opt. Lett. 141071–1073 (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]

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]

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


Sort:  Journal  |  Reset  

References

  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]
  3. 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]
  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, 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), <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]
  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, 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]
  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), <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]
  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), <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>.
  11. D.L. Miles, Nonlinear Optics (Springer, Berlin, 1998). [CrossRef]
  12. V.M. Agranovich, V.I. Rupasov, and V.Y. Chernyak, "Self-induced transparency of surface-polaritons,�?? JETP Lett. 33, 185-188 (1981).
  13. 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]
  14. C.M. de Sterke and J.E. Sipe, "Coupled modes and the nonlinear Schrodinger-equation,�?? Phys. Rev. A 42, 550-555 (1990). [CrossRef]
  15. G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).
  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]
  18. F. Biancalana, D.V. Skryabin, 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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1. Fig. 2. Fig. 3.
 
Fig. 4.
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited