Coupled core-surface solitons in photonic crystal fibers

Coupled core-surface solitons inphotonic crystal fibers

Dmitry V. SkryabinDepartment of Physics, University of Bath, Bath BA2 7AY, UK

[email protected]

http://staff.bath.ac.uk/pysdvs/

Abstract: We predict existence and study properties of the coupledcore-surface solitons in hollow-core photonic crystal fibers. These solitonsexist in the spectral proximity of the avoided crossings of the propagationconstants of the modes guided in the air core and at the interface betweenthe core and photonic crystal cladding.

© 2004 Optical Society of America

OCIS codes: (060.5530) Pulse propagation and solitons

References and links1. P.St.J. Russell, ”Photonic crystal fibers,” Science 299, 358-362 (2003).2. C.M. Smith, N. Venkataraman, M.T. Gallagher, D. Muller, 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).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).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).5. D.G. Ouzounov, F.R. Ahmad, D. Muller, 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).

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

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

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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394

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

10. G. Humbert, J.C. Knight, G. Bouwmans, P.St.J. Russell, D.P. Williams, P.J. Roberts, andB.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

11. D.L. Miles, Nonlinear Optics (Springer, Berlin, 1998).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).14. C.M. de Sterke and J.E. Sipe, ”Coupled modes and the nonlinear Schrodinger-equation,” Phys. Rev. A 42, 550-

555 (1990).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).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).

(C) 2004 OSA 4 October 2004 / Vol. 12, No. 20 / OPTICS EXPRESS 4841#5156 - $15.00 US Received 31 August 2004; revised 22 September 2004; accepted 23 September 2004

mailto:[email protected]

http://staff.bath.ac.uk/pysdvs/

http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-835




18. F. Biancalana, D.V. Skryabin, A.V. Yulin, ”Theory of the soliton self-frequency shift compensation by the reso-nant radiation in photonic crystal fibers,” Phys. Rev. E 70, 016615 (2004).

1. Introduction

Optical fibers with hollow cores and photonic crystal cladding [1] are fabricated now with thelevel of losses ∼ 10dB/km [2] and below. These fibers are already used in various nonlinearoptical applications and have potential to find their niche in communication technologies [1].For example, hollow-core photonic crystal fibers (PCFs) filled with gases allow enhancementof the efficiency of the stimulated Raman scattering and four-wave mixing by many orders ofmagnitude [3, 4]. Another interesting nonlinear application of such fibers is delivery of mega-watt optical solitons over long distances [5, 6].

An important feature of the guided modes in the hollow-core PCFs is existence of the avoidedcrossings of the propagation constants of the modes guided in the fiber core with the surfacemodes guided at the interface between the core and photonic crystal cladding [2, 7, 8, 9, 10].An elegant coupled mode theory describing linear coupling between the modes localized insidethe core and at the core walls has been reported in [7, 9]. This theory explains experimentallymeasured variations of the PCF losses with wavelength [2, 7]. Note, that the surface modes arethe guided modes, not the leaky ones. Excitation of the surface modes increases the loss levelbecause their overlap with the leaky cladding modes is greater than the overlap between thecore and cladding modes.

Of course, optical effects at interfaces have been an active research area well before advent ofPCFs [11]. One of the interesting objects, which can exist at nonlinear interfaces is the opticalsurface soliton, see e.g., [12, 13]. Unfortunately, existence of these structures still lucks its clearexperimental verification. Main and obvious practical difficulties preventing their observationare 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 betweenair and photonic crystal have been observed in hollow-core PCFs [10] and, as it is demonstratedbelow, nonlinear coefficient of the surface mode is large enough to support solitons at the pumppowers reachable with available laser sources.

The aim of this paper is to study nonlinear regimes of pulse propagation and solitons inhollow-core PCFs pumped in the proximity of the avoided crossing. It’ll be demonstrated thatcombined action of the linear coupling between the core and surface modes on one side and Kerrnonlinearity of the interface between the core and cladding on the other can support coupledcore-surface solitons. Peak powers required for observation of these structures can be one totwo orders of magnitude less than the mega-watt powers required for excitation of the coresolitons with the central frequency detuned far from the avoided crossing [5, 6].

2. Model

Model we use for theoretical and numerical analysis is the extension of the existing lineartheory [7, 9]:

∂ZAc +αc∂T Ac − iκ As = iDc(i∂T )Ac + iNc, (1)

∂ZAs +αs∂T As − iκ Ac = iDs(i∂T )As + iNs −ΓAs. (2)

Here Ac and As are the slowly varying envelopes of the core and surface states. The referencefrequency ωre f 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 notcoupled linearly. However, here, as in many other physical contexts [14], introduction of thelinearly 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 isthe coupling between the core and surface states. Γ is the loss arising from the coupling of thesurface state with cladding modes. αc,s are the slopes of the graphs of the propagation constantsof the core and surface states as functions of frequency. Nc,s are the nonlinear responses of thesurface and core states. Operators Dc,s(i∂T ) describe dispersions of the second order and higher.For large detunings from ωre f each of the supermodes asymptotically tends either to the puresurface or to the pure core mode, see Fig. 1(a).

Equations (1-2) are quite general and can be substantially simplified after the balance ofdifferent terms is taken into consideration for the realistic values of the PCF parameters. Toestimate the values of parameters in Eqs. (1,2) we use data from Refs. [2, 5, 7, 8, 9], which allstudy very similar PCFs. We take λre f = 2πc/ωre f = 1580nm, αc = 1.01/c, αs = 1.4/c andκ = 103m−1, where c = 3×108m/s. Effective refractive indices for the two supermodes and cor-responding group velocity dispersions (GVDs) derived from Eqs. (1,2) for Dc,s = 0 are shownin Fig. 1. According to [5, 8] there is a zero GVD point of the core mode at λ0 � 1425nm,which can be matched by taking Dc(i∂T ) = −iβ2∂ 2

T + β3∂ 3T with β2 � −0.036ps2/m and

β3 = 0.0001ps3/m. One can show that the Dc term becomes important only for detunings fromωre f approaching 2π× 15THz. In this frequency range the single mode generalized nonlinearSchrodinger equation for the amplitude of the core mode can be used to model propagation offemto-second solitons at mega-watt powers [5]. The focus of this paper, however, is the spectralproximity of the avoided crossing, where the Dc,s terms can be safely neglected. This is becausevalues 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 β2 entering into Dc,s

terms.

Fig. 1. (a) Black lines (full and dashed) show the effective refractive indices of the twosupermodes undergoing avoided crossing at 1580nm. Straight red, blue and green linesshow effective refractive indices of the Fourier components of the coupled core-surfacesoliton for q = 0 and different values of w. (b) Group velocity dispersion parameter β2 asfunction of the wavelength for the same supermodes.

Now we turn our attention to the nonlinear properties of PCFs in the proximity of the avoidedcrossing. Typical Kerr nonlinearity of gases inside the fiber core, e.g., air is n2c � 3 ·10−23m2/W,which is 3 orders of magnitude lower than n2 for silica, n2s � 2.4 · 10−20m2/W. Assuming theeffective area of the core mode is Sc � 60µm2, we find that the nonlinear parameter γ [15] ofthe core mode is γc � 2π

λre f

n2cSc

∼ 10−6W−1m−1, which matches the value reported in [5, 6]. 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 γs � 2π

λre f

n2sSs

∼ 10−1W−1m−1. After some

algebra one can estimate the nonlinear cross-coupling of the core mode to the surface modeas γcs � εcγ0, where γ0 � 2π

λre f

n2cSc+n2sSsScSs

∼ 10−3W−1m−1. Here εc is the phenomenological

coefficient characterizing the ratio of the intensity of the core mode at the interface to theintensity maximum of the core mode. The estimate for the nonlinear cross-coupling of the


surface mode to the core mode is γsc � εsγ0, where εs characterizes the ratio of the intensityof the core mode at the interface to the intensity maximum of the surface mode. Values ofεc,s depend on the fiber design [10] and should be evaluated on the case by case basis. Here weconsider the typical situation, when the spread of the core mode to the silica is reasonably smalland εc,s are order or less than 0.1, i.e., γcs,sc ∼ 10−4W−1m−1. Thus, γc,cs,sc/γs � 1 and thereforewe can safely assume in our calculations that Ns = γs|As|2As and Nc = 0. Note, however, thatwe have verified robustness of the numerical results presented below by introducing slightlyexaggerated, upto 0.05γs, coefficients of the nonlinear cross-coupling between the core andsurface modes.

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

∂zFc − sgn(v)∂tFc − iFs = 0, (3)

∂zFs + sgn(v)∂tFs − iFc = i|Fs|2Fs − ΓFs. (4)

Here v = 2/[αs −αc], t = [T −α Z]/τ , τ = 1/[|v|κ ], α = [αc +αs]/2, z = κ Z, Fc,s = Ac,s/√

P,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], which gives Γ = 4×10−3.

Fig. 2. (a) Temporal profiles of the amplitudes of the core (full line) and surface (dashedline) components of the coupled core-surface soliton for q = −0.7 and w = −0.5. (b) De-pendencies of the peak power vs FWHM for the core surface solitons calculated for q = 0and different values of w. Full green lines correspond to the core and dashed red lines tothe surface components, respectively. For q = w = 0 the amplitude profiles of the core andsurface 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 modesbelong to the general class of models exhibiting the wavenumber band-gap [14, 16, 17], asopposite to the frequency band-gap for the coupled counter-propagating waves [11, 14]. Note,that the nonlinear parts of Eqs. (3,4) are substantially different from the symmetric with respectto the permutation of the two fields nonlinear response occurring in the conventional fibersystems [14, 16, 17].

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

i[w+1]∂ξ fc = fs −q fc, i[w−1]∂ξ fs = fc −q fs + | fs|2 fs. (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 q2 + w2 < 1. We have confirmed numerically that the latter inequality gives theexistence boundary for the soliton solutions. For −

√1−q2 < w < 0 the surface component

of the soliton contains less power than the core component and it is vise versa for 0 < w <√1−q2, 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 Fourierharmonics. Propagation constant of the harmonic with frequency ω is given by βsol(ω) = β0 +qκ +{ω−ωre f }{wκ − [αc +αs]/2}, where β0 is the propagation constant at the central pointof the gap. Then by plotting the effective refractive index of the soliton, ne f f/sol = βsolc/ω, asfunction of λ , we find that ne f f/sol tends to approach the effective index of the surface mode ifw > 0 and of the core mode if w < 0, see Fig. 1(a). Therefore, if w > 0, then the larger portionof energy is concentrated inside the glass. This reduces the peak powers required to supportsolitons down to 10kW and below, see Fig. 2(b).

Note, that an important condition for the soliton solutions reported above to exist in their idealform is that neither of the two supermodes in question should have the second avoided crossingin the spectral proximity of the first one. More precisely the frequency detuning between the twoavoided crossings of the same supermode should be at least greater than characteristic spectralwidth of the soliton. Presence of the second crossing will imply existence of the resonancebetween the linear dispersive wave and soliton. These kind of resonances do not usually destroysolitons, but lead to the emission of radiation [18]. Detail study of this effect, however, goesbeyond our present scope. Typical spectral width of the solitons reported above is less or orderof 10nm and therefore PCFs with dispersion characteristics shown, e.g., in Fig. 2 in [8] and inFig. 7 in [7] are suitable for observation of the core-surface solitons.

010

2030

4050 0

2040

60

0

50

100 (a)

propagation distance [mm]

time delay, [ps]

|Ac|2 ,

[kW

]

010

2030

4050 0

2040

60

0

20

40 (b)


time delay, [ps]

|As|2 ,

[kW

]

Fig. 3. Results of the numerical modelling of Eqs. (3,6) showing z evolution of the squaredamplitudes 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 thered lines in (a) and (b). Peak pump power is 100kW, pump wavelength is 1580nm and pulseduration is 1ps.

4. Excitation of solitons

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

∂zFs + sgn(v)∂tFs − iFc = iFs

∫ +∞

−∞R(t ′)|Fs(t − t ′,z)|2dt ′ − ΓFs, (6)


010

2030

4050 0

2040

60

0

200

400


(a)

time delay, [ps]|A

c|2 , [k

W]

010

2030

4050 0

2040

60

0

50

100 (b)


time delay, [ps]

|As|2 ,

[kW

]

Fig. 4. The same as Fig. 3, but with 5ps pump pulse.

where R(t) is the response function: R(t) = [1−θ]∆(t)+ θαΘ(t)exp(−t/τ2)sin(t/τ1). Hereα = [τ 2

1 + τ 22 ]/[τ1τ 2

2 ], ∆(t) and Θ(t) are, respectively, delta and Heaviside functions, θ = 0.18,τ1 = 12.2 f s/τ , τ2 = 32 f s/τ [15]. For θ = 0, i.e. without the Raman effect, Eq. (6) transformsinto Eq. (4). To solve Eqs. (3), (6) we used split-step method. At the first step we solved de-coupled linear equations ∂zFc,s ± ∂tFc,s = 0. At the second step, we took account of the linearcoupling 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 theo-rem. We have also used absorbing boundary conditions in order to minimize reflection of theradiation 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 confi-dence in reliability of our numerical approach. Then, to check possibility of the experimentalexcitation of the core-surface solitons we have taken simplest and probably most practical ini-tial conditions, when the pump pulse couples only to the core state, i.e. Fs = 0 for z = 0. InFig. 3 we present results of the numerical modelling with the 1ps pump pulse having 100kWpeak power. One can see, that the initial pulse quickly couples to the surface mode. The latteracquires well pronounced localized component and some dispersive radiation. It is clear thatalready after about 1cm of propagation the coupled core-surface soliton is formed and prop-agates further without significant distortion of its shape. For longer pulses we have observedexcitation of two or more solitons. An example of propagation with 5ps pump pulse is shown inFig. 4. We have checked that all the excited solitons retain the pump frequency, which indicatesthat the characteristic length at which solitons are formed is much shorter than the length atwhich Raman effect becomes noticeable.

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


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Coupled core-surface solitons in photonic crystal fibers