## 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] are fabricated now with the level of losses ~ 10dB/km [2] and below. These fibers are already used in various nonlinear optical applications and have potential to find their niche in communication technologies [1]. 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, 4]. Another interesting nonlinear application of such fibers is delivery of megawatt optical solitons over long distances [5, 6].

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, 7, 8, 9, 10]. 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, 9]. This theory explains experimentally measured variations of the PCF losses with wavelength [2, 7]. 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]. One of the interesting objects, which can exist at nonlinear interfaces is the optical surface soliton, see e.g., [12, 13]. 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] 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, 6].

## 2. Model

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

Here *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], 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. *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).

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, 5, 7, 8, 9], which all study very similar PCFs. We take *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, 8] there is a zero GVD point of the core mode at *λ*
_{0} ≃ 1425nm, which can be matched by taking *D*_{c}
(*i∂*_{T}
) = -*i$\tilde{\beta}$*
_{2}
${\mathit{\partial}}_{T}^{2}$ + *$\tilde{\beta}$*
_{3}
${\mathit{\partial}}_{T}^{3}$ with *$\tilde{\beta}$*
_{2} ≃ - 0.036ps^{2}/m and $\tilde{\beta}$
_{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]. The focus of this paper, however, is the spectral proximity of the avoided crossing, where the *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 *$\tilde{\beta}$*
_{2} entering into *D*_{c,s}
terms.

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^{-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 ${\gamma}_{c}\simeq \frac{2\pi}{{\lambda}_{\mathit{ref}}}\frac{{n}_{2c}}{{S}_{c}}~{10}^{-6}{W}^{-1}{m}^{-1}$, which matches the value reported in [5, 6]. The realistic estimate for the area of the silica interface between the core and cladding is *S*_{s}
≃ 1*μ*m^{2}. Thus, the nonlinear parameter for the surface mode is ${\gamma}_{s}\simeq \frac{2\pi}{{\lambda}_{\mathit{ref}}}\frac{{n}_{2s}}{{S}_{s}}~{10}^{-1}{W}^{-1}{m}^{-1}$. After some algebra one can estimate the nonlinear cross-coupling of the core mode to the surface mode as *γ*_{cs}
≃ *ε*_{c}
*γ*
_{0}, where ${\gamma}_{0}\simeq \frac{2\pi}{{\lambda}_{\mathit{ref}}}\frac{{n}_{2c}{S}_{c}+{n}_{2s}{S}_{s}}{{S}_{c}{S}_{s}}~{10}^{-3}{W}^{-1}{m}^{-1}$. Here *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] 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 *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.

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

Here *v* = 2/[*α*_{s}
- *α*_{c}
],*t* = [*T* - *αZ*]/*τ*, *τ* = 1/[|*v*|*κ*], *α* = [*α*_{c}
+ *α*_{s}
]/2, *z* = *κZ*, ${F}_{c,s}=\frac{{A}_{c,s}}{\sqrt{P}}$, *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], which gives $\tilde{\Gamma}$
= 4 × 10^{-3}.

## 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, 16, 17], as opposite 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 respect to the permutation of the two fields nonlinear response occurring in the conventional fiber systems [14, 16, 17].

We seek localized solutions of Eqs. (3,4) in the form *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

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 $-\sqrt{1-{q}^{2}}<w<0$ the surface component of the soliton contains less power than the core component and it is vise versa for $0<w<\sqrt{1-{q}^{2}}$, 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 *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).

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]. 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] and in Fig. 7 in [7] are suitable for observation of the core-surface solitons.

## 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

where R(t) is the response function: *R*(*t*) = [1 - *θ*]Δ(*t*) + *θα*Θ(*t*) exp(- *t*/*τ*
_{2}) sin(*t*/*τ*
_{1}). Here *α* = [${\tau}_{1}^{2}$ + ${\tau}_{2}^{2}$]/[*τ*
_{1}
${\tau}_{2}^{2}$], Δ(*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.

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. *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.

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.

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