1. Introduction

One of the initial stimuli driving research into hollow-core photonic crystal fibers (HC-PCFs) has been development of waveguides where the guided light has only small overlap with solid materials and therefore nonlinearity and loss are minimized [1]. However, opportunities offered by HC-PCFs for research into new nonlinear and quantum optical effects are becoming a dominant current theme. Indeed instead of using gas cells or capillaries one can fill a HC-PCF with a required gas and conduct experiments on nonlinear [2, 3, 4, 5, 6] and quantum optics [7] in an environment where a constant intensity of the diffraction free beam is sustained along the desirable length. Atmospheric air naturally exists within HC-PCFs unless it has been removed or replaced with another gas. Nonlinear optical effects accompanying high energy pulse propagation in air have of course been well studied over the last decade [8]. HC-PCFs, however, have strong waveguide dispersion, which ensures existence of a wide spectral range of anomalous group velocity dispersion (GVD) on the long wavelength side of the fiber transparency window. Anomalous GVD in combinations with weak focusing Kerr nonlinearity (two-three orders of magnitude less, than in telecom fibers) makes possible formation of femtosecond solitons with megawatt peak powers [2, 3, 4, 5, 6]. The high power ultrashort pulses are not suited for transmission through the solid-corewaveveguides due to low material damage threshold and/or pulse splitting into weaker solitons. Thus HC-PCFs make possible practical use of solitons, e.g., in medical and micro-machining applications, where delivery of sufficiently high energy (upto hundreds of nJ) pulses is required. An important feature of the air-core PCFs is the Raman response of air. The Raman gain of air is stronger than the one of silica and the cumulative Raman gain line, mainly composed of quasi-continuum of rotational resonances in nitrogen and oxygen, is narrower [9], cf. Fig. 1(a) and 1(b). These differences lead to emergence of interesting quantitative and qualitative changes in the soliton dynamics in air-core fibers, relative their silica-core counterparts. These effects constitute the main subject of this work.

2. Model

The model we use to describe femtosecond pulse propagation in air-core PCFs includes nonlinear responses of silica in the cladding and of the air filled core. Each of the materials is assumed to have instantaneous Kerr nonlinearity and delayed Raman response. The dimensionless generalized nonlinear Schrödinger equation used by us is

and it is an extension of the model known for the silica only fibres [11, 12]. Subscripts a and s correspond to air and silica, respectively. t is the dimensionless time normalized to τ=200fs and measured in the coordinate system moving with the group velocity at the pump frequency. This time normalization makes one unit of the dimensional frequency δ used in the spectral plots equal to 0.79THz in physical units. z is the distance normalized to the dispersion length L=τ²/|$\tilde{\beta }$ ₂|, where $\tilde{\beta }$ ₂ is the physical GVD (group velocity dispersion) parameter at the pump frequency. Choosing -57ps²/km as a realistic value of $\tilde{\beta }$ ₂, gives as L=70cm The field amplitude A is measured in the units of $\frac{1}{\sqrt{\gamma L}}$ , where we take γ=10^-5/W/m. β(i∂_t) accounts for the higher order dispersions. The latter become important if the pulse spectrum is either close to zero GVD point or to the boundary of the fiber transparency window, where GVD and linear losses acquire a very steep gradient [1, 2, 10]. Though it can be useful to include these effects into consideration, while analyzing experimental results, we prefer to neglect them here for the sake of clarity of our arguments focusing on the interplay of various physical effects originating from the different properties of the Raman response of air and silica glass. Thus we assume that the spectral evolution described below happens away from the regions of small or zero and sharply rising GVD values and take β(i∂_t)=0. Possible, but unlikely significant, effects such as self-steepening, two-photon absorbtion and plasma generation have also been neglected in the above model.

The Raman response functions of silica glass and air R_a,s can be presented in the form

Θ(t) is the Heaviside function. The Raman effect on solitons in silica fibres is well known [11, 13] and the corresponding Raman parameters are well established: τ^(s) ₁=12.2fs/τ, τ^(s) ₂=32 fs/τ, f_s=0.18 [11]. Though the Raman response of air over the relevant bandwidth consists of a set of discrete spectral lines associated mainly with rotational transitions in nitrogen and oxygen, it also can be well fitted by the function like (2) with τ^(a) ₁=62 fs/τ, τ^(a) ₂=77 fs/τ, f_a=0.5 [9].

Nonlinear coefficients γ_a,s are defined as: γ_i=[1/γ]·[n ⁽ⁱ⁾ ₂ ω ₀]/[cA ⁽ⁱ⁾ _eff], where A ⁽ⁱ⁾ _eff are the effective area parameters, characterising light overlap with silica and air [10, 11]. Apart from varying the values of γ_a,s by changing the fiber geometry, γ_a can also be increased/decreased by increasing/decreasing the gas pressure in the core. Two main cases to be compared below are the ones when either air or silica glass nonlinearity is dominant. γ_s=1, γ_a=0.1 corresponds to the silica-dominant fiber and γ_a=1, γ_s=0.1 to the air-dominant one. This comparison clearly illustrates new effects in the soliton propagation introduced by the Raman scattering in air. Note, that our choice of numbers for γ_a,s provides identical power normalization in the two cases.

3. Soliton self-frequency shift and spectral cut-off

We start our analysis describing dynamics of the single soliton initial conditions: $A(t,z=0)=N\sqrt{\frac{2q}{\left({\gamma }_a+{\gamma }_g\right)}}\mathrm{sech}\left(t\sqrt{2q}\right)$ , where N=1 and q>0 is the soliton parameter. For higher order dispersions neglected the above is the exact soliton amplitude, providing |A(t-t′)|² is replaced with |A(t)|²(∫dtR_i(t)=1).

 

Fig. 1. Raman gain functions of air (a) and silica (b) plotted together with spectra of q=2 (180fs FWHM, 0.5MW peak power) and q=20 (60 fs FWHM, 5MW peak power) solitons. Spectrum outside the gray interval is mainly transformed into non-solitonic radiation (spectral cut-off effect), see Section 3 for details.

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It is well known that in the silica fibers the Raman effect can be treated as a perturbation to the exact soliton solution and the resulting frequency shift can be worked out analytically [11]. There are two reasons for success of the text-book approach. First is the relative smallness of f_s and second is that spectra of solitons with pico- to femto-second durations feel only the linearly rising part of the Raman gain, so that its maximum is at 2π/τ^(s) ₁=13THz and subsequent decay can be ignored, see Fig. 1(b). In air f_a=0.5 and the Raman scattering is equally important with Kerr nonlinearity. Therefore, the perturbation analysis breaks down. Apart from that the spectral maximum of the Raman gain in air is at [2πτ^(a) ₁]^-1=2.6THz and, therefore, is much closer to the spectral maximum of the soliton, see Fig. 1(a). At first glance these factors suggest that the Raman induced soliton self-frequency should be enhanced by the air response. One of the central results of this work is that the latter assertion turns out to be true only for the relatively spectrally narrow input pulses, i.e. for those which are sufficiently wide in time. As soon as the spectral width of the pump becomes comparable with the distance between the maximum and minimum of the Raman gain line, it starts to suffer significant energy transfer into non-solitonic radiation. This readily results in notable reduction of the soliton self-frequency shift in the air dominant fiber, relative to the shift achieved in the silica-dominant fibers. This is explained by the fact that the Raman gain line in air is about 5 times narrower than the one in silica.

Figs. 2 (a–c) demonstrate sequence of cross-correlation spectrograms, S(δ,t, z)=ln|∫^+∞ _-∞ dt′G(t′-t)A(t′, z)e ^-iδt′|, for soliton evolution with q=20 (corresponding to 60fs of the full width at half maximum (FWHM) and to 5MW of the peak power) in the air-dominant fiber. G(t) is the reference Gaussian pulse with suitably chosen duration. One can see that after 1m of propagation the significant part of the initial spectrum splits from the soliton part of the signal. We call this effect spectral cut-off. In Figs. 2(a–c) the solitonic content of the pulse is easily distinguished from the non-solitonic one. The latter experiences dispersive spreading, while its central frequency remains at δ=0 and hence its trace in the (δ,t)-plane rotates. Contrary, GVD is suppressed for the soliton, hence all its frequency content propagates with the same group velocity, while the entire pulse undergoes the self-frequency shift, which leads to the soliton traces remaining vertical and shifting downwards.

Dramatic energy loss into non-solitonic radiation, clearly seen in Fig. 2(b), notably shrinks the soliton spectrum, which implies drop in its peak intensity, temporal broadening and reduction in its frequency shift. The bandwidth Δ_s of the surviving solitonic component can be roughly estimated as being less than the distance between the Raman gain extrema, i.e. Δ_s<[πτ^(a) ₁]^-1THz. Thus for the initial pulse to avoid the spectral cut-off its spectrum should be narrower than [πτ^(a,s) ₁]^-1. τ^(s) ₁ is five time larger than τ^(a) ₁. Therefore for the same initial conditions the spectral cut-off in the silica-dominant fiber is far less dramatic and the soliton spectrum is notably broader, cf. Figs. 3(b,c) and 2(b,c). This naturally leads to the temporally narrower and more intense solitons acquiring larger frequency shifts, see Fig. 4(a). This is despite the fact that the Raman effect in silica accounts only for 18% of the overall nonlinearity. Taking weaker and hence temporally wider initial soliton with q=2 (180fs FWHM) we find that the spectral cut-off effect in the air dominant case becomes significantly less important, resulting in the larger soliton self-frequency shift in the air-dominant fiber (f_a=0.5), then in the silica-dominant one (f_s=0.18), see Fig. 4(b). Throughout this paper spectral intensities are computed numerically as |Ã(δ)|=|∫^∞ _-∞ dtA(t)e ^-iδt|.

 

Fig. 2. Spectrograms (logarithmic scale) showing the effect of the Raman induced spectral cut-off in the air-dominant fiber. Initial soliton parameters are q=20 (60fs FWHM), N=1. Relatively narrow (cf. Fig. 3(b,c)) soliton spectra, indicate strong spectral cut-off and resulting energy loss into non-solitonic radiation.

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Fig. 3. The same as Fig. 2 but for the silica-dominant fiber. Relatively broad (cf. Fig. 2(b,c)) soliton spectra, indicate that the spectral cut-off effect is weak and the resulting energy loss into non-solitonic radiation is small.

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4. Airy tails

Another important observation, which one can make from Figs. 2, 3 is that, apart from the dispersive radiation created by the cut-off of the initial spectrum, there is a pronounced radiation tail emerging in front (negative delays) of the soliton and reproducing its trajectory in the (δ,t)-plane. This tail consists of the radiation, which is continuously emitted by the soliton during its propagation. Unlike the radiation lost due to cut-off effect the frequency content of this tail is changing with propagation and fills the spectral interval between the cut-off peak and the soliton. This tail is particularly strong in the air-dominant fibers, see Fig. 4. For short propagation distances, as in Figs. 2,3,4, the radiative tail of the soliton strongly overlaps with the cut-off radiation and not very clearly identifiable as a new effect in the spectrograms. This overlap and associated interference also cause the spectral roughness in the spectral interval between the soliton and pump peaks, see Fig. 4. However, presence of the two different components in the non-solitonic signal becomes obvious for longer distances, see Figs. 5(a,b).

 

Fig. 4. Direct comparison of the soliton self-frequency shifts in the air-dominant (AD) and silica-dominant (SD) fibers. (a) shows spectra corresponding to the spectrograms in Fig. 2(c) (black line) and Fig. 3(c) (red line). Thus for q=20 (60fs FWHM) the larger soliton self-frequency shift is achieved in the silica-dominant fiber. (b) shows spectra calculated for q=2 (180fs FWHM) and propagation distance Lz=10m, other parameters are the same as in (a). Thus for q=2 the larger frequency shift is achieved in the air-dominant fiber.

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Existence of the new radiation is related to the fact that the solitons in the presence of intrapulse Raman scattering move with negative acceleration. Indeed, the soliton frequency is continuously red shifting, which for anomalous GVD implies continuous decrease of the group velocity. The deceleration is fixed by the pulse duration and the Raman response parameters. Making transformation from the reference frame moving with a constant group velocity, as in Eq. (1), to the reference frame moving with a constant acceleration g, one can demonstrate that the soliton tails asymptotically behave as the tails of the Airy function [14, 15]. Substituting A(t, z)=B(ξ)exp(iξgz+ig ² z ³/6+iqz), where ξ=t-gz ²/2 and g>0 is the soliton deceleration, we find the linear part of the equation for the amplitude B to be ∂² _ξB=2(q+gξ)B+…. Asymptotically B[x≡(2_g)^1/3(ξ+q/g)]≈Ai[x]≈1/(2√π)x ^-1/4 exp[-2x ^3/2/3] for ξ>0, which corresponds to the exponentially damped soliton tail, and B[x]≈Bi[x]+iAi[x]≈1/√πx ^-1/4 exp[i2x ^3/2/3+iπ/4] for ξ<0, here Ai(x) and Bi(x) are the first and second order Airy functions. The radiation tail, or Airy tail, emitted in front (negative delays) of the decelerating soliton originates from the algebraically decaying and strongly oscillatory tail of the [Bi(x)+iAi(x)] function.

Taking the soliton parameters from Fig. 5 (a), we construct the Airy tail B(x), see Fig. 5(c). The corresponding spectrum shown in Fig. 5(d) has a typical “flat top” shape with the spectral width being proportional to the temporal length of the tail (the longer the Airy tail is the more frequencies it contains). This explains formation of expanding spectral tails on the highfrequency side of the soliton spectrum, see Fig. 4 and Fig. 5(b). Pronounced spectral tails adjacent to the soliton spectra have been seen in many experiments with air-core PCFs [2, 4, 5]. Our results strongly suggest that these are the Airy tails, see above and Figs. 4,5. Note here, that transformation into the accelerating reference frame can be done even in the absence of the intrapulse Raman scattering with g being a free parameter. Then arranging initial conditions in the form of the Airy function results in the Airy wave excitation, see, e.g. [16] for the spatial analog of this effect. However, when Raman effect is present the Airy wave emerges as a system attractor and a sech-like localized initial condition acquires the Airy tail during propagation.

 

Fig. 5. (a) Spectrogram as in Fig. 2, but calculated for the distance Lz=5m. The difference between the cut-off radiation and Airy tail is obvious. (b) Spectrum corresponding to (a). (c) is the Airy function given by B[x≡2g ^1/3(t+q/g)]=Bi[x]+iAi[x] multiplied by the suitable super-gaussian envelope function exp[-(x/w)³²] and (d) is its spectrum. Soliton parameters g and q, as well as the envelope function width w are taken from the data in (a): w=30ps.

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Fig. 6. (a) XFROG diagram of the output signal after 40cm of propagation for the case of multi-solitonic input condition with N=3 and q=20. Higher order dispersions are neglected. (b) Output spectra after 50cm of propagation for the case of multi-solitonic input pulse with N=1 and N=1.35, q=215.

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5. Self-induced transparency of very short solitons

In silica core fibers the Raman driven absorption of the photons from the solitonic part of the signal and their transfer into the non-solitonic part is practically negligible. As we have demonstrated above this transfer becomes important in the air-core fibers. This rises the question, whether one can observe the self-induced transparency effect in presence of the intra-pulse Raman scattering, i.e. the effect, that for the solitons shorter than the ones inducing strong non-solitonic radiation, the radiative energy losses in the air-dominant fibers should again become negligible. The reason for this is that the pulse spectrum is now so broad that the relatively narrow material resonance under its umbrella, does not significantly influence evolution of the pulse. This formulation of the problem is similar to the existence of the special soliton solution realizing transparency of media of nonlinear Lorentz oscillators [17]. In the context of the inter-pulse Raman scattering between two sufficiently long pulses in the gas filled hollow-core PCF the transparency regimes have been analyzed in [18].

In order to observe the self-induced transparency, i.e. significant reduction of the Raman induced non-solitonic radiation related to both cut-off effect and growth of Airy tails, we have carried out a dedicated series of numerical experiments. Fig. 6(a) shows the spectrogram calculated for the multi-solitonic initial pulse with N=3 and q=20 (60fs FWHM) after 40cm of propagation in the air-dominant fibre. One can see, that two solitons are formed, one of which is spectrally narrow and has strong Airy and cut-off radiation tails, which agrees with the observations described in the previous sections. However, the other pulse has spectral width ≈ 50THz, which is much larger than one should expect providing that the spectral cut-off concept is applied and its Airy tail is also far less pronounced. Analysis of the corresponding data in time domain gives that the temporal width of this pulse corresponds to q≈215 (~10fs FWHM), however its amplitude does not match the expected $\sqrt{\frac{2q}{\left({\gamma }_a+{\gamma }_g\right)}}$ and is approximately 1.4 higher. This discrepancy is easily explained by noting that for such short pulses the sin function in the Raman response of air in Eq. (2) can be approximately replaced with 0 and the exp function with 1 on the time scale of the pulse. The same approximation can not be applied to the Raman response of silica, because τ^(s) _1,2 are still comparable to the pulse duration, while τ^(a) _1,2≫10fs. In the absence of the Raman integral of air the approximate soliton amplitude is given by $\sqrt{\frac{2q}{\left({\gamma }_a\left(1-f_a\right)+{\gamma }_g\right)}}\mathrm{sech}\left(t\sqrt{2q}\right)$ and its Raman shift together with associated radiation effects are essentially due to silica only and therefore much weaker. The factor 1-f_a=0.5 in the denominator gives a very good match of the new amplitude with the numerical result.

To confirm this, we now take q=215 in our original initial condition $N\sqrt{\frac{2q}{\left({\gamma }_a+{\gamma }_g\right)}}\mathrm{sech}\left(t\sqrt{2q}\right)$ and monitor the resulting pulse propagation for different values of the input parameter N. Fig. 6(b) illustrates output spectra for N=1 and $N=N_{*}=\sqrt{\frac{\left({\gamma }_a+{\gamma }_g\right)}{\left({\gamma }_a\left(1-f_a\right)+{\gamma }_g\right)}}\approx 1.35$ after propagation distance of 50cm. N=N _* corresponds to the single soliton amplitude with R_a ≡ 0. One can see that for N=1 a part of the initial pulse forms a soliton and the other part is lost into radiation at δ=0 due to strong spectral cut-off effect. For N=N _*, however, both cut-off radiation and Airy tail are not visible because the pulse amplitude is now adjusted for the self-induced transparency effect and propagation happens as the Raman gain of air is simply not there. The effect is sufficiently robust and persists for the range of N values upto N=1.5, above which features of multi-soliton dynamics start to appear.

6. Summary

We have reported series of numerical experiments on femtosecond pulse propagation using generalized nonlinear Schrödinger equation accounting for the anomalous group velocity dispersion and for mixture of Kerr and Raman nonlinearities of the silica glass and air. The model parameters are chosen close to the ones typical for experiments with hollow-core photonic-crystal fibers. We have found that if the air contribution into material response dominates over silica, then the soliton pulses with durations less than or close to 100fs suffer from strong energy losses into non-solitonic radiation. In particular all initial spectrum, which is wider than the Raman gain line of air is lost into radiation. In addition, the emerging soliton emits a strong radiation tail in the form of an Airy wave. The latter has a broad spectrum, which likely explains several recent observations of similar spectral tails in experiments with solitons in air-core fibers. We have also found that the soliton pulses with duration around 10fs demonstrate self-induced transparency, which makes negligible all radiation losses associated with Raman scattering in air.