1. Introduction

During the last two decades impressive progress has been achieved in the generation of intense few-cycle pulses, which led to dramatic achievements in high-field physics and extreme nonlinear optics, time-resolved laser spectroscopy and others [1] such as e.g. in the generation of soft X-ray radiation and isolated attosecond pulses by high-order harmonics generation [2]. A well-established and proven technique for few-cycle pulse generation is based on the application of noble-gas-filled dielectric hollow waveguides [3] using the combination of spectral broadening by self-phasemodulation and subsequent pulse compression by negative group delay dispersion provided by an external optical element [4, 5, 6, 7]. Several other methods have also demonstrated few-cycle pulse compression with energies in the millijoule range [8, 9, 10, 11, 12].

An alternative technique for compression of ultrashort pulses relies on the method of soliton compression in fibers which does not require additional chirp compensation elements. In this method the periodic evolution of the temporal shape and spectrum of higher-order solitons in the anomalous dispersion region is used for efficient compression of a pulse at the output of a fiber with appropriate length. In early experiments standard single-mode fibers (for an review see [13]) have been used which allow the required anomalous dispersion only for wavelengths exceeding 1.3 µm. The introduction of photonic crystal fibers (PCF) [14] and nanowires permit an extension of the anomalous dispersion region to the near-IR and visible spectral range. Recently these types of fibers have been used for pulse compression at 800 nm with sub-nJ energy pulses from 70 to 6.8 fs [15]. On the other hand, hollow-core photonic bandgap fibers (PBGF) also exhibit anomalous dispersion at these wavelengths with diameters in the range of 10 µm and consequently a higher pulse energy in the 10- to 500-nJ range. Compression of 225 nJ pulses from 120 fs to 50 fs in hollow-core PBGF has been demonstrated in Ref. [16]. However, the allowed bandwidth in hollow-core PBGF is limited by the bandgap, which sets strict limits for the shortest achievable pulse durations. Soliton-effect compression in standard fibers with inclusion of higher -order dispersive effects has also been studied [17, 18]. Recently soliton-effect compression in PCFs, nanowires and hollow-core PBGF have been theoretically investigated in Ref. [19, 20].

The purpose of the present paper is to study soliton compression in an alternative waveguide type which has low loss in a broad spectral range, for waveguide diameters which simultaneously yield anomalous dispersion in the optical and near-IR frequency range even at a relatively high gas pressure. Dielectric hollow waveguides provide tolerable levels of losses only for diameters larger than 100 µm. For these parameters the waveguide contribution to dispersion is relatively small and anomalous dispersion in the range of 800 nm can only be achieved for a very small gas pressure.

Recently we have shown that dielectric-coated metallic hollow waveguides represent a suitable option for the realization of dispersion control in a broadband hollow waveguide [21]. Such waveguides can be fabricated by chemical vapor deposition [22, 23] or other methods. The dielectric coating of the metallic wall reduce the waveguide loss and also the scattering loss in comparison to purely metallic or dielectric waveguides. Therefore tolerable loss in the optical and near-IR region can be achieved even for small waveguide diameters in the range of 20–80 µm with a broad range of anomalous GVD in the visible even for high pressures of the gas filling. These findings can have interesting applications in ultrafast nonlinear optics and in particular for pulse compression. Recently we have also shown that the same waveguide type with different design parameters can be used for the generation of soliton-induced super-continua with about two octave broad spectra and unprecedentedly high power, five orders of magnitude higher than in standard PCFs [24].

As we will show, the anomalous dispersion combined with the higher nonlinearity due to a higher pressure enable high-order soliton self-compression of µJ pulses up to the single-cycle region with a pulse duration (FWHM) of 1.7 fs without technically sophisticated chirp compensation by external elements with anomalous GVD. It is obvious that such compression can not be described by the nonlinear Schrödinger equation, but higher-order dispersion and effects arising by the violation of the slowly varying envelope approximation play a crucial role and determine the shortest achievable pulse durations. We will examine the physical limitations for the largest compression factor caused by these effects.

2. Model and numerical approach

We consider a straightwaveguide consisting of a circular gas-filled hollow core with a refractive index around unity and a metallic cladding coated by a layer of a dielectric from the inner side. The optical properties of this system were studied in [21] by using the transfer matrix formalism developed in [25]. For each mode, the dependence of all field components on time t and longitudinal coordinate z are given by the same factor exp(-iωt)exp(iβ (ω)z), where ω is the frequency and β(ω) is the (generally complex-valued)wavenumber of themode. Themodal fields are solutions of the Helmholtz equation given as a linear combination of Bessel functions of the radial coordinate r with different parameters within each of the regions. The boundary conditions at the interface between two regions can be expressed in terms of a transfer matrix which relates the coefficients before the Bessel functions in one region to those in another region. The characteristic equation for the eigenvalue β(ω) can be obtained requiring finiteness of the field in the core and absence of the incoming waves in the outermost layer. The resulting equation has the form F(β(ω))=0 where F is a complex analytical function of β(ω). The solution of this equation yields β(ω), which can be complex even in a waveguide made from lossless materials, with an imaginary part accounting for the mode loss α=(λ/π)Im(β). To calculate the total waveguide loss, we add a contribution form the roughness loss. The latter was calculated by using a model in which roughness loss is described by the action of many uncorrelated pointlike scatterers [26]. The roughness loss then can be calculated by the formula α_r=ω ⁴ V ² _S ρ _s<I>_s Δε ²/(3πRI ₀ a_Mc ⁴) with the surface density ρ_s and the volume V_S at the interface between the two layers of the waveguide. In this expression, Δε is the difference of the dielectric constants of the materials on the two sides of the interface, <I>_s is the intensity of the mode profile near the interface averaged over the scatterer volume, I ₀ is the intensity of the mode profile in the center of the waveguide, and a_M is a dimensionless mode factor which is equal to 0.269 for the fundamental HE ₁₁ mode. In the calculation, we have assumed hemispherical scatterers with V_S=(1/12)πσ ³, ρ_s=σ^-2 where s is the mean scatterer size. In the following calculations we assumed a mean scatterer size σ=100 nm. Details of the model for the calculation of the waveguide properties, full discussion of the dispersion control and waveguide losses, including roughness losses, can be found in Ref. [21]. The group velocity dispersion of argon is described by the Sellmeier-type formula with parameters found in [27].

For the numerical simulations we use a model based on the unidirectional equation of Ref. [28]. The Fourier transform E⃗(z,x,y,ω) of the electromagnetic field E⃗(z,x,y,t) can be represented as the product of the transverse fundamental-mode distribution T(r) and the function E(z,ω) which describes the evolution of the field with propagation, z being the propagation coordinate. We consider linearly-polarized input fields exciting only the fundamental mode HE ₁₁ which has azimuthal symmetry. Substituting this ansatz into the wave equation and neglecting weak backreflected field components the following first-order differential equation can be derived

where β(ω) and α(ω) are the wavenumber and the loss of the fundamentalmode and P_NL(z,ω) is the nonlinear polarization. This equation is solved in the time frame moving with the velocity c/n_g with n_g close to unity in the considered case. In the previous study [24] it was shown that energy fraction coupled to higher-ordermodes does not exceed 20% even for very high intensities around 100 TW/cm², therefore we disregard higher-order modes in this study which deals with much lower intensities. In deriving this equation we disregard terms which originate from the combined effects of the nonlinearity and the longitudinal components of the modal fields. This is justified in hollow waveguides because the energy fraction carried by the longitudinal components is roughly (2λ/D)²~10^-3 and the nonlinear modification of the refractive index is also small, on the scale of 10^-5. For the input intensities considered here, photoionization can also be neglected.

Note that this approach does not use the slowly-varying-envelope approximation which allows to describe adequately the generation of single-cycle pulses. The quantities β(ω), α(ω) and F(r,ω) are calculated by the transfer-matrix approach assuming a circular waveguide structure described above.

The Fourier transform P_NL(z,ω) of the nonlinear polarization is calculated from

where χ ₃=(4/3)cε ₀ n ₂ is the third-order polarizability of the gas filling and t is the moving time. The above radius-dependent relation (2) is multiplied by the mode profile T(r) and integrated over the waveguide cross-section, to obtain the propagation equation for the field in the fundamental mode E(z,t), relying on the orthogonality of the modes. For details of this procedure see e.g. [13]. For argon we have n ₂=1×10^-19 cm²/W at 1 atm [29]. We neglected the nonlinearity of the waveguide walls, since the intensity of the modal field is roughly 10^-3 of the intensity at r=0, and since any nonlinear polarization of the wall material does not influence the pulse evolution due to its small overlap with the mode field. To solve the Eq. (1) numerically, we have used the split-step Fourier method with fourth-orderRunge-Kuttamethod for nonlinear steps.

3. Results and discussion

First we consider a waveguide with a cladding made from silver and a SiO₂ inner coating. Silver has a relatively low loss in the visible range and is characterized by a dielectric function ε=1-ω ² _p/[ω(ω+i ν)] with ω_p=12.0 fs^-1 and ν=0.076 fs^-1.

 

Fig. 1. Cross-section (a), loss and GVD (b) of an argon-filled metal-dielectric hollow waveguide with a diameter of 60 µm. The silver cladding is coated by a fused-silica layer with thickness a of 40 nm. The loss is shown by the green curve and the GVD is presented for a pressure of 13 atm by the red curve.

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In Fig. 1(a) the cross-section of the considered waveguide is shown, and in Fig. 1(b) the calculated loss and the GVD is presented for silica-coated silver waveguide with a diameter of 60 µm and a coating thickness of 40 nm. The loss has values in the range of 1 dB/m and therefore the linear transmission for a 50 cm long waveguide is T=95%. Despite the high pressure of 13 atm, the group-velocity dispersion, indicated by the green curve in Fig. 1, is anomalous in a broad range for wavelengths larger than 648 nm.

The method of soliton-effect compression is based on the combined action of anomalous GVD and self-phase modulation in pulse propagation (see e.g. [13]). For longer pulses in the ps and sub-ps region this phenomenon can be described by the nonlinear Schrödiger equation (NSE) predicting the excitation of higher-order solitons which undergo a periodic temporal compression followed by temporal spreading or correspondingly a spectral broadening followed by a spectral narrowing. However in the range of few-cycle pulses or for small GVD parameters in relation to third-order dispersion (TOD) a crucial modification of this scenario comes into play during pulse propagation in PCFs. Due to higher-order dispersion under typical conditions only the first period of temporal compression is realized, while after that at a certain propagation step the pulse splits into many fundamental solitons accompanied by super-continuum generation due to the emission of non-solitonic (dispersive) radiation [28]. Recently we predicted a similar phenomenon also in metallic hollow waveguides coated by a dielectric [24]. In spite of this strong violation of the prediction of the nonlinear Schrödiger equation the question arise, up to which extend the first period of temporal compression can be used for the generation of shortest self-compressed pulses, what is the limit of the shortest pulse duration and under which optimal conditions this limit is reached. In the following these questions will be addressed.

 

Fig. 2. Evolution of the temporal shape (a),(c),(e) and spectrum (b),(d),(f) for a 17-TW/cm², 20-fs pulse in a waveguide with D=60 µm and a=40 nm. The input wavelength is 1037 nm, the propagation length is 1.9 cm (a),(b), 3.8 cm (c),(d), and 4.74 cm (e),(f).

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We first study the propagation of pulses with a duration of τ ₀=20 fs centered at 1037 nm with input intensity I ₀=17 TW/cm² in a waveguide with parameters given in the caption of Fig. 1. As can be seen in Fig. 2(a),(c) the initial gaussian pulse transforms into a compressed one and after an optimal propagation of L _opt=4.74 cm an optical pulse in the single-cycle regime [Fig. 2(e)] is formed, which has a FWHM duration of 1.7 fs, a low pedestal, and an energy of 6 µJ. The peak intensity is more than four times larger than the input intensity. For a longer propagation length than L _opt the pulse splits into several spikes (here not shown). The evolution of spectra in Fig. 2(b),(d),(f) demonstrates a gradual broadening, covering at the optimum length a broad range from 600 to 1500 nm. In comparison, the nonlinear Schrödinger equation predicts a compression factor which depends only on the soliton number N, which is given by N=0.433τ₀[I ₀ n ₂/(cβ″)]^1/2. For the considered pulse and fiber parameters the soliton number is N=2.75 and the corresponding compression factor is F_c=10 (see Fig. 6.8 in Ref. [13]). Despite the fact that the NSE is not valid in the single-cycle region as considered here, its prediction of the compression factor is still in approximate agreement with the results of Fig. 2 based on the evolution equation (1) yielding a compression factor of 11.75. However, the results in Fig. 2 represent the physical limit of the shortest possible pulse duration, which is critically influenced by higher-order dispersive and nonlinear effects. Besides the compression factor, the quality factor defined as the fraction of the total energy contained in the compressed pulse is an important measure for the quality of the compression. The quality factor is roughly 0.4 in Fig. 2(e).

To get a more comprehensive understanding of the compression process in the single-cycle region and the physical factors determining the limitations in the shortest pulse duration we calculated the dependence of the compressed pulse duration (FWHM) on the duration of the input pulse, as shown in the red curve of Fig. 3. The input intensity was also varied to keep the soliton number N constant at the value of 2.75. As can be seen, the output duration decreases with decreasing input duration, but for an input pulse durations below 15 fs the compression factor becomes smaller and finally the output duration saturates at the value of 1.7 fs in the single-cycle region. In comparison, in the case of the unperturbed NSE the compression factor F_c=τ ₀/τ_out with F_c=10 is constant and no limitation is predicted. It should be noted that the definition of full width at half maximum (FWHM) for the characterization of the pulse duration is a measure which is not precisely defined in the single-cycle region since it discontinuously depends on the strength of side maxima. For fixed input intensity and duration the relative output peak intensity F=I_max/I_in is another measure of the pulse compression factor which is more precisely defined. (For a quality factor equal to 1 we have the relation F=F_c).

 

Fig. 3. Dependence of the output pulse duration on the input pulse duration. The soliton number and the input wavelength, as well as the waveguide parameters are the same as in Fig. 2. The propagation distance is optimized separately for each input duration. The gray curve corresponds to the compression factor F_c=10 predicted by the NSE.

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Fig. 4. Dependence of the compression factor F on the input wavelength. The input-pulse soliton number and duration, as well as the waveguide parameters are the same as in Fig. 2. The propagation length is optimized separately for each input wavelength. The green curve shows the wavelength dependence of the TOD parameter.

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Let us now study the dependence of soliton-effect pulse compression in the considered hollow waveguide on the input wavelength. We fix the soliton number N=2.75 and the input pulse duration of 20 fs but correspondingly vary the input intensity. The NSE predicts for this case a constant relative output intensity of around 5. However, as shown in Fig. 4, the numerical solutions of the Eq. (1) show a dramatic deviation from this behavior. It can be seen that the relative output peak intensity presented by the red curve exhibits a pronounced maximum at 1037 nm. Since the soliton number and input pulse duration was kept constant for all wavelengths, such behavior can only be explained by the influence of higher-order effects. The major perturbation occurs due to third-order dispersion (TOD) which we describe by the relative coefficient ε=β ⁽³⁾/(β″τ ₀) similar to the one used in the perturbed nonlinear Schrödinger equation. In the green curve in Fig. 4 the inverse of this coefficient is presented for the considered waveguide with the same parameters as in Figs. 2 and 3. It can be seen that $\genfrac{}{}{0.1ex}{}{1}{\epsilon }$ exhibits a maximum at 960 nm which is near the wavelength with the highest compression factor F at 1037 nm. The pronounced maximum of the compression factor can be understood by the detrimental influence of third-order dispersion for pulse compression if the relative TOD parameter ε is too large.

 

Fig. 5. Dependence of the compression factor F on the TOD parameter.

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On the first view one would expect that the compression factor is highest where ε has its smallest value. In order to explain the deviation of the extrema of both curves in Fig. 4 we study pulse propagation in a fiber which includes only the GVD and the TOD effect: β(ω)=β ₀+β′(ω-ω ₀)+(1/2)β″(ω-ω ₀)²+(1/6)β(3)(ω-ω ₀)³. This allows to vary the TOD parameter without changing the GVD or other parameters. Fig. 5 shows the relative output intensity as function of the TOD parameter. It can be seen that the relative output intensity reach a maximum at ε=0.1, but for larger ε the compression factor decreases to a value about 5 times lower than the maximum. The existence of a nonzero optimum third-order dispersion was also predicted in [17, 18] for the combined action of TOD and Raman self-scattering in standard fibers. This sensitive dependence of pulse compression on the TOD parameter and the threshold-like degradation of compression for larger TOD explains both the pronounced maximum of the red curve in Fig. 4 and the shift of the maximum from ε=0 in Fig. 5.

In Fig. 6 we study the dependence of the output pulse shape on the carrier-envelope offset (CEO) of the input pulse. The temporal position of the peak is monotonously shifted with increasing CEO. It can be seen that the CEO is preserved during compression in the process based on the Kerr nonlinearity, despite the sub-cycle output pulse duration. It should be noted that the shortest pulse duration does not depend on the CEO, but the optimum propagation length depends slightly on the CEO.

 

Fig. 6. Temporal shapes of the pulse after 4.74 cm propagation for an input CEO of 0 (red solid curve), π/4 (green long-dashed curve), π/2 (blue dash-dotted curve) and 3π/4 (magenta dotted curve).

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4. Conclusion

In conclusion, we have predicted the possibility to compress µJ pulses to a sub-cycle FWHM of 1.76 fs without external dispersion control in dielectric-coated metallic hollow waveguides. We have shown that TOD is the main physical factor determining the limitations in the shortest pulse duration in the single-cycle region in the method of pulse compression by soliton-effect compression. The dependence of the compression factor on the TOD coefficient is very sensitive. In microstructure fibers it is difficult to find an optimum geometry with small TOD parameter since GVD and TOD are simultaneously changed. This complication is lifted by the independent control by the pressure in hollow waveguides. A possible experimental realization of pulse compression up to the single-cycle region by the effects considered here can lead to a simplification and improvement of the pulse compression technique in the single-cycle region.