Influence of the Carrier–Envelope Phase on the Generation of the Multioctave Supercontinuum and Ultrashort Pulses in Antiresonant Hollow Waveguides

The influence of the carrier–envelope phase on the spectrum of the supercontinuum and on the characteristics of ultrashort pulses, which are formed by the nonlinear optical transformation of pump pulses in an argon-filled antiresonant hollow waveguide has been demonstrated. The experimental and theoretical analysis has shown that the soliton self-compression of pump radiation with a central wavelength of about 2 μm forms a pulse with a duration of nearly one optical cycle and with a spectrum broadened to the region of 400‒800 nm, where interference with the broadband third harmonic generated by the same pulse is observed. The interference pattern is sensitive to the carrier–envelope phase of the laser pulse. The analysis of the interference pattern provides information on the difference of the spectral phases of the soliton and third harmonic in the spectral range wider than an octave and allows one to control the duration of pulses formed in the process of soliton self-compression.

1 INTRODUCTION

To analyze phenomena induced by ultrashort laser pulses with a duration of nearly one field cycle, it is necessary to take into account the influence of the carrier–envelope phase (CEP), which is critically important both in the generation of ultrashort pulses [1–3] and in phenomena induced by them such as high- harmonic generation [4–6], attosecond [7] and terahertz pulses [8], and ultrafast electron dynamics in solids at petahertz frequencies [9–11]. The physics of ultrashort pulses formed due to the complex combination of nonlinear optical processes such as phase self-modulation, self-steepening of the back edge of the pulse, and soliton self-compression can be described in the classical approximation [12, 13] and disregarding the influence of the carrier–envelope phase on the propagation regimes of pulses with durations down to several optical cycles [14, 15]. However, this approximation obviously has limits of applicability. In particular, physical effects such as the generation of harmonics and sum frequencies, ultrafast tunnel ionization that depend on the carrier–envelope phase of the phase inevitably occur in the case of the multioctave broadening of the spectrum of pump pulses and the formation of single- and sub-cycle pulses at extremely high intensities [16–18]. These effects lead to the dependence of the spectral and temporal characteristics of the resulting pulse on the carrier–envelope phase [19].

The influence of the CEP of the laser pulse on the supercontinuum generation is poorly studied. The carrier–envelope phase dependence of the spectral superbroadening in the process of filamentation of ultrashort laser pulses in gases and solids was analyzed in [20–23], whereas this dependence at the cascade intrapulse generation of difference frequencies on the χ⁽²⁾ nonlinearity in crystals was examined in [24, 25]. In this work, we study the influence of the carrier–envelope phase on the spectrum of the supercontinuum, which is formed due to the soliton self-compression of pump pulses in an argon-filled antiresonant hollow fiber. The superbroadened spectrum of the soliton compressed to ultrashort durations interferes with the generated broadband third harmonic covering the entire visible range, and this interference pattern is sensitive to the carrier–envelope phase of the laser pulse. The interference pattern makes it possible to reconstruct the difference of the spectral phases of the soliton and third harmonic fields in the visible range and to control the formation of pulses with cycle and sub-cycle durations.

2 METHODS

The experiments were carried out with a laser system [26] (see Fig. 1), which consisted of a Ti:sapphire generator and a chirped-pulse amplifier. The system provided 50-fs pulses with an energy of 2.4 mJ at a wavelength of 810 nm was used to pump a two-channel optical parametric amplifier, which gave 50 fs idler pulses with an energy of 180 μJ with a central wavelength of about 2000 nm. A small fraction of this radiation with an energy of several microjoules was guided by changing its polarization to an f–2f interferometer. The main part of the radiation passed through an optical attenuator to control the radiation energy and was focused by a CaF₂ lens with a focal length of 75 mm into a gas cell with 2-mm-thick sapphire input and output windows, which was filled with argon at a pressure of 4 atm and contained a 20-cm-long antiresonant hollow waveguide. The cross section of the waveguide is shown in Fig. 1 and includes a hollow core 70 μm in diameter surrounding by six hollow capillaries each 36 μm in diameter with w ≈ 580-nm-thick walls. Correspondingly, the wavelengths of resonant mixing of core and shell modes can be estimated by the formula

$${{\lambda }_{k}} = 2w{{({{n}^{2}} - 1)}^{{1/2}}}{\text{/}}k,$$

(1)

where $k =$ 1, 2, 3, 4, 5 and $n \approx 1.44$, as λ₁ ≈ 1200 nm, λ₂ ≈ 600 nm, λ₃ ≈ 400 nm, λ₄ ≈ 300 nm, and λ₅ ≈ 240 nm, respectively.

Fig. 1.

(Color online) Layout of the experimental setup: (Ti:Sapphire) Ti:sapphire laser system, (OPA) two-cascade optical parametric amplifier, (W) wedge, (HWP) half-wave plate, (P) polarizer, (L) CaF₂ lens, (W1, W2) sapphire windows of the gas cell, (Ar) argon, (AR HC PCF) antiresonant hollow photon crystal fiber, (PM) parabolic mirror, (DL) delay line, (X-SEA-F-SPIDER) interferometry system for the spectral phase with the spatial resolution of the studied and reference pulses for the direct reconstruction of the field in the near infrared range, and (Spectrometers) Si and InGaAs spectrometers.

To characterize the output radiation spectrum, an image from the output of the waveguide was transformed using 4 – f systems on the slits of Si and InGaAs spectrometers having working ranges of 200‒1100 nm and 1000–2500 nm, respectively. The temporal profile of the pulse was characterized using the spatially resolved spectral phase interferometry of the studied and reference pulses (X-SEA-F-SPIDER) [27]. This technique allows one to reconstruct the spectral phase of the pulse in the wavelength range of 1000–2500 nm.

To measure the phase shift of the field with respect to the pulse envelope and to control this phase, we used a nonlinear Mach–Zehnder f–2f interferometer. The spectrum of the supercontinuum generated in a 4.8-mm-thick yttrium aluminum garnet plate by an idler pulse with an energy of about 7 μJ and a central wavelength of about 2000 nm was split by a dichroic mirror with a cutoff wavelength of 1500 nm into two parts. One of the parts passed through a 500-μm-thick β-barium borate crystal to generate the second harmonic from the long-wavelength part of radiation. The spectral interference of the short-wavelength part of the supercontinuum and the second harmonic of the long-wavelength part was observed near 730 nm by a silicon spectrometer. Since the positions of interference peaks are determined by the CEP of the idler pulse, the phase shift can be reconstructed by measuring the frequency shift of the peaks.

The f–2f interferometer was connected through a feedback system to a high-precision N-565 Physik Instrumente linear stage, which controls the delay at the final cascade of the parametric amplifier with an accuracy of 0.5 nm and a minimum step of 2 nm. Fast fluctuations of the CEP were stabilized passively during the parametric transformation, an active stabilization system implemented with the feedback system could stabilize slow phase mismatches (up to 200 Hz). The standard deviation of phase jumps of the idler wave in 100 s was 555 mrad without active stabilization and 106 mrad with active stabilization [26] due to the exclusion of the slow drift of the CEP. Furthermore, the feedback system allows one to vary the CEP within several periods, which is necessary for studying phase-sensitive effects.

The spatiotemporal field dynamics was studied theoretically using the solution of the generalized nonlinear Schrödinger equation for the instantaneous complex electric field [28, 29, 19]

$$\frac{{\partial A(\omega ,z)}}{{\partial z}} = iD(\omega )A(\omega ,z) - \alpha (\omega )A(\omega ,z)$$

$$ + \frac{{i\omega }}{{4{{c}^{2}}n_{0}^{2}{{\epsilon }_{0}}}}\tilde {F}[3\chi _{{\omega :{{\omega }_{1}},{{\omega }_{2}}, - {{\omega }_{3}}}}^{{(3)}}{{A}^{2}}(\eta ,z)A{\text{*}}(\eta ,z)$$

$$ + \,\chi _{{\omega :{{\omega }_{1}},{{\omega }_{2}},{{\omega }_{3}}}}^{{(3)}}{{A}^{3}}(\eta ,z)]$$

(2)

$$ - \tilde {F}\left[ {\frac{{{{U}_{i}}W(I)({{\rho }_{0}} - \rho (\eta ,z))A(\eta ,z)}}{{2I}}} \right]$$

$$ - \left( {\frac{{i\omega _{0}^{2}\omega }}{{2c{{n}_{0}}{{\rho }_{c}}({{\omega }^{2}} + \tau _{c}^{{ - 2}})}} + \frac{{\sigma (\omega )}}{2}} \right)\tilde {F}\left[ {\rho (\eta ,z)A(\eta ,z)} \right],$$

where $E(\eta ,z) = [2c{{n}_{0}}{{\epsilon }_{0}}{{]}^{{ - 1/2}}}(A(\eta ,z) + A{\text{*}}(\eta ,z))$ is the electric field, $A(\eta ,z)$ is its complex representation, $A(\omega ,z)$ is the Fourier transform of $A(\eta ,z)$, $\omega$ is the frequency, $\eta$ is the time, z is the propagation direction, $D(\omega ) = \beta (\omega ) - 1{\text{/}}{{{v}}_{s}}$ is the dispersion operator, ${{{v}}_{s}}$ is the velocity of the coordinate system moving with the pulse, $\beta (\omega ) = \omega n(\omega ){\text{/}}c$ is the propagation constant, $n(\omega )$ and $\alpha (\omega )$ are the effective refractive index and the mode loss coefficient calculated with the model presented in [30], ${{n}_{0}}$ is the refractive index at the central frequency of the laser pulse ${{\omega }_{0}}$, $\tilde {F}$ is the inverse Fourier transform operator, and $\chi _{{\omega :{{\omega }_{1}},{{\omega }_{2}}, - {{\omega }_{3}}}}^{{(3)}}$ and $\chi _{{\omega :{{\omega }_{1}},{{\omega }_{2}},{{\omega }_{3}}}}^{{(3)}}$ are the third-order nonlinear susceptibilities for the generation of the difference and sum frequencies, respectively. The model includes photoionization, which complicates the observation of spectra depending on the phase of the laser pulse. The electron density was calculated using the rate equation

$$\frac{{\partial \rho }}{{\partial \eta }} = W(I) + \frac{{\sigma ({{\omega }_{0}})}}{{{{U}_{i}}}}\rho I,$$

(3)

where I(η, z) = 2Re[A(η, z)]² is the instantaneous intensity; $\rho$ is the electron density; ${{U}_{i}} = {{U}_{0}} + {{U}_{{{\text{osc}}}}}$ is the ionization potential; ${{U}_{{{\text{osc}}}}}$ is the energy of the field-induced oscillatory motion of the electron; $W(I)$ is the photoionization rate calculated by the Keldysh formula [31] modified by Popov, Perelomov, and Terent’ev [32]; ${{\rho }_{{\text{c}}}} = \omega _{0}^{2}{{m}_{e}}{{\epsilon }_{0}}{\text{/}}{{e}^{2}}$ is the critical plasma density; ${{\rho }_{0}}$ is the density of neutral particles; σ(ω) = ${{e}^{2}}{{\tau }_{{\text{c}}}}{{[{{m}_{e}}{{\epsilon }_{0}}{{n}_{0}}c(1 + {{\omega }^{2}}\tau _{{\text{c}}}^{{\text{2}}})]}^{{ - 1}}}$ is the cross section for back bremsstrahlung calculated with the Drude model; e and ${{m}_{e}}$ are the charge and mass of the electron, respectively; and ${{\epsilon }_{0}}$ is the permittivity of free space. Pulse propagation in the argon-filled waveguide was simulated with the following model parameters [28, 29]: $\chi _{{\omega :{{\omega }_{1}},{{\omega }_{2}}, - {{\omega }_{3}}}}^{{(3)}} = \chi _{{\omega :{{\omega }_{1}},{{\omega }_{2}},{{\omega }_{3}}}}^{{(3)}} = 1.8 \times {{10}^{{ - 21}}}(p{\text{/}}{{p}_{{{\text{atm}}}}})$ cm²/V², which corresponds to the nonlinear refractive index n₂ = $3\chi _{{\omega :{{\omega }_{1}},{{\omega }_{2}}, - {{\omega }_{3}}}}^{{(3)}}{\text{/}}(4c{{\epsilon }_{0}}n_{0}^{2}) = 1.7 \times {{10}^{{ - 19}}}(p{\text{/}}{{p}_{{{\text{atm}}}}})$ cm²/W, U₀ = 15.76 eV, τ_c = 190(p_atm/p) fs, where p is the pressure and p_atm is the atmospheric pressure.

3 RESULTS AND DISCUSSION

The propagation of the 50-fs laser pulse with an energy of 19.5 μJ and a central wavelength of 2 μm in the waveguide filled with argon at a pressure of 4 atm is accompanied by soliton self-compression due to the joint effect of the anomalous dispersion in the argon-filled waveguide and the Kerr nonlinearity; as a result, the spectrum of the pulse is expanded to the range of 200–2500 nm, which covers more than 3.5 octaves (see Fig. 2) and includes several resonances of the waveguide structure near wavelengths of 1200, 600, 400, 300, and 240 nm (gray vertical fringes in Fig. 2). The broadening of the spectrum in the short-wavelength region is limited by the energy transfer to dispersion waves due to their phase matching with the soliton (240-nm peak in Fig. 2).

Fig. 2.

(Color online) Typical (solid line) measured and (dashed line) calculated supercontinuum at the output of the antiresonant fiber filled with argon at a pressure of 4 atm, which appears under the propagation of a 19.5-μJ 50-fs laser pulse with the spectrum shown by the dotted line. Gray vertical fringes mark the regions of resonances of the waveguide and DW is the spectral peak at the wavelength of phase matching of dispersion waves.

Soliton self-compression results in the supercontinuum at the output of the fiber, which periodically depends on the carrier–envelope phase of the laser pulse with a period of π (see Fig. 3). Such a periodicity indicates the interference of the coherent blue wing of the supercontinuum and the third harmonic generated by the long-wavelength part of the soliton. Indeed, the observed interference of these fields has the form

$$\begin{gathered} S(\omega ,{{\psi }_{{{\text{CEP}}}}}) = {\text{|}}{{A}_{{{\text{sol}}}}}(\omega )\exp [i{{\phi }_{{{\text{sol}}}}}(\omega )] \\ + \,{{A}_{{{\text{THG}}}}}(\omega )\exp [i{{\phi }_{{{\text{THG}}}}}(\omega )]{{{\text{|}}}^{2}} \\ = {\text{|}}{{A}_{{{\text{sol}}}}}(\omega {\text{)}}{{{\text{|}}}^{2}} + \;{\text{|}}{{A}_{{{\text{THG}}}}}(\omega ){{{\text{|}}}^{2}} \\ + \;{{A}_{{{\text{sol}}}}}A_{{{\text{THG}}}}^{*}\exp [i{{\phi }_{{{\text{sol}}}}}(\omega ) - i{{\phi }_{{{\text{THG}}}}}(\omega )] \\ + \;A_{{{\text{sol}}}}^{*}{{A}_{{{\text{THG}}}}}\exp [i{{\phi }_{{{\text{THG}}}}}(\omega ) - i{{\phi }_{{{\text{sol}}}}}(\omega )]. \\ \end{gathered} $$

(4)

Fig. 3.

(Color online) (a–d) Measured and (e–h) calculated spectra at the output of the antiresonant hollow fiber versus the carrier–envelope phase (CEP) of the laser pulse with an energy of (a, e) 18.5, (b, f) 19.5, (c, g) 21.0, and (d, h) 22.0 μJ. The dashed lines in panel (b) show the approximated spectral profile of the phase difference ψ_dif/2 + π n, where n is an integer.

Here, A_sol(${{\phi }_{{{\text{sol}}}}}(\omega )$) and A_THG(${{\phi }_{{{\text{THG}}}}}(\omega )$) are the spectral amplitudes (phases) of the soliton and third harmonic, respectively. In terms of the intensities of these fields ${{I}_{{{\text{sol}}}}}(\omega ) = {\text{|}}{{A}_{{{\text{sol}}}}}(\omega ){{{\text{|}}}^{2}}$ and I_THG(ω) = ${\text{|}}{{A}_{{{\text{THG}}}}}(\omega ){{{\text{|}}}^{2}}$, Eq. (4) is represented in the form

$$\begin{gathered} S(\omega ,{{\psi }_{{{\text{CEP}}}}}) = {{I}_{{{\text{sol}}}}}(\omega ) + {{I}_{{{\text{THG}}}}}(\omega ) \\ \, + 2({{I}_{{{\text{sol}}}}}(\omega ){{I}_{{{\text{THG}}}}}(\omega {{))}^{{1/2}}}{\text{cos}}[{{\phi }_{{{\text{sol}}}}}(\omega ) - {{\phi }_{{{\text{THG}}}}}(\omega )], \\ \end{gathered} $$

(5)

i.e., the interference pattern is determined by the intensities of the components and by the difference of their phases. The substitution of both the soliton phase in the form ${{\phi }_{{{\text{sol}}}}}(\omega ) \approx {{\psi }_{{{\text{sol}}}}}(\omega ) + {{\psi }_{{{\text{CEP}}}}}$, where ψ_CEP is the carrier–envelope phase and ψ_sol(ω) is the spectral phase of the soliton that due to effects independent of ψ_CEP, and the phase of the third harmonic of the soliton in the form ${{\phi }_{{{\text{THG}}}}}(\omega ) \approx {{\psi }_{{{\text{THG}}}}}(\omega ) + 3{{\psi }_{{{\text{CEP}}}}}$, where ψ_THG(ω) is the phase of the third harmonic of the soliton independent of ψ_CEP into Eq. (5) gives

$$\begin{gathered} S(\omega ,{{\psi }_{{{\text{CEP}}}}}) = {{I}_{{{\text{sol}}}}}(\omega ) + {{I}_{{{\text{THG}}}}}(\omega ) \\ + \,2({{I}_{{{\text{sol}}}}}(\omega ){{I}_{{{\text{THG}}}}}(\omega {{))}^{{1/2}}} \\ \times \cos [{{\psi }_{{{\text{sol}}}}}(\omega ) - {{\psi }_{{{\text{THG}}}}}(\omega ) - 2{{\psi }_{{{\text{CEP}}}}}]. \\ \end{gathered} $$

(6)

According to Eq. (6), the period of the interference pattern of the soliton and its third harmonic is half the period of ψ_CEP; i.e., it is equal to π, which coincides with the period of the interference pattern in Fig. 3. In addition, the lines of constructive interference maxima on these maps are determined by the equality ψ_dif(ω) – 2ψ_CEP = 2πl, where ψ_dif(ω)= ψ_sol(ω) – ψ_THG(ω) and l is an integer; consequently, the phase difference ψ_dif(ω) between the soliton and its third harmonic can be determined by approximating the positions of band maxima on maps in Fig. 3 (see Figs. 3b, 4b).

Fig. 4.

(Color online) (a) Visibility of measured interference maps, some of which are shown in Figs. 3a–3d, versus the energy of the pulse. (b) Spectral profile of the phase difference ${{\psi }_{{{\text{dif}}}}}$ approximated from Fig. 3b. (Inset) Transmission coefficient of the fiber versus the input energy of the pulse.

The phase difference ψ_dif(ω) is important for the characterization of the spectral phase of the supercontinuum in the visible range. As shown below, calculations predict a sub-cycle duration of the output pulse; unfortunately, the experimental characterization of such ultrashort pulses is a very difficult problem. In this work, the spectral phase of the pulse in the spectral range of 1–2.5 μm was characterized using the X-SEA-F-SPIDER method. It is difficult to measure the spectral phase of the supercontinuum in the range below 1 μm; dispersion effects in the window of a chamber and in the bulk of nonlinear crystals complicate the reconstruction of the spectral phase in a wide spectral range. The phase difference ψ_dif(ω) determined in this work is free of these dispersion constraints because the third harmonic is generated in the last centimeter of the path of the pulse in the waveguide and in front of the output window of the chamber. Under the assumption on the phase of the third harmonic, the spectral phase of the soliton in the visible range at the output of the fiber can be determined from ψ_dif(ω), which supplements the measurement of the duration of the ultrashort pulse.

The dependences of the spectra on the carrier–envelope phase at various energies of the laser pulse indicate that the spectrum of the soliton at an energy of 18.5 μJ (see Figs. 3a, 3e) does not yet reach the region of the third harmonic; for this reason, their interference is observed only in the 730–870-nm part of the third harmonic. The optimal energy ensuring the octave range of field interference is 19.5 μJ (see Figs. 3b, 3f). In addition to broadband interference, these maps exhibit a narrow-band interference component near 710 nm, which is due to the third harmonic generation at the initial stage of soliton self-compression [33], where the rapid broadening of the pulse spectrum is yet absent (see Fig. 5c). Nearly vertical interference fringes of the third harmonic indicate an insignificant frequency modulation (chirp) of the compressed soliton and the field of the broadband third harmonic induced by the soliton at the end of the fiber. On the contrary, tilted fringes of the narrow-band third harmonic near a wavelength of 710 nm indicate its frequency modulation collected in the process of propagation in the fiber. A further increase in the energy to 21 and 22 μJ (see Figs. 3с, 3d, 3g, 3h) results in the increase in the spectral intensity of short-wavelength components of the spectrum and, finally, the energy of the soliton is partially transferred to the energy of 200–300 nm dispersion waves (see Fig. 3d), where the spectrum becomes independent of the carrier–envelope phase of the laser pulse.

Fig. 5.

(Color online) (a, b) Calculated dynamics of (а) the spectrum and (b) the temporal profile of the field at the initial stage of pulse propagation. (c, d) Fourier analysis of the maps (a, b): the spectrum and the temporal profile of the component of the third harmonic from calculations presented in panels (a, b) cut by a Fourier filter with a passband of 0.37–0.9 PHz.

Dispersion waves clearly seen in Figs. 3d and 3h at wavelengths below 300 nm were generated by the soliton on the nonlinearity $\chi _{{\omega :{{\omega }_{1}},{{\omega }_{2}}, - {{\omega }_{3}}}}^{{(3)}}$; hence, their spectrum is independent of the carrier–envelope phase.

Figure 4 presents the dependence of the visibility of the interference pattern given by the formula

$$V(\lambda ) = \frac{{{{I}_{{\max }}}(\lambda ) - {{I}_{{\min }}}(\lambda )}}{{{{I}_{{\max }}}(\lambda ) + {{I}_{{\min }}}(\lambda )}},$$

(7)

where

$${{I}_{{\max }}}(\lambda ) = \mathop {\max }\limits_{{{\psi }_{{{\text{CEP}}}}}} I({{\psi }_{{{\text{CEP}}}}},\lambda ),$$

(8)

$${{I}_{{\min }}}(\lambda ) = \mathop {\min }\limits_{{{\psi }_{{{\text{CEP}}}}}} I({{\psi }_{{{\text{CEP}}}}},\lambda ),$$

(9)

on the maps shown in Figs. 3a–3d on the initial pulse energy. It is seen that the dependence of the output radiation spectrum on the carrier–envelope phase ψ_CEP is manifested at energies above 18 μJ at wavelengths of 0.7–0.8 μm, which are reached by the blue wing of the soliton spectrum. As the energy increases, the spectrum of the soliton is broadened towards the visible region and broadband interference in the entire visible range is observed at an energy of about 19.5 μJ. The transmittance of the fiber at this energy reaches 95% (see Fig. 4b). The length of the fiber is no longer optimal for the self-compression of pulses with higher energies; as a result, the spectral intensity of the supercontinuum at the output of the fiber decreases in the visible range and the visibility of the broadband interference pattern of the soliton and the third harmonic generated by them also decreases (see Fig. 4a). In this case, only the interference region near a wavelength of 710 nm remains, where the classical narrow-band third harmonic is generated in the entire considered energy range of the initial pulse.

The solution of the propagation equation (2) of the laser pulse in the waveguide at the initial stage (see Fig. 5) demonstrates a classical scenario of the third harmonic generation in the presence of a large phase mismatch. The coherence length defined as

$${{L}_{{{\text{THG}}}}} = \pi /{\text{|}}\Delta \beta ({{\omega }_{{\text{p}}}}{\text{)|}}{\text{,}}$$

(10)

where

$$\Delta \beta ({{\omega }_{{\text{p}}}}) = 3\beta ({{\omega }_{{\text{p}}}}) + 3{{n}_{2}}I\omega {\text{/}}c - \beta (3{{\omega }_{{\text{p}}}})$$

(11)

is the phase mismatch, at the wavelength λ_p = 2πc/ω_p = 2 μm at low intensities is L_THG ≈ 2.5 mm, which coincides with half the distance between the peaks of the vertical fringes of the third harmonic in Figs. 5c and 5d. This component of the third harmonic is narrow-band: its spectrum is 400–500 ТHz (see Figs. 5c, 6a, 6b). The phase mismatch of the third harmonic has a noticeable frequency dependence such that the blue and red spectral winds of the harmonic at frequencies of 505 and 445 ТHz are generated in opposite phases already at the distance z = 6 cm (see Figs. 5с). For this reason, fringes of the signal of the narrow-band third harmonic at a significant propagation length of z ≈ 18 cm in Figs. 6b are strongly tilted, and the harmonic itself propagating in the waveguide collects frequency modulation (chirp).

Fig. 6.

(Color online) (a, b) Calculated dynamics of (а) the spectrum and (b) the temporal profile of the field at the final stage of pulse propagation. (c, d) Fourier analysis of the maps (a, b): the spectrum and the temporal profile of the component of the third harmonic from calculations presented in panels (a, b) cut by a Fourier filter with a passband of 0.37–0.9 PHz.

The simulation indicates that the peak intensity of the compressed laser pulse at the final stage of its propagation in the waveguide (see Figs. 6) increases sharply to 100 TW/cm² due to soliton self-compression, inducing the intense broadband third harmonic generation with a spectrum of 400–900 ТHz wider than an octave (see Figs. 6a, 6c). In the time representation, the compressed pulse has a steep back edge caused by the self-steepening effect, and the Fourier analysis of the broadband third harmonic shows that it is generated in the form of an ultrashort pulse. The broadest spectrum is observed at z = 20.3 cm, i.e., in the first centimeter of the broadband third harmonic generation. Further, a fraction of radiation of the soliton is transferred to a dispersion wave, which is clearly seen at z > 20 cm at a frequency of 1.2 PHz (see Fig. 6a) and is also observed as interference at 20 fs < η < 40 fs in Fig. 6b.

Figure 7 presents temporal profiles of the compressed pulse. The pulse profile measured by the X‑SEA-F-SPIDER method had durations of 7.3 and 9.8 fs at the phases ψ_CEP = 0 and π/2, respectively. These durations correspond to 1.1 and 1.5 optical cycles at a central wavelength of 1920 nm of the measured spectrum. The X-SEA-F-SPIDER system implemented in our experiments has a limited working spectral range of 1.0–2.5 μm, which correspondingly limits the measured duration of pulses. For comparison with experiments, the same spectral range was separated in the calculated field by means of a band Fourier filter. The durations of pulses calculated with the filtered field in good agreement with the measured durations and are presented by paint bucket in Fig. 7a. The calculations of the total width of the compressed pulse without filtration predict the formation of ultrashort pulses with durations of 1.2 and 3.5 fs at the phases ψ_CEP = 0 and π/2, respectively, i.e., shorter than a half field cycle. However, to experimentally confirm this calculation result, it is necessary to use other measurement techniques, which are beyond the scope of this work. Figure 7 shows the maximum intensity of the compressed pulse at the output of the fiber and the temporal profile of the electron density generated by the pulse with this intensity. The maximum intensity is I_m ≈ 50 TW/cm², which corresponds to the Kerr addition to the refractive index δn_K = n₂I_m ≈ 3.5 × 10^–5. The corresponding maximum electron density is ρ_m ≈ 5 × 10¹⁵ cm^–3, which corresponds to the plasma addition to the refractive index δn_p = ρ_m/ρ_c ≈ 1.3 × 10^–5, where ρ_c ≈ 2.8 × 10²⁰ cm³ is the critical electron density at a wavelength of 2 μm. The Kerr addition δn_K exceeds the plasma one δn_p even at the maximum intensity of the propagating pulse, which explains an insignificant role of plasma effects in this work.

Fig. 7.

(Color online) (а) (Solid lines) Measured waveforms of the compressed pulse by means of spatially resolved spectrometry X-SEA-F-SPIDER of the spectral phase of the studied and reference pulses at two phases ψ_CEP. The paint bucket indicates the temporal profile of the calculated pulse at z = 20.3 cm filtered by a Fourier filter with a passband of 1.0–2.5 μm. (b) (Left axis) Unfiltered temporal profile of the calculated pulse and (right axis) the temporal profile of the pulse-induced electron density.

To demonstrate these estimates, Fig. 8 presents the dependences of the output spectra on the phase ψ_CEP and the evolution of the spectra of the pulse propagating in the fiber calculated in three models: (i) disregarding the ionization and the sum-frequency generation, (ii) disregarding the ionization and including the sum-frequency generation, and (iii) including the ionization and sum-frequency generation. The comparison of the results shows that ionization at a pulse energy of 19.5 μJ still weakly affects the spectra of the pulse, whereas dependences on the carrier–envelope phase are already pronounced. It is also seen that the supercontinuum is generated even when the sum-frequency generation is disregarded, but dependences on the carrier–envelope phase are not observed because one of the interfering fields is absent.

Fig. 8.

(Color online) (a–c) Spectra at the output (at z = 20 cm) of the antiresonant hollow fiber versus the carrier–envelope phase and (d–f) the dynamics of the spectrum of the 19.5-μJ pulse propagating in the waveguide. (a, d) Calculation with the model given by Eq. (2), where ionization and generation of sum frequencies are disregarded ($\rho = 0$, W = 0, $\chi _{{\omega :{{\omega }_{1}},{{\omega }_{2}},{{\omega }_{3}}}}^{{(3)}} = 0$). (b, e) Calculation with the model given by Eq. (2), where ionization is disregarded ($\rho = 0$, W = 0). (c, f) Calculation with the complete model given by Eq. (2) with all terms.

4 CONCLUSIONS

To summarize, the influence of the carrier–envelope phase on the spectrum of the supercontinuum and on the characteristics of ultrashort pulses, which are formed by the nonlinear optical transformation of pump pulses in an argon-filled antiresonant hollow fiber has been demonstrated. The experimental and theoretical analysis has shown that the soliton self-compression of pump pulses with a central wavelength of about 2 μm forms a pulse with a duration of nearly one optical cycle and with a spectrum broadened to the region of 400–800 nm, where interference with the broadband third harmonic generated by the same pulse is observed. The interference pattern is sensitive to the carrier–envelope phase of the laser pulse. The analysis of the interference pattern provides information on the difference of the spectral phases of the soliton and third harmonic in the spectral range wider than an octave and allows one to control the duration of pulses formed in the process of soliton self-compression of pulses. The minimum duration of the compressed pulse measured by the X-SEA-F-SPIDER method is 7.3 fs, i.e., 1.1 optical cycles at the central wavelength of the pulse of 1920 nm. Although constraints imposed by the experimental method prevent the measurement of shorter durations of pulses, the presented theoretical analysis indicates that the generation of pulses with a duration shorter than half the field cycle is possible.