Abstract

We have experimentally demonstrated the generation of sub-half-cycle phase-stable pulses with the carrier wavelength of 10.2 µm through two-color filamentation in nitrogen. The carrier-envelope phase (CEP) of the MIR pulse is passively stabilized and controlled by the attosecond time delay between the two-color input pulses. The duration of the MIR pulse is 13.7 fs, which corresponds to 0.402 cycles. The absolute value of the CEP of the generated sub-half-cycle pulse is consistent with a simple four-wave difference frequency generation model. We have also found that the 10 kHz repetition rate of the light source causes the fluctuation of the pulse energy on a few hundred millisecond time scale.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Carrier-envelope phase (CEP)-stable few-cycle optical pulses play an irreplaceable key role to investigate ultrafast dynamics from femtosecond to attosecond time scale. Recently, the generation of few-cycle mid-infrared (MIR) pulses has received much more attention because of their various potential applications, such as X-ray generation [1,2], high-harmonic generation [37], field-driven photo-electron emission [810] and pump-probe spectroscopy for molecular structure and carrier dynamics in solids [1116], sensitive MIR absorption spectroscopy for biological tissues [1719].

So far, several methods have been experimentally demonstrated to generate MIR pulses, such as intra- or inter-pulse difference frequency generation (DFG) in a conventional nonlinear crystal [2024], DFG through the quasi-phase-matched waveguides [2527], optical parametric chirped pulse amplifier (OPCPA) [2830], single-color supercontinuum generation in a solid [3135] or gas [3638], and multi-color DFG through the laser-induced filamentation in gases [39,40].

The formation and accompanied phenomena of the laser-induced plasma filament have been well-known in nonlinear optics [4144], such as spectrum broadening, self-guided, intensity clamping, self-compression, and acoustic density wave. These phenomena contribute to form a plasma channel which is much longer than the Rayleigh length [45]. This is very effective for the frequency conversion with a gaseous medium, which has a smaller group velocity dispersion (GVD) and higher transparency than liquid or solid media in general.

Another benefit of using a gas is that the medium is naturally exchanged on a millisecond time scale [46,47], therefore, a sort of the damage of the medium (due to the ionized atoms or molecules) does not remain permanently. This fact allows us to use much higher input power, which would compensate for the low nonlinear index of the gaseous medium.

These unique properties can provide a high potential to generate a strong coherent supercontinuum [4850]. Moreover, the relatively high order electric susceptibility $\chi ^{(\mathrm {odd})}$ of the isotropic medium creates the probability of frequency conversion through four-wave-mixing or higher order nonlinear processes [42,51]. Mixing a few numbers of harmonics [52,53] can enhance the efficiency by avoiding additional phase mismatch due to Gouy phase shift [51,54]. One of the typical examples is the two-color filamentation scheme where the fundamental and second harmonic (SH) pulses are mixed and focused together into a nonlinear medium. Such a two-color scheme can provide a super wide range of the coherent light from terahertz (THz) [55,56], infrared (IR) [40,57], visible, ultraviolet [5860] and even extreme ultraviolet [6163].

When a Ti-sapphire based amplifier laser is used as the light source, the two-color system (800 nm and 400 nm) have much more efficient down-conversion than the one-color system (only 800 nm) [40,55]. The mechanism of the coherent down-conversion to MIR or THz through these two-color filamentation setup can be explained by the four-wave difference frequency generation (FW-DFG) through the interaction of the fundamental pulse and the coherent harmonic pulses [64].

In this work, we have constructed a sub-half-cycle MIR source through the FW-DFG between two-color pulses in nitrogen gas at the standard temperature and pressure. We have characterized the waveforms of the sub-half-cycle pulses using frequency-resolved optical gating capable of CEP determination (FROG-CEP) [65]. The duration of the MIR pulse is 13.7 fs at the carrier wavelength of 10.2 µm, which corresponds to 0.402 cycles. Based on the pulse characterization results, we also discuss the half-cycle pulse generation process. We have found out that the spectral phase and the absolute value of the CEP of the generated pulse are consistent with a simple four-wave difference frequency generation model. We have also found that the 10 kHz repetition rate of the light source causes the fluctuation of the pulse energy on a few hundred millisecond time scale.

2. Experimental setup

The system setup is shown in Fig. 1. The light source is a Ti:sapphire laser system based on a regenerative amplifier and a single-pass amplifier (Spitfire Ace, 804 nm, 1.4 mJ, 35 fs, 10 kHz). Here, we use a compact design for generating two-color filamentation [66,67]. The vertically polarized fundamental pulse ($\omega _1$, 870 µJ, 804 nm) from the Ti:sapphire laser is sent to a doubling crystal ($\beta$-BBO, type-I, $\theta$ = 29°, $t$ = 100 µm) to generate a horizontally polarized SH pulse ($\omega _2$, 36 µJ, 403 nm). The collinear two-color pulses pass through a birefringent crystal (DP, calcite, $t$ = 1.7 mm) and the relative delay between them is aligned by controlling the angle of the crystal with an attosecond accuracy by using a piezoelectric inertia actuators (PIA13, Thorlabs). After that, a dual wave plate (DWP, $\lambda$ for 400 nm, $\lambda$/2 for 800 nm) is used for rotating the polarization of the fundamental pulse so that the fundamental and SH pulses have the same polarization and the efficiency of the FW-DFG process is optimized. The FW-DFG process is described as

$$\omega_1 + \omega_1 - \omega_2 \rightarrow \omega_0,$$
where $\omega _1$, $\omega _2$ and $\omega _0$ are the angular frequencies of the fundamental pulse, SH pulse, and MIR pulse generated through the FW-DFG, respectively.

 figure: Fig. 1.

Fig. 1. Experimental setup of the MIR pulse generation and characterization. BBO: $\beta$-BaB$_2$O$_4$, DL: delay plate (calcite), DWP: dual wave plate, CM1: dielectric concave mirror (high reflection for $\omega _1$ and $\omega _2$), CM2: aluminium-coated concave mirror with a hole of 7 mm diameter, BF: short pass filter (FGB37, Thorlabs)

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To generate a filament in air, the co-linear two-color beam is gently focused by using a concave mirror to obtain a strong field and ionize the gaseous medium. After the filament, the beam is collimated with an aluminium-coated concave mirror with a hole ($\phi$=7 mm). Since the generated MIR beam has much larger divergence than the input beam, the residual fundamental and SH beams are efficiently extracted in this way. Next, the MIR pulse goes to the pulse diagnostics part. The detail of the setup is explained in the section of the pulse characterization.

3. MIR generation through two-color filamentation

To optimize the convergent angle of the fundamental beam, we employed concave mirrors with several radii, $R$ = 100, 150, and 200 cm. The output power of the corresponding MIR pulses were 1.6, 4.2, and 3.2 mW, respectively. The MIR spectrum did not show a clear difference among these radii. In the end, we have chosen the concave mirror ($R$ = 150 cm) which provides the highest power. The length of the filament was estimated as $\sim$2 cm from the intensity distribution of the plasma fluorescence.

The optimum length of the filament is consistent with the following theory. The wave vector mismatch of the FW-DFG can be described as follows [68],

$$\Delta \boldsymbol{k} = \boldsymbol{k}_0 - 2 \boldsymbol{k}_1 + \boldsymbol{k}_2$$
$\boldsymbol {k}_1$, $\boldsymbol {k}_2$, and $\boldsymbol {k}_0$ are the wave vectors of the fundamental pulse, SH pulse, and MIR pulse generated through the FW-DFG, respectively. When the wave vector mismatch $\Delta \boldsymbol {k}$ is proportional to the density of the $\chi ^{(3)}$ medium, the optimized confocal parameter is close to four times of the coherence length ($\pi /\left |\Delta \boldsymbol {k}\right |$) [51]. The non-zero wave vector mismatching is required to be compensated for the Gouy phase of the fundamental pulse [51,54]. The wave vector mismatch in 1 atm nitrogen gas at 10 µm MIR generation is around 630 rad/m by assuming the free electron density in the filament is $2.93\times 10^{16}$ cm$^{-3}$ ($\omega _1$ = 788 nm, $\omega _2$ = 410 nm, $\omega _0$ = 10.1 µm) [69,70]. The length of the filament corresponds to the estimated phase mismatch value.

3.1 Power stability

Although the pulse-to-pulse stability of the Ti:sapphire laser system was 0.8% at 10 kHz repetition rate measured by using a fast silicon-based photo-diode (Thorlabs, DET10A), the pulse-to-pulse stability of the MIR pulses at 10 kHz was measured as 6.7% by using a fast MCT detector (Thorlabs, PDAVJ10). Figure 2 shows the results of the power stability measurement. The horizontal axis in Figs. 2(a) and 2(b) is the group delay dispersion (GDD) of the grating compressor, which corresponds to the dispersion of the fundamental pulse. The sign of the horizontal axis corresponds to the direction of the chirp. Although the power of the generated MIR pulse does not show a perfect mirror symmetry along the chirp axis [50,71,72], the efficiency of the MIR generation is the maximum when GDD is around zero. The standard deviation changes from 0.7 to 1.5 mV at 10 kHz repetition rate. We cannot explain what is the reason for the periodic oscillation of the standard deviation. Indeed, there are some points where the MIR pulse is stable, however, it is not very practical to use the system at 10 kHz for a several hour long experiment since the chirp of the Spitfire system drifts on such a time scale.

 figure: Fig. 2.

Fig. 2. The group delay dispersion (GDD) of the input pulse dependence of the average (a) and standard deviation (b) of the MIR pulse-to-pulse intensity, which were measured by using the MCT detector. The zero position of GDD was determined by the maximum output of MIR pulse. (c) Typical MIR single-pulse photodiode signals from a boxcar gated integrator at 10 kHz and 5 kHz repetition rate. (d) A typical pattern of the He-Ne laser beam passing through the filament and (e) that without filament as a reference.

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By reducing the repetition rate of the system down to 5 kHz by changing the frequency of the Pockels cell in the regenerative amplifier, the standard deviation of the power of the MIR pulse drops down to 0.8 mV and is independent of the grating position. The spectra of the fundamental and second harmonic at 5 kHz are basically the same as those at 10 kHz.

The single-pulse-induced relaxation time of the ionized gases is a few hundred microsecond [73,74]. The time scale is very close to the period of the repetition rate of the Spitfire system. As a result, the refractive index hole still remains in the air, and the following pulse would be affected by the hole created by the last pulse. If all the experimental parameters are ideally fixed, the filamentation will reach a dynamic equilibrium state even at such a high repetition rate [75]. For example, the density hole of plasma will reach to an equilibrium state after 250 pulses train come at 1 kHz repetition rate [75]. However, in our experiment, we observed some repeatable power fluctuations in a sub-second time scale as is shown in Fig. 2(c). When the long-term fluctuation happened, the photodiode signal of the MIR pulse was clearly different from the normal stable case, and the unstable state remained for $\sim$500 ms. We believe that the fluctuation is a kind of stochastic event rather than a periodic event. The long-term fluctuations seem to be synchronized with some blinking of the fluorescence from the filament. This implies that the fundamental reason of the long-term fluctuation is the instability of the filament caused by the high repetition rate of the input pulses. In our experimental condition, the nonlinear medium does not reach an equilibrium state at 10kHz but reaches at 5kHz.

In addition, we monitored the pulse-to-pulse instability of the filament by measuring the transmission of a parallel HeNe laser beam (0.8 mW, $\phi =4.5$ mm) through the filament from the side. When the filament exists, there is a shadow in the He-Ne laser beam due to the scattering of the filament (see Figs. 2(d) and 2(e)). The time series of the images at each repetition rate are shown as a supplement movie (Visualization 1). It clearly shows the stability difference of the filament between the two cases.

3.2 Beam profile

When all of the input three pulses at the FW-DFG process are the lowest order Gaussian beams and these three beam waists are also at the same position with the same confocal parameter [51], the far-field of the FW-DFG signal will be in a symmetric ring-shaped due to the phase matching condition [51,76]. Figure 3 shows typical beam profiles of the MIR pulses on the concave mirror with a hole for the collimation. These were measured with a pyroelectric camera (Pyrocam III, Spiricon). The half of divergent angles $\theta$ (the radius of the ring) are 1.41° and 1.93°, and the FWHM of the half divergent angle $\Delta \theta$ (the width of ring) are 0.97° and 0.75° at 10 kHz and 5 kHz, respectively. The divergent angle of the 10 kHz system is similar to that of the MIR pulse generated using our previous Femtopower system [67] ($\theta =$1.5° half divergent angle). The larger divergent angle of the 5 kHz system is probably due to the smaller size of the filament.

 figure: Fig. 3.

Fig. 3. The intensity distribution of the MIR pulse on the concave mirror with a hole (a) at 10 kHz and (b) 5 kHz. The exposure time of the camera at 5 kHz is twice longer than that at 10 kHz. (c) is the intensity distribution when the beam is focused by a parabolic mirror($f$ = 15 cm) at 10 kHz.

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To check the spatial uniformity, we focused the 17 mm diameter MIR beam by using a 90° off-axis parabolic mirror (gold coated, $f$ = 150 mm). As is shown in Fig. 3, the beam diameter of the MIR pulse at the focus was 25.6 µm, which was measured by the same pyroelectric camera with an additional magnifier (the magnification was 17.6). These beam profiles show that the MIR beam remained well radially symmetric in terms of not only the intensity- but also phase-spatial distribution.

3.3 Spectrum

It is not very straightforward to record the spectrum which covers the frequency range from MIR to THz. Fourier-transform spectrometers are not suitable because it is very difficult to have a proper beamsplitter and detector. In addition, the kilohertz repetition rate is too slow for such a spectrometer.

Here the spectrum of the MIR pulse was characterized by using cross-correlation frequency-resolved optical gating (XFROG) with FW-DFG through a gaseous medium. Because of the low dispersion of the gaseous medium, the available phase matching bandwidth is also as broad as that of the down-conversion FW-DFG. The inverse FW-DFG process is described as follows,

$$\omega_1 + \omega_1 - \omega_0 \rightarrow \omega'_2,$$
where $\omega _1$, $\omega _0$, and $\omega '_2$ are the angular frequencies of the reference pulse, the MIR pulse, and the up-conversion signal which has slightly longer wavelength than the SH pulse. The up-conversion field $E'_2(t)$ through the FW-DFG can be written as follows,
$$E'_2(t) \propto \chi^{(3)}E_1^2(t-\tau)E_0^*(t),$$
where $\tau$ is the time delay between the reference beam $E_1$ and the MIR pulse $E_0$. By recording the spectrum of the signal by scanning the delay $\tau$, the XFROG traces are measured and we can reconstruct the electric field of the MIR pulse from the trace through the XFROG retrieval algorithm [77].

The experimental setup for the XFROG is also shown in Fig. 1. To control the dispersion of the reference pulse, we use a compressor based on a pair of transmission gratings (PCG-1250-800-989, Ibsen). Since the compressor introduces a much larger negative dispersion than necessary, we insert a dispersive substrate (anti-reflection coated N-BK7, $t$=12 mm) and let the beam pass through it four times before the compressor. The reference pulse was characterized by using an SHG-FROG device (FROG-FC, FemtoEasy).

The measured XFROG traces are shown in Figs. 4(a) and 4(b). The retrieved spectra are also shown in Fig. 4(c). The spectrum of the Spitfire system at 10 kHz is spread from 4.90 to 43.1 µm (from 2040 to 232 cm$^{-1}$), which corresponds to more than three octave with full width at tenth maximum. The peak of the spectrum of the Spitfire system is at 14 µm (730 cm$^{-1}$) whereas the center-of-mass of the spectrum is 10.2 µm (977 cm$^{-1}$). The pulse duration of the transform limited pulse is 13.4 fs, which corresponds to 0.393 cycles. The peak wavelength is much longer and the bandwidth is much narrower than those of the spectrum generated using the Femtopower. Although the parameters of the fundamental pulses from the two lasers are very similar, we have obtained the distinct spectra of the MIR pulses. In each case, the carrier wavelength of the MIR pulse does not depend on either the pulse energy or even the dispersion of the fundamental pulse very much. The detailed comparison between the two Ti:sapphire laser systems is shown in Appendix A.

 figure: Fig. 4.

Fig. 4. The experimental XFROG traces of the MIR pulse (a) at 10 kHz and (b) at 5 kHz. (c) MIR spectra normalized by the integrated area. The red and green solid lines are the retrieved XFROG spectra at 10 kHz and 5 kHz repetition rate, respectively. The blue solid line is the retrieved XFROG spectrum with our previous Femtopower laser system [78].

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3.4 Carrier-envelope phase

3.4.1 Passive stabilization of the CEP

The CEP value is a very important physical parameter especially for sub-cycle pulses. Based on the scheme of the FW-DFG and the corresponding Eq. (1), the phase of the MIR pulse should follow the relation $\phi _0 = \pi /{2} + \phi _1 + \phi _1 - \phi _2$ [79]. Additionally, the phase of the coherent SH pulse $\omega _2$ is determined by the fundamental pulse $\omega _1$, the phase of the SH pulse is $\phi _2 = \pi /{2} + \phi _1 + \phi _1$. According to these relations, the phase of the MIR pulse should be automatically stabilized even though the phase of the fundamental pulse is not fixed. The CEP of the MIR pulse is only related to the phase shift between the $\omega _1$ and $\omega _2$ pulses. This implies that we can control the passively stable CEP of the MIR pulses by changing the phase delay between the two-color pulses [66,80,81].

3.4.2 Stability measurement of the CEP

We measured the stability of the CEP of the MIR pulse by using the experimental setup shown in Fig. 5(a). A BBO ($\theta$ = 29°, $t$ = 50 µm) on a 2-mm thick fused silica substrate was inserted after the focus where the FW-DFG signal (the XFROG signal) was generated. The reference beam passed through the crystal from the substrate side, which provides a temporal delay $\sim$310 fs between the reference and FW-DFG pulses, and the SH of the reference pulse was generated in the BBO crystal. The drift of the interference signal between the SH and FW-DFG signals corresponds to the stability of the CEP of the MIR pulse. The intensity ratio between the SH and FW-DFG signals were optimized by using a half-wave plate and a polarizer which was inserted behind the crystal. To confirm the pulse-to-pulse stability of the CEP, we used a high speed spectrometer (OCEAN-FX-UV-VIS, Ocean Insight) synchronized with the laser pulse train.

 figure: Fig. 5.

Fig. 5. (a) The experimental setup of the single-shot CEP measurement. The CEP modulation in frequency domain in (b) single-shot measurement and (c) multi-pulses long time monitor with 10 kHz repetition rate. (d) The relative CEP value based on the multi-pulses diagram results.

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Figures 5(b) and 5(c) show the time evolution of the fringe pattern in a short and long time scale, respectively. The SH spectrum is subtracted as the background so that the interference fringe is clearly shown up. Due to the cut-off wavelength of the SH spectrum, the fringes of the interference are only clear up to $\sim$415 nm, which corresponds to 17 µm of the MIR pulse. The period of the fringes was 1.7 nm (3.1 THz = 1/320 fs$^{-1}$), which is consistent with the estimated temporal delay, 310 fs.

At the short time scale measurement, the sampling number of the single pulse measurement was 1000 pulses within 2 second. The standard deviation of the phase value was 0.0397$\pi$ (124 mrad) and 0.0490$\pi$ (154 mrad) for the 10 kHz and 5 kHz system, respectively. At the long time scale measurement, the standard deviation of the phase value was 0.0611$\pi$ (192 mrad) and 0.0748$\pi$ (235 mrad) in 1 and 8 hours respectively for the 10 kHz system. For the 5 kHz system, the standard deviation of the phase value was 0.0875$\pi$ (275 mrad) and 0.0961$\pi$ (302 mrad) in 1 and 4 hours. The drift of the phase value is also shown in Fig. 5(d). The results prove that the fringe of CEP was reasonably stable for hours without any feedback control inside the system. The phase drift is mainly due to the instability of the delay between the two-color input pulses. The stability can be improved by using a well-closed chamber for the MIR pulse generation.

4. Waveform of the sub-half-cycle pulse

4.1 FROG-CEP

4.1.1 Experimental results

We applied FROG-CEP to measure the complete waveform of the sub-cycle MIR pulse. The brief explanation of the FROG-CEP concept is as follows; By performing the simultaneous measurement of the XFROG and air-biased coherent detection (ABCD [82]), the power spectrum and (relative) spectral phase of the MIR pulse are obtained from the XFROG measurement, and the absolute value of the CEP is obtained from the ABCD measurement.

The experimental setup is also shown in Fig. 1. In addition to the XFROG measurement described in section 3.3, we applied an electric bias vertically by using Rogowski-type electrodes with a distance of 3 mm by two high voltage amplifiers (HEOPS-5B6, Matsusada) with an amplitude of $\pm$4 kV, and the polarization of the SH pulse was also in the vertical direction, which was orthogonal to the polarization of the FWM signal. Because of the different polarization of the SH and FWM signals we are able to control the intensity ratio between these two signals.

We recorded the spectra with and without the high voltage bias and subtracted the spectra to carry out the interference of the EOS signal. The results are shown in Fig. 6. The interference signal appears in the wavelength from 400 to 415 nm, which means the shortest detectable signal is 17 µm (17.5 THz) with the current condition.

 figure: Fig. 6.

Fig. 6. The frequency-resolved ABCD measurement results at (a) 10 kHz and (b) 5 kHz. (c) The integrated signals of the frequency-resolved ABCD traces along the wavelength.

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We can recover the complete electric field of the MIR pulse by setting the offset of the spectral phase of the retrieved XFROG based on the absolute value of the CEP from the ABCD measurement [65]. Figure 7 shows the retrieved electric fields with four different phase delays between the fundamental and SH input pulses. The blue-shift of the MIR spectra was observed when the power was lower (we assume that it is when the phase delay lengths are 100 and 300 nm). This phenomenon is consistent with our previous work [67]. The intensity in the lower frequency components is more sensitive to the phase delay of the two-color pulses.

 figure: Fig. 7.

Fig. 7. (a) The spectra and phases of the retrieved XFROG traces at the different phase delay length of the two colors $\xi$. The filled areas are the corresponding numerical simulation results. (b) The waveforms at the maximum intensity (when the phase delay lengths are 0 and 200 nm). (c) The waveforms at the minimum intensity (when the phase delay lengths are 100 and 300 nm). The dashed lines are the corresponding simulation results. The color of the lines in (b) and (c) correspond to the color in (a).

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As can be seen in Fig. 7(a), the phase is almost flat in the whole frequency region, in particular for the cosine shaped waveforms. This means that the pulses are nearly transform-limited. The pulse widths of the cosine and sine shaped waveform are estimated as 13.7 and 13.4 fs, respectively. The number of cycles of the cosine shaped waveform becomes 0.402 cycles at the carrier wavelength of 10.2 µm, whereas that of the sine shaped wavelength becomes 0.459 cycles at the carrier wavelength of 8.7 µm. Each carrier wavelength is defined by the center-of-mass of each spectrum. To our knowledge, the duration is the shortest in the pulses at the carrier wavelength of $\sim$10 µm and the number of cycles is the fewest for MIR pulses. When the power is higher (when the phase delay lengths are 0 and 200 nm), the shape of the MIR waveform is clearly cosine, namely, the field is even function. On the other hand, when the power is lower the shape of the MIR waveform is sine, namely, the field is an odd function.

4.1.2 Discussion

One might wrongly assume that the pulse with the cycle number of less than 0.5 should not exist because it sounds that the wavelength of the pulse cannot be determined or the pulse contains some zero frequency (DC) component. We believe that such a misunderstanding comes from the traditional definition of the cycle number. The cycle number is defined as the product of the pulse duration (FWHM of the intensity envelope) and the carrier frequency of the pulse, however, it always becomes smaller than the number of local maximum and minimum of the oscillating field. The oscillation of the electric field can easily exist in $4\sqrt {2}$ times the standard deviation of the intensity distribution in time, $\sigma _t = \sqrt {\int t^2 \left |E_{0}(t)\right |^2 dt / \int \left |E_{0}(t)\right |^2 dt}$. In our case, $\sigma _t=$8.43 fs, and the full width of $1/e^{2}$ level (13.5%) of $\left | E_{0}(t) \right |$ is 47.7 fs, which is 1.40 times longer than the period of the carrier wavelength (34.1 fs). That means that the time period is enough for “sub-half-cycle” pulse to physical oscillation.

4.2 Phase control of the sub-half-cycle pulse

4.2.1 Experimental results

To investigate the phase control of the MIR pulse, we performed several individual FROG-CEP measurements with different phase delays of the two-color (fundamental and SH) input pulses, and reconstructed each waveform. The phase delay dependence of the MIR waveform is shown in Fig. 8(a). The $x$-axis is the time of the retrieved MIR waveform while the $y$-axis is the phase delay length of the two-color pulses. It can be seen that the period of the intensity modulation is 200 nm whereas the period of the CEP modulation is 400 nm. The phase delay dependence of the up-conversion spectra, which are obtained by integrating the XFROG trace along the delay axis, is shown in Fig. 8(b). The intensity modulation depth is $\sim$2:1, which is also close to our previous work [67]. The phase delay dependence of the normalized retrieved spectra, which are obtained with the XFROG algorithm, is shown in Fig. 8(c). The FROG errors of all the retrieved waveforms are less than 0.4% on a 512$\times$512 grid.

 figure: Fig. 8.

Fig. 8. Experimental results of the phase control of the MIR pulse. (a) Phase delay length dependence of the retrieved electric fields. (b) Phase delay length dependence of the up-conversion spectrum, which is obtained by integrating each XFROG trace along the delay axis. The upper axis ($\omega '_2$ wavelength) shows the wavelength of the up-conversion spectrum. The bottom axis (wavenumber) is the approximated MIR frequency $\omega _0$ obtained by calculating $\omega _0 = \omega _{\mathrm {ref}} - \omega '_2$, where $\omega _{\mathrm {ref}}$ corresponds to the angular frequency of 400 nm. (c) Phase delay dependence of normalized retrieved XFROG spectra.

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4.2.2 Numerical simulations

To explain the phase delay dependence of the power and CEP of the MIR pulse, we have performed numerical simulations based on 1-D FW-DFG model. We assume that only the FW-DFG processes contribute to the MIR pulse generation and the phase mismatch is negligible because of the low dispersion of the nonlinear medium. This model is basically the same as our previous work [81]. The third-order nonlinear polarization $\tilde {P}^{(3)}(\omega )$ for the FW-DFG can be written as

$$\tilde{P}^{(3)}(\omega) = \iint d\omega^{\prime} d\omega^{\prime\prime}\epsilon_0 \chi^{(3)} \tilde{E}_1(\omega^{\prime\prime} \tilde{E}_1(\omega') \tilde{E}^*_2(\omega^{\prime\prime}+\omega^{\prime}-\omega),$$
$\epsilon _0$ and $\chi ^{(3)}$ are the electric permittivity constant of free space and the third order electric susceptibility, respectively. The $\tilde {E}_1(\omega )$ and $\tilde {E}_2(\omega )$ are the Fourier-transform of the complex electric fields of fundamental $E_1(t)$ and SH $E_2(t)$, respectively. Assuming that the nonlinear polarization is the point source for the MIR generation, the far-field electromagnetic wave of the MIR pulse is explained as the second derivative of the nonlinear polarization in time domain, or the product of the nonlinear polarization and the square of the angular frequency in frequency domain. Thus, the complex electric field of the MIR pulse generated by the FW-DFG process is described as follows,
$$\tilde{E}_{0}(\omega) \propto \omega^{2} \tilde{P}^{(3)}(\omega),$$
where $\tilde {E}_0(\omega )$ is the Fourier transform of the complex electric field of the MIR pulse $E_0(t)$. As a result, the power spectrum of the MIR pulse becomes the product of the nonlinear polarization and the fourth power of the angular frequency. As is described in the section {2}, we obtain the intensity and phase of the fundamental and SH pulses from the SHG-FROG result in this simulation. The phase delay length $\xi$ dependence of the MIR waveform can be obtained by replacing $\tilde {E}_2(\omega )$ by $\tilde {E}_2(\omega ) \exp {\left ( i \omega \xi /c \right )}$ in Eq. (5) and calculating $E_0(t)$ with scanning $\xi$. Figure 9 shows the results which are plotted in the same way as the experimental results.

 figure: Fig. 9.

Fig. 9. The numerical simulation diagram of two-color FW-DFG. (a) is the waveform of electric field $E_0(t)$ in the different phase delay length $\xi$. (b,c) are the spectra of the MIR pulses in the different phase delay length $\xi$. (c) is the normalized results to show the blue-shift of the spectra.

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4.2.3 Discussion

Comparing between the experimental and simulation results shown in Fig. 8 and 9, the periodic change of the CEP and power spectrum is very well reproduced. Therefore, we claim that the 1-D FW-DFG model well explain the mechanism of the phase-controlled MIR generation with the Spitfire system. The detailed physics of the periodic change of the CEP and power spectrum is explained in [67,78,81]. The brief explanation is as follows; The phase shift due to the delay between the fundamental and SH pulses directly affects the CEP of the generated FW-DFG signal, namely, the phase term $\exp \left (i \omega _2 \xi /c \right )$. The period of the CEP change corresponds to the wavelength of the SH. On the other hand, the periodic change of the power spectrum is caused by the interference between the components which originate $\omega _1 + \omega _1- \omega _2 \to \omega _0$ and $\omega _2 - \omega _1 - \omega _1 \to \omega _0$ processes. Since the two processes have the opposite phase with each other, the interference term is explained as

$$\left| \exp\left(i \omega_2 \xi/c \right) + \exp\left(-i \omega_2 \xi/c \right) \right|^2 = 2 + 2 \cos \left( \omega_2 \xi/c \right) = 4 \cos \left( 2 \omega_2 \xi/c \right).$$
Therefore, the period of the power spectrum change is the half of the wavelength of the SH.

There is some difference between the spectra based on $\omega _1 + \omega _1- \omega _2 \to \omega _0$ and $\omega _2 - \omega _1 - \omega _1 \to \omega _0$ processes. When the interference is destructive, the difference between the spectra appear as the generated MIR spectrum. Since the residual components have more intensity in the higher frequency region in principle, the center of mass of the MIR spectrum at the destructive intereference shows some blue-shift. The blue-shift effect is clearly observed at the experiment and reproduced with the numerical simulations as is shown in the Figs. 8(c) and 9(c). The depth of the intensity modulation of the simulation results is $\sim$10:1 whereas that of the experimental results is $\sim$2:1. We believe that the difference comes from the spectral broadening in the filament, which is not included in the numerical simulations.

We believe that the interference effect also causes the generation of transform-limited sub-half-cycle pulse. When $\omega _2 \xi /c$ is integer multiple of $\pi$, the spectral phase of the input pulses which contribute to the interference completely cancel out with each other and $\tilde {E}_0(\omega )$ becomes real number. As a result, $E_{0}(t)$ becomes transform-limited and symmetric in time. The oscillation of the waveform is only defined by the second derivative of the nonlinear polarization envelope.

Concerning the CEP modulation, it should be noted that not only the modulation period but also the absolute value is reproduced by the numerical simulations. The experimentally obtained absolute value of the CEP should be the phase value of the MIR pulse at the focus where the XFROG signal is generated. We believe that the phase value is the same as that at the MIR pulse generation point for the following reasons. First, the chromatic dispersion of the MIR pulse during the propagation is negligible. The estimated CEP shift is $\sim$19 mrad per meter when a 10 µm pulse propagates through nitrogen at atmospheric pressure [83]. The path between MIR generation and detection is 60 cm, therefore, the corresponding group delay 11 mrad is small enough for precise phase determination. Second, the MIR pulse is focused at both the generation and detection points. If the pulse is not focused at one of the points, the Gouy phase shift should be seriously considered because of the concentric spatial distribution in frequency domain [84]. However, we do not expect such a phase shift because we produced and detected the MIR pulse at the focus of the beam. We have $\pi$ phase ambiguity because the period of the intensity modulation is 200 nm whereas the CEP modulation period is 400 nm. Based on the above discussions, we claim that the detected MIR waveform represents the realistic situation of the MIR generation at the laser-induced filament.

5. Conclusion

We have experimentally demonstrated the generation of sub-half-cycle MIR pulses through two-color filamentation in nitrogen. The CEP of the MIR pulse is passively stabilized and controlled by the delay between the two-color input pulses. The waveform was characterized by using FROG-CEP and the spectral phase was almost flat, which means that the pulse was almost transform-limited. The duration of the MIR pulse was 13.7 fs, which corresponds to 0.402 cycles. The waveform can be clearly switched from cosine to sine shape and vice versa by controlling the CEP of the MIR pulse. The absolute value of the CEP of the generated sub-half-cycle pulse is consistent with an 1-D FW-DFG model.

It is still not clear why the generated MIR spectrum is so different from what we had with the Femtopower system. Although the MIR spectrum from the Spitfire system is not as broad as the Femtopower system, the cycle number is smaller and the pulse is nearly transform-limited. The higher density in the spectral region from 5 to 20 µm can be useful for the study of the structural dynamics of protein, which has amide bands in the wavelength range.

It is very important to know the pulse energy fluctuation at 10 kHz repetition rate. Although the instability of the pulse energy will limit the application of the light source with 10 kHz repetition rate, the system at 5 kHz repetition rate is stable enough for MIR spectroscopy.

The nearly transform-limited MIR pulse is a very attractive light source for the study of high harmonic generation in solids. In our previous experiment with the sub-cycle pulses, namely the pulses generated by using Femtopower, the effect of the chirp was very dominant [3]. It would be possible to observe clearly the CEP dependence of the plateau and cutoff of the high harmonic spectra by using the current light source.

Appendix A: Difference between the two Ti:sapphire laser systems

A.1. Characteristics

Here we discuss the difference between the input pulses of the current and previous systems. Our previous system was the Ti-sapphire based multi-pass amplifier with a Dazzler, which provides the pulse with the duration of 30 fs at 1 kHz repetition rate. The input pulse energy for the two-color filamentation was 790 µJ at the center wavelength of 798 nm and the bandwidth of 39 nm at FWHM. The beam was focused by using a concave mirror with the curvature of 1 m. The spectrum of the optimized SH pulse was centered at 404 nm with the bandwidth of 9 nm. On the other hand, the new Ti:sapphire laser system consists of a regenerative amplifier (750 µJ) and a single-pass amplifier (1.4 mJ, 35 fs, 10 kHz). The input pulse energy is 850 µJ at the center wavelength of 804 nm and the bandwidth of 35 nm. The spectrum of the optimized SH pulse is centered at 403 nm with the bandwidth of 7 nm. At both the systems, we shift the spectrum of the SH pulse to slightly longer wavelength by tilting the SHG crystal to have a better conversion efficiency. There is no obvious difference between the specifications of the two laser systems.

Figure 10 shows the SHG frequency-resolved optical gating (SHG-FROG) traces of the fundamental pulses from the two laser systems. We used a commercial SHG-FROG device (FROG-FC, FemtoEasy) to characterize the temporal and spatial dispersion of the fundamental pulse. Thanks to the high dynamics range of the FROG measurement, the difference between the traces of the Spitfire and the Femtopower can be clearly seen. The trace of the Femtopower is more symmetric and less complex than that of the Spitfire.

 figure: Fig. 10.

Fig. 10. The SHG-FROG traces of the fundamental pulses in logarithmic scale. (a) is the trace of the pulse from the Spitfire 10 kHz system. The pulse duration is 35.1 fs with the third order dispersion (TOD) of $-$2466 fs$^3$ and the retrieved error of 3.05$\times$10$^{-2}$. (b) is the previous Femtopower 1 kHz system. The pulse duration is 29.5 fs with the TOD of $-$287 fs$^3$ and the retrieved error of 3.68$\times$10$^{-2}$.

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 figure: Fig. 11.

Fig. 11. (a) The power spectra (solid curves) and spectral phases (open squares) of the MIR pulses with the Spitfire and Femotpower systems. The filled curves are the simulation results based on the spectra of the input two-color pulses. (b) The electric waveforms of the MIR pulses generated with the Spitfire and Femtopower systems.

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A.2. Discussion

To clarify what is the reason for the difference between the MIR spectra generated using the two lasers, we have performed a simple numerical simulation based on the ideal FW-DFG model. In brief, we have calculated the Fourier-transform of the products of two fundamental and SH fields, namely, $E_1^2(t)E^*_2(t)$, which is proportional to the nonlinear polarization for the FW-DFG. We obtained the intensity and phase of the fundamental field from the SHG-FROG result. Concerning the SH field, We have obtained the power spectrum of the SH pulse by using a spectrometer and the spectral phase from the Fourier-transform of the square of the fundamental field in time-domain. Afterwards, we have multiplied $\omega ^2$ to the spectrum of the nonlinear polarization, which corresponds to the second derivative in time-domain. We assume that the nonlinear polarization is the far-field point source.

The simulation results are also shown in Fig. 11(a). The simulated MIR spectra from the Spitfire and Femtopower systems are similar to each other. This fact indicates that the difference between the two input pulses is rather small in terms of temporal and spectral features. The experimental MIR spectra of the Spitfire system are very close to the simulation results and the spectral phase of the XFROG result is flat in the spectral range.

On the other hand, the MIR spectrum from the Femtopower system is very far from the simulation result. As can be seen in Fig. 11(a), the MIR spectrum from the Femtopower system consists of two components (those peaks are at 1200 and 3000 cm$^{-1}$, respectively). In addition, there was a phase shift between the two peaks and the positive chirp in the high frequency region [67,81]. The source of the low frequency component might be FW-DFG as well as the Spitfire system, however, the source of the high frequency component could be different. As a result, the wave form of the MIR pulse from the Femtopower system is more complex than that from the Spitfire system (see Fig. 11(b)).

Based on the above discussions, the extremely broadband MIR spectrum, in particular the high frequency components, from the Femtopower system would be due to the stronger interaction between the three pulses (fundamental, SH, and MIR), something like cross-phase modulation and cascaded FW-DFG.

Funding

Core Research for Evolutional Science and Technology (JPMJCR17N5); Japan Society for the Promotion of Science (17H02801).

Disclosures

The authors declare no conflicts of interest.

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73. Y.-H. Cheng, J. K. Wahlstrand, N. Jhajj, and H. M. Milchberg, “The effect of long timescale gas dynamics on femtosecond filamentation,” Opt. Express 21(4), 4740–4751 (2013). [CrossRef]  

74. J. Annaloro, V. Morel, A. Bultel, and P. Omaly, “Global rate coefficients for ionization and recombination of carbon, nitrogen, oxygen, and argon,” Phys. Plasmas 19(7), 073515 (2012). [CrossRef]  

75. N. Jhajj, Y.-H. Cheng, J. K. Wahlstrand, and H. M. Milchberg, “Optical beam dynamics in a gas repetitively heated by femtosecond filaments,” Opt. Express 21(23), 28980–28986 (2013). [CrossRef]  

76. V. Blank, M. D. Thomson, and H. G. Roskos, “Spatio-spectral characteristics of ultra-broadband THz emission from two-colour photoexcited gas plasmas and their impact for nonlinear spectroscopy,” New J. Phys. 15(7), 075023 (2013). [CrossRef]  

77. S. Linden, H. Giessen, and J. Kuhl, “XFROG - A new method for amplitude and phase characterization of weak ultrashort pulses,” Phys. Status Solidi B 206(1), 119–124 (1998). [CrossRef]  

78. T. Fuji, Y. Nomura, and H. Shirai, “Generation and characterization of phase-stable sub-single-cycle pulses at 3000 cm −1,” IEEE J. Sel. Top. Quantum Electron. 21(5), 8700612 (2015). [CrossRef]  

79. A. Baltuška, T. Fuji, and T. Kobayashi, “Controlling the carrier-envelope phase of ultrashort light pulses with optical parametric amplifiers,” Phys. Rev. Lett. 88(13), 133901 (2002). [CrossRef]  

80. J. Dai and X.-C. Zhang, “Terahertz wave generation from gas plasma using a phase compensator with attosecond phase-control accuracy,” Appl. Phys. Lett. 94(2), 021117 (2009). [CrossRef]  

81. Y. Nomura, Y.-T. Wang, A. Yabushita, C.-W. Luo, and T. Fuji, “Controlling the carrier-envelope phase of single-cycle mid-infrared pulses with two-color filamentation,” Opt. Lett. 40(3), 423–426 (2015). [CrossRef]  

82. I.-C. Ho, X. Guo, and X.-C. Zhang, “Design and performance of reflective terahertz air-biased-coherent-detection for time-domain spectroscopy,” Opt. Express 18(3), 2872–2883 (2010). [CrossRef]  

83. R. J. Mathar, “Refractive index of humid air in the infrared: model fits,” J. Opt. A: Pure Appl. Opt. 9(5), 470–476 (2007). [CrossRef]  

84. P. Klarskov, A. C. Strikwerda, K. Iwaszczuk, and P. U. Jepsen, “Experimental three-dimensional beam profiling and modeling of a terahertz beam generated from a two-color air plasma,” New J. Phys. 15(7), 075012 (2013). [CrossRef]  

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2020 (2)

T.-T. Yeh, C.-M. Tu, W.-H. Lin, C.-M. Cheng, W.-Y. Tzeng, C.-Y. Chang, H. Shirai, T. Fuji, R. Sankar, F.-C. Chou, M. M. Gospodinov, T. Kobayashi, and C.-W. Luo, “Femtosecond time-evolution of mid-infrared spectral line shapes of Dirac fermions in topological insulators,” Sci. Rep. 10(1), 9803 (2020).
[Crossref]

I. Pupeza, M. Huber, M. Trubetskov, W. Schweinberger, S. A. Hussain, C. Hofer, K. Fritsch, M. Poetzlberger, L. Vamos, E. Fill, T. Amotchkina, K. V. Kepesidis, A. Apolonski, N. Karpowicz, V. Pervak, O. Pronin, F. Fleischmann, A. Azzeer, M. Žigman, and F. Krausz, “Field-resolved infrared spectroscopy of biological systems,” Nature 577(7788), 52–59 (2020).
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2019 (2)

N. Ishii, P. Xia, T. Kanai, and J. Itatani, “Optical parametric amplification of carrier-envelope phase-stabilized mid-infrared pulses generated by intra-pulse difference frequency generation,” Opt. Express 27(8), 11447–11454 (2019).
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T. D. Phan, D. Vu, and T. Imasaka, “Vacuum-ultraviolet stimulated emission generated via four-wave Raman mixing in molecular hydrogen,” Appl. Phys. B 125(7), 128 (2019).
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2018 (4)

2017 (7)

H. Park, A. Camper, K. Kafka, B. Ma, Y. H. Lai, C. Blaga, P. Agostini, L. F. DiMauro, and E. Chowdhury, “High-order harmonic generations in intense MIR fields by cascade three-wave mixing in a fractal-poled LiNbO3 photonic crystal,” Opt. Lett. 42(19), 4020–4023 (2017).
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T.-T. Yeh, H. Shirai, C.-M. Tu, T. Fuji, T. Kobayashi, and C.-W. Luo, “Ultrafast carrier dynamics in Ge by ultra-broadband mid-infrared probe spectroscopy,” Sci. Rep. 7(1), 40492 (2017).
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S. Türker-Kaya and C. Huck, “A review of mid-infrared and near-infrared imaging: principles, concepts and applications in plant tissue analysis,” Molecules 22(1), 168 (2017).
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A. Cartella, T. F. Nova, A. Oriana, G. Cerullo, M. Först, C. Manzoni, and A. Cavalleri, “Narrowband carrier-envelope phase stable mid-infrared pulses at wavelengths beyond 10 μm by chirped-pulse difference frequency generation,” Opt. Lett. 42(4), 663–666 (2017).
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U. Elu, M. Baudisch, H. Pires, F. Tani, M. H. Frosz, F. Köttig, A. Ermolov, P. St. J. Russell, and J. Biegert, “High average power and single-cycle pulses from a mid-IR optical parametric chirped pulse amplifier,” Optica 4(9), 1024–1029 (2017).
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A. Dubietis, G. Tamošauskas, R. Šuminas, V. Jukna, and A. Couairon, “Ultrafast supercontinuum generation in bulk condensed media,” Lith. J. Phys. 57(3), 113 (2017).
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M. Cassataro, D. Novoa, M. C. Günendi, N. N. Edavalath, M. H. Frosz, J. C. Travers, and P. S. Russell, “Generation of broadband mid-IR and UV light in gas-filled single-ring hollow-core PCF,” Opt. Express 25(7), 7637–7644 (2017).
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2016 (3)

2015 (9)

C. Hu, T. Chen, P. Jiang, B. Wu, J. Su, and Y. Shen, “Broadband high-power mid-IR femtosecond pulse generation from an ytterbium-doped fiber laser pumped optical parametric amplifier,” Opt. Lett. 40(24), 5774–5777 (2015).
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O. Kfir, P. Grychtol, E. Turgut, R. Knut, D. Zusin, D. Popmintchev, T. Popmintchev, H. Nembach, J. M. Shaw, A. Fleischer, H. Kapteyn, M. Murnane, and O. Cohen, “Generation of bright phase-matched circularly-polarized extreme ultraviolet high harmonics,” Nat. Photonics 9(2), 99–105 (2015).
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M. Baudisch, H. Pires, H. Ishizuki, T. Taira, M. Hemmer, and J. Biegert, “Sub-4-optical-cycle, 340 MW peak power, high stability mid-IR source at 160 kHz,” J. Opt. 17(9), 094002 (2015).
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F. C. Cruz, D. L. Maser, T. Johnson, G. Ycas, A. Klose, F. R. Giorgetta, I. Coddington, and S. A. Diddams, “Mid-infrared optical frequency combs based on difference frequency generation for molecular spectroscopy,” Opt. Express 23(20), 26814–26824 (2015).
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K. Iwaszczuk, M. Zalkovskij, A. C. Strikwerda, and P. U. Jepsen, “Nitrogen plasma formation through terahertz-induced ultrafast electron field emission,” Optica 2(2), 116–123 (2015).
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A. A. Lanin, A. A. Voronin, A. B. Fedotov, and A. M. Zheltikov, “Time-domain spectroscopy in the mid-infrared,” Sci. Rep. 4(1), 6670 (2015).
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H. Shirai, T.-T. Yeh, Y. Nomura, C.-W. Luo, and T. Fuji, “Ultrabroadband midinfrared pump-probe spectroscopy using chirped-pulse up-conversion in gases,” Phys. Rev. Appl. 3(5), 051002 (2015).
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T. Fuji, Y. Nomura, and H. Shirai, “Generation and characterization of phase-stable sub-single-cycle pulses at 3000 cm −1,” IEEE J. Sel. Top. Quantum Electron. 21(5), 8700612 (2015).
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Y. Nomura, Y.-T. Wang, A. Yabushita, C.-W. Luo, and T. Fuji, “Controlling the carrier-envelope phase of single-cycle mid-infrared pulses with two-color filamentation,” Opt. Lett. 40(3), 423–426 (2015).
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2014 (7)

B. Piglosiewicz, S. Schmidt, D. J. Park, J. Vogelsang, P. Groß, C. Manzoni, P. Farinello, G. Cerullo, and C. Lienau, “Carrier-envelope phase effects on the strong-field photoemission of electrons from metallic nanostructures,” Nat. Photonics 8(1), 37–42 (2014).
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J. Weisshaupt, V. Juvé, M. Holtz, S. Ku, M. Woerner, T. Elsaesser, S. Ališauskas, A. Pugžlys, and A. Baltuška, “High-brightness table-top hard X-ray source driven by sub-100-femtosecond mid-infrared pulses,” Nat. Photonics 8(12), 927–930 (2014).
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T. Horio, R. Spesyvtsev, and T. Suzuki, “Generation of sub-17 fs vacuum ultraviolet pulses at 133 nm using cascaded four-wave mixing through filamentation in Ne,” Opt. Lett. 39(20), 6021–6024 (2014).
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C. R. Petersen, U. Møller, I. Kubat, B. Zhou, S. Dupont, J. Ramsay, T. Benson, S. Sujecki, N. Abdel-Moneim, Z. Tang, D. Furniss, A. Seddon, and O. Bang, “Mid-infrared supercontinuum covering the 1.4–13.3 μm molecular fingerprint region using ultra-high NA chalcogenide step-index fibre,” Nat. Photonics 8(11), 830–834 (2014).
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I. Kubat, C. Rosenberg Petersen, U. V. Møller, A. Seddon, T. Benson, L. Brilland, D. Méchin, P. M. Moselund, and O. Bang, “Thulium pumped mid-infrared 0.9–9μm supercontinuum generation in concatenated fluoride and chalcogenide glass fibers,” Opt. Express 22(4), 3959–3967 (2014).
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N. Jhajj, E. W. Rosenthal, R. Birnbaum, J. K. Wahlstrand, and H. M. Milchberg, “Demonstration of long-lived high-power optical waveguides in air,” Phys. Rev. X 4(1), 011027 (2014).
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O. Lahav, L. Levi, I. Orr, R. A. Nemirovsky, J. Nemirovsky, I. Kaminer, M. Segev, and O. Cohen, “Long-lived waveguides and sound-wave generation by laser filamentation,” Phys. Rev. A 90(2), 021801 (2014).
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2013 (10)

T. Horio, R. Spesyvtsev, and T. Suzuki, “Simultaneous generation of sub-20 fs deep and vacuum ultraviolet pulses in a single filamentation cell and application to time-resolved photoelectron imaging,” Opt. Express 21(19), 22423–22428 (2013).
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Y. Nomura, H. Shirai, and T. Fuji, “Frequency-resolved optical gating capable of carrier-envelope phase determination,” Nat. Commun. 4(1), 2820 (2013).
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T. Fuji and Y. Nomura, “Generation of Phase-Stable Sub-Cycle Mid-Infrared Pulses from Filamentation in Nitrogen,” Appl. Sci. 3(1), 122–138 (2013).
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Y.-H. Cheng, J. K. Wahlstrand, N. Jhajj, and H. M. Milchberg, “The effect of long timescale gas dynamics on femtosecond filamentation,” Opt. Express 21(4), 4740–4751 (2013).
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N. Jhajj, Y.-H. Cheng, J. K. Wahlstrand, and H. M. Milchberg, “Optical beam dynamics in a gas repetitively heated by femtosecond filaments,” Opt. Express 21(23), 28980–28986 (2013).
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V. Blank, M. D. Thomson, and H. G. Roskos, “Spatio-spectral characteristics of ultra-broadband THz emission from two-colour photoexcited gas plasmas and their impact for nonlinear spectroscopy,” New J. Phys. 15(7), 075023 (2013).
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K. Ramasesha, L. De Marco, A. Mandal, and A. Tokmakoff, “Water vibrations have strongly mixed intra- and intermolecular character,” Nat. Chem. 5(11), 935–940 (2013).
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A. B. Seddon, “Mid-infrared (IR) - A hot topic: The potential for using mid-IR light for non-invasive early detection of skin cancer in vivo,” Phys. Status Solidi B 250(5), 1020–1027 (2013).
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Y. Yu, X. Gai, T. Wang, P. Ma, R. Wang, Z. Yang, D.-Y. Choi, S. Madden, and B. Luther-Davies, “Mid-infrared supercontinuum generation in chalcogenides,” Opt. Mater. Express 3(8), 1075–1086 (2013).
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P. Klarskov, A. C. Strikwerda, K. Iwaszczuk, and P. U. Jepsen, “Experimental three-dimensional beam profiling and modeling of a terahertz beam generated from a two-color air plasma,” New J. Phys. 15(7), 075012 (2013).
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2012 (6)

C. Calabrese, A. M. Stingel, L. Shen, and P. B. Petersen, “Ultrafast continuum mid-infrared spectroscopy: probing the entire vibrational spectrum in a single laser shot with femtosecond time resolution,” Opt. Lett. 37(12), 2265–2267 (2012).
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T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
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G. Herink, D. R. Solli, M. Gulde, and C. Ropers, “Field-driven photoemission from nanostructures quenches the quiver motion,” Nature 483(7388), 190–193 (2012).
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J. Annaloro, V. Morel, A. Bultel, and P. Omaly, “Global rate coefficients for ionization and recombination of carbon, nitrogen, oxygen, and argon,” Phys. Plasmas 19(7), 073515 (2012).
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K.-Y. Kim, J. H. Glownia, A. J. Taylor, and G. Rodriguez, “High-Power Broadband Terahertz Generation via Two-Color Photoionization in Gases,” IEEE J. Quantum Electron. 48(6), 797–805 (2012).
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S. L. Chin, T. J. Wang, C. Marceau, J. Wu, J. S. Liu, O. Kosareva, N. Panov, Y. P. Chen, J. F. Daigle, S. Yuan, A. Azarm, W. W. Liu, T. Seideman, H. P. Zeng, M. Richardson, R. Li, and Z. Z. Xu, “Advances in intense femtosecond laser filamentation in air,” Laser Phys. 22(1), 1–53 (2012).
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2011 (3)

J. Dai, B. Clough, I.-C. Ho, X. Lu, J. Liu, and X.-C. Zhang, “Recent Progresses in Terahertz Wave Air Photonics,” IEEE Trans. Terahertz Sci. Technol. 1(1), 274–281 (2011).
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S. Ghimire, A. D. Dichiara, E. Sistrunk, P. Agostini, L. F. Dimauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
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G. Andriukaitis, T. Balčiūnas, S. Ališauskas, A. Pugžlys, A. Baltuška, T. Popmintchev, M.-C. Chen, M. M. Murnane, and H. C. Kapteyn, “90 GW peak power few-cycle mid-infrared pulses from an optical parametric amplifier,” Opt. Lett. 36(15), 2755–2757 (2011).
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2010 (4)

F. Théberge, M. Châteauneuf, G. Roy, P. Mathieu, and J. Dubois, “Generation of tunable and broadband far-infrared laser pulses during two-color filamentation,” Phys. Rev. A 81(3), 033821 (2010).
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I.-C. Ho, X. Guo, and X.-C. Zhang, “Design and performance of reflective terahertz air-biased-coherent-detection for time-domain spectroscopy,” Opt. Express 18(3), 2872–2883 (2010).
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T.-J. Wang, J.-F. Daigle, Y. Chen, C. Marceau, F. Théberge, M. Châteauneuf, J. Dubois, and S. Chin, “High energy THz generation from meter-long two-color filaments in air,” Laser Phys. Lett. 7(7), 517–521 (2010).
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2009 (7)

J. Dai and X.-C. Zhang, “Terahertz wave generation from gas plasma using a phase compensator with attosecond phase-control accuracy,” Appl. Phys. Lett. 94(2), 021117 (2009).
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T. Fuji, T. Suzuki, E. E. Serebryannikov, and A. Zheltikov, “Experimental and theoretical investigation of a multicolor filament,” Phys. Rev. A 80(6), 063822 (2009).
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J. Dai and X.-C. Zhang, “Terahertz wave generation from gas plasma using a phase compensator with attosecond phase-control accuracy,” Appl. Phys. Lett. 94(2), 021117 (2009).
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V. P. Kandidov, S. A. Shlenov, and O. G. Kosareva, “Filamentation of high-power femtosecond laser radiation,” Quantum Electron. 39(3), 205–228 (2009).
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J. Dai, N. Karpowicz, and X.-C. Zhang, “Coherent Polarization Control of Terahertz Waves Generated from Two-Color Laser-Induced Gas Plasma,” Phys. Rev. Lett. 103(2), 023001 (2009).
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R. A. Ganeev, H. Singhal, P. A. Naik, I. A. Kulagin, P. V. Redkin, J. A. Chakera, M. Tayyab, R. A. Khan, and P. D. Gupta, “Enhancement of high-order harmonic generation using a two-color pump in plasma plumes,” Phys. Rev. A 80(3), 033845 (2009).
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A. D. DiChiara, E. Sistrunk, T. A. Miller, P. Agostini, and L. F. DiMauro, “An investigation of harmonic generation in liquid media with a mid-infrared laser,” Opt. Express 17(23), 20959–20965 (2009).
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2008 (1)

2007 (6)

T. Fuji, T. Horio, and T. Suzuki, “Generation of 12 fs deep-ultraviolet pulses by four-wave mixing through filamentation in neon gas,” Opt. Lett. 32(17), 2481–2483 (2007).
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A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2-4), 47–189 (2007).
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L. Bergé, S. Skupin, R. Nuter, J. Kasparian, and J.-P. Wolf, “Ultrashort filaments of light in weakly ionized, optically transparent media,” Rep. Prog. Phys. 70(10), 1633–1713 (2007).
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M. Kaku, Y. Oishi, A. Suda, F. Kannari, and K. Midorikawa, “Generation of extreme ultraviolet continuum radiation driven by sub-10-fs two-color field,” Springer Ser. Opt. Sci. 132, 413–419 (2007).
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T. Fuji and T. Suzuki, “Generation of sub-two-cycle mid-infrared pulses by four-wave mixing through filamentation in air,” Opt. Lett. 32(22), 3330–3332 (2007).
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2005 (1)

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I. Koprinkov, “Ionization variation of the group velocity dispersion by high-intensity optical pulses,” Appl. Phys. B 79(3), 359–361 (2004).
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C. Hauri, W. Kornelis, F. Helbing, A. Heinrich, A. Couairon, A. Mysyrowicz, J. Biegert, and U. Keller, “Generation of intense, carrier-envelope phase-locked few-cycle laser pulses through filamentation,” Appl. Phys. B 79(6), 673–677 (2004).
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2003 (1)

2002 (2)

A. Couairon, S. Tzortzakis, L. Bergé, M. Franco, B. Prade, and A. Mysyrowicz, “Infrared femtosecond light filaments in air: simulations and experiments,” J. Opt. Soc. Am. B 19(5), 1117–1131 (2002).
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A. Baltuška, T. Fuji, and T. Kobayashi, “Controlling the carrier-envelope phase of ultrashort light pulses with optical parametric amplifiers,” Phys. Rev. Lett. 88(13), 133901 (2002).
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1998 (1)

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C. R. Petersen, U. Møller, I. Kubat, B. Zhou, S. Dupont, J. Ramsay, T. Benson, S. Sujecki, N. Abdel-Moneim, Z. Tang, D. Furniss, A. Seddon, and O. Bang, “Mid-infrared supercontinuum covering the 1.4–13.3 μm molecular fingerprint region using ultra-high NA chalcogenide step-index fibre,” Nat. Photonics 8(11), 830–834 (2014).
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Agostini, P.

Alisauskas, S.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
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Ališauskas, S.

Amotchkina, T.

I. Pupeza, M. Huber, M. Trubetskov, W. Schweinberger, S. A. Hussain, C. Hofer, K. Fritsch, M. Poetzlberger, L. Vamos, E. Fill, T. Amotchkina, K. V. Kepesidis, A. Apolonski, N. Karpowicz, V. Pervak, O. Pronin, F. Fleischmann, A. Azzeer, M. Žigman, and F. Krausz, “Field-resolved infrared spectroscopy of biological systems,” Nature 577(7788), 52–59 (2020).
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Andriukaitis, G.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
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G. Andriukaitis, T. Balčiūnas, S. Ališauskas, A. Pugžlys, A. Baltuška, T. Popmintchev, M.-C. Chen, M. M. Murnane, and H. C. Kapteyn, “90 GW peak power few-cycle mid-infrared pulses from an optical parametric amplifier,” Opt. Lett. 36(15), 2755–2757 (2011).
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J. Annaloro, V. Morel, A. Bultel, and P. Omaly, “Global rate coefficients for ionization and recombination of carbon, nitrogen, oxygen, and argon,” Phys. Plasmas 19(7), 073515 (2012).
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Apolonski, A.

I. Pupeza, M. Huber, M. Trubetskov, W. Schweinberger, S. A. Hussain, C. Hofer, K. Fritsch, M. Poetzlberger, L. Vamos, E. Fill, T. Amotchkina, K. V. Kepesidis, A. Apolonski, N. Karpowicz, V. Pervak, O. Pronin, F. Fleischmann, A. Azzeer, M. Žigman, and F. Krausz, “Field-resolved infrared spectroscopy of biological systems,” Nature 577(7788), 52–59 (2020).
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Arpin, P.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
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Azarm, A.

S. L. Chin, T. J. Wang, C. Marceau, J. Wu, J. S. Liu, O. Kosareva, N. Panov, Y. P. Chen, J. F. Daigle, S. Yuan, A. Azarm, W. W. Liu, T. Seideman, H. P. Zeng, M. Richardson, R. Li, and Z. Z. Xu, “Advances in intense femtosecond laser filamentation in air,” Laser Phys. 22(1), 1–53 (2012).
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Azzeer, A.

I. Pupeza, M. Huber, M. Trubetskov, W. Schweinberger, S. A. Hussain, C. Hofer, K. Fritsch, M. Poetzlberger, L. Vamos, E. Fill, T. Amotchkina, K. V. Kepesidis, A. Apolonski, N. Karpowicz, V. Pervak, O. Pronin, F. Fleischmann, A. Azzeer, M. Žigman, and F. Krausz, “Field-resolved infrared spectroscopy of biological systems,” Nature 577(7788), 52–59 (2020).
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T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
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G. Andriukaitis, T. Balčiūnas, S. Ališauskas, A. Pugžlys, A. Baltuška, T. Popmintchev, M.-C. Chen, M. M. Murnane, and H. C. Kapteyn, “90 GW peak power few-cycle mid-infrared pulses from an optical parametric amplifier,” Opt. Lett. 36(15), 2755–2757 (2011).
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Baltuska, A.

T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Alisauskas, G. Andriukaitis, T. Balciunas, O. D. Mucke, A. Pugzlys, A. Baltuska, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernandez-Garcia, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the keV X-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012).
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Baltuška, A.

Bang, O.

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C. R. Petersen, U. Møller, I. Kubat, B. Zhou, S. Dupont, J. Ramsay, T. Benson, S. Sujecki, N. Abdel-Moneim, Z. Tang, D. Furniss, A. Seddon, and O. Bang, “Mid-infrared supercontinuum covering the 1.4–13.3 μm molecular fingerprint region using ultra-high NA chalcogenide step-index fibre,” Nat. Photonics 8(11), 830–834 (2014).
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Baudisch, M.

U. Elu, M. Baudisch, H. Pires, F. Tani, M. H. Frosz, F. Köttig, A. Ermolov, P. St. J. Russell, and J. Biegert, “High average power and single-cycle pulses from a mid-IR optical parametric chirped pulse amplifier,” Optica 4(9), 1024–1029 (2017).
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M. Baudisch, H. Pires, H. Ishizuki, T. Taira, M. Hemmer, and J. Biegert, “Sub-4-optical-cycle, 340 MW peak power, high stability mid-IR source at 160 kHz,” J. Opt. 17(9), 094002 (2015).
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Supplementary Material (1)

NameDescription
» Visualization 1       Series of the images of the HeNe laser beam passing through the filament at each repetition rate of the Ti:sapphire laser. The camera is synchronized with the 10 kHz RF signal from the laser system.

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Figures (11)

Fig. 1.
Fig. 1. Experimental setup of the MIR pulse generation and characterization. BBO: $\beta$ -BaB $_2$ O $_4$ , DL: delay plate (calcite), DWP: dual wave plate, CM1: dielectric concave mirror (high reflection for $\omega _1$ and $\omega _2$ ), CM2: aluminium-coated concave mirror with a hole of 7 mm diameter, BF: short pass filter (FGB37, Thorlabs)
Fig. 2.
Fig. 2. The group delay dispersion (GDD) of the input pulse dependence of the average (a) and standard deviation (b) of the MIR pulse-to-pulse intensity, which were measured by using the MCT detector. The zero position of GDD was determined by the maximum output of MIR pulse. (c) Typical MIR single-pulse photodiode signals from a boxcar gated integrator at 10 kHz and 5 kHz repetition rate. (d) A typical pattern of the He-Ne laser beam passing through the filament and (e) that without filament as a reference.
Fig. 3.
Fig. 3. The intensity distribution of the MIR pulse on the concave mirror with a hole (a) at 10 kHz and (b) 5 kHz. The exposure time of the camera at 5 kHz is twice longer than that at 10 kHz. (c) is the intensity distribution when the beam is focused by a parabolic mirror( $f$ = 15 cm) at 10 kHz.
Fig. 4.
Fig. 4. The experimental XFROG traces of the MIR pulse (a) at 10 kHz and (b) at 5 kHz. (c) MIR spectra normalized by the integrated area. The red and green solid lines are the retrieved XFROG spectra at 10 kHz and 5 kHz repetition rate, respectively. The blue solid line is the retrieved XFROG spectrum with our previous Femtopower laser system [78].
Fig. 5.
Fig. 5. (a) The experimental setup of the single-shot CEP measurement. The CEP modulation in frequency domain in (b) single-shot measurement and (c) multi-pulses long time monitor with 10 kHz repetition rate. (d) The relative CEP value based on the multi-pulses diagram results.
Fig. 6.
Fig. 6. The frequency-resolved ABCD measurement results at (a) 10 kHz and (b) 5 kHz. (c) The integrated signals of the frequency-resolved ABCD traces along the wavelength.
Fig. 7.
Fig. 7. (a) The spectra and phases of the retrieved XFROG traces at the different phase delay length of the two colors $\xi$ . The filled areas are the corresponding numerical simulation results. (b) The waveforms at the maximum intensity (when the phase delay lengths are 0 and 200 nm). (c) The waveforms at the minimum intensity (when the phase delay lengths are 100 and 300 nm). The dashed lines are the corresponding simulation results. The color of the lines in (b) and (c) correspond to the color in (a).
Fig. 8.
Fig. 8. Experimental results of the phase control of the MIR pulse. (a) Phase delay length dependence of the retrieved electric fields. (b) Phase delay length dependence of the up-conversion spectrum, which is obtained by integrating each XFROG trace along the delay axis. The upper axis ( $\omega '_2$ wavelength) shows the wavelength of the up-conversion spectrum. The bottom axis (wavenumber) is the approximated MIR frequency $\omega _0$ obtained by calculating $\omega _0 = \omega _{\mathrm {ref}} - \omega '_2$ , where $\omega _{\mathrm {ref}}$ corresponds to the angular frequency of 400 nm. (c) Phase delay dependence of normalized retrieved XFROG spectra.
Fig. 9.
Fig. 9. The numerical simulation diagram of two-color FW-DFG. (a) is the waveform of electric field $E_0(t)$ in the different phase delay length $\xi$ . (b,c) are the spectra of the MIR pulses in the different phase delay length $\xi$ . (c) is the normalized results to show the blue-shift of the spectra.
Fig. 10.
Fig. 10. The SHG-FROG traces of the fundamental pulses in logarithmic scale. (a) is the trace of the pulse from the Spitfire 10 kHz system. The pulse duration is 35.1 fs with the third order dispersion (TOD) of $-$ 2466 fs $^3$ and the retrieved error of 3.05 $\times$ 10 $^{-2}$ . (b) is the previous Femtopower 1 kHz system. The pulse duration is 29.5 fs with the TOD of $-$ 287 fs $^3$ and the retrieved error of 3.68 $\times$ 10 $^{-2}$ .
Fig. 11.
Fig. 11. (a) The power spectra (solid curves) and spectral phases (open squares) of the MIR pulses with the Spitfire and Femotpower systems. The filled curves are the simulation results based on the spectra of the input two-color pulses. (b) The electric waveforms of the MIR pulses generated with the Spitfire and Femtopower systems.

Equations (7)

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ω 1 + ω 1 ω 2 ω 0 ,
Δ k = k 0 2 k 1 + k 2
ω 1 + ω 1 ω 0 ω 2 ,
E 2 ( t ) χ ( 3 ) E 1 2 ( t τ ) E 0 ( t ) ,
P ~ ( 3 ) ( ω ) = d ω d ω ϵ 0 χ ( 3 ) E ~ 1 ( ω E ~ 1 ( ω ) E ~ 2 ( ω + ω ω ) ,
E ~ 0 ( ω ) ω 2 P ~ ( 3 ) ( ω ) ,
| exp ( i ω 2 ξ / c ) + exp ( i ω 2 ξ / c ) | 2 = 2 + 2 cos ( ω 2 ξ / c ) = 4 cos ( 2 ω 2 ξ / c ) .

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