Temporal self-compression is a universal behavior of ultrashort optical field waveforms exposed to anomalous dispersion and optical nonlinearity [1]. In its pure form, such behavior is manifested in anomalously dispersive optical fibers [2], where the temporal evolution of an optical field is decoupled from its spatial dynamics. With the latest generation sources of ultrashort pulses in the mid-infrared (mid-IR) range [3–5], where many solid-state materials exhibit anomalous dispersion, some of the soliton pulse self-compression scenarios, earlier observed primarily in their one-dimensional (1D) version in optical fibers, have been extended to optical beams freely propagating in the bulk of transparent solids [6–11]. If extended to atmospheric air, these regimes would open unique opportunities for a long-distance transmission of high-peak-power laser pulses and remote sensing of the atmosphere [12].

Dispersion properties of molecular gases are, however, drastically different from those of solids. For molecular gases, general causality arguments allow dispersion anomalies only near or within molecular absorption bands. Because of the complex behavior of air refractivity near molecular absorption bands [13], analysis of dispersion properties in these regions is difficult both conceptually and technically. With full calculations of air refractivity using the high-resolution transmission molecular absorption (HITRAN) database [14] being time- and labor-consuming, a useful polynomial-fit approximation has been developed [15] to facilitate refractive-index analysis. As a significant, visionary step in the quest for optical solitons in air, dispersion of air in the mid-IR range has been analyzed [16] in order to identify appropriate regions of anomalous group-velocity dispersion (GVD). In this analysis, the polynomial-fit approximation was used for GVD calculations, leading to a prediction of a continuous GVD anomaly of air within the range of wavelengths from $\approx 3.0$ to 3.3 μm. However, the full model of air refractivity, including all the pertinent HITRAN-database molecular transitions, yields a significantly different GVD profile in the 3.0–3.3 μm region. Instead of a continuous GVD anomaly, the full model dictates a much more complicated, rapidly varying, sign-alternating GVD profile [Figs. 1(a) and 1(b)]. Full-model analysis, on the other hand, shows that a reasonably broadband continuous GVD anomaly exists in the wavelength range of $\approx 3.5$ to 4.2 μm, i.e., near the carbon dioxide rovibrational band. Here, we show, both experimentally and theoretically, that this GVD anomaly can provide, when combined with optical nonlinearity of air, a self-compression of high-peak-power pulses, enabling the generation of subterawatt few-cycle mid-IR pulses. Because of strong high-order dispersion (HOD), which grows toward the edge of the ${\mathrm{CO}}_2$ absorption band, the dynamics of self-compressing pulses in this regime is drastically different from that predicted by the earlier modeling [16,17] based on the polynomial-fit approximation of air dispersion.

 

Fig. 1. (a) Refractive index and (b) GVD calculated as functions of the wavelength using the full model of air refractivity including the HITRAN-database manifold of atomic and molecular transitions. Absorption spectrum of air $\kappa (\lambda )$ is shown by gray shading. Atmospheric air is modeled as a mixture of molecular and atomic gases at a temperature of 296 K with ${\mathrm{N}}_{{\mathrm{N}}_2}=1.971·{10}^{19}{\mathrm{cm}}^{-3}$, ${\mathrm{N}}_{{\mathrm{O}}_2}=5.285·{10}^{18}{\mathrm{cm}}^{-3}$, ${\mathrm{N}}_{\mathrm{Ar}}=2.357·{10}^{17}{\mathrm{cm}}^{-3}$, ${\mathrm{N}}_{{\mathrm{H}}_2\mathrm{O}}=3.408·{10}^{17}{\mathrm{cm}}^{-3}$, and ${\mathrm{N}}_{{\mathrm{CO}}_2}=1.32·{10}^{16}{\mathrm{cm}}^{-3}$, $n_0=1.000273$. (c) Experimental setup: GS, grating stretcher; GC, grating compressor, $T$, telescope; $W$, wedge.

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Our analysis of air dispersion is based on the full model of air refractivity [13,14], which includes the entire HITRAN-database manifold [15] of atomic and molecular transitions. Dispersion of air within the 3.5–4.2 μm wavelength range is dominated by the asymmetric-stretch rovibrational band of atmospheric ${\mathrm{CO}}_2$. In Figs. 1(a) and 1(b), we plot the spectral profiles of the refractive index of air, $n(\omega )$, and the GVD $k_2={\partial }^{2}k(\omega )/\partial {\omega }^2$, where $k(\omega )=\omega n(\omega )/c$, $c$ is the speed of light in vacuum and $\omega$ is the frequency, near and within this ${\mathrm{CO}}_2$ absorption band. Within the absorption band, i.e., for wavelengths $\lambda >4155\mathrm{nm}$, individual rovibrational transitions of ${\mathrm{CO}}_2$ give rise to rapid oscillations in the refractive index [Fig. 1(a)] and the GVD [Fig. 1(b)].

Of particular interest for our study is the region that stretches from $\approx 3.5$ to 4.2 μm. Here, unlike the central part of the ${\mathrm{CO}}_2$ absorption band, where the attenuation length is as short as $l_a\approx 1\mathrm{m}$ at $\lambda =4.26\mathrm{\mu m}$, atmospheric air is highly transparent, with the attenuation length at our laser wavelength of 3.9 μm estimated as $l_a\approx 40\mathrm{km}$. Remarkably, in contrast to most of the high-transmission regions in gas media, the GVD is anomalous in this region [Fig. 1(b)]. When combined with a positive-$n_2$ Kerr-type optical nonlinearity, anomalous GVD opens parameter space for pulse self-compression.

All the experiments have been performed at our laboratory at the Russian Quantum Center [18], using a laser system [Fig. 1(c)] that consists of a solid-state ytterbium laser with an amplifier, a three-stage optical parametric amplifier (OPA), a grating–prism (grism) stretcher, a Nd:YAG pump laser, a three-stage optical parametric chirped-pulse amplifier (OPCPA), and a grating compressor. The 1 kHz, 200 fs, 1–2 mJ, 1030 nm regeneratively amplified output of the $\mathrm{Yb}\text{:}{\mathrm{CaF}}_2$ laser system pumps the three-stage OPA, which generates 1460 nm pulses at its output. These 1460 nm pulses are then stretched with a grism stretcher to seed a three-stage OPCPA, consisting of three KTA crystals [Fig. 1(c)], pumped by 100 ps Nd:YAG laser pulses with energies 50, 250, and 700 mJ, respectively. The stretched-pulse idler-wave output of the OPCPA system has a central wavelength ${\lambda }_0\approx 3.9\mathrm{\mu m}$ and an energy up to 50 mJ. Compression of these pulses with a grating compressor yields $\approx 80\mathrm{fs}$ mid-IR pulses with an energy up to 35 mJ.

Spectral measurements in the mid-IR range are performed with a homebuilt spectrometer consisting of a scanning monochromator and a thermoelectrically cooled HgCdTe detector [Fig. 1(c)]. Temporal envelopes and phases of mid-IR pulses are characterized using frequency-resolved optical gating (FROG) based on second-harmonic generation (SHG) in a 0.5-mm-thick ${\mathrm{AgGaS}}_2$ crystal [Fig. 1(c)]. FROG traces are measured using two identical beam replicas produced with a thin-film beam splitter, which travel equal propagation paths before reaching a near-IR spectrometer accurately calibrated up to 2130 nm. Beam-profile measurements were performed with a pyroelectric camera. To avoid photoionization along the entire beam propagation path, we choose to work in a loose-focusing beam geometry. To this end, the mid-IR OPCPA output is focused with an adjustable telescope consisting of a 22 cm focal length convex lens and a 20 cm concave lens. Beam-profile analysis shows that the minimum beam diameter in the beam-waist region never goes below 1–2 mm even for the highest peak powers in our experiments, confirming no beam filamentation.

In Fig. 2(a), we present a typical FROG trace measured without any diaphragm for a mid-IR pulse with an initial pulse width ${\tau }_0\approx 100\mathrm{fs}$, a pulse energy $W_0\approx 20\mathrm{mJ}$ ($\approx 18\mathrm{mJ}$ after the telescope $T$), and an initial chirp $\alpha \approx 1200{\mathrm{fs}}^2$ (the transform-limited pulse width is ${\tau }_l\approx 80\mathrm{fs}$) at a distance $z\approx 6.5\mathrm{m}$ from the telescope $T$ in Fig. 1(c). The mid-IR pulse undergoes spectral broadening [Fig. 2(b)], yielding a spectrum whose long-wavelength wing falls way beyond the low-frequency edge of the ${\mathrm{CO}}_2$ absorption band [gray shading in Fig. 2(b)]. The pulse envelope retrieved from the FROG trace in Fig. 2(a) shows that the pulse undergoes self-compression, giving rise to a field waveform with a pulse width (defined as the FWHM of field intensity) of $\approx 39\mathrm{fs}$ [Fig. 2(c)]. The energy of the pulse at $z\approx 6.5\mathrm{m}$ is about 16 mJ, i.e., only 11% lower than the energy of the pulse after telescope $T$. The main part of this energy loss ($\approx 8.5\%$) is due to ${\mathrm{CO}}_2$ absorption, with $\approx 2.5\%$ lost through the Raman excitation of ${\mathrm{O}}_2$ and ${\mathrm{N}}_2$. With the chosen beam-focusing geometry, high-quality pulse self-compression is achieved within the range of input pulse energies from $\approx 18$ up to at least 25 mJ (Fig. 3).

 

Fig. 2. (a) FROG trace of the compressed mid-IR pulse. (b) Spectra of the compressed mid-IR pulse: (solid line) experiment and (blue shading) simulations. The spectrum of the input pulse is shown by the dashed line. Gray shading is the absorption spectrum of air $\kappa (\lambda )$. The dashed–dotted line is the GVD of air. (c) Temporal envelope of the compressed mid-IR pulse: (solid line) experiment, (blue shading) simulations. Temporal envelope of the input pulse is shown by the dashed line. (d) The Wigner distribution of the compressed mid-IR pulse. The input pulse has an initial pulse width ${\tau }_0\approx 100\mathrm{fs}$, an input pulse energy $W_0\approx 20\mathrm{mJ}$, and an initial chirp $\alpha \approx 1200{\mathrm{fs}}^2$. The compressed pulse is characterized at $z=6.5\mathrm{m}$.

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Fig. 3. (a)–(d) Temporal envelopes and (e)–(h) Wigner distributions of the mid-IR pulse at (a), (e) $z=6.5\mathrm{m}$ for $W_0\approx 15\mathrm{mJ}$, $\alpha \approx 0$; (b), (f) $W_0\approx 20\mathrm{mJ}$, $\alpha \approx 1200{\mathrm{fs}}^2$, (c), (g) $W_0\approx 22\mathrm{mJ}$, $\alpha \approx 1650{\mathrm{fs}}^2$, and (d), (h) $W_0\approx 25\mathrm{mJ}$, $\alpha \approx 2100{\mathrm{fs}}^2$: (solid line) experiment and (blue shading) simulations. Temporal envelope of the input pulse is shown by the dashed line. The dashed–dotted line in the Wigner maps is the dispersion-induced group delay ${\tau }_g=z(\partial k/\partial \omega )$ as a function of the wavelength. The dashed–dotted line in panel (d) is the compressed field waveform on the beam axis.

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To understand the physics behind this self-compression dynamics, it is instructive to start with the 1D soliton pulse self-compression scenario as a reference point for our analysis. This scenario is understood in terms of a breathing soliton solution to the nonlinear Schrödinger equation (NSE) [1,2]. For the parameters of our experiments, such a 1D soliton breathing dynamics is illustrated in Fig. 4(a). With $W_0=25\mathrm{mJ}$, ${\tau }_0=100\mathrm{fs}$, and ${\lambda }_0=3.9\mathrm{\mu m}$, the soliton number $\mathrm{N}={(l_2/l_{nl})}^{1/2}(l_2={({\tau }_l/1.76)}^2/|k_2|)$ is the dispersion length and $l_{nl}=c/(\omega n_2{I}_0)$ is the nonlinear length ($I_0$ is the field intensity) is $\mathrm{N}\approx 4.3$. Pulse self-compression is identified in this approximation as a universal initial stage of soliton breathing dynamics [$z<5\mathrm{m}$ in Fig. 4(a)].

 

Fig. 4. (a)–(c) 1D dynamics of a laser pulse with ${\tau }_0\approx 100\mathrm{fs}$, $W_0\approx 25\mathrm{mJ}$, and ${\lambda }_0=3.9\mathrm{\mu m}$ in air: numerical solution of (a) the 1D NSE, (b) the 1D GNSE with HOD disabled, and (c) the full 1D GNSE. (d)–(g) Three-dimensional dynamics of a laser pulse with ${\tau }_0\approx 100\mathrm{fs}$, $W_0\approx 25\mathrm{mJ}$, and ${\lambda }_0=3.9\mathrm{\mu m}$ in air: (d)–(f) the Wigner distributions of the pulse at (d) $z=0$, (e) 3.5 m, and (f) 6.5 m, and (g) the beam dynamics. The solid line shows the beam radius $a_0$ as a function of $z$. The dashed line is the $a_0(z)$ dependence in the linear regime. (h) The beam profile at $z=5\mathrm{m}$ and 6.5 m: (solid line) experiments and (dashed line) 3D simulations. (i) Simulated temporal envelope of the mid-IR pulse with ${\tau }_0\approx 100\mathrm{fs}$, $W_0\approx 35\mathrm{mJ}$, and ${\lambda }_0=3.9\mathrm{\mu m}$ in air at $z=6.5\mathrm{m}$.

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To quantify non-NSE effects, we numerically solve the 1D generalized nonlinear Schrödinger equation (GNSE) [2,19,20] for the parameters of our experiments. The Raman-induced soliton self-frequency shift tends to decouple solitons from each other, giving rise to isolated soliton transients in the pulse envelope [Fig. 4(b)]. It is, however, HOD that leads to the most significant deviations from the canonical soliton breathing dynamics within the considered parameter space. Indeed, the $k_2(\omega )$ profile is so steep near the edge of the ${\mathrm{CO}}_2$ absorption band [Fig. 1(b)] that the GVD cannot be even defined as a single constant $k_2$ for the entire bandwidth of $\sim 100$- fs laser pulses [Fig. 2(b)] used in our experiments. The HOD is so strong near the edge of the ${\mathrm{CO}}_2$ absorption band that it in no way can be treated perturbatively, as a small correction to the soliton breathing dynamics. For ${\tau }_l\approx 80\mathrm{fs}$ and ${\lambda }_0=3.9\mathrm{\mu m}$, the dispersion length $l_3=2{({\tau }_l/1.76)}^3/|k_3|$, related to the third-order dispersion, is $l_3\approx 20\mathrm{m}$, i.e., only slightly longer than the GVD length, $l_2\approx 17\mathrm{m}$. The soliton transients are shattered in this regime [Fig. 4(c)]. Pulse self-compression, however, is still possible, albeit not as efficient as for NSE soliton breathers in Fig. 4(a).

For a full three-dimensional (3D) analysis, we numerically solve the 3D GNSE [19,20]. Simulations were performed using parallel-programming supercomputation algorithms [21] on computer clusters of Moscow State University.

3D simulations reproduce all the key tendencies observed in pulse self-compression experiments, providing a reasonably accurate description of the output spectra, pulse profiles, and beam shapes (Figs. 3, 4). Since $l_2\gg l_{nl}$ for the parameters of the input pulses, the initial stage of pulse evolution [Figs. 4(d) and 4(e)] is dominated by spectral broadening due to self-phase modulation (SPM). At around $z\approx 3.5\mathrm{m}$, the spectrum of the pulse becomes so broad that its long-wavelength wing reaches the high-frequency edge of the ${\mathrm{CO}}_2$ absorption band [Fig. 4(e)]. Beyond $z\approx 6.5\mathrm{m}$, strong HOD, which rapidly grows toward the edge of the absorption band [Figs. 1(b), 2(b)], gives rise to a large group delay [Fig. 4(f)]. The dashed lines in Figs. 3(e)–3(h) shows the group delay due to air dispersion, ${\tau }_g=z(\partial k/\partial \omega )$. Near the ${\mathrm{CO}}_2$ absorption band edge, HOD effects are so strong that the overall group delay [Figs. 3(e)–3(h)] is totally dominated by the dispersion-induced group delay ${\tau }_g$. This limits the minimum pulse width in whole-beam pulse compression (Fig. 3) to $\approx 35-40\mathrm{fs}$. HOD effects are thus the main factor that limits pulse intensity growth in this scenario of pulse compression. This is in a stark contrast with mid-IR filaments, where the field intensity growth may be limited by linear loss due to ${\mathrm{CO}}_2$ absorption [22].

The pulse at $z\approx 6.5\mathrm{m}$ features an intense compressed peak with an FWHM width of $\approx 35\mathrm{fs}$ against a long pedestal [Fig. 3(d)]. This pedestal is mainly due to HOD. In addition, since the high-frequency wing of the pulse spectrum, which carries $\approx 6\%$ of pulse energy, falls beyond the zero-GVD point, dispersive broadening of this fraction of the pulse also contributes to the pulse pedestal. Pulse compression on the beam axis occurs with the maximum efficiency, yielding an on-axis FWHM pulse width of $\approx 13\mathrm{fs}$ at $z\approx 6.5\mathrm{m}$ [dashed–dotted line in Fig. 3(d)]. 3D simulations for the beam profile of this pulse agree very well with the measured beam profile [Fig. 4(h)]. Defining the peak power of a complex-shaped pulse as $P(z)={\max }_t{\int }_0^{\infty }I(r,t,z)2\pi r\mathrm{d}r$, where $I(r,t,z)$ is the field intensity, we find for the compressed pulse in Figs. 3(d) and 3(h): $P(z\approx 6.5\mathrm{m})\approx 240\mathrm{GW}$, which is $\approx 1.4$ times higher than the peak power of the input pulse after the telescope $T$, $P(z=0.01\mathrm{m})\approx 175\mathrm{GW}$. The energy of the compressed 35 fs peak is about 9.2 mJ, which corresponds to 46% of the total energy of the pulse at $z\approx 6.5\mathrm{m}$.

Spatial self-action effects are quantified in Fig. 4(g), where the field intensity integrated over the pulse, $F(r,z)={\int }_{-\infty }^{\infty }I(r,t,z)\mathrm{d}t$, is plotted as a function of $z$ and $r$. The solid line in Fig. 4(g) shows the beam radius $a_0(z)={({\int }_0^{\infty }F(r,z)2\pi r^3\mathrm{d}r)}^{1/2}{({\int }_0^{\infty }F(r,z)2\pi r\mathrm{d}r)}^{-1/2}$. The dashed line shows the behavior of $a_0(z)$ for a beam propagating in vacuum, where $n_2=0$, and beam divergence is entirely due to diffraction. Self-focusing is seen to slightly shift the beam focus, reduce the beam-waist size, and increase the beam-waist length. No filamentation is possible for the chosen input beam parameters. The beam-waist diameter in Fig. 4(g), $d_w\approx 1.35\mathrm{mm}$, agrees well with experimental measurements [Fig. 4(h)], which yield $d_w\approx 1.4\mathrm{mm}$. Such a beam diameter is an order of magnitude larger than the typical size of mid-IR laser filaments in air. For higher input peak powers, in a stark contrast with filamentation-assisted pulse compression [19,23], photoionization is a limiting, detrimental factor for the considered pulse-compression scenario, rather than an enabling tool, as it gives rise to radiation loss and scattering, eventually leading to a breakup of self-compressing laser pulses [Fig. 4(i)].

Analysis performed by numerically solving field-evolution equations [24] shows that field-dependent nonlinear effects, including carrier shock buildup in the first place, do not have any noticeable impact on pulse-compression dynamics in our experimental conditions. However, for much higher peak powers and much longer propagation paths, these effects become very significant, exactly as predicted in Ref. [24].

To summarize, we have identified and experimentally demonstrated a physical scenario whereby high-peak-power mid-IR pulses can be compressed as a part of their free-beam spatiotemporal evolution within the regions of anomalous dispersion in air to yield few-cycle subterawatt field waveforms. When compared to mid-IR pulse compression in anomalously dispersive solids, pulse self-compression in anomalous-GVD regions of air provides much higher output beam quality, enabling better beam focusability of pulse-compressed mid-IR output.