1. Introduction

High-power long-wave IR (LWIR) lasers emitting at 10 µm wavelengths, producing relativistic pulses enable the exploitation of new regimes in utilizing wavelength scaling of the ponderomotive force for acceleration of electrons and ions from laser-ionized plasmas [1,2] or in the generation of keV X-rays [3]. For 10 µm light, the laser-matter interaction becomes relativistic at laser intensities I ≥ 10¹⁶ W/cm², at which the normalized field strength parameter, a₀=eE/mωc ≥ 1. Several numerical studies indicate that a 5–10 TW, 200–300 fs laser-wakefield drivers capable of reaching a₀∼2–4 at these long wavelengths will allow for the generation of high quality electron bunches with a low energy spread, high charge and an extremely low <50 nm normalized emittance, critical for X-ray FELs [4–6]. CO₂ lasers operating at 10 µm are currently the most viable candidates for reaching multi-terawatt peak powers in the LWIR portion of the spectrum, but have a 10x longer than required pulse duration limited by the gain bandwidth to ∼ 2–3 ps [7,8] at best. It is the aim of this work to show that self-compression in a suitable medium is a promising avenue to achieve 300 fs pulse durations while providing high peak power and good beam quality.

It is known that a laser pulse with a power above the critical power for self-focusing can be self-guided in a gas generating a frequency chirp due to self-phase modulation (SPM) that can then be removed by pulse recompression with different dispersive elements [9,10]. In the near-IR range it became a technology for producing low energy pulsed wavepackets with a bandwidth close to a single optical cycle [11]. However, even though pulse compression through SPM is well documented for wavelengths ranging from the UV to the near IR with recent high impact works in the literature [12], typical compression setups that work for these wavelengths cannot be directly applied to the nonlinear regime found at 10µm, mainly due to key material dispersion/absorption considerations. Recent studies, inspired by demonstration of nonlinear propagation of a TW-power 10 µm pulses in air [13], have identified that for km-scale distances anomalous dispersion of atmospheric CO₂ can self-compress a spectrally broadened LWIR pulse reaching very high peak powers [14,15]. Self-compression by a factor of 3 due to the dispersion anomaly of air was experimentally observed for 100 fs laser pulses with a wavelength of ∼4 µm close to the strongest fundamental band of CO₂ molecule at 4.3 µm [16]. However, for compression of multi-terawatt 10 µm CO₂ laser pulses, propagation in air would require very long distances and resonant absorption losses hinder their propagation in a CO₂ gas filled cell.

In this paper we consider and study numerically a novel scheme in which a currently available 10.3 µm, 3.5 ps, ∼1 TW CO₂ laser pulse [13], can be self-compressed to ∼300 fs pulse duration reaching a relativistic power above 7 TW (a₀∼4). Such a remarkable ∼12x compression is achieved by choosing ¹³CO₂ molecule as a compression-mediating medium in a several meter-long cell. Combining SPM in a self-focused beam (a peak power P/P_cr∼2.7) with anomalous dispersion of the weakly absorbing 10.8 µm, 100-001 band of a ¹³CO₂ isotope [17] allows one to reach a multi-terawatt peak power in a femtosecond pulse.

2. Methodology and numerical model

Simulations are conducted in radial symmetric geometry (r, t, z), using the electric field resolved Unidirectional Pulse Propagation Equation (UPPE) solver [18,19]. The UPPE propagator is sourced by a nonlinear polarization and current. The model incorporates all relevant physical effects such as diffraction, Kerr nonlinear response [18], tunnel ionization [20], two-temperature avalanche memory effects [21], as discussed in previous works [22]. The ionization potential of CO₂ is 13.8 eV [23]. Linear dispersion and absorption of ¹³CO₂ are modeled with use of the HITRAN database [24] for the whole computational spectral window for 1 atm pressure and temperature of 22.5 C, as depicted in Fig. 1.

 

Fig. 1. Blue curve: real (a) and imaginary (b) linear susceptibility of ¹³CO₂. Black curve: input pulse spectrum of a 10.3 µm Fourier limited 3.5 ps (FWHM) laser pulse. Inset: zoom-in of imaginary χ of ¹²CO₂ (red curve) and ¹³CO₂ (blue curve) gases. In this work only ¹³CO₂ was used.

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For our numerical experiment, we launch a 10.3 µm Fourier limited 3.5 ps at FWHM (Full width at Half Maximum) Gaussian spatio-temporal laser pulse [12] in a ¹³CO₂ filled gas cell maintained at 1 atmosphere pressure. We add a 1 cm thick NaCl window at the entrance of the gas cell. This window performs a dual function: 1) to confine the ¹³CO₂ gas and 2) to impart a weak positive lens on the beam before it interacts with the gas. The latter proves important in ensuring a clean well-formed pulse spatial and temporal profile of the output [see Fig. 3]. The input pulse has 4.1 J of energy, corresponding to ∼2.7 critical powers (Pcr) for CO₂. The choice of ¹³CO₂ gas as the compression medium is motivated by the significant offset of its spectral response relative to the 10.3 µm central wavelength of the ¹²CO₂ laser source. The red curve in Fig. 1(b) compares the ¹²CO₂, absorption relative to ¹³CO₂ over the relevant spectral window showing a strong reduction in absorption while retaining a wide spectral window of anomalous dispersion. A propagation regime with relatively low losses while experiencing weak anomalous dispersion is crucial for optimal compression.

Pulse compression through SPM is well documented in the literature [12,25,26] for various materials and laser wavelengths. However, to our knowledge it has never before been successfully applied to 10 µm picosecond CO₂ lasers. Compression is typically achieved by first broadening of the pulse spectrum through nonlinear SPM, which is followed by chirp compensation through various means such as mirrors or fibers, in order to reach shorter pulse durations. A simplified formula for describing the newly generated frequencies from SPM can be written in the form of $\omega (t )= {\omega _0} + {\omega _0}z/2{n_0}c{\rho _C} \cdot \frac{{\partial \rho (t )}}{{\partial t}} - {n_2}{\omega _0}z/c \cdot \frac{{\partial I(t )}}{{\partial t}}$, where n₀, and n₂ are the linear and nonlinear refractive indices, c is the speed of light in vacuum, ω₀ is the central frequency of the laser, and ρ_C is the critical plasma density at which plasma becomes opaque. The derivative terms on the right-hand side account for the newly generated frequencies from plasma and Kerr respectively. For typical Gaussian shaped pulses, Kerr will produce lower frequencies in the leading part of the pulse (redshift) and higher frequencies in the trailing part (blueshift). At the onset of filamentation plasma will generally produce a blueshift in the pulse’s time frame. The goal in this work is to maximize spectral broadening from Kerr while at the same time minimize plasma generation that builds up during filamentation while exploiting the very broad window of anomalous dispersion in CO₂.

3. Results and discussion

Figure 2 shows the propagation of the 3.5 ps (Gaussian at FWHM) – 4.1 J and 1 cm waist at 1/e² radius, 10.3 µm wavelength initially collimated pulse when launched through a NaCl window into a ¹³CO₂ gas cell over a propagation distance of 10 m. The pulse has an initially imposed weak positive lens after passing through the cell window due the high Kerr nonlinearity of NaCl. Propagating now in ¹³CO₂ gas, the wavepacket self-focuses and forms a single on-axis filament at approximately 6.6 m inside the cell. The filament extends up to the end of the 10 m long gas column reaching a peak on-axis intensity of 3-4 × 10¹⁷ W/m². The filament diameter (FWHM) remains stable at ∼2–3 mm. Significant plasma generation is generated through tunneling and avalanche ionization reaching peak electron densities of 8 × 10¹⁶ cm⁻³ as depicted in Fig. 2(a) (red dashed curve).

 

Fig. 2. Propagation dynamics of the 3.5 ps – 4.1 J – 1 cm – 10.3 µm collimated wavepacket in ¹³CO₂ gas. (a) Peak intensity (black curve) and peak plasma density (rad dashed curve), (b) beam waist at FWHM diameter calculated on intensity (black curve) and fluence (blue dashed curve).

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In order to better understand the compression dynamics, we separate the propagation inside the gas cell into three distinct stages: the pre-filament stage (0–6.6 m), the onset of filamentation (6–6.6 m), and finally the filamentary propagation stage (6.6 m – 10 m). Figure 3(a) shows the on-axis pulse duration vs propagation distance. The end points of each of the above three stages are denoted by the blue diamond, red square and black star respectively.

 

Fig. 3. (a) On-axis pulse duration at FHWM. Blue diamond (z=6 m), red square (z=6.6 m) and black star (z=10 m) denote the end points of each of the three propagation stages, and are potential gas cell extraction points. (b), (c) On axis electric field and radial-temporal intensity distribution at z = 6 m. (d), (e) at z = 6.6 m, (f), (g) at z = 10 m. Energy content: (c), (e) ∼2.2 J, (g) ∼1.8 J. Movies of (b) - (g) as a function of z can be found in the supplementary material (Visualization 1, Visualization 2). (h) Pulse spectrum at various positions along z.

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In the first stage (pre-filamentation) we see that the pulse is compressed from 3.5 ps down to ∼1 ps duration due to the balance between SPM and the anomalous GVD of ¹³CO₂. We can see the on-axis electric field and spatio-temporal intensity distribution at the end of the first stage at z = 6 m in Fig. 3(b), (c) respectively. Both on-axis electric field and spatio-temporal intensity profiles are of excellent quality resembling a steepened Gaussian, and devoid of any pulse splitting or beam breakup. The energy content of the wavepacket as depicted in Fig. 3(c) is ∼2.2 J (∼55% of initial) which corresponds to ∼2.2 TW of power. Notably, the spectral broadening up to this point is exclusively due to Kerr induced SPM as depicted in Fig. 3(h) (blue dashed curve), since plasma related effects from tunneling are negligible. While pulse duration is predicted to get much shorter downstream, extracting the pulse from the gas cell at the end of the first stage (z = 6 m) is much easier since no plasma related complications have to be accounted for.

Filamentation is abruptly initiated beyond z = 6 m, resulting in a spike in intensity and plasma density through tunnel ionization of CO₂. The high plasma density causes the trailing edge of the pulse to be defocused and at the same time strong envelope steepening in the leading part of the pulse. As a result, the pulse is shortened in an asymmetric fashion as depicted in Fig. 3(d), (e) at z = 6.6 m, having an on-axis duration of roughly 300 fs right after the onset of filamentation. Spectral broadening is greatly enhanced compared to the first stage due higher intensity and significant plasma generation, as can be seen in Fig. 3(h) in the red dotted curve. Due to the rapid collapse of the pulse into a filament, overall energy losses are negligible in this stage, with the pulse carrying ∼2.2 J corresponding to ∼7 TW of power.

Beyond this point we enter the third stage, at which the wavepacket is propagating in the form of a filament, accompanied by typically high intensity, plasma generation and nonlinear losses. Spatially the beam is driven by the dynamic equilibrium between Kerr self-focusing and plasma defocusing, maintaining a narrow diameter of ∼3mm for the rest of the propagation in the gas cell. In the temporal domain we have a similar equilibrium, this time between SPM and anomalous dispersion resulting in a stable sub-300fs pulse duration for the rest of the propagation inside the gas cell. As can be seen in Fig. 3(f), (d), the spatial and temporal steepening in the trailing part, started in the previous stage, is further reshaping the wavepacket. As expected, the sustained plasma generation will lead to increased spectral broadening (gray dash-dotted curve in Fig. 3(h)). The energy in the pulse at the end of the gas cell as depicted in Fig. 3(g) is 1.8 J.

While each of the above described stages is driven by different dynamics, there are some key takeaways that apply to the setup as a whole. When considering the pulse’s on-axis electric field and radial-temporal intensity profiles at z = 6 m, z = 6.6 m, and z = 10 m (depicted in Fig. 3(b)-(g)), we can clearly see that both pulse and beam shapes show a gradual compression/reshaping without signs of splitting throughout this process. In addition, since the pulse is continuously and gradually compressing throughout all three stages, it should in principle be possible to control the output pulse duration by extracting at the appropriate distance. This process is fairly efficient in terms of energy loss in the temporal domain since the higher and lower newly generated frequencies do not walk-off (higher harmonic radiation is left in the wake of the pulse after the underlying optical carrier wave self-steepens) as would be the case in normal dispersive media, but instead are redirected towards the pulse temporal center. This leads to an increase of peak intensity and temporal gradients which in turn leads to further compression. While overall energy in the compressed wavepacket is roughly around 50% of the input depending on the pulse extraction point, we are predicting near 300 fs 10 µm pulses that carry multi-TW of power over laboratory scale distances.

Our goal is to extract the pulse on reaching a duration of ∼300 fs which would correspond to a cell length of around z = 6.6 m. This represents an optimal strategy for extracting a high power/energy, short duration pulse before nonlinear losses associated with filamentation will deplete energy throughout the rest of the cell. The relatively high energy fluence at the exit of the cell will require the introduction an alternative means of pulse extraction. Although this is not the focus of the present work an option could be to extract the pulse through a pressure gradient at the end of the cell and use of an aerodynamic window [27].

Figure 4 depicts a more desirable scenario where a 1.1 ps pulse (the shortest pulse that can be produced by CO₂ lasers) with a 1 cm beam waist at 1/e² radius carrying of 1.1 TW propagates in a ¹³CO₂ filled cell. In a similar manner to the 3.5 ps case, the pulse initially compresses due to action of Kerr induced SPM and anomalous dispersion by the same factor of ∼3.5 but now down to ∼300 fs (FHWM) while avoiding filamentation. As can be seen in Fig. 4(a) the plasma density is negligible at ∼4 × 10⁷ cm⁻³ (red dashed curve). The spatio-temporal intensity profile is depicted in the inset of Fig. 4(a) exhibiting a clean symmetric bell shape, carrying 0.8 J (∼58% of the starting energy) corresponding to a power of ∼2.7 TW. Figures 3 and 4 prove that our proposed approach is able to compress a pulse by a factor of 3 before the onset of filamentation for a variety of input pulse durations.

 

Fig. 4. Propagation dynamics of the 1.1 ps – 1.38 J – 1 cm – 10.3 µm collimated wavepacket in ¹³CO₂. (a) Peak intensity (black curve) and peak plasma density (rad dashed curve) vs z. Inset: Spatio-temporal intensity at z = 5 m. Energy contained in the spatio-temporal box is 0.8J (58% of starting energy), corresponding to 2.7 TW. (b) On-axis electric field at z = 0 m (red curve) and z = 5 m (black curve). (c) Pulse duration at FHWM as a function of radius and z. Movies of the spatio-temporal intensity and on-axis E-field vs z can be found in the supplementary material (Visualization 3, Visualization 4).

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In summary, we have shown an overall 12x compression of a multi-picosecond 10.3 µm CO₂ laser pulses inside spectrally offset ¹³CO₂ gas cells resulting in stable 300 fs pulse durations carrying ∼7 TWs of peak power. This extreme compression is possible by combining the well-known spectral broadening through SPM with the broad anomalous dispersion landscape of weakly absorbing ¹³CO₂ in the vicinity of the 10.8 µm (100-001 band), followed by onset of filamentation in the final compression stage. Importantly, our compression scheme is scalable to shorter input pulses, and is able to compress 1 ps input pulses in a similar fashion to generate sub-300 fs pulse while avoiding filamentation, plasma generation and spatiotemporal distortion of the final pulse. One can envision a two-cell configuration where the pulse compression to ∼300 fs without filamentation is realized for ∼1 ps pulses generated in the first cell. The strong wavelength scaling of the ponderomotive driving of electrons and ions in laser generated plasmas is expected open up many exciting new applications including laser-wakefield generation. The demonstrated approach can also be used to scale the peak power of high-repetition rate picosecond ≥100GW CO₂ lasers to TW level necessary for self-guiding in the atmosphere [13].