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It was recently proposed that ionization-induced self-compression could be used as an effective method to further compress femtosecond laser pulses propagating freely in a gas jet [He et al., Phys. Rev. Lett. 113, 263904 2014]. Here, we address the question of the homogeneity of the self-compression process and show experimentally that homogeneous self-compression down to 12fs can be obtained by finding the appropriate focusing geometry for the laser pulse. Simulations are used to reproduce the experimental results and give insight into the self-compression process and its limitations. Simulations suggest that the ionization process induces spatio-temporal couplings which lengthen the pulse duration at focus, possibly making this method ineffective for increasing the laser peak intensity.
© 2016 Optical Society of America
1. Introduction
Despite tremendous progress in high-intensity laser technology, researchers are still exploring new frontiers and attempting to reduce the pulse duration down to a few optical cycles. While relativistic intensities, I > 1018 W/cm2, have opened new fields of research such as laser-plasma based electron acceleration [1] or ion acceleration [2], few-cycle laser pulses have paved the way to attosecond science [3]. Achieving laser-matter interaction at relativistic intensities and with few-cycle laser pulses would permit new interaction regimes, such as the control of collective electron dynamics at the level of the optical cycle [4,5] and new applications, such as kHz laser plasma accelerators and their use for femtosecond electron diffraction [6, 7]. However, at kHz repetition rate, it has proven difficult to generate few cycle laser pulses with more than a few millijoules. Indeed, while chirped pulse amplification in Ti:Sapphire crystals [8] is able to provide femtosecond pulses with several joules of energy, gain narrowing typically limits the pulse duration to τ ≥ 20fs. Other than resorting to OPCPA [9], which requires a complete change of the laser architecture, post-compression is necessary to decrease the pulse duration down to a few cycles. Post-compression relies on the generation of strong variations of the optical index n by means of nonlinear effects, such as Kerr nonlinearity, ionization induced nonlinearity [10, 11] or even relativistic nonlinearities in plasmas [12, 13]. Indeed, an ultrafast variation of the index of refraction results in variations of the instantaneous frequency and the spectral broadening of the pulse as [10,14]:
where ω0 is the central frequency, r the transverse coordinate and z the propagation axis. Because the nonlinearity depends on the transverse distribution of the laser intensity, spectral broadening and compression are both spatially inhomogeneous in nature. Therefore, post-compression is usually performed in a guiding device in order to wash out the inhomogeneities and provide a homogeneous spectrum and pulse duration. For instance, post-compression to sub-10 fs was demonstrated by coupling laser pulses into a gas-filled hollow core fiber in which the spectrum is broadened by self phase modulation [15–18]. Post-compression using ionization induced nonlinearity was also demonstrated in hollow core fibers [19, 20] and produces homogeneous ∼ 10 fs compressed pulses. Guiding the pulse in the fiber ensures the homogeneity of the pulse but it has the disadvantage of a poor transmission, typically < 50%, mostly because of imperfect coupling to the fiber mode. In addition, a major drawback is that the technique is limited by laser damage to the fiber and therefore, it is not scalable to highly energetic laser systems, explaining why post-compression beyond 5 mJ pulse is not common.Recently, in [21], He et al. showed that ionization induced self-compression in free space (i.e. without guiding) leads to self-compression with an excellent transmission efficiency of ~ 90%. In addition to its excellent efficiency, the method is very interesting because of its potential scalability to energetic laser systems. In [21], the 36 fs laser pulse was tightly focused in a short gas jet, resulting in spectral broadening and self-compression down to 16 fs. Although the homogeneity of the process was briefly mentioned, no data concerning this crucial point was presented.
In this article, we present a complete characterization of ionization induced self-compression of 4 mJ, 25 fs laser pulses propagating freely, in a regime similar to [21]. In particular, we present the first study of the spectral and temporal homogeneity of ionization induced self-compression. The paper is organized as follows: section 2 describes the experimental results, where it is shown that in the near-field, homogeneous spectral broadening and compression can be obtained for very specific experimental conditions. Section 3 describes the numerical modeling with Particle-In-Cell (PIC) simulations. The latter reproduce the results very well and give insight into the underlying physics, while exposing some of the limitations to this technique. Finally, section 4 discusses the spatio-temporal couplings that appear during ionization and shows that such couplings are detrimental to the compression of the laser pulse at focus, in the far-field.
2. Experimental results
The experiment was performed using a 1-kHz laser system delivering up to 4 mJ and 25 fs at full width at half maximum (FWHM) pulses at λ0 = 800nm. The laser pulse was focused in vacuum down to a spot size of ≤ 4 µm onto a 100 µm gas jet of continuously flowing Nitrogen. The Rayleigh length was measured to be zR = 33 µm. The peak intensity at focus IL ≈ 9 × 1016 W/cm2, was high enough to ionize Nitrogen five times as , where is the ionization threshold for creating N5+ by barrier suppression.
After interaction with the gas jet, the laser pulse was collimated using a f/2 off-axis parabolic mirror at 90° and sent toward various diagnostics located outside the vacuum chamber, see Fig. 1. The homogeneity of the intensity distribution was measured in the near-field by imaging the light transmitted by a diffuser. The homogeneity of the spectrum and temporal duration were assessed by performing measurements every 4 mm across the near-field. Temporal characterization was achieved using a dispersionless second harmonic generation frequency-resolved optical gating (SHG-FROG) [22]. The dispersion due to the propagation in air and in the exit window was compensated by compressing the pulse when no gas was injected in the chamber using chirped mirrors and 2 wedges placed before the FROG. The laser beam was also sent into an imaging spectrometer which provides a spatially resolved spectrum in the far-field. This diagnostic gives information on the re-focusability of the laser pulse after self-compression in the jet and can also reveal spatio-spectral correlations. Finally, the plasma electron density was determined interferometrically using a transverse probe beam. The plasma density profile was Gaussian with a FWHM of ≈ 120μm.
2.1. Demonstration of self-compression
Spectral broadening and self-compression were optimized experimentally by scanning the jet density and its position with respect to the laser focus. The best self-compression was observed for a peak electron density of ne = 8×1019cm−3. Figure 2 shows typical FROG traces obtained in vacuum, 2(a), and after interaction with the gas, 2(b). The FROG traces show that the laser pulse is blue shifted due to ionization [11, 23] and that the pulse is compressed from 25 fs to 12 fs, i.e. decreased by a factor of two. Figure 2(c) displays the retrieved temporal profile with and without compression. The Fourier Transform Limit (FTL) of the broadened pulses is 8 fs, indicating that the laser pulse is slightly chirped. This is confirmed by the spectral phase plotted in Fig. 2(d): the blue part of the spectrum has a strongly increasing phase. This is consistent with the mechanism of ionization-induced blue-shifting: the front of the pulse gets blue shifted by the abrupt decrease in the index of refraction, resulting in the blue wavelengths being located at the front of the pulse whereas the red part of the spectrum remains at the back of the pulse. In addition, the blue shifted radiation experiences large Group Velocity Dispersion (GVD) as it propagates into a sharp density gradient where the medium is a gas at the front and a plasma at the back. These effects explain the fact that the duration is not Fourier transform limited. Having demonstrated self-compression as in Ref. [21], we now address the crucial point of the homogeneity of the process.
Fig. 2 Self-compression of the laser pulse. Panels a) and b) show the raw FROG traces with and without gas. c): retrieved temporal profiles of the initial pulse (red line), after self-compression (blue line), and FTL (dashed line). d): measured spectrum (red dashed line), retrieved spectrum (blue line), and spectral phase (green line).
2.2. Study of the homogeneity of the self-compressed pulses
Our measurements indicate that the spatial and spectral homogeneity of the pulse depend strongly on the position of the focus in the gas jet: an homogeneous pulse is only obtained for a very narrow range of focusing. Figure 3 illustrates this effect on the spatial homogeneity. When the laser is focused at the entrance of the gas jet (as in position 1, where the focus is at z = −50 µm with respect to the center of the jet), the beam profile is relatively smooth and homogeneous and is comparable to the profile without gas, see Figs. 3(a) and 3(b). On the other hand, when the laser is focused before the entrance of the jet, as in position 2 (z = −150 µm), the intensity distribution exhibits some hot spots and is rather inhomogeneous, see Fig. 3(c).
Fig. 3 Spatial homogeneity for 2 different focusing positions. a): near-field image of the laser pulse intensity in position 1, z = −50 µm, b): without interaction with the gas jet, and c): in position 2, z = −150 µm (c). The red dots represent the 8 locations where the spectral and temporal measurements are made.
We found that the dependence on the focal position also impacts the homogeneity of spectral broadening. Indeed, Figs. 4(a)–4(b) show laser spectra recorded at eight different positions in the near-field (indicated by the red dots in Fig. 3(a)). After interaction with the gas, the spectra are broadened and blue shifted down to 600–650nm, as expected from ionization. The striking result is that in position 1, spectral broadening is quite homogeneous along the beam whereas in position 2, it is clearly inhomogeneous. Translating the focal position by a Rayleigh length is enough to destroy the homogeneity of the beam.
Fig. 4 Results on spectral homogeneity. a–b): the blue shifted laser spectra at the 8 locations across the beam in the case of position 1 (homogeneous) and position 2 (inhomogeneous). The red curve represents the initial laser spectrum. c–d): spatially resolved spectra in the far-field.
Spatially resolved laser spectra in the far-field are shown in Figs. 4(c) and 4(d). The images show that the transmitted laser beam has excellent focusability, down to a few microns at FWHM. It also appears that the spatial homogeneity of the spectrum is not perfect, indicating some spatio-spectral couplings. These defects are relatively small but appear to be larger in the inhomogeneous case of position 2. Note also that the bandwidth in the far-field is not as broad than it is in the near-field which could be due to inopportune couplings (see section 4).
Most importantly, the temporal homogeneity was measured by recording FROG traces at the different transverse locations, see Fig. 5. The data for the homogeneous case (position 1) is shown in Figs. 5(a) and 5(b). In this case, the FROG traces are symmetric so that the pulse envelopes could be retrieved with a FROG error lower than 1%. In addition, Fig. 5(b) shows that the retrieved pulse duration and the FTL are fairly uniform across the beam, with variations of the of the order of 1 fs. The data for position 2 is shown in Figs. 5(c) and 5(d): the FROG traces are asymmetric which is a sign of inhomogeneity in the laser intensity distribution [22]. Consequently, the pulse durations could not be retrieved with accuracy. It is also clear that the FTL is not homogeneous across the beam: it ranges from 8 fs at the center of the beam to 13 fs at the edges, see Fig. 5(d).
Fig. 5 Experimental measurements of the temporal homogeneity. a–b) Results for the homogeneous case: a) symmetric FROG trace recorded at a given location in the beam, b) Pulse duration and FTL duration across the beam. c–d) inhomogeneous case. c): the FROG traces are not symmetric, and the pulse duration cannot be retrieved. d): the transform limited pulse duration across the beam.
To conclude on the experiment, measurements in the near-field show that there is indeed a focusing position for which the spectral broadening and self-compression to 12 fs is homogeneous. In the far-field, the data shows that although the focusability of the pulses is very good, there might be spatio-temporal couplings that decrease the beam quality. We tackle the modeling of this process and the question of spatio-temporal couplings in the next sections.
3. PIC simulations
Ionization and laser-plasma interaction was modeled with PIC simulations using the code Calder-Circ, a fully electromagnetic, quasi-3D code based on cylindrical coordinates (r, z) and Fourier decomposition in the poloidal direction [24]. Calder-Circ solves the Maxwell equations for laser propagation and the equations of motion for the electrons of the plasma and models ionization via the ADK model [25, 26]. These equations were solved in a moving window of size Δz × Δr = 61 × 84µm2, meshed with δz = 0.15/k0 and δr = 1/k0 (where k0 = 2π/λ0) for the 5 first Fourier modes. Each cell contains 10 macro-particles representing the neutral Nitrogen Molecules. In the course of the simulation, the molecules are ionized five times, resulting in 50 macro-electrons and 10 Nitrogen ions per cell. Non-relativistic effects such as Kerr nonlinearity or self-phase modulation do not need to be included in the code because at these laser intensities, the leading edge of the pulse ionizes the first electrons very rapidly so that a negligible amount of energy interacts with the unionized gas.
To reproduce the experimental results, simulations use the experimental parameters as input, including the experimentally measured neutral gas density profile. In addition, we found that the simulation results were sensitive to the exact laser mode in the interaction region. In consequence, we used the experimental laser mode and spatial phase as inputs to the simulations. The experimental laser wavefront was retrieved using the Gerchberg-Saxton algorithm [27] following the same procedure as in [28]. We also performed PIC simulations using a perfect transverse Gaussian mode, but we were not able to reproduce the observed blue shift and the spectral homogeneity across the pulse. Therefore, we only report here on simulations using the experimental laser mode which reproduce more accurately the experimental results.
To fully model the experimental diagnostics, we also computed the electric field in the near-field and the far-field by propagating the output of the PIC simulation using a 3D code based on the plane wave decomposition method. Such a code is necessary because regular Fresnel integrals are no longer valid due to the tight focusing geometry and the very broad bandwidth of the laser pulses. The full wave equation in vacuum can be solved exactly in Fourier space: a field Ei(x,y,t,z = 0) is first transformed to the Fourier space Ẽ(kx,ky,ω,z = 0) and then propagated by a distance z by applying
where . The spatio-temporal field at distance z, E(x,y,t,z), can then be computed by inverse Fourier transform in the time domain. To model the near-field after a collimating optics, such as the parabola in the experiment, we simply multiply the field by a spherical phase so that the near-field is computed as where k = ω/c is the wave vector in vacuum.3.1. Ionization and self-compression
Figure 6 depicts the result of a PIC simulation obtained when the laser is focused at position 1 (homogeneous case). Figures 6(a)–(c) illustrate the evolution of the laser intensity I(x,y,z) during propagation and Figs. 6(d)–(f) show the corresponding on-axis longitudinal profiles. Figures 6(a) and 6(b), taken respectively in the middle and at the exit of the gas jet show that the leading edge of the pulse is progressively depleted by ionization-induced defocusing [23]. This results in pulse self-compression to 12 fs at the output of the gas jet, see Figs. 6(b) and 6(e). The simulations also confirm that the front of the pulse is blue shifted and undergoes strong GVD in the ionization region (not shown). This effect also contributes to decreasing the intensity at the leading edge of the laser pulse and to the shortening of the overall temporal envelope. In Fig. 6(c), we computed the near-field at z = 1mm ≫ zR = 33 µm. There, the pulse duration is 10 fs and the pulse is relatively homogeneous temporally, in agreement with experiments. Interestingly, the simulation reveals that in the near-field, the laser energy front is curved even though the phase fronts are flat, see dotted line in Fig. 6(c). Most likely, this curvature is the signature of spatio-temporal couplings as we will discuss below. Evidently, this has the unwelcomed effect that while the local pulse duration is short, the global (i.e. radially integrated) pulse duration is longer [29].
Fig. 6 PIC simulations of self-compression. a–c): cut of the laser intensity distribution I(y,z) at different times in the simulation. In a), the envelope in the middle of the jet, in the region where the density is the highest. b) shows the laser envelope at the exit of the jet, and c) shows the envelope in the near-field after z = 1 mm of propagation. d–f) show the corresponding on-axis longitudinal profiles. The on-axis FWHM pulse duration is marked in red.
Figures 7(a)–7(c) show a detailed comparison between the experimental and simulated on-axis spectrum, spectral phase and temporal profile in the near-field. The agreement is remarkable, showing that the simulation is able to fully capture the physics of ionization-induced self-compression. In particular, the spectral phase on the blue side of the spectrum is adequately reproduced, showing that the complex 3D dynamics of ionization and pulse propagation is very well modeled.
Fig. 7 Comparison between the experimental and simulated on-axis spectrum (a), spectral phase (b) and temporal profile (c) in the near-field.
3.2. Homogeneity of the self-compression process
The PIC simulations are also able to reproduce the (in)homogeneity of the experimental results. Figure 8 shows the results of two different PIC simulations obtained by focusing the laser at position 1, Figs. 8(a)–8(c), and at position 2, Figs. 8(d)–8f). The beam profile in the near-field is represented, along with maps of the local pulse duration and the FTL duration. For the simulation at position 1, the near-field is rather homogeneous although some hotspots are still visible in the distribution, see Fig. 8(a). The local pulse duration is about 10–12 fs and it is rather uniform in the region where the laser intensity is high, see Fig. 8(b). Similarly, the FTL duration is also quite uniform, around 5 – 6 fs, see Fig. 8(c). On the other hand, the simulation corresponding to position 2 shows that the intensity distribution, pulse duration and FTL are very inhomogeneous. In Figure 8(e), it is also clear that in this case, the pulse is not really self-compressed as the pulse duration is around 25 fs. These results reproduce very well the trends of the experiment and underline the crucial role of laser focus position within the gas jet in order to obtain homogeneous results.
Fig. 8 PIC simulation in the near field: a–c) Maps of the intensity distribution I(x,y), local pulse duration and FTL duration, in the case of focusing the laser in position 1. d–f) Same results when the laser is focused at position 2.
3.3. Discussion
From all of the experimental results and simulations we performed, we conclude that several conditions need to be satisfied in order to achieve homogeneous self-compression at the 10 fs level. First, the gas jet length should be comparable to the Rayleigh length so that the intensity remains in the same range throughout the gas jet (in our case Ljet = 120 µm ≳ zR = 33 µm). Experiments with longer gas jets (we tried 200 µm) or shorter Rayleigh lengths resulted in more blue shifted spectra but less homogeneous results. Second, the laser intensity should be only slightly higher than the last accessible ionization level. Indeed, if the laser intensity is too high , only the very front of the pulse is blue shifted and there is no significant spectral broadening. On the other hand, if the laser intensity is too close to the ionization threshold , the bulk of the laser pulse is affected by ionization, resulting in large pulse distortions and inhomogeneities. Typically, in this case the density gradient induced by ionization covers the whole pulse enveloppe and the associated GVD prevents self-compression. This explains the sensitivity to the focus position: homogeneous self-compression is achieved by tuning the position of the focus so that the laser intensity is only slightly higher than the ionization threshold throughout the jet, . In this case, a substantial part of the pulse leading edge is strongly affected by ionization, resulting in the etching of the front of the pulse and self-compression. Presumably, in this case, the homogeneity of the back of the pulse is preserved explaining the homogeneous self-compression.
4. Limitations due to spatio-temporal couplings
Post-compression is relevant only if it provides (i) shorter pulses but also (ii) higher peak intensities at focus. This can be achieved when the ionization process does not introduce spatio-temporal couplings which would hinder the pulse focusability. Spatio-temporal couplings appear through couplings in amplitude or in phase [29,30]. Let us consider a complex laser field: E(x,y,t) = |E(x,y,t)|eiφ(x,y,t). A spatial coupling in amplitude occurs when the laser enveloppe cannot be factorized, i.e. |E(x,y,t)| ≠ f (x,y)g(t). This is the case for example if the laser spectrum is spatially inhomogeneous, leading to an inhomogeneous pulse duration. When it comes to the phase, spatio-temporal couplings appear when the phase cannot be expressed as a sum φ(x,y,t) ≠ φa(x,y) + φb(t). A more convenient way to look at spatio-temporal distortions in phase is to express them in the frequency domain by considering the spatio-spectral phase φ(x,y,ω) ≠ φa(x,y) + φb(ω).
In the previous sections, we demonstrated that we were able to achieve homogeneous self-compression, indicating that most spatio-temporal couplings in amplitude were minimized. However, spatio-temporal couplings in phase were not accessible experimentally. Therefore, we rely here on PIC simulations to assess the spatio-temporal couplings in phase. Figure 9(a) shows the spatially resolved spectrum in the far-field whereas Fig. 9(b) shows the corresponding spectral phase φ(y,λ). Clearly, the phase displays spatio-spectral correlations as the bluest components have a radius of curvature when the red part of the spectrum have a rather flat phase. This result can be explained by the fact that initially, ionization acts as a negative lens for the blue shifted front of the pulse only, while the back of the pulse “sees” a more uniform plasma. Then, the front of the pump is progressively depleted as represented in Fig. 6(a) and the pulse duration decreases. As the ionization front moves backwards, a complex dynamics of the ionization results in a wavelength-dependent radius of curvature, with dramatic consequences in the far-field. Indeed, Fig. 9(d) shows that while the pulse can be well focused, the pulse duration is not shorter than 20–25 fs in the focal plane and the effect of self-compression is lost. In consequence, although the pulse focusability is good (as in the experiment), the peak intensity is not increased: it reaches only 6 × 1017 W/cm2 which is less than the initial vacuum intensity of 9 × 1017 W/cm2.
Fig. 9 PIC simulations of spatio-temporal couplings. a) and b): Maps of the spatially resolved spectrum and spectral phase in the far-field. The spectral phase shows that the radius of curvature depends on the wavelength. c) and d): maps of the intensity distribution and local pulse duration in the far-field.
A simple way to grasp this effect is to write the spatio-spectral phase as φ(x,y,ω) = α(x2 + y2)(ω − ω0), which corresponds to the well-known spatio-temporal coupling referred to as “pulse front curvature”. It is the spatio-temporal equivalent of the well-known chromatic effect: the different frequencies have a different radius of curvature and are focused at different longitudinal positions. The effect of this type of coupling is depicted in Fig. 10: in the near-field, it is characterized by a curved energy front [31] (see also the curved front in Fig. 6(c)). In the far-field, it results in a lengthening in the pulse duration and a reduction of the peak intensity [32]. This lengthening originates from the fact that the different spectral components are focused at different locations, resulting in a smaller bandwidth at focus and thus, a longer pulse duration.
Fig. 10 Effect of a “pulse front curvature” type of coupling: φ(x,y,ω) = α(x2 + y2)(ω − ω0). For the calculation, the broadened spectrum was used as well as the focusing geometry of the experiment. a) Pulse intensity I(x,t) in the near-field showing a curved energy front. The on-axis pulse duration is 7 fs. b) Far-field pattern showing a lengthening of the pulse duration to 21 fs at focus.
The spatio-spectral phase computed in our PIC simulation reveals that, due to ionization, the laser pulse suffers from a coupling which resembles pulse front curvature. A possible evidence of this spatio-temporal coupling might be found in Fig. 4 which shows that the spectral bandwidth in the far-field is narrower than in the near-field, as expected from a pulse front curvature type of coupling.
5. Conclusion
We demonstrated the post-compression of 25 fs laser pulses down to 12 fs using ionization-induced self-compression of pulses freely propagating in a gas medium. We showed that this post-compression technique can provide homogenous results for specific focusing geometries. However, our simulations indicate that spatio-temporal couplings might lengthen the pulse duration and significantly decrease the intensity at focus. Therefore, to confirm the applicability of ionization induced self-compression in free space, more sophisticated measurements addressing spatio-temporal couplings and/or pulse duration at focus will have to be performed. This will be the subject of future work.
Acknowledgments
The authors acknowledge fruitful discussions with F. Quéré on spatio-temporal couplings as well as the use of his imaging spectrometer. This work was funded by the European Research Council under Contract No. 306708, ERC Starting Grant FEMTOELEC and the Agence Nationale pour la Recherche (under contracts ANR-11-EQPX-005- ATTOLAB and ANR-14-CE32-0011-03 APERO) as well as the Région Ile-de-France (under contract SESAME 2012-ATTOLITE).
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