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Abstract
Large diameter, flying focus driven ionization waves of arbitrary velocity (IWAV’s) were produced by a defocused laser beam in a hydrogen gas jet, and their spatial and temporal electron density characteristics were measured using a novel, spectrally resolved interferometry diagnostic. A simple analytic model predicts the effects of power spectrum non-uniformity on the IWAV trajectory and transverse profile. This model compares well with the measured data and suggests that spectral shaping can be used to customize IWAV behavior and increase controlled propagation of ionization fronts for plasma-photonics applications.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Ionization fronts with precisely controlled characteristics could help to overcome fundamental limitations in laser-plasma-based photonics applications by improving phase matching, extending interaction lengths, and facilitating better control of plasma conditions. These capabilities are particularly useful in plasma-based light manipulation processes such as photon acceleration [1–6], Raman amplification [7–14] and THz generation [15–17]—processes that could lead to a new generation of exotic, compact and versatile radiation sources.
Highly controllable ionization fronts can be driven using a recently developed method called the flying focus, in which a chirped laser pulse is focused by a hyperchromatic optic such as a diffractive lens [18,19]. The chromatic aberration causes different frequencies in the laser pulse to come to focus at different positions along the propagation axis. For a fixed focal geometry, the temporal delay between when each frequency reaches its focal position is determined by the chirp, which can be adjusted to cause the point of the maximum laser intensity to move at any velocity over distances that can greatly exceed the Rayleigh length. For a linearly chirped pulse of duration $\delta t$ much greater than the transform limited duration, propagating in vacuum, the velocity of the intensity maximum $v_{f}$ is given by
where $L = (\delta \omega / \omega _{0}) f_{0}$ is the length of the extended focal region for a diffractive optic that causes the central frequency $\omega _{0}$ to come to focus at a distance $f_{0}$, $\delta \omega$ is the bandwidth and $c$ is the speed of light [18]. The $\pm$ on the right-hand side takes the sign of the chirp, so this equation can yield any velocity from $- \infty$ to $+ \infty$.When the instantaneous intensity of a flying focus pulse exceeds the ionization intensity threshold of a background gas, $I_{i}$, an ionization front is produced that tracks the propagation of an intensity isosurface at $I_{i}$ [20]. These ionization waves of arbitrary velocity (IWAV’s) were experimentally demonstrated to have predictable and easily adjustable velocities ( Eq. (1) ) when driven by a laser pulse with a highly uniform power spectrum in the laser far field [21].
While modification of the power spectrum was proposed as a means to increase control of IWAV propagation [20], previous theoretical and experimental investigations were limited to the laser far field and mainly considered flat power spectra, i.e., high order super-Gaussians. In this parameter regime, all frequencies in the bandwidth have just enough power to ionize near their minimum spot size. Experimental observation of channels formed by IWAV propagation indicated radii $\sim 10 \ \mathrm {\mu m}$, close to the measured far field laser spot size. Such small diameter IWAV’s would have limited usefulness in applications due to the difficulty of coupling another beam into the IWAV, the small available cross section for interaction, and strong refraction due to the short transverse electron density scale lengths.
It is possible to increase the diffraction-limited minimum spot size by increasing the $f / \#$ so that larger IWAV’s can be driven in the laser far field. It may be experimentally favorable, however, to simply increase the total pulse power so that all wavelengths ionize before they reach their minimum spot size. Operation in this so called quasi-far field (QFF) regime provides control of the IWAV radius without changing the focusing geometry. Here we define QFF to include all axial positions for which $z_{R} \,< \,z \,<< \,f$, where $z$ is the distance away from the beam waist for frequency $\omega$, and $z_{R}$ and $f$ are the Rayleigh length and focal length that also correspond to $\omega$. Furthermore, it offers the possibility of using non-uniform power spectra to control the dynamic behavior of the IWAV’s. QFF IWAV’s may be the only path forward for applications that require a significant pump intensity to exist behind the ionization front, such as flying focus driven plasma-Raman amplification [9,10].
In this paper we present the first experimental demonstration of large diameter IWAV’s ($\sim 10 \ \times$ larger than in previous experiments) driven in the QFF of a flying focus beam, and develop new theory to predict their behavior. In Sec. 2, we describe a simple model of IWAV propagation when the flying focus power spectrum is non-uniform and use it to demonstrate the possibility of enhanced control of IWAV characteristics through spectral shaping. In Sec. 3, we describe an experimental setup incorporating a novel, spectrally resolved interferometry diagnostic that allows for the inference of IWAV characteristics such as velocity, radius and temporal electron density gradients. In Sec. 4, we present experimental results and compare to the theory developed in Sec. 2 and to the previously published flying focus and IWAV theory.
2. QFF IWAV theory
Increasing the laser pulse power makes it possible to ionize a larger area transverse to the propagation axis, since a particular ionization intensity threshold can be reached prior to best focus. For a Gaussian beam focused by a doublet consisting of a diffractive lens that focuses the frequency $\omega _{0}$ at a distance $f_{D}$ and an ideal refractive lens that focuses all frequencies at a distance $f_{R}$, the beam radius for a single frequency $\omega$ evolves according to
The electric field evolves in space and time according to
where $\phi (t)$ is an arbitrary phase function. In vacuum, the temporal and spectral energy densities are given byThe axial position at which a particular frequency $\omega$ will ionize can be found by solving the ionization condition,
for $z_{\omega }$. The IWAV radius at this position is assumed to be the beam radius, $r_{\omega }(z_{\omega },\omega )$.Finally, the time that it takes for a specific frequency to reach the axial distance at which it will ionize is given by
Figure 1 shows the results of this model assuming three different spectral energy densities, $\varepsilon (\omega )$. The spectral energy densities are shown in Fig. 1(a). The IWAV radial profiles produced by each $\varepsilon (\omega )$ are shown in Fig. 1(b) and the the corresponding IWAV velocities are shown in Figs. 1(c) and 1(d). The color scale of the lines in all plots in Fig. 1 directly corresponds to the wavelength on the horizontal axis of Fig. 1(a). The focusing geometry $( r_{0} = 25.5 \ \mathrm {mm} \ , \ f_{D} = .51 \ \mathrm {m} \ , \ f_{R} = -1.49 \ \mathrm {m} )$, bandwidth $(\delta \lambda \cong 5.5 \ \mathrm {nm})$ and central wavelength $(\lambda _{0} = 1053 \ \mathrm {nm})$ are the same for all three calculations. These parameters were chosen for consistency with the experimental parameters described in Sec. 3. In all cases the chirp is chosen such that the IWAV velocity is $\dot {z}_{\omega } = v_{IWAV} = -c$.
Fig. 1. Calculations of the IWAV radius (b) and velocity (c) and (d) are shown for the spectral energy densities in (a). Cases 1) and 2) show the simple far field and QFF IWAV propagation respectively. Case 3) shows that the spectrum can be adjusted to change the IWAV radius as it propagates so that the $f / \#$ of a separate beam ( (b), black line) can be matched, but still maintain a constant velocity. Single frequency radii for the edges (blue and red) and center (green) of the bandwidth are shown as solid-color dashed lines in (b). The rainbow color scale shown in all plots directly corresponds to the wavelength axis in (a). Note that the laser pulse propagates from left to right in (b)–(d), but since $v_{IWAV} = -c$, the IWAV propagates from right to left.
In case 1), $\varepsilon (\omega )$ is flat, and the power in each frequency slice of the bandwidth is just high enough to induce ionization at the minimum spot size for all frequencies. We obtain an IWAV that moves across a distance equal to the length of the extended focal region $L$ at the expected focal velocity $v_{f}$. This case was previously demonstrated in [20,21]. In case 2) $\varepsilon (\omega )$ is also flat and the chirp is the same, but the total pulse energy is increased such that each frequency can ionize in a larger spot size. The IWAV still moves across a distance $L$ at $v_{f}$, however the ionized region is shifted closer to the focusing optic and the radius of the IWAV is increased to $\sim 30\times$ its original radius. This case is the simplest QFF IWAV propagation, where all properties of the far field IWAV are retained except that the radius is increased. Cases 1) and 2) have been verified using the pseudo-spectral split-step Fourier propagation algorithm described in [20], which simulates the propagation of a flying focus pulse through an ionizable gas.
Case 3) represents a true extension of IWAV theory from the direct correspondence to flying focus theory. A quadratic $\varepsilon (\omega )$ was chosen to show that the IWAV radius can be matched to the $f / \#$ of a separate, achromatic beam (shown in black in Fig. 1(b), with $f / \# = 25.5$ ), while maintaining a constant velocity. Far outside of the Rayleigh length for each frequency the radius of the beam is approximately proportional to the propagation distance, therefore the area changes quadratically with propagation distance. Selecting the chirp and the curvature of the quadratic spectrum appropriately produces an IWAV that has a linearly increasing radius, at any nearly constant velocity. Notably, this allows for the IWAV propagation distance to greatly exceed the length of the already extended focal region $L$.
Spectral shaping can also be used to accelerate the IWAV and produce more complex trajectories, since a non-zero slope in $\varepsilon (\omega )$ will lead to spreading or bunching of the positions $z_{\omega }$ at which adjacent frequencies in the pulse ionize. If the shape of the spectrum is not chosen to be a quadratic with the correct curvature in the QFF, as described above, then this non-linear spreading of $z_{\omega }$ will lead to a change in $v_{IWAV}$. This effect can be seen on the far right side of case 3) in Fig. 1(d). In the far field the spot area does not change quadractically with propagation distance, so the variation in beam intensity due to the quadratically shaped spectrum improperly compensates for the variation in beam intensity due to the changing beam radius, which results in acceleration. This change in velocity is always related to a change in radius for linear chirps.
3. Experimental setup
QFF IWAV’s were measured experimentally using the setup shown in Fig. 2. Pulses from an Nd:YLF laser with energies ranging from $\sim 100 \ \textrm {mJ} - 5 \ \textrm {J}, \ \lambda _{0} = 1053 \ \textrm {nm}, \ \delta \lambda \cong 5.5 \ \textrm {nm}$ and $\delta t = 10 - 60 \ \textrm {ps (FWHM)}$ were split into pump and probe beams. Spectra and autocorrelation traces were collected for the initial beam, allowing for direct measurement of the pulse duration and bandwidth on every shot. The pump was incident on a diffractive optic with $f_{D} = .51 \ \textrm {m}$ and an achromatic refractive optic with $f_{R} = -1.414 \ \textrm {m}$, which together focused the beam into a hydrogen gas jet of length $5 \ \textrm {mm}$ capable of producing molecular $H_{2}$ densities of $\sim 1.75 e 19 \ \mathrm {cm^{-3}}$.
Fig. 2. The Multi-Terawatt (MTW) laser was split into a $1 \omega$ pump beam that drove IWAV’s in a hydrogen gas jet and a $2 \omega$ probe beam that passed through the interaction region perpendicular to the IWAV propagation with variable timing allowing conventional 2D (a) and 1D-spectrally resolved (b) interferograms of the IWAV propagation to be collected.
The probe beam was frequency doubled and passed through a variable delay stage that allowed its timing to be adjusted relative to when the pump interacted with the gas jet. The probe passed through the interaction region perpendicular to the pump axis of propagation, then through an interferometer with the output incident on the slit of a spectrometer that imaged the plane of the interaction and had its slit aligned parallel to the pump axis of propagation. The pump focal position with respect to the gas jet was changed by moving the position of the focusing lenses with a linear translation stage. This allowed for control over how far into the QFF the beam was when interacting with the gas jet.
The interferometer used a Nomarski prism to split the probe beam into two beams after traversal of the interaction region. The two beams were overlapped such that the part of one beam containing the image of the plasma coincided on the spectrometer slit at a slight angle with the part of the other beam not containing the plasma. This caused interference fringes in the overlapped region. Two different kinds of interferograms were collected, with at least one of each collected at the same laser settings (chirp, energy and focal position) for all settings investigated. With the probe beam delayed such that it passed through the interaction region just after the pump beam had completely passed, the spectrometer slit was fully opened and the grating was set to not spectrally disperse the signal (zero order mode). This produced conventional interferograms (2D data) of the plasma channel immediately after it was formed. An example of raw 2D data is shown in Fig. 2(a) . Both axes represent space, with the vertical axis transverse to the pump propagation and the horizontal axis parallel.
To spectrally resolve the probe wavelengths, the spectrometer slit was closed to $\sim 10 \ \mathrm {\mu m}$ around the image of the pump axis and the grating was set to disperse at the probe wavelength. The probe beam was then co-timed with the pump beam. This produced interferograms like the one shown in Fig. 2(b) , where the horizontal axis is still distance parallel to the pump axis, but the vertical axis corresponds to the probe wavelength. Since the probe beam had a known, linear chirp, the spectral resolution on the vertical axis could be converted to temporal resolution by scaling the spectral axis by the ratio of the pulse duration to the bandwidth, $\delta t / \delta \lambda$. These spectrally resolved interferograms (1D data) give a time history of the accumulated probe phase at every point along the pump axis with $\sim \mathrm {ps}$ resolution since the relative time at which each probe wavelength passed through the interaction region is directly proportional to the instantaneous wavelength.
The magnitude of fringe shifts observed in the interferograms are proportional to the integral of the refractive index along the path traversed by the probe beam. The extracted phase can be converted to plasma electron density after making some assumptions about its radial distribution. This measurement technique is in contrast to the one in [21], where a Schlieren imaging technique was used. In that case no quantitative information about the electron density could be extracted from the data. Furthermore, the interferometric technique used here is, in principle, more sensitive to refractive index variations.
Further differences between this setup and the one used in [21] are as follows: 1) These experiments were performed in vacuum with a hydrogen gas jet target, whereas the previous experiments were performed in air, 2) pulse energies were scanned over a large range in these experiments, whereas all data in [21] were collected for pulse energies of $\sim 25 \ \mathrm {mJ}$, 3) additional amplification stages in this setup led to a smaller bandwidth and pronounced spectral non-uniformity, and 4) the focusing geometry was changed to maintain a similar extended focal range, $L$, even with the smaller bandwidth.
4. Data analysis and results
Probe beam phase was deduced from the interferograms using a typical Fourier phase extraction algorithm for both 1D and 2D data. The electron density distribution was assumed to be radially uniform within the channel, so that the electron density could be extracted from the phase by simply dividing out the chord length traversed by the probe. With this assumption, electron densities for all 2D interferograms were obtained that are consistent with full ionization of the expected neutral gas density, with small fluctuations about that value ( Fig. 3(b) ). The transverse sizes of channels observed were much larger than those previously produced by a flying focus pulse in the laser far field [21]. As expected, increasing the laser energy by about $100 \ \times$ produced channels with radii about $10 \ \times$ larger.
Fig. 3. The experimental spectral energy density (a) was used to calculate an expected radial profile ((b) rainbow line) and trajectory ((c) rainbow line) which are overlaid on the electron density data extracted from 2D interferometry ((b) colorbar) and 1D spectrally resolved interferometry ((c) colorbar) respectively. The FWHM bandwidth is shown in rainbow in (a), with the rest of the spectral energy density shown as dashed black lines that extend out from the rainbow section.
Analysis of the 1D data relied on the chord lengths at each position along the closed spectrometer slit that were extracted from 2D data with the same laser settings. The electron density was found to rise in time from zero to values near the densities extracted from the 2D data for all axial positions, which can be seen in Fig. 3(c). This provides a self-consistency check on the analysis technique. Shot-to-shot variations in laser parameters are likely the main contribution to the small differences observed between densities extracted from the 2D data and the final (late time) densities obtained in the 1D data. Agreement between 2D densities and final 1D densities were consistent over the entire range of parameters investigated and in general standard deviations from the 2D mean density and 1D mean final density within each data set were small $(\lessapprox 10-15 \ \%)$.
The measured experimental spectral energy densities were used to calculate the expected radial profile and trajectory using the method described in Sec. 2 for all data sets. Note that $\delta \omega \delta t=93.4\gg 1$ for the shortest pulse duration used in the experiments, so the model is valid for all experimental pulse durations. In contrast to the highly uniform spectra observed in previous experiments, the additional amplification stages used to obtain higher laser pulse energies lead to exaggerated spectral non-uniformity [21].
Calculations using the spectral energy density shown in Fig. 3(a) are overlaid on the electron density data in Figs. 3(b) and 3(c). The rainbow portion of Fig. 1(a) corresponds to the FWHM bandwidth, which represents the main contribution to the IWAV formation and propagation observed in the data. Only this portion of the spectrum was used in the calculations shown in rainbow in Figs. 3(b) and 3(c), but the rest of the spectrum is shown as dashed black lines extending away from the FWHM section in Fig. 3(a) for completeness. The rainbow color scale shown in all plots in Fig. 3 corresponds to the wavelength axis shown in Fig. 3(a). Although perfect agreement is not reached, the analytic calculations reproduce the overall behavior of the data well. Spectral non-uniformity causes variations in IWAV radius and trajectory observed in the data. Smaller scale variations in both kinds of data could be due to a non-uniform transverse beam profile, which is not accounted for in the model.
The small shift in ionization position between the radial profile in Fig. 3(b) and the trajectory in Fig. 3(c) occurred because the measured pulse energy and duration were slightly different for the two shots. The spectrum was almost identical, but the shot shown Fig. 3(b) was lower energy and longer duration (this amounts to simply scaling the vertical axis of Fig. 3(a) slightly before performing the calculation), which caused the expected ionization positions to shift towards the far field.
The velocity of an IWAV driven by a “flat” part of the spectrum is expected to agree with the flying focus velocity, which is demonstrated in Fig. 3. This flat part of the spectrum, where $\overline {d \varepsilon (\omega ) / d \omega } \cong 0$, with the overbar indicating an average over $\omega$, and the portion of the trajectory
driven by it are demarcated by vertical, dashed black lines in Figs. 3(a) and 3(c) respectively. A linear fit to this section of the 1D electron density data, gives a velocity (-.71c) consistent with the analytic flying focus velocity calculated using Eq. (1) (-.75c). The average velocity predicted by the IWAV model (-.73c) in this region is in agreement with both values, so consistency between both theories and the data is obtained.
An estimate of the temporal gradient of the electron density (ionization rate) is easily obtained from the 1D data by taking a numerical derivative along the time axis. The results of this calculation are compared to simulation results in Fig. 4. The simulation uses a pseudo-spectral split-step Fourier algorithm to solve the paraxial wave equation in a frame moving at the speed of light with a flying focus laser pulse [20]. Rates of field ionization, electron impact ionization, radiative recombination and three body recombination are calculated to update the electron density, which in turn causes refraction and absorption of the laser pulse due to the resulting plasma. Inverse bremsstrahlung, atomic absorption and collisional cooling contribute to the loss of laser energy. The experimental spectral energy density measured for the shot shown in Fig. 4(a) was directly used in the simulation shown in Fig. 4(b). These results demonstrate the power of the spectrally resolved interferometry diagnostic to make novel measurements of the electron density with relative experimental ease. They also lend experimental validation to recent theoretical predictions of the extreme frequency upshifts achievable by co-propagating a witness laser pulse with a flying focus driven IWAV [6].
Fig. 4. The measured temporal density gradient (ionization rate) (a) has values and trajectory that are close to those predicted by simulations of ionization due to flying focus pulse propagation (b). An outline of the simulated data is shown in both (a) and (b) as a green line.
The values of the maximum ionization rates observed in the data are remarkably similar to those obtained from the simulation $(\sim 6 \times 10^{18} \ \mathrm {cm^{-3} / ps})$, and the trajectory is comparable. The ionization rate, which is closely related to the IWAV axial density scale length, is a critical parameter that dictates the rate at which photons co-propagating with the IWAV are up-shifted in frequency. This data helps to further experimentally validate the code used to simulate the ionization rates that are presented in Fig. 4(b). This code was used recently to simulate IWAV electron density profiles, which were then used in a computational study of photon acceleration in [6]. The close agreement between ionization rates in this experiment and the simulation further increase our confidence in the accuracy of results presented in [6], and lay an experimental groundwork for possible future experiments aimed at producing high frequency, coherent, table top radiation sources based on IWAV photon acceleration and related techniques.
Discrepancies between the data and the simulation are again likely due to non-uniformity in the transverse beam profile. The code allows the beam profile to be specified in the near field, where an ideal super-Gaussian beam profile was used. Intensity hot-spots in the experiment, which are not accounted for in this model, could cause ionization to occur at different positions and times in the data, leading to different observed trajectories. Furthermore, this could change the ionization rates at different axial positions due to different rates of intensity-dependent field ionization, ionization induced refraction and absorption. Another possible explanation is that the spectrometer slit sampled transverse positions off the IWAV axis, which could have led to changes in the ionization rates and trajectory observed in the data.
5. Summary and conclusion
Large diameter, flying focus driven IWAV’s have been demonstrated in the laser quasi-far field. The energy of the flying focus drive pulse can be increased and the beam defocused to produce arbitrarily wide IWAV’s without changing the focusing geometry. Characteristics of such IWAV’s driven in a hydrogen gas jet were measured using a novel, spectrally resolved interferometry diagnostic which allows the temporal and spatial dependence of the electron density to be inferred.
A simple analytic model was developed to describe IWAV characteristics when the drive pulse has a non-uniform power spectrum, and broadly agrees with the data. This new description of IWAV propagation facilitates exploration of the effects of intentionally shaped spectra on IWAV characteristics. In particular, acceleration and modification of the radius can be induced by spectral shaping to produce complex trajectories and radial profiles that may be useful for the proposed applications. Spectral shaping could be used in conjunction with a nonlinear chirp to more completely control the IWAV by counteracting the inherent coupling of acceleration with a change in IWAV radius induced by spectral non-uniformity alone.
Funding
National Nuclear Security Administration (DE-NA0003856); Fusion Energy Sciences (DE-SC0016253, DE-SC0019135).
Acknowledgments
This material is based upon work supported by the Department of Energy National Nuclear Security Administration under Award Number DE-NA0003856, the Fusion Energy Sciences under Award Numbers DE-SC0019135, DE-SC0016253, the University of Rochester, and the New York State Energy Research and Development Authority.
This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacture, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.
Disclosures
The authors declare no conflicts of interest.
References
1. S. C. Wilks, J. M. Dawson, W. B. Mori, T. Katsouleas, and M. E. Jones, “Photon accelerator,” Phys. Rev. Lett. 62(22), 2600–2603 (1989). [CrossRef]
2. J. T. Mendonça, Theory of Photon Acceleration (IOP, 2001).
3. J. M. Dias, N. C. Lopes, L. O. Silva, G. Figueira, J. T. Mendonça, C. Stenz, F. Blasco, A. Dos Santos, and A. Mysyrowicz, “Photon acceleration of ultrashort laser pulses by relativistic ionization fronts,” Phys. Rev. E 66(5), 056406 (2002). [CrossRef]
4. N. C. Lopes, G. Figueira, J. M. Dias, L. O. Silva, J. T. Mendonça, P. Balcou, G. Rey, and C. Stenz, “Laser pulse frequency up-shifts by relativistic ionization fronts,” Europhys. Lett. 66(3), 371–377 (2004). [CrossRef]
5. L. Oliveira e Silva and J. T. Mendonça, “Photon acceleration in superluminous and accelerated ionization fronts,” IEEE Trans. Plasma Sci. 24(2), 316–322 (1996). [CrossRef]
6. A. J. Howard, D. Turnbull, A. S. Davies, P. Franke, D. H. Froula, and J. P. Palastro, “Photon acceleration in a flying focus,” Phys. Rev. Lett. 123(12), 124801 (2019). [CrossRef]
7. V. M. Malkin and N. J. Fisch, “Backward raman amplification of ionizing laser pulses,” Phys. Plasmas 8(10), 4698–4699 (2001). [CrossRef]
8. D. S. Clark and N. J. Fisch, “Regime for a self-ionizing raman laser amplifier,” Phys. Plasmas 9(6), 2772–2780 (2002). [CrossRef]
9. D. Turnbull, S. Bucht, A. Davies, D. Haberberger, T. Kessler, J. L. Shaw, and D. H. Froula, “Raman amplification with a flying focus,” Phys. Rev. Lett. 120(2), 024801 (2018). [CrossRef]
10. D. Turnbull, S.-W. Bahk, I. A. Begishev, R. Boni, J. Bromage, S. Bucht, A. Davies, P. Franke, D. Haberberger, J. Katz, T. J. Kessler, A. L. Milder, J. P. Palastro, J. L. Shaw, and D. H. Froula, “Flying focus and its application to plasma-based laser amplifiers,” Plasma Phys. Controlled Fusion 61(1), 014022 (2019). [CrossRef]
11. V. M. Malkin, G. Shvets, and N. J. Fisch, “Fast compression of laser beams to highly overcritical powers,” Phys. Rev. Lett. 82(22), 4448–4451 (1999). [CrossRef]
12. V. M. Malkin, G. Shvets, and N. J. Fisch, “Detuned raman amplification of short laser pulses in plasma,” Phys. Rev. Lett. 84(6), 1208–1211 (2000). [CrossRef]
13. Y. Ping, W. Cheng, S. Suckewer, D. S. Clark, and N. J. Fisch, “Amplification of ultrashort laser pulses by a resonant raman scheme in a gas-jet plasma,” Phys. Rev. Lett. 92(17), 175007 (2004). [CrossRef]
14. Y. Ping, R. K. Kirkwood, T.-L. Wang, D. S. Clark, S. C. Wilks, N. Meezan, R. L. Berger, J. Wurtele, N. J. Fisch, V. M. Malkin, E. J. Valeo, S. F. Martins, and C. Joshi, “Development of a nanosecond-laser-pumped raman amplifier for short laser pulses in plasma,” Phys. Plasmas 16(12), 123113 (2009). [CrossRef]
15. K. Y. Kim, J. H. Glownia, A. J. Taylor, and G. Rodriguez, “Terahertz emission from ultrafast ionizing air in symmetry-broken laser fields,” Opt. Express 15(8), 4577–4584 (2007). [CrossRef]
16. P. Sprangle, J. R. Peñano, B. Hafizi, and C. A. Kapetanakos, “Ultrashort laser pulses and electromagnetic pulse generation in air and on dielectric surfaces,” Phys. Rev. E 69(6), 066415 (2004). [CrossRef]
17. C. D’Amico, A. Houard, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and V. T. Tikhonchuk, “Conical forward thz emission from femtosecond-laser-beam filamentation in air,” Phys. Rev. Lett. 98(23), 235002 (2007). [CrossRef]
18. D. H. Froula, D. Turnbull, A. S. Davies, T. J. Kessler, D. Haberberger, J. P. Palastro, S.-W. Bahk, I. A. Begishev, R. Boni, S. Bucht, J. Katz, and J. L. Shaw, “Spatiotemporal control of laser intensity,” Nat. Photonics 12(5), 262–265 (2018). [CrossRef]
19. A. Sainte-Marie, O. Gobert, and F. Quere, “Controlling the velocity of ultrashort light pulses in vacuum through spatio-temporal couplings,” Optica 4(10), 1298–1304 (2017). [CrossRef]
20. J. P. Palastro, D. Turnbull, S.-W. Bahk, R. K. Follett, J. L. Shaw, D. Haberberger, J. Bromage, and D. H. Froula, “Ionization waves of arbitrary velocity driven by a flying focus,” Phys. Rev. A 97(3), 033835 (2018). [CrossRef]
21. D. Turnbull, P. Franke, J. Katz, J. P. Palastro, I. A. Begishev, R. Boni, J. Bromage, A. L. Milder, J. L. Shaw, and D. H. Froula, “Ionization waves of arbitrary velocity,” Phys. Rev. Lett. 120(22), 225001 (2018). [CrossRef]
