We provide a theoretical description of the spatio-temporal dynamics of sequential filamentation in noble gases that can lead to pulse compression down to nearly single-cycle pulses. We show that the strong pulse compression occurs as a result of serially-generated on-axis filaments and spectral filtering of an extensive blue-shifted compressible spectra. We show that the dynamics of this sequential filamentation can be readily tuned by varying the gas pressure and can be scaled to various pulse energies.
©2008 Optical Society of America
Light can be confined over distances several orders of magnitude greater than its confocal parameter by the combined effects of self-focusing and light-induced gas ionization . The formation of light filaments has been shown to be robust to perturbations [2, 3], and the onset of filamentation and multiple filamentation spatial patterns are controllable by several means [4, 5, 6, 7, 8, 9]. Experiments involving high-power filaments contribute to our understanding of fundamental physical processes and offer an array of potential applications [10, 11, 12, 13].
More recently, filamentation has provided a mechanism for compressing pulses to only a few optical cycles with millijoule energies , wherein the sustained balance of self-focusing and ionization produces extensive spectral broadening. Although several other methods have demonstrated few-cycle pulse compression with energies well over a millijoule [15, 16, 17, 18], these methods require serial processes, where spectral broadening or dispersion compensation are required using separate stages. Hollow-core capillaries have been used to produce few-cycle pulse duration compression above a millijoule without the need for a secondary stage of dispersion compensation , however it is unclear how pulse compression could scale to higher energies in waveguiding configurations, even with large-core photonic band gap fibers . Although filamentation as a mechanism for pulse compression is not without challenges, this approach appears scalable to higher pulse energies. Furthermore, recent experiments and numerical simulations demonstrate self-compression or compression without the need for additional negative dispersion compensation [21, 22].
In this paper, we provide a theoretical description of optimal pulse compression via filamentation and we propose that the alternating dynamics between beam focusing and defocusing and the production of visible sequential plasma filaments is the primary mechanism that leads to the extensive blue-shifted spectrum which enables pulse compression. With noble gases, optimizing spectral broadening and sequential filamentation involves simply varying the gas pressure, and a distinct two-plasma-filament structure provides the visual signature of optimal spectral reshaping. While there have been extensive numerical investigations examining pressure trends [23, 24, 25, 26] and a description of serial filamentation, [27, 28, 29], to our knowledge this is the first work that ascribes the extensive spectral broadening to frequency-dependent linear and nonlinear spatial dynamics. This understanding is necessary for properly characterizing filament pulses and provides guidance for improving compression techniques.
Plasma filaments are associated with self-guiding or self-channeling of laser pulses, and only partial information can be extrapolated about the spatial or temporal profile simply by the time-integrated plasma densities. One difficulty in interpreting the nonlinear dynamics of extended filamentation is that the electric-field background reservoir, which is not directly observable in experiment, plays an enormous role in the nonlinear propagation dynamics . In fact, simulations and precise measurements of meter-long single-plasma-filament densities indicate a corrugated electron-density structure [29, 30], with intermediate self-focusing stages within isolated filaments. In this paper, we refer to a filament as the presence of ionized gas or plasma, and we believe that the mechanism described here of sequential filamentation provides a building block for understanding the more complicated evolution of high-power ultrashort laser pulses.
Previous experimental results have demonstrated pulse compression via sequential filamentation with initial input pulses of 0.7 mJ at 800 nm in which initial input pulses of 35 fs were compressed to 11 fs in argon gas  and with 0.33 mJ pulses at 2 µm  from 50 fs to 18 fs in xenon. Here we show how the dynamics scale for different input pulse geometries, input pulse durations, input pulse energies, and wavelengths, and different gas media. The goal of this paper is twofold: to provide a better framework for optimizing pulse compression via filamentation techniques and to synthesize previously published results to arrive at a more comprehensive paradigm for filamentation in gases.We provide predictions that describe how pulse compression via sequential filamentation can scale to higher pulse energies.
2. Nonlinear envelope equation
The nonlinear propagation equation we use to model the evolution of the electric-field amplitude E(r, τ, z) for the case of a radial symmetry in a transparent medium is given by,
where u(ρ, t, ζ)=E(r, τ, z)/E 0 is the electric field amplitude envelope initially centered at angular frequency w0 and normalized by E 0=E(r=0, τ=0, z=0), ρ=r/w 0, ζ=z/2Ldf, Ldf=k 0 w 2 0/2 is the diffraction length or confocal parameter, k 0=2π n 0/λ, t=(τ-k1z)/τp is the retarded time, and the operator T=(1+i∂t/ω 0τp) follows from the slowly-varying envelope approximation. The dispersion length is Lds=τ2 p/k 2, L -1 nl=αPpk/2Pcr, Ppk is the initial peak power, and the critical power for self-focusing is defined Pcr=αλ2/4πn 0 n 2 where n 2 is the nonlinear refractive index coefficient and α=1.8962 is a constant associated with the initial gaussian beam profile . The first term in Eq. 1 represents linear diffraction with the inclusion of space-time focusing. The second term accounts for dispersion k(ω)=k 0+k1(ω-ω 0)+k 2(ω-ω 0)2 and the third term describes self-steepening.
The remaining terms in Eq. 1 represent m-photon absorption and changes in the refractive index due to the formation of plasma and are derived via the Maxwell equation for the bound-electron and free-electron current densities, respectively, in a manner similar to and described in  and . The dimensionless plasma density η is scaled by the laser-induced breakdown (LIB) threshold, N 0=β (m) I m 0τp/mħω 0, where I 0 is the gas-specific and pulse-duration dependent LIB intensity , and β (m) is the Keldysh coefficient . We define the m-photon absorption length as (Lmp)-1=β (m) I (m-1) 0 and the plasma length as (Lpl)-1=σ 0 N 0/2n 2 0 ω 0τc. We assume a constant inverse bremsstrahlung cross-section derived in  so that σ 0=σ(ω 0)=k 0 ω 0τc/[Np(1+(ω 0τc)2)]. Np=me ω 2 0/4πe 2 is the electron plasma density when the plasma frequency equals ω 0. The last term of Eq. 1 in square brackets is a first-order chromatic dispersion correction. It is assumed that recombination and avalanche ionization effects occur at picosecond timescales  and can be neglected so that the nondi-mensionalized plasma density η is entirely formed by m-photon ionization,
3. Compression and pressure-dependent sequential filamentation at 800 nm
In this section, we consider the regime of material parameters where sequential filamentation is observed in experiments. By tuning the gas pressure, we change several physical parameters and observe trends that are repeatable under numerous experimental conditions. Although changing the pressure shifts the relative contributions of both linear and nonlinear effects in Eq. 1, due to the robust dynamics, sequential filamentation can still be observed within a range of input pulse powers. We also show that sequential filamentation is a template for understanding seemingly isolated and longer single plasma filaments.
The dispersion and nonlinear self-focusing coefficients scale linearly with pressure, that is,
since k 2,n 2∝p. The higher-order processes of plasma defocusing, multiphoton absorption, and ionization are more sensitive to changes in pressure. Since both the Keldysh rate, β (m), and LIB intensity I 0 scale linearly with pressure, it follows that the multiphoton ionization length scales to the power of m, that is,
The electron collision time τc scales inversely with pressure, and in the limit ω 0τc≫1,σ 0~1/τc, and the inverse bremsstrahlung cross-section scales linearly with pressure (i.e., σ 0∝p). Consequently, plasma defocusing scales at a different rate,
Different plasma structures arising from changes in the pressure near the optimal regime of sequential filamentation are shown in Fig. 1. These simulations correspond to experimental parameters described in , where compression from 30 fs to 10 fs was demonstrated at an optimum pressure at 0.75 atm. The experimental input pulse with an energy of 0.7 mJ is collimated to a 1/e 2 diameter of 0.5 cm and focused with a 100-cm lens, where the linear focus is approximately 40 cm within an argon gas cell. To find matching numerics, we use the following length scales: Ldf/Lds=4×10-3, Ldf/Lnl=1.3, Ldf/Lmp=3×10-7, and Ldf=Lpl=4×10-6, which are consistent with the experimental parameters. Argon has an ionization energy of 15.7 eV, corresponding to m=11, and at a pressure of 1 atm, k 2=2.5×10-8 fs2/nm and n 2=1×10-19 W/cm2. The calculated Keldysh coefficient of β (11)=10-140 cm19/W10 yields an intensity I 0=2×1013 W/cm2, which leads to an LIB density of N 0=7×1010/cm3. If we use an electron-neutral collision time at p=1 atm of τc=375 fs, then σ 0~5×10-20 cm2.
Several trends in Fig. 1 are also evident in experiment and follow intuitively. At low pressures, we observe one plasma filament with a length approximately equal to one confocal parameter, as shown in Fig. 1(a). As we increase the pressure, the onset of filamentation occurs earlier due to increased contributions from self-focusing and plasma formation. At 0.7 atm, shown in the Fig. 1(b), we observe a light plasma trail after the first plasma filament, which increases in density as we increase pressure. When we tune to the optimum pressure at 0.75 atm, shown in Fig. 1(c), the double plasma structure is most distinct. Continuing to increase the pressure above this optimum results in the second filament both lengthening and drawing closer to the first, as shown in Fig. 1(d) at 0.83 atm. At 0.88 atm, the second structure nearly merges with first. In experiments , we observe off-axis multiple filamentation when the pressure exceeds approximately 1 atm.
Since the sequential filament pattern is not easily observed when the pressure decreases 10% below the optimum and since 10% above the optimal pressure the sequential filaments have the appearance of one longer filament, one might expect that the double-plasma filament structure is difficult to access, yet we find that the dynamics are robust and observable over a range of beam and material parameters. As we increase the input pulse energy, the optimum double-filament structure occurs at slightly lower pressures, which increases the self-focusing to balance plasma effects. The greater contributions of self-focusing balance the higher pulse energies and yield more extended sequential plasma filament patterns.
Pulses at longer wavelengths and gases with higher ionization energies contribute to reduced ionization rates , which favors higher energy pulse compression. At longer wavelengths and lower input pulse powers, higher pressures are necessary to achieve the same ionization rates, in agreement with . However, at higher pressures the dispersion-dominated post-shock dynamics  limit the length of plasma filaments, and a more tightly-focused geometry is necessary to observe sequential filamentation.
Plasma defocusing affects the development and evolution of a background field reservoir, and we propose that pulse compression via sequential filamentation scales to higher pulse energies at lower gas pressures, especially when plasma defocusing can be decreased accordingly. We recall that the relative contributions of plasma defocusing to multiphoton ionization scale inversely with τ2 c, that is,
The electron collision time increases with lower gas pressure but also increases with atomic number [43, 44] due to Coulomb screening. We analyze how sequential filamentation could vary in different media in Fig. 2. The double-structured on-axis peak-intensity curves coincide with two distinct plasma filaments with P=Pcr=[0.95;1.35;1.8;2.2], Ldf/Lmp=[3×10-8;3×10-7,3×10-6,3×10-5] and Ldf/Lpl=[5.7×10-8,1.2×10-6,2.1×10-5,2.7×10-4], respectively.
Longer plasma filaments with lower peak intensities are observed in materials with higher ionization rates and increased plasma defocusing, while shorter sequential plasma filaments occur at higher peak intensities with lower ionization rates and decreased plasma defocusing. To achieve similar dynamics at lower pressure, it is more favorable to use gases with smaller inverse bremsstrahlung cross-sections or higher atomic number to minimize the effects of plasma defocusing. Such materials also result in higher plasma densities in the sequential filament and may demonstrate additional plasma filament stages.
4. Nonlinear dynamics of sequential filamentation
The sequential plasma filament structure we study here is formed by a two-peak temporal profile, where on-axis plasma defocusing of the front peak results in a background reservoir that feeds the rear peak [27, 29, 40]. To emphasize the spatial dynamics here, we show in Fig. 3(a) the contour plot of the fluence or time-integrated intensity as a function of propagation, which illustrates that each plasma filament has a distinctly different spatial beam profile. Figure 3(b) is the spatial beam profile at z=40 cm, where the time-integrated spatial beam profile has a sink in the center due to multi-photon absorption. Approximately 5% of the initial pulse energy is lost to the production of plasma within the first filament structure. The resulting spatial beam profile with off-axis maxima requires higher powers for self-focusing , and with the combined effects of plasma defocusing, a substantial fraction of the pulse energy also diffracts after the first plasma filament.
As a consequence of plasma absorption and diffraction, lower peak-fluence values are observed during the second filament, and in contrast to the first filament, the spatial beam profile during the second plasma filament has an on-axis maxima. In spite of lower fluence, ionization occurs due to the higher intensities resulting from self-compression [29, 40, 41] and increased shock-wave-induced self-focusing dynamics [41, 42]. At z=70 cm, towards the end of the plasma filament, the spatial beam profile approaches a Townes profile [45, 46], as shown in Fig. 3(c), which has a narrower peak and wider wings than a Gaussian profile. This profile is the 2-D self-focusing attractor in the absence of plasma and is responsible for filamentation self-filtering of the spatial-mode .
When the ionization rate is low, the off-axis maximum in the spatial beam profile during the first plasma filament has a narrower radius, which suggests a higher power threshold for multiple filamentation. Since the self-focusing propagation dynamics are determined by the input spatial beam profile [7, 8], it is not surprising that shaping the input spatio-temporal profile and changing the relative contributions to the filament and background field reservoir in refocusing events can dramatically affect the propagation dynamics. It was recently demonstrated how the use of a circular mask on the input spatial beam profile improved the filamentation pointing stability and increased the spectral broadening . Understanding the interplay between plasma defocusing and spatial replenishment in sequential filamentation may also explain pointing instabilities associated with polarization .
The significance of the sequential plasma filament in pulse compression can be observed in the spatial-spectral distribution, which indicates how the spectrum would change if apertured at the designated propagation distance. In Fig. 4 we plot the power spectrum as a function of radius at several distances of propagation, corresponding to the optimally pressure-tuned scenario of sequential filamentation shown in Fig. 1(c). Before the first filament at z=36 cm, depicted in Fig. 4(a), we observe that there has been slight broadening on-axis due to the higher intensities on-axis, but the broadening is generally symmetric and centered on-axis. At z=43 cm, we see that the spectra has broadened most significantly off-axis [Fig 4(b)], due to the higher off-axis fluence as previously described in Fig. 3(b). At z=51 cm [Fig. 4(c)], towards the end of the first plasma filament, the power spectra is centered on axis, and we see that the spectrum has broadened by approximately a factor of two. The spectra is slightly asymmetric with a sharper edge on the longer-wavelength edge. We note that after the first filament, the spectral broadening is not sufficient to explain the pulse compression observed at the output in experiments.
The sequential plasma filament is the signature of a dramatic spatial redistribution of spectral components. Between the end of the first plasma filament and the beginning of the second at z=62 cm [Fig. 4(d)], we observe a highly asymmetric spectral distribution, where on-axis the power spectrum exhibits an extended blue-tail, which coincides with a red-wavelength-shifted background off axis. This trend continues towards the end of the second plasma filament, as shown in Fig. 4(e). At low pressures when there is only a single plasma filament [i.e., Fig 1(a)], no spatial-spectral reshaping occurs.
We confirm that spatial-spectral reshaping accompanies the propagation dynamics when the second filament merges with the first to produce the appearance of one longer filament, such as that observed at higher pressures shown in Fig. 1(e). In , the extensive blue-shifted spectrum is crucial to pulse self-compression, which suggests that the spatial-spectral redistribution we observe with sequential filamentation plays a general role in the pulse compression dynamics of filamentation. Similar observations of spatial-spectral reshaping in serial plasma filaments have been made  and were attributed to plasma-induced changes in the refractive index.We note that the electron density results in a lower refractive index and beam defocusing, which is incompatible with the explanation in .
Between the end of the first filament and the onset of the second, that is, between z=51 and z=63 cm, we observe comparable peak intensities, but relatively lower fluence, and negligible contributions from plasma. Therefore, in contrast to previous assumptions, we cannot attribute the sudden extensive on-axis spectral broadening that occurs between the first and second filaments to self-phase modulation or plasma formation. Fluence and peak intensities are relatively low, and as we recall from Fig. 2, the sequential filament structure is observed even when initial peak powers are below the threshold for critical self-focusing.We extrapolate that the blue-compressible spectrum  is a combined effect of frequency-dependent space-time focusing and self-steepening. A comparison of the relative contributions in the nonlinear envelope equation [Eq. 1],
which indicates that higher frequencies experience greater self-focusing. Equation 8 is not qualitative; it is a direct asymptotic limit for the coefficients as defined in Sec. 2. Consequently, it is the bluer frequencies that refocus after the first plasma filament to ionize and produce the second plasma stage. Although it is difficult to isolate the contribution of the sequential filament from that of the extended spatial confinement of light, our observations suggest that the two separated plasma channels not only coincide with the longest filament structures but result in minimal loss of pulse energy due to plasma formation and multiphoton ionization.
The resulting output beam is spatially inhomogeneous, as similarly described in , and in Fig. 4(f), we show the aperture-dependent power spectrum when 20%, 40%, 60%, 80%, and all of the power is transmitted.We observe the widest blue-shifted spectra on-axis, which also corresponds to the highest degree of compression. The total spatially-integrated power spectrum has only twice the initial input bandwidth, whereas on-axis, the spectrum has broadened by almost a factor of four. We have approximately 30% power efficiency pulse compression to 10 fs, measured by the full-width half-maximum of the temporal profiles in Fig. 4(g). In the experiment, negatively-chirped output pulses are observed [22, 31, 32], which remains unexplained by the simulation model.
An illustration of spatial-spectral reshaping due to linear frequency-dependent propagation effects is shown in Fig. 5, where we provide contour plots of the spatial-spectral distribution for the two-filament structure corresponding to Fig. 4 at z=74 cm [Fig. 5(a)] and the far-field spatial-spectral distribution found in the focal plane of an achromatic 100-cm lens, assuming aberration-free linear propagation after z=74 cm [Fig. 5(b)] in which linear space-time focusing is included. Contours are equally-spaced on a linear scale. A comparison of the near and far-field plots both show similar blue-shifted spectra on-axis and red-shifted background, however the spatially-varying structures differ markedly due to linear space-time focusing effects. Spatial-spectral reshaping in the linear and nonlinear domains provides rigorous challenges for pulse characterization and measurement in both numerical and experimental investigations. In the far-field, we observe that the shortest pulse duration is no longer precisely on axis, and that the shortest self-compressed pulses are observed when integrating over a finite aperture size, which agrees with experiments.
In our simulations [Fig. 1] in which the pressure is varied, we observe that distinct sequential filamentatiion coincides with a maximal blue spectral tail, which we believe to be another indication of optimal spectral reshaping with serial filaments. This observation is in agreement with the experimental results provided in Ref. . In Fig. 6 we plot the on-axis spectra with a maximal norm of one, which corresponds to the pressure tuning in Fig. 1. The red curve, which corresponds to the maximal blue-tail, coincides with the most distinct sequential plasma filament pattern [Fig. 1(c)].
Longer sequential filamentation patterns have wider on-axis blue-shifted spectral tails, despite lower peak filament intensities [Fig. 2] and indicate that longer filament structures favor higher degrees of pulse compression. Our claim that higher-energy pulse compression via sequential filamentation occurs with shorter sequential filaments [Sect. 3] suggests that there exists a conjugate balance between pulse energy and pulse compression. In order to achieve both high energy pulses and a higher degree of pulse compression, precise control, tuning, and characterization of experimental parameters are required, which makes the spatial-spectral effects described in this paper highly relevant.
It is important to clarify that the spatial-spectral dynamics described here and associated with the sequential filament are entirely separate from conical emission. In fact, the radial dependence of frequencies in the output far-field of normal-dispersion filamentation processes due to conical emission generally has a trend opposite to that of the processes described here and for larger bluer radii [30, 40, 49, 50], which is explained by phase-matched wave mixing . Although dispersion is negligible in our configuration, subtle evidence of conical emission can be seen by comparing the change in lobe angle of the background spectra between Figs. 4(d) and 4(e). In Fig. 4(d) the off-axis spectral maxima align at 1-o’clock and 5-o’clock, and evolve in Fig. 4(e) to align more vertically. This angular shift indicates that the longer background wavelengths have a smaller divergence. While conical emission is governed by the interplay of dispersion and self-phase modulation, the spatial redistribution of the spectral components emphasized in this article is most dramatic in the serial refocusing of the field background.
Sequential filamentation underlines the fundamental propagation dynamics associated with more extended plasma filament production. The formation of a second plasma channel is the signature of maximal blue-shifted wavelengths in the background refocusing on-axis, extended spatial confinement of light with minimal pulse energy loss to multiphoton absorption and is the result of frequency-dependent space-time focusing and self-steepening effects. We show by simulation that two distinct sequential plasma filaments are the signature of optimal pulse compression [31, 32] and a maximally blue-shifted tail in the on-axis power spectrum . Dramatic spatial-spectral reshaping occurs with the formation of the second filament in a process entirely separate from conical emission. We also describe the general dynamics that accompany the gas pressure tuning to optimize sequential filamentation and provide predictions for how pulse compression via filamentation can scale to higher powers. We anticipate that pulse compression efficiency and pointing stability can be improved with input spatial beam pulse shaping.
We thank G. Steinmeyer and M.A. Foster for helpful discussions, A. L’Huillier and L. Di-Mauro for their help during the early stages of this work, funding from AT&T Research Labs, the National Science Foundation under Grant No. PHY-0244995, the Army Research Office under Grant No. 48300-PH and the Air Force Office for Scientific Research.
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