Reversibility of laser filamentation

We investigate the reversibility of laser filamentation, a self-sustained, non-linear propagation regime including dissipation and time-retarded effects. We show that even losses related to ionization marginally affect the possibility of reverse propagating ultrashort pulses back to the initial conditions, although they make it prone to finite-distance blow-up susceptible to prevent backward propagation.

Deterministic chaos clearly highlights that determinism does not ensure the time invertibility of physical systems [1].Attractors or dissipation induce a loss of memory of initial conditions within a finite time, which prevents to recover the initial conditions by e.g.reversing time in the dynamics.While the invertibility of linear differential systems is well understood [2,3], its non-linear counterpart remains an open question in many areas of non-linear physics and mathematics [4].
Optical systems provide a well-suited framework to study invertibility because of the availability of efficient numerical models, as well as the possibility to perform tabletop experiments with a wide variety of free experimental parameters.Time invertibility has been demonstrated in several non-linear optical systems, where it is generally termed as time-reversibility.In this work, we shall stick to this common terminology, although strictly speaking reversibility is a stronger condition.It depicts systems with full inversion symmetry with regard to time, like, e.g. a pendulum.Time-reversibility has been reported in optical fibers [5], solid focusing [6] and defocusing [7] Kerr media, chaotic systems [8] and coupled-resonator optical waveguides [9].However, these systems are lossless and only consider effects featuring an instantaneous response, i.e., their non-linearities do not contain any explicit temporal dependencies.Conversely, such explicit temporal dependence in the system can be expected to induce a loss of memory, therefore jeopardizing time reversibility.
Here, by investigating the case of laser filamentation and the associated ionization as well as molecular alignment, we show that time reversibility can be maintained even in lossy non-linear system including time-retarded effects.Filamentation [10][11][12][13][14] is a propagation regime typical of high-power, ultrashort-pulse lasers.It stems from a dynamic balance between Kerr self-focusing and self-defocusing non-linearities of higher orders, including defocusing by the laser-generated plasma and the saturation of the medium polarisability under strong-field illumination [15][16][17].
This dynamic balance tends to stabilize the intensity at a so-called clamping intensity (≈ 50 TW/cm 2 in air at 800 nm) [19], where all non-linear effects balance each other.It can therefore be seen as an attractor for this observable, which could lead one to expect the propagating electrical field to loose memory of its initial conditions during propagation.The complex time-dependent effects like plasma accumulation or intra-pulse electron dynamics, that play a significant role in the filamentation process [17], and the propagation through the laser-generated plasma, that causes a loss of coherence of the beam, can also induce memory losses.
We numerically demonstrate that filamentation is actually time-reversible in the sense that the initial conditions can be recovered by retro-propagating the final electric field obtained at the end of laser filaments.This result is obtained in spite of the losses, decoherence and time-dependent effects intrinsic of filamentation.In a preliminary step, we shall briefly recall the formal reversibility of an instantaneous (cubic-quintic) non-linear Schrödinger equation (NLSE) where the saturation of the χ (3) focusing term is described by a defocusing χ (5) term [18].We then demonstrate numerically the reversibility of more realistic filamentation models.Finally, we show that the time-reversibility still holds within a defined confidence interval even when adding noise to the pulses before retro-propagating them.
Let us first consider the NLSE in a general form, applied to the field envelope of an electric field ε(r, z): The operator ∆ ⊥ is the transverse Laplacian r −1 ∂ r r∂ r up to a normalization constant.If one sets f (|ε| 2 ) = γ|ε| 2 −δ|ε| 4 , this equation describes the paraxial propagation of a linearly polarized laser beam in a cubic-quintic nonlinear medium.
Let us rewrite this equation by introducing the real fields u and v such that ε = u + iv: Projecting the fields u = j u j (z)φ j (r) and v = j v j (z)φ j (r) on a Hilbert basis {φ j } j which diagonalizes the Laplacian with eigenvalues −k 2 j leads to the infinite set of equations indexed by j These equations can be recast as Hamilton equations where the dots denote a derivative with respect to z, and where one has introduced the Hamiltonian H given by where F is the antiderivative of f .Equivalently : The equations of motion ( 6) can then be rewritten under the compact form d dz involving the Poisson bracket {•, H} which reads: The flow of any observable A[ε] can then readily be obtained by The evolution operator acting on A in Eq. ( 11) is invertible by changing z − z 0 to z 0 − z in the exponential.It is now clear that the dynamics arising from ( 1) is reversible, in the sense that one can recover the initial conditions starting from any propagated solution by simply integrating Eq. ( 1) with respect to −z.Such a formal reversibility can be illustrated by a numerical simulation of a cubic-quintic propagation equation describing the evolution of the complex electric field − → E is linearly polarized along − → u ⊥ , which is orthogonal to the propagation direction − → u z .The propagation equation then reads: in which the electrical field ε(z, r, t) now also depends on time, and where the tilde denotes a Fourier-Hankel transform Here J 0 is the zeroth-order Bessel function, v g the group velocity, ǫ 0 the vacuum permittivity, c the speed of light, and , n(ω) being the refractive index at frequency ω.We numerically integrated the Unidirectional Pulse Propagation Equation (UPPE) (12) in the approximation of cylindrical symmetry [20,21].We considered a Gaussian input pulse of 1.7 mJ energy, 60 fs duration (FWHM), centered at 800 nm, with a beam radius of 3 mm (FWHM), slightly focused by an f = 1m lens.After propagating up to z f = 2 m, the electric field was then retro-propagated back to z = 0 by changing the sign of the z-derivative in (12).The results are shown in Figure 1.Although the pulse temporal shape (Panel a) and spectrum (Panel b) are strongly modified by the non-linear propagation along the filamentation, the retro-propagated pulse perfectly overlaps with the initial one in spite of the complex propagation dynamics (Panel d).The residual error does not exceed 0.06%, as displayed in Panel c. Considering the formal reversibility of the cubic-quintic model, this error is mainly numerical.This demonstrates that the theoretical time-reversibility of filamentation in a cubic-quintic medium can effectively be achieved numerically with realistic parameters.
As our purpose is to investigate the effects of losses and time-retarded effects on the time-reversibility, we turn to a more complete description of filamentation, including the losses, the complex temporal dynamics induced by molecular alignment (i.e., the Raman effect) [22], as well as ionization, a delayed effect that accumulates over the whole pulse duration.Replacing the quintic contribution in (12) by these effects yields: where ∆n r (t) corresponds to the modification of the refractive index by molecular alignment [22] and τ (ω) = (ν en + iω)/(ν 2 en + ω 2 ), ν en being the collision frequency between free electrons and neutrals atoms.The last term in equation ( 14) accounts for plasma losses, and is calculated using When neglecting recombination effects over the very short pulse durations at stake here, the free-electron density ρ follows where W(|ε| 2 ) describes the ionization probability, σ the inverse Bremsstrahlung cross section, U i = K ω 0 the ionization potential, featuring K as the number of photons required for ionization.Since this time-integrating effect couples with the z-propagation, one could expect the reversibility of the cubic-quintic phenomenological model ( 12) to be lost.We investigated this question, focusing first on the role of ionization by simulating the propagation in argon, an atomic gas in which the Raman term vanishes.We used the same initial conditions as for the cubic-quintic case discussed above except for an energy of 1.5 mJ (Figure 2).Consistent with the introduction of plasma that induces non-instantaneous response times coupling temporal slices of the pulse with each other, the pulse propagation dynamics is more complex.In particular, a refocusing cycle is clearly visible in Panel d, while pulse splitting is observed in the temporal domain in the filamentation region.In spite of these effects, both retro-propagated pulse shape and spectrum are almost indistinguishable from their initial counterparts (Panels a and b).The residual error between them stays below 0.08% (Panel c) over the whole pulse duration.The temporal asymmetry of the error can be directly related with the plasma generation.The largest error corresponds to the of the pulse at the propagation distance where the intensity is maximal (z = 97 cm, see panel a), hence where the plasma generation is the most efficient.
We then included an additional time-retarded effect, namely molecular alignment [22], by considering nitrogen at a pressure of 4 bar.Figure 3 shows the result of propagation and retro-propagation over 2 m.The energy was limited to 0.5 mJ, and the pulse duration stretched to 100 fs in order to to remain below the multi-filamentation regime that would be incompatible with the radial symmetry of our code.The consideration of the Raman effect further complexifies the propagation dynamics, resulting in a very clear refocusing cycle visible on both beam diameter and on-axis fluence (Panel d).The associated pulse splitting is so strong that it results in a triple pulse that is retained even after the pulse has propagated over 1 m beyond the filamenting region (Panel a).Correspondingly, its spectral counterpart is pretty much structured around the fundamental wavelength.Again, even in the presence of delayed effects, one observes a very good overlap between the retro-propagated pulse and the initial condition.However, the residual error in this case is one order of magnitude higher (up to 0.5%) than what we obtained with less or no retarded effects.This indicates that in this regime deviations from the propagating pulse mainly arise from physical effects (i.e., by the temporal dependence of molecular alignment and ionization) rather than from numerical errors.This is also evidenced by the marked asymmetry of the error plot (Figure 3c), that we associate with the asymmetry of the non-instantaneous process.
Beyond the numerical reversibility of filamentation, we investigated its chaoticity by introducing noisy perturbations in the propagated beam, before retro-propagating it.Such perturbations are set to model, at least qualitatively, the possible effects of air turbulence, and to probe the stability of the numerical reversed scheme including the different physical phenomena.
Starting from the final pulse as obtained in Figure 2 (Argon, plasma), we added to ε(r, z f , t) a uniformly distributed white noise ξ(r, t), namely ξ(r, t) = 0, (17) ξ(r, t)ξ(r ′ , t ′ ) ∝ δ(r − r ′ )δ(t − t ′ ), (18) with an amplitude of a few percents of the initial electric field envelope.As Figure 2a displays, the error on intensity integrated over the pulse, defined by mainly develops in the filamentation region.The error first decrease as the diffraction tends to smooth out the added noise.The error then mainly develops in the filamentation region, where the rise in intensity induces the highest nonlinear effects, especially the plasma generation.The major contribution of the non-linear effects is further evidenced by the fact that the relative error is concentrated in the center of the beam (Figure 4b).The difference between the initial and retro-propagated pulses increase non-linearly (Figure 4c).However, this difference stays reasonable as long as the noise level is restricted to a few percent of the local intensity, showing the robustness of the time-reversibility of filamentation.As a conclusion, our work illustrates the unexpected resilience of time-reversibility to time-retarded effects in laser filamentation, in spite of the key role they play during the filamentation process.It constitutes a clear evidence that practical time-reversibility of physical systems can expand beyond hamiltonian systems.

FIG. 1 :
FIG. 1: (Color Online) Retro-propagation of a laser filament within the frame of the cubic-quintic model.(a) On-axis intensity and (b) spectrum of the initial (red solid line), propagated (blue dash-dotted line) and retro-propagated pulses (black dashed line); (c) Normalized error between the initial and retro-propagated pulses; (d) Fluence distribution along the propagation distance.The solid white line indicates the beam radius.

FIG. 2 :
FIG. 2: (Color Online) Retro-propagation of a laser filament in argon when including the modeling for ionization.(a) On-axis intensity and (b) spectrum of the initial (red solid line), propagated (blue dash-dotted line), retro-propagated pulses (black dashed line) and at z = 97 cm (green solid line); (c) Normalized error between the initial and retro-propagated pulses; (d) Fluence distribution along the propagation distance.The solid white line indicates the beam radius.

FIG. 3 :
FIG. 3: (Color Online) Retro-propagation of a laser filament in nitrogen, taking into account ionization as well as molecular alignment.(a) On-axis intensity and (b) spectrum of the initial (red solid line), propagated (blue dash-dotted line) and retro-propagated pulses (black dashed line); (c) Normalized error between the initial and retro-propagated pulses; (d) Fluence distribution along the propagation distance.The solid white line indicates the beam radius.

FIG. 4 :
FIG. 4: (Color Online) (a) Evolution of the pulse-integrated Error function of the propagation distance; (b) On-axis intensities of retro-propagated beams after addition of noise on the propagated beam; (c) integrated error at the end of the retro-propagation