Onset of self-steepening of intense laser pulses in plasmas

The self-steepening of laser pulses with intensities in excess of 1018 W cm-2 and with typical durations shorter than 30 fs propagating in underdense plasmas is examined by resorting to the framework of photon kinetics. Thresholds for self-steepening at the back/front of short laser pulses are determined, along with the self-steepening rates, and the connection between self-steepening, self-compression and frequency chirps is established. Our results are illustrated with particle-in-cell simulations, revealing the key physical mechanisms associated with the longitudinal laser dynamics, critical for the propagation of intense laser pulses in underdense plasmas.


Photon kinetic model for intense laser-plasma interactions
In photon kinetics, the laser pulse is represented by a distribution function of quasi-particles N (k, r, t) [19]- [21]. The early self-steepening of a laser pulse can be examined using the 1D photon distribution function, which, in underdense plasmas, where the ratio of the photon frequency to the plasma electron frequency is ω ω p , is given by [20,21], [23]- [25] ∂N ∂t + dz dt where t is the propagation time normalized to the inverse of the plasma frequency ω −1 p , z is the propagation distance normalized to k −1 p = c/ω p , and k is the wavenumber of the photons in the z-direction normalized to k p . The right-hand side of equation (1) represents corrections to the geometric optics approximation that can be readily neglected for ω ω p . In addition, this analytical framework is valid independently of the laser pulse duration and intensity, as long as the envelope approximation can be employed. The initial photon distribution function of a Gaussian laser pulse with normalized envelope of the vector potential a ≡ q A/(m e c 2 ) = a 0 exp(−z 2 /L 2 0 ) is given by N 0 = √ π/(8 √ 2)ω 0 L 0 a 2 0 exp − 2z 2 /L 2 0 + (k − k 0 ) 2 L 2 0 /2 , where k 0 and ω 0 are the central laser wavenumber and frequency, respectively, and L 0 is the initial laser pulse length. The photon density is given by n ph (z, t) = N 0 (z, k, t)dk/ N 0 (z, k, t)dk dz (in this paper, all integrations extend from −∞ to +∞) [20,21]. For ω ω p , equation (1) describes the photon number (or classical wave action) conservation. Thus, formally, the solution of equation (1) where z i , k i and t i are the initial position, wavenumber and propagation time of each photon. The phase-space trajectory of each photon, (z(t), k(t)), is determined according to the equations for the characteristics of equation (1), which correspond to the 1D ray-tracing equations (RTEs)ż = k/ω anḋ k = −(1/2ω)∂ zn , whereq = d t (q) with q being a generic quantity, andn = n/(n 0 γ ) is the normalized plasma density, n is the local plasma density, n 0 is the background plasma density and γ is the relativistic mass factor. The photon frequency and the wavenumber are related by the dispersion relation for a forward propagating EM wave in an underdense plasma, ω(k) = k 2 +n 1/2 . As long as the laser pulse duration and central frequency are such that the envelope approximation can be employed, the laser dynamics is mainly determined by photons with frequency ω ω 0 (and wavenumber k k 0 ). Thus, the RTEs can be expanded in Taylor series around k = k 0 − k. In the co-moving frame variables where v glinear 1 − 1/2k 2 0 , and the resulting RTEs becomė wheren ≡ ∂n/∂ξ . Equations (2) and (3) describe the phase-space trajectory of a photon initially at (ξ i , k i ). Equation (2) determines the group velocity of each photon. The first term on the right-hand side of equation (2) leads to corrections of the laser linear group velocity and to self-steepening [17]. The second term contributes to self-steepening if L 0 λ p (i.e. ifn is asymmetric with respect to the centroid of the laser), and is responsible for pulse self-compression in the limit of long pulses [18]. Equation (3) describes the blue/red shifts of the photons due to the gradients ofn.
The ratio α between the first and the second terms of equation (2), α ∼ φk 0 / k φk 0 L 0 , where φ = (n − 1) /n is the electrostatic potential [31], and k L −1 0 can be used to examine the range of validity of the previous models [17,18]. The results in [18] are valid for α 1, while the results in [17] are valid in the limit α 1, which corresponds to lower plasma densities and/or stronger plasma responses. However, for photons located at ξ ∼ L 0 /2-which are the most relevant ones for the evolution of the front of the laser-and for state-of-the-art lasers with a duration of 10 fs, and intensities exceeding 10 19 W cm −2 , propagating in plasmas with 10 17 -10 18 cm −3 , α 1. Thus, the inclusion of both terms of equation (2) is required to examine self-steepening and the global laser pulse dynamics in conditions which are directly relevant for state-of-the-art experiments with short laser pulses.

Onset of self-steepening and optical shock formation
The trajectories in phase space for early times are determined using a polynomial approximation for ξ and k up to O(τ 2 ). In order to retrieve the early longitudinal evolution of the laser the solutions of equations (2) and (3) , and then inserted in N . In the variables (ξ, k, τ ), the photon density is then n ph (ξ, τ ) where A vg = 4ξ(1 −n) + L 2 0n is the dominant contribution for the early propagation in the limit α 1. The term n is a selfcompression term which, in combination with P disp , recovers the known self-compression theory in the limit α 1 [18] and B vg = L 2 0 /(2k 2 0 ) 4 + 3k 2 0 L 2 0 n 2 . Equation (4) provides a global description of self-compression, self-amplification and selfsteepening. It predicts self-steepening either at the front or at the back of the laser, for arbitrary laser intensities and durations in the limits k 0 1 and k/k 0 1.
The key features associated with the asymmetric evolution of the laser can be retrieved by determining the skewness of the distribution in equation (4). The skewness, 3 , quantifies the asymmetry of a distribution [32], where ξ c = n ph (ξ, τ )ξ dξ/ n ph (ξ, τ )dξ is the laser pulse centroid and L(τ ) 2 = n ph (ξ, τ )(ξ − ξ c ) 2 dξ/ n ph (ξ, τ )dξ is the square of the laser pulse length. For a laser moving to the right, self-steepening at the back (front) of the pulse leads to γ skew > 0 (γ skew < 0). Considering terms O(τ ) in equation (4), and expanding A vg up to O(ξ 3 ) reveals that the direction of the early selfsteepening of a short laser pulse (L 0 λ p ) is determined by γ skew ∝ −n (ξ = 0). The average wavenumber along the laser pulse is k = n ph kd k/ n ph d k ∝n (ξ = 0), which means that if the back of the laser is down-shifted (up-shifted) with respect to the front, self-steepening occurs at the back (front) of the laser pulse. An illustration of self-steepening at the back of the pulse is shown in figure 1 with results from a 3D PIC simulation in QuickPIC [29].
The onset of self-steepening can be determined analytically. The quasi-static plasma response for a linearly polarized laser pulse is given by In the weakly relativistic regime, φ WR (1/4) ∞ ξ a 2 (x)sin[x − ξ ]dx, which can be determined for a Gaussian laser profile provided that L 0 λ p [31], by expanding sin[x − ξ ] for x λ p . The onset of self-steepening γ skew < 0 can then be written as the threshold condition κ t : In order to illustrate self-steepening at the front/back of the laser we have designed a set of 1D PIC simulations in OSIRIS [30] according to the threshold condition using a 0 L 0 = 1.7 < κ t and a 0 L 0 = 17 > κ t . The length of the simulation window, which moves at the speed of the light, is 25 c/ω p long, and is divided into 4000 cells, with 200 particles per cell. The results comparing the laser pulse length, peak vector potential, and skewness evolution between the   (4) the numerical solution of the quasi-static plasma response. The square of the vector potential of the laser is estimated according to a 2 (ξ, τ ) = n ph (ξ, τ )/ ω(ξ, τ ) , where ω(ξ, τ ) = ωN dk/ N dk is the average local frequency of the laser. In figure 2(a), a 0 L 0 < κ t and therefore the skewness is positive and the laser beam is compressed while a 2 increases. In figure 2(b), a 0 L 0 > κ t , and thus the blowout of electrons, is strong enough to cause self-steepening at the front of the laser (γ skew < 0). Although the full laser dynamics results from an interplay between the transverse and longitudinal motion of the laser pulse photons, our model, given by equation (4), can also give insights into conditions that are directly relevant for experiments. State-of-the-art experiments typically use laser pulses with initial peak intensities a 0 = 1, with a central wavelength λ 0 = 800 nm, and typical duration of 30 fs. In addition, typically, n 0 4 × 10 18 cm −3 , for which k 0 /k p = 20. In this situation, equation (4) predicts that the laser pulse self-compresses at a rate d(L 2 /L 2 0 )/dτ −0.06k 2 p /k 2 0 . Thus, at each propagation mm the laser becomes 7.25 fs shorter, a result that is consistent with recent experimental observations [10]. At the same time, the skewness of the laser decreases at a rate dγ sk /dτ −0.3k 2 p /k 2 0 , which indicates that the front of state-of-the-art laser pulses self-steepens. This is also consistent with the numerical modeling of the seminal experiments on the generation of quasi-monoenergetic electron bunches in the laser wakefield accelerator [33]- [36].

7
It is also interesting to compare the results from equation (4) in conditions that are directly relevant for the generation of single-cycle laser pulses [11]. To this end, a laser with a 0 = 4, with L 0 = π and with k 0 /k p = 5 was considered. The early self-compression rates are d(L 2 /L 2 0 )/dτ −0.12k 2 p /k 2 0 . Therefore, a single-cycle laser pulse is formed for τ k 2 0 /0.12 200, consistent with the previous results [11]. The skewness of the laser pulse intensity profile also decreases as dγ sk /dτ −0.6k 2 p /k 2 0 , which indicates that the self-compression mechanism is closely associated with the compression of the front edge of the laser pulse, and was already observed in the simulations in [11].
The early longitudinal dynamics of the laser beam can be physically illustrated by examining the trajectories of single photons. The distance between any two photons of the laser located initially at ξ 1 and ξ 2 is is the difference between the group velocity of each photon. According to equations (2) and (3), . For a linearly polarized long laser pulse, in the weakly relativistic regime, such thatn ≡n WR = 1 − (a 2 0 /4) exp(−2ξ 2 /L 2 0 ) [31], the distance between two photons at the front (back) of the laser, L + (L − ) (cf figure 1), initially separated by half the pulse length and located at ξ 1 = L 0 /2 and ξ 2 = 0 (ξ 1 = 0 and ξ 2 = −L 0 /2), is This indicates that the front of the pulse compresses while the back stretches, leading to the steepening at the front. In addition, an optical shock can occur at the laser pulse front when L + = 0, or equivalently in qualitative agreement with [17]. If the plasma response associated with a linearly polarized and short (L 0 λ p ) laser pulse is used, both self-steepening and optical shock formation can occur at the front or at the back of the laser. The condition for self-steepening at the front is L + /L − < 1, or equivalently, and using equation (6), a 0 L 0 > 2.9 κ t . Moreover, the onset of optical shock formation at the front (τ + ) and at the back (τ − ) of the laser occurs after In equations (8) and (9), the numerical factors correspond to approximations of the constants which are functions of π , and e, and cumbersome roots, which are thus avoided. It is straightforward to extend these arguments to describe self-compression by including terms of the order of O(τ 2 ) in the photon trajectories ξ 1,2 . In the light of a similar interpretation for selffocusing [37], this suggests that self-steepening can be viewed as a longitudinal asymmetric self-focusing. A rigorous quantification of self-steepening is required for practical purposes. According to equation (4), and usingn WR , ξ c = a 2 0 τ/(8 matches the nonlinear relativistic shifts to the laser linear group velocity of [37] including finite pulse length effects. Inserting ξ c into the expression for the skewness then yields If L 0 λ p , the early rates for the skewness evolution can only be retrieved by coupling equation (4) with the numerical quasi-static plasma response. The results are presented in figure 3. We observe that for early times (τ/k 2 0 1), the rate depends on k 2 0 only (cf A vg term in equation (4)). The parameters that correspond to γ skew = 0 were fitted to a Laurent series yielding a 0 L 0 = 2.6 + 2.5L 2 0 − 1.6L 3 0 + O(L 4 0 ), L 0 < L m with L m = 1.9, corresponding to a mildly relativistic regime where φ 1. We note that the condition a 0 L 0 > κ t closely matches the fit in the limit L 0 c/ω p . If L 0 > L m , the skewness is negative for any initial a 0 because γ skew ∝ −n (ξ = 0) can become negative (regardless of the laser initial intensity) solely by increasing the laser pulse length. For a sinusoidal wake,n (ξ = 0) > 0 if L 0 λ p /4 L m . State-of-the-art laser wakefield acceleration experiments typically use lasers with L 0 ∼ λ p /2 and self-steepening should occur at the laser pulse front. However, it is possible to conceive experimental scenarios where the opposite can be observed (for instance, by using shorter laser pulses). Examining the dependence of the rate with a 0 , keeping the laser pulse length fixed, shows that increasing a 0 leads to higher self-steepening rates at the front of the laser pulse up to a 0 L 0 < 16.7 − 0.122L 2 0 + 0.34L 3 0 − 0.08L 4 0 + O(L 4 0 ). From then on the initial rates for γ skew decrease. This occurs because for such high a 0 , the blowout of the plasma electrons is so strong [5] that most of the laser effectively travels in a vacuum-like region. 9 The global dynamics of the laser pulse is also affected by the transverse dynamics associated with self-focusing. The relative importance between self-focusing and selfsteepening can be illustrated in the case of a linearly polarized long laser pulse, in the weakly relativistic regime. In this case, and usingn WR , the evolution of the pulse front length is where P/P c = a 2 0 W 2 0 /32 is the ratio between the power of the laser and the critical power for self-focusing. Thus, unlike self-focusing, compression of the front of the laser pulse will occur, regardless of the laser power and intensity. In addition, for early propagation, the longitudinal compression rateL + /L 0 is higher than the transverse focusing ratesẆ 2 /W 2 0 , as long as τ < (2 − √ 2)(P/P c )W 3 0 4 √ π (P/P c − 1)L 0 .
Moreover, in matched propagation regimes (P = P c ), ideally suited for plasma-based accelerators, all the dynamics of the laser occurs in the longitudinal direction.

Conclusions
In conclusion, we have examined for the first time early laser pulse longitudinal evolution with the inclusion of self-steepening and self-compression for initial arbitrary laser intensities and pulse durations, as long as the envelope approximation is valid. Conditions and early rates for the onset of self-steepening and optical shock formation at the front/back of the pulse were derived. Our results show that experiments can be designed such that longitudinal modulations are enhanced in order to increase the plasma wave amplitude, to facilitate self-injection and to further increase the energy gain of electron beams. Furthermore, in this work, we have identified the conditions for self-steepening to occur both at the front and at the back of the laser pulse, and that can also be used to identify new regimes to laser pulse amplification beyond the limits of state-of-the-art laser technology via optical shock formation.