A perturbative approach to self-phase modulation and self-steepening of short laser pulses propagating in nonlinear media

TL;DR

This paper develops a first-order perturbative method to analyze self-phase modulation and self-steepening of short laser pulses in nonlinear media, clarifying the effects of various nonlinear terms on pulse evolution.

Contribution

It introduces a closed-form, perturbative solution for laser pulse propagation accounting for Kerr effect, ionization, and absorption, providing new insights into nonlinear pulse dynamics.

Findings

Quantitative analysis of femtosecond pulse propagation in air

Clarification of nonlinear effects on pulse amplitude and frequency

Scaling parameters for nonlinear effects

Abstract

The solution of the wave equation in the envelope approximation with temporal corrections for a laser pulse propagating in a medium where the Kerr effect, field ionization, and associated absorption take place, is obtained through a first-order perturbative approach. The closed-form expressions so obtained clarify the influence of the various terms of the equation on the laser amplitude and on the frequency generation as a function of the retarded time. Furthermore, they allow extracting scaling parameters which size the nonlinear effects. The results are illustrated quantitatively on the case of a femtosecond pulse focused in the air with typical parameters.

Equations

$$i{\partial }_{z}A+\frac{1}{2k_0}{\mathfrak{T}}^{-1}{\nabla }_{\perp }^{2}A-\frac{1}{2}{k}^{\prime \prime }{\partial }_{\tau }^{2}A+k_0{n}_2\mathfrak{T}\left(|A{|}^{2}A\right)-\frac{k_0}{2n_0^2{\rho }_c}{\mathfrak{T}}^{-1}\left(\rho A\right)$$

$$+i\frac{\beta }{2}|A{|}^{2K-2}A+i\frac{\sigma }{2}\rho A=0$$

$$A(z,r,\tau )=\mathfrak{A}(z,\tau )\left[1-\left(\frac{1}{w^2(z)}-i\frac{k_0}{2}{\partial }_z\ln (w(z))\right)r^2+O(r^4)\right]$$

$$i{\partial }_z\mathfrak{A}-\left(\frac{2}{k_0{w}^2}-i{\partial }_z\ln (w)\right)\mathfrak{A}+\lambda |\mathfrak{A}{|}^2\mathfrak{A}-\eta {\mathfrak{I}}_K\mathfrak{A}+i\gamma {\partial }_{\tau }\left(|\mathfrak{A}{|}^2\mathfrak{A}\right)+i\delta {\partial }_{\tau }\left({\mathfrak{I}}_K\mathfrak{A}\right)$$

$$+i\frac{\beta }{2}|\mathfrak{A}{|}^{2K-2}\mathfrak{A}+iϵ{\mathfrak{I}}_K\mathfrak{A}=0$$

$${\mathfrak{I}}_K(z,\tau )={\int }_{-\infty }^{\tau }|\mathfrak{A}(z,{\tau }^{\prime }){|}^{2K}d{\tau }^{\prime }$$

$$\lambda =n_2{k}_0,\eta =\frac{k_0{\alpha }_K{\rho }_m}{2n_0^2{\rho }_c},\gamma =\frac{2n_2}{c},\delta =\frac{{\alpha }_K{\rho }_m}{2c{n}_0^2{\rho }_c},ϵ=\frac{1}{2}\sigma {\alpha }_K{\rho }_m$$

$${\partial }_z\left(w^2(z){\mathfrak{I}}_1(z,\infty )\right)=-w^2(z)\left(\delta {\mathfrak{I}}_{K+1}(z,\infty )+\beta {\mathfrak{I}}_K(z,\infty )+ϵ{\int }_{-\infty }^{\infty }\rho |\mathfrak{A}{|}^{2}d\tau \right)$$

$$\frac{\delta {\mathfrak{I}}_{K+1}(z,\infty )}{\beta {\mathfrak{I}}_K(z,\infty )}=\frac{{\bar{\Phi }}_p}{U_I}{\left(1+\frac{1}{K}\right)}^{-1/2}$$

$$\left(1+\frac{i}{{\omega }_c}{\partial }_{\tau }\right)\left(1-\frac{{\nu }_e}{{\omega }_c}\right)\rho A\leftarrow {\mathfrak{T}}^{-1}(\rho A)$$

$$\begin{array}{rl}{\partial }_z\left(w^2(z){\mathfrak{I}}_1(z,\infty )\right)& =-w^2(z)(-\delta {\mathfrak{I}}_{K+1}(z,\infty )+2\delta {\nu }_e{\int }_{-\infty }^{\infty }|\mathfrak{A}{|}^2{\mathfrak{I}}_{K}d\tau \\ & +\frac{2\delta {\nu }_e}{{\omega }_c}{\int }_{-\infty }^{\infty }|\mathfrak{A}{|}^{2K+2}{\partial }_{\tau }\theta d\tau +\beta {\mathfrak{I}}_K(z,\infty ))\end{array}$$

$$\mathfrak{A}(z,\tau )=\psi (z,\tau )e^{i\theta (z,\tau )}$$

$${\partial }_z\theta +\frac{2}{k_0{w}^2}-\lambda {\psi }^2+\eta {\mathfrak{I}}_K+\gamma {\psi }^2{\partial }_{\tau }\theta +\delta {\mathfrak{I}}_K{\partial }_{\tau }\theta$$

$${\partial }_z\psi +{\partial }_z\ln (w)\psi +3\gamma {\psi }^2{\partial }_{\tau }\psi +\delta \left({\psi }^{2K+1}+{\mathfrak{I}}_K{\partial }_{\tau }\psi \right)+\frac{\beta }{2}{\psi }^{2K-1}+ϵ{\mathfrak{I}}_k\psi$$

$$\psi ={\psi }_0+\gamma {\psi }_{\gamma }+\delta {\psi }_{\delta }+\beta {\psi }_{\beta }+ϵ{\psi }_{ϵ}$$

$${\psi }_0(z,\tau )$$

$${\psi }_{\gamma }(z,\tau )$$

$${\psi }_{\delta }(z,\tau )$$

$${\psi }_{\beta }(z,\tau )$$

$${\psi }_{ϵ}(z,\tau )$$

$${\varphi }_N(z_0,z)={\int }_{z_0}^z{\left(\frac{w_0}{w(z^{\prime })}\right)}^{2N}d{z}^{\prime },{\xi }_N(\tau )={\int }_{-\infty }^{\tau }{F}^{2N}({\tau }^{\prime })d{\tau }^{\prime }$$

$$F(\tau )=A_0{e}^{-{\tau }^2/T^2}$$

$$\begin{array}{rl}\psi (z,\tau )=A_0\frac{w_0}{w(z)}{e}^{-{\tau }^2/T^2}& \{1+\frac{6\gamma A_0^2}{T}{\varphi }_1(z_0,z)\frac{\tau }{T}{e}^{-2{\tau }^2/T^2}\\ & -\delta A_0^{2K}{\varphi }_K(z_0,z)\left[e^{-2K{\tau }^2/T^2}-\sqrt{\frac{\pi }{2K}}\frac{\tau }{T}\left(1+\text{erf}\left(\sqrt{2K}\frac{\tau }{T}\right)\right)\right]\\ & -\frac{\beta }{2}{A}_0^{2K-2}{\varphi }_{K-1}(z_0,z)e^{-\left(2K-2\right){\tau }^2/T^2}\\ & -ϵA_0^{2K}{\varphi }_K(z_0,z)\sqrt{\frac{\pi }{8K}}T\left(1+\text{erf}\left(\sqrt{2K}\frac{\tau }{T}\right)\right)\}\end{array}$$

$$\psi =A_0\exp \left(-{\left(\tau -3\gamma (z-z_0){\psi }^2\right)}^2/T^2\right)$$

$$\psi =A_0{e}^{-{\tau }^2/T^2}\left[1+2\mu \frac{\tau }{T}{e}^{-2{\tau }^2/T^2}-{\mu }^2\left(1-10\frac{{\tau }^2}{T^2}\right)e^{-4{\tau }^2/T^2}+O\left({\mu }^3\right)\right]$$

$$3\gamma A_0^2{\varphi }_1(z_0,z)/T,\delta A_0^{2K}{\varphi }_K(z_0,z),\beta A_0^{2K-2}{\varphi }_{K-1}(z_0,z)/2,ϵA_0^{2K}T{\varphi }_K(z_0,z)\sqrt{\pi /2K}$$

$$\theta ={\theta }_0+\gamma {\theta }_{\gamma }+\delta {\theta }_{\delta }+\beta {\theta }_{\beta }+ϵ{\theta }_{ϵ}$$

$${\theta }_0(z,\tau )$$

$${\theta }_{\gamma }(z,\tau )$$

$${\theta }_{\delta }(z,\tau )$$

$$+\eta (2K-1){\Phi }_{K,K}(z_0,z){\xi }_{2K}(\tau )+2\eta {\Phi }_{K,K}(z_0,z)F^{2K}(\tau ){\xi }_K(\tau )$$

$${\theta }_{\beta }(z,\tau )$$

$${\theta }_{ϵ}(z,\tau )$$

$${\Phi }_{i,j}(z_0,z)={\int }_{z_0}^z{\left(\frac{w_0}{w(z^{\prime })}\right)}^{2i}{\varphi }_j(z^{\prime })d{z}^{\prime }$$

$$\Delta {\omega }_0=\frac{4\lambda A_0^2}{T}{\varphi }_1(z_0,z)\frac{\tau }{T}{e}^{-2{\tau }^2/T^2}+\eta A_0^{2K}{\varphi }_K(z_0,z)e^{-2K{\tau }^2/T^2}$$

$${\partial }_z^{2}w{|}_{w=w_0}\approx \frac{4}{k^2{w}_0^3}\left(1-\frac{P_0}{P_{cr}}\right)+\frac{2K}{(K+1{)}^2{w}_0}\frac{{\rho }_{peak}}{{\rho }_{cr}}$$

Full text

I Introduction

The propagation of intense short laser pulses in optically transparent media involves several linear and nonlinear effects which modulate the pulse’s amplitude and phase. Identifying these effects and their influence on the laser pulse characteristics is of great inteterst due to the real or potential use of such laser pulses in various fields of science and technology such as communications Knox (2000), medicine Juhasz et al. (1999), material science Gattass and Mazur (2008), sensors Mihailov et al. (2017), high-harmonic generation Krause et al. (1992), X-ray production Döpp et al. (2018), etc. In the past decades, this topic has also brought much attention from a theoretical point of view Couairon and Mysyrowicz (2007); Bergé et al. (2007); Couairon et al. (2011); Kolesik and Moloney (2014). While the numerical approach is the only universal means to deal with fully developed nonlinear effects, it provides only case-by-case solutions and it requires numerical schemes whose accuracy must be carefully controlled. It is often possible, however, to derive closed-form solutions of nonlinear problems in the perturbative regime, that is when the nonlinear effects are not fully developed. Such analytical solutions, besides providing quick answers and reference points for numerical calculations, though in restricted conditions, allow a better understanding of the specific influence of the various terms and effects included in the basic nonlinear equations. In addition, they can provide dimensionless scaling parameters which help to evaluate the magnitude of the nonlinear effects. In this work, we present such closed-form expressions for the problem of interest in the perturbative regime and we illustrate them on the particular case of a femtosecond laser pulse focused in ambient air.

II Model equation

The basic equation for the field amplitude in the envelope approximation has been comprehensively discussed by Bergé et al. Bergé et al. (2007). A simplified form of this equation reads Ward and Bergé (2003):

[TABLE]

where $z$ is the propagation axis coordinate, $\tau=t-z/v_{g}$ is the retarded time with $v_{g}$ the group velocity, $\mathfrak{T}=1+\frac{i}{\omega_{c}}\partial_{\tau}$, with $\omega_{c}$ the central frequency, and $\rho(\tau)=\alpha_{K}\rho_{m}\int_{-\infty}^{\tau}|A|^{2K}d\tau^{\prime}$ is the electron density generated by the laser pulse propagating in a medium where $\rho=0$ initially, $\rho_{m}$ is the particle density of the propagation medium, here considered to be much larger than $\rho$. The field amplitude $A(\mathbf{r},\tau)$ is normalized such that $|A|^{2}$ is the intensity (irradiance) in Wm*-2*. In order to allow analytical treatment, we approximate the ionization rate by a function of the form $W=\alpha_{K}|A|^{2K}$ around some value of laser intensity $|\tilde{A}|^{2}$, where $\alpha_{K}$ and $K$ are fitted to $W(|A|^{2})$ around $|\tilde{A}|^{2}$. For Keldysh parameters $\gamma_{K}>1$, $K$ is the number of photons $N_{p}$ required to overcome the ionization potential, otherwise $K$ is a real number such that $K<N_{p}$. This approximation overestimates the ionization rate for $|A|^{2}$ higher or lower than $|\tilde{A}|^{2}$ since $K$ is a decreasing function of $|A|^{2}$ Schwarz et al. (2012).

In Eq. (1), the meaning of the terms following the first one are: diffraction, dispersion, Kerr effect, field ionization, absorption due to field ionization, and collisional (inverse bremsstrahlung) absorption by free electrons, respectively. The constants have their usual meaning: $k_{0}$ is the wave number in vacuum, $n_{0}$ and $n_{2}$ are the linear and nonlinear indices of refraction, respectively, $\rho_{c}$ is the plasma critical density, $k^{\prime\prime}$ is the dispersion coefficient and $\sigma=e^{2}\nu_{e}/(m_{e}\varepsilon_{0}\omega_{c}^{2})$ in the limit $\nu_{e}/\omega_{c}\ll 1$ (with $e$ the electron charge, $m_{e}$ the electron rest mass, and $\varepsilon_{0}$ the vacuum permittivity) is the collisional absorption cross section, which depends on the electron collision frequency $\nu_{e}$. The operator $\nabla_{\perp}^{2}$ is the Laplacian in the transverse direction.

To keep the problem tractable analytically, we make the following simplifications. First, as dispersion introduces considerable complications in the perturbative solution, it will not be considered. In fact, dispersion can be neglected in weakly dispersive media such as gases for sufficiently short propagation distances, as exemplified in Section V. Second, we consider only the first order Taylor expansion $\mathfrak{T}^{-1}\approx 1-\frac{i}{\omega_{c}}\partial_{\tau}$. Third, in order to decouple the transverse coordinates from $z$ and $\tau$, we assume that the beam behaves as follows close to the radial position $r=0$ Max (1976),

[TABLE]

We assume that the beam waist $w(z)$ is a known function and we consider only the position $r=0$, so that the problem becomes one-dimensional in the axial coordinate $z$. Several simple approaches can be used to estimate $w(z)$ Couairon and Mysyrowicz (2007). Consistently with this approximation, we neglect the non-paraxial correction $-i\omega_{c}^{-1}\partial_{\tau}\nabla_{\perp}^{2}$ (space-time focusing) in Eq. (1). One notes, however, that in the special case where $w(z)$ is constant, the latter correction can be get rid of simply by redefining the retarded time as $\tau\leftarrow\tau-\frac{2z}{k_{0}\omega_{c}w^{2}}$.

With these simplifications, substituting Eq. (2) in Eq. (1) gives:

[TABLE]

where

[TABLE]

and

[TABLE]

We also use the estimate $\beta=\alpha_{K}\rho_{m}U_{I}$, where $U_{I}$ is the ionization potential.

III Energy balance

From Eq. (3), it is straightforward to show that

[TABLE]

where $w^{2}(z)\mathfrak{I}_{1}(z,\infty)$ is proportional to the pulse energy at the position $z$. Equation (6) indicates that the $\delta$ term of Eq. (3) induces an energy leak. The relative importance of the first two terms on the right hand side can be estimated by using the Gaussian form $|A|=A_{0}\exp(-\tau^{2}/T^{2})$, where $T$ is the pulse duration. Using the above definitions, one obtains:

[TABLE]

where $\bar{\Phi}_{p}$ is the ponderomotive potential averaged over a laser cycle. The $\delta$ term is thus related to the oscillation energy of the free electrons in the laser field and this explains the energy leak induced by this term. For $K>1$, the energy leak due to the $\delta$ term becomes comparable to that of the $\beta$ term when $\bar{\Phi}_{p}\approx U_{I}$. For the typical value $U_{I}\approx 10$ eV, this happens for the common parameters $A_{0}^{2}\approx 10^{18}$ Wm*-2* and $\lambda_{c}\approx 1$ $\mu$m, where $\lambda_{c}$ is the central wavelength.

It is worth noting that an alternative model to Eq. (1), involving the substitution

[TABLE]

has also been used Gaeta (2000); Zia (2018). With the latter substitution, Eq. (6), without the $\epsilon$ term now somehow included in the left hand side of (8), becomes

[TABLE]

The first term on the right hand side has a sign opposite to that in Eq. (6), implying that the $\delta$ term is now a source of energy. In that model, this source term can, however, be compensated by the negative second term for an appropriate choice of $\nu_{e}$. The third term has an undetermined sign due to the $\partial_{\tau}\theta$, as discussed below.

IV Perturbative solution

To obtain a perturbative solution of the wave equation (3) we set

[TABLE]

where $\psi$ and $\theta$ are real functions. Substituting Eq. (10) in Eq. (3) one obtains

[TABLE]

for the real and imaginary parts, respectively. Neglecting dispersion allowed decoupling $\psi$ from $\theta$ in Eq. (12). Note that dispersion was taken into account by Tzoar and Jain in their perturbative treatment of the propagation of laser pulses in optical fibres, but when retaining only the additionnal $\gamma$ term Tzoar and Jain (1981).

We seek a perturbative solution for $\psi$ by substituting the first order series expansion

[TABLE]

in Eq. (12). The resulting equation holds for arbitrary values of $\gamma$, $\delta$, $\beta$ and $\epsilon$ provided

[TABLE]

In these expressions, $w_{0}=\min(w(z))$ within the range of $z$ considered, and

[TABLE]

where $z_{0}$ is the coordinate $z$ where the initial condition is defined.

Assuming that the zero order temporal pulse shape is the Gaussian function

[TABLE]

one finds the simple closed-form expression for the pulse amplitude

[TABLE]

In the particular case where $w(z)=w_{0}$, and only the $\gamma$ term is retained, Anderson and Lisak found that the exact solution of Eq. (12) for the zero order temporal profile, Eq. (20), is the solution of the algebraic equation Anderson and Lisak (1983):

[TABLE]

The solution of Eq. (22) has a maximum of $\psi=A_{0}$ at $\tau=\mu T$, where $\mu=3\gamma(z-z_{0})A_{0}^{2}/T$ is the scaling parameter, but tends to $\psi_{0}$ for small values of $\psi$. It thus becomes very steep past the maximum when $\mu$ is on the order of 1 or greater Anderson and Lisak (1983). The series expansion of the solution of Eq. (22) up to the order $\mu^{2}$ is:

[TABLE]

Consistently with the exact solution, the first order term in $\mu$ in Eq. (23), which coincides with that of Eq. (21), shifts the peak to the back of the pulse ($\tau>0$). The $\mu^{2}$ correction lowers the amplitude around $\tau=0$ and shifts the peak somewhat further to the back.

From Eq. (21), one can identify the following dimensionless scaling parameters for the four perturbation terms

[TABLE]

The time dependence of the five terms of Eq. (21) are illustrated in Fig. 1 for $K=5$ with the scaling parameters (24) set equal to 1. As expected, the $\beta$ term (absorption due to optical field ionization) induces a symmetric depletion at the initial peak of the pulse where the ionization rate is maximum. The $\gamma$ term (Kerr effect) discussed above produces an antisymmetric contribution consistently with its general energy conserving property (although this property holds only to the first order in $\gamma$ in the current level of approximation). The $\epsilon$ term (collisional absorption) produces an off-centred asymmetric depletion as this effect is most efficient near the peak of the pulse and where the electron density is highest. More interestingly, the $\delta$ term (time derivative of optical field ionization) brings an asymmetric contribution, with a sharp depletion at $\tau=0$, and a shift of the peak toward the back of the pulse. With reference to the discussion of Section III, this asymmetric shape can be interpreted as an absorption of the pulse energy by the free electrons at the front and the center of the pulse, and a partial restitution of the electron energy at the back of the pulse as the amplitude decreases. In general, one thus expects a shift of the peak toward the back of the pulse and a steepening effect due to both the $\gamma$ and the $\delta$ terms. This effect is however hindered by the $\epsilon$ term, which tends to deplete the amplitude at the back of the pulse.

The phase $\theta(z,\tau)$ is obtained by substituting the first order series expansions

[TABLE]

and (13) in Eq. (11). One obtains

[TABLE]

where

[TABLE]

The full expression of $\theta(z,\tau)$ using Eq. (20) is quite lengthy and will not be developed here for that reason. One notes that the first term of $\theta_{0}(z,\tau)$ is the Gouy phase for the selected waist function $w(z)$.

More interesting than the phase itself is the instantaneous frequency shift $\Delta\omega(z,\tau)=\omega(z,\tau)-\omega_{c}=-\partial_{\tau}\theta(z,\tau)$ where $\omega(z,\tau)$ is the frequency generated by the nonlinear effects Boyd (2003). For the time profile (20), one finds for the component corresponding to $\theta_{0}$,

[TABLE]

In this expression, the first term produces a red shift at the front of the pulse due to the increase of the Kerr component of the index of refraction $n_{Kerr}=n_{0}+n_{2}|A|^{2}$ and a blue shift at the back due to the subsequent decrease of $n_{Kerr}$, resulting in the well-known linear chirp around $\tau=0$. The second term brings a narrower blue shift contribution around the center of the pulse due to the generated electrons, which decrease the index of refraction as $n_{elec}\approx 1-\rho/2\rho_{c}$. The effect of the remaining components of $\Delta\omega$ will be illustrated in the following example.

V Example: laser pulse focused in air

As an example, we consider a short laser pulse, with a duration $T=50$ fs (or a full width at half maximum of the intensity of $T\sqrt{2\ln{2}}=59$ fs) and a central wavelength $\lambda_{c}=800$ nm, focused in ambient air, where the molecule density is $\rho_{m}=2.7\times 10^{25}$ m*-3*, with $20\%$ O2, and $n_{0}=1$. The waist will be assumed of the form $w(z)=w_{0}\left(1+\left(z/z_{R}^{eff}\right)^{2}\right)^{1/2}$, where $w_{0}=50$ $\mu$m, and $z_{R}^{eff}$ is the effective Rayleigh length. We consider $A_{0}^{2}\leq 1.4\times 10^{18}$ Wm*-2* so that the peak intensity $\psi^{2}$ at focus $z=0$ is lower than that.

For these parameters, the energy of a Gaussian pulse is $E_{0}\lesssim 0.34$ mJ, and its power $P_{0}\approx A_{0}^{2}\frac{\pi}{2}w_{0}^{2}$ $\lesssim 5.5$ GW is below the critical power for self focusing $P_{cr}\approx\lambda^{2}/2\pi n_{2}\approx 12.7$ GW, where we used $n_{2}=0.8\times 10^{-23}$ m2 W*-1* Wahlstrand and Milchberg (2015). We estimate the importance of dispersion from the relative time broadening $\Delta T/T\sim 2\left(k^{\prime\prime}z/T^{2}\right)^{2}$ of the pulse. Using $k^{\prime\prime}=2\times 10^{-29}$ s2 m*-1* for ambient air Couairon and Mysyrowicz (2007), one finds that, for the propagation distance $z=1$ m, $\Delta T/T$ is on the order of $10^{-4}$, which is negligible as assumed above. Ionization of O2 is more effective than that of N2 since their ionization potentials $U_{I}$ are 12.1 eV and 15.6 eV, respectively. The Keldysh parameter $\gamma_{K}=\omega A_{0}\sqrt{U_{I}/c\varepsilon_{0}}\approx 10^{-4}$ indicates that ionization is in the tunnel regime. The coefficients $K$ and $\alpha_{K}$ of the ionization rate, fitted as $W=\alpha_{K}|\psi|^{2K}$ for O2, can be estimated from the PPT model as $K\approx 5$ and $\alpha_{K}\approx 5\times 10^{-79}$ m10 W*-5* s*-1* around $\psi^{2}=10^{18}$ Wm*-2* Schwarz et al. (2012) producing an ionization rate $W\approx 5\times 10^{11}$ s*-1* and an electron density near the peak of the pulse $\rho_{peak}\approx 4\times 10^{22}$ m*-3*, which is much smaller than the critical plasma density $\rho_{c}=1.7\times 10^{27}$ m*-3*. Considering few eV electrons and an electron-neutral geometrical cross section of $\sim 10^{-19}$ m2, one obtains the ratio $\nu_{e}/\omega_{c}\sim 10^{-3}$. Therefore, $\sigma\approx 4.5\times 10^{-24}$ m2.

We estimate the effective Rayleigh length as $z_{R}^{eff}\approx\left(w_{0}/\partial^{2}_{z}w|_{w=w_{0}}\right)^{1/2}$, where Schwarz et al. (2012)

[TABLE]

Using the above parameters, one finds for $\psi^{2}=10^{18}$ Wm*-2*, $z_{R}^{eff}\approx 1$ cm, which is nearly the same value as the Rayleigh lenght in vacuum $z_{R}=kw_{0}^{2}/2=0.98$ cm, which is the value used in the following. This result can be understood as a near cancellation of Kerr self-focusing and plasma defocusing since their scalelengths are $1/k_{0}n_{2}A_{0}^{2}\approx 1.3$ cm and $2\rho_{c}/k_{0}\rho_{peak}\approx 1.1$ cm, respectively Couairon and Mysyrowicz (2007).

We assume that, well before the focus ($z_{0}\ll-z_{R}$), $\mathfrak{A}(z_{0},\tau)=\psi_{0}(z_{0},\tau)$, where $\psi_{0}$ is given by Eqs. (14) and (20), and we are interested in the field envelope $\mathfrak{A}(z,\tau)$ well past the focus ($z\gg z_{R}$). All the nonlinear effects take place around the focus where the intensity $\psi^{2}$ is maximum.

For the selected function $w(z)$, the integrals $\phi_{i}$ and $\Phi_{i,j}$ evaluated between $-\infty$ and $\infty$ can be calculated exactly. They are on the order of $z_{R}$ and $z_{R}^{2}$, respectively. Using the above parameters, one finds that the dimensionless scaling parameters identified in Eq. (24) are, for$A_{0}^{2}=1.2\times 10^{18}$ Wm*-2*: $3\gamma A_{0}^{2}\phi_{1}/T=0.118$, $\delta A_{0}^{10}\phi_{5}=0.054$, $\beta A_{0}^{8}\phi_{4}/2=0.052$, and $\epsilon A_{0}^{10}\phi_{5}\sqrt{\pi/10}=0.018$. Since all these parameters (and in particular the sum of the $\delta$ and $\beta$ parameters) are much smaller than 1, the laser pulse can be considered in the perturbative regime. For $A_{0}^{2}=1.4\times 10^{18}$ Wm*-2*: $3\gamma A_{0}^{2}\phi_{1}/T=0.138$, $\delta A_{0}^{10}\phi_{5}=0.117$, $\beta A_{0}^{8}\phi_{4}/2=0.097$, and $\epsilon A_{0}^{10}\phi_{5}\sqrt{\pi/10}=0.038$. In that case the scaling parameters are larger and might depart from the perturbative regime.

The shape of the pulse intensity $\psi^{2}$ normalized as $\left(\frac{w(z)}{w_{0}}\frac{\psi}{A_{0}}\right)^{2}$ is illustrated in Fig. 2 for the zeroth order $\psi=\psi_{0}$ ($z\ll-z_{R}$) and for the full perturbative solution at $z\gg z_{R}$ for $A_{0}^{2}=1.2\times 10^{18}$ Wm*-2* and $1.4\times 10^{18}$ Wm*-2*. It is worth stressing that the results depend on $w(z)$ only through the integrals $\phi_{i}$ and $\Phi_{i,j}$ evaluated between $-\infty$ and $\infty$. According to Fig. 1 and to the scaling parameters evaluated above, the $\gamma$, $\delta$, $\beta$, and $\epsilon$ terms have comparable magnitudes but act at different times. Consistently with Fig. 1, the perturbative solution, as compared to the zeroth order, is depleted at the front and at the centre of the pulse due to electron generation and their energy absorption, and enhanced at the back mostly due to the Kerr effect ($\gamma$ term). As $A_{0}^{2}$ is increased, the shoulder around $\tau=0$ becomes more and more pronounced and, for $A_{0}^{2}\gtrsim 1.4\times 10^{18}$ Wm*-2*, forms a hole in the intensity profile. The intensity profiles obtained share much resemblance with those of the numerically calculated light bullets propagating in dielectric materials Zia (2018); Smetanina et al. (2013), where the Kerr effect is however much larger than in air. In the present case, the normalized pulse shape stops changing due to the divergence of the laser pulse past the focus.

The relative frequency shift $\Delta\omega/\omega_{c}$ is shown in Fig. 3 as well as the zeroth order solution $\Delta\omega_{0}/\omega_{c}$, given by Eq. (32), for $z\gg z_{R}$. For $A_{0}^{2}=1.2\times 10^{18}$ Wm*-2* the full frequency shift is clearly dominated by the zeroth order contribution for all values of $\tau$. The perturbative terms $\delta$, $\beta$, and $\epsilon$ flatten the frequency shift around $\tau=0$. When $A_{0}^{2}\gtrsim 1.4\times 10^{18}$ Wm*-2*, $\Delta\omega/\omega_{c}$ shows complicated oscillations due to the sharp negative contributions of the perturbative terms near $\tau=0$. The total frequency shift is nevertheless always dominated by the $\lambda$ term (Kerr effect) in $\Delta\omega_{0}$ at the front (red shift) and at the back (blue shift) of the pulse due to the large broadeness of $F^{2}(\tau)$. Note that, for $A_{0}^{2}=1.2\times 10^{18}$ Wm*-2*, the dimensionless scaling parameters of the zeroth order phase component $\theta_{0}$ are $\lambda A_{0}^{2}\ \phi_{1}=2.33$ and $\eta A_{0}^{10}\phi_{5}T\sqrt{\pi/10}=3.56$, and thus cannot be considered as perturbative components.

Finally, we calculate the intensity spectrum of the pulse, here normalized as

[TABLE]

since it is a directly measurable quantity. Figure 4 shows the intensity spectra as a function of wavelength for $z\gg z_{R}$ as well as the zeroth order spectrum ($z\ll-z_{R}$). The full perturbative spectra are strongly broadened and depleted as compared to the zeroth order spectrum. Consistently with the discussion around Fig. 3, the spectrum for $A_{0}^{2}=1.2\times 10^{18}$ Wm*-2* undergoes mostly a blue shift and a lower peak appears on the red side. The peak shift obtained is $-10$ nm from the initial value $\lambda_{c}=800$ nm. One notes that a similar double-humped spectrum for a tightly focused laser pulse in nitrogen was recently measured in a regime which is however not clearly perturbative Clerici et al. (2019). When $A_{0}^{2}$ is increased, the two peaks become more and more comparable in height and the shorter wavelength peak broadens and shifts further to the blue side.

VI Summary

We have employed a perturbative approach to derive closed-form expressions for the solution of the wave equation discussed by Bergé et al. Bergé et al. (2007), describing the envelope amplitude and phase of a laser pulse propagating in optically transparent media. The the waist $w(z)$ of the laser pulse was assumed to be known so that the problem becomes one-dimensional in the axial coordinate $z$. The perturbative approach employed in this work allowed to display the contribution of the various terms of the basic model equation to the temporal shape of the pulse in the limit where the nonlinear effects are modest. The most interesting term is likely $\partial_{\tau}\left(\rho A\right)$ in Eq. (3) (the $\delta$ term), associated with field ionization, which was shown to produce a depletion of the initial peak amplitude and its displacement toward the back of the pulse. This term was also shown to produce an overall energy transfer toward the free electrons oscillating in the laser field. Moreover, dimensionless parameters enabling to size the nonlinear effects have been extracted from the perturbative solution. The solution obtained was illustrated on the specific example of a short laser pulse focused in air using common laser parameters. Experiments prepared in appropriate perturbative conditions could be performed to verify some conclusions of this work.