## Abstract

We develop a one-dimensional model of THz emissions induced by laser-driven, time-asymmetric ionization and current oscillations in a hydrogen gas. Our model highlights complex scalings of the THz fields with respect to the laser and gas parameters, in particular, a non-monotonic behavior against the laser parameters. Analytical expressions of the transmitted and reflected fields are presented, explaining the THz spectra observed in particle-in-cell and forward-pulse propagation codes. The backward-propagating THz wave is mainly driven by the electron current oscillations at the plasma frequency, and its resulting spectrum operates below the plasma frequency. The transmitted THz wave is emitted from both plasma current oscillations and photo-ionization. Their respective signal presents a contribution below and around the plasma frequency, plus a contribution at higher frequencies associated to the photo-induced current. The interplay between these two mechanisms relies on the ratio between the propagation length and the plasma skin depth.

© 2014 Optical Society of America

## 1. Introduction

Laser-gas interaction is an elegant way of energy conversion to higher frequencies through up-conversion mechanisms [1], as well as to lower frequencies through down-conversion processes [2]. The reasonable size and high-repetition rates of the latest laser-based frequency converters make them highly promising for next-generation THz pulse devices [3, 4]. Spanning from the infrared to microwaves, the THz radiation spectrum is attractive for many applications in various fields of physics [5], biology and medicine [6], industry [7], security, communication, remote sensing [8] and basic science with, for instance, molecular dynamic spectroscopy [9, 10].

Initially attributed to optical rectification and four-wave mixing via a third-order nonlinearity [11, 12], THz emission has been widely discussed in the optics and plasma literature [13–16]. Recently, a consensus has been reached on the so-called two-color scheme, which allows to noticeably increase, by around two orders of magnitude, the THz conversion efficiency in laser-gas interaction above the ionization threshold [17]. The principle of this scheme is to combine a fundamental frequency carried by an ultrashort (femtosecond) pulse with its second harmonic within a highly nonlinear plasma spot. As a result of the subtraction of broad optical bandwidths, new electromagnetic components are generated with spectra centered around THz frequencies. Remarkably, similar THz emissions over large distances have also been reported from long-range filamentation of femtosecond pulses in air or in noble gases (e.g., argon) [18–22]. Recent works also pinpoint local plasma-induced photocurrents as key players for THz generation in the filamentation regime [23, 24]. THz pulses are then emitted during and after the gas ionization. Firstly, bursts of attosecond currents are produced around each ionization event. For a specific dephasing between the two laser beams, these bursts interfere constructively to generate THz pulses of the order of the ionization duration [25]. Moreover, transverse current oscillations are driven at the plasma frequency, which prolongs the THz emission after the laser interaction [26]. The resulting THz field reaches amplitudes of the order of several MV/cm, much higher than those produced by four-wave mixing and comparable to those usually attained in tightly-focused geometries. In two-color filamentation experiments, the pulse intensity is currently limited to 50 – 100TW/cm^{2}. Under these conditions, the THz yield appears to increase with respect to the pump energy, up to some saturation [18, 19]. THz yields up to 100 *μ*J were recently reported for pump energies at the Joule level [27]. The enhanced THz conversion efficiency (∼ 10^{−4}) was attributed to meter-range filaments that maintain long plasma channels.

Out of a filamentation scenario, Particle-In-Cell (PIC) simulations [26, 28], solving Vlasov-Maxwell equations by means of macro-particles, have shown that rising the laser intensity beyond the previous values does not necessarily increase the THz amplitude. While essentially driven by ionization currents, the THz signal also depends on the noble gas serving as the interaction medium and on the pulse duration and carrier-envelope phase, which impose a saturation value for the ionization current. For instance, the large ionization potential in helium was exploited to obtain THz field amplitudes as high as 1.28 GV/m with moderate laser energies. A careful choice of the plasma and laser parameters is thus required to achieve efficient THz production. When computed inside a plasma of constant density, the radiated field is dominated by ionization currents [28]. By contrast, the contribution of the plasma current oscillations seems to prevail outside the plasma region and/or at high pump intensities [29, 30].

In this regard, the THz spectral signature attributed to photocurrents may be different from that produced by plasma current oscillations along the propagation direction. While the former operate over a broad range of low frequencies [16, 31], the latter more specifically operate around and below the plasma frequency [30], that spans the THz region for electron densities < 10^{19} cm^{−3}. Thus, a clear distinction between the potential sources of THz radiation involved in the dynamics of the free electrons still remains to be drawn. Another open issue concerns the THz energy partition between the transmitted and reflected waves. Several studies have assumed that the largest fraction of the THz energy in the photocurrent model is carried by the forward wave component [23–25,31]. Straightforward comparisons between backward and forward spectra computed from a Unidirectional Pulse Propagation Equation (UPPE) model and two-dimensional (2D) Finite-Difference Time-Domain (FD-TD) Maxwell code have validated the unidirectional approach for an argon gas [32]. For pump intensities less than 100 TW/cm^{2}, 2D FD-TD Maxwell simulations predict mostly forward THz emission. This forward radiation is generated predominantly at the ionization front and is not affected by plasma opacity, unlike the backward spectrum. However, when a single laser pulse of much higher intensity (up to 10^{17} W/cm^{2}) interacts with a 100 *μ*m-thick hydrogen plasma, a noticeable fraction of the THz energy, produced by the plasma current oscillations, can be conveyed by the backward wave [30]. In this regime, alternative key agents for THz generation may be plasma wakefields and the laser wavelength. In this respect, several studies predict strongly enhanced THz yields in the case of single- or two-color pulses operating at long central wavelengths (e.g., 2*μ*m or 4*μ*m) [23, 26]. Here, several effects superimpose, such as the respective influences of the pump laser wavelength and the effective pulse duration, as well as the interference between the low-frequency tail of the pump spectrum with the inner THz spectrum at long enough pump wavelengths.

All of the previously published models of laser-driven THz sources have failed to clarify the connection between the two main “plasma” mechanisms, i.e., ionization-induced photocurrents and current oscillations, particularly the link between forward and backward waves, and their dependencies upon the laser and plasma parameters. The goal of the present work is thus to propose a unified framework accounting both for photocurrents and current oscillations, in order to extend the existing models to laser intensities ≲ 10^{17} W/cm^{2}. To this purpose, we shall revisit the well-known photocurrent mechanism [17] by separating the THz component from the laser pump field and by including the plasma current oscillation in a self-consistent way. For the sake of simplicity, we shall introduce two semi-analytical 1D models, either discarding or retaining propagation effects, and restrict our analysis to a hydrogen gas. Section 2 introduces our analytical model, which assumes decoupled THz and laser fields. In section 3, we derive scalings of the THz emission against the laser and plasma parameters within the assumption of negligible propapagation effects. Despite this approximation, our scalings perfectly reproduce the tendencies observed in PIC simulations [26]. The choice of optimal parameters for THz emissions is demonstrated in the case of one and two laser colors. In section 4, propagation effects are included. Analytical formulas for the reflected and transmitted THz waves are derived. We demonstrate that the forward wave carries most of the THz energy, provided that the plasma length exceeds the plasma skin depth. Besides, the dependence of the THz amplitude upon the photocurrent amplitude is clearly evidenced and its limitations are discussed. The competition between photocurrents and plasma current oscillations is analyzed, and shown to depend on the ratio between the propagation length inside the gas and the plasma skin depth. In section 5, the backward and forward spectra inferred from our general model are found in remarkable agreement with both PIC and UPPE simulation results.

## 2. 1D model for laser-driven THz pulses

In this section, we present a simple semi-analytical model predicting the THz emission in gases for a large set of parameters. Drawing upon Refs. [30, 31], it involves two mechanisms, both attributed to the transverse electron current oscillations induced during and after gas ionization by one- or two-color laser pulses. For the laser intensity range scanned in the present article (∼ 10^{14} − 10^{17} W/cm^{2}), both the nonlinear polarization (bound electron response) of the medium and the relativistic effects are neglected. Instead, we focus on the interplay of laser-driven photoionization [4, 17, 31] and transverse current oscillations [26, 30].

In the following, the time and space coordinates, (*t*, *x*), are normalized to 1/*ω*_{0} and *λ*_{0}/2*π*, respectively, where *ω*_{0} is the pump laser frequency and *λ*_{0} is the pump central wavelength. Let us consider a semi-infinite gas of neutral density *n _{a}*, filling the positive

*x*-region and a transverse electric field, ${E}_{y}^{L}$, composed of one- or two-color laser pulses propagating toward the gas. The gas density and the laser electric field are normalized to the critical density ${n}_{c}={m}_{e}{\omega}_{0}^{2}{\epsilon}_{0}/{e}^{2}$ and the Compton field

*m*

_{e}ω_{0}

*c/e*, respectively. For low gas densities,

*n*≪ 1, we can neglect the laser dispersion and energy loss during the ionization processes. We hence assume the laser electric field, generally embedding two colors, to maintain a constant shape as

_{a}*H*is the Heaviside function,

*β*

^{+}=

*x*−

*t*,

*τ*is the number of laser cycles in the pulse,

*a*

_{0}is the laser potential vector normalized to

*m*

_{e}c^{2}/

*e*(the physical laser intensity is evaluated as ${I}_{0}=1.38{a}_{0}^{2}{\lambda}_{0}^{-2}[\mu \text{m}]$),

*r*is the ratio of the second harmonic amplitude to the fundamental, and

*ϕ*and

*ϕ*

_{2}are the phase shifts of the

*ω*

_{0}and 2

*ω*

_{0}components, respectively. Various experimental setups are nowadays available and they allow to fix rather freely the second harmonic duration and its relative delay compared to the fundamental pulse length [2, 12, 27]. Therefore, for technical convenience, we shall use equal pulse durations as in Refs. [28, 30] in our theoretical analysis. Second harmonic pulse lengths being twice smaller than the pump duration will instead be employed in section 5, when confronting theoretical predictions to direct numerical simulations. Initially, the finite laser pulse is located in front of the gas, between

*x*= −2

*πτ*and 0. Writing the full electric field as we propose to calculate the vector potential

*δa*, and the induced electric field, −

_{y}*∂*, containing the THz signal. Given the weak fields considered, the ions can be assumed motionless. In the non-relativistic limit (

_{t}δa_{y}*∂*≫

_{t}*v*), the 1D cold-fluid model reduces to the electron continuity equation, the transverse electron current equation and the wave equation: where

_{x}∂_{x}*J*is the transverse electron current,

_{y}*ν*is the total (electron-ion plus electron-atom) collision rate,

*n*is the electron density, and

_{e}*ν*[ ${E}_{y}^{L}$] is the normalized Ammosov-Delone-Krainov ionization rate of an atom by an oscillating field in linear polarization [33–35]. For hydrogen, the latter is defined as

_{E}*ω*=

_{a}*eE*is the characteristic atomic frequency, ${E}_{a.u.}={e}^{2}/4\pi {\epsilon}_{0}{a}_{B}^{2}{m}_{e}{\omega}_{0}c$ is the atomic field normalized to the Compton field, and

_{a.u.}a_{B}/h̄*a*= 4

_{B}*πε*

_{0}

*h̄*

^{2}

*/m*

_{e}e^{2}is the Bohr radius. Equations (4) and (5) recast as

## 3. Non-propagating THz pulses

In this section, we wish to obtain scaling laws for the THz emissions in terms of the plasma and laser parameters. For simplicity, we first neglect the wave propagation ( ${\partial}_{x}^{2}\delta {a}_{y}\to 0$), which reduces the above equation to an ordinary differential equation:

*δE*= −

_{y}*∂*. The assumption

_{t}δa_{y}*∂*≪

_{x}*∂*can be justified in some regions of the (

_{t}*x*,

*t*) plane (see next section). The laser field is here a function of time: ${E}_{y}^{L}(x-t)\equiv {E}_{y}^{L}(-t)$.

Equations (3) and (8), with the initial condition

can be readily solved numerically. A solution*δE*(

_{y}*t*) (black line) is presented in Fig. 1(a) for a single-color laser pulse (

*a*

_{0}= 0.02,

*τ*= 10,

*λ*

_{0}= 1

*μ*m) and negligible collisions (

*ν*= 0). During the ionization process (

*t*< 2

*πτ*), the newborn electrons oscillate in the laser pulse, producing the field at frequency

*ω*

_{0}. In contrast to Refs. [4, 17, 31], the remaining current oscillates at the plasma frequency ${\omega}_{\text{pe}}=\sqrt{{n}_{e}}$ (normalized to

*ω*

_{0}), as described in [30] (note the absence of damping mainly due to the absence of propagation). This is illustrated in the field spectrum (black line) displayed in Fig. 1(b). For the selected gas density,

*n*= 0.0011, complete ionization takes place, so that the main mode emerges at the plasma frequency

_{a}*ω*

_{pe}≃ 0.033. We have applied a low-frequency filter,

*ω*< 0.3, to calculate the THz field amplitude. As an example, the filtered field and its spectrum are plotted in red dotted line in Fig. 1(a) and (b), respectively. This methodology has been used to explore the dependencies of the THz field maximum upon the laser and/or gas parameters.

The maximum of the THz field normalized to
$\sqrt{{n}_{a}}{a}_{0}$, versus the laser intensity and for different set of parameters is presented in Fig. 2. On panel (a), the THz field maximum is plotted for different laser wavelengths, keeping constant the number of laser cycles in the pulse. For small laser intensities, the gas is weakly ionized, yielding negligible THz emission. When the laser field is strong enough to fully ionize the gas (*I* > 2×10^{14} Wcm^{2}), the THz field increases. At very high intensities, ionization occurs at the very beginning of the pulse and thus, due to weakened photocurrents, the THz field amplitude drops. At leading order, the normalized THz amplitude appears weakly sensitive to the laser wavelength, obeying the scaling

*n*

_{gas}is the gas density. As already reported in Ref. [26], the THz field amplitude, proportional to

*a*

_{0}, “oscillates” when increasing the laser intensity. A similar plot is presented in Fig. 2(b) for other parameters. The black solid line corresponds to a reference case (

*λ*

_{0}= 1

*μm*,

*τ*= 10,

*n*= 0.0011). For a fixed number of laser cycles, the normalized THz field is weakly changed when increasing the gas density by a factor 4 (red dashed line) or when doubling the laser frequency (dashed-dotted blue line). This trend confirms Eq. (10).

_{a}In addition, the THz field amplitude is substantially reduced when increasing the number of laser cycles in the pulse, as observed for *τ* = 40 (green solid line). This suggests (at least) a linear decrease in the THz amplitude with the number of optical cycles *τ*. To further illustrate this dependency, the THz field amplitude is plotted for the case *λ*_{0} = 2*μm*, *τ* = 5, and *n _{a}* = 0.0044. This situation corresponds to doubling the laser wavelength, while keeping constant the gas density and the laser pulse duration. The resulting pattern (cyan solid curve) does differ from the reference case (black solid line). The number of oscillations noticeably decreases and their amplitude grows from smaller laser intensities, whereas the peak values of
$\delta {E}_{y}/{a}_{0}\sqrt{{n}_{a}}$ is multiplied by about a factor 2. This result suggests a strong sensitivity of the THz amplitude to the effective number of optical cycles, more pronounced in the case of few-cycle pulses owing to strengthened time-asymmetry and net ionization currents, in agreement with Ref. [26]. Therefore, when considering a single color, we can propose from the previous behaviors a scaling of the THz amplitude as

Apart from this leading-order scaling, the oscillations seen in Fig. 2 introduce a margin of uncertainty of 2 to 4 when doubling the central laser wavelength. This additional refinement completes the above scaling by a factor > 2, which then renders it comparable to the variations reported in Ref. [26]. Here, the THz field amplitudes were seen to drop by a factor ∼10 when passing from 2 *μ*m to 1 *μ*m wavelength using a pump pulse of constant (50 fs) duration and intensity. Similar variations are also compatible with the 14-fold growth in the THz yield reported in Ref. [23] when comparing two-color filaments with 0.8*μ*m and 2*μ*m pumps of same duration but for different atomic densities (local pressure = 6.44 bars resulting in a net scaling rate of about 15.9).

We have also examined the THz field behavior with the phase shifts *ϕ* and *ϕ*_{2}. For one laser color (Fig. 3), the THz field amplitude (black solid line) is obviously *π*-periodic as
${E}_{y}^{L}(t,\varphi +\pi )=-{E}_{y}^{L}(t,\varphi )$. However, a more complex behavior arises when using two colors (red dashed line). The THz field amplitude is now 2*π*-periodic, with a maximum value reached for *ϕ*_{2} ≈ 4.1 rad in hydrogen with *r*^{2} = 0.2. Importantly, the THz field amplitude is strongly enhanced when using two colors, as illustrated in Fig. 4(a), where, typically, an order of magnitude increase in intensity is found compared to Fig. 2(a). The optimum THz intensity is obtained for the laser intensity *I*_{0} ∼ 3 × 10^{14} W/cm^{−2} for *τ* = 10 and *ϕ*_{2} = 0. By comparison with Fig. 2(b), the THz intensity attained with one color at similar pump intensity is at least 100 times weaker, thereby confirming the two orders of magnitude often mentioned in gases [18–20]. Note that the two-color configuration also gives rise to an oscillatory intensity dependence of the THz field. This dependence is sensitive to the phase *ϕ*_{2}, as illustrated by the red dashed line (*ϕ*_{2} = 0.5) and the cyan solid line (*ϕ*_{2} = 1.5). By increasing the number of optical cycles to 40, the oscillations relax (see the dashed-dotted green line), while the optimum laser intensity is shifted. Although a scaling in 1/*τ* may not apply, the THz field still remains sensitive to the number of laser cycles. The interplay between the harmonic amplitude ratio *r* and the phase *ϕ*_{2} is shown in Fig. 4(b). Here again, the THz field amplitude exhibits non-monotonic variations with *r*. The maximum THz field is obtained for *r* ≃ 0.85 and *ϕ*_{2} ≃ 4.7.

The non-monotonic behavior of the THz field and the previous scalings can be explained with a qualitative analysis of Eqs. (3) and (8). In the short-pulse limit, $2\pi \tau \ll 1/\sqrt{{n}_{a}}$, the solution of Eq. (8) is approximated as

*n*

_{ef}is the final electron density after gas ionization. For weak collisions ( $\nu \ll 2\sqrt{{n}_{\text{ef}}}$), it is then straightforward that the THz field oscillates at plasma frequency and scales as $\sqrt{{n}_{a}}{a}_{0}$.

The oscillatory dependence of the THz field upon the laser intensity, and its sensitivity to the laser duration then follows from estimating the source term

According to [31], the electron density evolution can be calculated knowing that the ionization events are localized around the electric field extrema. The electron density is then approximated as where*N*is the number of ionization events occurring at time

_{f}*t*and

_{i}*δn*is the increase in the electron density following each ionization event. Around an extrema

_{e,i}*E*, the laser electric field expresses as ${E}_{y}^{L}\approx {E}_{m,i}\left[1-\frac{{g}_{i}}{2}{(t-{t}_{i})}^{2}\right]$, with ${g}_{i}=-\frac{1}{{E}_{m,i}}{\frac{{d}^{2}{E}_{y}^{L}}{d{t}^{2}}|}_{{t}_{i}}>0$. Using this expression in Eq. (3), we obtain the recurrence relation

_{m,i}*i*> 1 and

*δn*

_{e,0}= 0.

As the laser electric field derives from a potential ${E}_{y}^{L}=-{\partial}_{t}{A}_{y}^{L}$ with a mean value equal to 0 [ $\int {E}_{y}^{L}dt={A}_{y}^{L}(0)-{A}_{y}^{L}(+\infty )=0$], the THz source term becomes

where ${A}_{y,i}^{L}={A}_{y}^{L}({t}_{i})$. The scaling of the source term*G*depends on the time-symmetry of the ionization events.

*For one laser color*, Eq. (1) recasts as
${E}_{y}^{L}(t)=-{a}_{0}g(t/\tau )\text{sin}(t-\varphi )$. As *τ* ≫ 1, the laser envelope derivative is small *g′/g* ∼ 1/*τ* ≪ 1, and the field extrema are reached at
${t}_{i}=\frac{\pi}{2}+\varphi +k\pi $, *k* ∈ **N**. Since *g*(0) = 0, the vector potential is
${A}_{y,i}\approx -{a}_{0}{\int}_{0}^{{t}_{i}}{g}^{\prime}(u)\text{cos}[u-\varphi ]\text{d}u\propto {a}_{0}/\tau $, and the THz source term is thus proportional to *G* ∝ *n _{a}a*

_{0}/

*τ*, thereby yielding Eq. (11). For one laser color, the higher the number of laser cycles, the weaker the THz field amplitude. The THz amplitude behavior against laser intensity can be explained as follows. For small laser intensities, such as

*ν*[

_{E}*E*] ≪ 1, ionization occurs around the laser pulse maximum. As the laser field is anti-symmetric around the maximum, the source term is exactly equal to 0. At higher intensities, there exists a time,

_{m,k}*t*<

_{k}*πτ*, such that

*ν*[

_{E}*E*] > 1. In this case, ionization occurs stepwise at several field extrema.

_{m,k}For simplicity, let us assume a two-step ionization:
$G=-\delta {n}_{e,k-1}{A}_{y,k-1}^{L}-\delta {n}_{e,k}{A}_{y,k}$. We look for the periodic zeroes of the THz source. The THz field vanishes if *G* = 0. As *δn*_{e,k−1} + *δn _{e,k}* =

*n*and $\delta {n}_{e,k}=\sqrt{\pi {g}_{k}/3}{\nu}_{E}[{E}_{m,k}]{n}_{a}$, one finds the relation

_{a}*E*| =

_{m,k}*a*

_{0}

*g*[(

*π*/2 +

*kπ*+

*ϕ*)/

*τ*], we obtain

_{−1}(

*x*) is the Lambert function W [36] and

*k*∈

**N**, there exists a laser amplitude,

*a*

_{0,k}, such that the above relation is fulfilled. As

*W*

_{−1}(

*K*) ∼ log(

*K*), the condition (18) is weakly sensitive to the laser wavelength, consistently with Fig. 2(a), but depends on

*g*

^{−1}( $1/\sqrt{I}$) (we remind that

*E*∝

_{a.u.}*λ*

_{0}). This variation with the inverse function of the laser envelope for, e.g., a Gaussian or sin

^{2}profile, accounts for the increase of the oscillation wavelength with the laser intensity seen in Fig. 2. For stronger laser intensities, the ionization events shift from the extremum

*k*to the extremum

*k*− 1 until reaching the first extremum (

*k*= 1). Hence, the number of oscillations is proportional to the number of extrema included in half of the laser duration, as observed in Fig. 2(b) (green solid line) where the number of oscillations has increased by a factor ∼ 4.

When the ionization process is asymmetric, as in the *two-color configuration*, the scaling of the source term *G* changes. Assuming *ϕ* = 0, one has

*t*solutions of cos

_{i}*t*+ 2

_{i}*r*cos[2

*t*−

_{i}*ϕ*

_{2}] = 0. Using the previous relation, the potential vector recasts as

*A*≈ −

_{y,i}*a*

_{0}

*g*(

*t*)3cos

_{i}*t*/4 ∝

_{i}*a*

_{0}. The THz emission reaches its maximum when the gas is fully ionized during the whole laser duration. In Fig. 4(a), this event corresponds to

*I*∼ 5 × 10

^{14}W/cm

^{2}for

*τ*= 10. Maximum THz emission varies with the number of laser cycles: for

*τ*= 40, it is shifted down to the value

*I*∼ 2 × 10

^{14}W/cm

^{2}. Reversely, at fixed intensity (e.g.,

*I*= 10

^{14}W/cm

^{2}) and wavelength, a 4-fold increase in the pulse duration raises the THz intensity by a minimum factor of ∼36. Up to propagation and multidimensional aspects discarded here, such impressive growths are comparable with those reported in Refs. [18, 23].

In conclusion to this section, our simplified 1D model predicts that the ionization-induced THz emission is utterly defined by the electron current amplitude left after the ionization process, *J _{y}* ∼

*G*. As already known, the stronger the ionization asymmetry, the higher the THz field amplitude. However, the origin of the asymmetry and its influence change between the one- and two-color configurations. Besides, as we neglect the THz field propagation, the THz spectrum is mainly located around the plasma frequency. This anomalous result is corrected in the next section, where propagation aspects are taken into account.

## 4. Propagating THz pulses

This section is devoted to the forward and backward THz emissions, and related modifications of the spectrum. For that purpose, we analytically and numerically solve the interaction of a laser pulse with a semi-infinite gas assuming that the laser pulse remains unperturbed while propagating.

The laser pulse defined by Eq. (1), initially located in the vacuum region, *x* ∈ [−2*πτ*, 0], interacts with a semi-infinite gas of density profile *n _{a}*(

*x*) =

*n*[

_{a}H*x*]. The THz field equation is given in Eq. (7), with the electron density

*n*described by the continuity equation, Eq. (3). The latter equation is numerically integrated using a 4

_{e}^{th}-order Runge Kutta method, while the THz wave equation is solved using a 2

^{nd}-order explicit centered scheme. A solution is presented in Fig. 5 in black solid line in the panels (a) and (b) for a single color. The THz field presents a small reflected wave, visible in panel (b) at the distance

*x*∼ −900, with an amplitude ∼ 10

^{−4}for the chosen parameters, and an intense forward wave of amplitude ∼ 10

^{−2}centered around

*x*= 900 [panel (a)]. The reflected wave is similar to that observed in PIC simulations [26, 30]. Basically, the laser field ionizes the gas, hence producing a net transverse current. As discussed in section 3, this current oscillates at the plasma frequency, which maintains the THz emission during several plasma periods. Inside the gas, the laser propagates and pursues the ionization process. The resulting field

*δE*propagates also forward and increases quasi-linearly with the distance.

_{y}For a more quantitative description, we now derive an approximate formula for the propagating field *δE _{y}*. Without collisions (

*ν*→ 0), the wave equation, Eq. (7), recasts as:

*β*

^{+}=

*x*−

*t*only, the solution of Eq. (3) expresses as

*s*=

*t*and

*β*

^{+}=

*x*−

*t*, which we find convenient to clearly separate backwarded and forwarded THz components, the wave equation becomes

*the beam head*,

*β*

^{+}∈ [−2

*πτ*, 0], we neglect the THz field compared to the laser field $\delta {E}_{y}\ll {E}_{y}^{L}$. This approximation is valid for pulse duration and time such that $(2\pi \tau ,s)\ll 1/\sqrt{{n}_{a}}$, i.e., such that the THz dispersion is negligible during the ionization process and the propagation distance. In the second region,

*behind the laser pulse*, the electron plasma density is constant and Eq. (22) is solved using the Laplace transform.

In the beam head, the wave equation is approximated as

*x*> 0 and

*x*−

*t*> −2

*πτ*, with $F(x-t)={\int}_{0}^{x-t}G({\beta}^{+\prime})d{\beta}^{+\prime}$ and $G({\beta}^{+})={\int}_{0}^{x-t}{n}_{e}{E}_{y}^{L}d{\beta}^{+\prime}$. In Eq. (24),

*F*is the current integral while

*G*is related to the current density. As

*F*∼ 2

*πτG*, the second quantity, affected by a linear growth factor in

*x*, prevails in the forward component for

*x*≫ 2

*πτ*.

Behind the laser pulse ( ${E}_{y}^{L}=0$), Eq. (22) reduces to

*β*

^{+}+

*s*=

*x*> 0. As the laser is fully inside the gas at time

*t*= 2

*πτ*, we introduce the Laplace transform $\widehat{\delta f}={\int}_{2\pi \tau}^{\infty}f(s){e}^{-ps}ds$. Using the initial conditions

*δE*(

_{y}*s*= 2

*πτ*,

*β*

^{+}) =

*∂*(

_{s}δE_{y}*s*= 2

*πτ*,

*β*

^{+}) = 0, Eq. (25) becomes

*β*

^{+}= −2

*πτ*. In the Laplace space, formula (24) becomes $\widehat{\delta {E}_{y}}\left({\beta}^{+}=-2\pi \tau \right)=\frac{1}{2{p}^{2}}G-(\pi \tau G+F/4)\frac{1}{p}$. The solution is thus

*t*> 2

*πτ*and 0 <

*x*<

*t*− 2

*πτ*.

Expression (28) reproduces the main functional dependencies derived earlier in Ref. [28], but without the *F*-function. Here, we go one step beyond. Knowing the solution *δE _{y}*(

*x*,

*t*) inside the plasma, the reflected wave $\delta {E}_{y}^{\text{ref}}$, satisfying the equation ${\partial}_{x}\delta {E}_{y}^{\text{ref}}-{\partial}_{t}\delta {E}_{y}^{\text{ref}}=0$, is deduced following the characteristic

*x*+

*t*=

*t*

_{0}, i.e., $\delta {E}_{y}(x=0,{t}_{0})=\delta {E}_{y}(0,x+t)=\delta {E}_{y}^{\text{ref}}(x,t)$, where

*δE*(

_{y}*x*,

*t*) is given by Eqs. (24) and (28). For 0 <

*x*+

*t*< 2

*πτ*and

*x*< 0, the reflected wave is

*x*< 0 and

*x*+

*t*> 2

*πτ*, we find

Comparing this expression with Ref. [37], we can infer that Eq. (30) describes the backward propagation of a distortionless wave oscillating at the plasma frequency with an electron density fixed by *n*_{ef}. Owing to the bi-directional character of the 1D propagator, the THz backward emission solely follows from the current oscillations over the plasma skin depth. At large time *t*, the backscattered solution behaves as
$\propto 1/\sqrt{t}\text{sin}(\sqrt{{n}_{\text{ef}}}t)$; its spectrum should convey a dominant mode at plasma frequency accompanied by damped, thus smaller frequencies. In the limit
$2\pi \tau \sqrt{{n}_{a}}\ll 1$, the first right-hand side term of Eq. (30) is dominant, hence confirming the scaling
$\delta {E}_{y}~G/\sqrt{{n}_{\text{ef}}}$ suggested in section 3. Moreover, at late times *t* ≫ *x* + 2*πτ* and near the plasma border for *x* > 0, the field simplifies to

*G*and increases linearly with the distance. As we neglect the field dispersion in the beam head, this linear increase is valid for $x\ll 1/\sqrt{{n}_{a}}$. Behind the laser pulse, where the dispersion is included, the electric field scaling, at a given distance

*β*

^{+}from the beam head, turns into in the limit

*t*≫ max(1/2

*n*

_{ef}

*β*

^{+},

*β*

^{+)}and

*β*

^{+}≫ 2

*πτ*(see Appendix B). As a result, THz scalings can vary with respect to the location of the propagated THz wave.

To validate our analytical formulae, we plot the solution in Fig. 5 in red solid line. The agreement in the plasma region is remarkable, especially in the beam head [Fig. 5(a)], where the small difference is ascribed to the THz wave dispersion, i.e., the *n _{e}δE_{y}* term in Eq. (20). To illustrate this effect, we have numerically solved Eq. (7) neglecting the term

*∂*in the laser region

_{t}δa_{y}n_{e}*x*−

*t*∈ [−2

*πτ*, 0]. The resulting curve (dashed black line) fairly agrees with the analytical solution both in the beam head and in the plasma region far from

*x*= 0. Although capturing the right oscillation frequency, the analytical expression of the reflected field yields small discrepancies in the amplitude values, which we attribute to the fact that Eq. (28) is valid for an infinite plasma only.

As for the reflected wave, the forward wave oscillates at the plasma frequency behind the beam head. Yet, far from the laser pulse, the wavelength of the emitted field increases. This effect is illustrated in Fig. 6 where the numerical solution is plotted against *x* and *t*. In vacuum, the reflected wave propagates at the velocity −1, while in the plasma, the emitted wave propagates at the velocity +1 right behind the laser pulse. Far from the pulse, the longitudinal wave number relaxes to *k _{x}* = 0. Indeed, according to Eq. (31) and since

*x*≫

*t*, the space derivative is

*∂*∼

_{x}δE_{y}*O*(1/

*t*), hence supporting the approximation

*∂*≪

_{x}*∂*used in section 2.

_{t}To visualize the propagation effect on the THz spectra compared with the “ideal” spectrum of Fig. 1(b), we have run a similar case to Fig. 5 but with a finite rescaled plasma length of 40*π*. The spectra of the reflected and transmitted waves are plotted in Fig. 7 in red and black solid lines, respectively. The intensity spectrum of the reflected wave is approximately a step function with a cut located at the plasma frequency
$\propto \sqrt{{n}_{a}}$. Below this frequency, photocurrents weakly contribute to THz emission. Given the plasma thickness, the source term, Eq. (24), mainly depends on *xG*(*x* − *t*) ≫ *F*(*x* − *t*) inside the plasma. As the reflected wave is driven by *F*(*x* + *t*) outside the plasma, the energy carried by the transmitted wave is higher and mainly determined by the ampitude ratio 2*xG/F*. Besides, its spectrum now extends above the plasma frequency. This THz field is attributed to both the constructive interferences during the ionization process and the plasma current oscillation. The electron current during ionization presents frequencies around 2/*τ*, which broadens the THz spectrum above the plasma frequency. The spectra deduced from the analytical expressions Eqs. (24), (28) and (30) are plotted in red and black dashed lines for the transmitted and reflected waves, respectively. A good agreement is obtained with the numerical solution, especially concerning the transmitted wave. The reflected wave presents a plateau between 0 and
$\sqrt{{n}_{a}}$, with a peak pronounced around the plasma frequency. Above this frequency, the spectrum dramatically decreases by many orders of magnitude. The intensity spectrum of the transmitted wave, in black dashed line, presents similar features to the numerical solution. In particular, the contribution of the frequencies higher than *ω*_{pe}, typical of ionization-induced THz emission, has increased. However, for the gas thickness of 40*π*, the electron current oscillation at the plasma frequency is the dominant process. Indeed, the spectrum above
$\sqrt{{n}_{a}}$, mainly depends on the source term *xG*(*x* − *t*), while the low-frequency spectrum depends on the plasma current oscillations, driven by the term
$~G(-2\pi \tau ){x}^{1/4}/\sqrt{{n}_{\text{ef}}}$.

Hence, for propagation length much larger than
$1/\sqrt{{n}_{\text{ef}}}$, the photocurrent mechanism becomes dominant, whereas for propagation length comparable to, or smaller than the plasma wavelength, as in Fig. 7, the main mechanism is the plasma current oscillation. To confirm this behavior, we have calculated the transmitted spectrum for a thicker gas of *L* = 100*π*. The numerical and semi-analytical solutions are plotted in green dashed line and green solid line, respectively. In this regime, the propagation length is such that
$L\gg 1/\sqrt{{n}_{a}}$, which leads to a spectrum contribution higher for
$\omega >\sqrt{{n}_{a}}$, as a result of dominant photocurrents. This tendency is reproduced by the semi-analytical model.

In summary, we have demonstrated within a unified formalism that laser-driven THz radiation can originate from different sources, namely, photoionization [23, 31] and residual current oscillations [26,30]. Our model establishes the missing link between the extreme nonlinear optics and plasma physics communities. The nontrivial interplay between these two mechanisms appears to be highly sensitive to the laser and medium parameters. For a propagation length shorter than the plasma wavelength, THz emission mainly results from plasma current oscillations. For longer gases, THz emission mostly originates from the constructive interferences during gas ionization.

## 5. Simulations

To further validate our model, we have run 1D PIC simulations using the code calder [38]. The code resolves the wave propagation inside the plasma, the particle trajectories including relativistic effects and field ionization [39]. To reproduce the conditions used in Ref. [26], collisions are neglected. The initial density profile of the hydrogen gas is trapezoidal, with a 90 *μ*m (700*c/ω*_{0})-long plateau bordered by 5 *μ*m-long linear ramps. The maximum atomic density is *n _{a}* = 0.0044, yielding an electron density of

*n*= 1.2 × 10

_{e}^{18}cm

^{−3}. The initial ion temperature is set to 1 eV. The laser beam has a Gaussian intensity profile with a 35 fs FWHM duration. The laser wavelength is

*λ*

_{0}= 2

*μ*m and the maximum intensity is 10

^{17}W/cm

^{2}. The simulation box is discretized in 9000 cells, each one containing 1000 electrons and 1000 atoms of hydrogen. The numerical resolution is Δ

*x*= 16 nm and Δ

*t*= 0.048 fs. Calculations have been performed using 3

^{rd}-order weight factors, with absorbing conditions for the fields.

The spectra of the transmitted and reflected waves are shown in Fig. 8(a) and compared to those obtained by the simplified wave equation, Eq. (7), assuming an unperturbed laser pulse. At the intensity of 10^{17}W/cm^{2}, both reflected spectra present a plateau at frequencies < *ω*_{pe} (*f* ≡ *ω*/2*π* < *ω*_{pe}/2*π* ≃ 10THz), due to the dominant contribution of the plasma current oscillations. In contrast, the transmitted wave presents a broadband spectrum in the range [*ω*_{pe}, *ω*_{0}] (10 ≤ *f* ≤ 150THz). Despite the single-color setup considered, the ionization occurring at the beginning of the pulse is asymmetric and produces a net transverse current. As discussed previously, this photoinduced current contributes to the spectrum well above the plasma frequency. Since the gas length (∼ 180*π*, corresponding to 90*μ*m) exceeds the plasma skin depth (15, corresponding to 5*μ*m), this radiation mechanism prevails, so that the THz energy is mainly contained at frequencies
$>\sqrt{{n}_{a}}$. Figure 8(b) compares the resulting THz fields computed below *ω* = 0.2 (f = 30 THz) from the transmitted and reflected THz waves obtained from the 1D calder code and from our simplified wave equation. This figure displays an excellent agreement between PIC simulations and our theoretical approach.

We have also compared our reduced numerical model to PIC and uppe1d simulations [40] in a two-color configuration for much weaker intensities. The numerical code solving the Unidirectional Pulse Propagation Equation model in one-dimensional (1D) geometry, uppe1d, is generally used to simulate the forward propagation of ultrashort laser pulses through gas-filled capillaries, accounting for Kerr optical effect and gas ionization. This code neglects relativistic and diffraction effects, but it includes the transverse current oscillation at the plasma frequency. In our simulations the laser intensity is set to 10^{14} W/cm^{2} or 10^{15} W/cm^{2} with *r*^{2} = 0.1, *ϕ*_{2} = 0 and *λ*_{0} = 1*μ*m. The fundamental and frequency-doubled pulses have Gaussian intensity profiles with ~30 fs and ~15 fs FWHM durations, respectively. A 100 *μ*m-long gas is considered. The other parameters are those used in the one-color case. Here, the Kerr polarisation, having a small impact in 1D geometry, is switched off in uppe1d. The transmitted wave spectra are presented in Figs. 9(a) and 9(b) from weak to strong pump intensity. Albeit overestimated by the simple model, which neglects laser dispersion, the agreement is satisfying. The contributions of both the current oscillation (
$\omega <\sqrt{{n}_{a}}$) and the non-symmetric ionization (
$\sqrt{{n}_{a}}<\omega <1$) are correctly reproduced. A striking feature is the increase of the THz spectrum at high intensities reproducing a spectral pattern comparable with that of Fig. 8(a), i.e., THz emission induced through photocurrents assures the merging with the pump field spectrum.

To end with, we have tested the robustness of our theoretical model by performing three-dimensional (3D) simulations that now include transverse diffraction and Kerr self-focusing. The blue dashed-dotted curves in Figs. 9(a) and 9(b) show the spectra computed at 10^{14} W/cm^{2} or 10^{15} W/cm^{2} pump intensities, respectively, using the uppe3d model of Ref. [23]. Under similar initial conditions, the two-color field propagates over 100*μ*m with a Gaussian transverse amplitude profile of 20*μ*m radius (1/*e*^{2}). We can observe that the 3D spectra remain close to their 1D counterparts and, thereby, they still remain strongly connected to the physics captured by our simplified model. With a 10^{14} W/cm^{2} pump, the laser field slowly diffracts and, in turn, the THz spectrum decreases in intensity. With a 10^{15} W/cm^{2} pump, the laser field starts to experience Kerr self-focusing, which contributes to enhance the THz spectrum to some extent.

## 6. Conclusions and perspectives

In conclusion, we have developed two models of THz emissions induced by laser-gas interaction. The simplest model describes the plasma current oscillation induced during gas ionization. Some scaling laws for the THz emission in terms of the main laser and gas parameters have been derived and found in good agreement with previous numerical studies [26,31]. A more sophisticated model has been developed, which demonstrates that the previous scalings still mostly hold when accounting for propagation effects. This model is able to reproduce the reflected and transmitted wave spectra obtained by self-consistent numerical simulations. Whereas the reflected spectrum is mainly determined by the current oscillation at plasma frequency, the transmitted spectrum is a subtle mix between plasma current oscillation- and ionization-induced THz emissions. For plasma lengths shorter than the plasma skin depth, the former contribution is dominant, while the latter becomes significant in the opposite case. At leading order, the THz field scaling is
$\propto \sqrt{{n}_{\text{gas}}I}{\lambda}_{0}/\tau $ for one laser color and
$\propto \sqrt{{n}_{\text{gas}}I}{\lambda}_{0}$ for two colors, for which the number of optical cycles in the overall pulse matters in a seemingly more complex dependency. Therefore, because the actual THz emission amplitude is oscillating and therefore not monotonic against laser intensity and the number of laser cycles, a simple, systematic dependence of the THz emission upon the laser wavelength may not be achievable. The present analysis is valid in the non-relativistic regime, upon 1D symmetry assumption. A more detailed model is, however, required to prospect the effect of the *J* × *B* force and the induced plasma wave on the THz emissions. Furthermore, one should explore different gases from hydrogen, as the THz field strength and related spectra are expected to vary with the ionization (binding) energy and occurence of multiple ionization.

## A. Computing Eq. (24)

We look for a solution *δE _{y}*, in the region

*β*

^{+}∈ [−2

*πτ*, 0]. Introducing

*B*=

*∂*, Eq. (23) becomes

_{s}δE_{y}*B*(

*s*= 0,

*β*

^{+}) =

*δE*(

_{y}*s*= 0,

*β*

^{+}) = 0 and

*B*(

*s*,

*β*

^{+}= 0) =

*δE*(

_{y}*s*,

*β*

^{+}= 0) = 0. Starting from

*s*

_{0}= 0, the characteristic is ${\beta}^{+}-{\beta}_{0}^{+}=-2s$, along which the solution reads

*s*> −

*β*

^{+}, we split the above integral in three intervals: ${\int}_{0}^{-{\beta}^{+}/2}d{s}^{\prime}$, ${\int}_{-{\beta}^{+}/2}^{-{\beta}^{+}}d{s}^{\prime}$, and ${\int}_{-{\beta}^{+}}^{s}d{s}^{\prime}$. As ${\beta}_{0}^{+}\ge -{\beta}_{0}^{+}$ for

*s*≥ −

*β*

^{+}/2, the first term is equal to zero, and the solution becomes:

## B. Asymptotic behavior Eq. (32)

Behind the beam head, the scaling for the THz amplitude with the crossed thickness can be obtained in the limit *t*(= *s*) ≫ *β*^{+} ≫ 2*πτ*, with *β*^{+} = *cste*. As
$\frac{G}{2\sqrt{{n}_{\text{ef}}}}\sqrt{\frac{t+x+2\pi \tau}{t-x-2\pi \tau}}\gg \pi \tau G+F/4$, Eq. (28) recasts as

*z*≫ 1, the Bessel function turns into ${\text{J}}_{1}[z]~\sqrt{\frac{2}{\pi z}}\text{cos}\left[z-\frac{3\pi}{4}\right]$ [36]. Hence, the THz field amplitude at given distance

*β*

^{+}from the laser pulse scales as

## Acknowledgments

This work was granted access to the HPC resources of TGCC and CINES under the allocation 2013-x2013052707 made by GENCI (Grand Equipement National de Calcul Intensif).

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