1. INTRODUCTION

A particular subject of plasma science is to explain various kinds of electromagnetic radiation ranging from terahertz to x-ray frequencies induced by high-power laser pulses interacting with photoinduced plasma [1, 2]. It has been observed that a broadband electromagnetic pulse (EMP) of terahertz radiation can be emitted from a plasma filament produced by an ultrashort (less than 100 fs) high-intensity $\left({10}^{12}–{10}^{15}\text{\hspace{0.17em} W}/{\text{cm}}^2\right)$ laser pulse propagating in a gas medium [3, 4, 5]. Earlier published theories suggest that the basic mechanism behind the terahertz radiation relies on the effect of a ponderomotive force (PF), which separates electrons from ions spatially within the plasma filament [5, 6] and, consequently, generates a nonlinear electric current. Recently, it has been noted that the emitted terahertz power increases manifold if a two-color scheme is used where a fundamental optical wave with the frequency ω is mixed with its second harmonic (SH) satellite at the doubled frequency $2\omega$ [7, 8, 9].

The nonlinear electric current gives rise to the plasma filament polarization. Phenomenological models, formulated in terms of nonlinear polarization susceptibilities, attribute the terahertz emission to a four-wave mixing (FWM) rectification process [7, 8, 9, 10, 11, 12]. The phenomenology describes polarization properties of the terahertz field reasonably well [9, 13] and the fact that the terahertz field is subject to the phase difference φ between the fundamental and the SH waves. Specifically, it predicts that $E_{THz}\propto E_{\omega }^2{E}_{2\omega }\text{\hspace{0.17em} cos}\phi$ [7, 11, 14, 15] or $E_{THz}\propto E_{\omega }^2{E}_{2\omega }\text{\hspace{0.17em} sin}\phi$ [9]. Within the phenomenological approach it is hardly possible to distinguish whether the $\text{cos}\phi$ or the $\text{sin}\phi$ dependency is correct but the latter has been confirmed by experimental studies [9, 16]. Furthermore, it has been shown that $E_{THz}\propto E_{2\omega }$ and $E_{THz}\propto E_{\omega }^2$ for sufficiently low fundamental wave energy (less than $100\mu \text{J}$) [8, 13].

The understanding of plasma formation through optical breakdown has a paramount value for the unveiling of the physical mechanism of the terahertz emission from an optically generated plasma filament. Earlier dynamic theories of the emission assumed that the photoelectrons, pulled away from gas atoms by strong laser fields, are born with zero macroscopic (i.e., average) velocities [5]. This natural contemplation was disputed in [16, 17]. It was suggested that the terahertz emission at a microscopic level originates from a transient photocurrent created by the photoelectrons produced in a symmetry-broken laser field composed of fundamental and phase-shifted SH optic waves.

In this paper we present a simple theory based on classical assumptions of the terahertz emission, which takes into account the FWM process. It has led us to the conclusion that the manifold increase of the radiated power, observed in a number of experiments, occurs mainly due to transient photocurrent effect and also due to enhanced plasma production in the process of photoionization, since the multiphoton ionization (MPI) at doubled frequency goes on faster than at the fundamental frequency. Additional enhancement occurs due to the effect of the plasma pressure force [18]. For the first time, to our knowledge, we compute the radiation pressure force (RPF) for a laser pulse that contains both the first harmonic and the SH. We then argue that this newly computed force can produce a larger terahertz emission even if it is smaller than the PF. At a phenomenological level we also add to the theory the effect of the photocurrent by allowing a nonzero initial macroscopic velocity of the photoelectrons. Although we are not ready at the moment to compute the initial velocity from a strict quantum theory, we present a simple classical consideration of the photoionization process in a two-color laser field, which might hopefully elucidate major properties of the terahertz emission originated from the photocurrent. In particular, we note that the terahertz emission induced by the photocurrent scales as $\text{sin}\phi$ while that induced by the RPF or the PF varies in proportion to $\text{cos}\phi$.

In our model the electrons and ions are initially created by the MPI within laser string, and a dipole moment is subsequently induced in the plasma filament via the RPF acting on the electrons and resulting from the velocity-dependent Lorentz force and from the PF resulting from spatial inhomogeneity of the laser fields. These forces separate the electrons from the heavy ions both longitudinally and transversely regarding the plasma filament axis on the short time scale of the laser pulse. Behind the laser pulse head the electron fluid oscillates at plasma frequency and generates an EMP propagating away from the plasma filament as first described in [18]. Since plasma frequency is proportional to the square root of the local electron density, and the density varies from point to point within the plasma filament, the frequency spectrum of the terahertz emission ranges from a maximal value down to almost zero.

The paper is organized as follows. In Section 2 we compute the RPF and then in Section 3 we compute the PF. Slow motion of the plasma electrons under the action of these forces that spatially separates the electrons from the ions is analyzed in Section 4 and the resulting terahertz emission is computed in Section 5. Section 6 adds to our consideration of the effect of the transient photocurrent. Section 7 specifies experimental apparatus used to validate the theory and our conclusions are summarized in Section 8.

2. RADIATION PRESSURE FORCE

To calculate the RPF we extend a model first elaborated in [18], where the force was computed for a single-color laser field. In contrast to the PF to be analyzed in Section 3, the RPF does not explicitly depend on the gradient sizes of the laser beam and, therefore, it dominates for a relatively wide beam. In a single-color scheme, the RPF is directed along the laser pulse path [18], whereas a transverse component of the force emerges when a laser pulse at the fundamental frequency ω is mixed with that at the doubled frequency $2\omega$.

Following [3, 4, 18, 19], we employ an electron-fluid model of the plasma in the presence of the electric and magnetic fields E and B, which is governed by the equation

The RPF indicated below as G formally originates from a velocity-dependent part of the Lorentz force, i.e., from the second term on the right-hand side (RHS) of Eq. (1). Since an oscillatory velocity of an electron in the field of a laser pulse is of the order of $v_{\omega }\sim e{E}_{\omega }/m\omega$, and $B_{\omega }\sim E_{\omega }$, one formally finds that $G\sim m{v}_{\omega }^2/ƛ$, where $ƛ=c/\omega$ is the reduced laser radiation wavelength.

The PF indicated below as F formally originates from the second term on the left-hand side of Eq. (1), which is evaluated as $m{v}_{\omega }^2/a$, with a being a gradient length of the laser pulse envelope. Since $a⪢ƛ$, one might deduce that $G⪢F$ in any circumstances, but this would be a rushed conclusion because the main term $m{v}_{\omega }^2/ƛ$ in G turns out to be zero due to phase cancellation. One can suggest although that the next term in an expansion of G over $v_{\omega }$ can be as large as F, alternatively, collisional friction can prevent complete phase cancellation.

At the first stage we ignore spatial derivatives in Eq. (1) and drop the term $m\left(\mathbf{v}\cdot \nabla \right)\mathbf{v}$; it will be retained over again in Section 3. Furthermore, spatial dimensions of the laser string are assumed in this section to be so enormous that the laser electromagnetic fields $\mathbf{E}={\mathbf{E}}_L$ and $\mathbf{B}={\mathbf{B}}_L$ can be approximated as spatially uniform over the transverse profile of the plasma filament.

In the rest of this section we sequentially consider a case of mutually parallel harmonics, ${\mathbf{E}}_{2\omega }\parallel {\mathbf{E}}_{\omega }$, and that of perpendicular harmonics, ${\mathbf{E}}_{2\omega }\perp {\mathbf{E}}_{\omega }$. After having completed the calculations, we shall see that the treatment of these two geometries is sufficient for a comprehensive characterization of the RPF.

2A. Case ${\mathbf{E}}_{2\omega }\parallel {\mathbf{E}}_{\omega }$

Let a laser pulse be linearly polarized along the x axis while propagating in air along the positive direction of the z axis so that

The parameter $1/c$ in Eq. (1) formally tracks the order of perturbation theory and reflects the fact that the velocity-dependent Lorentz force is of the order of $v/c⪡1$ smaller than the dominant force due to the electric field under nonrelativistic conditions appropriate here. Then, to zeroth and first orders in the expansion parameter $1/c$, Eq. (1) yields the triplet of equations,

Next, the inspection of the term $e\left(v_x/c\right)E_L$ in the equation for $v_z$ reveals that it has the low frequency part,

Formula (4) has been derived in [18]. It should not be confused with the light pressure force $f_T={\sigma }_T\left({\left|E_{\omega }\right|}^2+{\left|E_{2\omega }\right|}^2\right)/8\pi$, where ${\sigma }_T=\left(8\pi /3\right){\left(e^2/m{c}^2\right)}^2$ is the Thomson cross section of an electromagnetic wave scattering by a free electron. The light pressure force can be derived from Eq. (1) with the radiative reaction force $\left(2e^2/3c^3\right)\ddot{\mathbf{v}}$ substituted instead of the friction force $-\gamma m\mathbf{v}$. In most circumstances, the light pressure force is negligibly small, $f_T⪡G_z$.

Straightforward integration of Eq. (3) for $v_z$ yields five harmonics, from zeroth to fourth, varying as $e^{\pm i0\omega t}$, $e^{\pm i\omega t}$, $e^{\pm i2\omega t}$, $e^{\pm i3\omega t}$, and $e^{\pm i4\omega t}$, respectively,

2B. Case ${\mathbf{E}}_{2\omega }\perp {\mathbf{E}}_{\omega }$

Now let the SH be polarized perpendicularly to the first harmonic so that

The longitudinal force $G_z$ is small as compared with $G_y$ provided that the SH is strong enough, $E_{2\omega ,y}⪢4\gamma mc/e$. To evaluate γ we assume that the electron–atom collisions dominate over the electron–ion collisions, which is the case for sufficiently low ionization of the ambient gas in the plasma filament. Then $\gamma =n_a{\sigma }_{ea}{v}_e$, where ${\sigma }_{ea}\sim {10}^{-16}{\text{cm}}^{-3}$ is the cross section of the electron scattering by neutral atoms, $n_a$ is the atoms' density, and the average velocity $v_e$ of the electrons is of the order of their oscillatory velocity $v_{\omega }\sim e{E}_{\omega }/m\omega$.

With these assumptions used, the condition $E_{2\omega }⪢4\gamma mc/e$ is equivalent to the inequality $E_{2\omega }/E_{\omega }⪢4c{n}_a{\sigma }_{ea}/\omega \sim 0.2n_a/n_{\text{atm}}$, where $n_{\text{atm}}\approx 3\times {10}^{19}{\text{cm}}^{-3}$ is the air density at normal conditions and $\omega =2\times {10}^{15}{\text{s}}^{-1}$. The condition can be met in a rarefied gas. Furthermore, if it is not satisfied, the transverse force can still produce a more powerful radiation from the plasma filament as shown in Section 5.

Following [20, 21], many papers refer the estimation $\gamma \approx {10}^{12}–{10}^{13}{\text{s}}^{-1}$ to the collision rate in the air at atmospheric pressure. This is consistent with the estimation above for the electron energy in the range from 0.1 to 10 eV.

3. PONDEROMOTIVE FORCE

We can now retain the term $m\left(\mathbf{v}\cdot \nabla \right)\mathbf{v}$ in Eq. (1) neglected earlier. Furthermore, we relax the assumption that the laser field is spatially uniform over the transverse profile of the plasma filament and retain spatial derivatives of the laser pulse envelope. Specifically, for a linearly polarized laser pulse with a given amplitude of the first harmonic,

The PF can be neglected if $F_z⪡G_z$ and $F_y⪡G_y$. The first condition is equivalent to the inequality $v_{\omega }^2/l⪡e{v}_{\omega }{E}_{\omega }/mc$, where $l=c{\tau }_L$ is the coherence length of the laser pulse with the time duration ${\tau }_L$. Putting here $v_{\omega }\sim e{E}_{\omega }/m\omega$, one immediately concludes that the longitudinal PF $F_z$ is small as compared with $G_z$ if $l⪢c/\gamma$. The latter inequality is feasible even for short laser pulses.

The condition $F_y⪡G_y$ leads to the inequality

The GW laser power is currently feasible, which provided the initial motivation for introducing into account the RPF and looking for a prospective physical mechanism that could create a dipole moment in the plasma filament [18]. It is worth noting, however, that the transverse part ${\mathbf{G}}_{\perp }$ of the RPF can produce larger emission from the plasma filament than that of the PF ${\mathbf{F}}_{\perp }$ even if the ordering Eq. (12) fails. Since ${\mathbf{G}}_{\perp }$ has a predominant direction across the entire section of the plasma filament, it produces a dipole polarization and, hence, a dipole radiation. On the contrary, ${\mathbf{F}}_{\perp }$ is directed mainly toward the center of the filament. It produces therefore a dipole radiation only due to the unavoidable ellipticity of the filament cross section. Ceteris paribus, a quadrupole radiation is less powerful and it is therefore usually neglected in comparison with the dipole radiation caused by the longitudinal part of the PF $F_z$ [5].

4. SLOW MOTION

The combined force $\mathbf{f}=\mathbf{F}+\mathbf{G}$ causes the plasma electrons to execute relatively slow motion that can be characterized by the electron–ion separation vector ξ. The latter is associated with the averaged velocity $⟨\mathbf{v}⟩=\partial \mathit{\xi }/\partial t$ and obeys the equation

The polarization electric field E in Eq. (13) is related to the electron density perturbation $\delta n_e$ by the equation

The driving force $\mathbf{f}=\mathbf{f}\left(\mathbf{r},t-z/v_g\right)$ is assumed below to be a function of the radius vector $\mathbf{r}=\left(x,y,z\right)$ and the time $t^{\prime }=t-z/v_g$ in a comoving frame of reference. Within a plasma filament, a small difference between the group velocities of the fundamental harmonic, $v_g\left(\omega \right)\approx c\left(1-{\omega }_p^2/2{\omega }^2\right)$, and that of the second one, $v_g\left(2\omega \right)\approx c\left(1-{\omega }_p^2/8{\omega }^2\right)$, can be neglected. Indeed, the mistiming length in the plasma,

On the contrary, in some circumstances the misphasing of the two harmonics should be taken into account. By misphasing we imply the phase factor $\text{exp}\left[i2k\left(\omega \right)z-ik\left(2\omega \right)z\right]$ that enters the term $E_{\omega }^2{E}_{2\omega }^{\ast }$ in ${\mathbf{G}}_{\perp }$. Within the plasma filament,

Misphasing in the gas determines a relative phase difference φ between the first harmonic and the SH at the “entrance” to the plasma filament, whereas the misphasing in the plasma causes modulation of the driving force f within the filament. The modulation period,

Integrating Eq. (16) over time with the initial conditions $\mathit{\xi }=0$ and $\partial \mathit{\xi }/\partial t=0$ at $t=-\infty$ and a constant electron density gives

If force f varies slowly on the time scale ${\omega }_p^{-1}$, i.e., ${\omega }_p{\tau }_L⪢\pi$, Eq. (21) reduces to

5. ELECTROMAGNETIC EMISSION FROM THE PLASMA FILAMENT

In a dipole approximation, the energy spectral intensity of the low frequency radiation generated by the slow motion of the plasma electrons per unit solid angle can be calculated directly from the well-known expression

When taking the spatial Fourier transform of the current, we assume the radius of the plasma filament a to be smaller than the emitted wavelength $2\pi c/\Omega$ so that

To proceed further, we approximate the dependency of the force $\mathbf{f}\left(\mathbf{r},t^{\prime }\right)$ on the retarded time $t^{\prime }=t-z/c$ in the head of the laser pulse by a tablelike peak with the duration ${\tau }_L$ so that $\mathbf{f}\left(\mathbf{r},t^{\prime }\right)=\mathbf{f}\left(\mathbf{r}\right)H\left(t^{\prime }+{\tau }_L/2\right)H\left({\tau }_L/2-t^{\prime }\right)$, where H stands for the Heaviside step function. Then,

Performing the integration in Eq. (25) and putting the result in Eq. (24) gives an expression that is a bit cumbersome expression,

This constitutes our principal result describing the spectral and angular distribution of the radiation. The frequency spectrum has a maximum at the plasma frequency, corresponding to ${\omega }_p/2\pi \sim 1\text{\hspace{0.17em} THz}$ for $n_e={10}^{16}{\text{cm}}^{-3}$. Experimental data exhibit a broadband frequency spectrum rather than a resonantlike one. It would mean either that the friction coefficient γ is comparable with ${\omega }_p$ or that the variation ${\omega }_p$ across the plasma filament cross section should be taken into account. This can be done by averaging $\mathbf{f}/\left[\gamma /2\pm i{\omega }_p-i\Omega \right]$ in Eq. (25) rather than just the force f alone. In our opinion, both these effects are equally important.

Although wide, the frequency spectrum in most experiments with femtosecond lasers is narrower than one could evaluate from the spectral width $\pi /{\tau }_L$ of the seed laser pulse. It means that the exact shape of the seed pulse is not very important, and the factor ${\text{sinc}}^2\left(\Omega {\tau }_L/2\right)$ can be dropped from Eqs. (28, 29) as it is approximately equal to 1 within the main part of the spectrum. We utilize this observation in Section 6.

As it is seen from Eqs. (28, 29), the emitted terahertz power is affected by the phase difference φ between the fundamental and the SH pulses caused by the dispersion of the refractive index along the path of these optical pulses in the gas to the foci, as reported by Cook and Hochstrasser [7] and later by Kress et al. [9] and Chen et al. [27]. Furthermore, Eq. (28) also describes the effect of the dispersion inside the plasma filament, and the latter limits the effective length of the filament that contributes to the terahertz emission. In particular, Eq. (29) predicts that the total emitted power oscillates from zero to a maximal value as the phase shift φ varies with the gas pressure. The more accurate Eq. (28) excludes lowering the emitted power down to zero as the result of the phase modulation within the plasma filament. Instead of being zero, the total radiated power falls down to a minimal value, which depends on the ratio of the plasma filament length L to the modulation length ${\ell }_p$. Since $\phi =0$ both for the PF and longitudinal RPF, detecting the power oscillations would certainly indicate that the transverse RPF plays a dominating role in the emission of the terahertz waves from the plasma filament. It is remarkable that the maximal emitted power is proportional to the square of the total number $N_{e}L$ of the free electrons in the plasma filament.

The angular distribution also provides crucial information regarding which of the two forces plays a dominating role. According to Eq. (29), the angular distribution contains multiple lobes, the angular positions of which are defined by the condition $2L{\text{sin}}^2\theta /2=N\lambda$ for the emitted wavelength $\lambda =2\pi c/\Omega$ and integer number $N=0,1,2,\dots$.

The lobe $N=0$ does not exist if the force ${\mathbf{f}}_0$ is directed along the plasma filament. This occurs when the longitudinal force is greater than the transverse one, i.e., $f_z⪢f_{\perp }$. In this case, the angular distribution is proportional to ${\text{sin}}^2\theta {\text{sinc}}^2\left(\Omega L{\text{sin}}^2\left(\theta /2\right)/c\right)$, and the main lobe, $N=0$, is empty inside resembling an inverse Cherenkov cone as shown in Fig. 1a for the case $L/\lambda =50$. The lobe $N=1$ is the strongest and corresponds to the cone opening angle

On the contrary, if the transverse force is dominant, i.e., $f_{\perp }⪢f_z$, the angular distribution is proportional to $\left(1-{\text{sin}}^2\alpha {\text{sin}}^2\theta \right){\text{sinc}}^2\left(\Omega L{\text{sin}}^2\left(\theta /2\right)/c\right)$, where α is the azimuthal angle (around the filament axis z) counted from the direction ${\mathbf{f}}_{\perp }$. In this case, the main lobe is filled. It is also slightly distorted in the azimuthal direction but this distortion is very weak since $\left(1-{\text{sin}}^2\alpha {\text{sin}}^2\theta \right)\approx 1$ for small θ. In addition, the contribution of the longitudinal force is suppressed by the small factor ${\text{sin}}^2\theta \sim 2\lambda /L$. It means that the domination of the transverse force over the longitudinal one requires a less severe condition $f_{\perp }⪢f_z\left(\lambda /L\right)$ to be satisfied instead of the inequality $f_{\perp }⪢f_z$ specified above from superficial consideration.

Comparing Figs. 1a, 1b shows that measuring the angular distribution of the radiated power allows determining the dominant direction of the driving force f. However, it is important to note that the contribution of the transverse PF ${\mathbf{F}}_{\perp }$ to the emitted power is substantially suppressed and might seem to be lower than that of the RPF ${\mathbf{G}}_{\perp }$ of the same magnitude. It follows from the fact that $\int \int n_e{\nabla }_{\perp }{\left|E_{\omega }\right|}^2\text{d}x\text{d}y=0$ for an axisymmetric distribution of both $n_e$ and ${\left|E_{\omega }\right|}^2$.

The angular distribution becomes even more complicated if ${\ell }_p\sim L$, when the exact formula (28) should be used instead of Eq. (29). In this case, not only the total emitted power varies with the gas pressure (through the phase shift φ) but also the angular distribution. A more certain conclusion that follows from Eq. (28) is that it predicts a drastic decrease in the emitted power when the plasma filament length greatly exceeds the modulation length, $L⪢{\ell }_p$. In this case, the terms in the curly brackets drop as ${\left({\ell }_p/L\right)}^2$, and the total power occurs to be proportional to ${\left(N_e{\ell }_p\right)}^2$ instead of ${\left(N_{e}L\right)}^2$. Thus, ${\ell }_p$ is an effective length of the plasma filament that effectively contributes to the radiation.

6. EFFECT OF THE TRANSIENT PHOTOCURRENT

We now relax the assumption that photoelectrons are born with zero average velocity. Although the MPI is essentially a quantum process, we employ a simple classical model in the hope that it catches such a crucial feature of the photoionization as the dependency on the phase shift φ between the first harmonic and the SH of the laser field.

Let us consider the motion of a free electron in the two-color laser field,

Employing the classical model of the photocurrent generation [16, 17], we make one more proposition, the validity of which we will discuss later. Namely, we assume that due to the extremely large nonlinearity of the MPI process tearing-off a bound electron from an atom occurs exclusively at the instants of time, corresponding to the maxima of the absolute magnitude of the electric field strength. Making use of the smallness of the parameter $\alpha =E_{2\omega }/E_{\omega }$ in realistic experimental conditions, we find the phase of the fundamental harmonic at these instants of time,

Generally speaking, this difference might be very huge even for a very small ratio α since a photon with doubled frequency has doubled energy and, thus, a two times smaller number of such photons is required to complete the act of ionization. However, for the sake of simplicity we assume that the parameter α is so small that the photoionization is produced exclusively by the fundamental harmonic of the laser field. It is then reasonable to think that the number of the electrons produced in the odd maxima differs from that produced in the even ones by the value of the order of α. If it is not so and the ionization by the SH dominates over the fundamental harmonic, one needs to think that the electrons are predominantly born at the maxima of the SH but we will not consider this case. Since the number of the electrons produced at different moments differs by a small amount of the order of α according to our assumption, the difference can be neglected when calculating the total (macroscopic) initial momentum of the electrons because their average velocity Eq. (35) itself contains the small parameter α. We note also that any initial spread of the electron velocities does not appear in our model. It means that a hydrodynamic approximation with zero temperature can be used to describe further the motion of the electron fluid.

To include the effect of the initial electron velocity, we extend the equations of slow motion from Section 4. First, we need to add a source producing $\dot{n}$ electrons per unit of time in a unit of volume and take into account that the electrons acquire an initial velocity ${\mathbf{v}}_0$ when they are born. Respectively, we add the term $\dot{n}$ to the RHS of the continuity equation,

To proceed further, we linearize Eqs. (37, 38), assuming the velocity v and the perturbation of the electron density $\delta n_e$ to be equally small quantities. By an unperturbed electron density we shall imply the quantity

Since Eq. (40) exactly coincides with Eq. (14) from Section 4, expression (15) for the polarization electric field E also remains valid. Putting it in Eq. (41) gives the final equation

Integrating Eq. (42) over time should be done with the initial conditions $\mathit{\xi }=0$ and $\partial \mathit{\xi }/\partial t=0$ at $t=-\infty$. Since the plasma density n, the plasma frequency ${\omega }_p$, the driving force f, and the initial velocity ${\mathbf{v}}_0$ depend on time, the integration can be generally done only numerically. There is, however, a practically important case, when one can present an approximate solution.

As has been noted at the end of Section 4, the period $2\pi /{\omega }_p$ of the plasma oscillations in present-day experiments with femtosecond lasers does not exceed the duration of the laser pulse. Restricting ourselves to the solution of Eq. (42) at larger times, we can say that the separation vector ξ in an arbitrary point with the coordinate z starts to differ from zero only after the laser pulse head passes through the point, i.e., at $t>z/v_g$. Behind the pulse head the unperturbed electron density remains constant as well as the plasma frequency. As a result, Eq. (42) is transformed into the ordinary differential equation

Elementary comparison of the summands in the integrand in Eq. (45) shows that the effect of the initial electron momentum acquired by the electrons at the act of the ionization causes a stronger plasma polarization if the momentum exceeds the momentum gained by the electrons from the driving force for the entire duration of the laser pulse, i.e., $m{v}_0⪢f{\tau }_L$. A formal estimation $v_0\sim \alpha v_{\omega }$, following from the classical treatment in the beginning of this section, leads to the conclusion that the condition $m{v}_0⪢f{\tau }_L$ is satisfied practically for any laser power feasible to the date. However, available quantum theories predict a substantially lower value of $v_0$, especially in the case of the laser pulse with the duration significantly exceeding the period of the laser field, $\omega {\tau }_L⪢1$.

Indeed, the notion of the instant of time of the ionization has no evident sense in quantum physics. According to the uncertainty principle, the instant of time of the ionization is determined at most with the accuracy of the order of the period of the bound electron gyration around the atomic core $2\pi /{\omega }_a$ (if the frequency ${\omega }_a=J/\hslash$, corresponding to the ionization potential J can be interpreted so) or with the accuracy of the ionizing field period $2\pi /\omega$. As a quantum theory of the photoionization [28, 29] says, the ionization occurs at times close to the absolute maximum of the laser field (taking into account the field envelope time profile) in a so called limit of low frequency, when $v_{\omega }⪢v_e=\sqrt{2J/m}$ [30]. According to the results of [29], the velocity distribution of the produced electrons has a maximum near $v_{\text{max}}={\int }_0^{\infty }\text{d}teE\left(t\right)/m$. Although the computation in [29] was performed for a special case, where $E\left(-t\right)=E\left(t\right)$, which corresponds to the phase shift φ, multiple of π, its result assumes that $v_{\text{max}}$ drops quickly as the pulse duration increases since the integral ${\int }_0^{\infty }\text{d}teE\left(t\right)/m$ decreases in proportion to $1/\omega {\tau }_L$ for a pulse with a tablelike time envelope and even more quickly for smoother envelope profiles. For example, it gives $v_{\text{max}}\propto \text{exp}\left(-1/{\omega }^2{\tau }_L^2\right)$ for the Gaussian profile. For a monochromatic wave, the theory yields $v_{\text{max}}=0$.

On the other hand, a numerical solution of the Schrödinger and experimental data, reported in [31], supports the dependency of the average velocity on the phase shift proportional to $\text{sin}\phi$, as predicted by our simple classical model. However, the same result contradicts to the results of [29] as the latter gives nonzero $v_{\text{max}}$ for $\phi =\pi N$.

7. EXPERIMENTAL RESULTS AND DISCUSSION

Experimental evidence is needed to establish the veracity of the model presented above. We used an experimental scheme analogous to that described in [7].

We employed an amplified Ti:sapphire laser system providing fundamental 120 fs and 800 nm pulses at the repetition rate of 1 kHz. The SH is generated in a beta barium borate crystal (BBO, type-I, $300\mu \text{m}$ thick). Both the intense fundamental laser pulses and their SH descendants are focused into a gas cell by a lens with a focal length of $f=18\text{\hspace{0.17em} cm}$. Radiated terahertz waves are collected by a set of crystal quartz lenses and then focused by an off-axis parabola $\left(f=25\text{\hspace{0.17em} mm}\right)$ onto $300\mu \text{m}$ thick $⟨110⟩$ oriented ZnTe crystal for electro-optical detection. The metal gas cell of 100 mm length has a window with a diameter of 25 mm. The front window is made of a $300\mu \text{m}$ thick fused quartz plate and a rear window is fabricated from Teflon. The gas cell is filled with air or with a noble gas (argon, neon, xenon, or krypton) at adjustable pressure in the range from 0 to 1.5 bar. Pure gas can be pumped into or evacuated from the cell trough a gas diluting system. To separate and reduce the contribution of the terahertz wave from the BBO crystal, we used an adjustable special filter placed inside the gas cell in the focus area.

According to basic principles of the terahertz time domain measurements, the temporal profile of the emitted terahertz wave is measured by scanning the time delay of a probe beam at the fundamental frequency ω relative to the terahertz pulse. The phase shift φ between the fundamental and the SH pulses can be changed [7, 9, 11] by varying the distance $\Delta l$ between the SH generation BBO crystal and the laser focus. In addition, φ varies with the gas pressure in the cell since both $n_{\omega }-1$ and $n_{2\omega }-1$ are proportional to the pressure but with different coefficients of proportionality. In both cases, the modulation of the terahertz wave amplitude has been detected in accordance with Eq. (28). Similar behavior has also been observed by Kress et al. [9] who moved a SH nonlinear crystal (BBO) away from to the focus. In this case the phase shift can be written as $\phi =\left(2\omega /c\right)\left(n_{\omega }-n_{\omega }\right)\Delta l$.

The variation in the phase shift leads also to the changes in the temporal profile of the terahertz field and its polarity [11, 16] in agreement with our discussion at the end of Section 5. In our experiments, the phase shift changes by π as the gas (Xe) pressure changes by an amount in the range from 0.6 to 0.7 bar. Figure 2 illustrates this statement.

Using indices of refraction from [32, 33, 34, 35], we have estimated the pressure periods to be 1.1, 0.6, 0.25, and 10 atm for Ar, Kr, Xe, and Ne, respectively. These estimates agree fairly well with experimentally observed values, especially at lower pressures.

8. CONCLUSION

In this paper we discussed basic principles of the terahertz wave generation initiated by the optical breakdown in air of a noble gas. We focused mainly on the two-color scheme of the terahertz wave generation. We presented a simple classical model of the multiple photoionization in a two-color laser field that hopefully catches some peculiar features of the process. In particular, we emphasize that the contribution of the radiation pressure force (RPF) to the emitted terahertz power varies with the phase shift φ due to the optical phase difference in the air to the laser focus in proportion to $\text{cos}\phi$, whereas the contribution of the transient photocurrent is proportional to $\text{sin}\phi$.

We have distinguished the RPF from the ponderomotive force (PF) and, for the first time, we have calculated the transverse component of the RPF. We have shown that the effect of the second harmonic (SH) is stronger when it is polarized differently from the first harmonic. In addition, the effect of the transverse PF is suppressed as compared with the effect of the transverse RPF because the PF varies across the section of the laser beam. Furthermore, the effect of the longitudinal driving force is suppressed as compared with the transverse force by the factor $\lambda /L$.

Both empty and filled angular diagrams of the terahertz emission (see Fig. 1) are allowed by our theory and both types of the diagram were observed experimentally although no special care has been taken so far to quantitatively validate experimental data against our theory (as it was not available earlier). By inspecting the angular diagram of the terahertz radiation, one can deduce which component—the longitudinal or the transverse—makes a dominating contribution to the emitted terahertz power.

We showed that the effective length ${\ell }_p$ [given by Eq. (20)] of the emitting part of the plasma filament is related to the angle spread of the terahertz radiation angular diagram if ${\ell }_p⪡L$. We explained that the effective length shortens due to the optical dispersion within the filament in inverse proportion to the electron density. We attributed the broadband character of the terahertz radiation to the spatial inhomogeneity of the electron density within the plasma filament.

If we compare our theoretical speculations with the complete experimental data available so far, we conclude that they do not allow validating our theory without doubts since some experimental results are mutually exclusive. We hope that future experiments will resolve this discrepancy.

ACKNOWLEDGMENTS

The authors are grateful to V. Davydenko for consulting us regarding properties of the laser-induced plasma, X.-C. Zhang and A. B. Savelev for useful discussions of various aspects of the laser-induced terahertz radiation, and M. Nazarov for the assistance in the experiments. This work was supported by the Russian Foundation for Basic Research (RFBR) in the frameworks of grants RFBR 08-02-00869 and 09-02-12198 and by the Russian Federal Agency for Science and Innovation (Rosnauka), the state contract 02.740.11.0223.

 

Fig. 1 Angular distribution of the terahertz radiation computed by Eq. (29) for $L/\lambda =50$: (a) driving force is directed along the plasma filament; (b) driving force is perpendicular to the plasma filament.

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Fig. 2 Dependence of the terahertz field amplitude generated in the Xe versus the gas pressure. In the figure, open squares correspond to 980 mW and the dark triangles correspond to the 600 mW of the overage power of the ω fundamental field.

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