The spectrum of coherent Thomson scattering (CTS) induced by a periodic ponderomotive perturbation in a low-density low temperature plasma is considered. The analysis is performed for the case when the period of the resulting optical lattice is less than the Debye screening length in the plasma by solving an electron Boltzmann equation, where the total force is the sum of the periodic force due to the optical lattice and the electrostatic force due to self-consistent electric field in the plasma. An analogy between the CTS spectra calculated here and coherent Rayleigh scattering spectra in a neutral gas is established. For relatively low intensity for the optical lattice, the calculated CTS spectra are nearly Gaussian with widths slightly wider than the incoherent Thomson widths. We demonstrate that at higher intensities the line shape narrows and saturates to a width approximately half of that found at low lattice intensities. The proportionality of the spectral width to the square root of the electron temperature allows one to extract the electron temperature from the saturated spectra. Possible application of CTS for remote measuring the electron temperature in plasma is discussed.

## I. INTRODUCTION

The study of the complex behavior of the charged species in various plasma sources and devices requires substantial improvement of non-intrusive plasma diagnostic techniques. Incoherent Thomson scattering (ITS) of radio waves and laser radiation has been used for the noninvasive measurement of the electron density and temperature in plasmas from the 1960s up to today.^{1,2} ITS is characterized by the scattering parameter $\alpha =hrD\u22121$, where $h=k\u2212k\u2032$, $rD$ is the Debye length, and the wave vectors $k$ and $k\u2032$ describe incident and scattered waves, respectively. If $\alpha >1$, electron fluctuations in the plasma are strongly coupled to those of ions, giving rise to the so-called “collective” scattering. If $\alpha <1$, the electrostatic interactions become non-important, and thus, the incoherent scattering due to thermal fluctuations occurs as if the charged particles are free. Laser ITS experiments in this regime have shown that the frequency of photons from a narrow linewidth laser is Doppler shifted due to the scattering from the electrons in the plasma.^{3,4} In the collisionless limit, the spectrum of ITS is similar to that of the Rayleigh scattering from neutral particles.^{5} However, since the electrons have a much smaller mass and their translational temperature in a low-density, low temperature plasma is usually much higher than that of heavy species, the Thomson scattering spectral linewidth is orders of magnitude broader than the Rayleigh linewidth.

ITS is mainly used for diagnostics of dense fusion plasma in state-of-the-art plasma devices.^{2,6–10} In this technique, the incident probe laser power is scattered to 4π steradians, while the scattering power is proportional to the number of electrons in the scattering volume, decaying as *r*^{−2} with the distance *r* from the scattering volume. This gives rise to a low ratio of the scattered power to that of the probe laser and limits the application of ITS for the diagnostics of low temperature plasma sources with electron densities of 10^{12} cm^{−3} or lower.^{4,11–17} To perform ITS measurements under such low densities, it is paramount to attenuate stray light, collect statistics, accumulate measurements for thousands or tens of thousands of laser pulses, or significantly increase the intensity of laser radiation. These approaches are problematic, since accumulating statistics over a long time (tens of minutes or several hours) is impractical and meaningless under non-stationary plasma conditions, while an increase in laser intensity may only lead to large disturbances in the plasma under study.^{18}

In this paper, we propose a novel, alternative approach to measuring the electron temperature and density in a low-density plasma. This approach relies on the observation of Bragg scattering or induced coherent Thomson scattering (CTS) from the electron gas trapped by traveling optical lattice potentials and is a variation of four-wave mixing techniques. Following this approach, it is anticipated that the scattered signal will be largely increased, due to the constructive interference of the waves reflected from a refractive-index grating induced in a weakly ionized plasma by the ponderomotive interaction of the electron gas with an moving optical interference pattern. The resulting signal beam is another *coherent* laser beam, maintaining all the characteristics of the probe. The diagnostic scheme presented here is not to be confused with the collective Thomson scattering mentioned above, which is sometimes referred to as “coherent Thomson scattering” in the literature.^{19,20}

In the case considered here, the electron density perturbation $\delta ne$ is a traveling wave that oscillates at the beat frequency of pump beams $\Omega =\omega 1\u2212\omega 2$, with a grating wave vector $q=k1\u2212k2$, where $k1$ and $k2$ are wave numbers of two pump beams, which interfere within the plasma. The value of $\delta ne$ at different beat frequencies $\Omega $ determines the intensity of the scattered signal. The character of electron density perturbation depends on the value of parameter $qrD\u22121$, where $q=q$. If $qrD\u22121>1$, the electron density perturbations in the plasma are strongly coupled to those of ion density via self-consistent electric field in plasma.^{21,22} In contrast, at $qrD\u22121\u226a1$, electrostatic interactions are weak, and the electrons are, in fact, subject only to the external optical field. Here, we focus on the latter case and check the unimportance of the electrostatic interactions by direct numerical simulation. The condition $qrD\u22121\u226a1$ can be met in a plasma with a not too large electron density $ne$; for example, if it is assumed that the electron temperature is $Te=1$ eV and the optical lattice produced by two counter-propagating laser beams has a wavelength of λ = 266 nm, then $ne\u226a1.2\xd71017$ cm^{−3}. Here, we calculate the line shape of the resulting coherent Thomson scattering signal and demonstrate that line narrowing occurs for high intensities of the pump lasers. This is in analogy to similar phenomena observed and studied in the past for atomic and molecular gases, where the four-wave mixing technique has been used to determine temperature and neutral atomic and molecular gas composition from the coherent Rayleigh and Rayleigh–Brillouin scattering line shape.^{23–26}

## II. FORMULATION OF THE PROBLEM

Consider the motion of charged particles in an electric field of two interfering laser beams, termed the *pumps*, that cross at angle $\theta $ and have frequencies $\omega 1$ and $\omega 2$ and wave numbers $k1$ and $k2$. We choose the *x*-axis along the vector $q=\u2009k1\u2009\u2212\u2009k2$, which is perpendicular to the fringes of the optical lattice. The period of the lattice $d$ is determined by the angle $\theta $, $d=\lambda /[2\u2009sin(\theta /2)]$, where $\lambda $ is the pump wavelength. For two almost counter-propagating ($\theta \u2248\pi $) pump beams (see Fig. 1), $d=\lambda /2$ and $q=4\pi /\lambda $, where $q=q$. In an experiment, the beams must be crossed within a plasma at an angle close to but not equal to π, to avoid backward propagation of the laser beams. This crossing angle results in the interference pattern to be two-dimensional. However, as work in gases with the beams crossing at an angle of 178° shows, the theory based on the one-dimensional approximation describes the experiment well.^{23–25}

The electric field of the two interfering pump beams produces a traveling optical potential, which moves at a phase velocity $\xi =\Omega /q$. In a weakly ionized plasma, this potential will trap electrons in the low-intensity nodes, due to the ponderomotive force. To induce noticeable electron density perturbations, the phase velocity $\xi $ has to fall within the range of electron velocities $v$ for which the value of the one-dimensional velocity distribution function (VDF) of electrons $f\u221dexp(\u2212v2/2veT2)$ is significant, where $veT=kTe/m$, where $Te$ is the electron temperature and $m$ is the electron mass. Similarly, the one-dimensional VDF of ions or neutral species with velocities $vh$ will be given by $fh\u221dexp(\u2212vh2/2vT2)$, where the characteristic thermal velocity now is $vT=kT/M$, where $T$ is the species' temperature and $M$ is their mass. Since $M\u226bm$, it follows that the electronic VDF will be much wider than the one for the heavy species, even if assuming that $Te=T$. Consequently, if $\xi $ is greater than the velocities seen in the heavy species VDF, then one cannot expect that the motion of ions and neutral particles will be greatly perturbed by the optical lattice. On the other hand, a relative shift of electrons with respect to ions in the plasma leads to the generation of a self-consistent electric field, which tends to slow down the electrons and drag the ions. However, if the charge separation occurs at a distance of about one optical lattice period, which is much lower than the Debye length, then this electric field is relatively small and the motion of ions under its action can be neglected. Thus, the electrostatic field is taken into consideration to maximize its effect on the electron motion, while the ions are assumed to form a fixed neutralizing background.

An electron placed in the electric field generated by the interference of two pump beams is accelerated by the ponderomotive force $F=\u2212\u2202U(x,t)/\u2202x$ with an effective optical potential $Ux,t=e2E01E022m\omega 2cos\Omega t+qx$, where $e$ is the electron charge, $E01$ and $E02$ are the electric fields amplitudes of the two pump beams, and $\omega $ is the laser frequency, which is close to both $\omega 1$ and $\omega 2$; we assume that $\Omega =\omega 1\u2212\omega 2\u226a\omega 1,\omega 2$. The formula for $U(x,t)$, which corresponds to the traveling optical potential, has been obtained by us by modifying slightly the derivation (given, e.g., in Ref. 27) of the stationary ponderomotive potential $U(x)$ for a standing light wave.

Since the mean free path of the electrons with respect to various types of collision, such as electron–neutral particle and electron–ion, is higher than the length scale of the optical force gradients (typical dimensions of $d=\lambda /2=266$ nm, assuming counter-propagating pump beams with a wavelength of 532 nm) the perturbation of the electron density $\delta ne$ is found by a kinetic approach. To calculate the electron distribution function $f=f(x,v,t)$, we solve the Boltzmann equation, subject to the assumption that the total force acting on electrons in the plasma is a sum of the periodic ponderomotive force $F$ and the self-consistent electrostatic force $\u2212eE$:

Here, $f0=1/(2\pi veT)exp(\u2212v2/2veT2)$ is the Maxwellian electron distribution function, the collision integral is written in the Bhatnagar–Gross–Krook approximation^{28} and $\tau $ is the characteristic time between collisions. If electron–neutral atom collisions are dominant, the relaxation time $\tau $ in (1) can be calculated as $\tau =lvem$, where *l *=* *1/(*N*σ_{tr}) is the electron mean free path for electron–neutral atom collisions with gas of a density $N$, $vem=8kTe/\pi m$ is the electron mean velocity and σ_{tr} is the effective momentum transfer cross section.

To illustrate the validity of the operating regime considered here, without loss of generality, we consider the perturbation of electron density in a quasineutral weakly ionized plasma in a gas of argon, which is at a temperature of $T=293\u2009K$ and at a pressure of 216 Torr, having an electron density of $ne=1011$ cm^{−3} at a temperature of *T*_{e} = 1–3 eV. These conditions are typical for a low-temperature plasma source. For $Te=1$ eV, we have $\tau =10\u221211$ s and $l=6.7\xd710\u22124$ cm, which is 25 times higher than the period of the optical lattice *d* = λ/2 = 266 nm, hence demonstrating the validity of the kinetic approach followed here.

To determine the electric field $E$ in Eq. (1) self-consistently, we use the Poisson equation:

where $\rho x,t=eni\u2212\u222bf(x,v,t)dv$ is the space charge density and $ni$ is a fixed density of ions.

Thus, the perturbation of the electron density at each lattice phase velocity $\xi =\Omega /q$ is determined by solving the initial value problem for Eqs. (1) and (2) with the initial conditions $f(x,\u2009v,\u2009t=0)=f0$ and $E(x,\u2009t=0)=0$. The intensities of the two pump beams are assumed to be equal to each other $I=I1=I2$. Even for the highest pump intensities considered in this paper, $I=1012\u20131013$ W/cm^{2}, we have the parameter $U/(kTe)=0.1$, so that the modification of the electron distribution function by the optical lattice is relatively small, as compared to the Maxwellian one.

The computational domain is rectangular in the $(x,\u2009v)$ space, with the domain width along the $x$-direction equal to the optical lattice period $d$. Equation (1) is solved numerically by LeVeque's unsplit wave propagation method.^{29} As in Ref. 30, Eq. (2) is transformed to the current conservation equation and solved for $E(x,t)$ by an explicit scheme. As in Refs. 31 and 32 the assumption of the optical potential which is one-dimensional and periodic allows the use of the periodic boundary condition $f\u2009(0,\u2009v,\u2009t)=f\u2009(d,v,\u2009t).$ The same condition is applied for the electric field $E(0,\u2009t)=E(d,\u2009t)$. In addition, we have $fx,v\u2192\u221e,t=0$. With regard to the latter condition, in the implemented numerical method we have restricted the absolute value of electron velocity to values not exceeding $5veT$. We seek the solution of Eqs. (1) and (2) for the potential $U(x,t)$ created by the pump laser field pulse with a rectangular shape and temporal width not greater than 0.1 ns.

The mean electron density perturbation squared is computed as $\delta ne2=ne2d\u22121\u222b0ddx|\u222b\u2212\u221e+\u221efx,v,t\u2212f0dv|2,$ and it takes a time of the order of 10 ps for $\delta ne2$ to approach the steady state in the numerical simulations. The fact that the steady state is reached quickly confirms the initial assumption that electrostatic plasma oscillations can be neglected in the case considered here. The value of $\delta ne2$ ultimately determines the scattering signal at each phase velocity of the optical lattice $\xi $. Indeed, the intensity of the generated signal laser beam is described by the Bragg reflection coefficient $R$ from the periodic structure with the modulation of the refractive index $\delta n$. The refractive index $\delta n$ is expressed via the electron density perturbation $\delta ne$ induced by the optical lattice, $\delta n=\u2212\omega p/\omega 32\delta ne/(2ne)$, where $\omega p$ is the plasma frequency and $\omega 3$ is the frequency of the probe beam. The reflection coefficient from such a grating with $\delta n\u226a1$ is given by $R\u2248tanh22\delta nKd/\lambda 3,$^{33} where $K=L/d$ is the number of periods in the optical lattice having a length $L$ and $\lambda 3=2\pi c/\omega 3$. The argument of hyperbolic tangent can only be small by virtue of $\delta n$ being very small in our case. Hence, we have $R\u22482\delta n2K2d/\lambda 32$ and the intensity of the scattered signal $I3\u221dR\u221d\delta ne2$.

## III. RESULTS AND DISCUSSION

The resulting induced coherent Thomson scattering spectral profiles, in relative units, calculated as outlined above are presented in Fig. 2. As checked by direct numerical calculations, the electrostatic electric field $E$ given by (2) is negligible and dropping it from Eq. (1) does not affect the results presented below. With *n*_{e} = 10^{11} cm^{−3} and a numerically established amplitude of electron density perturbation (δ*n*_{e})_{a}/*n*_{e} = 1.5 × 10^{−2} for the high pump intensities assumed here (amplitude of *I *=* *10^{12} W cm^{−2}), the corresponding maximum electric field strength would be 0.013 V cm^{−1}.

For the intensity of the pump beams of $I=109$ W cm^{−2}, which corresponds to $U/kTe=10\u22124$, the periodic density perturbation induced on the electrons by the optical lattice is very small. The electron distribution function remains approximately equal to the Maxwellian one. However, as shown on Fig. 2(a), this weak perturbation provides the broadened Thomson scattering spectral profile, which is 10% wider than the spontaneous scattered profile. This effect is analogous to that predicted and observed in neutral gases.^{23,32}

With increased pump intensity $I>109$ W cm^{−2}, the obtained spectral profile first begins to narrow and then reaches saturation. The variation of the line shape with the pump intensity is shown in Fig. 2(b) for the electron temperature *T*_{e} = 1 eV, where the considered pump intensity is not higher than 10^{12} W cm^{−2}. The comparison of the spectral lines at *I *=* *10^{11} and 10^{12} W cm^{−2} clearly demonstrates the saturation of the narrowing with pump intensity. Such a narrowing is again similar to that predicted and observed in neutral gases.^{32,34}

For an increased electron temperature, *T*_{e} = 3 eV, we again find the spectral profiles undergoing the saturation of the broadening when decreasing the pump intensity and the saturation of the narrowing when increasing the pump intensity. The broad and narrow spectra are shown in Fig. 2(c).

Although an analytical theory of the narrowing is not developed yet, qualitatively the effect is related to the trapping of the particles (electrons) within the moving potential.^{32} Additionally, we note that the pump intensity *I *=* *10^{12} W cm^{−2} corresponds to $U/kTe=0.1$. Unlike the case with low pump intensity, at high pump intensity the perturbation of the electron distribution function due to its interaction with the optical lattice is noticeable. As a result, after interaction of the electronic gas with intense optical fields, a plateau forms within the distribution function due to the oscillation of the particles within the potential well. This effect is counteracted by electron–neutral collisions trying to restore the Maxwellian distribution function. The resulting perturbed distribution function is shown in Fig. 3. A significant fraction of the electron distribution function is trapped by the optical potential only when optical lattice velocities are not too high $\xi <2.5veT$. For this reason, the calculated saturated spectra of Figs. 2(b) and 2(c) at the high pump intensity strongly deviates from that found at low intensity, where the trapping effect is negligible [Fig. 2(a)]. Moreover, at $\xi >2.5veT$, the saturated spectrum at the high pump intensity approaches that found at low intensity [Figs. 2(b) and 2(c)].

It is expected that a broad Gaussian spectrum at low pump intensities would be difficult to observe, due to the low strength of induced electron density perturbations and the scattered signal. However, the scattered signal should be detectable at higher pump intensities. From the results of Fig. 2, it is observed that the width of the line shape to which the signal saturates at the high pump intensity is approximately two times lower than that of the Gaussian spectrum.

The width of the Gaussian spectrum is proportional to square root of *T*_{e}. The narrow, saturated signal possesses approximately the same property for the above numerical examples with *T*_{e} = 1 and 3 eV in argon. In these calculations, we took into account the variations of the electron mean free path with *T*_{e}, *l *=* l*(*T*_{e}) through the dependence of the momentum transfer cross section on the electron energy.^{35} Let us calculate full-width-at-half-maximum (FWHM) Δω = *qξ*_{1/2} of the narrowed spectra of Figs. 2(b) and 2(c), where *q *=* *4π/λ and *ξ*_{1/2} is FWHM measured in terms of the lattice velocity. For the narrowed spectrum of Fig. 2(b) (black solid line), we have $\xi 1/2\u22481.2veT$, $veT=kTe/m$, *T*_{e} = 1 eV, and Δω ≈ 1.2 × 10^{13} s^{−1} or 1.9 × 10^{3} GHz. For the narrowed spectrum of Fig. 2(c) (black solid line), with *T*_{e} = 3 eV and $\xi 1/2\u2009\u2248\u20091.24veT$ found from this figure, we obtain Δω ≈ 2.1 × 10^{13} s^{−1} or 3.4 × 10^{3} GHz. We see that within the accuracy of 3% (the constant 1.2 replaced by 1.24) FWHM is proportional the square root of the electron temperature *T*_{e}, $\Delta \omega =q\xi \u221dTe$.

If it is assumed that *l *=* l*(*T*_{e}) = const, as it is in helium in the considered range of *T*_{e}, it would have been obtained that the above dependence of FWHM of the narrowed spectrum on *T*_{e} is exact. The reason for this behavior is clear: in the case of a negligible electrostatic field *E *≈* *0, Eq. (1) can be rewritten, introducing the dimensionless variables $v\u0303=v/veT$, $x\u0303=x/d$, and $t\u0303=tveT/d$:

with $f0\u0303=1/2\pi exp\u2212v2\u0303/2$, $f\u0303=fveT$, $q\u0303=qd$, and $Fa=e2E02q2m\omega 2$ (in the last expression we assume that $E0=E01=E02$). Equation (3) shows that if $l=l(Te)=const$ the solution $\delta f=\u222b\u2212\u221e+\u221ef\u2212f0dv=\u222b\u2212\u221e+\u221e(f\u0303\u2212f0\u0303)dv\u0303$ as a function of $\xi /veT$ depends only on the value $Fad/mveT2\u221dI/Te$. Therefore, with $\xi /veT$ fixed, the variations in both $I$ and $Te$ such that the ratio $I/Te$ remains constant would not change the perturbation $\delta f$, and thus, the calculated signal lies on the universal curve $Is\u2009vs\u2009\xi /veT$.

To demonstrate the improvement offered by our approach, let us estimate the intensity of the coherent scattered signal for relatively high pump beam intensities $I1=I2=I=1012$ W cm^{−2}, and probe beam intensity *I*_{3} = *I*. According to the numerical simulations, with these values of $I1$ and $I2$, the perturbation of the electron density of $ne=1011$ cm^{−3} due to the optical lattice is $\delta ne/ne\u2248\u200910\u22122$. For a probe beam with wavelength $\lambda 3=532$ nm, this results in the perturbation of the index of refraction $\delta n=\u2212\omega p/\omega 32\delta ne/2ne=\u22121.3\xd710\u221213$. Assuming a typical length for the optical lattice of *L *=* *1 cm, the number of lattice periods will be $K=3.8\xd7104$ and thus the reflection coefficient $R\u22481.2\xd710\u221217$. Therefore, the intensity of the scattered signal will be $I4=RI3=1.2\xd710\u22125$ W cm^{−2}. The number of photons in the reflected signal per a probe laser pulse of a duration $\delta t$ is $Nph=I4\delta t\lambda 3S/(hc)$, where $S=\pi Rb2$ is the area of the laser beam and *h* is Plank's constant. With $\delta t=5$ ns and $Rb=100$*μ*m, we have $Nph$ ≈ 50, which is acceptable for reliable measurements.^{4} At the low pump intensities $I1$ and $I2=109$ W cm^{−2}, we have the amplitude of perturbations $\delta ne/ne\u2009\u2248\u20094\xd710\u22125$ and the number of photons *N*_{ph} would be significantly lower.

In the case of a standard incoherent Thomson scattering, number of scattered photons *N*_{s} is given by^{4}

where $V=SLS$ is the scattering volume, $LS$ is its length, $r0$ is the classical electron radius, $\Delta \Omega $ is the solid angle of observation, and $\eta $ is the transmission coefficient of the optical system. Let us estimate $Ns$ for a typical experimental case. Using the same values of $I3=1012$ W cm^{−2}, $\delta t$ = 5 ns, $\lambda 3$ = 532 nm, *n*_{e} = 10^{11} cm^{−3}, *S* = $\pi Rb2$, *R _{b}* = 100

*μ*m, and $LS$ = 1 cm as we have taken above in evaluating

*N*

_{ph}for the coherent scattering, substituting

*r*

_{0}= 2.82 × 10

^{−13 }cm, and assuming $\Delta \Omega $ = 10

^{−3 }sr and η = 0.1 as in Ref. 4, we obtain

*N*

_{s}≈ 3. Comparing this value with

*N*

_{ph}≈ 50 estimated for the coherent Thomson scattering, we can conclude that the proposed coherent Thomson scattering technique can be preferable to a standard incoherent Thomson scattering technique for diagnostics of low-temperature plasma sources.

## IV. CONCLUSION

At the core, the proposed four-wave mixing Thomson scattering scheme is the utilization of optical lattices for the creation of the periodic perturbation of electron density in plasmas via a periodic optical dipole force. The traveling optical potential perturbs the motion of a group of electrons whose velocities are close to the speed of the interference pattern of the crossed pump fields. By changing the frequency difference between the two pump laser beams, or the speed of the interference pattern, a perturbation of electron distribution function centered at the particular velocity is created. The relative magnitude of the induced electron density perturbation at each velocity can be determined by the measurement of the relative intensity of a third probe laser beam, Bragg scattered from the induced electron density perturbations.

This scheme is capable of by-passing the Rayleigh signal contributions from neutrals and ions in the plasma—only keeping the Thomson scattering signal from the electrons. More importantly, this scheme results in a coherent Thomson signal beam, which maintains all the beam characteristics of the probe beam. This enables the placement of the collection optics far from the point of measurement without any loss of signal, in comparison with, for example, incoherent Thomson scattering where the signal scales with 1/*r*^{2} with respect to the distance *r* where the collection optics are placed. It is envisioned that this capability will enable accurate and non-intrusive measurements in low-density plasmas, where the current state-of-the-art is mechanical probes.

The four-wave mixing nature of the proposed technique renders it ideal for application in optically noisy environments, such as those encountered in plasmas—while the necessary angled crossing of the laser beams provides with a high degree of localization and spatial resolution. Furthermore, if one utilizes a chirped lattice approach, where the range of optical lattice velocities is scanned in a single laser shot (as experimentally demonstrated and theoretically studied for neutral gases in Ref. 25), it is envisioned that the induced coherent Thomson scattering scheme will have single shot spectral acquisition, making it an ideal diagnostic for highly dynamic systems.

## ACKNOWLEDGMENTS

This work was partially supported by the Princeton Collaborative Research Facility (PCRF) and supported by the U.S. Department of Energy (DOE) under Contract No. DE-AC02–09CH11466. A.G. also received support from the U.S. DOE Office of Science Award No. DE-SC0021183.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts of interest to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.