Analysis of an induced Langmuir wave by ponderomotive forces and its applicability for plasma diagnostics

The characterization of thermodynamic properties of low-temperature plasmas (LTPs) such as the temperature of electrons (T_e), ions (T_i), and neutrals (T) and their density (n_e, n_i, and N, respectively) is crucial for a wide array of applications, including surface cleaning, etching, thin film deposition, aerospace propulsion, and biomedical and environmental applications.^1–6 Nowadays, a great variety of plasma diagnostic methods exist for determining these plasma parameters; the selection of an appropriate diagnostic method varies and greatly depends on factors such as the specific properties of the plasma being studied or the desired parameters to be measured. However, state-of-the-art diagnostics often come with limitations, such as perturbing the plasma state they aim to observe or being restricted by specific operating conditions. For example, Langmuir probes are introduced into the plasma, being in direct contact with it and thus perturbing it. Additionally, mechanical probes can only be applied at the edge of the plasma since they cannot withstand the high temperatures at their core, and thus spatially resolved measurements across the whole plasma extent are not possible. In contrast, laser-based, non-intrusive diagnostics can probe plasma properties with minimal disruption while they require limited assumptions to be made for the plasma state. From an experimental perspective, one of the main challenges of these non-intrusive diagnostics is the coupling of plasma radiation with background noise. In many instances, to improve the signal-to-noise ratio (SNR), it is necessary to collect photons over multiple laser pulses;^7–9 thus, these measurements have the inherent assumption that the plasma conditions remain exactly the same on a shot-to-shot basis. A single laser shot acquisition approach alleviates this assumption and enables measurements in highly transient plasmas.

In this Letter, a novel non-intrusive diagnostic approach to measure electron temperature and density in LTPs is proposed. This approach consists in observing the scattering of a laser beam from Langmuir density perturbation waves driven by ponderomotive forces, which arise from the interaction and beating of two laser beams within the plasma. Ponderomotive forces in optical lattices have been already discussed by Pan et al. to probe plasmas,¹⁰ although the focus there was on the translational neutral temperature rather than electron temperature. We anticipate significant enhancement in the scattered signal as the frequency detuning approaches a frequency difference given by the Bohm–Gross dispersion relation,¹¹ indicating a resonance with an excited Langmuir wave.¹² 

The density perturbation on electrons and ions caused by the beating of the laser beams is studied in the hydrodynamic regime, i.e., when the grating period is larger than the Debye length given by $λD=ε0kBTen0e2$ and collective processes such as plasma oscillations manifest Jackson. In the opposite case, i.e., at low-pressure plasmas with a grating period shorter than the Debye length, the density perturbation must be studied in the kinetic regime by solving the electron Boltzmann equation with the Bhatnagar–Gross–Krook approximation.¹⁴ Scattering from this collective periodic perturbation is referred to as coherent Thomson scattering (CTS).¹⁶ 

The radial motion of electrons and ions can be neglected since the axial ponderomotive forces are much larger than the radial ponderomotive forces for Gaussian beam profiles with beam width $r0$⁠. The radial ponderomotive forces become significant when $Δk=4πλsin(θ2)∼12r0$⁠, which is not the case for laser beams with radii $r0∼(100−200)μ$m. Therefore, assuming laser beams with relatively large radii, the density perturbation on electrons and ions is modeled only along the axial direction (z-axis) with constant amplitudes $E(r)=E0$⁠.

Figure 1 shows the maximum electron density perturbation as a function of time for various frequency differences detuning, which is achieved by fine-tuning one laser beam at $ω1=ω0+2πΔf$ and fixing the second one at $ω0=2πcλ$⁠. The electron density perturbation reaches an oscillating steady state around 0.4 ns, with an average value of $∼3.33×1017$ m⁻³ at the Bohm–Gross frequency given by Eq. (9) taking a value of $Δf∼0.23$ THz for the plasma parameters mentioned above. For the other frequency differences considered, the electron density perturbation reaches an oscillating steady state around 0.13 ns. The induced electron density perturbation is calculated as $δne(t)=Max(ne(x,t)−n0)$⁠.

In Fig. 1, the red curve represents the amplitude of an exited Langmuir wave for a relatively high-pressure low-temperature plasma. In these conditions, the constraint $d≫λD$ is easily achieved since the Debye length scales as $λD∝Ten0$⁠, and the main damping mechanism is due to electron–neutral collisions. The numerical simulations show the excitation of Langmuir waves even for $λD>d$⁠, although several authors have shown that the limit wavelength for a plasma wave is the Debye length.^11,12,23,24 Additionally, in both cases, at the long and short wavelengths, the Langmuir wave amplitude remains constant without any indication of wave braking or soliton formation.^25–27 

The ion density perturbation is shown in Fig. 2 and reaches an oscillating steady state faster than electrons around 0.2 ns, with an average value of $∼8.97×1012$ m⁻³ for the Bohm–Gross frequency (0.23 THz). The ion density perturbation reaches an oscillating steady state slower than electrons around 0.3 ns for the other frequency differences considered. Importantly, the ion density perturbation is four orders of magnitude less than electrons due to their heavy mass compared to the electron mass.

Figure 3 shows the electron density perturbation in the limiting case for a pump wavelength of 1064 nm and a crossing angle of 1°, in which low-pressure plasma sources $(ne∼1016$ $m−3)$ could be probed with $d∼λD$⁠. Here, the Langmuir wave is subject to collisional and Landau damping.^15,28 The electron density perturbation is greatly enhanced when the frequency detuning approaches the Bohm–Gross frequency independently of the power density used. By increasing one order of magnitude of the power density, the amplitude of the exited Langmuir wave is also increased by one order of magnitude indicating a linear trend. The maximum of each curve is centered at a frequency provided by the Bohm–Gross dispersion relation, represented by the vertical lines in the figure. For each frequency detuning, the simulation time was $4πωpe−1$ s, and the electron density perturbation has been calculated as $δne=(Max(δne(t))+Min(δne(t)))/2$⁠.

Our findings suggest that the resonances shown in Fig. 3 hold potential for plasma diagnostics since the resonance shift directly correlates with the unperturbed electron density and electronic temperature. This correlation is further demonstrated by comparing the three curves in Fig. 3:

1. With the unperturbed plasma density fixed at $n0=4.9×1016$ m⁻³, and electron temperatures of 1 and 1.5 eV considered, a shift toward higher frequency differences is observed (red and blue curves).

2. When the electron temperature remains fixed at $Te=1.5$ eV, and unperturbed plasma densities of $4.9×1016$ and $6.6×1016$ m⁻³ are examined, the resonance shifts toward a higher frequency detuning (blue and green curves).

Thus far, only the two crossing pump laser beams that induce this density grating have been considered. By introducing a third laser beam, termed the probe, with polarization orthogonal to that of the pumps, we can utilize its scattering from the electron grating for plasma characterization based on the frequency value $ΔfMax$ where the perturbation reaches its maximum value. Moreover, if the probe is scattered at a Bragg angle $θB$ and fulfilling phase matching conditions, the resulting signal is coherent due to constructive interference of scattered waves from a moving electron grating. The coherence of the signal has been demonstrated experimentally with optical lattices in molecular and atomic gases.^29–33 The intensity $Is$ of the signal can be estimated as $Is=RI3$⁠, where $R=tanh2(2δnKd/λ3)$ is the reflection coefficient of the periodic grating with modulation of the refractive index $δn≪1$⁠.³⁴  $K=L/d$ is the number of periods of the grating having a length L. $I3$ and $λ3$ are the intensity and wavelength of the probe, respectively. The relation between the probe wavelength and the Bragg angle is given by $mλ3=2d sin(θB)$⁠, where m is the diffraction order.³⁵ 

In a weakly ionized plasma, the refractive index is influenced by the presence of electrons, ions, and neutrals.³⁶ This modulation can be described in terms of the density perturbations of electrons $δne$⁠, ions $δni$⁠, and neutrals $δN$ given by $δn=12πε0(α0δN+αiδni)−(ωpe/ω3)2(δne)/2n0$⁠. Here, $ω3$ is the angular frequency of the probe, $ωpe$ is the electron plasma frequency, and $α0$ and $αi$ are the polarizability of neutrals and ions, respectively. Due to the lighter mass of electrons compared to ions and neutrals, their response time is faster, resulting in larger density perturbations on such timescales, as numerically checked (Fig. 2). Additionally, the density perturbation of neutral argon atoms can be neglected, as its perturbation is expected to be comparable to that of ions due to their similar mass. Consequently, the modulation on the refractive index can be mainly attributed to the electron density perturbation $δn≈−(ωpe/ω3)2(δne)/2n0$⁠. For a small density perturbation and small refractive index modulation, the spectrum of the scattered signal is proportional to the probe intensity, square of the grating length, probe wavelength, and electron density perturbation $Is∝(Lλ3δne)2I3$⁠.

From the resonance value $Δf≈0.23$ THz shown in Fig. 1 and using Eq. (10), an electron number density of $∼6.6×1020$ m⁻³ is obtained with a percentage error of $4%$ for an unperturbed density of $∼6.3×1020$ m⁻³. The electron temperature estimation relies on the peak height, which correlates with electron temperature variations. When the electron temperature increases the electron–neutral collision frequency increases, and the amplitude of the Langmuir wave decreases due to collisional damping with neutrals, as illustrated in Fig. 4. Here, the unperturbed plasma density is $n0=7.5×1020$ m⁻³ with background neutral densities $N0=1.3×1025$ m⁻³. The ion temperature $Ti=0.05$ eV and crossing angle $θ=10°$ remain constant. Additionally, the grating period lies in between $∼10−25$ Debye lengths.

For the resonant frequencies specified in Fig. 3, electron temperatures of 1.08, 1.58, and 1.61 are obtained with percentage errors of $8%,5%$⁠, and 7%, respectively. Additionally, the relative electron number density can be estimated from the height of the peaks at the resonance frequency. Increasing the unperturbed plasma density increases the electron density perturbation as shown in Fig. 5. Here, a pressure of 10 Torr has been considered while the ionization degree was varied from $1×10−7$ to $5×10−7$⁠. Electron and ion temperatures are fixed at $Te=1$ eV and $Ti=0.05$ eV, respectively. The crossing angle is $θ=1°$ and the grating period lies between ∼1 and 4 Debye lengths. Lower plasma densities require pump wavelengths larger than 1064 nm.

We investigated the electron density perturbation within typical parameters of LTPs under the influence of the beating of two laser beams by varying their frequency difference in the hydrodynamic regime, which is valid for a lattice period larger than the Debye length. We have found a noticeable enhancement in the electron density perturbation when the frequency detuning approaches the Bohm–Gross frequency, indicating a resonance with a Langmuir wave. The numerical simulations show resonance with Langmuir waves at low-pressure and high-pressure plasmas and its limits should be tested experimentally. We considered and proposed the applicability of this resonance as a new tool for plasma characterization, by observing the spectrum of a Bragg-scattered signal from electrons trapped in the traveling wave oscillating at the beat angular frequency. The intensity of the scattered signal is ultimately determined by the square of the induced electron density perturbation, which is proportional to the intensity of the pumps, and laser–plasma heating and multiphoton ionization should be considered when using high power densities.^37,38 The signal presents its maximum at the Bohm–Gross frequency, and the shift and height of this peak offer the possibility of determining both electron temperature and density.

The main advantage of this approach is the fact that the resulting signal is a coherent laser beam, which in an experimental setup will allow for the detector to be placed at a considerable distance from the measurement point, mitigating any interference from plasma radiation. In contrast with non-coherent optical diagnostics, this is anticipated to enhance the SNR of the measurement by several orders of magnitude. Moreover, if chirped optical lattices³³ are utilized, where the frequency detuning is performed in a single laser shot, this diagnostic method would be ideal for characterizing non-stationary and highly transient plasma environments.

Finally, our study suggests the potential for utilizing similar methods to excite various plasma waves, offering further advanced plasma diagnostics methods. For instance, ion–acoustic waves could be excited following the same approach to characterize electron and ion plasma parameters.

G.F.A and A.G. are supported by the Luxembourg National Research Fund 15480342 (FRAGOLA). M.N.S. acknowledges partial support by the Princeton Collaborative Research Facility (PCRF) supported by the U.S. DOE under Contract No. DE-AC02-09CH11466.