One of the most common plasma diagnostics methods is based on radiation scattering on electron density fluctuations.1 The fluctuation phenomenon involved here relates to both chaotic thermal motion and the collective excitations1 that are the focus of the present paper. Collective radiation scattering on electron density fluctuations was predicted by Akhiezer et al.2 A few years later, Salpeter3 obtained a fluctuation spectral characteristic of a nonmagnetic equilibrium plasma that is integral to the description of the plasma. In the same decade, a number of authors developed both classical as well as quantum mechanical approaches for describing fluctuation spectra for various types of plasmas.4–8

Along with the advances in the theory of radiation and plasma interaction, experimental applications of scattering theory were actively developed. As the scattering radiation spectra contain information on plasma temperature and density,9 many papers devoted to scattering plasma diagnostics have appeared since the 1960s. Thus, Hughes10 proposed a method for measuring plasma temperature and density based on Thomson scattering. Decades later, this approach became a classical method of plasma diagnosis in tokamaks.11–13

In a series of papers, attention was drawn to the interaction of radiation with plasma particles involved in collective motion. For instance, Mooradian and McWhorter14 experimentally investigated radiation scattering on plasmons in a solid-state plasma. Glenzer et al.15 demonstrated a new x-ray scattering technique for studying the properties of warm dense matter. The theory of coherent radiation collective scattering in plasmas is described and supported by experimental data in the book by Froula et al.1

The last two decades have been marked by significant advances in techniques for generating ultrashort laser pulses (USLPs). One of the most significant achievements in this area is the development of attosecond extreme ultraviolet (XUV) lasers, which have now became a widely used experimental tool.16–19 With the aid of high-harmonic generation (HHG) spectroscopy,20,21 it is now possible to observe the dynamics of atomic-scale systems with sub-angstrom spatial and attosecond time resolution.22,23 This makes it possible to measure such fundamental characteristics as phases of quantum-mechanical wave-packets.24 Along with this, there are more practical applications of pulses with sub-femtosecond duration, such as chemical reaction control at the molecular level25 and study of charge migration in biologically relevant molecules.26 As lasers generating pulses with duration of the order of hundreds of attoseconds are no longer exotic research instruments and, moreover, methods have been proposed for the generation of x-ray free electron laser (XFEL) pulses with duration of ∼40 as,27 we are faced with the challenge of reconsidering standard models of the interaction of radiation and matter, which were developed for pulse durations much greater than the characteristic time of atomic processes. Following this paradigm, we focus on the interaction of ultrashort laser pulses (USLPs) with a Maxwellian plasma and, in particular, on electron collective effects, which are a rather important aspect of experimental plasma diagnosis.

There have been a number of previous works devoted to USLP interaction with plasmas. For example, Glover et al.28 experimentally obtained spectra of sub-picosecond x-ray pulses scattered from the plasma. Later work by Ta Phuoc et al.29 was devoted to experimental investigation of nonlinear Thomson scattering of femtosecond x-ray pulses on relativistic electrons in a helium plasma. Nevertheless, in our literature review, we did not find any descriptions of models predicting USLP scattering probability during all the time of pulse action. Thus, the development of such a model is the main purpose of the current paper. In Sec. II we describe the basic relationships between the spectral characteristics of plasma fluctuations and the scattering cross-section. Section III presents our extension of Fermi’s golden rule to USLPs and provides the bridge between the conventional theory of plasma fluctuations and the scattering probability of USLPs. In Sec. IV, we apply our new approach to the description of USLP scattering on plasmons and demonstrate some particular features of this interaction. In this work, we consider plasmas with temperature in the range of 1–20 eV and electron densities of 10¹⁶–10¹⁸ cm⁻³, which are typical conditions found in industrial plasmas.1

The character of radiation interaction with plasmas depends on the relation between the incident light wavelength λ and the Debye radius λ_D. In the case when λ < λ_D, the incident photons interact with individual electrons.30,31 Otherwise, when λ ≥ λ_D, the situation is different. The incident pulse interacts with all electrons constituting the Debye sphere and photoprocesses have a collective character.32

The scattering process in a plasma is due to the existence of electron density fluctuations. Otherwise, in a plasma with uniformly distributed electron density, for each scattered component the opposite component could always be found, and their superposition would give zero field strength. The fluctuations are caused by electron motion.30,33 Besides thermal motion, plasma elections are involved in collective oscillations. It is possible to identify two types of oscillations in unmagnetized plasmas: transverse and longitudinal. Transverse oscillations are represented by waves that are similar to electromagnetic waves in vacuum, but, in contrast to the vacuum case, these waves correspond to a dispersion relation that makes the plasma opaque for waves with frequencies lower than the plasma frequency ω_pe.34 Longitudinal waves are represented by ion-acoustic and electron oscillations. Ion-acoustic wave generation requires the electron temperature to be much higher than the ion temperature,35 which is not the case for the plasma parameters in the present work, and so we leave aside the issue of ion-acoustic waves. Longitudinal electron waves occur owing to oscillations of the electrons shielding each electron and each ion.

As mentioned above, the process of radiation scattering in a plasma is governed by the character of the electron density fluctuations. For a detailed spectral description of the whole system of interacting particles, we use a dynamic plasma form factor that is defined as the time and space Fourier transform of the electron density pair correlation function S(k, ω) (for details, see Refs. 1, 30, and 36).

In the framework of the Salpeter approximation,3 a general expression for the dynamic form factor of a plasma with charge number Z is37S(ω,k)=Se+Si=2πk1−χeε2fe0ωk+2πZkχeε2fi0ωk.(2.1)The four-vector (ω/c, k) indicates the changes in the radiation frequency and the wave vector as a result of the scattering process. The difference between the frequencies ω_i and ω′ of the incident and scattered field components, respectively, is denoted by ω.

As we consider a Maxwellian plasma, the electron (ion) velocity distribution function f_e(i)0 has the form1,38fe(i)0ωk=me(i)2πTe(i)exp−me(i)ω22Te(i)k2,(2.2)where m_e(i) and T_e(i) are the electron (ion) mass and temperature in eV. The following dimensionless values are used:xe(i)=ωkme(i)2Te(i),α=1kλD.The plasma dielectric permittivity ɛ = 1 + χ_e + χ_i contains the electron and ion components of the electric susceptibility, given respectively by1χe=α2F(xe),(2.3)χi=α2ZTeTiF(xi),(2.3′)whereF(x)=1−πx⁡exp(−x2)erfi(x)+iπx⁡exp(−x2).(2.4)

The contributions of the electron and ion components to the scattering process described by (2.1) are significantly different in various frequency ranges. Thus, at frequencies much greater than the ion plasma frequency (ω ≫ ω_pi), ions cannot respond to the excitation, and the ion component in (2.1) can be neglected. In other words, the incident radiation is completely scattered at plasmon and thermal fluctuations, as described in more detail in Sec. IV. In the opposite case, when ω ≪ ω_pi and the plasma is almost in equilibrium, the ion component dominates,1 and the initial radiation is scattered by the electrons shielding the ions. In the present work, we do not consider this case and focus only on the parts of the S(k, ω) spectrum that respond to plasmon excitations. Furthermore, for simplicity, we consider Z = 1.

According to Fermi’s golden rule, for monochromatic radiation, the differential scattering probability per unit time can be expressed through the scattering cross-section σ_sc and the incident photon flux jωi with frequency ω_i:37d2wscdΩ′dω′=d2σscdΩ′dωjωi.(3.1)The following expression relates the scattering cross-section per electron and the dynamical form factor of the plasma:30d2σsc(k,ω)dΩ′dω=1+cos2⁡θ2re2S(ω,k),(3.2)where θ and Ω′ are the plane and solid scattering angles, respectively.

All of the theory presented in Sec. II is sufficient for the description of the interaction of quasi-monochromatic pulses. The situation is entirely different, however, in the case of ultrashort pulses, because their spectrum is rather broad (Δω ∼ 1/τ). Thus, the formula (3.1) no longer provides an adequate description.

A formula describing the scattering probability of ULSPs by atoms in second-order quantum-mechanical perturbation theory is derived in Ref. 39, and the following extension of this formula to scattering in a plasma is presented in Ref. 40:d2WscdΩ′dω′=c4π2∫0+∞d2σscdΩ′dω|E(ωi)|2ℏωidωi,(3.3)where c is the speed of light and E(ω_i) is the Fourier transform of the incident pulse.

To explain the meaning of the formula (3.3), it is convenient to refer to the phenomenological Fermi-equivalent photon concept.41 For calculation of the USLP scattering probability, the contribution of each spectral component of the incident pulse should be taken into account. Thus, a USLP could be presented as a composition of photon fluxes, the densities of which are proportional to the spectral density of the incident pulse. To calculate the full USLP scattering probability during the entire time of pulse action, it is necessary to integrate over all spectral components of equivalent photons fluxes multiplied by the corresponding cross-section, and this is reflected in the formula (3.3).

To model the spectral and time dependences of the laser pulse electric field strength, a Gaussian approximation could be used. Nevertheless, this gives an adequate description only for quasi-monochromatic pulses. In the case when ω_cτ < 1, which corresponds to a USLP, E(ω_i = 0) ≠ 0. The presence of the constant field component is physically incorrect, and, moreover, such an approximation could cause artifacts.17 For the description of generated laser pulses, we use the model of a corrected Gaussian pulse,42 which does not suffer from these drawbacks. The Fourier transformation of such a pulse isE(ωi,ωc,τ)=iE0τπ2ωi2τ21+ωc2τ2exp−(ωi−ωc)2τ22−exp−(ωi+ωc)2τ22,(3.4)where E₀ is the electric field magnitude of the incident pulse and φ is the initial phase of the pulse. Hereinafter, we consider that E₀ = 1 a.u. (corresponding to the strength of the field at the first Bohr orbit of the hydrogen atom).

The formula (3.3) is at the core of the present work. From this formula, we have a model for the calculation of the USLP scattering probability in a Maxwellian plasma, and this is what Sec. IV is devoted to.

In as much as xi/xe=mi/me, it follows that x_i ≫ x_e. According to (2.2), the function S_e(x_e) has an exponential decay, and we can consider values of x_e ∼ 1. Thus, taking into account the fact that F(x) ≈ 0 for x ≫ 1, we can neglect the ion component of the dielectric permittivity, χ_i. The ion part of the dynamic form factor according to (2.2) can also be neglected. The following simplified expression for the dynamical form factor for an equilibrium plasma then follows from (2.1)–(2.4):S(ω,k)=1emen¯eα⁡exp(−xe2)(1+α2Re{F(xe)})2+(α2Im{F(xe)})2,(4.1)where n¯e is the mean electron density and e is the elementary charge. The condition for plasmon resonance excitation isRe{ε}=0⇔1+(αres)2−(αres)2πxres⁡exp[−(xres)2]erfi(xres)=0,(4.2)where α^res and x^res are the values of α and x_e corresponding to the resonance conditions.

Figures 1(a)–1(c) demonstrate how the dependence of the plasma form factor on the frequency of scattered radiation changes with changes in the parameters that influence the value of α. It can be seen that for some changes in the problem parameters, the peak related to the thermal fluctuations is split into two plasmon peaks.

For a clearer understanding of the dependency dynamics, it is useful to consider the spectrum as a function of the dimensionless parameter x_e. Figure 1(d) shows the dependence of the dynamic form factor on x_e on a logarithmic scale for different plasma densities. It can be seen that the dynamic form factor has a bell-shaped form, with peaks corresponding to plasmon resonances. With increasing density, the plasmon peaks appear to slide down the side of the bell. Thus, at low concentration, only thermal motion peaks are seen. As the density increases, the order of magnitude of the plasmon peaks becomes similar to the peak magnitude of thermal motion peak, and it becomes possible to observe dense fluctuation spectra of different types: only a thermal peak, only plasmon peaks, and simultaneous thermal and plasmon peaks. However, with further increase in density, the effect of thermal motion completely suppresses the plasmon resonance. This description of the form-factor spectral dynamics S_e(x_e) can be generalized: any changes in the parameters that lead to growth in α (an increase in density or a decrease in temperature or scattering angle) will lead to shifts of the plasmon peaks from zero. It should be noted that the magnitudes of the two plasmon peaks are not equal in the general case.

Both of the spectral maxima result from satisfaction of the Bohm–Gross relation43,44 for Langmuir plasma waves. Let us define the plasmon energy as ℏω_pl = ℏ(ω_i − ω′) = ℏω. The peak with lower frequency corresponds to photon absorption by the plasmon with transition to the virtual state, after which photons are re-radiated with lower frequency, transferring part of their energy to the plasmon. The peak with higher frequency corresponds to the reverse process: on re-radiations, photons take away part of the plasmon energy.

In the case when the plasmon peaks are narrow enough and do not overlap with maxima of the thermal fluctuations, the Lorenz approximation can be used for description of a particular peak Γ_m(x_e, α):1,3Γm(xe,α)=12kπme2Te(xmres)2⁡exp[−(xmres)2](xe−xmres)2+12πα2(xmres)2⁡exp[−(xmres)2]2→πme8neδ(xe−xmres)αe,(4.3)where m = {1, 2}. The full width at half maximum (FWHM) of the peak, γ_m, is given byγm=π4(αmresxmres)2⁡exp[−(xmres)2].(4.4)

The resonance condition isxe=xmres⇔ωi−ω′aωi2+ω′2−2ωiω′⁡cos⁡θ=ωimres−ω′a(ωimres)2+ω′2−2ωimresω′⁡cos⁡θ.(4.5)

The equality (4.5) leads to the following relation between the two resonant frequencies and the frequency of scattered radiation:ωi1resωi2res=ω′2.(4.6)

When the peak magnitude of thermal fluctuations is negligibly small in comparison with the magnitude of the plasmon peaks, the dynamic form factor tends to the sum of two delta functions:S(ω,k)→1αeπme8ne[δ(xe−x1res)+δ(xe−x2res)]=G1δ(ωi−ωi1res)+G2δ(ωi−ωi2res),(4.7)withGm=πλDαmresc1kmresc−(ωimres−ω′)(ωimres−ω′⁡cos⁡θ)(kmresc)3−1.(4.8)On combining (3.2), (3.3), and (4.7) we derive a simple formula for the scattering probability:d2WscdΩ′dω′=cre2(1+cos2⁡θ)8π2×G1|E(ωi=ωi1res)|2ℏωi1res+G2|E(ωi=ωi2res)|2ℏωi2res.(4.9)

Let us consider a scattering process with parameters T_e = 2 eV, n_e = 10¹⁷ cm⁻³, ω′ = 2.1 eV, and θ = π/4. The scattering cross-section spectrum for this case is illustrated in Fig. 2, together with the spectra of incident pulses with various durations. For the parameter values considered here, the plasmon peaks do not overlap, and their width is small enough that the approximations (4.7) and (4.9) apply. The resonant frequencies in this case are ωi1res=2.087 eV and ωi2res=2.113 eV. These frequency values correspond to the Bohm–Gross dispersion relation with high accuracy:44(ωires−ω′)2=ωpe2+3Te(kres)2me.(4.10)

Figure 3 presents the dependence of the differential scattering probability on the pulse duration (the τ dependence) for various values of the carrier frequency. The τ dependence is calculated by two methods: numerical integration using the formula (3.3) and the analytical approximation (4.9). The τ dependence shows different trends at different carrier frequencies. Its shape is determined by the relative positions of the pulse spectrum and the scattering cross-section spectrum. Figure 4 illustrates how the type of curve depends on the relative positions of the carrier frequency ω_c and the plasmon resonance frequency ω_i.

When the carrier frequency is close to the plasmon resonance frequency (the green areas in Fig. 4), the τ dependence is monotonically increasing and becoming linear as the pulse duration growths [Fig. 3(b)]. If ωc<ωi1res and ωc>ωi2res (the blue areas in Fig. 4), the curves are bell-shaped [Fig. 3(a)]. With increasing τ, the probability becomes almost zero, because the pulse spectrum becomes narrow and does not overlap with the peaks of the cross-section (see the red curves in Fig. 2). Thus, this effect is inherent in pulses with rather broad spectrum. There have been a number of studies describing the interaction of nonmonochromatic pulses with plasmas in terms of probability. For example, Gorbunov and Salikhov45 considered quasi-stationary nonmonochromatic radiation. Thus, it is correct to describe the problem in terms of probability per unit time assuming incident energy flux to be constant. Hence, the interaction can be described in terms of the scattering cross-section corrected by the spectral width of the nonmonochromatic radiation. Nevertheless, such considerations are not related to the pulse duration τ. In our approach, the stationary radiation flux is absent, and we therefore deal with the scattering probability for the whole time, which is the novel aspect of our approach. The revealed nonlinear dependence is an effect that appears when the ratio of the spectral width to the carrier frequency of the pulse is of the order of 10⁻² and more.

When ωi1res<ωc<ωi2res [Fig. 3(c)], the τ dependence is not monotonic for low durations, a clear maximum and minimum appear. As the pulse duration increases, the τ dependence becomes linear. A similar trend is found in some regions ωc∈Uδ1(ωi1res−Δω1)∪Uδ2(ωi2res+Δω2) [Fig. 3(d)], where Δω_1,2 are small detunings from the resonant frequencies (the pink areas in Fig. 4). In contrast to the case illustrated in Fig. 3(a), as τ increases, the probability grows. This is due to scattering on the thermal fluctuations. In spite of the fact that in the case considered here, the magnitude of this process is several orders lower than the plasmon peaks, it is nevertheless not zero, and the integral (3.3) provides linear growth. The same applies for ωc∈Uδ1∪Uδ2, owing to the tails of the plasmon peaks.

With regard to the approximations (4.7) and (4.9), this approach does not predict linear growth of the τ dependence for long durations for the cases illustrated in Figs. 3(c) and 3(d), because it does not take scattering on thermal fluctuations into account. In spite of the magnitude discrepancy, the probability values obtained in both ways are of the same order. The values of τ_max corresponding to the probability maximum have a discrepancy of ∼10%.

The value of τ_max tends to infinity in the neighborhoods of the resonant frequencies (Fig. 5), which corresponds to monotonic growth of the τ dependence [Fig. 3(b)].

We have generalized the theory of radiation scattering on electron collective excitations in a Maxwellian plasma to the case of USLPs. Drawing on formulas obtained in the framework of second-order quantum-mechanical perturbation theory, we have extended Fermi’s golden rule and derived a formula for the scattering probability that takes into account the contribution of each spectral component of an incident USLP.

Scattering of a USLP on Langmuir excitations (plasmons) in an equilibrium plasma has been described in terms of the differential scattering probability during all the time of pulse action. We have studied the trends according to which plasmon components appear in the plasma form-factor spectrum and have assessed the behavior of the components corresponding to plasmon excitation and thermal motion fluctuations for different parameters of the problem. For the case in which the plasmon components dominate over the thermal motion and there is no overlap between the two plasmon peaks, an expression connecting the plasmon energy and the incident radiation frequency has been obtained. Moreover, in the framework of this approximation, we have derived an analytical expression for the scattering probability.

The way in which the scattering probability depends on the pulse duration (the τ dependence) has been investigated for different values of the carrier frequency by examining the spectrum of the plasma form factor S(ω_i). We have demonstrated that τ dependence can be nonmonotonic and that, in particular, for some values of the carrier frequency it can exhibit a maximum and a minimum. The value of τ_max corresponding to the local maximum has been studied as a function of the carrier frequency. It has been shown that τ_max tends to infinity in the vicinity of the plasma resonant frequencies, which corresponds to monotonic growth. An important feature of USLPs has been demonstrated, namely, that in contrast to quasi-monochromatic pulses, the scattering probability of USLPs can be nonzero for a carrier frequency lying beyond the scattering cross-section band, owing to the broad pulse spectrum.

Acknowledgment. The study reported here was funded by MIPT in the framework of the 5-Top-100 Program. This work has also been supported by RFBR, Project No. 19-32-90016.