#### 4.1. First Discovery of the Ion Acoustic Turbulence in Relativistic Laser–Plasmas Using X-ray Spectroscopy

Let us consider the experimental profiles of Si XIV Ly

_{γ}, produced in an interaction with a single laser pulse with initial laser intensity at the surface of the target estimated as 1.01 × 10

^{21} W/cm

^{2} (black trace in

Figure 1b), and Si XIV Ly

_{β}, produced in an interaction with a single laser pulse with initial laser intensity at the surface of the target estimated as 0.24 × 10

^{21} W/cm

^{2} (blue trace in

Figure 1b) [

12]. For these profiles the peak-intensity contrast ratio was 10

^{9}. We mention that L-dips were not observed in the experimental profile of Si XIV Ly

_{β} line in the black trace

Figure 1b. It seems that a process of self-absorption of radiation prevented L-dips from being visible in this line. We note that without the self-absorption the Ly

_{β} line would show a relatively deep central minimum (related not to the the L-dips, but to the structure of the Stark splitting of this line) and shallow L-dips. The self-absorption completely washes out the L-dips but does not completely washes out the central minimum: it just makes it shallow instead of being relatively deep. As for the Ly

_{γ} line in the black trace

Figure 1b, at the possible locations of the L-dips, the experimental profile already merged into the noise.

The experimental black trace profile Si XIV Ly

_{γ} shows a pair of two L-dips nearest to the line center (

q = ± 1,

s = 1) separated from each other by 28 mÅ yielding an electron density of

N_{e} = 3.6 × 10

^{22} cm

^{−3}. In addition to this, a pair of L-dips, separated from each other by 56 mÅ, were also observed, representing a superposition of two pairs of the L-dips:

q = ± 2,

s = 1 and

q = ± 1,

s = 2. From the Equation (2) this yields the same

N_{e} = 3.6 × 10

^{22} cm

^{−3}, thus reinforcing the interpretation of the experimental dips as the L-dips—caused by the resonant interaction of the Langmuir waves, developed at the surface of the relativistic critical density, with the quasi-static electric field. Let us remark that at the initial laser intensity at the surface of the target estimated as 1.01 × 10

^{21} W/cm

^{2}, the relativistic critical density would be

N_{cr} = 2.3 × 10

^{22} cm

^{−3} according to Equation (4). However, due to several physical effects [

26,

27] (self-focusing of the laser beam, Raman and Brillouin back scattering), the actual intensity of the transverse electromagnetic wave in the plasma can be significantly higher than the intensity of the incident laser radiation at the surface of the target. In the present experiment for the relativistic critical density to be approximately equal to the density

N_{e} = 3.6 × 10

^{22} cm

^{−3} deduced by the spectroscopic analysis, it would require the enhancement of the initially estimated intensity of the transverse electromagnetic wave at the surface of the target due to the above physical effects just by a factor of two. In both cases, L-dips separated by 28 mÅ and for the L-dips separated by 56 mÅ, the mid-point between the two dips in the pair practically coincides with the unperturbed wavelength

λ_{0}. This is a strong indication that the quasi-static field

F was dominated by the LET. If the LET would be absent, then according to Equation (1), the mid-point of the pair of the L-dips separated by 28 mÅ should have been shifted by 5.8 mÅ to the red with respect to

λ_{0} and the mid-point of the pair of the L-dips separated by 56 mÅ would have been similarly shifted by 10.7 mÅ to the red with respect to

λ_{0}, these shifts being due to the spatial non-uniformity of the ion micro-field reflected by the second term in Equation (1). Another strong indication of the presence of the LET observed from the above result comes from the analysis of the broadening of this spectral line that was performed using code FLYCHK [

34]. This code, which does not take into account the Stark broadening by the LET (and the presence of L-dips), yielded

N_{e} = 0.9 × 10

^{23} cm

^{−3}, i.e., almost three times higher than the actual

N_{e} = 3.6 × 10

^{22} cm

^{−3} for the best fit to the experimental profile as shown in

Figure 2a. This result is not reasonable.

The analysis of the experimental profile of Si XIV Ly

_{β} with the initial laser intensity at the surface of the target estimated as 0.24 × 10

^{21} W/cm

^{2} (blue trace in

Figure 1b), shows a situation similar to Si XIV Ly

_{γ}. There is a pair of the L-dips separated from each other by 43 mA. The electron density deduced from the separation within the pair of these L-dips, is

N_{e} = 1.74 × 10

^{22} cm

^{−3} (assuming

|q|s = 2 in Equation (2)). This pair of dips corresponds either to the Stark component

q = 1 for a two-quantum resonance

s = 2 or to the Stark component

q = 2 for the one-quantum resonance

s = 1. The superposition of two different dips at the same location results in a L-super-dip with a significantly enhanced visibility. The location of the would-be L-dips, corresponding to |

q|s = 1, is too close to the central, most intense part of the experimental line profile so that they are not observed either due to a self-absorption in the most intense part of the profile, or because the relatively small values of the field

F, corresponding to the central part of the profile, are not quasi-static. The mid-point between the two dips in the pair practically coincides with the unperturbed wavelength

λ_{0}. This is again a strong indication that the quasi-static field

F was dominated by the LET. If the LET would be absent, then according to Equation (1), the mid-point of this pair of L-dips should have been shifted by 6.1 mÅ to the red with respect to

λ_{0}. Another strong indication of the presence of the LET in the profile of Si XIV Ly

_{β} comes from the analysis performed using code FLYCHK [

34] (

Figure 2b). An electron density of

N_{e} = 3 × 10

^{23} cm

^{−3}, was obtained, i.e., 17 times higher than the experimentally verified

N_{e} = 1.74 × 10

^{22} cm

^{−3}. This is yet another strong indication of an additional Stark broadening by the LET (not accounted for by FLYCHK). At the initial laser intensity at the surface of the target estimated as 0.24 × 10

^{21} W/cm

^{2}, the relativistic critical density would be

N_{cr} = 1.1 × 10

^{22} cm

^{−3} according to Equation (4). However, again due to the self-focusing, as well as Raman and Brillouin backscattering, the actual intensity of the transverse electromagnetic wave in the plasma can be significantly higher. In the present case, for the relativistic critical density to be approximately equal to the density

N_{e} = 1.74 × 10

^{22} cm

^{−3} deduced by the spectroscopic analysis, again it would require the enhancement of the intensity of the transverse electromagnetic wave due to the above physical effects just by a factor of two.

From both analysis of the Si XIV Ly_{γ} and Si XIV Ly_{β} profiles, it can be confirmed that the LET developed simultaneously with the Langmuir waves at the relativistic critical density surface and thus it is most likely to be an ion acoustic turbulence. The most probable and the best studied mechanism for developing Langmuir waves at the surface of the relativistic critical density is a parametric decay, which is a nonlinear process where the pump wave (t_{1}) excites both the Langmuir wave (l) and an ion-acoustic wave (s): t_{1} → l + s.

#### 4.2. Robust Computations for the Analysis of the Experimental Si XIV Ly_{γ} and Si XIV Ly_{β} Profiles

The robust computations including L-dips and the spectral line broadening by LET (

Section 3.3) performed for the analysis of the Si XIV Ly

_{γ} and Si XIV Ly

_{β} profiles are given in

Figure 3a,b. The comparisons with the experimental spectra is fruitful as confirmed in the following.

From the experimental profile of Si XIV Ly

_{γ} (

Figure 3a), it follows that the root-mean-square value of

F_{t} was

F_{t,rms} = 2.1 GV/cm. For comparison, the characteristic ion microfield

F_{i,typ} = 2.603

eZ^{1/3}N_{e}^{2/3} was 1.0 GV/cm. From the half width of the experimental L-dips, by using Equation (3), we found the amplitude of the Langmuir wave to be

E_{0} = 0.6 GV/cm. The resonant value of the quasi-static field

F_{res}, determined by the condition of the resonance between the separation of the Stark sublevels and the plasma frequency 3

nħF_{res}/(2

Z_{r}m_{e}e) =

ω_{pe}, was 3.1 GV/cm, so that the validity condition for the existence of L-dips

E_{0} < F_{res} (

Section 3.2) was satisfied.

From the experimental profile of Si XIV Ly

_{β} (

Figure 3b), it follows that the root-mean-square value of

F_{t} was

F_{t,rms} = 3.9 GV/cm. For comparison, the characteristic ion microfield was

F_{i,typ} = 0.6 GV/cm. From the halfwidth of the experimental L-dips, by using Equation (3), we found the amplitude of the Langmuir wave to be

E_{0} = 1.0 GV/cm. The resonant value of the quasi-static field was

F_{res} = 5.8 GV/cm, so that the validity condition for the existence of L-dips

E_{0} < F_{res} was again satisfied.

The good agreement between experimental spectra and simulations reinforces the discovery of the simultaneous production of LET with the Langmuir waves. We note that the electron densities involved turned out to be much lower than the densities deduced using FLYCHK simulations, which ignored the LET and the L-dips. The densities used for the robust computations are in perfect agreement with the densities deduced experimentally from the dips separations.

We also performed experiments where the spectrometer viewed the laser-irradiated front surface of the target. As an example,

Figure 4 shows the experimental spectrum of Al XIII Ly

_{β} line (4 µm Al foil coated by 0.45 µm CH), which was obtained in a single laser shot with duration of 0.9 ps and the laser intensity at the surface of the target theoretically estimated as 6.7 × 10

^{20} W/cm

^{2}. The peak-intensity contrast ratio was 10

^{9} as for the experimental spectra Si XIV Ly

_{β} and Ly

_{γ}. This spectrum exhibits two pairs of L-dips: one pair—at ± 16.8 mA from the line center, another pair—at ± 33.6 mA from the line center. The two dip pairs (two pairs of dips) are symmetrical with respect to the line center, confirming the production of LET with the Langmuir waves. Also, the density used for the robust computation is in perfect agreement with the density deduced experimentally from the dips separations.

#### 4.3. In-Depth Study of ISS in the X-ray Range Emission from Relativistic Laser–Plasmas

Recently new experiments have been performed at RAL [

17] allowing a peak-intensity contrast ratio exceeding 10

^{−11} (instead of 10

^{−9} for the experimental results discussed in the paper up to now) due to a plasma mirror allowing an amplified spontaneous emission. An in-depth study was then undertaken for the case Si XIV Ly

_{β} allowing a reliable reproducibility of the Langmuir-wave-induced dips at the same locations in the experimental profiles as well as of the deduced parameters (fields) of the Langmuir waves and ion acoustic turbulence in several individual 1 ps laser pulses and of the peak irradiance of 1 to 3 × 10

^{20} W/cm

^{2}.

Figure 5 shows the comparisons of the robust computations, allowing in particular, for the LET and L-dips, with the corresponding experimental profiles from shots A, B, C. The L-super-dip is observed twice in the experimental profiles: one in the blue part and the other in the red part. These L-super-dips are located practically symmetrically at the distance

Δλ_{dip}(N_{e}) = 24 mÅ from the unperturbed wavelength. According to Equation (3) with |

q|

s = 2, this translates into the electron density

N_{e} = 2.2 × 10

^{22} cm

^{−3} The theoretical profiles were calculated for this density 2.2 × 10

^{22} cm

^{−3} and the temperature

T = 600, 550, and 600 eV for shots A, B, and C, respectively. The comparison demonstrates a good agreement between the theoretical and experimental profiles, and thus reinforces the good interpretation of these experimental profiles. Moreover, it reinforces that it was the PDI at the surface of the relativistic critical density that produced simultaneously the Langmuir waves and the ion acoustic turbulence in shots A, B, and C. The modeling of the experimental profiles A, B, C using the code FLYCHK [

34] (

Figure 6) confirmed once more the need to introduce the LET for the interpretation. This innovative code yielded

T = 500 eV and

N_{e} = 6 × 10

^{23} cm

^{−3}. This value of

N_{e} is one and a half orders of magnitude higher than the electron density

N_{e} = 2.2 × 10

^{22} cm

^{−3} deduced from the experimental L-dips.

The determination of the fields of the Langmuir waves and ion acoustic turbulence can be deduced from the experimental Si XIV Ly

_{β} profiles A, B, C (

Figure 5) and their robust computations. The values of the amplitude of the Langmuir waves using (3) have been obtained:

E_{0} = 0.7, 0.5, and 0.6 GV/cm for shots A, B, and C, respectively. The resonant value of the quasi-static field

F_{res} responsible for the formation of the L-dips, can be determined from the resonance condition

ω_{F} = s ω_{pe}(

N_{e}). For shots A, B, and C, it yields:

F_{res} = 6.5 GV/cm for

s = 2 and

F_{res} = 3.25 GV/cm for

s = 1. These values of

F_{res} are about 10 and 5 times higher than the Langmuir wave amplitude

E_{0}, respectively. Thus, the condition

E_{0} <<

F_{res}, necessary for the formation of the L-dips, was fulfilled. These values for

F_{res} are coherent with the root-mean-square field values of the LET introduced for the robust computations:

F_{t,rms} = 4.8, 4.4, and 4.9 GV/cm for shots A, B, and C respectively. For comparison, the characteristic ion micro-field

F_{i,typ} = 2.603 eZ1/3Ne2/3 was 1.5 GV/cm.

Now we proceed to analyze the experimental profile of the Si XIV Ly

_{β} line in shot D (

Figure 5). The experimental profile does not show bump–dip–bump structures—in distinction to shots A, B, and C. In shot D, the incident laser intensity was

I = 8.8 × 10

^{19} W/cm

^{2}, significantly lower than in shots A, B, and C. The corresponding relativistic critical density is

N_{cr} = 6.6 × 10

^{21} cm

^{−3}. The modeling of this spectra using the code FLYCHK yielded

N_{e} = 1.7 × 10

^{23} cm

^{−3}, which is one and a half order of magnitude higher than the relativistic critical density and by an order of magnitude higher than the region of the electron density

N_{e} = 2.2 × 10

^{22} cm

^{−3} from which the experimental profiles were emitted in shots A, B, and C is not reliable for the interpretation of shot D profile. The most probable interpretation of the experimental profile in shot D is the following.

In shot D the electron density was significantly lower than in shots A, B, and C. Therefore, the damping of the Langmuir waves was significantly lower, which could allow the Langmuir waves to reach a significantly higher amplitude.

Figure 5d shows the comparison of the experimental profile from shot D with robust computing based on the code allowing for the LET and the Langmuir waves at

N_{e} = 6.6 × 10

^{21} cm

^{−3},

T = 550 eV,

F_{t,rms} = 2.0 GV/cm,

E_{0} = 2.0 GV/cm. It is seen that this theoretical profile is in a good agreement with the experimental profile and it does not exhibit bump–dip–bump structures. For high Langmuir wave amplitudes—i.e., when the ratio

E_{0}/

F_{res} > 0.5—the L-dips cannot form (see

Section 3.3) and the resonant value of the quasi-static field is given by (5). For

N_{e} = 6.6x10

^{21} cm

^{−3} and

E_{0} = 2.0 GV/cm, Equation (5) yields

F_{res} = 1.7 GV/cm for

s = 1 (so that

E_{o}/

F_{res} = 1.2) and

F_{res} = 3.5 GV/cm for

s = 2 (so that

E_{o}/

F_{res} = 0.6). Thus, both for the one-quantum resonance (

s = 1) and for the two-quantum resonance (

s = 2), we get

E_{0}/

F_{res} > 0.5, so that the L-dips were not able to form. We note that, while for shots A, B, and C the electron density could be deduced from the locations of the L-dips and from the robust computation, in shot D the only one possibility is by modeling the entire experimental profile using the code that allows for the interplay of the LET and the Langmuir waves. Up to now, it seems to be the first experimental profile diagnosed in relativistic laser–plasma interactions and explained with the simultaneous production of Langmuir waves and ion acoustic turbulence, but with no dips exhibited.