Characterizing extreme laser intensities by ponderomotive acceleration of protons from rarified gas

Since the ponderomotive force (4) of the laser is in the opposite direction to the intensity gradient, in the field of a tightly focused laser pulse, protons are pushed predominantly in the direction perpendicular to the laser propagation. Figure 3(a) shows φ-integrated proton spectra as a function of ϑ-angle, which is the polar angle measured from the direction of laser propagation (the positive direction of the z-axis). The angle φ is the azimuthal angle in the plane perpendicular to the z-axis. It is measured from the direction of the laser polarization, i.e. the positive direction of the x-axis. The majority of the ponderomotively accelerated protons move in the $\vartheta =90^\circ$ direction. This angle also refers to the emission direction of the most energetic particles (see figure 3(a)).

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Here we analyze the impact of laser parameters on the proton energies. For a proton initially at rest close to the laser focus where the phase fronts are approximately planar, the laser intensity may be separated into $I={I}_{0}\,g(\vec{r})\,f(\xi )$, where I₀ is the peak laser intensity, $g(\vec{r})$ is the spatial distribution and f(ξ) is the pulse shape with respect to the coordinate ξ = t − z/c, which was introduced in the previous section. Assuming the proton moves a negligible distance during the laser pulse, which is valid provided ${a}_{0{\rm{p}}}\ll 1$, the energy gain of the protons, ε_p, can be roughly estimated from ponderomotive approximation in the following way:

$$\begin{eqnarray}&&{\varepsilon }_{{\rm{p}}}=\displaystyle \frac{2{\pi }^{2}{q}^{4}}{{m}^{3}{c}^{2}{\omega }^{4}}\displaystyle \frac{{\tau }_{\mathrm{FWHM}}^{2}}{{w}_{0}^{2}}{I}_{0}^{2}{\left({\vec{{\rm{\nabla }}}}_{* }g({\vec{r}}_{* })\right)}^{2}{\left({\int }_{-\infty }^{\infty }f({\xi }_{* }){\rm{d}}{\xi }_{* }\right)}^{2},\end{eqnarray} \tag{ 5 }$$

where ${\xi }_{* }=\xi /{\tau }_{\mathrm{FWHM}}$ and ${\vec{r}}_{* }=\vec{r}/{w}_{0}$, ${\vec{{\rm{\nabla }}}}_{* }={w}_{0}\vec{{\rm{\nabla }}}$ are normalized variables. This formula shows that the particles with the highest energy, ε_max, are initially located in the area of the maximum intensity gradient (${[\vec{{\rm{\nabla }}}g(\vec{r})]}_{\max })$, which agrees with the results obtained from the numerical modeling of the interaction with a focused Gaussian laser pulse (see figure 2(a)). Thus, the proton spectra has a maximum value ε_max of the achieved energies (see numerically calculated spectrum with ε_max ≈ 350 keV in figure 3(b)) that hereinafter will be named the 'cutoff energy'.

This formula allows comparing energies, ε_p, of protons accelerated by the ponderomotive force with energies, ε_C, gained through the 'Coulomb explosion', if the proton concentration would be rather high. The problem of ion acceleration by electrostatic field of a volume of charge has been studied in detailed in the paper [49], where linear dependence of the energy ε_C on the laser intensity has been demonstrated. Comparison of the formula (5) with the result of that paper gives the following relation: ${\varepsilon }_{{\rm{p}}}\sim {a}_{0{\rm{p}}}^{2}{\varepsilon }_{{\rm{C}}}$, where ${a}_{0{\rm{p}}}\lt 1$ for the intensity range considered in this paper. It means that the regime of 'Coulomb explosion' results in higher proton energies.

The formula (5) shows that energies of protons are proportional to the square of the peak intensity, if the other parameters introduced in the equation are fixed. Figure 4 illustrates the cutoff energies (dots in the form of circles, triangles and squares) obtained in numerical calculations as a functions of the peak intensity for different values of the focal spot size. The lines represent theoretical estimations based on the formula (5) and the assumption of the Gaussian spatial-temporal form of the laser pulse near the focal spot. For the laser intensities in the range from 10²¹ to 10²³ W cm⁻², the results obtained with theoretical and numerical approaches are in a good agreement with each other for different values of the focal spot size. Thus the cutoff energy of the proton spectra can be estimated by the following analytical formula: ${\varepsilon }_{\max }\approx 1.7\times {10}^{-10}{I}_{{\rm{L}}18}^{2}{({\tau }_{\mathrm{FWHM}}[\mathrm{fs}]/{D}_{{\rm{F}}\lambda })}^{2}$(keV), where ${I}_{{\rm{L}}18}={I}_{{\rm{p}}}/{10}^{18}$ W cm⁻² for the laser parameters (wavelength and spatial-temporal profile) similar to ones considered in this paper.

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The formula (5) has been deduced under the assumption that the shift of protons during the interaction with the laser pulse is negligible as compared with the focal spot size, i.e. the intensity gradient retains its own value in the position of each particle. However, higher laser intensities leads to substantial shifting the particles, therefore the effect of the gradient variation through the interaction time changes the dependence of proton energies on the laser peak intensity. Figure 4 shows deviations of the results, that have been calculated for the laser diameters D_F = 1.3λ and D_F = 2λ, from the fitting lines. The smaller focal spot and the higher laser intensity requires proton shifting to be taken into account. Net proton energies also depend on the time τ_int of the interaction with the laser beam, which is equal to the pulse duration (${\tau }_{\mathrm{int}}\approx {\tau }_{\mathrm{FWHM}}$, see formula (5)) in the case of lower intensities. For higher intensities, the interaction time is determined by the dynamics of accelerated protons (so that ${\tau }_{\mathrm{int}}\sim {w}_{0}/\sqrt{{I}_{p}}$). By using this ratio instead of τ_FWHM in formula (5) results in the net proton energies to be linearly proportional to the laser intensity and independent of the focal spot size. That is why, the points, corresponding to I_p = 10²⁴ W cm⁻² for D_F = 1.3λ and 2.0λ, turn out to be close together.

The purple line in figure 4 shows the energies gained during proton acceleration by pulses with the same power but different values of the focal spot size and peak intensity. In this case, the expression ${I}_{{\rm{p}}}{D}_{{\rm{F}}}^{2}$ should be held constant, which results in the proton energy being proportional to the cube of the intensity. The slope coefficient of the purple line has been obtained after fitting the numerical results as 3, which is also in agreement with the theory. Thus, the cutoff energy of the protons and the laser intensity are connected and are a function of both the spot size and the laser pulse power. The proton cutoff energy differs from the electron energy characteristics, where expansion of the focal size at first leads to growth of their energies and then to their reduction [39].

The estimation of the final energy shows its value to be proportional to the square of the laser duration. However this estimate is only valid when the drift of the protons is negligible compared with the spatial scale of the laser pulse throughout the interaction. This restricts the range of valid laser durations, which can be estimated by the following approximate relation: ${a}_{0p}c{\tau }_{\mathrm{FWHM}}\lt {D}_{{\rm{F}}}$. Figure 5 shows the cutoff energy as a function of the laser duration on a double logarithmic scale. Results from numerical calculations are shown with red dots and a linear fit of the form y = kx + b on this scale is indicated by the blue line. The fit coefficient is k ≈ 2, which is in agreement with the theoretical hypothesis of squared dependence of the cutoff energy. It means that the simulation parameters indeed refer to a regime of proton dynamics with negligible spatial drift during the interaction time.

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In this regime of the laser-particle interaction, the accelerated proton energy depends on the integral of the temporal distribution of the laser pulse (5). It results in the same energy distribution of protons for laser pulses with the same energy W and focal spot size D_F, where the laser pulse energy is $W=\int I(x,y,t){\rm{d}}{x}{\rm{d}}{y}{\rm{d}}{t}$. This assumption refers to the negligible drift of protons as well as the case discussed above. We have carried out a number of numerical calculations for laser pulses with different temporal profiles (see in figure 6(a)) and equal energies, peak intensities and focal spot sizes. We have considered Gaussian, 6-order Gaussian distributions and their different configurations. The results show that the temporal form of a pulse with fixed pulse energy does not impact the cutoff energies of the proton spectra for the peak laser intensity 10²² W cm⁻², and therefore this characteristic is not uniquely determined by the accelerated protons spectrum (figure 6(b)). A recent paper [39] shows that the electron distributions are sensitive to the temporal profile of the laser pulse, and so can be used as a complementary diagnostic in this case. Hence, if the laser pulse is free of femtosecond features and has a high level of energy contrast, the laser duration can be estimated with a good accuracy by the proposed method. Our calculations have shown that the protons gain 1% higher energy during the interaction with the laser pulse, which has additional 1% of its energy in picosecond prepulse. Therefore, this introduces an insignificant contribution to the results of the proposed diagnostic.

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As it was shown in section 3.2, the displacement of the protons becomes significant relative to the focal spot size for a laser intensity of 10²⁴ W cm⁻². We have performed a series of numerical calculations for the same laser profiles and given peak laser intensity. Similar to the case of the laser interaction with electrons, the temporal form of the laser pulse impacts the proton energies in this regime, as shown in figure 6(c). A laser pulse with a steeper leading edge results in greater values of the maximum proton energy: 32 MeV for $f(t)=\exp (-{t}^{6}/{\tau }_{0}^{6})$ (blue) and 23 MeV for a double Gaussian laser pulse with a femtosecond prepulse (green).

In the previous section we discussed the impact of the laser parameters on the proton energy characteristics. The ponderomotive approximation also allows us to analyze the angular distributions of accelerated particles. From the expression (4) introduced to the equation of motion, the relation between components of the proton momentum and laser intensity distribution is given by:

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{d}}{p}}_{i}}{{\rm{d}}{t}}=-\displaystyle \frac{2\pi {q}^{2}}{m{\omega }^{2}c}\displaystyle \frac{\partial I(\vec{r},t)}{\partial {r}_{i}},\end{eqnarray} \tag{ 6 }$$

where the dummy index i indicates either the x, y or z coordinate. Because the emission angles (ϑ and φ) of the particle are determined by the values of the momentum projections, ${p}_{x}=p\cos \varphi \sin \theta$, ${p}_{y}=p\sin \varphi \sin \theta$, p_z = p cos θ, features of the laser focal distribution will correspond to properties of the angular distributions of protons.

Let us consider two laser pulses with different spatial profiles in the focal plane (figure 7) and the same peak intensity, I_p = 5 × 10²² W cm⁻², and duration, τ_FWHM = 36 fs. One of them is a laser beam with the Gaussian initial spatial profile studied in the previous sections, $g(x^{\prime} ,y^{\prime} )=\exp (-({x}^{{\prime} 2}+{y}^{{\prime} 2})/{w}_{0}^{2})$, the other is a half-Gaussian with non-zero values for x' > 0 with initial profile proportional to $\exp (-({x}^{{\prime} 2}+{y}^{{\prime} 2})/{w}_{0}^{2}){\rm{\Theta }}(x^{\prime} )$, where Θ(x') is the Heaviside step function. Thus, in the focal plane, the first pulse is axially symmetric while the second one turns into an ellipse with the major axis oriented along the x-axis with ${D}_{{\rm{F}}x}\approx 2.94\lambda$, ${D}_{{\rm{F}}y}$ along the minor axis y is approximately equal to 1.57λ. Such change of the spatial distribution can be explained because the far-field distribution is related to the Fourier transform of the near-field distribution, so introducing an aperture in one dimension will increase the diameter in reciprocal space in that dimension.

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These laser beams allow us to analyze the effect of the spatial inhomogeneity on the angular-energy distributions of protons. Figure 8 illustrates ϑ-integrated spectral distributions of protons, the results were received by the numerical calculations. Although the f_# of the focusing systems were the same, the smallest diameter of the elliptical focal spot is larger than one of the φ-symmetrical laser beam (1.57λ and 1.3λ, respectively). It leads to the smallest energies of protons accelerated by the half-Gaussian laser pulse for the same peak laser intensities, as shown in section 3.2. The formula (6) keeps a similar form in cylindrical coordinates for the radial component of proton momentum, which means the φ-energy distributions should be φ-symmetric for particles accelerated by a Gaussian beam, since the intensity does not depend on φ-angle. This assumption is in agreement with results obtained from our numerical calculations (see in figure 8(a)). For the elliptical laser beam, we consider two orthogonal directions, φ = 0° and φ = 90°, for particles propagating in the ϑ = 90° direction, which correspond to the momentum components p_x and p_y respectively. Since ${p}_{x}/{p}_{y}\simeq {D}_{{\rm{F}}y}/{D}_{{\rm{F}}x}$, which can be evaluated from formula (6), the ratio between maximum energies detected along the φ = 0° and 90° directions will be proportional to the square of this expression. Figure 8(b) shows the φ-angular spectral distribution of protons accelerated by the laser pulse pictured in figure 7(b). As predicted by the theoretical hypothesis, the maximum energy detected along the x-axis achieves lower values than the same characteristic of the particles propagating along y-axis. The ratio of ε_p(φ = 0°) to ε_p(φ = 90°) is equal to 0.3, which agrees with the estimate ${({D}_{{\rm{F}}y}/{D}_{{\rm{F}}x})}^{2}\approx 0.3$. Thus, this laser diagnostics method also enables us to evaluate the ratio of the beam focal widths. The ponderomotive force is independent of the laser polarization (see formula (4)). It means that replacing the x'-coordinate by y' in the function Θ(x') only leads to rotating the proton distribution in figure 8(b) to 90°, i.e. maximum energies will be detected along φ = 0°and 180°.

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The diagnostic method considered in this paper is not limited to a Gaussian spatial form of the laser pulse. Based on formula (5), the maximum energy of protons can be calculated for the laser pulses with different focal distributions, including profiles with peripheral rings. In this case, the maximum energy of protons will be defined by the intensity at the main (central) peak of the spatial profile, whereas peripheral rings should lead an increase in the number of protons with intermediate energies, as it has been shown for electrons in [39]. At the same time our approach is restricted by the intensity gradient of the rings, which should be less than that of the main peak. Let consider a pathological example; the laser pulse has an on axis intensity minimum. In this case, the assumption that the proton energy is correlated with the diameter and intensity of the main peak, which does not exist, is erroneous. It means that we should know some a priori information about the focal spot distribution. However, this method can be supported by a similar method based on electrons [39].

Figure 3(a) indicates an angular width, ${\rm{\Delta }}\vartheta$, measured at the half-maximum level, in the angularly resolved energy spectrum. To estimate how the laser intensity profile relates to this characteristic, we start from formula (6) under the same assumption ${a}_{0{\rm{p}}}\,c\,{\tau }_{\mathrm{FWHM}}\lt {D}_{{\rm{F}}}$ as before, which results in the following relations in cylindrical coordinates, $r=\sqrt{{x}^{2}+{y}^{2}}$ and z:

$$\begin{eqnarray}&&\begin{array}{l}{p}_{r}(r,z)\propto \displaystyle \frac{\partial I(r,z)}{\partial r},\quad {p}_{z}(r,z)\propto \displaystyle \frac{\partial I(r,z)}{\partial z},\\ \tan \alpha (r,z)=\displaystyle \frac{{p}_{z}}{{p}_{r}}=\displaystyle \frac{\partial I(r,z)}{\partial z}{\left(\displaystyle \frac{\partial I(r,z)}{\partial r}\right)}^{-1},\end{array}\end{eqnarray} \tag{ 7 }$$

where α is the angle between the direction of particle emission and ϑ = 90°. The final energy of the protons is proportional to the sum of squared components of their momenta:

$$\begin{eqnarray}&&{\varepsilon }_{{\rm{p}}}(r,z)\propto {p}_{r}^{2}+{p}_{z}^{2}\propto {\left(\displaystyle \frac{\partial I(r,z)}{\partial r}\right)}^{2}+{\left(\displaystyle \frac{\partial I(r,z)}{\partial z}\right)}^{2}.\end{eqnarray} \tag{ 8 }$$

The results produced above show the highest-energy particles propagating along ϑ = 90° and that the longitudinal component of their momentum, p_z, is zero. If the initial position of such particles is $\{{r}_{0},{z}_{0}\}$, then their energy can be expressed in the form:

$$\begin{eqnarray}&&{\varepsilon }_{\max }\propto {p}_{r}^{2}\propto {\left({\left.\displaystyle \frac{\partial I(r,z)}{\partial r}\right|}_{{r}_{0},{z}_{0}}\right)}^{2}.\end{eqnarray} \tag{ 9 }$$

Hence, the initial position of protons with energy ε_max/2 may be determined as an implicit function r of z:

$$\begin{eqnarray}&&{\left(\displaystyle \frac{\partial I(r,z)}{\partial r}\right)}^{2}+{\left(\displaystyle \frac{\partial I(r,z)}{\partial z}\right)}^{2}=\displaystyle \frac{1}{2}{\left({\left.\displaystyle \frac{\partial I(r,z)}{\partial r}\right|}_{{r}_{0},{z}_{0}}\right)}^{2}.\end{eqnarray} \tag{ 10 }$$

Introducing the substitution of variables: Z = z/z_R and R = r/D_F, where z_R corresponds to the FWHM intensity along the z-axis, i.e. the Rayleigh range, and using the expression for tanα, we can rewrite (10) in the following form:

$$\begin{eqnarray}&&{\left(\displaystyle \frac{\partial I(R,Z)}{\partial R}\right)}^{2}\quad [1+{\tan }^{2}\alpha ]=\displaystyle \frac{1}{2}{\left({\left.\displaystyle \frac{\partial I(R,Z)}{\partial R}\right|}_{{R}_{0},{Z}_{0}}\right)}^{2}.\end{eqnarray} \tag{ 11 }$$

In our calculations, the squared tangent of the emission angle α of the protons with energies ${\varepsilon }_{{\rm{p}}1/2}={\varepsilon }_{\max }/2$ was small, α ≪ 1, and so it can be neglected in formula (11). Thus, using normalized scales {R, Z} the initial positions of the particles, which gain approximately ε_p1/2 energies, have almost no dependence on the focal spot size (see figure 9(a)). Let this area be determined by the function R = R_i(Z).

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Based on the introduced substitution, the formula (7) for $\tan \alpha (R,Z)$ is represented in the following form:

$$\begin{eqnarray}&&\displaystyle \frac{\tan \alpha (R,Z)}{{D}_{{\rm{F}}}/{z}_{R}}=\displaystyle \frac{\partial I(R,Z)}{\partial Z}{\left(\displaystyle \frac{\partial I(R,Z)}{\partial R}\right)}^{-1}.\end{eqnarray} \tag{ 12 }$$

To find the value of Δϑ/2 for given focal spot size, which refers to the maximum α-value for particles initially located in the area R = R_i(Z), we should maximized the value attained by the right part of the equation (12) under the condition R = R_i(Z):

$$\begin{eqnarray}&&\displaystyle \frac{\tan ({\rm{\Delta }}\vartheta /2)}{{D}_{{\rm{F}}}/{z}_{R}}=\max {\left[\displaystyle \frac{\partial I(R,Z)}{\partial Z}{\left(\displaystyle \frac{\partial I(R,Z)}{\partial R}\right)}^{-1}\right]}_{R={R}_{{\rm{i}}}(Z)}.\end{eqnarray} \tag{ 13 }$$

Since both the maximized function and the condition do not depend on the focal spot size, if the maximum value (i.e. the right-hand side of the equation) exists, that was observed in the numerical simulations, it also does not depend on the focal spot size. As a result, the left part of the equation (13) remaining fixed for different values of the focal diameter. Consequently, tan (Δϑ/2) is inversely proportional to z_R/D_F. Since z_R is nonlinearly related with D_F (e.g. the square dependence for paraxial approximation [50]), the ratio z_R/D_F takes on a different value for various focal spot sizes. The considered characteristic of proton spectra does not depend on the laser peak intensity in the ranges of the laser parameters discussed at the beginning of this section. This fact was verified by a series of numerical calculations. At the same time the results demonstrated here were received for I_p = 5 × 10²² W cm⁻².

Table 1 contains the spatial characteristics of the laser pulse (D_F, z_R) and corresponding angular widths of the proton spectra. The numerical calculations show that the ratio of $\tan ({\rm{\Delta }}\vartheta /2)$ to the expression (D_F/z_R) is maintained for different spot sizes, being approximately equal to 0.8 in the case of the laser pulse with the Gaussian initial spatial profile. The points corresponding to these results are shown in figure 9(b) by the red dots. The green squares represent semi-theoretical results, that have been obtained by two sequential steps: (i) numerical calculation of the Rayleigh length for the different focal diameters and (ii) transition from the value of D_F/z_R to the angular width of proton spectra by formula (13). Figure 9(b) also illustrates with the blue line the result of data approximation, this curve is converted into the dashed line for $\tan ({\rm{\Delta }}\vartheta /2)$ exceeding 0.14. It means that for the latter range of the angular widths the determination of the focal spot size by this plot should take into account error associated with no smallness of ${\rm{\Delta }}\vartheta /2$ (or equivalently α). If the error has an unacceptably high value, the numerical calculations can be used to specify the result.

Table 1.  Numerically calculated angular width of proton spectra for different values of the focal spot size for the Gaussian laser pulse.

f#	DF	zR	Δϑ, deg.	$\tan ({\rm{\Delta }}\vartheta /2)$	$\tan ({\rm{\Delta }}\vartheta /2)/({D}_{{\rm{F}}}/{z}_{{\rm{R}}})$
1	1.3λ	7.8λ	15.4	0.14	0.84
1.5	2.0λ	18λ	9.8	0.086	0.77
3	3.9λ	73.6λ	4.95	0.043	0.81

Therefore, the proposed method of the laser diagnostics allows estimating the focal spot size by means of corresponding calculations, based on the technique discussed above, or the graph in figure 9(b) (for Gaussian laser pulses). Using the angular widths of the proton spectra for the laser diagnostics requires minimizing the effect of multiple proton scattering due to collisions of protons with hydrogen nuclei (ion–ion collisions). It leads to both distortion of the proton angular distribution and energy losses. The effect of proton interactions with ambient gas is considerably weaker. With regard to the angular spreading, the limit of the gas thickness, l, (proton propagation distance) and density, n, can be roughly estimated by the following expression [51]: ${nl}\lt 1.5\times {10}^{19}{({\rm{\Delta }}{\vartheta }_{\mathrm{er}})}^{2}{(2{\varepsilon }_{{\rm{p}}}/{m}_{{\rm{e}}}{c}^{2})}^{2}[$ cm⁻²], where Δϑ_er is the acceptable angular error in degrees, that should be as low as the accuracy of the angular measurement, m_e is the electron mass. The energy losses of the accelerated protons can be also neglected for small angular widening. Thus for Δϑ_er = 1° and ε_p = 2 keV, the gas parameters are limited by nl < 10¹⁵ [cm⁻²]. It can be fulfilled in experiments, for instance, by using a gas jet of low concentration (<10¹⁶ cm⁻³) and 1 mm thickness. For more accurate estimations of the proton elastic scattering and energy losses, commonly used computer software [52, 53], which model ion propagation though matter using a Monte Carlo method can be used.

Using the results discussed above, we propose the following scheme of laser pulse diagnostics (see figure 10). The laser pulse is focused by the off-axis parabolic mirror with an effective focal length F_eff and an off-axis angle ψ_off into a rarefied hydrogen gas. The concentration of the gas should be sufficiently low to satisfy three conditions: (i) the insignificance of plasma effects (section 2); (ii) stopping distance for protons with energy E ∼ ε_max/10, where ε_max corresponds to the peak laser intensity (figure 4), should be much larger than the distance to the detector (section 3.5); (iii) the laser power is below the critical power for the relativistic self-focusing P_L < P_crit = 17(ω/ω_pe)² [GW]. The most demanding restriction is related to proton scattering, it is overcome by using a gas jet [with centimeter-thickness and low concentration(<10¹⁶cm⁻³)]. The number of particles in the interaction area exceeds 10⁴ for this gas density and the smallest focal spot considered in the paper.

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For the detection of the accelerated protons, the chevron type MCP detectors working in the TOF mode or nuclear track detectors CR-39 can be used. The MCPs have a detection efficiency of ∼10% for protons with E ∼ 0.1 keV and of ∼70% for protons with E ≥ 1 keV [54], providing intensity measurements above 10²¹ W cm⁻² (figure 4). For the MCP operation, a vacuum better than 0.1 mTorr is required. If the pressure in the experimental chamber is above this value, differential pumping of the MCP can be used. The time of arrival to the detector situated at a distance L_det and the proton energy are connected as $t={L}_{\det }/\sqrt{2E/m}$. By measuring the time of arrival of the accelerated protons, their spectrum can be reconstructed. Ideally, it is necessary to use several MCP detectors positioned at different ϑ-angles relative to the laser propagation direction as indicated on figure 10. This will enable collecting the information about the angular and spectral distribution of protons.

The CR-39 detectors have ∼100% detection efficiency for protons above 30 keV [55]. Such energy detection threshold for the CR-39 sets a lower boundary for a peak laser intensity of (2−5) × 10²² W cm⁻², depending on the focal spot size. CR-39 detectors can work in an accumulation mode, collecting information for hundreds or even thousands of laser shots. Multiple CR-39 detectors can also be placed at the different ϑ-angles and at different φ-angles. The later will provide information about the axial asymmetry of the focal distribution (see section 3.4). This configuration of the experiment allows the laser focal spot size and a peak intensity to be inferred.