2. EXPERIMENTAL SETUP AND METHOD

The experiment was carried out using the short-pulse high-power J-KAREN laser of Japan Atomic Energy Agency (JAEA), which emits 25 TW pulses of 40 fs duration. The use of a double chirped-pulse amplifier (CPA) with an optical parametric chirped-pulse preamplifier, a main amplifier, and a saturable absorber results in extremely high temporal contrast exceeding 10¹⁰ on sub-nanosecond time scales before the main pulse (Kiriyama et al., Reference Kiriyama, Mori, Nakai, Shimomura, Sasao, Tanoue, Kanazawa, Wakai, Sasao, Okada, Daito, Suzuki, Kondo, Kondo, Sugiyama, Bolton, Yokoyama, Daido, Kawanishi, Kimura and Tajima2010). The wavelength of the laser is 800 nm. Figure 1 shows a schematic setup of the experiment. The laser passed through a laser beamline and was guided to an experimental chamber, where it was focused on a Xe cluster target using an F/9 off-axis parabolic mirror. The beamline and chamber were kept under a vacuum condition using vacuum pumps. The best focal diameter (4 · σ) defined by the four-sigma method [ISO, 2005] was 35 µm × 36 µm.

Fig. 1.

Experimental setup.

A conical nozzle was utilized to produce the Xe cluster target. The throat diameter of the nozzle was 0.5 mm and the output diameter was 1 mm. The Xe gas was supplied by a gas cylinder outside the experimental chamber and the backing pressure was controlled by a gas regulator. The Xe clusters were prepared by the adiabatic expansion of the Xe gas flowing inside the nozzle. The peak laser intensity on the cluster was changing the X position of the nozzle as shown in the figure. Y and Z positions of the nozzle were fixed to Y = 0 mm and Z = 1.5 mm.

The X-rays were measured with a PI-LCX400 hard X-ray charge-coupled device (CCD) manufactured by Princeton Instruments. According to the datasheet, the CCD has a quantum efficiency of about 1.5% for Xe K-shell X-rays with an energy of approximately 30 keV. The energy resolution of the CCD for 30 keV X-rays is about 0.32 keV (Hayashi et al., Reference Hayashi, Pirozhkov, Kando, Fukuda, Faenov, Kawase, Pikuz, Nakamura, Kiriyama, Okada and Bulanov2011). The CCD can be treated as a multielement detector since it contains 5.2 × 10⁵ silicon elements. Therefore, the CCD can be utilized in the single-photon counting mode (Issac et al., Reference Issac, Vieux, Ersfeld, Brunetti, Jamison, Gallacher, Clark and Jaroszynski2004). A 200 µm aluminum filter was set in front of the CCD. The filter removes not only the scattered laser light but also low-energy X-rays (≤10 keV), which is important for the single-photon counting mode. The CCD was oriented at an angle of 35° from the laser propagation direction in the horizontal plane (xy plane) and 45° from the vertical axis. The distance from the CCD to the point of laser-cluster interaction was about 0.9 m and the solid angle of the CCD was 2.65 × 10⁻⁴ sr.

3. ESTIMATION OF CLUSTER RADIUS

In this section, we discuss a formula for estimating the cluster radius. Hagena carefully measured cluster radii using several types of nozzle and found an empirical equation for estimating the cluster radius involving the Hagena parameter Γ* (Hagena, Reference Hagena1992):

(1)$$\overline N=33 \cdot \left({\Gamma^{\ast} \over 1000}\right)^{2.35} \quad \lpar 10^3\lt \Gamma ^ {\ast} \lt 10^4 \rpar \comma \;$$

(2)$$\Gamma^{\ast} ={5554 \cdot P \over T_0 ^{2.29}} \cdot \left({0.74 \cdot \Phi \over \tan{\rm \alpha} } \right)^{0.85}\comma \;$$

where $\overline N $ is the average number of atoms comprising a cluster, P is the backing pressure in mbar, T ₀ is the initial gas temperature in K, Φ is the throat diameter of the nozzle in μm, and α is the half-angle of the region of gas expansion. The mean cluster radius R _cl is calculated by the equation $R_{cl}=R_{ws} \cdot \mathop {\overline N }\nolimits^{1/3}$ for Wigner-Seitz radius R _ws, which is 0.273 nm [Krainov & Smirnov, Reference Krainov and Smirnov2002] for Xe gas.

Eqs. (1) and (2) can be used when the parameter Γ* is smaller than 10⁴. For Γ* ≥ 10⁴, the cluster radii can be measured by an electron diffraction approach, and the following relation between $\overline N $ and the slightly corrected parameter Γ_c* (Danylchenko et al., Reference Danylchenko, Kovalenko and Samovarov2008) has been obtained:

(3)$$\overline N=\exp [{-} 12.83+3.51 \times \lpar \ln {\Gamma }_c^ {\ast} \rpar ^{0.8}] \; \comma \;$$

(4)$${\Gamma }_c^ {\ast} =\displaystyle{{5554 \cdot P} \over {\mathop {T_0 }\nolimits^{2.3} }} \cdot \mathop {\left({\displaystyle{{0.74 \cdot \Phi } \over {\tan{\rm \alpha} }}} \right)}\nolimits^{0.8} \, \lpar {\Gamma }_c^ {\ast} \gt 10^4 \rpar \;.$$

When the Xe gas and clusters can be treated as a compressible fluid, the Mach number M is estimated by

(5)$$\displaystyle{A \over {A^ {\ast} }}=\displaystyle{1 \over M}\mathop {\left[{\displaystyle{{2+\lpar {\rm \gamma} - 1\rpar \cdot M^2 } \over {{\rm \gamma}+1}}} \right]}\nolimits^{{\textstyle{{{\rm \gamma}+1} \over {2\lpar {\rm \gamma} - 1\rpar }}}} \; \comma \;$$

where A is the output area of the nozzle, A* is the throat area of the nozzle, and γ is the specific heat ratio of the gas.

On the basis of fluid dynamics (Semushin & Malka, Reference Semushin and Malka2001), the angle α is obtained from the expression

(6)$$\sin{\rm \alpha}={1 \over M}.$$

From these formulas, it is possible to estimate the cluster radius. In our experiment, Φ = 500 µm and T ₀ = 297 K. The ratio of the output area to the throat area of the nozzle, A/A*, is 4, and we assume that the flow of the Xe clusters is supersonic. Cluster radii estimated from Eqs. (3)–(6) are shown in Table 1. The cluster radius is larger than 10 nm for pressures exceeding 2 × 10⁴ mbar.

Table 1.

Xe cluster size produced by conical nozzle

4. RESULTS AND DISCUSSION

We first placed the nozzle 250 µm from the best-focus point in the laser propagation direction. Figure 2 shows the relationship between the peak laser intensity I and the position X from the best-focus point in μm. The peak laser intensity is estimated from the measured laser beam profile, laser energy, and laser pulse duration. Usually, the peak laser intensity I is found by (ISO, 2005)

(7)$$I=\displaystyle{{2 \cdot P} \over {{\rm \pi} \cdot \Delta t}} \cdot \mathop {\left[{w^2+M^2 \times \mathop {\left({\displaystyle{{{\rm \lambda} \cdot X} \over {{\rm \pi} \cdot w}}} \right)}\nolimits^2 } \right]}\nolimits^{ - 1} \; \comma \;$$

where P is the laser energy, Δt is the FWHM pulse duration of the laser, w is a value corresponding to the focal spot size, λ is the laser wavelength, and M ² is the beam quality factor.

Fig. 2.

Peak laser intensity dependence on position from best-focus point. Squares: data; solid line: fitted curve.

By fitting Eq. (7) to the measured results in Figure 2, the peak laser intensity at 250 µm from the best-focus point is found to be 9 × 10¹⁸ W/cm². The experiment was carried out at a Xe gas backing pressure of 1.1 × 10⁴–4 × 10⁴ mbar. The cluster radius was found to be 8–17 nm, as shown in Table 1.

X-ray energy spectra averaged over 25 laser shots are shown in Figure 3. X-ray photon numbers at the source point are calculated by using the transmittance of the aluminum filter and the quantum efficiency of the CCD.

Fig. 3.

Energy spectra for cluster radii R _Cl at peak laser intensity of 9 × 10¹⁸  W/cm². (a) R _Cl = 8 nm; (b) R _Cl =12 nm; (c) R _Cl =13 nm; (d) R _Cl = 15 nm; (e) R _Cl = 16 nm; (f) R _Cl = 17 nm.

A peak with an energy of approximately 28.5 keV can be observed. We attribute this peak to the Xe K-shell X-ray, which is generated from the Xe K-shell upon excitation and ionization by the laser plasma electrons. The peak energy is slightly lower than the real energy of the Xe K_α X-ray. Since a lower energy X-ray source (Fe⁵⁵: 5.9 keV) was utilized for the energy calibration of the CCD, the peak energy was slightly underestimated. The broad energy spectrum from 10 to 40 keV is due to the bremsstrahlung X-ray.

The Xe K-shell X-ray photon number is calculated by subtracting the background counts from the integrated counts from 28.2 keV to 28.8 keV and is summarized in Figure 4. As the cluster radius increases from 8 to 12 nm, the Xe K-shell X-ray photon number increases from 2 × 10⁵ to 6.7 × 10⁵ photons/sr. Then, the photon number becomes saturated at around 7 × 10⁵ photons/sr, when the cluster radius increases from 12 to 17 nm.

Fig. 4.

Xe K-shell X-ray photon numbers and cluster radii. Squares: 9 × 10¹⁸ W/cm²; closed circles: 5 × 10¹⁸ W/cm².

A similar cluster radius dependence was found for Ar K-shell X-ray generation simulated by Deiss (Reference Deiss2009). Laser-cluster interaction was simulated with a mean-field classical transport approach. The Ar K-shell X-ray photon number increases sharply with increasing cluster radius, and saturates at the radius of 10 nm order. Then the photon number gradually decreases upon further increase of the radius.

We consider that skin depth is one of the reasons for the above-described behavior. The skin depth δ is the distance that the laser can penetrate into the overdense plasma and is estimated (Milosavljevic & Nakar, Reference Milosavljevic and Naker2006)

(8)$${\rm \delta}=\displaystyle{{c\sqrt {\rm \gamma} } \over {{\rm \omega} _p }}=5.31 \times 10^5 \sqrt {\displaystyle{{\rm \gamma} \over {n_e [{\rm cm}^{ - 3}] }}} \quad [{\rm cm}] \comma \;$$

(9)$${\rm \gamma}=\sqrt {1+3.613 \times 10^{ - 19} I [{\rm W}/{\rm cm}^2] {\rm \lambda} ^2 [{\rm \mu} {\rm m}^2] }\comma \;$$

where c is the light speed, ω_p is the plasma frequency, n _e is the electron density in cm⁻³, γ is the Lorentz factor, I is the peak laser intensity, and λ is laser wavelength. When the electron density n _e of 3.5 × 10²³ cm⁻³ is assumed in our experiment, the skin depth is estimated as 12 nm. Therefore, the energy absorption rate of the cluster increases with increasing radius up to the skin depth of 12 nm.

When K-shell electrons are excited or ionized, K-shell X-rays are emitted by the recombination of the K-shell. Excitation or ionization of K-shell electrons occurs when the electron energy is higher than the K-shell excitation energy or ionization potential. The ionization potential and excitation energy of the Xe K-shell are 40 and 35 keV, respectively. For this reason, electrons with energy exceeding 35 keV are required for Xe K-shell X-ray generation. The distribution function of the bremsstrahlung X-ray spectrum f is known to be related to electron temperature T _e (McCall, Reference McCall1982; Issac et al., Reference Issac, Vieux, Ersfeld, Brunetti, Jamison, Gallacher, Clark and Jaroszynski2004):

(10)$$\,f\lpar E\rpar =\displaystyle{{A_e } \over {\mathop {T_e }\nolimits^{1.5} }} \cdot \sqrt E \cdot \exp\lpar {-} E/T_e \rpar \; \comma \;$$

where E is the photon energy, A _e is the factor depending on the X-ray photon number, and T _e is the electron temperature.

By fitting Eq. (10) to the bremsstrahlung X-ray spectra, high electron temperatures T _e of 11.8–16.2 keV and A _e of (2.1–2.4) × 10⁸ are obtained for cluster radii of 12–17 nm (see Table 2). At the cluster radius of 8 nm, T _e and A _e are reduced to 9.6 keV and 0.9 × 10⁸, respectively. The lower X-ray photon number at the cluster radius of 8 nm is due to the decrement of A _e and T _e.

Table 2.

Parameters of X-ray energy spectrum under each condition

When the nozzle position was moved 500 µm from the position of best-focus in the laser propagation direction, the peak laser intensity on the target was 5 × 10¹⁸ W/cm² (see Fig. 2). Figure 5 shows energy spectra averaged over 25 laser shots for the cluster radii of 12 and 13 nm. The energy spectra are similar to each other, and the peak of the Xe K-shell X-ray is observed in both cases. The high electron temperatures and values of A _e are 8.2 keV and 1.5 × 10⁸, and 9.1 keV and 1.7 × 10⁸ for the cluster radii of 12 and 13 nm, respectively. These temperatures are slightly lower than that at a peak laser intensity of 9 × 10¹⁸ W/cm². The Xe K-shell X-ray photon number is 6.5 × 10⁵ photons/sr for the cluster radius of 12 nm and 6.2 × 10⁵ photons/sr for the radius of 13 nm (see Fig. 4).

Fig. 5.

Energy spectra at peak laser intensity of 5 × 10¹⁸ W/cm².

To confirm the dependence of the Xe K-shell X-ray photon number on the peak laser intensity, the cluster nozzle was moved 2 mm from the best-focus point. Figure 6a indicates the peak laser intensity at the position from the best-focus point in this measurement. From the fitted curve in Figure 6a, the peak laser intensity is found to be 2 × 10¹⁷ W/cm². The average energy spectrum of 50 laser shots for cluster radius of 13 nm is shown in Figure 6b. The X-ray photon number decreases sharply with increasing photon energy. The electron temperature T _e is clearly lower than those in the cases of Figures 3 and 5, and T _e of 4.1 keV is obtained. Almost no photons are measured around the energy of 30 keV, and no peak of the Xe K-shell X-ray is observed. Since the electron temperature is too low, most of the plasma electrons do not have sufficient energies to excite the Xe K-shell electrons.

Fig. 6.

(a) Peak laser intensity dependence on position from best-focus point. Circles: measured value; solid line: fitted curve. (b) Energy spectrum at peak laser intensity of 2 × 10¹⁷ W/cm² for cluster radius of 13 nm (solid line) and fitted curve (broken line).

It is also interesting to discuss the Xe K-shell X-ray photon number dependence on peak laser intensity. Figure 7 indicates the dependence for the Xe cluster radius of 13 nm. When the peak laser intensity is reduced from 9 × 10¹⁸ to 5 × 10¹⁸ W/cm², the photon number shows slight change. However, when the laser intensity is reduced to 2 × 10¹⁷ W/cm², the X-ray photon number decreases to the background noise level of the CCD. From the figure, it was found that the threshold laser intensity of the Xe K-shell X-ray generation exists between 2 × 10¹⁷ and 5 × 10¹⁸ W/cm².

Fig. 7.

Dependence of Xe K-shell X-ray photon number on peak laser intensity for cluster radius of 13 nm.

In order to generate a high K-shell X-ray photon number, it is important to understand the mechanism of K-shell X-ray generation. In this part, we discuss the mechanism of K-shell X-ray generation. Issac et al. (Reference Issac, Vieux, Ersfeld, Brunetti, Jamison, Gallacher, Clark and Jaroszynski2004) simulated the Kr K-shell X-ray generation using the nanoplasmas model. In the model, it is assumed that plasma electrons inside clusters are heated by inverse bremsstrahlung (Ditmire et al., Reference Ditmire, Donnelly, Rubenchik, Falcone and Perry1996). However, they have reported that the model explains only the production of electrons with a temperature of 2 keV, which is too low for Kr K-shell X-ray generation.

Another interpretation is that the high energy electrons production is due to the electron oscillation energy by the laser: ponderomotive potential U _p. Hence, we assume that plasmas electrons have energies equal to the ponderomotive potential U _p. U _p is given as 9.3 × 10⁻¹⁴I·λ² in eV by using the laser intensity I in W/cm² and laser wavelength λ in μm. The results can be seen in Table 3. Threshold laser intensities for K-shell X-ray generation are estimated under the condition that the energy of plasma electrons U _p is equal to the binding energy I _e. Two experimental results are also shown in the table. One experimental result indicates that the laser intensity of 2.9 × 10¹⁵ W/cm² is a threshold for Ar K-shell X-ray production with a 56 fs laser pulse width (Prigent et al., Reference Prigent, Deiss, Lamour, Rozet, Vernhet and Burgdorfer2008). The other result describes that the Kr K-shell X-ray is observed at the intensity of 2 × 10¹⁶ W/cm² (Issac et al., Reference Issac, Vieux, Ersfeld, Brunetti, Jamison, Gallacher, Clark and Jaroszynski2004). The laser intensities measured in the experiments are less than one-tenth of the intensities assumed from the binding energy I _e (see Table 3). This inconsistency suggests that different mechanisms are essential in X-ray generation.

Table 3.

Threshold laser intensity of K-shell X-ray generation

The threshold laser intensity for Ar K-shell X-ray generation was simulated by a mean-field classical transport approach by Deiss and coworkers (Deiss, Reference Deiss2009; Prigent et al., Reference Prigent, Deiss, Lamour, Rozet, Vernhet and Burgdorfer2008), who found that an optimum cluster radius was of 10 nm order for Ar K-shell X-ray generation. The threshold intensity of 2 × 10¹⁵ W/cm² was obtained by the simulation, and it is close to the 2.9 × 10¹⁵ W/cm² observed experimentally, as shown in Table 3. Unfortunately, Kr and Xe K-shell X-ray generation were not simulated by this approach. We anticipate that this approach will clarify the mechanism of Kr and Xe K-shell K-shell X-ray generation by laser-cluster interaction.