**6. Experimental results**

First, the gas jet and the effect of a background pressure on the resulting flow structure are investigated using the techniques described in the previous section. The derived results are compared to theoretical relations discussed in Section 3. Subsequently, the effect of the barrel shock on the plasma generation is explored and the brilliance improvement of the soft X-ray source is quantified.

#### **6.1. Characterization of the target gas jet**

Depending on the stagnation and background pressure, the gas jet may form various shapes, which are discussed in the following. In previous studies of laser-produced soft X-ray sources, the nozzle was operated in the range *p*<sup>0</sup> = 11–17 bar at a background pressure of *pb* = 10−4 mbar, i.e., practically without any background gas. In this case, the emerging flow is in the scattering regime and does not show any discontinuities. Independently of *p*0, the density distribution has a maximum value at the nozzle exit and rapidly falls off in all directions. The corresponding Schlieren images are taken with the knife edge aligned with the *y*-axis and can be found in **Figure 14** for the pressure range *p*0 = 11–17 bar.

**Figure 14.** Principle of plasma characterization by diode measurement.

**Figure 13.** Principle of plasma characterization by pinhole camera.

pinhole camera as described above.

**6.1. Characterization of the target gas jet**

**Figure 14** for the pressure range *p*0 = 11–17 bar.

**6. Experimental results**

90 High Energy and Short Pulse Lasers

source is quantified.

In good approximation, the lifetime of the plasma is assumed to be *τ* = 6 ns, which equals the duration of the exciting laser pulse. Finally, the luminescent area *A* is determined with a

First, the gas jet and the effect of a background pressure on the resulting flow structure are investigated using the techniques described in the previous section. The derived results are compared to theoretical relations discussed in Section 3. Subsequently, the effect of the barrel shock on the plasma generation is explored and the brilliance improvement of the soft X-ray

Depending on the stagnation and background pressure, the gas jet may form various shapes, which are discussed in the following. In previous studies of laser-produced soft X-ray sources, the nozzle was operated in the range *p*<sup>0</sup> = 11–17 bar at a background pressure of *pb* = 10−4 mbar, i.e., practically without any background gas. In this case, the emerging flow is in the scattering regime and does not show any discontinuities. Independently of *p*0, the density distribution has a maximum value at the nozzle exit and rapidly falls off in all directions. The corresponding Schlieren images are taken with the knife edge aligned with the *y*-axis and can be found in

With rising background pressure, particle collisions increasingly affect the gas jet and retard its free expansion. At a certain distance from the nozzle, this results in a shock that is directly connected to a local decrease of the Mach number *M*. At the same time, the local particle density increases. This becomes evident in regions of the Schlieren images that show strong changes in intensity, implying high-density gradients. As can be seen, e.g., in **Figure 14(b)**, the shape of the resulting shock structure resembles a barrel, why it is referred to as a barrel shock. In the downstream direction, the barrel shock is terminated by the Mach disk, which is indicated in the Schlieren image by an arrow. In the present Schlieren pictures, the Mach disk is reproduced only weakly because the knife edge was aligned perpendicular to the disk and only density gradients parallel to the disk were detected.

Increasing *pb*, as from **Figure 14(b)–(c)**, results in a confinement of the gas flow toward the nozzle axis—the lateral shocks approach each other and the Mach disk moves upstream. In contrast to this, increasing *p*<sup>0</sup> has the opposite effect, i.e., the radius of the barrel shock and the width *σ* of the density distribution increase and the Mach disk moves downstream, see **Figure 14(b)–(d)**. These two opposite effects allow generation of the same shock structure at different combinations of the pressures, provided that the ratio *p*0/*pb* stays constant.

From the Schlieren images as shown in **Figure 14**, the distance *lM* between the Mach disk and the nozzle throat is derived for pressure ratios in the range 18 ≤ *p*0/*p*b ≤ 340. In **Figure 15**, the resulting data set is compared to the empirical relation (11)

$$d\_M = 0.67 \cdot d\_\circ \left(\frac{p\_0}{b\_b}\right)^{1/2} \tag{16}$$

which has been derived by Ashkenas and Sherman [37] for a nondivergent nozzle. Apparently, the experimental results deviate from the depicted curve, especially for large pressure ratios. Most likely, this can be attributed to a different nozzle geometry as in the present situation. Here, a divergent orifice initially guides the supersonic expansion of the gas before it expands freely into the helium atmosphere. For that case, a relation of the form

$$d\_M = d\_s \left(\frac{p\_0}{p\_b}\right)^a \tag{17}$$

reveals good agreement with the measured shock distances as it is evident in **Figure 15**. By a least-squares fit routine, the exponent is derived to *a* = 0.4034.

**Figure 15.** Schlieren images indicating the supersonic flow structure of an N2 jet as a function of stagnation and back‐ ground pressure (flow direction: top → bottom). (a) Scattering regime, no internal structures evolve; (b) continuum re‐ gime with barrel shock structure, the Mach disk is indicated by the arrow; (b) → (c) shock structure contracts for increasing background pressure; (b) → (d) shock structure inflates for increasing stagnation pressure.

In **Figure 16**, wavefront and Schlieren measurements are compared with each other for a stagnation pressure of *p*<sup>0</sup> = 11 bar and a background pressure of *pb* = 170 mbar. The results of both techniques are well consistent. The particle density *N*(*x*, *y*) shows the mean gas distribu‐ tion inside the jet. In the downstream direction, along the nozzle axis, *N* first decreases to *N*min = 4.0 × 1018 cm−3 and then increases again up to a maximum value of *N*max = 9.8 × 1018 cm−3. Subsequently, the wave-like behavior of the particle density is repeated at lower density values. The observed maxima coincide approximately with the positions where the lateral shocks interfere, forming a Mach disk.

which has been derived by Ashkenas and Sherman [37] for a nondivergent nozzle. Apparently, the experimental results deviate from the depicted curve, especially for large pressure ratios. Most likely, this can be attributed to a different nozzle geometry as in the present situation. Here, a divergent orifice initially guides the supersonic expansion of the gas before it expands

> 0 *a*

= (17)

*b*

reveals good agreement with the measured shock distances as it is evident in **Figure 15**. By a

**Figure 15.** Schlieren images indicating the supersonic flow structure of an N2 jet as a function of stagnation and back‐ ground pressure (flow direction: top → bottom). (a) Scattering regime, no internal structures evolve; (b) continuum re‐ gime with barrel shock structure, the Mach disk is indicated by the arrow; (b) → (c) shock structure contracts for

In **Figure 16**, wavefront and Schlieren measurements are compared with each other for a stagnation pressure of *p*<sup>0</sup> = 11 bar and a background pressure of *pb* = 170 mbar. The results of both techniques are well consistent. The particle density *N*(*x*, *y*) shows the mean gas distribu‐ tion inside the jet. In the downstream direction, along the nozzle axis, *N* first decreases to *N*min = 4.0 × 1018 cm−3 and then increases again up to a maximum value of *N*max = 9.8 × 1018 cm−3. Subsequently, the wave-like behavior of the particle density is repeated at lower density values. The observed maxima coincide approximately with the positions where the lateral

increasing background pressure; (b) → (d) shock structure inflates for increasing stagnation pressure.

shocks interfere, forming a Mach disk.

*p* æ ö ç ÷ è ø

*M e*

*<sup>p</sup> l d*

freely into the helium atmosphere. For that case, a relation of the form

92 High Energy and Short Pulse Lasers

least-squares fit routine, the exponent is derived to *a* = 0.4034.

**Figure 16.** Distance between Mach disk and nozzle exit for various pressure ratios. The points are derived from the Schlieren images, the violet curve represents the empirical relation (11) from Ashkenas and Sherman [37] and the blue curve represents the modified relation given in Eq. (17) with *a* = 0.4034.

Employing the relations of gas dynamics introduced in Section 3, a rough theoretical estimate of the particle density ahead and behind the Mach disk is now compared to the results obtained with the wavefront sensor. Corresponding to the experimental situation, the density distribu‐ tion along the symmetry axis of the gas jet is shown in terms of its stagnation value *ρ*0 as shown in **Figure 6**. There, a normal shock induces a density increase from *ρ*min = 0.039 *ρ*0 to *ρ*max = 0.170 *ρ*0 in a distance of *lM* = 2.7 mm to the nozzle throat. For the current pressure ratio of *p*0/*pb* = 64.7, this coincides with the position of the Mach disk.

In order to derive absolute density values, the stagnation density *ρ*0 of the nitrogen jet is required. Following the ideal gas law, *p*0 = *ρ*<sup>0</sup> *R*sp *T*<sup>0</sup> results in *ρ*<sup>0</sup> = 12.65 kg/m3 with the specific gas constant for nitrogen *R*sp,N2 =296.8 J/(kg⋅K) [41], the temperature *T*<sup>0</sup> = 293 K, and pressure *p*0 = 11 bar inside the vessel. Finally, particle densities ahead and behind the Mach disk are evaluated with the molecular mass of nitrogen *mN*<sup>2</sup> =4.653×10−<sup>26</sup> kg [41]. A comparison between the theoretical estimation and the measured values is given in **Table 1**.


**Table 1.** Particle density ahead (*N*min) and after (*N*max) the barrel shock, given on the symmetry axis of the jet. Comparison between theoretical estimate and measurement.

The estimated values are of the same order of magnitude but larger than the experimental results. This discrepancy can be attributed to the spatial resolution of the wavefront sensor that is not able to resolve the high density value right behind a shock. Furthermore, the estimate provides an upper limit of the particle density since in a simplification a conical source flow has been assumed. In fact, the stream lines of the flow are bended in lateral direction, stronger than the cone geometry presumes. Consequently, this results in a higher rarefaction of the gas and the typical bulbous barrel shock. This explains why values of the estimated particle densities, both of the maximum and the minimum, are higher than the corresponding measured values.

#### **6.2. Characterization of the plasma enhancement**

The effect of an increase in target gas density on the plasma generation is illustrated in **Figure 17** for a stagnation pressure of *p*0 = 11 bar. Taking advantage of the barrel shock, obviously the brightness of the plasma is raised, whereas its size has decreased in the direction of the incident laser beam. Due to the increased target density, there are more emitters of soft X-ray radiation in the same volume. Besides, the absorption of laser energy is raised. Thus, the power density of the beam decreases more rapidly below its critical value and no further atoms are ionized. This confines the size of the plasma in the beam direction and explains its smaller size. Another mechanism causing the reduced size might be plasma defocusing [47]. Due to an increased plasma density, a stronger defocusing effect can be expected, limiting the ionization region. During the experiments, it turned out that generation of a plasma right below the Mach disk, where the density is expected to be at a maximum, is not the optimal position. It was found that even brighter and smaller plasmas occur when the laser is focused onto the edge of the jet at a location slightly above the Mach disk and after the barrel shock (see **Figure 17**). This behavior may be caused by reabsorption of soft X-rays by the surrounding nitrogen particles.

**Figure 17.** (a) Combination of quantitative wavefront and qualitative Schlieren image of the N2 jet expanding from *p*0 = 11 bar into an He atmosphere with *pb* = 170 mbar. (b) Density distribution *N*(*x*, *y*) of the N2 jet in the plane *z* = 0, which results from the wavefront as described in Section 5.2.

The barrel shock is enclosed by a thin supersonic compressed layer, which becomes thicker at the Mach disk [36], leading to increased reabsorption. In order to study the brilliance im‐ provement depending on the location of plasma generation with respect to shock structures in the jet, the latter were varied by changing the background pressure at a constant stagnation pressure (*p*<sup>0</sup> = 11 bar). By lowering *pb*, the radius of the barrel shock is increased; conversely, with increasing *pb*, the radius of the barrel shock decreases. Thus, with the location of the focus of the laser beam fixed, its relative location with respect to high-density regions behind the shock is changed. In **Figure 18**, intensity distributions of the plasma are shown for various background pressures *pb*. In this case, the location of plasma generation is kept constant. An optimum value is found at *pb* = 170 (see also **Figure 17**).

provides an upper limit of the particle density since in a simplification a conical source flow has been assumed. In fact, the stream lines of the flow are bended in lateral direction, stronger than the cone geometry presumes. Consequently, this results in a higher rarefaction of the gas and the typical bulbous barrel shock. This explains why values of the estimated particle densities, both of the maximum and the minimum, are higher than the corresponding

The effect of an increase in target gas density on the plasma generation is illustrated in **Figure 17** for a stagnation pressure of *p*0 = 11 bar. Taking advantage of the barrel shock, obviously the brightness of the plasma is raised, whereas its size has decreased in the direction of the incident laser beam. Due to the increased target density, there are more emitters of soft X-ray radiation in the same volume. Besides, the absorption of laser energy is raised. Thus, the power density of the beam decreases more rapidly below its critical value and no further atoms are ionized. This confines the size of the plasma in the beam direction and explains its smaller size. Another mechanism causing the reduced size might be plasma defocusing [47]. Due to an increased plasma density, a stronger defocusing effect can be expected, limiting the ionization region. During the experiments, it turned out that generation of a plasma right below the Mach disk, where the density is expected to be at a maximum, is not the optimal position. It was found that even brighter and smaller plasmas occur when the laser is focused onto the edge of the jet at a location slightly above the Mach disk and after the barrel shock (see **Figure 17**). This behavior may be caused by reabsorption of soft X-rays by the surrounding nitrogen particles.

**Figure 17.** (a) Combination of quantitative wavefront and qualitative Schlieren image of the N2 jet expanding from *p*0 = 11 bar into an He atmosphere with *pb* = 170 mbar. (b) Density distribution *N*(*x*, *y*) of the N2 jet in the plane *z* = 0, which

The barrel shock is enclosed by a thin supersonic compressed layer, which becomes thicker at the Mach disk [36], leading to increased reabsorption. In order to study the brilliance im‐ provement depending on the location of plasma generation with respect to shock structures in the jet, the latter were varied by changing the background pressure at a constant stagnation pressure (*p*<sup>0</sup> = 11 bar). By lowering *pb*, the radius of the barrel shock is increased; conversely, with increasing *pb*, the radius of the barrel shock decreases. Thus, with the location of the focus of the laser beam fixed, its relative location with respect to high-density regions behind the shock is changed. In **Figure 18**, intensity distributions of the plasma are shown for various

measured values.

94 High Energy and Short Pulse Lasers

**6.2. Characterization of the plasma enhancement**

results from the wavefront as described in Section 5.2.

**Figure 18.** Pinhole camera images of the plasma superimposed on the Schlieren images of gas jet at *p*<sup>0</sup> = 11 bar. Left: under vacuum conditions *pb* = 10−4 mbar. Right: with ambient He atmosphere at *pb* = 170 mbar. Both plasma images are an average of 30 single shots.

Unexpectedly, increasing both *p*0 and *pb* while preserving the pressure ratio *p*0/*pb*, does not lead to a considerable further increase in the brilliance of the source. Approaching high-pressure values (*p*0 = 17 bar), quite the reverse happens: the plasma appears even darker. It can be assumed that, in fact, more soft X-ray photons are generated since the target density is increased. However, the density of the background gas is increased as well, which leads to higher reabsorption of the generated photons. The latter effect seems to dominate the former. It is expected that further efforts in differential pumping can shorten the path length of the soft X-rays through the outer helium gas so that the brilliance of the source can further be increased.

Now, parameters characterizing the plasma in the optimal case are compared with those of a plasma produced near the nozzle exit with a jet in the scattering regime. In both cases, the same stagnation pressure of *p*0 = 11 bar is considered. Regarding the shape of the resulting plasma, which is represented by its luminescent area, it can be seen that the radiating area is reduced by a factor of 0.71 to *A* = 0.063 mm2 , and its eccentricity decreases slightly from *ε* = 0.91 to *ε* = 0.80 when a barrel shock is present. This results in a better brilliance and improves the coherence properties due to a smaller source size and a more uniform shape. The number of photons emitted per pulse and solid angle from the nitrogen plasma at a wavelength of *λ* = 2.88 nm is raised by a factor of 7.1 to a value of 1.2 × 1013 sr−1. Based on these values, the peak brilliance can be computed. One finds an improvement by a factor of 10 to a value of *Br* = 3.15 × 1016 photons/(mm2 mrad2 s). This clearly demonstrates the advantage of utilizing the density increase across a barrel shock system. An overview of the characteristic parameters of the plasma is given in **Table 2**.


**Table 2.** Comparison of plasma emission characteristics at *λ* = 2.88 nm obtained with a nitrogen jet issuing into vacuum (no barrel shock) and into a background gas (with barrel shock).
