**2. Physical properties of a plasma**

imaging of samples in aqueous environments [10–12]. Mostly, these applications are realiz‐ ed at large-scale facilities, such as synchrotron sources or free-electron lasers. However, the demand for beam time is always too large to be satisfied by these institutions, and thus, people endeavor to transfer experiments to their laboratories. This constitutes the need of compact beam sources as can be realized by the principle of laser-produced plasmas (LPP) that is the

In order to classify and compare the radiation of different soft X-ray beam sources, the brilliance *Br* is a commonly used quantity that is the number of photons within a narrow spectral range Δ*λ*/*λ* emitted into a solid angle *Ω* from an area *A* within the time scale *τ* (typically the wavelength range Δ*λ* is defined to be 0.1% of the central wavelength *λ*) [1]:

> *Nph Br* t

the order of several tens of μm (full-widths at half-maximum, FWHM).

. /

*<sup>A</sup>* <sup>=</sup> × ×W×D (1)

· 0.1%BW) with 0.1%BW indicating the

 ll

bandwidth of 0.1%. A distinction is made between the peak brilliance, where *τ* denotes the pulse duration, and the average brilliance, where *τ* is the inverse of the repetition rate.

In comparison to synchrotrons and free-electron lasers, the brilliance of laser-produced plasma sources is several orders of magnitude lower. However, there are strategies to increase their brilliance which involve, e.g., higher power densities of the generating laser pulse. On the other hand, the density of the target material has a strong impact on the achievable number of soft X-ray photons too, whereas basically the source brilliance scales with the density. Thus, the brightest plasmas can be achieved with solids. Respective target materials are deposited on rotating cylinders [13] or quickly moving tapes [14], which provide repetition rates of up to 1 kHz. Prominent elements are gold or tin for the production of radiation at a wavelength of 13.5 nm, which is applied in EUV lithography [15]. Furthermore, there are sources employing cold gases in a solid phase, such as an argon filament that emits soft X-rays in the wavelength range 2–5 nm [16]. Achievable plasma sizes with solid targets are comparably small and on

A plasma of similar brilliance and extent is obtained with liquid targets, e.g., xenon [17], methanol [18], or tin [15]. A fluid jet [19] provides high target densities but might lead to size and brightness fluctuations. Going one step further to individual microscopic droplets [20], the advantage is the mass limitation such that the entire target material is converted into a highly ionized plasma state, supporting source stability. However, the drawback of solid and liquid target concepts is the inevitable production of fast particles and ions with kinetic energies of up to several hundred keV [21], which severely damage optics in the beam path. There are mitigation strategies to slow down the debris material such as repeller fields [22] or localized gas jet shields [23], but still the collector optics has a limited lifetime [2]. Contrarily, gaseous targets are almost free from debris [24]. Short gas pulses with durations of μs to ms are expanded from a pressure of several 10 bar into vacuum by a piezomechanical or electro‐ magnetic nozzle, resulting in a supersonic jet. Different target gases feature individual spectra

·mrad2

subject of this chapter.

76 High Energy and Short Pulse Lasers

The value of *Br* is given in the unit 1/(s·mm2

Initially, the laser beam that irradiates the target material creates ions by multiphoton absorp‐ tion, tunneling, or field ionization [27]. The resulting free electrons are accelerated by the strong electric field leading to inverse bremsstrahlung and avalanche ionization. A hot dense plasma state is generated. In competition to the heating processes, deionization takes place in terms of diffusion and recombination [27]. Depending on the electron temperature, a continuum of electromagnetic radiation is produced due to bremsstrahlung and recombination of free electrons with ions. Additionally, bound–bound transitions within the ions contribute narrow lines to the emission spectrum. A corresponding scheme is depicted in **Figure 1**.

**Figure 1.** Scheme of the emission spectrum of a hot dense plasma.

The thermodynamics within a hot dense plasma can be approximated by the idealized state of a thermal plasma that is characterized by a single electron temperature *T* and a correspond‐ ing Maxwell velocity distribution. Within that simplification, the plasma may be treated as a blackbody that emits radiation with a continuous spectrum. The assumption that photons are emitted carrying discrete quanta of energy, with energy proportional to frequency, leads to the spectral energy density [1]

$$Br = 3.146 \times 10^{11} \left(\frac{k\_g T}{\text{eV}}\right)^3 \frac{\left(\frac{\hbar \alpha}{k\_B T}\right)^3}{\frac{\hbar \alpha}{e^{k\_g T}} - 1} \frac{Photons}{mm^2 \, mrad^2 \, s \, (0.1% BW)}\tag{2}$$

here, given in units of the brilliance with Planck's constant ℏ and the photon frequency *ω*. Typical electron temperatures for gas targets irradiated by nanosecond laser pulses are 20–200 eV [28, 29]. The corresponding spectral maxima of the Planck distribution are found at the photon wavelengths 2.2–22.0 nm with peak brilliances of 3.6 × 1015 to 3.6 × 1018 1/(s mm2 mrad2 ). In fact, a laser-produced plasma is far away from thermodynamic equilibrium and a thermal plasma rather is an upper limit for the spectral power density. However, mostly a twotemperature model is already sufficient to adequately describe the continuum radiation by a hot dense plasma, which is then called near-thermal plasma [1]. In addition to the thermal electrons, a suprathermal component is introduced which is raised by nonlinear interactions such as resonant absorption. When these electrons undergo bremsstrahlung or recombination, they give rise to a high photon energy tail in the emission spectrum as indicated in **Figure 1**. Line radiation is emitted when electrons change their energy state within an ion from an outer to an inner electron shell. The resulting photon energy corresponds to the transition energy of the electron as described by Moseley's law, which is an extension of the Rydberg formula [30]

$$\frac{1}{\lambda} = \frac{R\_o}{1 + m\_e/m\_{mc}} (Z\_{ar} - S\_{sh})^2 \left(\frac{1}{n\_1^2} - \frac{1}{n\_2^2}\right) \tag{3}$$

with the Rydberg constant *R*∞, the nuclear mass *m*nuc, and the atomic number *Z*at. The constant *S*sh describes the shielding due to electrons between the core and the considered electron. Furthermore, *n*1 and *n*<sup>2</sup> are the principal quantum numbers of the initial and final states of the electron. In plasmas of species with low atomic numbers like nitrogen (*Z*at = 7), comparatively few free electrons are produced and the emitted radiation is dominated by single spectral lines. In contrast, elements with high atomic numbers such as xenon (*Z*at = 54) yield much more free electrons, resulting in a spectrum of numerous closely packed lines and a significant thermal contribution.

Another important plasma parameter is the electron plasma frequency [1]

Brilliance Improvement of a Laser-Produced Soft X-Ray Plasma http://dx.doi.org/10.5772/64149 79

$$\alpha \rho\_{\rho} = \left(\frac{e^2 n\_e}{\epsilon\_0 m\_e}\right)^{1/2} \tag{4}$$

at which the free electrons tend to oscillate (where *e* is the electron charge, *ne* is the electron density, *me* is the electron mass, and 0 is the vacuum permittivity). As a consequence, an incident electromagnetic wave can propagate in the plasma only if its frequency *ω* is greater than *ωp* and it is totally reflected if *ω* = *ωp*. This yields a critical electron density [1]

$$m\_c = \frac{\epsilon\_0 m\_e \alpha^2}{e^2} \tag{5}$$

which is *nc* = 1 × 1023 cm−3 for a common Nd:YAG laser beam with a wavelength of 1064 nm. Thus, when the plasma reaches the critical electron density, it cannot further be heated to pose a limit especially on solid and liquid target concepts. In order to mitigate that limitation, a less intense prepulse can be used to heat the target material and decrease its density precedent to the main pulse [31].

#### **3. Gas dynamics of jet targets**

The thermodynamics within a hot dense plasma can be approximated by the idealized state of a thermal plasma that is characterized by a single electron temperature *T* and a correspond‐ ing Maxwell velocity distribution. Within that simplification, the plasma may be treated as a blackbody that emits radiation with a continuous spectrum. The assumption that photons are emitted carrying discrete quanta of energy, with energy proportional to frequency, leads to

3

here, given in units of the brilliance with Planck's constant ℏ and the photon frequency *ω*. Typical electron temperatures for gas targets irradiated by nanosecond laser pulses are 20–200 eV [28, 29]. The corresponding spectral maxima of the Planck distribution are found at the photon wavelengths 2.2–22.0 nm with peak brilliances of 3.6 × 1015 to 3.6 × 1018 1/(s mm2

( )<sup>2</sup>

with the Rydberg constant *R*∞, the nuclear mass *m*nuc, and the atomic number *Z*at. The constant *S*sh describes the shielding due to electrons between the core and the considered electron. Furthermore, *n*1 and *n*<sup>2</sup> are the principal quantum numbers of the initial and final states of the electron. In plasmas of species with low atomic numbers like nitrogen (*Z*at = 7), comparatively few free electrons are produced and the emitted radiation is dominated by single spectral lines. In contrast, elements with high atomic numbers such as xenon (*Z*at = 54) yield much more free electrons, resulting in a spectrum of numerous closely packed lines and a significant thermal

1 1 1 1 *at sh e nuc*

 *m m n n* ¥ <sup>æ</sup> <sup>ö</sup> = -- <sup>ç</sup> <sup>÷</sup> <sup>÷</sup> <sup>+</sup> <sup>è</sup> <sup>ø</sup>

*<sup>R</sup> Z S*

Another important plasma parameter is the electron plasma frequency [1]

l

2 2 1 2

). In fact, a laser-produced plasma is far away from thermodynamic equilibrium and a thermal plasma rather is an upper limit for the spectral power density. However, mostly a twotemperature model is already sufficient to adequately describe the continuum radiation by a hot dense plasma, which is then called near-thermal plasma [1]. In addition to the thermal electrons, a suprathermal component is introduced which is raised by nonlinear interactions such as resonant absorption. When these electrons undergo bremsstrahlung or recombination, they give rise to a high photon energy tail in the emission spectrum as indicated in **Figure 1**. Line radiation is emitted when electrons change their energy state within an ion from an outer to an inner electron shell. The resulting photon energy corresponds to the transition energy of the electron as described by Moseley's law, which is an extension of the Rydberg formula [30]

eV (0.1% ) <sup>1</sup> *<sup>B</sup>*

*mm mrad s BW <sup>e</sup>*

(2)

(3)

3

*B B*

2 2 3.146 10

*k T k T k T Photons Br*

h

w

æ ö w

h

11

ç ÷ <sup>æ</sup> <sup>ö</sup> è ø = ´ <sup>ç</sup> <sup>÷</sup> <sup>è</sup> <sup>ø</sup> -

the spectral energy density [1]

78 High Energy and Short Pulse Lasers

mrad2

contribution.

Supersonic gas jets employed as targets inherently exhibit strong density gradients. Here, the basics of supersonic nozzle flows and related shock phenomena are described theoretically, mainly based on [32, 33]. As a result, density estimations of the target gas are provided corresponding to the experimental situation at a laser plasma source.

**Figure 2.** Sketch of a de Laval nozzle. *A* denotes the local cross-sectional area with the minimum value *A*\* at its throat position.

Let us first consider the example of a compressible fluid that expands through a convergentdivergent nozzle, a so-called de Laval nozzle as shown schematically in **Figure 2**. In gas dynamics, the basic equations to describe that problem are the conservation laws of mass and energy, formulated for compressible and isentropic flows. It can be shown that these relations lead to the well-known area relation between the local cross-sectional area *A*, the throat area *A*\* , and the local Mach number *M* [33]

$$\frac{A}{A\_\*} = \frac{1}{M} \left[ \frac{2}{\kappa + 1} \left( 1 + \frac{\kappa - 1}{2} M^2 \right) \right]^{\frac{\kappa + 1}{2(\kappa - 1)}}.\tag{6}$$

Here, *κ* = *cp*/*cv* is the ratio of specific heats (*cp* at constant pressure and *cv* at constant volume) and the Mach number *M* is defined as the ratio between the local flow velocity and the local speed of sound. In the present example of a convergent-divergent nozzle, a gas is accelerated in the convergent part according to the continuity equation. If the critical Mach number *M\** = 1 is reached at the throat, this results in supersonic velocities *M* > 1 in the divergent part, and the thermal energy of the gas is efficiently converted into directed kinetic energy. Concurrently, the gas density decreases according to the relation [32]

$$\frac{\rho}{\rho\_0} = \left(1 + \frac{\kappa - 1}{2} M^2\right)^{-\frac{1}{\kappa - 1}}\tag{7}$$

**Figure 3.** State functions of a flow in a de Laval nozzle: density *ρ* in terms of its stagnation value *ρ*0, Mach number *M* and the local cross-sectional area *A* reaching *A*\* at its throat position. A diatomic gas with *κ* = 7/5 is assumed.

The shape of the cross-sectional area *A*/*A*\* of a typical de Laval nozzle is shown in **Figure 3** together with the resulting distribution of density *ρ*/*ρ*<sup>0</sup> (*ρ*0 stagnation density) and Mach number *M* under the assumption of a diatomic gas with *κ* = 7/5. Utilizing a supersonic gas jet as a target for laser-produced plasmas requires large particle densities for high conversions efficiencies of laser energy into soft X-ray energy. Thus, a compromise needs to be found between a directed, but rarefied flow at high Mach numbers and divergent but denser flow at low Mach numbers. This can be achieved by adapting the nozzle geometry [34].

Within this work, shock waves, as they can be observed in supersonic flows, are employed to further optimize the particle density in a jet target. Within very short distances on the order of the mean-free path of the molecules, this phenomenon leads to an increase in density, pressure, and temperature while the Mach number decreases. Based on the conservation laws of mass, momentum, and energy, it is possible to derive equations that relate the initial values of those properties with the conditions right behind a shock wave. Here, it is sufficient to consider the change of the initial density *ρ* and the Mach number *M* in the case of a normal shock relative to the flow direction. After passing through the shock structure, these properties are denoted as *ρ* ^ and *<sup>M</sup>* ^ , as indicated in **Figure 4**. The corresponding shock relations read [32]

lead to the well-known area relation between the local cross-sectional area *A*, the throat area

<sup>1</sup> . 1 2 <sup>1</sup> 1 2 *<sup>A</sup> <sup>M</sup>*

Here, *κ* = *cp*/*cv* is the ratio of specific heats (*cp* at constant pressure and *cv* at constant volume) and the Mach number *M* is defined as the ratio between the local flow velocity and the local speed of sound. In the present example of a convergent-divergent nozzle, a gas is accelerated in the convergent part according to the continuity equation. If the critical Mach number *M\** = 1 is reached at the throat, this results in supersonic velocities *M* > 1 in the divergent part, and the thermal energy of the gas is efficiently converted into directed kinetic energy. Concurrently,

k


1 2( 1)

(6)

(7)

+

k

k

2

1 <sup>1</sup> <sup>2</sup>

k

*A*\*

, and the local Mach number *M* [33]

80 High Energy and Short Pulse Lasers

\*

the gas density decreases according to the relation [32]

0

r

r

<sup>1</sup> <sup>1</sup> 2 *M*


**Figure 3.** State functions of a flow in a de Laval nozzle: density *ρ* in terms of its stagnation value *ρ*0, Mach number *M*

The shape of the cross-sectional area *A*/*A*\* of a typical de Laval nozzle is shown in **Figure 3** together with the resulting distribution of density *ρ*/*ρ*<sup>0</sup> (*ρ*0 stagnation density) and Mach number *M* under the assumption of a diatomic gas with *κ* = 7/5. Utilizing a supersonic gas jet as a target for laser-produced plasmas requires large particle densities for high conversions efficiencies of laser energy into soft X-ray energy. Thus, a compromise needs to be found between a directed, but rarefied flow at high Mach numbers and divergent but denser flow at

and the local cross-sectional area *A* reaching *A*\* at its throat position. A diatomic gas with *κ* = 7/5 is assumed.

low Mach numbers. This can be achieved by adapting the nozzle geometry [34].

 k

*A M*

k

$$\frac{\hat{\rho}}{\rho} = \frac{(\kappa + 1)\mathcal{M}^2}{2 + (\kappa - 1)\mathcal{M}^2} \tag{8}$$

$$\mathcal{M} = 1 - \frac{\mathsf{M}^2 - 1}{1 + \frac{2\mathsf{K}}{\mathsf{K} + 1}(\mathsf{M}^2 - 1)}. \tag{9}$$

**Figure 4.** Normal shock structure in a supersonic flow. Gas passing through the shock experiences a decrease from the initial Mach number *M* to *M* ^ and an increase in density from *ρ* to *ρ* ^.

Basically, high Mach numbers lead to a strong compression of the fluid when passing through a shock. However, relation (8) defines an upper limit for the density ratio that can be achieved in connection with a shock wave. This limit is approached if *M* → *∞*, and for diatomic gases it is *<sup>ρ</sup>* ^ *<sup>ρ</sup>* →6 (*κ* =7 / 5). At the same time, the Mach number behind the shock decreases to *M* ^ →1/7. Shock waves appear, e.g., when obstacles perturb a supersonic flow or vice versa, or when objects travel with Mach numbers *M* > 1 through a gas at rest. In the case of a supersonic jet that expands from a stagnation pressure *p*0 into an atmosphere with a sufficiently large background pressure *pb*, shock waves can also be observed. At a certain distance to the nozzle exit, the collision between the jet particles and the surrounding gas particles leads to a shock structure, which is called barrel shock (see **Figure 5**). With respect to that situation, Muntz et al. introduced the rarefaction parameter [35]

$$
\zeta' = d\_\bullet \frac{\sqrt{p\_0 \cdot p\_b}}{T\_0} \tag{10}
$$

where *d*\* is the throat diameter of the nozzle and *T*0 denotes the stagnation temperature. This parameter describes the interaction between jet and background particles, i.e., how strong the expansion flow is influenced by the surrounding gas. Muntz et al. propose a differentiation of the occurring flow into three regimes [35]:

**•** Scattering regime *ζ* ≤ *ζ<sup>s</sup>*

Molecules of the background gas interact with the freely expanding jet by diffusion only, no distinct shock waves evolve.

**•** Transition regime *ζs* < *ζ* < *ζ<sup>c</sup>*

Thick lateral shock waves develop and confine an undisturbed core of the jet that is surrounded by a mixing zone of jet and background particles.

**•** Continuum regime *ζc* ≤ *ζ*

The fully evolved barrel shock structure is present, as shown in **Figure 5**. The inner barrel shock waves and the Mach disk spatially delimit the influence of the background gas.

**Figure 5.** Typical structure of a barrel shock as apparent at supersonic jets in the presence of a background gas. Here, a fluid is expanded from a high pressure *p*0 through the conically diverging nozzle into an ambient atmosphere of rela‐ tively low pressure *pb*. The depicted shock system represents the continuum regime. Adapted from [36].

In the continuum regime, the extent of the shock structure scales with the nozzle pressure ratio *p*0/*pb*. In particular, within the range 15 < *p*0/*pb* < 17,000, the distance *lM* = *rM* − *r*\* between the nozzle throat and the Mach disk is given by [37]

$$d\_M = 0.67 \cdot d\_\* \sqrt{\frac{P\_0}{P\_b}} \tag{11}$$

where *de* is the exit diameter of the orifice. It should be noted that this relation has been derived for nozzles with a constant diameter, i.e., for a nondivergent geometry.

0 \*

z

diffusion only, no distinct shock waves evolve.

the occurring flow into three regimes [35]:

**•** Scattering regime *ζ* ≤ *ζ<sup>s</sup>*

82 High Energy and Short Pulse Lasers

**•** Transition regime *ζs* < *ζ* < *ζ<sup>c</sup>*

**•** Continuum regime *ζc* ≤ *ζ*

the background gas.

nozzle throat and the Mach disk is given by [37]

0 *<sup>b</sup> p p <sup>d</sup> T*

where *d*\* is the throat diameter of the nozzle and *T*0 denotes the stagnation temperature. This parameter describes the interaction between jet and background particles, i.e., how strong the expansion flow is influenced by the surrounding gas. Muntz et al. propose a differentiation of

Molecules of the background gas interact with the freely expanding jet by

Thick lateral shock waves develop and confine an undisturbed core of the jet

The fully evolved barrel shock structure is present, as shown in **Figure 5**. The inner barrel shock waves and the Mach disk spatially delimit the influence of

**Figure 5.** Typical structure of a barrel shock as apparent at supersonic jets in the presence of a background gas. Here, a fluid is expanded from a high pressure *p*0 through the conically diverging nozzle into an ambient atmosphere of rela‐

In the continuum regime, the extent of the shock structure scales with the nozzle pressure ratio *p*0/*pb*. In particular, within the range 15 < *p*0/*pb* < 17,000, the distance *lM* = *rM* − *r*\* between the

*b*

*<sup>p</sup>* = × (11)

tively low pressure *pb*. The depicted shock system represents the continuum regime. Adapted from [36].

<sup>0</sup> 0.67 *M e*

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

that is surrounded by a mixing zone of jet and background particles.

<sup>×</sup> <sup>=</sup> (10)

In the following, estimations are made for a gas jet with barrel shock structures as it is under experimental investigation in this work too. Nitrogen expands from a pressure of *p*<sup>0</sup> = 11 bar into a helium atmosphere with a pressure of *pb* = 170 mbar through a conically diverging nozzle (thickness *ln* = 1 mm, throat diameter *d*\*=300 μm, and exit diameter *de* =500μm). At rest, both gases are at room temperature *T*0 = 293 K. In a simplification, a source flow is assumed corresponding to the dotted cone in **Figure 5** with its apex in a distance of *r*\* = 1.5 mm to the nozzle's throat. According to Eq. (11), the Mach disk appears 2.7 mm behind the nozzle throat, i.e., *rM* = 2.7 mm. The dimensionless area of the assumed source flow is expressed in terms of the distance *r* to the virtual source as *A*/*A*\* = (*r*/*r*\*) 2 . Solving Eqs. (6) and (7) results in the state functions *ρ*/*ρ*0 and *M* along the symmetry axis of the nozzle from throat position to the Mach disk, i.e., in the range 1.5 mm < *r* < 4.2 mm. The conditions directly behind the Mach disk are determined by the shock relations (8) and (9). For *r* > 4.2 mm, the flow is assumed to be incompressible (*ρ* = const.) since the Mach number has decreased sufficiently below *M* = 1. Thus, subsequent behavior of *M* is approximated by the continuity equation *M* (*r*)=*M* ^ ⋅ (*r* /*rM* )2.

**Figure 6.** State functions along the symmetry axis of a barrel shock: density *ρ* in terms of its stagnation value *ρ*0, and Mach number *M*. The solid blue line for *A*/*A*\* represents the cross-sectional area of the orifice, whereas the dashed blue line indicates the subsequent conical source flow.

The corresponding state functions *ρ*(*r*)/*ρ*0 and *M*(*r*) are depicted in **Figure 6** with respect to the distance *r* to the virtual source. Usually, the nozzle is operated with a background pressure on the order of *pb* ≤ 10−4 mbar and the plasma is generated in a distance of 500 μm to the nozzle exit. The conditions at the usual plasma position, before and behind the Mach disk, are given in the diagram. It is revealed that due to the shock a two times higher density is achieved in a larger distance to the nozzle as compared to the typical plasma position. In practice, the density increase is even higher since plasma production takes place a few 100 μm besides the symmetry axis of the nozzle. Here, without ambient gas the jet is even more rarefied and with ambient gas the shock structure is present.
