**5. Gas jet and soft X-ray diagnostics**

In order to characterize the supersonic gas jet and the evolving shock structure, two different methods are employed: the Schlieren technique for highly resolved qualitative imaging of density gradients and wavefront measurements with a Hartmann–Shack sensor in order to quantify the density distribution, but at a lower resolution. Both methods are described in detail in Sections 5.1 and 5.2.

The plasma is imaged by a pinhole camera and the number of the resulting soft X-ray photons is determined with a calibrated photodiode. A description of these tools follows in Section 5.3.

**Figure 10.** Characteristic emission spectra of various target gases, captured with a soft X-ray spectrometer.

#### **5.1. Schlieren imaging**

Schlieren imaging is a common technique in fluid dynamics that enables the qualitative measurement of density gradients [42]. The experimental setup is schematically shown in **Figure 10**. A pinhole with a diameter of 100 μm is illuminated by white light, and a focusing lens collimates the resulting beam, which then travels in the *z*-direction through the gas distribution of the jet target. The *xy*-plane at *z* = 0 is imaged by a 4f setup to a CCD camera (Lumenera Lu160M) and captured with an exposure time of 50 μs. The camera is synchronized with the gas jet at a repetition rate of 10 Hz. Here, imaging lenses with focal lengths of *f*1 = 160 mm and *f*2 = 300 mm are used. A knife edge is moved close to the focal spot in between the two lenses, eliminating half of the spatial frequencies in the Fourier plane. The orientation of the blade determines which component of the density gradient will become visible. For example, as depicted in **Figure 10**, a knife edge aligned with the *x*-axis generates contrast proportional to the gradient of the refractive index *∂n*/*∂y* corresponding to the density gradient *∂ρ*/*∂y*. Note, however, that in the Schlieren images shown below, the knife edge is aligned with the *y*-axis, so that density gradients within the jet are visualized in radial direction, thus emphasizing the barrel shock.

#### **5.2. Wavefront monitoring**

the plasma can be generated further away from the nozzle exit, and degradation effects are

In order to characterize the supersonic gas jet and the evolving shock structure, two different methods are employed: the Schlieren technique for highly resolved qualitative imaging of density gradients and wavefront measurements with a Hartmann–Shack sensor in order to quantify the density distribution, but at a lower resolution. Both methods are described in

The plasma is imaged by a pinhole camera and the number of the resulting soft X-ray photons is determined with a calibrated photodiode. A description of these tools follows in Section 5.3.

**Figure 10.** Characteristic emission spectra of various target gases, captured with a soft X-ray spectrometer.

Schlieren imaging is a common technique in fluid dynamics that enables the qualitative measurement of density gradients [42]. The experimental setup is schematically shown in **Figure 10**. A pinhole with a diameter of 100 μm is illuminated by white light, and a focusing lens collimates the resulting beam, which then travels in the *z*-direction through the gas distribution of the jet target. The *xy*-plane at *z* = 0 is imaged by a 4f setup to a CCD camera

minimized.

86 High Energy and Short Pulse Lasers

**5. Gas jet and soft X-ray diagnostics**

detail in Sections 5.1 and 5.2.

**5.1. Schlieren imaging**

A Hartmann-Shack wavefront sensor [43, 44] is used to obtain quantitative information on the density distribution in the supersonic gas jet [34]. The experimental setup is mostly the same as that depicted in **Figure 10** for Schlieren imaging. However, the knife edge is removed and the CCD camera is replaced by the wavefront sensor. An initially plane wavefront of a test beam that travels through the target gas is deformed due to the spatial variation of the refractive index *n*(*x*, *y*, *z*). The sensor splits the test beam into many subbeams by an array of microlenses, each producing a spot on a CCD camera (Lumenera Lu160M). The position of the spots contains the information of the local wavefront gradient. Thus, the deformation of the wavefront can be recovered. The spatial resolution Δ*x* of a measured wavefront is equal to the pitch of the microlens array of 150 μm divided by the magnification factor *f*2/*f*1 = 1.88 of the 4f setup, yielding Δ*x* = 80 μm.

**Figure 11.** Experimental setup for Schlieren and wavefront measurements. The dotted lines represent the path of unre‐ fracted light. The dashed line indicates a light ray that is refracted by varying distribution of gas density below the nozzle, thus hitting the knife edge and darkening the image. In order to monitor wavefront deformations, the CCD camera is replaced by a Hartmann–Shack sensor and the knife edge is removed.

The particle density distribution *N*(*x*, *y*) in the nozzle plane *z* = 0 is recovered from a measured shape *w*(*x*, *y*) of a deformed wavefront as follows. The test beam integrates *n*(*x*, *y*, *z*) over the propagation direction *z* of the light beam, resulting in a difference *w*(*x*, *y*) in the optical path, as illustrated in **Figure 11**. Now, it is assumed that in a plane corresponding to a constant *y* = *y*0, *n*(*x*, *y*0, *z*) is approximated by a rotationally symmetric Gaussian shape with a maximum value *n*0(*y*0) = *n*(0, *y*0, 0). Then, the deformation of the wavefront reads

$$\begin{split} m(\mathbf{x}, \boldsymbol{\nu}) &= \int \left[ n\left(\mathbf{x}, \boldsymbol{\nu}, \boldsymbol{z}\right) - 1 \right] d\boldsymbol{z} \\ &= \int \left[ n\_0\left(\boldsymbol{\nu}\right) - 1 \right] \cdot \exp\left( -\frac{\boldsymbol{x}^2}{2\sigma\left(\boldsymbol{\nu}\right)^2} \right) \cdot \exp\left( -\frac{\boldsymbol{z}^2}{2\sigma\left(\boldsymbol{\nu}\right)^2} \right) d\boldsymbol{z} \\ &= \left[ n\_0\left(\boldsymbol{\nu}\right) - 1 \right] \cdot \sqrt{2\pi} \cdot \sigma\left(\boldsymbol{\nu}\right) \cdot \exp\left( -\frac{\boldsymbol{x}^2}{2\sigma\left(\boldsymbol{\nu}\right)^2} \right) .\end{split}$$

**Figure 12.** Wavefront deformation induced by the gas jet. The distribution of the optical density *n*(*x*, *y*, *z*) increases the optical path length, resulting in the indicated wavefront deformation.

The standard deviation *σ*(*y*) of *n*(*x*, *y*, *z*) is determined from the shape of the measured deformation of the wavefront *w*(*x*, *y*) by a Gaussian fit. The distribution of the refractive index in the plane *z* = 0 containing the jet axis is recovered by

$$m(\mathbf{x}, \mathbf{y}, \mathbf{0}) - \mathbf{l} = \frac{\mathbf{w}(\mathbf{x}, \mathbf{y})}{\sqrt{2\pi}\sigma(\mathbf{y})}.\tag{12}$$

Conversion of the refractive index *n*(*x*, *y*, 0) into a particle density *N* is done by using the Lorentz-Lorenz formula [45]

$$\frac{n^2 - 1}{n^2 + 2} = \frac{4}{3}\pi\,\alpha\,N\tag{13}$$

where *α*, the polarizability of the considered gas particles, is derived using the values *n* = 1.0002974 and *N* = 2.69 × 1019 cm−3 for nitrogen under normal conditions [41] (at a temperature of 273.15 K and a pressure of 1013.25 mbar). In these calculations, the surrounding helium atmosphere is neglected because of its low refractive index that amounts to only a few percent as compared to that of the nitrogen jet.

#### **5.3. Plasma characterization**

The particle density distribution *N*(*x*, *y*) in the nozzle plane *z* = 0 is recovered from a measured shape *w*(*x*, *y*) of a deformed wavefront as follows. The test beam integrates *n*(*x*, *y*, *z*) over the propagation direction *z* of the light beam, resulting in a difference *w*(*x*, *y*) in the optical path, as illustrated in **Figure 11**. Now, it is assumed that in a plane corresponding to a constant *y* = *y*0, *n*(*x*, *y*0, *z*) is approximated by a rotationally symmetric Gaussian shape with a maximum

( ) ( ) ( )

*x z n y dz*

1 2 exp . <sup>2</sup>

**Figure 12.** Wavefront deformation induced by the gas jet. The distribution of the optical density *n*(*x*, *y*, *z*) increases the

The standard deviation *σ*(*y*) of *n*(*x*, *y*, *z*) is determined from the shape of the measured deformation of the wavefront *w*(*x*, *y*) by a Gaussian fit. The distribution of the refractive index

> (, ) ( , ,0) 1 . 2 () *wxy nxy*

> > 1 4 2 3 *<sup>n</sup> <sup>N</sup> <sup>n</sup>* p a


2 2

ps

Conversion of the refractive index *n*(*x*, *y*, 0) into a particle density *N* is done by using the

*<sup>y</sup>* - = (12)

(13)

0 2 2

1 exp exp

æ æ ö ö = é - ù× - × - ç ç ÷ ÷ ë û è è ø ø

s

2 2

*y y*

2 2

2

 s

*y*

s

( ) ( ) ( )

0 2

*<sup>x</sup> n y <sup>y</sup>*

<sup>æ</sup> <sup>ö</sup> =é - ù× × -ç <sup>÷</sup> ë û <sup>ç</sup> <sup>÷</sup> <sup>è</sup> <sup>ø</sup>

p s

value *n*0(*y*0) = *n*(0, *y*0, 0). Then, the deformation of the wavefront reads

( ) ( )

ò

ò

88 High Energy and Short Pulse Lasers

optical path length, resulting in the indicated wavefront deformation.

in the plane *z* = 0 containing the jet axis is recovered by

Lorentz-Lorenz formula [45]

, ,, 1

= é -ù ë û

*w x y n x y z dz*

Qualitatively, the plasma is characterized by a pinhole camera system as sketched in **Fig‐ ure 12**. It consists of a phosphor-coated CCD camera (Lumenera Lu160M with three layers of phosphor P43 with a grain size of ≈ 1 μm) in combination with a titanium-filtered pinhole (100 μm diameter, Ti-layer 200 nm thick). This way, the intensity distribution of radiation at the wavelength *λ* = 2.88 nm is captured. Here, the luminescent area *A* is approximated by an ellipsoidal shape with the semiaxes *a* and *b*. Then, *A* = *π a b*, where *a* and *b* are defined as the full-widths at half-maximum of the intensity in the *x*- and *y*-directions. The uniformity of the plasma is characterized by its eccentricity *ε* = *a* <sup>2</sup> −*b* <sup>2</sup> / *a*. Examples of the intensity images are shown in **Figure 17** in combination with the corresponding Schlieren images of the gas jet for the case of both, gas issuing into vacuum and gas issuing into a background gas and thus forming a barrel shock.

Quantitatively, the peak brilliance *Br* of the plasma is derived by

$$Br = \frac{N\_{ph}}{\pi \,\Omega A} \tag{14}$$

with the pulse duration *τ*, the solid angle Ω, and the source area *A*. The number of photons *N*ph with a wavelength of *λ* = 2.88 nm is determined by a calibrated XUV photodiode (Inter‐ national Radiation Detectors, AXUV100), which is equipped with a titanium filter (thickness 200 nm, applied to a nickel mesh with a transmissivity of 0.89). As shown in **Figure 13**, only these photons reach the detector that propagate within a cone confined by a circular aperture with diameter *da* in a distance *la* to the plasma. Thus, the solid angle is defined by the corre‐ sponding opening angle *ωa* = 2 tan *da*/(2*la*) of the cone [46]

$$
\Omega = 4\pi \sin^2 \sin^2 \frac{\phi\_a}{4}.\tag{15}
$$

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

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 pinhole camera as described above.
