# Real-Time Tomography of Gas-Jets with a Wollaston Interferometer

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## Abstract

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## 1. Introduction

## 2. The Single Beam Wollaston Interferometer Setup

#### 2.1. Theory

#### 2.2. Estimation of Phase Shift Due to a Gas Jet Density Distribution

#### 2.3. Experimental Setup for Gas Density Measurements

#### 2.3.1. Wollaston Interferometer

#### 2.4. Experimental Set-Up for Non-Rotational Measurements

## 3. Data Analysis for Interferometry and Real-Time Tomography

#### 3.1. Numerical Tools

#### 3.2. Tomographic Reconstruction

#### 3.2.1. Maximum Likelihood-Expectation Maximization

#### 3.2.2. Convergence and Error Studies of ML-EM

#### 3.3. Real-Time Computation of the Density Reconstruction

## 4. Examples of Gas Jet Density Measurements

#### 4.1. Piezo Gas Jet for Free Electron Laser Beam Instrumentation

#### 4.2. Solenoid Gas Jet for LWFA

#### 4.3. Shock Front Characterization in an LWFA

- Height ${h}_{s}$ of the shock front: density difference between ramped and undisturbed distribution at the ramp,
- Ramp factor r: ramped peak density divided by undisturbed peak density,
- ${w}_{1}$: half-width (left) defined by the ramp peak density,
- ${w}_{2}$: half-width (right) defined by the height ${h}_{s}$.

#### 4.4. Error and Stability Analysis

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Working principle of the Wollaston prism (

**A**) and used prism (

**B**). (

**A**) Two parallel rays separated by distance d before the lens are brought to interference after the prism; (

**B**) Wollaston prism $30\times 30\times 5$ mm, $\u03f5=5.8$ mrad, Societe d’Optique de Précision Fichou.

**Figure 2.**Sketch of a conical nozzle. The quantities for the solenoid valve are $d=500\text{}\mathsf{\mu}$m, $L=250\text{}\mathsf{\mu}$m and $\theta ={45}^{\circ}$.

**Figure 3.**Estimation of the on-axis density $\rho $ for different backing pressures. This model is based on the ideal gas law, i.e., independent of the gas species.

**Figure 4.**Sketch of the interferometric setup using a Wollaston prism. The dashed line represents the breadboard with the shielding box. The Wollaston prism is mounted on a linear stage, such that the parameter b and therefore the fringe spacing S can be changed (see Section 2). A Nikon imaging lens is attached to the CCD camera.

**Figure 5.**Experimental setup for Wollaston interferometry. The expanded beam passes through the gas jet, the first polarizer, the lens, the Wollaston prism and the second polarizer. The lens images the gas jet onto the CCD by a 200-mm imaging lens (AF Micro–Nikkor 200 mm f/4D IF–ED).

**Figure 6.**Parker solenoid valve and interference fringes. (

**A**) Miniature high speed high vacuum dispensing valve with a conical outlet (Parker 009-0442-900 ); (

**B**) typical interference fringes of the Parker solenoid valve at 35 bar backing pressure with argon. Here, the gas flow is directed upwards. The two non-straight areas correspond to positive and negative phase differences $\Delta \varphi $ (see Equation (4)).

**Figure 7.**Tomography setup for shock front studies. A razor blade is positioned with a linear and a rotational stage.

**Figure 8.**Test of ML-EM on the undisturbed (rotationally symmetric) gas distribution. The distance from the nozzle is given in mm; the corresponding pixel row is placed in brackets. In general, good agreement is found between the Abel inversion (

**A**) and ML-EM after 15 iterations (

**B**). The problem due to the singularity of the Abel inversion at $r=0$ does not arise with ML-EM.

**Figure 9.**Reconstruction by ML-EM from seven projections with artificial Gaussian noise, $\sigma =0.01$.

**Figure 10.**Convergence studies of ML-EM for various degrees of Gaussian noise $\sigma $ and the number of projection angles ${N}_{a}$. Convergence is observed after 7–10 iterations.

**Figure 11.**Time line of density reconstruction with ${N}_{a}=7$ projection measurements. The angles are chosen such that more projections are acquired in the direction parallel to the shock front, maximizing the information about the shock. For each projection 20 images were acquired, 10 of which were recorded with the gas jet on and the other 10 with the gas jet off, as a reference.

**Figure 13.**Density measurements of the piezo valve used for SwissFEL instrumentation. The reconstruction is computed via an Abel inversion. The subplot indicates the FWHM of the gas density distribution as a function of the distance from the throat.

**Figure 15.**ML-EM reconstructed on-axis density of the solenoid valve without blade. (

**Left**) Dashed lines represent the estimate for the respective backing pressure. The data points with error bars are from the measurements. The resonant density is reached with backing pressures between 30 and 35 bar in the region of interaction for the Ti:Sa pulse ($h=$ 2.6–2.8 mm). (

**Right**) The density appears to increase for pressures up to 37.5 bar. For higher pressures, stagnation and even regression are observed.

**Figure 16.**Typical Wollaston phase image of a shock front generated by a razor blade inserted from the left to a gas jet. Gas flow is directed upwards.

**Figure 17.**

**Left**: ML-EM Reconstructed density distribution in a plane perpendicular to the gas flow at distance $h=2.6$ mm from the nozzle, i.e., 1 mm from the blade.

**Right**: Density profile along z direction for $y=0.55$ mm.

**Figure 19.**Ramp characterization with respect to different parameters.

**Left**: Widths ${w}_{1}$ and ${w}_{2}$ versus height h.

**Middle**: Widths ${w}_{1}$, ${w}_{2}$ and ratio r verus position of the blade ${L}_{b}$.

**Right**: Ratio r along y direction.

**Figure 20.**Error analysis of the reconstructed density (ML-EM, 15 iterations).

**Left**: Reconstructed density distribution.

**Right**: Noise of the reconstruction.

${\mathit{P}}_{\mathit{b}}$ (bar) | ${\mathit{L}}_{\mathit{b}}$ (mm) | h (mm) | h (pixel) | |
---|---|---|---|---|

min, max | 30.0, 40.0 | $-3.2,-2.5$ | 1.8, 3.0 | 1200, 1050 |

step | 2.5 | 0.1 | 0.2 | 25 |

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**MDPI and ACS Style**

Adelmann, A.; Hermann, B.; Ischebeck, R.; Kaluza, M.C.; Locans, U.; Sauerwein, N.; Tarkeshian, R.
Real-Time Tomography of Gas-Jets with a Wollaston Interferometer. *Appl. Sci.* **2018**, *8*, 443.
https://doi.org/10.3390/app8030443

**AMA Style**

Adelmann A, Hermann B, Ischebeck R, Kaluza MC, Locans U, Sauerwein N, Tarkeshian R.
Real-Time Tomography of Gas-Jets with a Wollaston Interferometer. *Applied Sciences*. 2018; 8(3):443.
https://doi.org/10.3390/app8030443

**Chicago/Turabian Style**

Adelmann, Andreas, Benedikt Hermann, Rasmus Ischebeck, Malte C. Kaluza, Uldis Locans, Nick Sauerwein, and Roxana Tarkeshian.
2018. "Real-Time Tomography of Gas-Jets with a Wollaston Interferometer" *Applied Sciences* 8, no. 3: 443.
https://doi.org/10.3390/app8030443