2.1

Setup of the double-pass shearing Wollaston interferometer

A sketch of the setup of the double-pass Wollaston prism interferometer with relay-imaging is shown in Figure 1. The optical source, a 405 nm monomode fiber laser, collimated by a singlet lens to a 20 mm laser beam diameter, enters the interferometer setup from the left. Due to the shorter wavelength of the laser compared to the conventionally used He-Ne laser (632.8 nm), a higher phase sensitivity is obtained due to larger wavenumber and higher refractive index of the Ar gas [19].

Figure 1.

Schematic setup of the double-pass shearing Wollaston interferometer with object relay imaging used for density characterization of gas jets.

The collimated laser beam is partially transmitted through a 50/50 beamsplitter and travels to the weakly refracting gas jet, placed in a vacuum chamber. After the gas jet, the beam travels to the relay-imaging optical system that consists of a 4-f telescope configuration of two doublet achromatic lenses (with 50 cm focal length) and a plane mirror. The relay-imaging system facilitates both the second pass of the collimated probing beam through the gas jet and, at the same time, preserving the spatial information in the beam undistorted by propagation. After the second pass through the gas jet, the laser beam is partially reflected by the beamsplitter and directed to the phase measurement system. This system consists of an imaging lens (with 30 cm focal length), a Wollaston prism (1-degree deviation angle), a polarizer, and a CMOS camera. Due to the characteristics of shearing interferometry with a Wollaston prism, a reference beam is not required for this setup, as the prism splits the incoming beam into two outgoing beams with different polarizations. The interference of these two beams after passing through the polarizer is detected on the camera.

For the comparison of the double-pass setup to the standard single-pass configuration, two turning mirrors are inserted in the optical setup – one after the gas jet and one before the phase measurement system. In this way, the laser beam is sent to the phase measurement system after only a single pass through the object.

With the presented interferometer setup with Wollaston prism, we can acquire individual interferograms of low-pressure gas jets, extract from them the phase accumulated by the probe beam and reconstruct the gas jet spatial density distribution. Each measurement of gas jet density can be divided into three parts:

For each measurement, we record two interferograms – a signal interferogram with the gas jet and a background interferogram without the gas jet, and we subsequently subtract the background one from the signal one. This allows us to minimize the noise and the systematic errors, e.g. alleviate the effect of the imperfections in the optical system elements. The phase is retrieved by a 2D Continuous Wavelet Transform (CWT) and a direct Maximum ridge detection algorithm. CWT allows to be tuned to each application case by choice of optimal wavelet function for the phase retrieval. The density reconstruction is done by either tomographic methods from multiple projections of the jet for complex gas distributions or by Abel transform from single projections for rotationally symmetric gas jets. Here, we use the Filtered-back projection method with a Hann filter [20]-[21] for the density characterization of a slit nozzle. We perform 18 projections of the jet with a step of 10 degrees between them.

2.2

Gas nozzles for LPA experiments

The gas jets are essential targets for LPA experiments as they offer the possibility to design different gas density distribution for the focused laser pulse. In particular, the gas jets that are employed as LPA targets include rotationally symmetric gas jet produced by cylindrical nozzles, non-rotationally symmetric jets generated by slit nozzles of various dimensions, jets with tailored shock fronts generated by blades or wires obstructing the gas flow [17], [22]-[25]. The diversity of gas jets results from the significant effect that the gas density spatial distribution has on laser-plasma interaction, e.g. it has been demonstrated that a plasma density longitudinal down-ramp can improve the stability of the generated electron beam [26]-[27]. Thus, the precise knowledge of gas density distribution is essential for the performance of the laser-plasma accelerators.

Here, two different types of gas jets are characterized – a rotationally symmetric gas jet produced by a cylindrical nozzle with de Laval shape, shown in Figure 2 (A) and an elongated jet produced by a 5 mm × 1 mm slit nozzle with de Laval shape along the long side and straight walls along the short side as shown in Figure 2 (B). The slit nozzle was fabricated from fused silica via the bottom-up milling approach using nanosecond laser [28],[29]. This technique allowed the precise 2.5D processing with high material removal rate reaching 0.07 mm³/s for the inner converging-diverging channel with only a 1.7 W average laser power. The average peak-to-valley distance of the milled surface does not exceed 10 μm using this technique [30].

Figure 2.

Sketches and images of the nozzles used as LPA targets – (A) cylindrical nozzle with de Laval shape and (B) slit nozzle 5 mm × 1 mm with de Laval shape along the long side.

The two jets are characterized by the proposed double-pass interferometer and also with the standard single-pass interferometer configuration. This comparison allows us to validate the performance of the double-pass interferometer and quantify its capabilities.

The gas nozzle is set on top of a rotation stage and two linear stages. The rotation stage provides the capability to turn the holder by a precise angle and to record multiple projections at these angles, This is required for the tomographic measurements. The two orthogonal translation stages allow for the precise alignment of the rotation axis of the rotation stage axis to the required position of the nozzles. The jets are operated in a pulsed regime using a pulsed valve synchronized with a delay generator with an opening time of the valves of 10 ms.

3.1

Comparison of single-pass and double-pass probing

In order to verify the capabilities of the proposed double-pass configuration, we have characterized the rotationally symmetric gas jet, produced by a cylindrical nozzle (see Figure 2 (A)) and the non-rotationally elongated gas jet, produced by a slit nozzle (see Figure 2(B)) with both the single-pass and double-pass configurations and compare the obtained results.

The gas jet phase map from the interferometric measurements of the symmetric gas jet obtained at the 1.6 bar of Ar is shown in Figure 3. We can see in Figures 3 (A) and 3 (B) that the double-pass configuration results in a two-fold increase in the phase acquired by the probe laser beam – maximum phase for double-pass is 0.88 rad, and maximum phase for single-pass is 0.44 rad. To quantify the sensitivity of the double-pass configuration in addition to the phase values, we also need the background noise value. We estimate it for both double and single-pass measurements as a standard deviation of the phase shift in a square area of 1 mm x 1 mm on the side of the interferograms where no gas density is expected. The background noise values estimated this way are 0.02 rad for both configurations. These results show that the double-pass configuration provides two times higher sensitivity than the standard single pass. This result is extremely important for the trend in LPA experiments of minimizing the gas jet dimensions and decreasing the gas jet densities, especially for low refractive index gas like helium, as the increased sensitivity can be vital for characterizing the gas interaction medium and its specific features.

Figure 3.

Phase maps comparison of a rotationally symmetric Ar gas jet measured by single (A) and double (B) pass interferometric configurations with valve backing pressure of 1.6 bar.

The comparison of phase profiles of the non-rotationally symmetric gas jet from a slit nozzle obtained with double-pass and single-pass setups is shown in Figure 4. The phase profiles obtained at 1 mm above the nozzle at 20 bar backing pressure of Ar (pressure measured before the valve) along the long side (5 mm) of the nozzle are shown. It is seen in the figure that the phase profiles have a characteristic shape with two density peaks near the walls and a lower plateau in the middle. The phase profile comparison shows clearly that the double-pass configuration has lower noise, which allows for better accuracy of the profile characterization.

Figure 4.

Phase profile comparison of a non-rotationally symmetric elongated gas jet, produced by a slit nozzle, measured by single and double pass interferometric configurations performed at 20 bar of Ar (measured before the valve) along the long side of the nozzle at the height of 1 mm above the nozzle edge.

3.2

Tomographic reconstruction of non-rotationally symmetric gas jets generated by a slit nozzle

We apply the double-pass configuration for density characterization of the slit 5 mm × 1 mm nozzle used for LPA experiments at 20 bar Ar. Here, we used 18 projections of the gas jet each taken at a step of 10 degrees apart.

The results from the tomographic reconstruction are shown in Figure 5. In the figure, we can see the 3D density map of the slit nozzle represented by iso-surfaces of equal gas density with different colors. It is seen in the figure that there are three separated density regions – two side density peaks and a lower central plateau and the ratio between the two regions is two-fold. The density ratio between the peak and the plateau is a function of the nozzle dimensions and the throat orifice size, which allows for obtaining different peak-plateau ratios with modification of the nozzle dimensions. The measured values are in good agreement with the simulation results in both the side peaks and the central plateau region, as shown in Figure 4. The noise visible in tomographic density plot can be further diminished by increasing the number of projections recorded for the reconstruction.

Figure 5.

Tomographic density reconstruction of a slit nozzle gas jet at 20 bar Ar for LPA applications: (A) tomography density reconstruction; (B) density profiles of interferometric measurements with double-pass configuration and of fluid simulations at the height of 1 mm above the nozzle.

In Figure 5 (B), a comparison is made between density plots from the tomographic density reconstructions and 3D gas density fluid simulations (for more information on the simulations see [31]). It is seen that there is a good agreement between the measurement and the simulation in both the side peaks and the central plateau region.

In LPA experiments, the slit nozzle with the density profile as shown in Figure 5 can be utilized for different purposes: the transverse profile can be used for limiting the acceleration length and thus controlling/limiting the electron beam energy; the longitudinal profile can be used for multi-stage acceleration taking advantage of the three separate density regions, first density ramp to inject the electron bunch, low-density plateu to boost the acceleration and the second ramp to enhance the X-ray generation.