EMPs versus laser energy and gas jet pressures

The maximum magnetic field $B_{\text{max}}$ corresponding to the varying nitrogen gas jet pressures at the two positions is shown in Fig. 2. The magnetic field B can be calculated using U(t)=-dϕ/dt and B = ϕ/S, where U(t) is the time domain signal, ϕ is the magnetic flux, and S is the loop area [31]. The results illustrate that the maximum magnetic field exhibits consistent trends at both locations. In previous studies, EMP energy was confirmed to be proportional to the laser energy and intensity [66, 67]. To eliminate the influence of laser energy fluctuations at different shots and make the conclusions more reliable, the mean laser energy for each nitrogen gas jet pressure is presented in Fig. 2. All error bars presented in this article were obtained by calculating the standard deviation over the sample set.

Fig. 2Full-screen View

(Color online) Evolution of the maximum magnetic field $B_{\text{max}}$ for different nitrogen gas jet pressures. (2.15(L) and 2.15(H) here represent the lower and higher laser energy at 2.15 MPa, respectively)

In Fig. 2, the maximum magnetic field B_max clearly changes when the nitrogen gas jet pressure varies from 1 to 3.15 MPa. Under the same nitrogen gas jet pressure of 2.15 MPa, the maximum magnetic field increases as the laser energy increases. Furthermore, in the range of 1 to 2.15(L) MPa, the maximum magnetic field increases with an increase in the nitrogen gas jet pressure. However, despite the increase in laser energy and nitrogen gas jet pressure from 2.15(H) to 3.15 MPa, a significant decrease was observed in the maximum magnetic field, indicating a downward trend. Finally, under various nitrogen gas jet pressures, the maximum magnetic field at a2 (30° from the laser beam) is higher than that at a1 (60° from the laser beam). This finding is consistent with previous experimental results, which are mainly attributed to the fact that a larger quantity of higher-energy electrons is produced at a location closer to the laser propagation direction [68-70].

Temporal and spectral characteristics of EMPs

To further elucidate the characteristics of the EMP arising from the interaction between the laser and nitrogen gas jet at varying pressures, a Fast Fourier Transform (FFT) was performed on the time-domain EMP signals to obtain the frequency domain. The amplitudes were then squared to obtain the power density spectra for the four different nitrogen gas jet pressures. Figure 3 illustrates that EMP frequencies generated by laser interaction with nitrogen gas jets at 1 and 1.5 MPa pressure are primarily concentrated within the frequency range of 0 to 2 GHz, while the spectra predominantly extended to the range of 7 GHz for the pressures of 2.15 and 3.15 MPa.

Fig. 3Full-screen View

(Color online) Power density spectra of EMPs for four nitrogen gas jet pressures obtained by FFT and squaring of the time-domain signals measured by B-dot antenna at position a2

In Fig. 3, there are six superimposed peaks appearing at 0.78, 0.96, 1.18, 1.32, 1.48, and 1.98 GHz. These are primarily attributed to the eigenfrequency radiation, which depends on the structure of the experimental chamber [71]. For an ideal cuboid resonant cavity, the wavelength ${\lambda }_{\text{r}}$ corresponding to the resonant frequency can be expressed as [72] ${\lambda }_{\text{r}}=\frac{2lh}{\sqrt{l^2+h^2}}\mathrm{.}$ (1) where l and h are the length and height of the cuboid resonant cavity, respectively.

The resonant frequency f_r for the cuboid resonant cavity can be expressed as $f_{\text{r}}=\frac{c}{{\lambda }_{\text{r}}}=\frac{c\sqrt{l^2+h^2}}{2lh}\mathrm{.}$ (2) where c=3×10⁸ m/s is the speed of light in a vacuum.

As depicted in Fig. 4a, the SILEX-II target chamber appears as a composite structure composed of a cube with side lengths of 1.2 m and a semi-cylinder with a base diameter of 1.2 m and height of 1.2 m [64]. The SILEX-II target chamber can be regarded as being situated between two cuboid resonant cavities. For the first cuboid resonant cavity, the length is l=1.8 m, width is w=1.2 m, and height is h=1.2 m. The fundamental resonant frequency of the corresponding cuboid cavity is f_r1=150.2 MHz. The second cuboid resonant cavity has a length of l=1.2 m, width of w=1.2 m, and height of h=1.2 m. The resonant frequency of the corresponding cuboid cavity is f_r2=176.8 MHz. Therefore, the resonant frequency of the empty SILEX-II target chamber can be calculated within the range of 150.2 to 176.8 MHz.

Fig. 4Full-screen View

(Color online) Tridimensional visualization of eigenmode electromagnetic distributions, numerically determined inside the vacuum chamber for the a b c empty resonant cavity and d e f actual resonant cavity (the electric field vector is indicated by the red arrow)

To elucidate the distribution of the electromagnetic field and eigenfrequencies in the actual SILEX-II target chamber, a simplified empty resonant cavity model of the SILEX-II target chamber was established using the COMSOL solver based on the finite element method [6, 73, 74]. Figure 4 shows the normalized electric fields associated with different modes inside the target chamber. As shown in Figs. 4b and 4c, there are two resonant frequencies: 152.58 and 153.31 MHz. These two typical resonant frequencies are significantly lower than those marked by the dashed magenta line in Fig. 3. This difference can primarily be attributed to the specific internal arrangements within the real target chamber [61, 72, 75, 76].

Considering the layout of the actual SILEX-II target chamber, multiple object models, as depicted in Fig. 4d, including mirror holders, the target holder, and electron spectrometers, were added to the COMSOL model, resulting in higher resonance frequencies. Figures 4e and 4f illustrate the two typical resonance frequencies of 780 and 1980 MHz, as also indicated in Fig. 3. The simulation results demonstrate an uneven distribution of the resonance frequencies, indicating certain variations in the resonance frequencies within different regions of the target chamber.

To obtain the temporal evolution of the spectra, a short-time Fourier transform (STFT) was applied to process the EMP signals [75]. The window width and overlap were set to 512 and 500, respectively. The results are presented in Fig. 5, where the main frequency band of the EMPs (0.6 to 0.96 GHz) appears from 0 to 5 ns.

Fig. 5Full-screen View

(Color online) Time-dependent spectrogram of EMPs for four nitrogen gas jet pressures (power here is in arbitrary unit)

Electron charge and temperature versus gas jet pressure

Figures 6a and 6b further illustrate the electron charge Q_e and electron temperature T_h generated by the petawatt laser irradiation of nitrogen gas jets with varying pressures during the experimental test. The experimental data of the electron energy spectra were fitted to a Maxwellian distribution with functional form $\exp \left(-E/k{T}_{\text{h}}\right)$, where E is the energy of the electron, k is the Boltzmann constant, and T_h is the hot electron temperature [77].

Fig. 6Full-screen View

(Color online) a Experimental electron charge and b corresponding experimental electron temperature for each nitrogen gas jet pressure. c Maximum magnetic field B_max as a function of electron charge Q_e

The maximum magnetic field B_max and laser energy are also shown on the right axis of Fig. 6. The EMP signal and electron charge show a high degree of consistency, approximately following the trend $B_{\text{max}}\left(\mu \text{T}\right)=7.47\times Q_{\text{e}}{\left(\text{nC}\right)}^{0.65}$. Overall, the maximum magnetic field is directly proportional to the electron temperature T_h. They all increase with an increase in laser energy and gas jet pressure until the density is higher than 2.15 MPa, when this trend reverses. These results suggest that accelerated electrons are likely the primary source of EMPs. Previous studies using laser interactions with solid-density targets have reached similar conclusions [19, 31, 40, 41, 44, 66, 68].

Simulation results and discussion

To further reveal the physical processes of petawatt laser ablation on nitrogen gas jets, the 2D PIC software EPOCH [78] was used. Considering that the width of the nitrogen gas jet nozzle employed in this experiment was 1 mm, the laser pulse was simulated to be incident directly onto a 1 mm-thick nitrogen gas jet. The laser pulse had a duration of 30 fs, diameter of 5 μm, and intensity of 5×10²⁰ W/cm². The central wavelength of the laser was 800 nm, which corresponded to a critical plasma density of n_c=1.7×10²¹ cm^-3.

As shown in Fig. 7, the electrons generated by the petawatt laser ablation of the nitrogen gas jet are closely linked to the pressure (plasma density). When the nitrogen gas jet pressure increased from 1 to 2.15 MPa, the self-focusing effect was enhanced, and a larger portion of the laser energy was captured by the plasma channel. This led to an increase in the length of the plasma channel; consequently, the number and energy of electrons captured and accelerated by the plasma channel also increased. However, when the nitrogen gas jet pressure increased to 3.15 MPa, more laser energy was lost during electron heating at the channel boundary owing to ionization and ionization defocusing. This resulted in a decrease in the length and width of the plasma channel, thereby reducing the number and energy of electrons captured by the plasma channel [52, 79, 80].

Fig. 7Full-screen View

(Color online) Electron density distribution diagrams generated by the interaction of the petawatt laser with nitrogen gas jets for various nitrogen gas jet pressures at time instance 1000 fs (the red dashed box delineates the approximate length and width of the plasma channel)

The simulated electron energy spectra and electron charges are shown in Fig. 8. In the optimum case of 2.15 MPa, the quantity and energy of electrons that could be captured and accelerated by the plasma channel were the highest, which is consistent with the experimental results shown in Fig. 6.

Fig. 8Full-screen View

(Color online) a Simulated electron energy spectra and b corresponding simulated electron charge induced by the interaction of the petawatt laser with nitrogen gas jet at various pressures

Investigation of the generation and spatiotemporal characteristics of EMPs

When the nitrogen gas jet is bombarded by the laser pulse, the nitrogen atoms are ionized by the main laser pulse to form a plasma. The corresponding plasma frequency at position r and time t can be expressed as [81] $f_{\text{p}}\left(r\mathrm{,}t\right)=\frac{{\omega }_{\text{p}}\left(r\mathrm{,}t\right)}{2\pi }=\frac{5.64\times {10}^4\sqrt{n_{\text{e}}\left(r\mathrm{,}t\right)}}{2\pi }\mathrm{,}$ (3) where n_e is the electron density in cm^-3.

In this experiment, the plasma electron density exceeded 3.9×10¹⁹ cm^-3; thus, the corresponding plasma frequency exceeded 56 THz. This is sufficient to suppress the electromagnetic waves in the GHz frequency range generated by electrons in the plasma channel [20, 25]. Therefore, the following investigations primarily focused on the EMPs generated after electrons escape from the gas target into the vacuum.

To further verify the correlation between the EMP intensity and the quantity and energy of electrons escaping from the gas target, we imported the parameters of the experimental electron beam into the 3D PIC simulation (coupled with a full EM solver) with the commercial code suite CST Particle Studio [25, 82]. The simulation model, as shown in Fig. 9a, is based on the actual target chamber model of SILEX-II. To decrease the computation time, we downsized the model by 10 times, including the target chamber, steel gas jet holder, and two steel conductors.

Fig. 9Full-screen View

(Color online) a Scheme of electron emission from the nitrogen gas target, b Gaussian pulse electron source with a duration of approximately 60 ps, c evolution of EMP intensity with the electron charge, and d electron temperature

The target material consisted of nitrogen gas and had dimensions of X (1 mm) × Y (4 mm) × Z (4 mm). A cylindrical target holder with a diameter of 2 mm and height of 60 mm was employed instead of the gas jet holder. As shown in Fig. 9a, the gas jet holder material was set to conductor steel. Moreover, to investigate the impact of metallic conductors inside the target chamber on the generation and distribution of EMPs, two identical cylindrical steel conductors with a diameter of 2 mm and height of 60 mm were positioned 30 mm away from the gas jet holder. These steel conductors served as simplified substitutes for the diverse metallic conductors typically found in the target chamber. One electric field probe was positioned 20 mm behind the target at an angle of 60° relative to the direction of electron escape. The electrons were ejected from a circular ring with a diameter of 5 μm located at the center of the nitrogen gas target towards the vacuum. In the simulation, the time required for electrons to escape from the gas target and eventually reach the target chamber walls was several hundred picoseconds, which is much longer than the 30 fs width of the Gaussian laser pulses used in the experiment. If the duration of escaping electrons in the simulation was set too short, not only would the computational time increase but the detailed observation of the underlying physical processes would be hindered. Therefore, as shown in Fig. 9b, compared with the 30 fs Gaussian laser pulse used in the experiment, we increased the duration of the escaping electron pulse by approximately 1000 times, with a Gaussian width of approximately 19 ps and duration of approximately 60 ps.

The simulation results are shown in Fig. 9c. Based on our experimental results, the electron temperature was maintained at 2.2 MeV, while the electron charge was set to vary from 31.9 to 245.5 nC. The simulation results suggest that the EMP intensity is directly proportional to the number of electrons escaping from the nitrogen gas target. Moreover, a position closer to the direction of the electron escape exhibits stronger EMPs. In Fig. 9d, the electron charge ejected from the nitrogen gas target was set to 31.9 nC. The corresponding electron temperature ejected from the nitrogen gas target was varied from 2.2 to 3.8 MeV. The simulation results demonstrate that as the escaping electron temperature increases, the EMP intensity increases. However, compared with the electron charge, the impact of the electron temperature on the EMPs was relatively insignificant.

A preliminary exploration of the sources of EMPs induced by the interaction of the laser with gas jets was performed using 3D PIC simulation with the commercial code suite CST Particle Studio. The simulation model and parameter setup were as shown in Figs. 9a and 9b. The time required for the electron beam to reach the target chamber wall can be approximately calculated as $T_{\text{2}}=D/2c$, where D is the diameter of the semi-cylindrical target chamber and 3×10⁸ m/s is the speed of light in the vacuum. In this simulation, D=0.12 m. At time instance T₁=60 ps, all escaping electrons escaped from the nitrogen gas target into the vacuum. At time instance T₂=200 ps, the fastest electrons reached the target chamber wall.

As shown in Figs. 10b, 10e, and 10h, at the time instance, due to the electron ejection, the nitrogen gas target becomes positively charged and a magnetic field envelops the path of electron transportation. The positively charged nitrogen gas target induces a charge displacement on the surface of the steel gas jet holder and then a neutralization current [83]. The neutralization current travels through the steel gas jet holder to the target chamber surface, while emitting intense EMPs into the vacuum. Furthermore, EMPs generated by the escaping electrons and neutralization currents reach the surrounding steel conductors, inducing currents on the surface of the steel conductors and producing EMPs. As shown in Figs. 10c, 10f, and 10i, at time instance t₂=220 ps, the fastest electrons reach and accumulate on the target chamber wall. Consequently, surface currents develop and spread to other sections of the target chamber wall, leading to the emission of EMPs into the vacuum. As shown in Figs. 10d, 10g, and 10j, at time instance t₂=800 ps, the current continuously oscillates on the gas jet holder, steel conductors, and target chamber wall, continuously radiating EMPs into the vacuum.

Fig. 10Full-screen View

(Color online) a Three-dimensional CST target chamber model. b c d Left view and e f g main view of the CST model for the temporal evolution of EMPs. h i j Temporal evolution of the surface current on the CST model for the ejection of electrons from the nitrogen gas target

Based on the aforementioned CST PIC simulation, we propose multiple sources of EMPs that arise from the interaction between the laser and gas jet, as illustrated in Fig. 11.

Fig. 11Full-screen View

(Color online) Schematic diagram of EMPs resulting from the interaction between the laser and gas jets

First, during the process of electrons escaping into the vacuum, transient currents are generated, and EMPs are produced around the transport path of the electrons.

Second, as some electrons escape from the gas target into the vacuum, the gas target becomes positively charged. A positively charged gas target induces displacement charges on the surface of the gas jet holder, generating neutralization currents and strong EMPs [83].

Third, the EMPs generated by the escaping electrons and neutralization currents induce transient currents on the surface of the surrounding metallic conductors inside the target chamber, generating EMPs.

Finally, the electrons escaping into the vacuum collide with the conductors and metallic target chamber wall along the transmission pathway, causing electrons to accumulate on the surface of the metallic conductors and metallic target chamber wall, resulting in transient currents and EMPs.

To further understand the temporal characteristics of EMPs from different sources, Fig. 12 presents the transient current waveforms at different positions during the simulation. Moreover, Fig. 12 shows the simulation results for the insulated Teflon gas jet holder. The following conclusions can be drawn from Fig. 12. First, in the time-domain profile, the transient currents at different locations exhibit a Gaussian distribution, and the width of the Gaussian pulse is approximately equal to the source current of the escaping electrons. Second, considering the start time of the transient current, when the electrons escape from the gas target, a transient current is generated behind the gas target and at the gas jet holder simultaneously. After 100 ps, EMPs generated by the escaping electrons and neutralization currents reach the steel conductor, inducing transient currents on the steel conductor. After 200 ps, the escaping electrons reach the target chamber wall, accumulate on it, and generate transient currents. Finally, compared with the steel gas jet holder, the Teflon-insulated gas jet holder can reduce the intensity of the neutralization and surface currents on the conductors.

Fig. 12Full-screen View

(Color online) Transient current time-domain waveforms for the source, gas jet behind, gas jet holder, steel conductor, and target chamber wall in CST simulation with a the steel gas jet holder and b Teflon gas jet holder

According to the Biot-Savart law, the magnetic field $\overrightarrow{B}$ generated by the dynamic current vector $\overrightarrow{I}$ at a distance of $\overrightarrow{r}$ can be expressed as [84] $\overrightarrow{B}=\frac{{\mu }_0}{4\pi }\frac{\overrightarrow{I}\times \overrightarrow{r}}{r^3},$ (4) where ${\mu }_0$ is the permeability of the vacuum and $\overrightarrow{r}$ is the distance vector.

According to the relationship between electric field strength $\overrightarrow{E}$ and magnetic field $\overrightarrow{B}$ in the vacuum when applying a plane wave approximation, the electric field strength $\overrightarrow{E}$ generated by the current $\overrightarrow{I}$ at $\overrightarrow{r}$ can be expressed as [12, 85] $\left|\overrightarrow{E}\right|\approx c\left|\overrightarrow{B}\right|=\frac{{\mu }_0}{4\pi }\left|\frac{\overrightarrow{I}\times \overrightarrow{r}}{r^3}\right|,$ (5) where c is the speed of light in the vacuum. The dynamic current vector $\overrightarrow{I}$ can be expressed as [31] $\left|\overrightarrow{I}\right|=q\left|\overrightarrow{v}\right|=Ne\left(\left|{\overrightarrow{v}}_0+\overrightarrow{a}t\right|\right),$ (6) where N is the number of electrons in a bunch, e is the charge of an electron, ${\overrightarrow{v}}_0$ is the initial velocity vector, and $\overrightarrow{a}$ is the acceleration vector of electrons in a bunch.

Eq. (5) describes the electric field strength $\overrightarrow{E}$ variation with the current $\overrightarrow{I}$ at fixed $\overrightarrow{r}$, indicating that $\overrightarrow{E}$ and $\overrightarrow{I}$ have the same time- and frequency-domain characteristics. Moreover, based on Eqs. (5) and (6), the electric field strength is proportional to the number of escaping electrons.

The typical time-domain waveform of a Gaussian pulse can be expressed as $g(t)=e^{-t^2/2{\sigma }^2},$ (7) where σ is the standard deviation. Based on $g\left(T_{\text{FWHM}}/2\right)$ =1/2, the full width at half maximum of the time-domain signal $T_{\text{FWHM}}=2\sqrt{2\ln 2}\sigma$.

The Fourier transform of the time-domain Gaussian pulse also follows a Gaussian pulse distribution, which can be expressed as $G(f)=\sqrt{2\pi }\sigma e^{-2{\pi }^2{f}^2{\sigma }^2}.$ (8) Based on $G\left(F_{\text{FWHM}}/2\right)=\sqrt{2\pi }\sigma /2$, the full width at half maximum of the frequency-domain signal $F_{\text{FWHM}}=\sqrt{2\ln 2}/\pi \sigma$.

According to Eqs. (7) and (8), the relationship between T_FWHM and F_FWHM can be expressed as $T_{\text{FWHM}}\cdot F_{\text{FWHM}}=4\ln 2/\pi .$ (9) According to Eqs. (5) and (9), controlling the pulse width of the escaping electrons allows the manipulation of the spectrum of the resulting EMPs. The escaping electrons with a narrower pulse width lead to a broader electromagnetic pulse spectrum.

To study the spatial distribution of EMPs, electric field probes were placed at different angles, 20 mm away from the gas target. In addition, to avoid the influence of echo oscillations inside the target chamber, we only recorded the peak intensity of the first Gaussian pulse signal of the time-domain waveforms of EMPs in CST simulation. The simulation results are presented in Fig. 13.

Fig. 13Full-screen View

(Color online) a Three-dimensional CST target chamber model. Spatial distribution of EMPs in the CST simulation: b top view, c left view, and d main view

According to Eq. (5), the distribution of EMPs mainly depends on two factors. The EMP intensity is directly proportional to the transient current intensity I and inversely proportional to the distance r² from the transient current. Figure 13 shows that the distribution of EMPs exhibits clear directionality, with EMPs mainly distributed around the escape path of electrons behind the gas target and gas jet holder. We preliminarily speculate that the EMPs generated by the gas target mainly originate from the electrons escaping from the gas target and neutralization current passing through the steel gas jet holder to the target chamber surface. Further experiments and simulations are required to validate the mechanism of EMP generation in gas targets. If confirmed, this conclusion will introduce novel strategies for shielding against EMPs induced by gas targets.

In addition, as shown in Figs. 12 and 13, compared with the steel gas jet holder, the Teflon-insulated gas jet holder can significantly reduce the neutralization current and indirectly decrease the EMP intensity driven by the neutralization current.