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Abstract
The temporal modulation of an electron bunch train accelerated from a foil target irradiated by an intense laser pulse is studied by measuring the coherent transition radiation (CTR) from the rear surface of a target. We experimentally obtained CTR spectra from a 1 µm thick foil target irradiated at a maximum intensity of 6.5 × 1019 W/cm2. Spectral redshifts of the emitted radiation corresponding to increases in laser intensity were observed. These measurements were compared with the theoretical calculation of CTR spectra considering ultrafast surface dynamics, such as plasma surface oscillation and relativistically induced transparency. Plasma surface oscillations induce a spectral redshift, while relativistic transparency causes a spectral blueshift. Both effects are required to find reasonable agreement with the experiment over the entire range of laser intensities.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, the interaction of ultra-intense short-pulse lasers with solid-density matter has been of great interest. This interaction helps us understand the fundamental physics of relativistic particle beam acceleration and transportation [1–4] and the extreme material properties relevant to planetary or stellar science [5–7]. They can also be applied to the development of advanced particle accelerators, ultrafast radiation sources, and fast-ignition fusion [8–11]. Particularly, the generation of ultrafast secondary radiation, such as extreme ultraviolet (EUV) and terahertz (THz) radiation, from laser-solid interactions at relativistic intensities is an active research topic [12,13]. Ultra-fast surface dynamics [14–17] (relativistically induced transparency (RIT) and plasma surface oscillation) and relativistic electron beam transportation [18–20] (micro-bunching electron and electron refluxing) play crucial roles in producing HHG and THz radiation from intense laser-matter interactions [21,22].
Relativistic plasma oscillations near the critical density surface occur when solid targets are irradiated by intense laser pulses with high contrast. This surface, which is in a state of relativistic motion, leads to a periodic temporal compression of the reflected light by the Doppler effect. Optical light contains several high-order harmonics because of the varying reflection points in each optical cycle [23]. EUV pulses from such processes can be used in attosecond science and time-resolved imaging experiments in chemistry and biology [24].
Electron bunches with relativistic energies are produced near the critical surface layer by various heating mechanisms and transported through targets [25,26]. Broadband transition radiation is produced when such electrons cross a solid–vacuum boundary. Radiation in the THz regime has recently been studied as a strong source for many applications in condensed matter research, bioimaging, chemistry, and wireless communications [27–29]. Transition radiation diagnostics in an optical regime could serve as simple but useful tools for studying the properties of relativistic electron beams [30,31]. Optical imaging can provide spatial distributions of beams, such as multiple-beam generation or filamentation. The spectrum of coherently enhanced transition radiation at the harmonic frequencies of the driving laser indicates that electron bunches in the relativistic beam are separated by a fraction of a single laser cycle [30,32].
In this study, we presented an experimental and theoretical analysis of the coherent transition radiation (CTR) spectra of 1 µm thick titanium foils irradiated with a high-contrast laser pulse at intensities exceeding 1019 W/cm2. The measured redshifts of the CTR spectra near the second harmonic of the laser frequency indicate that the temporal separation of the adjacent electron bunches deviates from the half cycle of the driving laser pulse. We developed a model for electron bunch generation from the oscillating plasma layer that interacts with the electric field of the laser pulse. We also included the motion of the critical density plane because of the effect of relativistically induced transparency (RIT). The temporal structure of the electron bunch train was affected by these dynamics, and the calculated CTR spectra were also sensitively modulated. By comparing the calculations with the measurements, it was observed that both dynamical processes were necessary to find reasonable agreement with the experiment. Although the oscillation effect showed a trend similar to that of the experiment, it was exaggerated at higher intensities. Linear motion of the critical density plane is required to mitigate the overall spectral shift to match the measurement. The speed of the critical density layer by RIT was estimated to be approximately∼ 0.015c, which is consistent with other studies.
2. Experiment
The experiment was performed using a 150 TW Ti:sapphire laser system at the Center for Relativistic Laser Science (CoReLS) [33]. A schematic of the experimental setup is shown in Fig. 1. Utilizing the double plasma mirror (DPM) system, a pulse contrast of 10−10 at 5 ps before the arrival of the main pulse can be achieved [34]. A pulse energy of up to 0.82 J was delivered within 30 fs at a central wavelength of 800 nm. The beam was focused by using an off-axis parabola (F/#3.7). A typical focal-spot image is shown in Fig. 1. The laser spot had a full-width-half-maximum (FWHM) diameter of 4 µm, yielding a maximum intensity of 6.5 × 1019 W/cm2. The target was a 1 µm thick titanium foil. The incident angle of the p-polarized laser beam was 30°.
Fig. 1. Schematic of the experimental setup. High pulse contrast (< 10−10) is achieved using a double plasma mirror (DPM) system. Optical emission between 300 to 500 nm range is measured using a spectrometer and an imaging system.
Optical emission from the rear side of the foil was collected and imaged simultaneously onto a spectrometer and charge-coupled-device (CCD) detector. Various neutral-density and bandpass filters were inserted to suppress the scattering of the main beam and to select a specific spectral window. Spectra with different laser intensities showed peaks near 400 nm. These are typical characteristics of a CTR obtained from a train of high-energy electron bunches accelerated by j × B heating [25].
The most interesting feature observed in the CTR spectra shown in Fig. 2(a) was the modulation of the emission spectra as the laser intensity varies. The coherent addition of the transition radiation at the second harmonic of the driving laser wavelength is due to the temporal separation between the electron bunches, which is an exact half-cycle of the laser field oscillation. Therefore, the intensity-dependent CTR spectra imply modulation of the temporal structure of the electron bunch train. As shown in Fig. 2(a), red spectral wings emerge at higher laser intensities. Each spectrum was normalized for clarity. As the laser intensity increases from 1.2 to 6.5 × 1019 W/cm2, the average wavelength of the CTR spectrum $\bar{\lambda } = \smallint \textrm{I}(\mathrm{\lambda } )\cdot \mathrm{\lambda }d\lambda /\smallint \textrm{I}(\mathrm{\lambda } )d\lambda $ shifted from 401 to 404 nm [Fig. 2(b)].
Fig. 2. (a) Examples of CTR spectrum from 1 µm thick targets. Laser intensities are 1.2, 4.3, and 6.5 × 1019 W/cm2 (from bottom to top, respectively). Spectra are normalized, and vertical offsets are applied. (b) Variation of the average wavelength of CTR to the laser intensity. Shadow is the standard deviation of 10∼15 measurements.
Before further discussion on the temporal structure of the electron bunch train, note that such changes in the CTR spectra have not been observed in previous investigations [30,31]. Although many experimental parameters are similar, the most significant difference between the present and earlier studies is the contrast of the laser pulse. In Refs. [30,31], pre-pulses (contrast ratio of ∼10−4) arrived 20–50 ps before the main pulse and generated a pre-plasma with a length of several micrometers. In a pre-plasma, the density of which varies gradually, the acceleration zone is elongated near the critical density surface. In this present study, a clean laser pulse (contrast ratio of 10−10) directly interacted with the sharp density gradient of the target surface, and the acceleration zone was accurately defined. Therefore, the acceleration process is susceptible to the micromotion of electrons perpendicular to the critical surface.
3. Theoretical calculations
3.1 Temporal modulation of electron bunch structure
Before further discussion about the temporal modulations of the electron bunch structure by the motion of the critical density (Nc) plane, we review the basic processes to find the oscillating current at the rear surface of the target. Further details can be found in Ref. [32]. Two electron bunches were injected into the target in a single laser cycle during the full width at half maximum (FWHM) pulse duration [ Fig. 3(a)]. Each bunch is generated at the peak of the oscillating field. Considering the Gaussian temporal profile of the laser pulse, the temperature of the micro-bunches varies according to the scale given by Wilks, ${T_e}(i )= 0.511\left( {\sqrt {1 + {I_{17}}({{t_i}} )\lambda_{\mu m}^2/13.7} - 1} \right)\,\textrm{MeV}$ [26]. I17(ti) is the instant laser intensity when ith bunch is generated at ti in units of 1017 W/cm2, and λµm is the laser wavelength in µm. Figure 3(b) shows some examples of the relativistic Maxwellian speed distributions f(β) of the electrons. Because electrons with β are close to 1, a single bunch results in a sharp burst of current with a long tail at the rear surface [Fig. 3 (c)]. The sharpness of the current burst also depends on Te.
Fig. 3. (a) Gaussian temporal profile of laser pulse. The peak intensity is set to be I0 = 1 × 1019 W/cm2. It is assumed that electron bunches are generated between the FWHM duration of the pulse. An instant laser intensity I(ti) is used to determine the temperature of each bunch born at ti using a scaling law given in the text. (b) Examples of speed distribution of electron bunches with given temperatures. (c) Examples of current burst by some e-bunches at the rear surface. all bunches travel the same distance, 1 µm.
We considered the temporal modulation of the electron bunch structure when the critical density (Nc) plane was oscillated by the laser field. As the p-polarized electric field of the driving laser oscillates, the electrons at Nc also oscillate with the same frequency νL in the normal direction of the target surface [ Fig. 4(a)]. The oscillation amplitude was scaled as $r = eE/\gamma {m_e}\omega _L^2$, where E is the laser electric field, γ is the Lorentz factor, me is the electron rest mass, and ωL is the laser frequency. Note that the relativistic Doppler effect should be considered for the incident laser parameters associated with the moving front surface [35]. The optical pulse duration, peak intensity, and hot-electron temperature were recalculated. Electron bunches born at each end of the oscillating plane travel distances of d + r or d-r, alternatively, and the currents at the rear surfaces are modulated, as shown in Fig. 5. In this example, the initial target thickness d was 1 µm. Two different peak laser intensities, 1 × 1019 W/cm2 (red curve) and 8 × 1019 W/cm2 (blue curve) were used. The maximum of each curve was normalized to 1, and t = 0 represented the moment when the first electron arrived at the rear surface. The Nc plane moved back and forth by 2r. The two adjacent bunches group together, and the time separations between the bunches deviate from the half-laser cycle. At higher intensities, this effect is more significant.
Fig. 4. Two plasma surface dynamics are considered in this work. (a) Oscillation of critical density (Nc) and (b) forward motion of the laser-cycle averaged Nc plane. At every half laser cycle, an electron micro bunch is generated at the Nc plane. It travels through the target, disperses, and produces transition radiation at the rear surface.
Fig. 5. Currents at the rear surface due to the oscillation of Nc plane for two laser intensities of 1 × 1019 W/cm2 (red) and 8 × 1019 W/cm2 (blue). λµ = 0.8 µm, tFWHM = 30 fs, d = 1 µm, and βs = 0. The maximum current values are normalized to 1. For clarity, the times corresponding to the half laser cycles are shown as vertical grids.
We also considered the temporal modulation of the electron bunch structure when the average position of the critical density (Nc) moves because of the RIT. Note that not only RIT but also other effects, such as plasma expansion or compression, might occur, and the motion of Nc might be quite complex. However, given the pulse contrast (<10−10) and duration (30 fs) of the laser in the experiment, the hydrodynamics can be neglected. The Nc plane may accelerate during the laser pulse. However, in this study, we focus on the relationship between the modulation of e-micro bunches in the time domain and the modulation of CTR spectra using a simple model. In this regard, a zeroth-order approximation for the motion is used; that is, Nc moves by the RIT along the laser propagation direction at a constant speed βs, which is typically known as a few percent of the speed of light [15,16] [refer Fig. 4(b)]. Electron bunches are generated in the later part of the laser pulses that travel shorter distances to the rear surface, where the transition radiation is emitted.
In Fig. 6, an example of the modulated current at the rear surface owing to the front surface motion is plotted. A peak laser intensity of I0 = 1 × 1019 W/cm2, central laser wavelength of 800 nm, and initial target thickness d0 = 1 µm were used. The speed of Nc was set to βs = 0.05 (blue curve). For comparison, the case of a stationary surface; that is, βs = 0 (red curve), is also shown. The maximum of each curve was normalized for ease of comparison. For the βs = 0 case, the time intervals between two adjacent peaks are identical. However, with a finite βs, it decreases as time progresses because the distance travelled by the electron bunch becomes shorter. The later bunches are also less dispersed. Consequently, a gradual increase in peak current was observed.
Fig. 6. Currents at the rear surface due to the front surface motion (βs = 0.05, blue curve). The βs = 0 case (red curve) is also shown for comparison. λµ = 0.8 µm, tFWHM = 30 fs, I0 = 1 × 1019 W/cm2, and d0 = 1 µm. The maximum current values are normalized to 1. For clarity, the times corresponding to the half laser cycles are shown as vertical grids.
Finally, both the oscillation and linear motion of the critical density are considered, as shown in Fig. 7. In this case, I0 = 8 × 1019 W/cm2, d0 = 1 µm, and βs = 0.03 were used. The current shows the separately examined effects; that is, (i) a gradual decrease in time intervals between peaks and increased peak heights by RIT and (ii) the grouping of two bunches by CSO. The modulated positions of the current peaks deviate from the half-laser cycle and infer the spectral modulation of the transition radiation, as examined in the following section.
Fig. 7. Normalized current at the laser intensity of 8 × 1019 W/cm2. (λ0 = 800 nm, tFWHM = 30 fs, d0 = 1 µm. CSO + RIT with vRIT = 0.03 c).
3.2 Modulation of CTR spectra and comparison with experimental data
The CTR spectra can be obtained using the Fourier transform of the obtained oscillating currents, as shown in Figs. 5–7. First, the effects of the RIT were examined separately. The βs-dependent spectral shift of CTR emission is shown in Fig. 8(a). The maximum value of each curve was normalized to 1. With the forward motion of the critical surface, the overall CTR emission shifted to shorter wavelengths. Assuming βs is typically a few percent of the speed of light, the amount of shift is on the order of several nanometers. However, note that this shift toward shorter wavelengths is opposed to the experimental data in Fig. 2. This suggests that the RIT may not be the primary mechanism. However, as discussed later, the RIT is still not negligible, and its effects must be considered in the intensity regime of over 1019 W/cm2.
Fig. 8. (a) Calculated CTR spectra with different critical surface velocities (d0 = 1 µm, I0 = 1 × 1019 W/cm2). (b) Front surface velocity (RIT)-dependent spectral shifts.
Second, the CSO effect was independently investigated (βs = 0), as shown in Fig. 9(a). When I0 was 1 × 1019 W/cm2, the average emission wavelength was close to the second harmonic of the driving laser (400 nm). However, at higher intensities, the spectrum shifts toward longer wavelengths. In Fig. 9(b), the intensity-dependent shift of the average CTR wavelength is compared with the measurement. The calculations with the CSO effect only (blue circles) showed the same trend as the experimental observations (red squares). However, in the high-intensity regime of over 6 × 1019 W/cm2, it overshoots the measurement. Finally, the combined effect of the CSO and RIT was examined. Because the two dynamics affect the CTR spectra in opposite directions, the RIT effect is adjusted to mitigate the strongly red-shifted spectra by the CSO at higher intensities. Note that if βs were on the order of 0.1, the blueshift caused by the RIT could be greater than 10 nm [Fig. 8(b)]. However, this is too much for the difference between the CSO-only calculation (blue) and the experiment (red). A reasonable agreement over all the laser intensity ranges [green squares] is obtained with βs = 0.015 [Fig. 9(b)], which is also similar to the results of other studies on the critical layer dynamics under similar conditions [15,16].
Fig. 9. (a) Calculated CTR spectra from 1 µm thick foil target with an oscillating Nc. Forward motion of Nc is not included. Three different laser intensities, 1, 2, and 8${\times} $1019 W/cm2 are used. (b) Variation of average wavelength of the calculated and the experimental spectra for different laser intensities. The CTR spectra was calculated from the critical surface oscillation with the speed of 0.015c to obtain the good agreement with experiment.
4. Conclusion
We presented an analysis of the ultrafast dynamics of a target surface irradiated by ultrashort high-intensity laser pulses by measuring coherent transition radiation. The CTR emission near the second harmonic of the laser is produced at the rear surface of the target by the oscillating current hot electron micro bunches driven twice per laser cycle from the target front. Gradual redshifts of the CTR spectra are observed with increasing laser intensity, inferring the modulation of the electron bunch structure in the time domain.
A series of calculations were performed to model the modulation of the electron bunch train. We considered two types of motion for the critical density while interacting with the laser pulse. The first is the oscillation of the critical density caused by the laser field. In a single laser cycle, two electron bunches are generated at different places, separated by twice the electron quivering amplitude. The two adjacent bunches group together, and the time separations between the bunches deviate from the half-laser cycle. As a result, the CTR spectrum shifted to longer wavelengths as the laser intensity increased. The second is the forward motion of the critical density by relativistically induced transparency with a light speed of a few percent. In this case, the effective target thickness from the Nc plane to the rear surface is reduced during the laser pulse. Consequently, the oscillating frequency of the hot electron current gradually increased, and the CTR emission spectra shifted to shorter wavelengths. Both processes are necessary to find a reasonable agreement with the gradual redshifts of the measured CTR spectra. Although the Nc oscillation exhibits a trend similar to that of the experiment, it is exaggerated at higher intensities. The forward motion of the Nc plane by the RIT is required to mitigate the overall spectral shift to match the measurement at an intensity greater than 4 × 1019 W/cm2. The average speed of Nc by the RIT was estimated to be approximately 0.015c, which is also consistent with other studies [15,16].
This study demonstrates that the CTR contains rich information on the ultrafast dynamics of electrons near the critical density interacting with ultra-intense laser pulses. In future studies, a simple model can be combined with PIC simulations for a more detailed analysis of the accelerating motion of the critical density plane. Note that CTR diagnostics in the optical regime are simple and easy to set up. Therefore, it can be applied to various laser-matter interaction experiments, such as the development of advanced radiation sources, ion beam acceleration, and laser-fusion research.
Funding
Institute for Basic Science (IBS-R012-D1); Ministry of Science and ICT, South Korea (524420-22); National Research Foundation of Korea (NRF-2019R1A2C2002864, NRF-2020K1A3A7A09080397).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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