## 1 Introduction

Terahertz (THz) waves^{[}
Reference Siegel
^{1}
^{]}, which are significantly less developed and utilized than microwaves and light waves, used to be regarded as the ‘THz gap’^{[}
Reference Sirtori
^{2}
^{]} in the electromagnetic spectrum due to the limitations of traditional electronic or optical methods in producing terahertz radiation during the early years. Such THz pulses have attracted significant interest for their potential applications in fields such as biology, medicine, material science, optical communication and the military^{[}
Reference Ulbricht, Hendry, Shan, Heinz and Bonn
^{3}
^{–}
Reference Zhang, Shkurinov and Zhang
^{8}
^{]}. To generate THz waves, several routine methods, such as cascade quantum laser^{[}
Reference Kazarinov and Suris
^{9}
^{,}
Reference Faist, Capasso, Sivco, Sirtori, Hutchinson and Cho
^{10}
^{]}, optical rectification^{[}
Reference Yang, Richards and Shen
^{11}
^{,}
Reference Auston, Cheung, Valdmanis and Kleinman
^{12}
^{]}, photoconductive^{[}
Reference Auston, Cheung and Smith
^{13}
^{]} and vacuum electronic^{[}
Reference Maestrini, Ward, Gill, Javadi, Schlecht, Tripon-Canseliet, Chattopadhyay and Mehdi
^{14}
^{,}
Reference Crowe, Mattauch, Roser, Bishop, Peatman and Liu
^{15}
^{]} methods, have been widely demonstrated. Due to the material damage threshold, these methods are incapable of generating extremely powerful THz pulses. Such pulses can be used as a powerfully driven pulse for probing and controlling material properties^{[}
Reference Kampfrath, Tanaka and Nelson
^{16}
^{,}
Reference LaRue, Katayama, Lindenberg, Fisher, Öström, Nilsson and Ogasawara
^{17}
^{]}, biological macromolecules^{[}
Reference Choi, Cheng, Huang, Zhang, Norris and Kotov
^{4}
^{]}, electron beam detection^{[}
Reference Zhao, Wang, Tang, Wang, Cheng, Lu, Jiang, Zhu, Hu, Song, Wang, Qiu, Kostin, Jing, Antipov, Wang, Qi, Cheng, Xiang and Zhang
^{18}
^{]} and charged particle acceleration^{[}
Reference Hibberd, Healy, Lake, Georgiadis, Smith, Finlay, Pacey, Jones, Saveliev, Walsh, Snedden, Appleby, Burt, Graham and Jamison
^{19}
^{,}
Reference Zhang, Fallahi, Hemmer, Wu, Fakhari, Hua, Cankaya, Calendron, Zapata, Matlis and Kärtner
^{20}
^{]}. For instance, the mechanisms of four-wave mixing and photoionization become saturated in under-dense plasma just around the power of
${10}^{15}\;\mathrm{W}/{\mathrm{cm}}^2$
^{[}
Reference Lu, He, Zhang, Zhang, Yao, Li and Zhang
^{21}
^{–}
Reference Bergé, Kaltenecker, Engelbrecht, Nguyen, Skupin, Merlat, Fischer, Zhou, Thiele and Jepsen
^{23}
^{]}. As laser intensity continues to improve and nanotechnology develops, terahertz sources based on ultra-intense laser irradiation of dense plasma have started to emerge^{[}
Reference Hamster, Sullivan, Gordon, White and Falcone
^{24}
^{,}
Reference Gopal, Herzer, Schmidt, Singh, Reinhard, Ziegler, Brömmel, Karmakar, Gibbon, Dillner, May, Meyer and Paulus
^{25}
^{]}.

During the past two decades, the quick development of relativistic laser systems, whose peak intensity exceeds
${10}^{18}\;\mathrm{W}/{\mathrm{cm}}^2$
, has opened a new door to obtain extremely powerful THz pulses^{[}
Reference Yi and Fülöp
^{26}
^{]}. However, the existing terahertz sources based on relativistic lasers lack the ability to tune their spectra, which limits their versatility in applications^{[}
Reference Zhang, Fallahi, Hemmer, Wu, Fakhari, Hua, Cankaya, Calendron, Zapata, Matlis and Kärtner
^{20}
^{,}
Reference Vicario, Ruchert, Ardana-Lamas, Derlet, Tudu, Luning and Hauri
^{27}
^{–}
Reference Yuan and Bandrauk
^{30}
^{]}. For example, two-color laser filaments can produce THz pulses with field amplitudes above 10 GV/m and a conversion efficiency of 2.36%^{[}
Reference Koulouklidis, Gollner, Shumakova, Fedorov, Pugžlys, Baltuška and Tzortzakis
^{31}
^{]}, but such pulses have a non-tunable center frequency and a large bandwidth around hundreds of GHz. Another novel method, coherent transition radiation (CTR), occurs when an over-dense electron bunch, produced by irradiating a relativistic laser on a solid target, passes through the vacuum–plasma interface and can emit THz pulses with an energy of 55 mJ and a peak electron field at 4 GV/m^{[}
Reference Liao, Li, Liu, Scott, Neely, Zhang, Zhu, Zhang, Armstrong, Zemaityte, Bradford, Huggard, Rusby, McKenna, Brenner, Woolsey, Wang, Sheng and Zhang
^{32}
^{]}. Numerous efforts have been dedicated to achieving tunable strong-field terahertz radiation using various approaches, such as plasma oscillation^{[}
Reference Pearson, Palastro and Antonsen
^{33}
^{–}
Reference Kwon, Kang, Song, Kim, Ersfeld, Jaroszynski and Hur
^{35}
^{]} and plasma slab methods^{[}
Reference Gupta, Jain, Kulagin, Hur and Suk
^{36}
^{]}. However, most of these solutions have not been able to reach high field strengths. Miao *et al*.^{[}
Reference Miao, Palastro and Antonsen
^{34}
^{]} demonstrated frequency-tunable terahertz radiation with weak field strength by manipulating the frequency of plasma waves. Kwon *et al*.^{[}
Reference Kwon, Kang, Song, Kim, Ersfeld, Jaroszynski and Hur
^{35}
^{]} developed a terahertz source based on plasma dipole oscillation formed by two laser pulses. One of the challenges lies in accessing terahertz frequencies through radiation from plasma waves, as it requires extremely low densities, making it difficult to obtain strong-field terahertz radiation simultaneously. In a significant breakthrough, Liao *et al*.^{[}
Reference Liao, Liu, Scott, Zhang, Zhu, Zhang, Li, Armstrong, Zemaityte, Bradford, Rusby, Neely, Huggard, McKenna, Brenner, Woolsey, Wang, Sheng and Zhang
^{37}
^{]} achieved 0.1–1 THz tunable radiation by transforming the terahertz generation mechanism in laser-irradiated solid targets. However, it should be noted that the radiation derived from different mechanisms exhibits distinct characteristics. The center frequency of strong-field terahertz radiations above 1 THz from CTR is also non-adjustable^{[}
Reference Liao and Li
^{38}
^{]}.

Here, we propose a novel plasma wiggler that uses a femtosecond (fs) laser pulse to produce frequency-tunable and extremely powerful terahertz radiations, improving the flexibility and accuracy of terahertz applications. Figure 1(a) draws the concept of the plasma wiggler. A laser pulse is used to irradiate a block-shaped near-critical density plasma, producing hot electrons^{[}
Reference Wilks, Kruer, Tabak and Langdon
^{39}
^{]}. The accelerated electrons move in two different trajectories. One kind of high-energy electrons gets rid of the potential barrier of the surface electric field and goes away from the plasma surface. The other group of electrons is trapped by the barrier acting as a wiggler. Figure 1(b) illustrates the laser–plasma interaction snapshot at around 273 fs. When this electron beam passes through the transverse interfaces of the plasma, THz radiation^{[}
Reference Liao and Li
^{38}
^{]} and sheath fields
${E}_{\mathrm{s}}$
^{[}
Reference Daido, Nishiuchi and Pirozhkov
^{40}
^{]} can be induced near the interfaces. Under the influence of
${E}_{\mathrm{s}}$
, the electrons are pulled back into the plasma and cross the transverse interface on the other side, exhibiting reciprocating motion along the plasma. The period of this motion is closely related to the plasma thickness and the size of
${E}_{\mathrm{s}}$
. As a result, the plasma with transverse sheath fields
${E}_{\mathrm{s}}$
can be used as a wiggler to manipulate the reciprocating motion of the electrons and regulate THz radiations. A theoretical model, which is verified by particle-in-cell (PIC) simulations, has been developed to describe the physical principles of the tunable THz pulse generation.

Based on the theoretical model and PIC simulations, we find that the center frequency of the THz pulse can be regulated by changing the thickness of the plasma. In simulations using a laser pulse with energy of approximately 430 mJ, the generated THz pulse has a divergence angle of approximately 20°, an ultra-strong-field strength of over 80 GV/m, a laser–THz conversion efficiency of over 2.0% and a center frequency tunable from 4.4 to 1.5 THz by varying the plasma thickness from 20 to
$80\;\mu \mathrm{m}$
. We also demonstrate that this plasma wiggler can work effectively over a wide range of plasma length, thickness, density and laser intensity. Therefore, this method could overcome significant scientific obstacles in the generation of high-quality THz sources and open up brand-new applications^{[}
Reference Vicario, Ruchert, Ardana-Lamas, Derlet, Tudu, Luning and Hauri
^{27}
^{–}
Reference Koulouklidis, Gollner, Shumakova, Fedorov, Pugžlys, Baltuška and Tzortzakis
^{31}
^{]}.

## 2 Results

### 2.1 Theoretical model and simulation setup

When an intense laser pulse irradiates on a plasma, as shown in Figure 1(b), hot electrons whose beam length is close to the laser duration can be generated by laser ponderomotive force^{[}
Reference Wilks, Kruer, Tabak and Langdon
^{39}
^{]}. The electron beam transports in the plasma and passes through its transverse interfaces. As a result, transverse sheath fields
${E}_{\mathrm{s}}$
(more details in the Supplementary Material) can be generated and the maximum strength^{[}
Reference Daido, Nishiuchi and Pirozhkov
^{40}
^{]} on the plasma surfaces can be expressed as follows:

where
$e$
is the Euler’s number and
${n}_{\mathrm{e}}$
and
${T}_{\mathrm{e}}$
are the density and temperature of the hot electrons, respectively. Here, transverse sheath fields
${E}_{\mathrm{s}}$
higher than
${10}^{12}$
V/m can be induced near the plasma transverse interfaces^{[}
Reference Tarkeshian, Vay, Lehe, Schroeder, Esarey, Feurer and Leemans
^{41}
^{]}, as shown in Figure 1(c). Under the action of
${E}_{\mathrm{s}}$
, electrons whose kinetic energy is less than
${\varepsilon}_{\mathrm{e}}$
could be pulled back into the plasma and pass through the transverse interface on the other side^{[}
Reference Sentoku, Cowan, Kemp and Ruhl
^{42}
^{]}, where
${\varepsilon}_{\mathrm{e}}$
could be calculated by the following:

Here,
$l$
is the transverse size of
${E}_{\mathrm{s}}$
,
$\phi$
is the angle at which the electron enters the transverse sheath fields
${E}_{\mathrm{s}}$
and
${l}_{\mathrm{e}}$
is the transverse location where the electron could be pulled back into the plasma. An electron field derived from the simulation was used (more details in the Supplementary Material) for calculation, and then the relation between
${\varepsilon}_{\mathrm{e}}$
and
${l}_{\mathrm{e}}$
could be plotted in Figure 2 by employing
${E}_{\mathrm{s}}$
as shown in Figure 1(c). In the case that the electron perpendicularly penetrates
${E}_{\mathrm{s}}$
(
$\phi ={90}^{\circ }$
), the threshold kinetic energy is about 6 MeV. Most of the electrons penetrate the transverse sheath fields with a much smaller
$\phi$
, and the threshold kinetic energy could be much higher than 6 MeV. Some electrons, which are located in the forefront of the electron beam and experienced much weaker
${E}_{\mathrm{s}}$
, could not be pulled back to the plasma even with a much smaller kinetic energy^{[}
Reference Mora
^{43}
^{]}. According to the large difference in the dynamical behavior of the hot electrons, as shown in Figure 1(c) with red and blue colors, we could separate the hot electrons into two groups: the electrons in group A located in the front of the electron beam are marked with a red color and the electrons in group B located in the beam rear express are marked with a blue color.

Here,
${E}_{\mathrm{s}}$
, which propagates along the plasma surface^{[}
Reference Li, Yuan, Xu, Zheng, Sheng, Chen, Ma, Liang, Yu, Zhang, Liu, Wang, Wei, Zhao, Jin and Zhang
^{44}
^{]} at a velocity close to light speed, can guide the hot electrons in group B to oscillate transversely and propagate longitudinally, as shown in Figure 1(c). Every time the electron passes through the transverse interfaces of the plasma, THz radiation can be emitted. This behaves like a wiggler in traditional light sources^{[}
Reference Bilderback, Elleaume and Weckert
^{45}
^{]}. Therefore, the plasma with transverse sheath fields
${E}_{\mathrm{s}}$
can be considered as a wiggler for the electrons in group B. The reciprocating (wiggler) period of the electron can be expressed as follows:

Here, ${l}_{\mathrm{s}}$ is the transverse distance between the plasma interface and the point where the electron turns around, ${l}_{\mathrm{t}}$ is the thickness of the plasma and ${v}_{\mathrm{t}}$ is the median transverse velocity of the electrons; ${l}_{\mathrm{s}}$ is determined by the transverse momentum of electrons and the strength of ${E}_{\mathrm{s}}$ . The radiation frequency $f$ is closely related to the wiggler frequency ${f}_{\mathrm{w}}=1/{\tau}_{\mathrm{r}}$ , and then we have the following:

where
$\gamma$
is the electron Lorentz factor and
$K$
is the wiggle (undulator) parameter^{[}
Reference Jackson
^{46}
^{]}. Normally,
${l}_{\mathrm{s}}$
is several micro-meters^{[}
Reference Mora
^{43}
^{]}; while the changes in
$\gamma$
,
$K$
and
${v}_{\mathrm{t}}$
can be ignored for the same laser intensity and plasma density, then
${l}_{\mathrm{t}}$
is the only parameter that could be varied. Hence, one can regulate the frequency of the THz radiation by changing the plasma thickness
${l}_{\mathrm{t}}$
according to Equation (4). Since the length of the electron beam is close to the laser pulse duration, which is only 32 fs here, the radiation satisfies the time coherence condition at the THz frequencies^{[}
Reference Liao, Li, Liu, Scott, Neely, Zhang, Zhu, Zhang, Armstrong, Zemaityte, Bradford, Huggard, Rusby, McKenna, Brenner, Woolsey, Wang, Sheng and Zhang
^{32}
^{]}.

### 2.2 Electron kinetics

To generate electrons in group B, one should use a plasma with limited transverse size to ensure the occurrence of reciprocating motion. Simulations, with plasma of different lengths ${l}_{\mathrm{p}}$ (fixing the plasma thickness ${l}_{\mathrm{t}}$ to $30\;\mu \mathrm{m}$ ), were performed to see the effect of plasma length on the electron dynamics. From the simulations with ${l}_{\mathrm{p}}\ge 50\;\mu \mathrm{m}$ , it was found that more than $75\%$ of the laser energy was converted to hot electrons (electron kinetic energy ${e}_{\mathrm{k}}\ge 0.5$ MeV). Meanwhile, in the case of ${l}_{\mathrm{p}}=10\;\mu \mathrm{m}$ , there were few electrons participating in reciprocating motion and most of the electrons could be classified into group A.

When
${l}_{\mathrm{p}}$
was increased to
$50\;\mu \mathrm{m}$
, one could easily distinguish the electrons in group B from those in group A, as plotted in Figures 3(a) and 3(b). Figure 3(a) shows a typical angular-spectra distribution of the electrons, with a large divergence angle, accelerated by the laser ponderomotive force^{[}
Reference Wilks, Kruer, Tabak and Langdon
^{39}
^{]}. This figure is very similar to the result of
${{l}_{\mathrm{p}}=10\;\mu \mathrm{m}}$
. Under the action of
${E}_{\mathrm{s}}$
, the reciprocating motion of the electrons is facilitated and obvious changes in the angular-spectra distribution can be seen in Figure 3(b). In this case, the electron beam had a divergence angle of approximately
${60}^{\circ }$
, which indicates that most of the electrons penetrate the transverse sheath fields with an angle of
${\phi \approx {60}^{\circ }}$
. Hence, the threshold kinetic energy of the electron
${\varepsilon}_{\mathrm{e}}$
is about 8 MeV, which is consistent with the theoretical value predicted by Equation (2), as shown in Figure 2. If
${l}_{\mathrm{p}}$
is further increased, such as
${l}_{\mathrm{p}}=200\;\mu \mathrm{m}$
, more electrons of relatively low energy in group A come into group B, as shown in Figure 3(c), and the number of electrons in group B increases with
${l}_{\mathrm{p}}$
. As a result, the electron number shown in Figure 3(d) is larger than that in Figure 3(b). It is noted that the electrons in group B were manipulated and collimated into a smaller divergence angle by
${E}_{\mathrm{s}}$
, and the electron threshold kinetic energy
${\varepsilon}_{\mathrm{e}}$
could be much higher, as predicted by Equation (2).

We used a semicircular receiving screen, whose radius was 250 μm with the center locating at the midpoint of the plasma right-hand interface, to collect the radiation field. Then, the angular-spectra distribution of the THz source could be obtained from the radiation field through Fourier transform^{[}
Reference Cai, Shou, Han, Huang, Wang, Song, Geng, Yu and Yan
^{47}
^{]}. In this work, the angular spectrum method^{[}
Reference Cai, Shou, Han, Huang, Wang, Song, Geng, Yu and Yan
^{47}
^{–}
Reference Ritter
^{49}
^{]} was employed to remove the near-field radiation, and then all the results of the THz source could be recognized as far-field radiation.

In the simulation with
${l}_{\mathrm{p}}=150\;\mu \mathrm{m}$
, the median longitudinal (transverse) velocity
${v}_{\mathrm{l}}$
(
${v}_{\mathrm{t}}$
) of the electrons in group B was 0.94*c* (0.85*c*). The transverse size of
${E}_{\mathrm{s}}$
was about
$2\;\mu \mathrm{m}$
(at full width at half maximum (FWHM)), and therefore we could assume
${l}_{\mathrm{s}}\approx 2\;\mu \mathrm{m}$
. According to Equation (3), one can get
${\tau}_{\mathrm{r}}=270$
fs and
${\lambda}_{\mathrm{w}}\approx {\tau}_{\mathrm{r}}\times {v}_{\mathrm{l}}=79.4\;\mu \mathrm{m}$
. The wiggler period
${\tau}_{\mathrm{w}}={\tau}_{\mathrm{r}}$
corresponds to a wiggle frequency
${f}_{\mathrm{w}}=3.7$
THz, and the Lorentz factor
$\gamma$
for the longitudinal velocity
${v}_{\mathrm{l}}$
was 2.93. By tracking the electromagnetic fields exerting on the electrons in group B, we got an average value of
$K\approx 6.0$
, which decreased to 5.5 when
${l}_{\mathrm{s}}$
increased to
$80\;\mu \mathrm{m}$
. According to Equation (4), the radiation frequency was
${f=3.34}$
THz, which is also the same as the center frequency of 3.35 THz from the simulation. Hence, the theoretical model can accurately predict the terahertz frequency obtained from the simulations.

### 2.3 Spectrum of terahertz pulses

Figures 4(a)–(c) show the angular-spectra distribution of the THz pulses from the plasma lengths
${l}_{\mathrm{p}}=50$
, 150 and
$300\;\mu \mathrm{m}$
. In the condition of
${l}_{\mathrm{p}}$
being shorter than
${\lambda}_{\mathrm{w}}$
, the role of the electrons in group B could be ignored and the generation of the THz source is dominated by target rear CTR^{[}
Reference Liao, Li, Liu, Scott, Neely, Zhang, Zhu, Zhang, Armstrong, Zemaityte, Bradford, Huggard, Rusby, McKenna, Brenner, Woolsey, Wang, Sheng and Zhang
^{32}
^{,}
Reference Liao, Liu, Scott, Zhang, Zhu, Zhang, Li, Armstrong, Zemaityte, Bradford, Rusby, Neely, Huggard, McKenna, Brenner, Woolsey, Wang, Sheng and Zhang
^{37}
^{,}
Reference Liao, Li, Li, Su, Zheng, Liu, Wang, Hu, Yan, Dunn, Nilsen, Hunter, Liu, Wang, Chen, Ma, Lu, Jin, Kodama, Sheng and Zhang
^{50}
^{]} from the electrons in group A. Hence, the THz source had a large divergence angle (pointed at 88°) and a small energy conversion efficiency of 0.45%, as shown in Figure 4(a).

Every time the electron passes through the transverse interfaces of the plasma, THz radiation can be emitted and becomes stronger in the case of the electrons in group B oscillating for several periods. When
${l}_{\mathrm{p}}$
was enlarged to
$150\;\mu \mathrm{m}$
, more than
$1.3\%$
of the laser energy was converted to THz radiation and the characteristics of the narrow spectrum became obvious, as shown in Figure 4(b). The simulation results indicated that
${l}_{\mathrm{p}}\ge 150\;\mu \mathrm{m}$
was more conducive to obtain THz pulses of a narrow spectrum, as shown in Figures 4(b) and 4(c). As the electrons in group B are guided to oscillate along the plasma by
${E}_{\mathrm{s}}$
, the resulting THz pulse could be better collimated than the other intense laser–plasma-based THz sources^{[}
Reference Yi and Fülöp
^{26}
^{,}
Reference Liao, Liu, Scott, Zhang, Zhu, Zhang, Li, Armstrong, Zemaityte, Bradford, Rusby, Neely, Huggard, McKenna, Brenner, Woolsey, Wang, Sheng and Zhang
^{37}
^{,}
Reference Liao, Li, Li, Su, Zheng, Liu, Wang, Hu, Yan, Dunn, Nilsen, Hunter, Liu, Wang, Chen, Ma, Lu, Jin, Kodama, Sheng and Zhang
^{50}
^{]}. It was found that the THz pulse that pointed at
$\theta ={37}^{\circ }$
could be collimated into a divergence angle of approximately
${20}^{\circ }$
, as shown in Figure 4(c). Since
$\theta >1/\gamma$
, we call it a wiggler instead of an undulator. Then we collected the radiation field at
${37}^{\circ }$
from the simulation of
${l}_{\mathrm{p}}=300\;\mu \mathrm{m}$
, and filtered the THz signal (frequency range: 0.1–10 THz) from the radiation field, as shown in Figure 4(d). The THz field was higher than 80 GV/m, which was much greater than those from THz sources driven by similar laser parameters^{[}
Reference Yi and Fülöp
^{26}
^{,}
Reference Liao, Liu, Scott, Zhang, Zhu, Zhang, Li, Armstrong, Zemaityte, Bradford, Rusby, Neely, Huggard, McKenna, Brenner, Woolsey, Wang, Sheng and Zhang
^{37}
^{]}, after the THz pulse propagated
$250\;\mu \mathrm{m}$
off the plasma. Because the THz radiation comes from the wiggler process, the THz electric field waveform of multi-cycles^{[}
Reference Tian, Liu, Bai, Zhou, Sun, Liu, Zhao, Li and Xu
^{51}
^{]} was completely different from the half-cycle THz sources driven by intense lasers and plasma^{[}
Reference Yi and Fülöp
^{26}
^{,}
Reference Liao, Liu, Scott, Zhang, Zhu, Zhang, Li, Armstrong, Zemaityte, Bradford, Rusby, Neely, Huggard, McKenna, Brenner, Woolsey, Wang, Sheng and Zhang
^{37}
^{,}
Reference Liao, Li, Li, Su, Zheng, Liu, Wang, Hu, Yan, Dunn, Nilsen, Hunter, Liu, Wang, Chen, Ma, Lu, Jin, Kodama, Sheng and Zhang
^{50}
^{]}.

### 2.4 Tunable frequency

From Figures 3(a) and 3(c), one can see that the electrons in group A had a large divergence angle. The electron density ${n}_{\mathrm{e}}$ decreases with the increase of transmission distance, and ${E}_{\mathrm{s}}$ becomes weaker for the longer plasma according to Equation (1), in which case the electrons in group B are more difficult to pull back to the plasma. As a result of enhancing ${l}_{\mathrm{p}}$ , ${l}_{\mathrm{s}}$ becomes larger and more electrons no longer participate in the wiggler motion. Hence, the center frequency decreases and the growth of energy conversion efficiency gradually slows down with the enhancement of ${l}_{\mathrm{p}}$ . Simulations by varying ${l}_{\mathrm{p}}$ from 50 to $900\;\mu \mathrm{m}$ have demonstrated the effect on the center frequency, as shown in Figure 5(a). From the figure, one can see that the center frequency of the THz pulse was tunable from 3.5 to 2.6 THz by changing the plasma length ${l}_{\mathrm{p}}$ from 150 to $900\;\mu \mathrm{m}$ . On the other hand, the laser–THz energy conversion efficiency, which increased with ${l}_{\mathrm{p}}$ and became saturated after $600\;\mu \mathrm{m}$ , as shown in Figure 5(a), was 2.05%. The saturation length may be longer than $600\;\mu \mathrm{m}$ since a small portion of the THz source came out of the simulation window, in the case of longer ${l}_{\mathrm{p}}$ , due to the limitation of computational ability.

Equation (4) indicates that we could regulate the center frequency of the THz pulse by changing the thickness of the plasma ${l}_{\mathrm{t}}$ . More simulations were performed by changing ${l}_{\mathrm{t}}$ from 20 to $80\;\mu \mathrm{m}$ (fixed ${l}_{\mathrm{p}}$ at $300\;\mu \mathrm{m}$ ). It was found that the laser–THz energy conversion efficiency was almost the same for different ${l}_{\mathrm{t}}$ , while the center frequency of the THz source decreased with the increase of ${l}_{\mathrm{t}}$ , as shown in Figure 5(b). For different ${l}_{\mathrm{t}}$ , ${v}_{\mathrm{t}}$ can be obtained from the corresponding simulations, and the simulation results also show ${l}_{\mathrm{s}}$ ranging from 0.5 to $5\;\mu \mathrm{m}$ . We assumed that $K$ decreased with the increase of ${l}_{\mathrm{s}}$ from 6.0 to 5.5 for the above cases, then we calculated the center frequencies of the THz pulse according to Equation (4) and plotted them in Figure 5(b). From the figure, one can see that the theoretical model developed here works very well in predicting the generation of center-frequency-tunable THz pulses with a laser-driven plasma wiggler.

## 3 Discussion

To validate the physical results, we conducted 3D PIC simulations using a box size of $1920\times 800\times 800$ and a resolution of $0.15\lambda \times 0.25\lambda \times 0.25\lambda$ (further details can be found in the Supplementary Material). The 3D simulation produced a spectrum in the cross-section that closely resembles that obtained from the 2D simulation. In addition, the spatial distribution exhibits a quasi-2D pattern, indicating similarities between the two simulations.

Parametric simulations were also carried out to see the effects of laser intensity, plasma density and density gradient at the boundary (more details in the Supplementary Material). It was found that the energy conversion efficiency was almost unchanged for the plasma of the same thickness under a relativistic intense laser ( ${a}_0\ge 1.0$ ), while the center frequency, which decreased from 3.35 to 1.7 THz, would further reduce under smaller ${a}_0$ due to the drop of electron ${v}_{\mathrm{t}}$ . Since a low-density plasma is more beneficial to improving the energy efficiency from the laser pulse to relativistic electrons, the laser–THz energy conversion efficiency was higher in the case of lower density plasma. However, the center frequency of the THz source was almost the same. In reality, the plasma has a density gradient at the boundaries. Simulations using a pre-expand plasma with a scale length varied from 0.1 to $1\;\mu \mathrm{m}$ showed that there were almost no changes in the THz angular-spectra distribution. The parametric scans demonstrated the validity of this novel laser-driven plasma wiggler for the generation of high-power and center-frequency-tunable THz pulses over a wide range of laser and plasma parameters. With this method, one could produce ultra-high-power THz pulses of approximately 200 mJ by using a laser pulse of approximately $10$ J. Such a state-of-the-art THz source would initiate future applications of high-power THz pulses based on a compact laser.

## 4 Conclusion

In summary, we propose a laser-driven plasma wiggler for efficiently generating a high-power, collimated, narrow-band and center-frequency-tunable THz pulse, by manipulating the electron reciprocating motion. A theoretical model is developed to describe the physical principle of realizing the center-frequency-tunable THz pulse. According to the model and PIC simulations, the center frequency of the THz pulse corresponds strictly to the reciprocating motion period of the electron beam. Simulations indicate that the center frequency of the THz pulses can be tuned from 4.4 to 1.5 THz as the plasma thickness changes from 20 to $80\;\mu \mathrm{m}$ . Meanwhile, the THz pulse is collimated into a divergence angle of approximately ${20}^{\circ }$ , enabling a laser–THz conversion efficiency of more than $2.0\%$ and an ultra-strong-field strength of over 80 GV/m, driven by a table-top laser of approximately $430$ mJ. This method could address a long-standing challenge in THz science and generate a state-of-the-art THz source over a wide range of laser and plasma parameters.

## Appendix A. Particle-in-cell simulation setup

Numerical simulations were performed by using the 2D PIC codes EPOCH^{[}
Reference Arber, Bennett, Brady, Lawrence-Douglas, Ramsay, Sircombe, Gillies, Evans, Schmitz, Bell and Ridgers
^{52}
^{]} and SMILEI^{[}
Reference Derouillat, Beck, Pérez, Vinci, Chiaramello, Grassi, Flé, Bouchard, Plotnikov, Aunai, Dargent, Riconda and Grech
^{53}
^{]} to verify this scheme. The simulation window
$X\times Y=600\;\mu \mathrm{m}\times 570\;\mu \mathrm{m}$
was divided into
$12,500\times 7125$
cells. A laser pulse, with Gaussian spatial and
${\sin}^2$
temporal profiles, wavelength
${\lambda}_0=800$
nm, waist
${w}_0=7.2\;\mu \mathrm{m}$
, normalized intensity
${a}_0=3$
, polarized in the *Y* direction and with the duration of 32 fs at FWHM, was used as the driving source. Such a laser pulse can be produced by a compact laser^{[}
Reference Danson, Haefner, Bromage, Butcher, Chanteloup, Chowdhury, Galvanauskas, Gizzi, Hein, Hillier, Hopps, Kato, Khazanov, Kodama, Korn, Li, Li, Limpert, Ma, Nam, Neely, Papadopoulos, Penman, Qian, Rocca, Shaykin, Siders, Spindloe, Szatmári, Trines, Zhu, Zhu and Zuegel
^{54}
^{]}. We used a block-shaped near-critical density plasma (electron density
${n}_{\mathrm{e}}=0.2{n}_{\mathrm{c}}$
, where
${n}_{\mathrm{c}}$
is the critical density) whose length and thickness were adjustable. Such a plasma could be made of carbon nanotube foams^{[}
Reference Ma, Song, Yang, Zhang, Zhao, Sun, Ren, Liu, Liu, Shen, Zhang, Xiang, Zhou and Xie
^{55}
^{,}
Reference Shou, Wang, Lee, Rhee, Lee, Yoon, Sung, Lee, Pan, Kong, Mei, Liu, Xu, Deng, Zhou, Tajima, Choi, Yan, Nam and Ma
^{56}
^{]}. Twenty-four (four) macro-electrons (C
${}^{6+}$
) were initialized into each cell. The simulation time was varied with plasma length
${l}_{\mathrm{p}}$
to make sure that we had more than 1 picosecond to collect the THz signal, which should be enough to cover the whole interaction process for the generation of THz sources. Meanwhile, the zero padding method^{[}
Reference Harris, Millman, Van Der Walt, Gommers, Virtanen, Cournapeau, Wieser, Taylor, Berg, Smith, Kern, Picus, Hoyer, van Kerkwijk, Brett, Haldane, del Río, Wiebe, Peterson, Gérard-Marchant, Sheppard, Reddy, Weckesser, Abbasi, Gohlke and Oliphant
^{57}
^{]} was used to further improve the resolution of the THz signal.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11921006 and 12175058), Beijing Distinguished Young Scientist Program and National Grand Instrument Project (Grant No. SQ2019YFF01014400). The PIC code EPOCH was in part funded by UK EPSRC (Grant Nos. EP/G054950/1, EP/G056803/1, EP/G055165/1 and EP/M022463/1). We give thanks for the helpful discussions with Prof. Z. Najmudin from Imperial College London and Prof. Wenjun Ma from Peking University.

## Supplementary Material

To view supplementary material for this article, please visit http://doi.org/10.1017/hpl.2023.78.