# Coherent diffraction radiation of relativistic terahertz pulses from a laser-driven micro-plasma-waveguide

###### Abstract

We propose a method to generate isolated relativistic terahertz (THz) pulses using a high-power laser irradiating a mirco-plasma-waveguide (MPW). When the laser pulse enters the MPW, high-charge electron bunches are produced and accelerated to 100 MeV by the transverse magnetic modes. A substantial part of the electron energy is transferred to THz emission through coherent diffraction radiation as the electron bunches exit the MPW. We demonstrate this process with three-dimensional particle-in-cell simulations. The frequency of the radiation is determined by the incident laser duration, and the radiated energy is found to be strongly correlated to the charge of the electron bunches, which can be controlled by the laser intensity and micro-engineering of the MPW target. Our simulations indicate that 100-mJ level relativistic-intense THz pulses with tunable frequency can be generated at existing laser facilities, and the overall efficiency reaches 1.

###### pacs:

^{†}

^{†}preprint: Preprint

^{†}

^{†}thanks: longqing@chalmers.se

High power terahertz (THz) pulses have attracted significant attention since they can serve as a unique and versatile tool in fields ranging from biological imaging to material science Tonouchi2007 (); Siegel2004 (); Cole2001 (); Hoffmann2011 (). In particular, at high intensities, such pulses allow manipulation of the transient states of matter, for example giving control over the electronic, spin and ionic degrees of freedom of molecules and solids Kampfrath2013 (). Several methods such as two-color laser filamentation Oh2014 (), optical reflection in lithium-niobate Hirori2011 (); Fulop2014 () or organic crystals Vicario2014 (), and relativistic laser irradiated plasmas Sheng2005 (); Gopal2013 (); Li2012 (); Liao2016 (); Tian2017 (); Chen2015 (); Herzer2018 (); Liao2019 (); Thiele2019 (), have been developed for generation of THz pulses with electric fields above 1 MV/cm. However, scaling up such methods towards higher intensities remains challenging, thus representing an active research field.

Relativistic electron beams have also been used to produce THz radiation through a variety of mechanisms that include synchrotron radiation Carr2002 (), transition radiation Ginzburg1982 (); Happek1991 (), and diffraction radiation Dnestrovskii1959 (); Shibata1995 (). Radiation emitted by these mechanisms is coherent if the bunch length is shorter than the radiated wavelength of interest. The radiated energy then scales as the square of the beam charge. Previous studies have also shown that the radiation power decreases significantly with the beam divergence, and the energy radiated in a small cone near-axis would strongly benefit from a high beam energy Schroeder2004 (). Therefore, choosing an electron source with desired qualities (high charge, high energy, and well-collimated) can be crucial for producing intense THz emission that is attractive to a range of applications Kampfrath2013 ().

Currently available sources of relativistic electron beams are either linear accelerators or compact sources based on laser-plasma acceleration. The THz radiation energy from linear accelerators has reached 600 J/pulse Wu2013 (), but such sources are expensive and large and thus can only offer limited accessibility. Laser wakefield acceleration in the nonlinear “bubble” regime can produce multi-GeV electron beams with small divergence (0.1 mrad), but only small charge (1-100 pC) Leemans2014 (). Self-modulated laser-wakefield acceleration can produce nano-Coulomb (nC) electron bunches Leemans2002 () but typically have a temperature of a few MeV, and the beam divergence is large due to direct laser acceleration Gahn1999 (). Last but not least, hot electrons that arise from laser-solid interaction can reach up to nC-C charge, but the electron temperature is typically only a few hundreds of keVs to a few MeVs, and the divergence is usually large () Liao2016 (). Recently, THz radiation energy above millijoule (mJ) level has been reported in laser-solid interaction Liao2019 (), but since a picosecond laser pulse is used, the coherent frequency range is below 1 THz, and the efficiency is .

In this letter, we propose a scheme to generate isolated THz pulses with electric fields beyond 1 GV/cm with high efficiency (). As illustrated in Fig. 1(a), an intense laser pulse is focused into a micro-plasma-waveguide (MPW), leading to electrons being extracted from the wall and accelerated by longitudinal electric fields of the transverse magnetic modes up to a few hundreds of MeVs Bulanov1994 (); Yi2016_1 (); Yi2016_2 (); Yi2016_3 (); Gong2019 (). The divergence is usually a few degrees and the duration is the same as the laser pulse. Typically the electron beam inherits the density of the plasma skin layer (, where is the critical density, and are the elementary charge and the electron mass, is the laser angular frequency) from which it is generated Naumova2004 (). Thus, a total charge of a few tens of nC can easily be obtained with contemporary -fs high-power laser systems. Such electron source is suitable for THz generation based on coherent transition radiation and/or coherent diffraction radiation (CDR) as noted above. The simulations demonstrate that when the electron beam exits the MPW, a substantial part of the electron energy () is transferred to electromagnetic energy through CDR, leading to relativistically strong THz pulses up to 10-100 mJ energy.

We demonstrate our scheme using 3-dimensional (3D) particle-in-cell (PIC) simulations with the EPOCH code Arber2015 (). A linearly polarised (in -direction) laser with intensity 1.4 10 W/cm (normalised intensity , where is the amplitude of laser electric field and is the speed of light) is focused on the entrance of a MPW, propagating along the -axis. The laser beam has a temporal Gaussian profile with FWHM duration of fs and a focal spot , where m is the laser wavelength. The MPW has a density of , the radius and length are m and m, respectively. The inner surface of the MPW (, where ) has a density gradient , and the scale length is m. This leads to an effective MPW radius to be m, where . The dimensions of the simulation box are and are sampled by cells with 8 macro particles for electrons and 2 for C ions. The algorithm proposed by Cowan et al. Cowan2013 () is used to minimise the numerical dispersion.

The electric fields in the simulation with frequency below 60 THz are presented in Fig. 1(a-b), where a 35-mJ THz pulse is obtained, and the radiated power reaches 0.7 TW. To show the polarisation of the CDR, we apply spherical coordinates with the origin at the exit of MPW on the laser propagation axis m, and convert the coordinates according to , , and as illustrated in Fig. 1(a).

The radiation fields are emitted simultaneously with electron propagation through the aperture at , mostly confined in a spherical shell. The THz emission is predominantly radially polarised in the plane determined by the observation line-of-sight and the laser propagation axis. The polar component, , contains 99 of the radiation energy. The preference of electron distribution in the laser polarisation direction results in a small quadrupolar azimuthal electric field . The radial component is negligible in the radiating shell.

The angular distribution of THz energy in the forward direction is shown in Fig. 1(c), with white lines in the centre representing the electron beam density. The electrons reach an cut-off energy of 100 MeV when they exit the MPW, and their total charge is 7.4 nC. The divergence of the electron beam is about . It is slightly elongated along the laser polarisation direction. A depleted area is observed within the electron beam, because the radiation fields add coherently and tend to cancel each other in this region. Since the electron energy is high, the CDR power is strongly peaked on the edge of the beam Carron2000 (). The intensity rises sharply forming a very thin layer () around . The radiation power in the direction perpendicular to the laser polarisation direction is higher due to the coherent sum of electric fields radiated by the elongated electron beam.

The radiation field seen at , is shown by the black line in Fig. 2(a). The red line shows the low-frequency component below 60 THz. The amplitude of the half-cycle THz pulse is 3 GV/cm, corresponding to a normalised amplitude of , reaching the relativistic intensity. Figure 2(b) shows the spectra of the radiation fields: most of the pulse energy concentrates in the desired THz frequency range of 1-10 THz. The inset in Fig. 2(b) shows the spectrum from 0 to 1000 THz for fs, with a small bump around the laser frequency at 300 THz and a peak at the double frequency 600 THz. The latter is the result of the modulation of the electron beam at , and can serve as an experimental signature of the CDR Schroeder2004 ().

The duration of the THz field coincides with the laser pulse, and the cut-off frequency is determined accordingly, as shown in Fig. 2, where the green and blue dashed lines represent the radiation field (after frequency-filtering) and spectra produced by laser pulses with of 36 fs and 72 fs, respectively. With currently available laser systems, this scheme is capable of generating relativistic pulses with frequencies ranging from near infrared to sub-THz.

The mechanism of the electron beam generation, i.e. the electron injected into the channel (vacuum core of the MPW), is crucial for understanding the THz radiation power. The production of electron bunches at a plasma-vacuum interface when irradiated by an intense laser pulse can be attributed to the counterstreaming electrons percolating through the laser nodes, as the laser pushes the surface electrons inwards Naumova2004 (). In the MPW, the underlying physics is similar, but the mechanism that pushes the surface electrons (towards the plasma cladding), and the associated radial counterstreaming, depends on the ratio of the laser focal spot size () and the effective MPW radius (), which results in different injection behaviour.

To show this, we perform 2D PIC simulations of lasers having different focal spot sizes propagating in a long waveguide (240 m). The laser and plasma parameters are the same as in the 3D simulation unless otherwise described. The resolution is 50 and 20 cells per laser wavelength in longitudinal and transverse directions, respectively. The third dimension is assumed to be 4 m when estimating the electron number. In Fig. 3, we plot the total electron number (above 10 MeV) produced in the laser-MPW interaction against the propagation distance for different ratios (the laser energy is fixed).

Figure 3 shows that when , the injection happens very fast, mainly at the entrance of the MPW. This is because the initial impact of the laser and MPW front surface is violent, which leads to strong diffracted light that pushes surface electrons into the plasma, thus results in significant counterstreaming. The electrons are more likely to be injected at this stage. In the cases where , the injection at the entrance is significantly reduced and the injection inside the MPW becomes important, which is due to the interaction between waveguide modes and the MPW wall. The photon momentum , associated with the transverse wave number , pushes the surface plasma radially as the light is bouncing between the walls.

Interestingly, despite different injection processes, all cases result in similar beam charges for sufficiently long MPW. This is because in order to be injected into the channel, the electrons percolating through the laser nodes must also overcome the electrostatic potential barrier near the MPW wall, which leads to saturation. The charge injected at the entrance suppresses the injection inside the channel. In the end, the maximum charge separation on the wall will be just sufficient to prevent the most energetic counterstreaming electrons inside the MPW from escaping. The energy of these electrons is determined by the fundamental waveguide mode, which is the same for all cases.

To estimate the energy of the electrons we use momentum conservation. Note that the ion response time is typically longer than the laser duration, therefore the photon momenta is first transferred to electrons. The number of plasma electrons streaming towards the MPW inner surface (counterstream due to charge separation) per unit time is , where and are the transverse and longitudinal wavenumber in the MPW, is the radial velocity of counterstreaming electrons normalised by . These electrons are reflected back on the plasma-vacuum interface due to the interaction with the photons ( per unit time). We assume the number of electrons percolating through the laser nodes as well as the number of the photons absorbed are negligibly small during this process. According to momentum conservation , where , and are the relativistic gamma factor and the normalised radial velocity of the electrons that being pushed back (after the interaction).

Here we are only interested in the maximum counterstreaming electron energy that can be achieved. Substituting due to quasi-neutrality, we find that when the longitudinal velocity of the surface electrons vanishes after interaction, reaches its maximum,

(1) |

where , , and is the normalised intensity of the waveguide mode. For , , and for , . We have assumed the radius of MPW is sufficiently large (), and only the fundamental mode exists inside the MPW, so that and is the first root of eigenvalue equation Shen1991 ().

The electrostatic field near the MPW wall can be estimated using Gauss’s law, , where is the charge that is lost from the wall (i.e. injected into the channel). Further injection can only happen when the kinetic energy of conterstreaming electrons overcomes the electrostatic potential within the skin layer, i.e. , which yields the saturation charge,

(2) |

As an order-of-magnitude estimate, for a micro-sized channel, is typically around unity. This means that a W/cm, 50-fs laser system could produce 10 nC electron beams, which agrees with simulations. From Eq. (2) the scaling of the charge with the normalised laser intensity can be estimated: for weakly-relativistic cases () , while for strongly-relativistic cases () . Note, that the energy of CDR scales as , it is therefore important to confirm these scalings by 3D PIC simulations to guide future experiments. In the analysis above we neglected the azimuthal dependence, which is strictly valid only for circular polarization. However, 3D PIC simulations presented in Fig. 4 show that the obtained scalings are valid also for linear polarization.

In Fig. 4(a), we plot the electron charge produced by the MPW and the total THz energy (below 60 THz) as functions of , where m and m are fixed, and the MPW length is m. The parameters are the same as in Fig. 1 unless otherwise stated. It is shown that the charge increases quadratically with the normalised laser intensity () when is small, and the scaling becomes linear () for , where exceeds unity. In addition, the simulation results indicate that the THz energy can be fitted by in the weakly-relativistic regime, where the transition efficiency increases linearly with the intensity. In the strongly-relativistic regime, the THz energy can be fitted by , where the conversion efficiency saturates at . Note, that this also demonstrates that the radiation is coherent (). Our numerical results suggest that TW-class, 100-mJ-strong THz emission can be produced by a 10-J/250-TW laser system, which is within reach of the existing laser facilities.

In Fig. 4(b), we consider the effects of varying the MPW radius variation when the laser parameters are fixed ( and m), and the MPW length is extended to m to ensure sufficient distance for injection. In this case, Eq. (2) leads to in the strongly-relativistic regime, and in the weakly-relativistic regime, which agrees with our simulations. Since the radiation is coherent, it results in a quenching effect: the radiation energy drops dramatically () as the effective radius exceeds a threshold near . This is verified by a sharp decrease of the THz energy at the separatrix of the two regimes around m, when the counterstreaming electrons become weakly-relativistic (i.e. ).

Finally, we note that Eq. (2) does not consider the effects of high-order waveguide modes (which give a higher transverse light pressure) and strong diffraction at the entrance (so that the charge produced at the entrance may already exceed the saturation limit suggested by Eq. (2), especially in large MPW). In fact, the value from Eq. (2) should be treated as the minimum charge that can be produced by laser-MPW interaction, as the interaction between the lowest-order mode and MPW is the weakest. A detailed study of these effects is left for future work.

In conclusion, we proposed a scheme to generate relativistic isolated THz pulses based on the interaction of a laser pulse with a micro-plasma-waveguide. 3D PIC simulations show the energetic electron beam with a few tens of nC charge can be produced inside the channel. As the beam exits the waveguide, a substantial part of the electron energy is transferred to an intense THz emission through coherent diffraction radiation. We demonstrated with 3D PIC simulations that the overall efficiency reaches 1, the radiation power 1 TW and the energy 100 mJ. We obtain scaling laws for THz generation energy in different regimes that are characterised by the maximum gamma factor of the counterstreaming electrons induced by the fundamental mode. The proposed scheme can be easily extended to other frequency ranges by varying the driving laser duration, allowing the generation of radiation from infrared to sub-THz range with relativistic intensities. This opens a new avenue towards high-power light matter interaction beyond the state of the art.

###### Acknowledgements.

The authors acknowledge fruitful discussions with I Thiele, S Newton, I Pusztai, E Siminos, and J Ferri. This work is supported by the Olle Engqvist Foundation, the Knut and Alice Wallenberg Foundation and the European Research Council (ERC-2014-CoG grant 647121). Simulations were performed on resources at Chalmers Centre for Computational Science and Engineering (C3SE) provided by the Swedish National Infrastructure for Computing (SNIC).## References

- (1) M. Tonouchi, Nat. Photonics 1, 97 (2007).
- (2) P. H. Siegel, IEEE Trans. Microwave Theory Tech. 52, 2438 (2004).
- (3) B. E. Cole, J. B. Williams, B. T. King, M. S. Shervin, and C. R. Stanley, Nature 410, 60 (2001).
- (4) M. C. Hoffmann and J. A. Fülöp, J. Phys. D 44, 083001 (2011).
- (5) T. Kampfrath, K. Tanaka, and K. Nelson, Nat. Photonics 7, 680 (2013).
- (6) T. I. Oh, Y. J. Yoo, Y. S. You, and K. Y. Kim, Appl. Rev. Lett. 112, 213901 (2014).
- (7) H. Hirori, A. Doi, F. Blanchard, and K. Tanaka, Appl. Phys. Lett. 98, 091106 (2011).
- (8) J. A. Fülöp, Z. Ollmann, Cs. Lombosi, C. Skrobol, S. Klingebiel, L. Pálfalvi, F. Krausz, S. Karsch, and J. Hebling, Opt. Express 22, 20155 (2014).
- (9) C. Vicario, B. Monoszlai, and C. P. Hauri, Phys. Rev. Lett. 112, 213901 (2014).
- (10) Z. M. Sheng, K. Mima, J. Zhang, and H. Sanuki, Phys. Rev. Lett. 94, 095003 (2005).
- (11) A. Gopal, S. Herzer, A. Schmidt, P. Singh, A. Reinhard, W. Ziegler, D. Brömmel, A. Karmakar, P. Gibbon, U. Dillner, T. May, H. G. Meyer, and G. G. Paulus, Phys. Rev. Lett. 111, 074802 (2013).
- (12) Y. T. Li, C. Li, M. L. Zhou, W. M. Wang, F. Du, W. J. Ding, X. X. Lin, F. Liu, Z. M. Sheng, X. Y. Peng, L. M. Chen, J. L. Ma, X. Lu, Z. H. Wang, Z. Y. Wei, and J. Zhang, Appl. Phys. Lett. 110, 254101 (2012).
- (13) G. Q. Liao, Y. T. Li, Y. H. Zhang, H. Liu, X. L. Ge, S. Yang, W. Q. Wei, X. H. Yuan, Y. Q. Deng, B. J. Zhu, Z. Zhang, W. M. Wang, Z. M. Sheng, L. M. Chen, X. Lin, J. L. Ma, X. Wang and J. Zhang, Phys. Rev. Lett. 116, 205003 (2016).
- (14) Y. Tian, J. S. Liu, Y. F. Bai, S. Y. Zhou, H. Y. Sun, W. W. Liu, J. Y. Zhao, R. X. Li, and Zhizhan Xu, Nat. Photonics 11, 242 (2017).
- (15) Z. Y. Chen and A. Pukhov, Phys. Plasmas 22, 103105 (2015).
- (16) S. Herzer, A. Woldegeorgis, J. Polz, A. Reinhard, M. Almassarani, B. Beleites, F. Ronneberger, R. Grosse, G. G. Paulus, U. Hübner, T. May and A. Gopal, New J. Phys. 20, 063019 (2018).
- (17) G. Q. Liao, Y. T. Li, H. Liu, G. G. Scott, D. Neely, Y. H. Zhang, B. J. Zhu, Z. Zhang, C. Armstrong, E. Zemaityte, P. Bradford, P. G. Huggard, D. R. Rusby, P. McKenna, C. M. Brenner, N. C. Woolsey, W. M. Wang, Z. M. Sheng and J. Zhang, PNAS 116, 3994 (2019).
- (18) I. Thiele, E. Simios, and T. Fülöp, Phys. Rev. Lett. 122, 104803 (2019).
- (19) G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, Nature 420, 153 (2002).
- (20) V. L. Ginzburg, Phys. Scr. T2/1, 182 (1982).
- (21) U. Happek, A. J. Sievers, and E. B. Blum, Phys. Rev. Lett. 67, 2962 (1991).
- (22) Yu. N. Dnestrovskii and D. P. Kostomarov, Dokl. Akad. Nauk 124, 792 (1959) [Sov. Phys. Dokl. 4, 132 (1959)]; 124, 1026 (1959) [4, 158 (1959)].
- (23) Y. Shibata, S. Hasebe, K. Ishi, T. Takahashi, T. Ohsaka, and M. Ikezawa, Phys. Rev. E 52, 6787 (1995).
- (24) C. B. Schroeder, E. Esarey, J. Tilborg, and W. P. Leemans, Phys. Rev. E 69, 016501 (2004).
- (25) Z. Wu, A. S. Fisher, J. Goodfellow, M. Fushs, D. Darancinang, M. Hogan, H. Loos, and A. Lindenberg, Rev. Sci. Instrum. 84, 022701 (2013).
- (26) W. P. Leemans, A. J. Gonsalves, H. S. Mao, K. Nakajima, C. Benedetti, C. B. Schroeder, Cs. Tóth, J. Daniels, D. E. Mittelberger, S. S. Bulanov, J. L. Vay, C. G. R. Geddes, and E. Esarey, Phys. Rev. Lett. 113, 245002 (2014).
- (27) W. P. Leemans, P. Catravas, E. Esarey, C. G. R. Geddes, C. Toth, R. Trines, C. B. Schroeder, B. A. Shadwick, J. van Tilborg, and J. Faure, Phys. Rev. Lett. 89, 174802 (2002).
- (28) C. Gahn, G. D. Tsakiris, A. Pukhov, J. Meyer-ter-Vehn, G. Pretzler, P. Thirolf, D. Habs, and K. J. Witte, Phys. Rev. Lett. 83, 4772 (1999).
- (29) S. V. Bulanov, F. F. Kamenets, F. Pegoraro, and A. M. Pukhov, Phys. Rev. A 195, 84 (1994).
- (30) L. Q. Yi, A. Pukhov, P. Luu-Thanh, and B. F. Shen, Phys. Rev. Lett. 116, 115001 (2016).
- (31) L. Q. Yi, A. Pukhov, and B. F. Shen, Phys. Plasmas 23, 073110 (2016).
- (32) L. Q. Yi, A. Pukhov, and B. F. Shen, Sci. Rep. 6, 28147 (2016).
- (33) Z. Gong, A. P. L. Robinson, X. Q. Yan, and A. V. Arefiev, Plasma Phys. Control Fusion 61, 035012 (2019).
- (34) N. Naumova, I. Sokolov, J. Nees, A. Maksimchuk, V. Yanovsky and G. Mourou, Phys. Rev. Lett. 93, 195003 (2004).
- (35) T. D. Arber, K. Bennett, C. S. Brady, A. Lawrence-Douglas, M. G. Ramsay, N. J. Sircombe, P. Gillies, R. G. Evans, H. Schmitz, A. R. Bell, and C. P. Ridgers, Plasma Phys. Control. Fusion 57, 113001 (2015).
- (36) B. M. Cowan, D. L. Bruhwiler, J. R. Cary, and E. Cormier-Michel, Phys. Rev. Accel. Beams 16, 041303 (2013).
- (37) N. J. Carron, Prog. Electromagnetics Res. 28, 147 (2000).
- (38) H. M. Shen, J. Appl. Phys. 69, 6827 (1991).