# Single-cycle attosecond pulses by Thomson backscattering of terahertz pulses

###### Abstract

The generation of single-cycle attosecond pulses based on Thomson scattering of terahertz (THz) pulses is proposed. In the scheme, a high-quality relativistic electron beam produced by a laser-plasma wakefield accelerator (LPWA), is sent through suitable magnetic devices to produce ultrathin electron layers for coherent Thomson backscattering of intense THz pulses. According to numerical simulations, single-cycle attosecond pulse generation is possible with up to 1 nJ energy. The waveform of the attosecond pulses closely resembles that of the THz pulses. This allows for a flexible waveform control of attosecond pulses.

## 1 Introduction

Attosecond pulses can be generated in a number of ways, such as high-harmonic generation [Krausz2009], undulator radiation [Zholents2004, Saldin2006, Marinelli2013, Dunning2013], or Thomson scattering [Wu2011, Paz-2012, Lu02013, Hack2017]. High-harmonic generation is capable of producing single-cycle attosecond pulses [Sansone2006]. Even subcycle pulses were generated by laser-generated ultrathin electron layers from double-foil targets [Ma-2014]. However, controlling the pulse shape has not been demonstrated so far. This is also true of radiation sources based on relativistic electrons [Dunning2013, Tanaka2014]. In order to circumwent this limitation, we proposed a technique to generate cerrier-envelope phase stable, waveform-controlled single-cycle [TibaiPRL, TibaiarXive] or few-cycle [Toth2016] attosecond pulses by coherent undulator radiation. In this scheme, ultrathin (of a few nm) relativistic electron layers, generated in a laser-assisted bunching process, emit attosecond pulses when passing through a radiator undulator. The magnetic field distribution of the undulator is transferred to the waveform of the emitted radiation [TibaiPRL, Toth2016, TibaiarXive], thereby offering a unique waveform tailoring capability. Nevertheless, this technique requires high-energy (GeV) electrons, as an extremely short undulator would be necessary for lower-energy (100 MeV) electrons, which are more easily avaialable even from laser electron accelerators.

A LPWA is able to generate electron beams over cm acceleration distances with parameters comparable to (and in some regards even better than) conventional sources. The pulse duration unique to LPWAs is intrinsically ultrashort, which is more than one order of magnitude shorter than those in X-ray free electron lasers (XFELs). The recent progresses in LPWA [Leemans2006, Lundh-2011, Esarey2009] allow efficient production of high quality electron beams in very high electric fields, moreover it has opened the possibility to design and conceive the compact setup [Couprie2016].

Alternatively, Thomson scattering is capable to generate attosecond pulses from low-energy electrons (see e.g. [Paz-2012]). Single-cycle attosecond pulse generation by Thomson scattering is possible using single-cycle laser pulses. However, single-cycle laser pulse generation is a big challenge in the visible spectral range. Therefore, alternative methods can be of significant interest.

In this paper, we propose a setup to generate waveform-controlled single-cycle attosecond pulses by Thomson scattering of intense THz pulses. In the terahertz (THz) spectral range, single-cycle pulse generation is straightforward [Hoffmann2011, Hirori2011, Vicario2014, Fulop2014, Oh2014]. The peak electric field of the most intense THz pulses today reaches 40 MV/cm [Vicario2014], and there are many suggestions to increase the efficiency of THz generation [Fulop-2016, Ofori-Okai2016, Palfalvi2016, Palfalvi2017, Vicario2015]. Here, we show that such THz pulses are a useful tool in generating single-cycle or waveform controlled attosecond pulses by Thomson scattering on 30–40 MeV electrons, conveniently delivered by present laser-driven electron sources.

The paper is organized as follows. Section 2 describes the basics of Thomson scattering. Section 3 introduces the setup proposed in this work. Section 4 describes the calculation method. The results and their discussion are presented in Section 5. The coclusion is drawn in the last section.

## 2 Scheme of the attosecond source

The proposed setup for the generation of single-cycle attosecond pulses is shown in Fig 1. Relativistic electrons are generated in a LPWA. The electron beam is propagated through the first triplet of quadrupoles to reduce the divergence of the beam. However, the typical energy spread of a LPWA beam is of a few percent. Importantly, the slice energy spread can be reduced by a chicane. The second triplet of quadrupoles focuses the electron beam to the focus of the THz radiation. In order to generate a spatially periodic energy modulation of the electrons, a high-power laser pulse is superimposed on them in the modulator undulator (MU). This energy modulation leads to the formation of ultrathin (20 nm) electron layers (nanobunches) in the drift space behind the MU. A counterpropagating strong-field THz pulse is focused to the position where the nanobunch has the smallest longitudinal size, where the electrons interact with the THz field.

Advantageously, a single laser system is used to drive the attosecond source, including the electron source, the nanobunching, and the THz source. A short-pulse pumped high-power few-cycle optical parametric chirped-pulse amplification (OPCPA) system can be an ideally fitting laser architecture [Fulop-2007, Major-2009, Fattahi2014]. Whereas the 10 to 100-TW power OPCPA output pulses are suitable to drive the LPWA and the nanobunching, the sub-ps to few-ps pump pulses can be used to efficiently drive the THz source.

## 3 Simulation methods

The simulation consists of two main parts. Firstly, we calculated a LPWA process, the transport and manipulation of the electron bunch, including nanobunch generation. Secondly, Thomson-scattering was calculated with the initial electron bunch parameters from the first part.

### 3.1 Generation and manipulation of relativistic electrons

In order to create a coherent attosecond source, a good quality electron beam with high charge and few tens of MeV energy is needed. According to the scaling law predicted by Gordienko and Pukhov [Gordienko2005], a high quality electron beam can be generated in the bubble regime by utilizing laser pulses with 800 nm central wavelength, a pulse duration of 8 fs, and 80 mJ pulse energy, corresponding to 10 TW power. Such a laser pulse can be focused in gas target to an optimized spot size of 3.7 m corresponding to a peak intensity of W/cm. As per scaling the optimum density range for efficient electron acceleration in bubble regime can be estimated as cm . Following the mentioned scaling law [Gordienko2005], an electron beam of 34 MeV energy and 442 pC charge can be obtained in an acceleration distance of 78 m. The main parameters of the used electron beam are listed in Table 1.

Parameter | Value |
---|---|

Energy () | 68 |

Energy spread () | 2 % |

Normalized emittance () | 0.078 mm mrad |

Transversal size () | 5 m |

Length () | 5 m |

Charge | 442 pC |

The General Particle Tracer (GPT) numerical code was used for the simulation of the electron beam transport and the ultrathin electron layer generation. One million macroparticles were used in the calculation, whereby one macroparticle consisted of 2760 electrons. The electron beam from a LPWA typically has a large transversal divergence, the reduction of which is important for further use. Furthermore, it was shown that the generation of coherent radiation is more efficient, when the transversal size of the nanobunch is smaller [TibaiPRL]. Therefore, the electron beam was focused by two quadrupole triplets to the THz beam waist. The gradients of the quadrupole triplets were optimized with a self-developed numerical code. In the best case 15 m and 25 m transversal sizes were calculated in the and directions, respectively. The first chicane decompresses the electron bunch longitudinally, and after the chicane the slice energy spread of the electron bunch is reduced from 2 % to 0.2 %. The main parameters of the quadrupoles and the chicane are listed in Table 2.

Parameter | Value |
---|---|

Distance between the gas jet and the 1st quadrupole tripled | 15 cm |

1st quadrupole triplet gradients | 10.1/-7.1/3.2 |

1st quadrupole triplet elements length | 10 cm |

Chicane dipole strength | 0.043 T |

Chicane dipole length | 15 cm |

2nd quadrupole triplet gradients | -2.5/5.6/-6.2 |

2nd quadrupole triplet elements length | 10 cm |

The relativistic electron beam is sent through the modulator undulator (MU), where a 17-TW power laser beam of 800 nm central wavelength is superimposed on it, in order to generate nanobunches. The MU used here has two periods. The magnetic field of the MU is trimmed in antisymmetric design () along the electron propagation direction. After modulating the electron energy with the laser, a series of nanobunches is being formed. The individual nanobunches are separated by the modulator laser wavelength. The shortest nanobunch with only 16 nm length, containing a charge of 0.6 pC, was used in the simulation of Thomson scattering. The modulator laser and MU parameters are listed in Table 3. The spatial distribution of the nanobunch is shown in Fig. 2, where the color represents the energies of the macroparticles.

Parameter | Value |
---|---|

Laser wavelength () | 800 nm |

Laser peak power | 17 TW |

Laser beam waist | 1 mm |

Laser beam Rayleight length | 3.9 m |

MU undulator parameter () | 0.5 |

MU period length () | 6.7 mm |

### 3.2 Thomson scattering

The interaction of a thin layer of relativistic electrons with a counterpropagating intense THz pulse results in Thomson scattering, schematically shown in Fig. 3. During the interaction, the electromagnetic field of the THz pulse affects the movement of the electrons according to the Lorentz-law. The resulting oscillatory electron motion generates a radiation field, which can be determined based on Lienard-Wiecherd potentials [Jackson] as:

(1) |

where is the charge of the electron, is the vacuum permeability, is the vector pointing from the position of the electron at the retarded moment to the observation point, is the velocity of the electron, , and is the speed of light.

When the THz pulse with central wavelength interacts with a relativistic electron, the wavelength of the backscattered radiation field becomes [Debus2010]

(2) |

where is the relativistic factor of the electron, is the angle between the electron propagation direction and the observation axis, is the interaction angle between the THz pulse and the electron, and is the normalized THz field strength, where is the peak electric field, is the angular frequency, and is the mass of the electron.

The scattered field was determined according to Eq. 1, whereby macroparticles, rather than single electrons, were taken into account. Accordingly, and was equal to the charge and the mass of one macroparticle, respectively. In order to use Eq. 1, first it was necessary to solve the equation of motion for the macroparticles in the electromagnetic field of the THz pulse. The equation of motion reads as:

(3) |

where is the electron momentum vector, is the mass of the macroparticle, is the charge of the macroparticle, is the radial coordinate, is the time. The electric and the magnetic fields of the THz pulse are given as follows, respectively:

(4) |

(5) |

In these equations, is the direction of the electric field polarization, is the THz beam waist radius, is the beam radius at position , is the Rayleigh length of the Gaussian beam, is the central wavelength of the THz pulse, is the full width at half maximum (FWHM) of the intensity in time, is the wave vector, is the Gouy phase. Because single-cycle THz pulses were supposed, the FWHM of the pulses were .

The energy of the electromagnetic pulse in the vacuum can be calculated according to [Diels_Rudolph], so the field amplitude in Eq. (4) was calculated as:

(6) |

where is the vacuum permittivity.

For linear Thomson scattering, the total output radiation yield is proportional to the intensity of the THz pulse [Debus2010, Esarey1993]. Therefore, the laser pulse must be focused to increase its intensity. In the calculations, was assumed, where is the central frequency of the THz pulse. Using this focusing and the single-cycle conditions, Eq. 6 gives the following energy- and frequency-dependent relationship:

(7) |

We note that in the calculations of the next Section, head-on collision between the electron and the THz beams was assumed and Eq. 2 can be simplified to:

(8) |

## 4 Results and discussion

We investigated the dependence of the scattered field energy on the frequency and energy of the THz pulse. The results are shown in Fig. 4. In accordance with previous theoretical predictions for laser pulses [Debus2010, Esarey1993], the radiated energy is proportional to the THz pulse energy. Therefore, the scattered radiation energy can be increased by increasing the energy of the THz pulse.

Figure 4 shows that the radiated energy, as function of the THz frequency, has a maximum at about 0.2 THz. The initial increase at lower THz frequencies can be explained as follows. The number of radiated photons per electron in case of a single-cycle THz pulse is given by [Ride1995], where is the fine structure constant. The THz pulse energy was kept constant along each curve in Fig. 4, whereby a single-cycle waveform and a diffraction-limited focal spot size was assumed at each value of the THz central frequency. The resulting scaling leads to . In addition, the energy of the radiated photon also increases with the THz frequency (see Eq. 2). It means that the energy emitted by one electron increases with increasing . As long as transversal effects can be neglected (see below), this leads to an increasing total emitted energy for an extended nanobunch.

However, as the THz frequency, and consequently the radiation frequency, increases, transversal effects for the extended electron sheet become significant. As we showed in [TibaiPRL], owing to coherent superposition, the solid angle of the emitted radiation for a nanobunch with a finite transversal size is smaller than that for a single electron. This also reduces the total radiated energy. Obviously, this effect becomes stronger for higher frequency (smaller wavelength) of the emitted radiation. This is the reason for the decreasing radiated energy with increasing THz frequency.

For a given THz frequency, the energy of the THz pulse is proportional to . Therefore, according to Eq. 2, increasing the THz energy at a constant THz frequency results in increasing radiation wavelength as for . This, together with the scaling mentioned above, causes a moderate variation of the radiation wavelength with the THz energy. By increasing the THz energy from 5 mJ to 10 mJ leads to about 2% (10%) shift in the radiation wavelength at 0.1 THz (1 THz) frequency.

A comparison of incoming THz and scattered attosecond pulse waveforms and spectra are shown in Fig. 5.a and 5.b, respectively. In Fig. 5.a the red curve shows a single-cycle THz pulse with 10 mJ energy and 0.4 THz central frequency. The blue curve shows the scattered field. The two waveforms are very similar. The pulse duration of the scattered pulse is less than 150 as with 48 nm central wavelength, consistent with the single-cycle waveform. The central frequency of the scattered field is about 17,000 times higher than the THz frequency, in good agreement with Eq. 2. This is clearly shown in Fig. 5.b, where the blue curve shows the spectrum of the attosecond pulse and the red curve shows the spectrum of the THz pulse on a 17,000 times larger wavelength scale.

We note that varying the interaction angle between the propagation direction of the electrons and the propagation direction of the THz pulse (see Fig. 3) allows to use an electron bunch with still higher energy. For example, allows for two times larger electron energy to be used. Because the radiated energy is proportional to the square of the electron energy, up to four times higher emitted pulse energy can be expected.

Furthermore, a tunable narrowband extreme ultraviolet source can also be constructed based on the proposed method, whereby a tunable THz source can be used. A suitable THz source can be a semiconductor contact-grating device [Fulop-2016], pumped by the periodically intensity-modulated pulses from a dual-chirped optical parametric amplifier [Toth2017].

## 5 Conclusion

A novel scheme of single-cycle attosecond pulse generation was introduced, and investigated by numerical simulations. The entirely laser-based scheme uses electrons of a few tens-of-MeV energy generated by a laser wakefield accelerator, subsequent nanobunching to ultrathin electron layers in a modulating undulator driven by TW laser pulses, and the Thomson scattering of intense THz pulses to generate nJ-level attosecond pulses in the few tens of nm wavelength range. The scheme can be driven for example by a short-pulse pumped few-cycle OPCPA system, whereby the strong THz pulses can be conveniently generated by the pump laser. The waveform of the generated attosecond pulses closely resembles that of the THz pulses and can be flexibly shaped by shaping the THz pulse.