Time-resolved measurements of sub-optical-cycle relativistic electron beams

We propose an all-optical technique to record the time information of relativistic electron beams with sub-optical-cycle duration. The technique is based on the interaction of the electron beam with the ponderomotive potential of an optical traveling wave generated by two counter-propagating circularly polarized optical fields at different frequencies in vacuum. One of the optical pulses is a vortex laser pulse, and the other is a normal Gaussian laser pulse. The time information of the electron beam is mapped into the angular information, which can be converted into a spatial distribution after a drift section. Thus, the temporal profile and arrival time of the electron beam can be retrieved from the spatial distribution of the electron beam. The measurement has a dynamic range comparable to the period of the optical intensity grating formed by two counter-propagating laser pulses. This technique may have wide applications in many research fields that require sub-optical-cycle electron beams.

The most widely applied method for characterizing the time information of an ultrashort electron beam is to map the time information into the angular information or energy information by an electromagnetic field. The time resolution varies with the wavelength of the electromagnetic waves used. In the field of electron accelerators, radio frequency (rf) deflecting cavities have been widely used to measure the temporal distribution of relativistic electron beams with a time resolution of tens of fs [15,16]. To further improve the time resolution, X-band deflection cavities with shorter wavelength can be used. In recent years, the electron beam length of sub-10fs has been successfully measured using a X-band deflection cavity [17]. However, the information of beam arrival time cannot be directly measured with a deflection cavity. Compared with rf deflecting, terahertz (THz) streaking has a higher time resolution [18][19][20][21][22][23], and can be used for characterizing the complete time information of MeV electron beams and x-ray pulses in FELs [24,25]. In addition, compared to deflection with a linearly polarized pulse, the dynamic range of the measurement can be significantly increased by using a circularly polarized pulse [22]. Optical streaking, which can realize attosecond time resolution, is widely used to characterize the time information of attosecond laser and electron beams [9,[26][27][28]. Additionally, this technique has also been adapted to characterize the time information of x-ray pulses in FELs, such as streaking of photoelectrons generated by x-ray pulses [29][30][31].
However, it can not be directly applied to a relativistic electron beam because of the high laser power requirements and Lawson-Woodward theorem [32]. Therefore, it is urgent to develop technique that can measure the time information of sub-fs electron beam.

Principle
Time information of electron beams with sub-optical-cycle duration is usually obtained by laser streaking. Free electron beams cannot gain energy by interacting with laser pulses in vacuum due to the phase slippage between the electrons and laser pulses. To break this limitation, a third body (e.g. a membrane) is introduced into the transport line of the electron beam for non-relativistic electron beams [9,26,27]. Besides, two linearly polarized laser pulses at different frequencies in vacuum are used to form a optical traveling wave, which is synchronized to the electron velocity, leading to the generation of an optical standing wave in the rest frame of the electron beam [33,34]. For the former method, it is difficult to apply it to relativistic electron beams due to the laser damage of the membrane material. For the latter, it can be applied to relativistic electron beams by selecting a suitable laser wavelength.
The laser streaking method usually imposes a time-energy related energy difference in the electron beam by using the optical electric field, and the time information of the electron beam can be mapped into the energy information. However, for MeV electron beams with a relative large intrinsic energy spread, measuring the time information through this principle requires very high-power laser pulses. Moreover, the time information of the electron beam can not be obtained visually from the measured spectrograms.
In this paper, we identify a regime to record the complete time information of a sub-optical-cycle relativistic electron beam based on the ponderomotive interaction of electrons with an optical traveling wave formed by circularly polarized vortex laser pulses rather than linearly polarized laser pulses. The time information of the electron beam can be mapped into the angular information. The schematic layout is shown in figure 1. An electron beam of several MeV interacts with the ponderomotive potential of an optical traveling wave formed by two counter-propagating circularly polarized optical fields at different frequencies in vacuum. One of the optical pulses is a vortex laser pulse and the other is a normal Gaussian laser pulse. The shorter wavelength pulse (λ 1 ) travels in the same direction as the electron beam, and the longer wavelength pulse (λ 2 ) comes from the opposite direction. The two counter-propagating laser pulses can originate from the same laser source, the jitter between them is negligible. To form a standing wave in the rest frame of the electron beam, the velocity of the traveling wave is matched to the normalized electron velocity β 0 (β 0 = v 0 /c, where v 0 is the electron velocity and c is the speed of light in vacuum), such as [35]: The period (λ gr ) of the intensity grating formed by the two counter-propagating laser pulses can be given by: As a result, the electron beam can exchange energy with laser pulses and the orbital angular momentum (OAM) of the vortex laser pulse can be transferred to the electron beam, which has been described in quantum picture [36]. Then the time information of the electron beam is mapped into the angular information, which can be converted into a spatial distribution after a drift section. Thus, the time information of the electron beam can be retrieved from the spatial distribution of the electron beam. The temporal resolution of this method can be written as δt = δθT/2π, where T is the time period of the intensity grating, δθ is the angle measurement error, which is proportional to the spatial resolution of the image measurement.

Simulation study
In the following, we demonstrate the proposed scheme by using a three-dimensional relativistic particle-tracing simulation with the General Particle Tracer code [37]. In the simulation, Laguerre-Gaussian (LG) laser pulses are used which carry a well-defined OAM along their propagation axis. The electric field of a circularly polarized LG pulse with wavelength λ and wave number k = 2π/λ can be written as [38]: where E 0 is the field amplitude, C p |l| stands for a normalization constant, L p |l| is a generalized Laguerre polynomial, c is the light velocity in vacuum, Z R = πw 2 0 /λ is the Rayleigh length, w = w 0 √ 1 + z 2 /Z 2 R and w 0 are the beam radius and the beam radius at the waist respectively, ω is the angular frequency, τ is the pulse duration, s is the helicity of the laser, l is the azimuthal index, p is the radial index and the phase Φ is given by: where R = z + Z R 2 /z is the radius of the curvature. According to Maxwell equations, the electric component E z can be obtained by using ∇ · E = 0 and the magnetic components can be obtained by using B = i ω ∇ × E. For circularly polarized laser pulses, s = ±1. To simplify, the radial index of the LG mode is set to 0. Thus, the generalized Laguerre polynomial L p=0 |l| (.) = 1. The expression of the Gaussian laser field can be obtained by setting l = 0. The notation [l 1 , l 2 , s 1 , s 2 ] will be employed in the following simulations. l 1 and s 1 are the azimuthal index and the helicity of the co-propagating laser pulse respectively. l 2 and s 2 are the azimuthal index and the helicity of the counter-propagating laser pulse respectively.
In all simulation results presented in this section, the central wavelength and the duration of the co-propagating circularly polarized laser pulse are 0.8 µm and 50 fs respectively. The beam radius is 50 µm and the laser energy is 0.833 mJ. For the counter-propagating circularly polarized laser pulse, the central wavelength and the duration are 306.12 µm and 1.5 ps respectively. The beam radius is 400 µm and the laser energy is 2.5 µJ. The actual bandwidth of the two laser pulses has been taken into account in the simulations. The electron beam follows uniform distributions in both longitudinal and radial directions within the beam. The energy of the electron beam is 4.5 MeV. The velocity of the intensity grating is matched to the velocity of the electron beam. And the period of the optical intensity grating is 0.798 µm which is close to the wavelength of the co-propagating laser pulse. Figure 2 shows the spatial distribution of the electron beam after interaction with two laser pulses. The initial radius of the electron beam is 10 µm. The motion of the electrons strongly depend on the LG laser mode and the handedness of the laser. As shown in figures 2(a)-(c), helical electron beams can be generated after interaction with two laser pulses. Different three-dimensional structure distributions of the electron beam can be obtained by changing laser parameters. Extending the results using a single circularly polarized LG laser pulse [38], here the number of helices N can be given by: In addition, after a drift section, the transverse distribution of the electron beam will be different from the initial set, as shown in figures 2(d)-(f). When the laser parameter is set as [−1, 0, 1, 1], the electron beam has a hollow ring distribution. And only the co-propagating laser pulse is a vortex laser pulse, which is convenient for practical application. The time information of the electron beam can be easily retrieved from the spatial distribution of the electron beam.

Discussion
To form a hollow ring distribution with high quality, the radius of the electron beam is limited. As shown in figure 4(a), the LG laser pulse has a hollow-structure intensity distribution. The transverse ponderomotive potential is proportional to intensity of the laser. The electron beam will be subjected to the ponderomotive force, which is proportional to the negative intensity gradient. Figure 4(b) shows the distribution of the pondermotive force along the x-axis. The pondermotive force has an extremum at 12.8 µm from the center of the electron beam. Therefore, as shown in figure 4(c), the transverse velocity (β x ) of the electron beam also has an extremum at 12.8 µm from the center of the electron beam. If the radius of the electron beam is greater than 12.8 µm when interacting with the laser pulses, some electrons will be scattered outside, as shown in figure 4(d). The radius of the electron beam should be smaller than the transverse coordinate value corresponding to the extremum of the pondermotive force.
To better illustrate the principle, an ideal electron beam is used in the simulation above. In this section, the effects of the electron beam emittance and space charge forces on the scheme are studied. To retrieve the time information from the spatial distribution of the electron beam, the divergence angle of the electron beam induced by the laser pulses needs to be much larger than the intrinsic divergence angle of the electron beam. Recently, an electron beam with a thermal emittance of 0.099 µm ·rad mm −1 can be obtained via Near-threshold photoemission from single-crystal Cu [39,40]. Suppose that the initial electron beam radius is 2 µm, the corresponding emittance is 0.099 nm·rad. The normalized emittance can be given by ε n = βγσ x σ ′ x , where β is the normalized electron velocity, γ = 1/ √ 1 − β 2 is the Lorentz factor, σ x is the transverse size of electron beam (rms), and σ ′ x is the transverse divergence angle of the electron beam (rms). When the space charge effect is ignored and the electron beam does not encounter an aperture, the normalized emittance will remain constant as followed by Liouville's theorem. The electron beam is expanded to 10 µm by a beam expander system. The energies of co-propagating and counter-propagating laser pulses are 6.5 mJ and 16.9 µJ respectively, which are readily obtained with today's technology [41][42][43]. Figures 5(a) and (c) shows the transverse distribution and the corresponding time distribution of the electron beam with initial emittance. The time distribution of the electron beam can be retrieved from the spatial distribution of the electron beam with high time resolution.
Space charge effects cause beam broadening in both transverse and longitudinal directions, which will affect the measurement results. Figures 5(b) and (d) shows the simulation results with the space charge  effect. The electron beam charge is 0.1 fC. Although the space charge effect introduces errors into the measurement, the time distribution of the electron beam can be basically retrieved from the spatial distribution. However, due to the fact that the electron beam is short (less than 1 fs) and strongly transversally focused, if the charge of the electron beam is too large, the space charge effects will have a significant impact on the measurement results. Figure 6 shows the streaked electron beam distribution at different electron beam charges. When the electron beam charge is 0.2 fC, the spatial distribution of the electron beam get worse due to the space charge forces. It is hard to retrieve the time information from the spatial distribution of the electron beam with high accuracy. Therefore, in this case, the maximum charge of the electron beam is about 0.15 fC. By increasing the electron beam radius, laser radius and laser intensity, the time information of electron beam with larger charge can be measured.
As for the electron beam energy, it is mainly limited by laser wavelength and energy. The central wavelength of the co-propagating circularly polarized laser pulse is 0.8 µm. The frequency range of the counter-propagating THz pulse is 0.2-10 THz, which can be achieved by optical rectification of LiNbO 3 and organic crystals [41][42][43][44][45]. Therefore, according to equation (1), the matched kinetic energy of the electron beam as a function of THz frequency is shown in figure 7. The maximum kinetic energy of the electron beam is about 10 MeV. As for larger electron beam energies, longer wavelengths and much larger laser intensities are required.

Conclusion
In conclusion, we have proposed a method for recording the time information of an sub-optical-cycle relativistic electron beam. Simulation results show that the temporal profile and the arrive time jitter of the electron beam can be measured with very high time resolution. This technique enables the optical streaking method to be applied to the measurement of relativistic electron beams. Together with THz streaking and rf deflector technologies, the complete measurement of the change process of electron beam from ps to sub-fs size can be realized. We hope that the proposed technique have wide applications in many ultrashort and jitter-free electron-beam-based facilities.

Data availability statement
The data generated and/or analysed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request.