Generating intense attosecond x-ray pulses using ultraviolet-laser-induced microbunching in electron beams

We propose a scheme that combines the echo-enabled harmonic generation technique with the bunch compression and allows to generate harmonic numbers of a few hundred in a microbunched beam through up-conversion of the frequency of an ultraviolet seed laser. Sending this beam through a short undulator results in an isolated sub-100 attoseconds pulse of x-ray radiation. Using a representative realistic set of parameters, we show that 1 nm x-ray pulse with peak power exceeding 100 MW and duration as short as 34 attoseconds (FWHM) can be generated from a 200 nm ultraviolet seed laser.

where an intense laser pulse was focused on an atomic gas jet and the high harmonic of the laser was generated and further used as the probe. However, it appears difficult to generate intense harmonic radiation with wavelength down to 1 nm or shorter with this technique.
An alternative scheme [8] which overcomes this problem is to use the high gain harmonic generation (HGHG) configuration. But due to the relatively low up-frequency conversion efficiency, the HGHG scheme requires an intense seed signal in the wavelength of a few nm which is in principle obtainable but does not exist today.
In this paper, we propose a novel scheme which allows to generate intense isolated attosecond x-ray in the wavelength ∼ 1 nm or shorter based on existed technologies. The scheme combines the echo-enabled harmonic generation (EEHG) technique [16,17] with the bunch compression technique and allows harmonic numbers of a few hundred to be accessible that eventually enables the generation of x-ray radiation from an ultraviolet (UV) seed laser. The required energy chirp in the bunch compression is provided by a few-cycle intense infrared (IR) laser which finally assists in generation of an isolated attosecond x-ray pulse.
Using a representative realistic set of parameters, we show that 1 nm isolated x-ray pulse with duration of about 34 as (FWHM) and peak power exceeding 100 MW can be generated in a short radiator with only 20 undulator periods using the proposed scheme. Since the generated attosecond x-ray pulse is in tight synchronization with the few-cycle laser, it is straightforward to use them for ultrafast pump-probe experiments.
Our scheme requires an ultrarelativistic electron beam, a UV seed laser, a few-cycle intense IR laser, together with four undulator sections and two dispersion sections. The wavelength of the UV seed laser is assumed to be 200 nm and that of the few-cycle IR laser is 800 nm. We further assume that the lasers originate from the same Ti:Sapphire oscillator which will allow tight synchronization between them. The schematic of the proposed scheme is shown in Fig.1.
The first part of the proposed scheme is similar to the EEHG FEL [16,17] in which the beam is energy modulated in the first modulator (M1) and then sent through a dispersion section with strong dispersion strength R (1) 56 after which the modulation obtained in M1 is macroscopically washed out while simultaneously complicated fine structures (separated energy bands) are introduced into the phase space of the beam. In the EEHG scheme, a second laser is used to further modulate the beam energy in the second modulator (M2).
After passing through the second dispersion section with dispersion strength R (2) 56 , the separated energy bands will convert to separated current bands and the echo signal then occurs as a recoherence effect caused by the mixing of the correlations between the modulation in the second modulator and the fine structures.
In our proposed scheme for generation of an isolated attosecond x-ray pulse, we introduce an extra modulator (M3) between M2 and the second dispersion section. The beam interacts in M3 with a few-cycle intense laser of which the wavelength is chosen to be much longer than that of the laser in M2, so that part of the electrons around the zero crossing of the few-cycle laser gets almost linear energy chirp. With this additional energy chirp, the beam is longitudinally compressed after passing through the second dispersion section and the harmonic number is increased by the compression factor. As we will show below, in addition to assisting in extension of the harmonic number to a few hundred, the few-cycle laser also offers a possibility to select an isolated attosecond pulse.
We assume an initial Gaussian beam energy distribution with an average energy E 0 and the rms energy spread σ E , and use the variable p = (E − E 0 )/σ E for the dimensionless energy deviation of a particle. The initial normalized distribution function of the beam is where N is the number of electrons per unit length of the beam.
After passing through M1, the beam energy is modulated with the amplitude ∆E 1 , so that the final dimensionless energy deviation p is related to the initial one p by the equation k is the wave number of the laser, and in our example we assume the laser wavelength of 200 nm. Sending the beam through the first dispersion section with the dispersion strength R (1) 56 converts the longitudinal coordinate z into z , z = z + R (1) 56 pσ E /E 0 , and the resulting distribution function is [16,17], The beam is then energy modulated in M2 with the dimensionless modulation amplitude A 2 . Assuming the laser wavelength to be the same, at the exit of M2 the beam longitudinal phase space evolves to, Before entering the second dispersion section, the beam is further energy modulated in M3 with an intense, few-cycle laser for which the wavelength is much longer than that in the upstream modulators. The carrier-envelope phase of the few-cycle laser pulse is set to π/2 so that the oscillating electric field is zero at the pulse peak [18]. A snapshot of the 800 nm few-cycle laser field normalized to the peak value is shown in Fig.2a.
To show how the few-cycle laser assists in generation of much higher harmonic and an isolated attosecond x-ray pulse, we add a linear energy chirp to the beam distribution before sending the beam to the second dispersion section. Assuming the linear energy chirp factor is h = dp/dζ, the resulting longitudinal phase space distribution at the exit from the second dispersion section is, where 56 kσ E /E 0 . Integration of Eq.(3) over p yields the final current distribution which can be expanded in Fourier series, The wave number of the echo signal is Cnk, where C = 1/(1 + hB 2 ) is the compression factor and the corresponding maximized bunching factor for this harmonic radiation is, The bunching factor at the harmonic number m is defined as e imkz , where mk is the wave number of the harmonic radiation and the brackets denote averaging over the longitudinal coordinate z. A comparison between this equation and Eq.(6) in Ref. [17] indicates that for given energy modulation amplitudes the bunch compression extends the harmonic number by a factor of C while keeping the value of the bunching factor unchanged. Assuming a compression factor of 5 and relatively small energy modulation amplitudes in the x-ray pulse. The current of the central slice is also enhanced by a factor of 5 from the bunch compression. It can be expected that intense 1 nm x-rays will be generated after sending this prebunched, high-peak-current beam slice through the radiator.
The output radiation field may be calculated by summing up the field of each electron in the radiator. The output x-ray pulse length is determined by both the electron bunch length and the slippage length in the undulator. For the SASE schemes [7,[9][10][11][12][13][14], the slippage length associated with the long FEL undulator typically limits the radiation pulse length to a few hundred attoseconds. Since in our scheme a short section of the beam is effectively prebunched at the radiator entrance, only a short undulator is required to generate intense x-rays which may allow pushing the pulse length to sub-100 as. In our example, we used a planar undulator with N u = 20 periods and the period length λ u = 4 cm to generate λ r = 1 nm x-rays. We further assumed that the relative longitudinal position of the electrons did not change during the passage through the short radiator, which was justified for an electron beam with 1 µm normalized emittance and a transverse rms size σ x = 20 µm. For these parameters, the single-electron undulator radiation has a rms transverse size √ 2λ r λ u N u /(4π) σ x and can be approximated as a plane wave. Thus the output power can be calculated as (see for example [19]) where Z 0 = 377 Ω is the vacuum impedance, K is the undulator parameter, [JJ] = J 0 (ξ) − J 1 (ξ) is the usual Bessel function factor associated with the planar undulator, ξ = K 2 /(4 + 2K 2 ), E 0 = γ 0 mc 2 is the beam energy, and the sum over jth electron is carried out within the radiation slippage length, i.e. 0 < c(t j − t) < N u λ r . Using the beam distribution at the radiator entrance, the power profile of the x-ray radiation at 1 nm wavelength generated in the radiator is shown in Fig.6. The peak power exceeds 100 MW, and the pulse length is 34 as (FWHM) approaching the atomic unit of time (24 as). It is worth pointing out that the bunching factor depends on the energy chirp factor which varies if the IR laser energy changes. In order to keep the fluctuation of the output x-ray power smaller than 10%, the laser energy fluctuation should be controlled to be within 1%.
In conclusion, we have proposed a scheme to extend the harmonic numbers to a few hundred that eventually enables the generation of an intense isolated attosecond x-ray pulse from a UV seed laser. It is capable of breaking the 100-as time barrier and may open a new regime of ultrafast sciences.
We thank A. Chao, Y. Ding, D. Ratner and J. Wu for helpful discussions. This work was