Carrier-Envelope-Phase Controlled Attosecond Pulse Generation by Undulator Radiation

Introduction

In recent years a few phenomena sensitive to the carrier-envelope-phase (CEP) of ultrashort laser pulses were recognized [1, 2]. Waveform-controlled few-cycle laser pulses enabled the generation of isolated attosecond pulses in the extreme ultraviolet (EUV) spectral range and their application to the study of electron dynamics in atoms, molecules, and solids [3]. The observation of inner-atomic (strong-field) phenomena and EUV pump—EUV probe measurements require intense CEP-controlled attosecond pulses [4–6]. EUV pump—EUV probe experiments can be carried out at free-electron lasers (FELs) [7, 8]; however, the temporal resolution is limited to the few fs regime.

Various schemes, such as the longitudinal space charge amplifier [9], emittance spoiler foil technique [10, 11], wave selection technique [12], or enhanced self-amplified spontaneous emission (E-SASE) [13, 14], were proposed for attosecond pulse generation at FELs. High current modulation technique were suggested to generate isolated attosecond pulses in X-ray region [15, 16]. Recently described schemes suggest the possibility of sub-attosecond pulse generation in the hard-X-ray region [17], and single-cycle FEL pulse generation in the THz [18] and in the X-ray region [19].

However, the stochastic pulse shape predicted for the pulses generated by these methods is disadvantageous. Furthermore, there are no reliable techniques available for CEP control of these or any other attosecond pulse sources. In contrast, recently we proposed and numerically investigated a robust method for producing waveform-controlled linearly and circularly polarized CEP-stable attosecond pulses in the EUV spectral range [20–22]. The analytical investigation of this concept was carried out by Shamuilov et al. [23]. It uses relativistic electron bunches from a linear accelerator (LINAC), and relies on ultrathin nanoscale-length electron layers (nanobunches) generated by laser-driven energy modulation, and on undulator radiation. In this setup waveform-controlled attosecond pulse generation is possible, unparalleled by other similar setups, where the predicted pulse energy and the short wavelength are imposing, but the waveform is stochastic [14].

In the present paper, a more detailed investigation is carried out on the feasibility of this technique. Practically important aspects were numerically studied. The dependence of the attosecond pulse energy on the radiation wavelength, and undulator parameter were also studied, together with a possible way for isolated attosecond pulse generation.

The Investigated Setup And The Simulation Methods

The scheme of the setup proposed in Ref. [20] and further investigated in this work is shown in Figure 1. This scheme is very similar to the ultrahigh harmonics generation scheme of Ref. [24], which is the simplest version of the high gain harmonic generation (HGHG) technique, however, there is no gain in the modulator. Recently such a scheme was further investigated [25], and it was shown that according to the calculation generation of harmonics to an order as high as 1000 is possible. In order to generate coherent attosecond pulses, nanometer scale electron modulation is needed. Thus, the FEL community investigated and proposed some sophisticated modulation techniques, such as the π shifter technique [26], echo-enabled harmonic generation (EEHG) [27] or cooled HGHG [28], which increase the frequency up-conversion efficiency.

FIGURE 1

Figure 1. Scheme of the investigated setup.

In our scheme the simplest combination, i.e., one modulator (MU) and one radiator (RU) undulator were used. Compared to Ref. [24, 25], our scheme uses a radiator undulator that consists only of a few periods (no exponential gain inside), and the generated radiation wavelength is arbitrary, i.e., independent from the wavelength of the modulator laser (which is usually called the seed). Furthermore, in our calculations we investigated the temporal shape of the radiation field (not only its spectrum).

In our modell the relativistic electron beam from a LINAC is sent through a modulator undulator where a TW-power laser beam is superimposed on it and introduces a periodic energy modulation of the electrons along the longitudinal (z) direction. This energy modulation leads to the formation of nanobunches at the end of the chicane behind the MU. Evidently, efficient generation of coherent and non-stochastic radiation pulses is possible only if the nanobunch length is shorter than the half period of the radiation. As our aim was to generate coherent EUV radiation in the 10 to 100 nm wavelength range, the minimization of the nanobunch length down to the sub-10-nm range was essential.

The nanobunched electron beam then passes through the radiator undulator consisting of a single period or of a few periods, depending on the desired waveform, and it emits electromagnetic radiation. The entrance of the RU is placed behind the MU at a position where the nanobunch length is the shortest. The wavelength of the generated radiation is determined by the well-known resonance condition [29]

$\begin{array}{ll}{\lambda }_r=\frac{{\lambda }_{RU}·\left(1+\frac{K_{RU}^2}{2}\right)}{2{\gamma }^2}.& (1)\end{array}$

Here, λ_r is the wavelength of the generated EUV radiation, λ_RU is the period of the RU, $K_{RU}=\frac{e{B}_{0,RU}{\lambda }_{RU}}{2\pi mc}$ is the undulator parameter, B_0,RU is the peak magnetic field of the RU, e and m are the electron charge and mass, respectively, γ is the relativistic factor, and c is the speed of light. The generated radiation waveform is mainly determined by the magnetic field distribution of the RU along the electron beam propagation direction z. For a better comparison of the EUV pulse parameters obtained in the different cases investigated below, the same RU magnetic field distribution

$\begin{array}{ll}{B}_{RU}=\{\begin{array}{ll}{{{\displaystyle B_{0,RU}}}^e}^{-\frac{z^2}{2w_{RU}^2}}\cos \left(\frac{2\pi }{{\lambda }_{RU}}z\right),\hfill & \text{if }-\frac{L}{2}<z<\frac{L}{2},\hfill \\ \hfill 0,\hfill & \text{otherwise}\hfill \end{array}& (2)\end{array}$

was used throughout this work (see also Figure 1). Here, w_RU is the width of the Gaussian envelope and L is the length of the RU. These parameters were set to w_RU = 1.5·λ_RU and L = 2.5·λ_RU.

In order to consider a realistic situation, the initial electron bunches (Table 1) were used according to the start to end (S2E) simulation of the accelerator of FLASH at DESY in Germany [30–34]. In the energy modulation we assumed a double-period MU with TW-level modulator laser, where ELEGANT [35] and CSRTrack [36] codes were used for the numerical simulation. In the radiator undulator, a similar theoretical model and simulation tools were used as in Ref. [20]. In our calculations we investigated 100 pC and 1000 pC charge scenarios of the FLASH accelerator, where 200,000 and one million macroparticles per bunch were used, respectively (see Table 1).

TABLE 1

Table 1. Parameters used in the simulations with 100 pC and 1000 pC bunch charges.

When the Coulomb interaction is neglected an approximate expression for the full width at half-maximum (FWHM) length of the nanobunch, Δz₀, can be given by analytical derivation [14, 37]:

$\begin{array}{ll}\Delta {\mathit{\text{z}}}_0\approx \frac{1}{2}\frac{{\lambda }_{\text{L}}}{A}.& (3)\end{array}$

The relative energy modulation is given by A = Δγ/σ_γ, where σ_γ = σ_γ* γ is the intrinsic energy spread (Table 1) and Δγ is the energy modulation acquired in the MU owing to energy modulation. At large relative modulation (A ≫ 1) the nanobunch becomes much shorter than the laser wavelength.

TW-class table-top light sources with pulse durations comprising only a few optical cycles were intensely developed during the last few years [38, 39]. For example, Herrmann et al. [38] reported a two-stage non-collinear optical parametric chirped-pulse amplification (OPCPA) system generating 16-TW, sub-three-cycle (130-mJ, 7.9-fs) pulses at an 805 nm central wavelength. Suitable light pulse sources are also being constructed elsewhere, for example in the Extreme Light Infrastructure (ELI) project [39]. Here we note that for λ_r ≳ 20 nm EUV wavelengths λ_L ≈ 800 nm gives higher EUV pulse energies. Only below ~20 nm is the choice of a shorter laser wavelength (i.e., 516 nm, Table 1] advantageous, because of the shorter nanobunches. Though TW-class few-cycle OPCPA systems around 500 nm central wavelength need yet to be demonstrated, promising techniques are being developed [40–42]. In section III.B we consider the possibilities for isolated attosecond EUV pulse generation, where laser pulses as short as two optical cycles are needed. For the generation of attosecond pulse trains longer laser pulses can be used.

We used the following handbook formula to calculate the electric field of the radiation generated in the RU [43]:

$\begin{array}{ll}\vec{E}\left(t,\vec{r}\right)=\sum {\left[\frac{q{\mu }_0}{4\pi }\frac{\vec{R}\times \left(\left(\vec{R}-R\vec{\beta }\right)\times \dot{\vec{v}}\right)}{{\left(R-\vec{R}·\vec{\beta }\right)}^3}\right]}_{ret},& (4)\end{array}$

where μ₀ is the vacuum permeability, q is the macroparticle charge, $\vec{R}$ is the vector pointing from the position of the macroparticle at the retarded moment to the observation point, $\vec{v}$ is the velocity of the macroparticle, $\vec{\beta }=\frac{\vec{v}}{c}$, c is the speed of light. The summation is for all macroparticles. During the radiation process the position, velocity, and acceleration of the macroparticles were traced numerically by taking into account the Lorentz force of the magnetic field of RU. The Coulomb interaction between the macroparticles was neglected during the undulator radiation process, because the transversal electron motion is by four orders of magnitude larger than the motion generated by Coulomb interaction.

Results

Nanobunch Generation

The energy modulation with a high power laser in the MU and the chicane was calculated with ELEGANT and CSRTrack codes, respectively. Despite the use of a TW-level modulator laser, in our calculation a chicane was also used, because it can control the position of the spatial focus of the nanobunch and allows to feed out the modulator laser pulses. The result of the nanobunches in case of 11 different simulations (gray) and their average (red) are shown in Figure 2 after the chicane for P_L = 10 TW laser power, in 100 pC and 1000 pC cases, respectively. The obtained energy modulation was Δ_γ = 134 at K_MU = 1.4 and λ_MU = 1.99 m. The charge of a single nanobunches are 1.1 pC and 1.0 pC and their lengths are as short as 6 nm and 4 nm at 1.25 m behind the end of the modulator undulator. These nanobunches were used in the further calculations to obtain the EUV pulses described in section III.B.

FIGURE 2

Figure 2. Macroparticle distribution of 11 different simulations (gray) and their average (red) along the z axis around the temporal focus and the distributions of γ (insets) for 100 pC (A) and 1000 pC cases (B).

EUV Pulse Generation

The temporal shape of the attosecond EUV pulses emitted by the extremely short electron nanobunches in the RU were calculated at a plane positioned 8 m behind the RU center. The EUV waveform can be conveniently set by the magnetic field distribution of the RU [20]. The distribution of the magnetic field of the RU (blue) and the average of the electric field of the attosecond pulses (red) are shown in Figure 3 for 100 pC and 1000 pC cases, respectively.

FIGURE 3

Figure 3. Example of a CEP controlled EUV waveforms for 20 nm radiation wavelength (gray) and their average (red) for K = 0.5 and 100 pC (A) and 1000 pC (B) cases, together with the spatial distribution of the RU magnetic field.

The emphasis was on exploring the dependence of the EUV pulse energy on important experimental parameters such as radiation wavelength and RU undulator parameter. The EUV pulse energy as function of the radiation wavelength (λ_r) in the range of 20÷200 nm is shown in Figure 4, and two different RU undulator parameter values (0.5 and 0.8). The larger EUV pulse energy is obtained with the larger K_RU. The undulator parameter can be set to the desired value by adjusting the magnetic field amplitude. The radiation wavelength, given by the resonance condition Equation (1), can be set by the choice of the RU period λ_RU. As seen in Figure 4, the pulse energy first increases with increasing wavelength, followed by saturation and subsequent energy decrease. The reason of the latter is the longer undulator period needed to generate longer wavelengths. Due to the associated longer path inside the RU the average nanobunch length and transversal size increases [21], thereby reducing coherence in the radiation process. As shown in Figure 4B for 20 and 120 nm cases, the nanobunch lengths are increased by 350 and 2,400% at the end of the radiator undulator (red and blue curves), respectively. We note that for the generation of waveform-controlled pulses at longer wavelengths longer nanobunches (and lower modulator laser power) can be used more advantageously.

FIGURE 4

Figure 4. (A) Dependence of the EUV pulse energy on the radiation wavelength λ_r for 100 pC and 1000 pC cases for K = 0.5 and K = 0.8. (B) shows the distribution of macroparticles along the z axis at the entrance (black curve) and the end of the RU for λ_r = 20 nm and 200 nm (red and blue curves).

In the second calculation series the electron bunch with 100 pC was used and the RU undulator parameter was varied in the range of K_RU = 0.1÷1.9 and the λ_RU undulator period was chosen such that for each value of K_RU the radiation wavelength was kept at 20 and 60 nm, respectively. For both wavelengths, the EUV pulse energy is proportional to $K_{\text{RU}}^2$ below about K_RU = 0.7, followed by saturation at larger K_RU (Figure 5A). The reason of the saturation is that a larger K_RU results in more pronounced harmonics of the radiation wavelength (see Figure 5B, where the spectra of the EUV pulses for three different K_RU values for the 20-nm case is shown). However, the spectral components of the shorter wavelengths are suppressed by the destructive interference, which results in the saturation of the energy.

FIGURE 5

Figure 5. (A) Dependence of the attosecond EUV pulse energy on RU undulator parameter at 20 nm (red) and 60 nm (blue) wavelengths. (B) EUV pulse spectra for three different K_RU values.

According to the calculations, single-cycle 20-as pulses with more than 10 nJ energy at 5 nm, 80-as pulses with more than 200 nJ energy at 20 nm, and 240-as pulses with more than 650 nJ energy at 60 nm can be generated.

Since in our setup the electron bunch consists of multiple nanobunches separated by the modulation laser wavelength, a pulse sequence is generated. The ratio of separation time to pulse duration is smaller than 40. This can be too small for certain applications. One possibility for increasing this ratio substantially is to use two modulation lasers with significantly different wavelengths [13, 44]. We investigated another possibility for isolated CEP stable attosecond pulse generation, namely the shortening of the modulator pulse duration. The modulator laser produces an energy modulation with a temporal profile that closely resembles the waveform of the laser electric field. We assumed a sine-like waveform, corresponding to zero CEP and zero energy modulation at the maximum of the laser pulse envelope, like in Ref. [13, 14]. The energy modulation produces current modulations, where a central and an even number of satellite peaks occure (the corresponding satellite peak amplitudes are substantially similar). In our simulations, the modulator laser pulses consist only of a few optical cycles (Figure 6A). Figure 6B shows the calculated waveform of the main attosecond pulse and the neighboring satellite pulses for a few different modulator laser pulse durations. As the modulating pulse duration is decreased below about 5 cycles or 8 fs the peak amplitude of the satellite attosecond pulses start to significantly decrease. Modulator pulses with less than about 2.0 cycles or 3.4 fs are able to generate CEP stable isolated attosecond pulses. In this case, the energy of the satellite pulses is only about 1–2% of the main attosecond pulse. We have to note that the CEP of the modulator pulse strongly influences the degree of isolation of the generated attosecond pulse. However, it does not determine the CEP of the attosecond pulse.

FIGURE 6

Figure 6. (A) Electric field of a 2.0-cycle (3.40 fs) modulator laser pulse. (B) Time dependence of the electric field of the generated attosecond pulse train for different modulator laser pulse lengths.

Finally, we compare our results to the experimental performance of attosecond EUV pulse sources based on high-order harmonic generation (HHG) in gas or plasma driven by high-intensity optical pulses [3, 45]. Isolated attosecond pulses (IAPs) can be generated by few-cycle driving pulses or by using gating techniques with longer pulses [45]. Typical IAP energies are in the sub-nJ to 10 nJ range. The shortest pulse duration reported to date was 67 as; the wavelength range was ~10 to 20 nm [46]. CEP-stable single-cycle 130-as IAPs were reported with an estimated CEP fluctuation of 140 mrad [43]. However, no pulse energy was given in these works [45–47]. Recently, the generation of 500-as IAPs with 1.3 μJ energy at ~30 eV was demonstrated [48]. Trains of attosecond pulses with an energy on the μJ scale can be generated by using many-cycle driving pulses. Significantly higher pulse and photon energies are expected from laser-produced plasmas [45, 49], though the full potential of this method needs yet to be demonstrated.

In comparison, as shown above, similar shortest pulse durations can be achieved with our method as by HHG, at comparable wavelengths. However, unparalleled by any other source reported so far, our method easily enables the full control of the attosecond pulse waveform.

Conclusion

In summary, practical aspects of the method proposed in our previous work [20] for stable arbitrary-waveform attosecond EUV pulse generation were investigated in detail by means of numerical simulations. The scaling of the generated attosecond EUV pulse energy with various parameters (λ_r, K_RU) was studied. For example, the nanobunches were predicted to emit 20-as EUV pulses at λ_r = 5 nm with 10 nJ energy, 80-as EUV pulses at λ_r = 20 nm with 90 nJ energy, and 240-as pulses at λ_r = 60 nm with 200 nJ energy for K_RU = 0.5. At longer generated wavelength or larger K_RU the generation of EUV pulses with more than 500 nJ energy can be possible. The shortening of the modulator laser pulse duration was discussed for the generation of isolated attosecond pulses.

The proposed scheme can enable the development of practical sources of CEP stable attosecond EUV pulses using existing LINACs. These unprecedented pulses can be used for example in EUV pump—EUV probe experiments in the near future.

Author Contributions

ZT, GT, AN, and AG contributed conception and design of the study. JF, JH, and GA wrote the sections of the manuscript.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

Financial support from Hungarian Scientific Research Fund (OTKA) grant No. 125808 and 129134 are acknowledged. The project has been supported by the European Union, co-financed by the European Social Fund Grant no.: EFOP-3.6.1.-16-2016-00004 entitled by Comprehensive Development for Implementing Smart Specialization Strategies at the University of Pécs; EFOP-3.6.2-16-2017-00005 entitled by Ultrafast physical processes in atoms, molecules, nanostructures and biology structures.

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