On the Usage of Tapered Undulators in the Measurement of Interference in the Intensity-Dependent Electron Mass Shift

In nonlinear Thomson scattering, the main emission line and its harmonics form a band-like structure due to the laser pulse shape induced ponderomotive broadening. We propose to use tapered undulators to mimic Thomson scattering and measure the intensity-dependent electron mass shift experimentally. We also numerically show that the effect is observable for realistic electron beams like in DESY or SKIF.


Introduction
An electron passing through a laser field emits radiation, the frequency of which is redshifted depending on the laser field strength. This frequency shift is attributed by some to the intensity-dependent increase of the electron's effective mass [1,2]. This effect is best seen in high-intensity laser pulses with a temporal profile, which leads to significant non-linear broadening [3][4][5][6][7][8]. There are some techniques to alleviate ponderomotive broadening, for instance, using laser pulses with flat-top profiles [4] or laser chirping techniques, where the laser frequency changes non-linearly to repeat the change of the temporal envelope [7,[9][10][11]. Recently, it was theoretically shown that it is possible to use only linear chirp to avoid ponderomotive broadening for high laser field intensities [12,13]. Furthermore, it was proposed to use laser pulses with temporally varying polarization to avoid ponderomotive broadening in the harmonics spectrum [14]. However, with ponderomotive broadening, the main Thomson line as well as its harmonics form a characteristic interference pattern, which, to the best of our knowledge, has not been measured experimentally so far.
Strong laser scattering systems typically have ω 0 ∼ 1.55 eV, which produces MeV photons when scattering off the ultrarelativistic electron beams (γ ∼ 500). Current detector technology is unable to resolve such a high-energy radiation spectrum with good quality, which is why we propose to mimic the Thomson scattering process with tapered undulators by constructing an appropriate electromagnetic field profile. Typical undulator frequency is several orders of magnitude smaller than that of a strong laser; namely, undulator wavelength λ u ∼ 1 cm corresponds to ω u ∼ 1.24 × 10 −4 eV and the radiation spectrum lies in the keV region. When γ 1, the Thomson and undulator radiation are essentially the same up to the scaling factor of 2 (in Thomson scattering, the initial laser frequency is upscaled with 4γ 2 while in an undulator, by 2γ 2 ). Taking this into account, it is appropriate to mimic one phenomenon through another.
In this paper, we show using numerical simulations of the nonlinear Thomson scattering process that it is possible to measure the band-like structure of the main emission line as well as its harmonics that are present due to the laser pulse shape induced ponderomotive broadening (called by others the intensity-dependent electron mass shift). We also propose a concept of an experiment for that purpose: by joining two tapered undulators, it is possible to form a field configuration from which the radiation spectrum would be the same as the Thomson spectrum from the laser pulse with a "roof-like" temporal envelope.
Throughout the paper, we useh = c = 1 units system, dimensionless spacetime (xω F → x), and energy (ω/ω F → ω) variables by rescaling with the frequency of the field ω F = 2πc/λ F , where λ F is either the undulator (λ u ), or the laser pulse wavelength (λ l ). The dimensionless undulator (laser) strength parameter is K = eB 0 λ u /2πm (= a 0 = eA/m), where B 0 is the amplitude of the magnetic field, A is the amplitude of the vector potential, e, m is the absolute value of electron charge and electron mass, respectively. We will use K and a 0 interchangeably. We consider the case of ultrarelativistic electrons γ 1 when the undulator slippage is negligible. Furthermore, we are interested in moderately strong K ∼ 1 and relatively short undulators when the nonlinear effects are essential and the energy loss of an electron bunch is very small.

Methods
In our problem setting, electrons are moving in the −z direction, counter-propagating a laser pulse with a temporal envelope, which is analogous to the case when the electron bunch is moving through a tapered undulator at rest (see Figure 1). Throughout the paper, we will use laser pulses with an undulator temporal envelope where τ = 2πN is the laser pulse length, N stands for the number of cycles, ∆ ∈ [0, 1] is the tapering rate. This temporal envelope corresponds to the tapered undulator field B = (0, B 0 g(z) cos z, 0), which may be achieved by "joining" positively (K is increasing) and negatively tapered undulators together. ∆ = 1 (triangle envelope) is a limit case of an examined function corresponding to an infinite initial transverse gap between magnets, while ∆ = 0 (rectangular pulse) corresponds to a regular (untapered) undulator. Blue ellipses correspond to electron bunches propagating in −z direction. In Thomson scattering, a laser pulse with a temporal envelope (red) is counter-propagating an electron bunch, while a tapered undulator (green) is at rest. Backscattered radiation is depicted by purple arrows along with expressions for emitted harmonic frequencies. K(z) = K g(z), for linear polarization A ⊥ (η) = 1 2 a 0 g(η), η = t − z is the light-front time, n is an odd integer standing for harmonic number. The emitted frequencies differ only by a scaling factor of 2.
The spectrum is obtained by numerical calculation of the following integral [15] where Ω is the solid angle, n is the unit vector pointing from origin to the observation point, u, r is the vector part of electron 4-velocity and 3-coordinate, respectively. Considering this equation as the Fourier transform in retarded time, one may use Fast Fourier Transform to efficiently calculate it [16]. Theoretical classical estimates of backscattered spectrum for symmetric laser pulse shapes can be found in [16]. From Equation (2), the Thomson and undulator emitted harmonic frequencies can be found and are given by where is the amplitude of a linearly polarized laser field vector potential, η = t − z is the light-front time, n is an odd integer standing for harmonic number, All figures in the Results section were obtained through numerical simulations. Scattering from one electron was simulated via the aforementioned Fourier method, while simulations involving electron beams were obtained with the code VDSR [17].

Results
In the Results section, we present figures and their discussion, which is organized in the following way. Scattering from one electron is presented in Section 3.1, where we numerically show (1) the similarity between the emission spectra from Thomson radiation and undulator radiation, (2) the dependence of the interference pattern on the tapering rate, (3) the dependence of the interference pattern on the laser strength and length (K, N). Scattering from a realistic electron beam is discussed in Section 3.2; namely, (1) how electron beam's angular and energy divergence affects the visibility of the interference pattern, (2) how increasing laser pulse strength leads to stronger nonlinear effects and how to observe the band-like structure in the harmonics spectrum, (3) how larger tapering rates result in a broader interference pattern. Figure 2 shows backscattered spectra from one electron (γ = 1000) off a laser pulse (K = 1, N = 40) with a triangular temporal envelope and tapered undulator (∆ = 1). As it was expected, taking into account different frequency normalization, for ultra-relativistic electrons the emitted spectra are the same. In Figure 3, the normalized vector potentials and corresponding backscattered spectra from one electron off a laser pulse with an undulator temporal envelope for various tapering rates (∆ = 1, 0.3, 0) are shown. The field intensity (K = 1.0, N = 20) is, on the one hand, large enough for the interference pattern from ponderomotive broadening to be distinctly seen and, on the other hand, small enough so the harmonics do not overlap with each other. As it was already mentioned, the main emission line for typical undulators resides in the keV range; therefore, the spectrum could be resolved in detail by modern detectors and one could measure this band-like structure in the experiment. We are interested in the band-like interference pattern, which is broader with larger ∆ (Figure 3). Typical tapered undulators for FEL applications have relatively low tapering rates around 1-5% or less [18,19], for which the interference pattern is quite small, while for our purposes, strongly tapered undulators are needed. Therefore, we make scan simulations with a triangle envelope for different laser pulse and electron beam parameters, and then for a selected set of parameters several tapering rates are modeled. We examined how the interference pattern behaves when varying tapering rates for fixed laser pulse strengths and lengths. Now, to give the intuition of how the interference pattern changes with laser pulse strength and length, let us fix the tapering rate. Figure 4 shows the main emission line in the backscattered spectrum off a laser pulse with an undulator temporal envelope with tapering rate ∆ = 0.7 for several laser pulse strength and length. It could be seen that increasing laser pulse strength leads to a broader interference pattern due to the higher redshift of the main emission line for stronger nonlinear effects. Furthermore, increasing the length of the pulse results in a more intense interference pattern and a higher number of sub-peaks due to the increase of the number of constructively interfering points in the electron's trajectory.

Scattering from an Electron Beam
To investigate how a non-ideal electron beam influences the observability of the bandlike structure, we conducted series of numerical simulations for various realistic laser and electron beam parameters. The electron beam is represented with 10 8 electrons, γ = 1000, energy divergence δE ∼ 10 −3 and normalized emittance n = σ p σ r ∼ 1.4 mm mrad where σ p , σ r are the angular and radial divergence, respectively. Such parameters are similar to electron beams from DESY FLASH [20] and SKIF [21]. The undulator wavelength λ u = 3 cm and all transverse beam size effects are negligible. Figure 5 shows the angular distribution of radiation spectrum from an electron beam (δE = 10 −3 , σ p = 0.15) off a laser pulse with K = 1.2, N = 40 for tapering rates ∆ = 1, 0.5. Close to the axis, the sub-peaks are distinctly seen, while further off axis they are more blurred. Moreover, for the ideal electron beam, only odd harmonics are emitted on-axis while, in our case, due to the broadening caused by non-ideal beam effects, a part of the second harmonic could be visible on-axis as well. It could also be expected and seen that for a larger tapering rate the interference pattern is more distinctly seen and every sub-peak is broader. For larger energy divergence δE and angular divergence σ p the interference pattern is less visible due to the larger range of frequencies emitted (Equation (2)). In order to estimate the visibility of band-like structure, we scanned over several values of δE, σ p while other parameters remained fixed. The top panel of Figure 6 shows the differential number of photons scattered from the electron beam (σ p = 0.15) off a triangle laser pulse with K = 1, N = 40 for different energy divergence δE = 10 −3 , 5 × 10 −3 , 10 −2 . The bottom panel of Figure 6 corresponds to a triangle laser pulse with K = 1.2, N = 40 and various electron beam (δE = 10 −3 ) angular divergence σ p = 0.1, 0.15, 0.2. The number of photons is obtained from the angular spectrum distribution where α ≈ 1/137 is the fine structure constant, by integration over the polar angle φ and collimation angle θ col = 0.2/γ. As expected, for greater angular and energy divergence, the band-like structure is more smooth but, for the chosen parameters (except δE = 10 −2 ), the band-structure is still visible. From Equation (3) we could see that on the axis the main emission line is broadened from ω ∼ 1 1+K 2 /2 up to ω ∼ 1. In other words, for larger K the nonlinearity effects are stronger, leading to a broader main emission line and broader harmonics. For large K harmonics may start to overlap due to both ponderomotive broadening and non-ideal electron beam effects. Figure 7 represents results for fixed angular and energy divergence (σ p = 0.15, δE = 10 −3 ) and increasing laser strength K = 0.8, 1.0, 1.2. The interference pattern is visible for all cases, and there are more sub-peaks for stronger pulses. The same band-like structure could be observed in harmonics as well; for instance, Figure 8 shows the differential number of photons in the harmonics region for a triangle envelope for two cases: For the first case due to a relatively large collimation angle and non-ideal beam effects, even off-axis harmonics overlap with odd on-axis ones, which spoils the overall picture. Still, we can consider more ideal electron beam parameters and a smaller collimation angle to make the interference pattern more visible.
Choosing smaller collimation angles and more ideal electron beam parameters leads to a more visible interference pattern in harmonics. Now, after we observed the influence of laser pulse and electron beam parameters on the interference pattern, it is interesting to model several tapering rates for some "optimal" parameters to make sure that a distinct band-like structure remains. Figure 9 shows the differential number of photons scattered from a realistic electron beam (δE = 10 −3 , σ p = 0.15) off a laser pulse (K = 1.2, N = 40) with various tapering rates ∆ = 0.4, 0.5, 0.6. As it was already discussed, for smaller tapering rates, the interference is less distinct. Furthermore, stronger tapered pulses contain less energy; therefore, the resulting spectrum is less intense. Speaking about experimental observation, for ∆ = 0.5, K = 1.2, λ u = 1 cm and γ = 1000, the main emission peak is at ω 2γ 2 = 0.6 (λ ∼ 8.6 nm) and the first subsidiary peak is at ω 2γ 2 = 0.7 (λ ∼ 7.1 nm). This difference is large enough to be measured experimentally. For larger λ u or lower γ the difference between these peaks increases, and it is easier to detect the interference pattern.

Conclusions
Overall, we proposed to use tapered undulators to mimic Thomson scattering and measure the intensity-dependent electron mass shift experimentally; namely, one may connect positively (K is increasing) and negatively tapered undulators to obtain a radiation spectrum similar to the Thomson spectrum off a laser pulse with an undulator temporal envelope. Firstly, we conducted a series of numerical simulations for a triangular temporal envelope (which has the most vivid interference pattern) scanning over the range of laser pulse and electron beam parameters. Secondly, for a chosen set of laser and electron beam parameters, we modeled several cases with different tapering rates to show that for modern realistic electron beam parameters, the effect is not completely smoothed out and still could be distinctly seen in the main emission line for a broad range of parameters. To observe this band-like structure in harmonics, one needs to choose smaller collimation angles and/or more ideal electron beams. Finally, the intensity-dependent electron mass shift can be observed experimentally by measuring the difference in wavelength of the subsidiary peaks. For a tapered undulator ∆ = 0.5, K = 1.2, λ u = 1 cm and an electron bunch γ = 1000, n = 1.4 mm mrad the difference is ∆λ ∼ 1 nm.