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Abstract
The properties of nonlinear inverse Thomson scattering (NITS) are investigated in the collision between a circularly polarized tightly focused intense laser pulse and a relativistic off-axis electron with numerical simulations. Due to the asymmetric effect of the laser field on the off-axis electrons, the electron trajectory is torqued to the off-axis direction, and the symmetry of the spatial radiation is also destroyed, which causes the concentrations of the radiation in the off-axis direction. With the increase of laser intensity, the torsion effect is more obvious, the radiation collimation improves, the direction turns to sideways. With the increase of electron’s initial energy, the direction turns back to backwards and the degree of off-axis effect decreases. In both cases, the power exponentially enhances, the pulse width shortens, the spectrum broadens and super-continuity appears. With the laser intensity, the duration of sideways X-ray pulse from the low-energy (2.61MeV) electron is only 0.2 as, and the normalized intensity reaches 109. While using ultra-high-energy (100MeV) electrons, the duration of backwards γ-ray pulse reaches 1.22 zs, and the normalized intensity reaches 1017. These results help the understanding of nonlinear Thomson scattering and provide important numerical references for the research of NITS as high-quality X-ray and γ-ray sources.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The past few decades witnessed the flourish of laser-matter interaction [1–3], spatially the interaction between lasers and electrons [1,4–21]. Due to the invention of chirped pulse amplification technology [9], the laser intensity greatly increased and reached ${10^{22}}\textrm{}W/c{m^2}$ [10], which pushed the laser-electron interaction into ultra-fast domain [5,8]. As a potential high-quality X-ray source [6], nonlinear inverse Thomson scattering (NITS) has become an important research direction in the field, which can be widely applied in biomedicine [11,12] and astrophysics [13].
NITS refers to the process that an intense laser pulse collides with a relativistic electron and emits electromagnetic radiation (Fig. 1), or the approximation of inverse Compton scattering under Thomson limit [7,14]. And it is of great significance to modulate X-rays through the research on radiation properties of NITS. Yoshitaka Taira et al. [15] studied the angular distribution of the radiative harmonic energy of NITS. Zhuang’s team [16,17] analyzed the spectrum of X-rays produced by NITS and proposed that Quasi-monochromatic X-rays can be obtained by changing the initial position of the electron. When Chang [18] and Chen [19] increased the intensity of the circularly polarized laser in NITS, they both found that the collimation of NITS radiation decreased, while the radiation power increased.
Fig. 1. Schematic diagram of the NITS process shows a laser pulse colliding with an off-axis electron.
However, obtaining highly collimated and higher power X-rays through NITS is of great research value. As a matter of fact, the off-axis collisions of electrons would break the symmetry of the NITS radiation [20,21], and change the properties of the scattered X-rays.
In this paper, the radiation properties of the collision between an off-axis relativistic electron and a tightly focused circularly polarized laser pulse are studied based on the framework of classical electrodynamics. A single-electron NITS model is established to study the asymmetry effects cause by off-axis electron. Through numerical simulations, the effects of laser intensity and initial energy of electron on the radiation properties are studied. When an off-axis electron colliding with the laser, its trajectory deflects to the off-axis direction, and the effect of deflection increases with the rise of laser intensity. The spatial symmetry of axial NITS [19] is broken due to the deflection effects, which improves the collimation of the radiation. The greater the laser intensity, the more concentrated the radiation, and the radiation direction changes from backward to sideways. However, when electron’s initial energy increases, radiation direction changes back to backward and the asymmetry is fixed. Further research shows that changing the initial deflection angle of the low-energy electrons can modulate the radiation direction. The research on the time domain and frequency domain shows that the intensity of NITS radiation increases exponentially with the increase of laser intensity, also with the initial energy of electron, the pulse width of the radiation peak decreases, the number of radiation frequency modes increases, and the frequency band broadens, showing supercontinuum characteristics. When electron’s initial energy is ultra-high, backward γ-rays can be generated from NITS. These has guiding significance for generating highly collimated ultrashort high-energy broad-spectrum X-ray and γ-ray pulses through NITS.
2. Theory and formula
2.1 Electron’s motion colliding with laser pulses
The model of NITS process is shown in Fig. 1. The laser pulse propagates forward along the z-axis, and an off-axis electron travels along the $- z$ axis with an initial velocity $- {v_0}$ and collides with the laser on the $xOy$ plane. A circularly polarized Gaussian laser beam with the wavelength $\lambda $ is used in this model, and the corresponding frequency is ${\omega _0}$, wave number $k = 2\pi /\lambda $. The Gaussian laser pulse with the spot size $2{w_0}$ and pulse width L can be described by the vector potential as [22]:
With the initial boundary conditions, solving the partial differential equations obtained by bringing Eq. (2) and (3) into (4) in the numerical simulation, the electron’s displacement, velocity, and acceleration at each time step during the process of collision were recorded, with the numerical 4-5 order Runge-Kutta-Fehlberg method (RKF45).
2.2 Angular distribution of spatial radiation
The time-dependent electromagnetic field generated by a moving charged particle can be deduced from Liénard-Wiechert potentials [25]. For NITS radiation, we use a polar coordinate system to describe the radiation at each unit angle. The direction vector in the polar coordinate system is ${\boldsymbol b} = \hat{{\boldsymbol x}}\sin \theta \cos \phi + \hat{{\boldsymbol y}}\sin \theta \sin \phi + \hat{{\boldsymbol z}}\cos \theta $, where $\theta $ and $\phi $ represent the polar angle and the azimuthal angle respectively, as is shown in Fig. 1. The radiation power per unit solid angle that is normalized by ${e^2}{\omega ^2}/4\pi c$ is given by
3. Numerical results
In our research, the laser pulse has the following parameters: the wave length $\lambda = 1\; \mu m$, the initial phase ${\psi _0} = 0$, the spot size $2{w_0} = 10\lambda $, and the pulse width $L = 5\lambda $. To study the difference between off-axis NITS and axial NITS at different laser intensities, the initial transverse distance of the electron is a constant that $D = \sqrt {x_0^2 + y_0^2} = \lambda $. The laser intensity I can be characterized by the normalized peak amplitude ${a_0}$ of the laser field, for instance, the peak amplitude ${a_0} = 1$ when the laser intensity $I = 1.384 \times {10^{18}}\; W/c{m^2}$. The radiation observation distance ${R_0}$ is set to $1\; m$ which means the observation point is far enough away from the collision center.
3.1 Trajectories of the electron
To analyze the process that the off-axis electrons accelerated in the laser field, we first fixed the laser peak amplitude at ${a_0} = 5$. We assume the case that the electron has an initial energy ${\kappa _0} = 2.61\; MeV$, so the normalized initial velocity ${u_0} = 0.981$.The trajectories of relativistic electrons are plotted in Fig. 2(a) under 12 different initial deflection angles, where $\alpha \in [{0,2\pi } )$ with an interval of $\pi /6$.
Fig. 2. The trajectories of off-axis electrons with different initial position when ${a_0} = 5$ in (a), and the transverse displacement of the electron when $\alpha = 0$ in (b). When the peak amplitude of laser ${a_0} = 3,\; 5,\; 7$, the trajectories of off-axis electrons are in (c).
Compared with axial collisions, the symmetry of electron trajectories [19] is broken in off-axis collisions. In the direction perpendicular to the optical axis (z-axis), the average ponderomotive force on the electron in off-axis direction is greater than that pointing to the axis, and the trajectories are torqued towards the off-axis direction, which also changes the direction of electron emission. That is to say, when the electron is in an off-axis position, the trajectory of the electron diverges to the surroundings, and the direction of divergence is centrosymmetric with respect to the initial deflection angle.
Figure 2(b) is the enlargement of the transverse displacement of the electron when $\alpha = 0$. When electrons move in an intense laser field, the light intensity of the laser pulse first increases and then decreases, so the trajectory radius also first increases and then decreases. The electrons experienced multiple oscillations and finally emit at a certain angle with the optical axis. The asymmetric pondermotive force on off-axis electrons by an intense laser field leads to the torsion effect of electron emit direction. Further research found that when changing the laser intensity, the degree of this divergence varies, as shown in Fig. 2(c). When the laser intensity increases, the torsion effect on off-axis electrons in the field is stronger, and the degree of electron emission divergence is greater.
3.2 Spatial distribution of NITS radiation with varied laser intensities
Next, the effects of laser intensities on spatial angular distribution of NITS radiation power are analyzed. The initial energy of the electron is the same with Section 3.1. When Chen and Chang et al. [18,19] modulated the x-rays and γ-rays generated by the axial NITS, they found that when the laser intensity is greater than the relativistic intensity $({a_0} > 1)$, the spatial radiation diffuses in a funnel shape, and the increase in the laser intensity will lead to the reduce of ray collimation. However, as shown in Fig. 3, the power of electron radiation increases exponentially with the increase of laser intensity, resulting in a contradiction between the power and collimation of high-energy x-rays.
Fig. 3. The spatial angular distribution of the NITS radiation power between an off-axis electron and an intense laser pulse of different peak amplitude ${a_0}$ in the left side (a), (c), (e), and the curves of radiation power peak and the corresponding polar angle in the right side (b), (d), (f). The initial deflection angle of the electron is $\alpha = 0$. Higher laser intensity means higher collimation and higher power of NITS radiation (see Visualization 1).
To describe the collimation of radiation, we define the degree of divergence of radiation as the full angular area of half-maximum (FAHM) of radiation, that is, the angular area S occupied by the radiation where the intensity is half of the radiation peak on unit sphere:
The smaller the FAHM, the smaller the divergence of the radiation beam, the more concentrated the radiation, and the better the collimation. In our research, the effects of laser intensity on the spatial angular distribution of electron radiation during off-axis collision at relativistic intensity is studied in the left side of Fig. 3, and the relationship between the peak radiation power and the corresponding polar angle is in the right side of Fig. 3. When ${a_0} = 1$ [Fig. 3(a)], the power distribution on azimuth angle $\phi $ is uniform, which is similar to the result of the axial collision. When ${a_0}$ increases, the electrons are subjected to asymmetrical average ponderomotive force that off the axis, its influence on trajectories is more significant, and so does its radiation. This causes the spatial range of electron radiation to shrink in the distribution on azimuth angle $\phi $, the original symmetry is destroyed, and the radiation becomes more concentrated [Fig. 3(c)]. The calculation results for FAHW when ${a_0} = 3,5,7,9$ is $1.3373\; ra{d^2}$, $1.0915\; ra{d^2}$, $0.4976\; ra{d^2}$, $0.2831\; ra{d^2}$, respectively, which confirmed that the collimation also becomes better with ${a_0}$ increasing. When ${a_0} = 9$, the radiation power of NITS is concentrated in the off-axis direction, and the spatial distribution of the radiation power shows a pattern of multi-lobe [Fig. 3(e)].
In terms of the distribution of radiation power on polar angle $\theta $, as ${a_0}$ increases from 1 to 9, $\theta $ corresponding to the maximum radiation direction decreases linearly from 166° to 89°, which indicates that when the laser intensity is high (${a_0} = 9$, i.e., $I = 1.121 \times {10^{20}}\; W/c{m^2}$), the NITS radiation energy is concentrated in the direction vertical to the optical axis [Figs. 3(b), 3(d) and 3(f)]. This result has important implications for experimentally modulating the direction of high-collimation and high-energy X-rays.
In view of the trajectories’ symmetry of off-axis collision electrons, it is feasible to modulate the direction of NITS radiation through the electron’s initial deflection angle. In Fig. 4, The laser’s peak amplitude ${a_0} = 9$ is selected to generate higher-energy spatial NITS radiation, and the spatial distribution is shown in the left side. From Figs. 4(a), 4(c) and 4(e), as $\alpha $ changes in a period of $[{0^\circ ,360^\circ } )$, the spatial radiation of the multi-lope shape surges like ocean waves. For a certain lobe, its motion in the direction of polar angle is continuously, and merge with the main lobe at $\theta = 89^\circ $, and the corresponding azimuth angle $\phi $ at this time [Figs. 4(e)] is $330^\circ $, which explains the downward spike of the blue curve in the right side of Fig. 4 [Figs. 4(b), 4(d) and 4(f)].
Fig. 4. The spatial angular distribution of the NITS radiation power between the laser pulse and off-axis electrons of different initial deflection angle $\alpha $ in the left side (a), (c), (e), and the curves of radiation power peak in the right side (b), (d), (f). The peak amplitude of the laser is ${a_0} = 9$. The direction of NITS radiation changes linearly with the initial deflection angle of the electron (see Visualization 2).
However, the influence of initial deflection angle on spatial NITS radiation power is very slight, and the main influence is manifested on the azimuth angle of the radiation. The initial deflection angle $\alpha $ has a linear correspondence with the azimuth angle of the spatial radiation peak, as the orange curve shows in the right side of Fig. 4. When $\alpha = 0^\circ $, $\phi ={-} 3.5^\circ $ in the figure, which is outside the domain of $\phi $, but it's just a change for the aesthetics of the curve. This value is actually equivalent to $\phi ={-} 3.5 + 360 = 356.5^\circ $, which is the same when $\alpha = 360^\circ $. That is to say, the change of the NITS radiation direction with the initial azimuth angle of the electron is a period of $2\pi $. During the off-axis collision, the NITS radiation direction can be modulated by the initial deflection angle of the electrons.
In a word, both the collimation and radiation power of NITS radiation under off-axis conditions are positively related to the laser intensity, and there is a negligible change in power when modulating the direction of radiation by the initial deflection angle of electrons.
3.3 NITS radiation properties in time and frequency domain with varied laser intensities
In this section, the effects on the time distribution of NITS radiation in the direction that the peak radiation power occurred are analyzed by laser intensity modulation, the initial energy of the electron remains the same with Section 3.1. We selected parameters of laser peak amplitude ${a_0} = 1,\; 3,\; 5,\; 7,\; 9$ for comparison, as shown in Fig. 5, which not only shows the power distribution over time, but also the main peaks of all radiation pulses are amplified.
Fig. 5. Radiation time distribution in the direction of maximum power of NITS at different laser peak amplitudes ${a_0} = 1,\; 3,\; 5,\; 7,\; 9$, and the amplification of main radiation pulses. The initial deflection angle of electron is $\alpha = 0$.
When the laser intensity increases, the number of radiation pulses gradually decreases, indicating that the radiation will be more concentrated, while the time interval between radiation pulses increases, which makes it easier to intercept the main peak of the pulse.
Comparing the intensities and pulse widths of the main radiation pulses in Fig. 5, it can be found that when the laser intensity increases, the pulse width of the ultrashort pulse generated by NITS further decreases, and the intensity continues to increase. When ${a_0} = 9$ [Fig. 5(e)], the intensity of the ultra-short ultra-intense pulse reaches ${10^9}$, and the full width at half maximum (FWHM) is only $0.2\; as$.
In Fig. 6(a), we normalized the time corresponding to the main peak of the pulse and compared the cases where ${a_0}\; = \; 1,\; 2,\; \ldots ,\; 9$. Due to the increase of the electron radiation intensity, we amplify the cases of ${a_0} = 1,\; 2,\; 3,\; 4$ in Fig. 6(b). It can be seen that the FWHM of the main radiation peak decreases continuously, while the radiation power increases exponentially. Figure 6(c) quantifies this change and plots the curve. The results show that increasing the laser intensity in off-axis NITS can generate shorter and more intense X-ray pulses.
Fig. 6. The normalized main radiation pulse in (a) and its amplification in (b). Curves in (c) show the relations between peak radiation power, the FWHM of main pulse and laser peak amplitude.
The frequency distribution of X-rays is also important in practical applications. Figure 7 shows the spectrum of X-rays generated by NITS in the direction corresponding to the peak radiation power when ${a_0} = 1,\textrm{}3,\textrm{}5,\textrm{}7,\textrm{ }9$, and the other parameters are the same as Fig. 5. When ${a_0} = 1$, the cutoff of the spectrum is very short, only $400{\omega _0}$, and it presents a multimodal distribution, that is, the frequency of electron radiation has only a few modes. When ${a_0}$ increases, it can be seen from the curves in Fig. 3 that the total power of radiation increases exponentially, but the peak value of the spectrum in Fig. 7 does not increase much. This is due to the fact that the increase in radiation power reflects in the increase of number of radiation frequency modes, which tends to infinity, and the radiation spectrum presents a phenomenon of super-continuity. The frequency distribution of off-axis NITS is mainly in the X-ray category, and with the increase of ${a_0}$, the obtained X-ray has a wider bandwidth and a higher frequency. This phenomenon facilitates tunable X-ray radiation by changing the laser intensity in off-axis NITS.
Fig. 7. The spectrum of NITS radiation at the direction of maximum radiation power when the laser’s peak amplitude ${a_0} = 1,\; 3,\; 5,\; 7,\; 9$. The initial deflection angle of electron is $\alpha = 0$.
3.4 NITS radiation properties with varied initial energies of electron
In this section, the properties of off-axis NITS radiation are studied with varied energies of electron. The normalized laser intensity ${a_0} = 9$ is a constant in this section. In the following, we first study the angular distribution of off-axis NITS radiation and the radiation duration at the radiation peak, and then explore the trend of radiation properties changing with the electron initial energy. Finally, the spectrum of high-energy electrons in the direction of radiation peak is studied.
As shown in Fig. 8, the angular distributions of NITS radiation with different initial energies of electron are illustrated. To show the spatial distribution of radiation clearly under each set of parameters, each graph is individually normalized by the maximum value. Firstly, the azimuthal distribution of the radiation is changed from non-uniform to uniform with electron’s initial energy increases. When the energy electron moving in the intense pulse in an off-axis position, although high-energy electron is subjected to the same asymmetric torching effects as low-energy electron, the duration of the effect is greatly reduced, which degenerates the distribution of spatial radiation azimuth angles into a uniform distribution. Secondly, the polar angle of the radiation peak increases as shown in Fig. 10(a), which means the radiation direction changes from sideways to backwards. The angle occupied by NITS radiation on the polar angle gradually becomes smaller, and when the electron’s initial energy is ultra-high (100 MeV), the spatial distribution is a thin line in the picture. This is actually a backward hollow circular distribution in space, due to the periodicity of the azimuth angle. Thirdly, the FAMH first increase and then increase, which means the radiation collimation first decreases and then increase as the electron’s initial increases, as is illustrated in Fig. 10(a). The rise of FAHM is because the effect of the asymmetric torching is weakened in the process of increasing the initial energy of electrons, which leads to the expansion of the azimuth angle distribution. Then, the FAHM begins to decrease after experiencing a peak of $0.84\; ra{d^2}$ when $\kappa = 5.6\; MeV$, caused by the shrink in the occupation of polar angle in space.
Fig. 8. The angular distribution of NITS radiation when the electron’s initial energy $\kappa = 5,\; 10,\; 100\; MeV$. The peak amplitude of laser ${a_0} = 9$, and each picture is individually normalized by the maximum value.
Figure 9 shows the duration and power of NITS radiation peak when the initial energy of electron $\kappa = 5,\; 10,\; 100\; MeV$. The pulse width of the radiation reaches the level of zeptoseconds, and decreases exponentially with the increase of the initial energy of electrons, as shown in Fig. 10(b), and the peak power of radiation increases also exponentially. In terms of radiation spectrum, the radiation generated by ultra-high-energy electron [ Fig. 11(b)] has a cut-off frequency higher then ${10^5}{\omega _0}$, which belongs to the category of γ-rays. This shows that ultra-high-energy electron off-axis NITS can be used as a tunable γ-ray source.
Fig. 9. Radiation time distribution of NITS radiation peak with different initial energies of electron $\kappa = 5,\; 10,\; 100\; MeV$. The peak amplitude of laser ${a_0} = 9$.
Fig. 10. Radiation properties varied with initial energy of electron. The polar angle and FAHM (collimation) of radiation peak in (a), the peak power and corresponding pulse duration in (b).
Fig. 11. The spectrum of NITS radiation at the direction of maximum radiation power when the initial energy of electron $\kappa = 2.61,\; 5,\; 10\; MeV$ in (a) and $\kappa = 50,\; 100\; MeV$ in (b). The peak amplitude of laser ${a_0} = 9$.
When the initial energy of electron $\kappa = 100MeV$, a highly-collimated γ-ray pulse with a FWHM of only $1.22\; zs$ and a normalized power of $2.65 \times {10^{17}}$ can be generated. However, the radiation is backwards, which points to the laser or the lens group and brings challenges to the design of the actual ray source.
4. Conclusions
In conclusion, we specifically studied the influence of laser intensity and electron’s initial energy on the spatial distribution, time domain and frequency domain characteristics of electron trajectories and radiation, under the condition of nonlinear Thomson inverse scattering of off-axis relativistic electrons colliding with circularly polarized tightly focused intense laser. The ultrashort broad-spectrum X-ray pulse with high collimation is generated by low-energy electron (2.61 MeV) when increasing the laser intensity. Firstly, the trajectories of off-axis electrons in the laser field diverge toward the off-axis direction, and this divergence becomes stronger as the laser intensity increases. Secondly, when the laser intensity is high ($I = 1.121 \times {10^{20}}\; W/c{m^2}$), the off-axis condition solves the problem of poor collimation [19] of the axial collision radiation of low-energy electrons, and the spatial distribution is concentrated in the sideways instead of backwards. By adjusting the initial deflection angle of low-energy electrons, the emission direction and radiation direction can be linearly modulated. The increasing initial energy of electron reduces the degree of asymmetry in the off-axis electron radiation, resulting in a highly collimated backward radiation pulse. Thirdly, the temporal and frequency spectrum in the direction of the radiation maximum power are investigated. As the laser intensity increases, the radiation pulse width decreases from $3.3\; as$ to $0.2\; as$, and the radiation pulse width further decreases to $1.22\; zs$ as the electron’s initial energy increases, while the intensity of radiation pulses rises exponentially. The mode number of the radiation frequency increases and tend to infinity, the spectrum changes from multi-peak to supercontinuum and the frequency band broadens. These results strengthen the prospect of nonlinear Thomson scattering as an ultrashort X-ray and γ-ray source of higher-collimation and higher-energy.
Funding
Qinglan Project of Jiangsu Province of China; Nanjing University of Posts and Telecommunications (NY221098); Natural Science Foundation of Shanghai (11ZR1441300); Natural Science Research of Jiangsu Higher Education Institutions of China (10KJB140006); National Natural Science Foundation of China (10947170/A05, 11104291).
Acknowledgments
This work is part of the STITP Project under Grant No. CXXZD2022148.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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