High-Energy Electron Emission Controlled by Initial Phase in Linearly Polarized Ultra-Intense Laser Fields

Abstract

Extensive numerical simulations were performed in MATLAB R2020b based on the classical nonlinear Thomson scattering theory and single-electron model, to systematically examine the influence of initial phase in tightly focused linearly polarized laser pulses on the radiation characteristics of multi-energy-level electrons. Through our research, we have found that phase variation from 0 to 2π induces an angular bifurcation of peak radiation intensity, generating polarization-aligned symmetric lobes with azimuthal invariance. Furthermore, the bimodal polar angle decreases with the increase of the initial energy. This phase-controllable bimodal distribution provides a new solution for far-field beam shaping. Significantly, high-harmonic intensity demonstrates π-periodic phase-dependent modulation. Meanwhile, the time-domain pulse width also exhibits 2π-cycle modulation, which is synchronized with the laser electric field period. Notably, electron energy increase enhances laser pulse peak intensity while compressing its duration. The above findings demonstrate that the precise control of the driving laser’s initial phase enables effective manipulation of the radiation’s spatial characteristics.

1. Introduction

In recent years, with the development of CPA [1,2] (Chirped Pulse Amplification) technology, it has been able to generate ultrashort and ultra-intense laser pulses with peak power up to the petawatt (PW) level [3], which provides an essential tool for strong-field physics research [4]. The ultra-intense pulses generated by CPA can be used to study the nonlinear interaction of lasers with plasma [5], atoms, molecules, etc., and to promote the development of high harmonic generation [6,7], attosecond pulse generation [4], etc. CPA is capable of generating ultrashort pulses at the femtosecond (fs) or even attosecond (as) level, which is widely used in the fields of ultrafast spectroscopy [8], chemical reaction dynamics, etc. In the future, it will continue to play a key role in the fields of particle acceleration, attosecond science [9], laser fusion, precision machining, ultrafast electron microscope [10], and medical science, etc. It will push forward the advancement of many disciplines.

Previous studies have found that the carrier envelope phase (CEP) affects the phase matching condition. The CEP is the phase difference between the peak position of the carrier electric field and the peak position of the pulse envelope. Previous experimental studies have shown that isolated attosecond pulses can be generated by proper CEP modulation [11], which enables a more precise temporal control of high harmonic radiation on attosecond time scales and provides experimental support for attosecond pulse applications. For example, Corkum [6] theorized from a plasma perspective that by changing the initial phase CEP, which affects the spatiotemporal tuning of electron ionization and acceleration, the phase relationship between the driving laser and the resulting high harmonics can be tuned to optimize the efficiency of high harmonic generation. Sansone’s [12] team further revealed the above mechanism by obtaining isolated attosecond pulses by the CEP locking technique for the first time. Ziegler et al. [13] found that the proton beam quality can be optimized by modulating the spectral phase of CEP in PW-class laser experiments. In addition, there are also Goulielmakis [14] and Kienberger [15] who further proved the phase matching mechanism of CEP by observing the CEP-dependent valence electron motions and recollisional phases in real time via the attosecond transient recording technique, respectively.

We analyze this process through classical Thomson scattering theory, where an incident plane wave accelerates a free electron, resulting in a dipole radiation emission [16]. The earliest Thomson scattering theory was started in 1907 after the discovery of electrons by J.J. Thomson, and has been strictly derived in modern electrodynamics textbooks [16,17]; there is also the single-electron model, which is the study of linear and nonlinear scattering processes [18] focused on a single electron and light. Nonlinear Thomson scattering dynamics in high-intensity laser fields exhibit strong parametric dependencies. For relativistic electrons interacting with ultra-intense laser pulses (I > 10¹⁸ W/cm²), the radiation spectra demonstrate pronounced nonlinear characteristics distinct from classical predictions [19]. This complexity arises from the non-perturbative nature of electron-light interactions at extreme field strengths. In attosecond γ-ray generation via aperiodic vortex-laser-driven nonlinear Thomson scattering, relativistic electron sheets impart both energy and angular momentum to γ-photons. Key radiation characteristics, including spectral density, photon energy, and brilliance, exhibit strong dependence on the electron ensemble’s kinetic energy (γ-factor) [20] and spatial charge distribution. This momentum-energy dual transfer mechanism enables controlled γ-ray vorticity generation.

Using MATLAB for numerical simulation and modeling, we describe the spatiotemporal and spectral characteristics of the nonlinear Thomson radiation emitted by relativistic electrons in a tightly focused linearly polarized laser field, with a focus on the emission modulation associated with the initial phase.

2. Materials and Methods

First, a few normalization parameters are introduced. The spatiotemporal coordinates are normalized by the laser wave number $k_0^{-1}={\lambda }_0/2\pi$, with the temporal scale defined via the laser angular frequency ${\omega }_0^{-1}={\lambda }_0/2\pi c$ (where c is the speed of light, $\lambda =1\mu m$).

This study examines the propagation dynamics of Laguerre–Gaussian (LG) laser modes in linear dielectric media (${\epsilon }_r>1,{\mu }_r=1,{\sigma }_{in}=0$), focusing on incidence along the z-axis. In a tightly focused Gaussian laser field, the electric and magnetic fields following Maxwell’s equations can be expressed as follows:

$$\mathrm{E}=\nabla \times \mathrm{A}$$

(1)

$$\mathrm{B}=\sqrt{\mathsf{ϵ}}\left[\left({\displaystyle \frac{i}{k}}\right)\nabla \left(\nabla ·A\right)+ikA\right]$$

(2)

where the vector potential A is the relevant solution of the Helmholtz equation, describing the distribution in the space in which it is located.

$${\nabla }^2\mathrm{A}+{\mathrm{k}}_0^2\mathrm{A}=0$$

(3)

For linearly polarized light in a laser field, the electric field oscillates in a fixed plane and can be resolved into two orthogonal components along the x-axis and y-axis. These components have a phase difference of either 0 or π, allowing the total electromagnetic field to be expressed as the vector sum $\mathrm{E}{=\mathrm{E}}_{\mathrm{x}\mathrm{p}}+{\mathrm{E}}_{\mathrm{y}\mathrm{p}}$, $\mathrm{B}{=\mathrm{B}}_{\mathrm{x}\mathrm{p}}+{\mathrm{B}}_{\mathrm{y}\mathrm{p}}$. The x-polarized component of the laser field was analytically obtained from Equations (1) to (3) in the work of Salamin et al. [21]. For the y-polarized component, equivalent field expressions were derived through a similar theoretical framework by Kane [22] and Barton [23].

Figure 1 shows the interaction between a tightly focused laser pulse and electrons. Since the motion of electrons in a high-intensity laser field is strongly nonlinear; an exact analytical representation of the electromagnetic field expression for a Gaussian beam is required to obtain the exact trajectory of the electrons. We can extend the diffraction angle to the fifth order to accurately describe the electromagnetic field.

$$E_x=E\{S_0+{\epsilon }^2[{\xi }^2-{\displaystyle \frac{{\rho }^4{S}_3}{4}}]+{\epsilon }^4[{\displaystyle \frac{S_2}{8}}-{\displaystyle \frac{{\rho }^2{S}_3}{4}}-{\displaystyle \frac{{\rho }^2({\rho }^2-16{\xi }^2)S_4}{16}}]\}$$

(4)

$$E_y=E\mathsf{\xi }\mathsf{\zeta }\{{\epsilon }^2{S}_2+{\epsilon }^4[{\rho }^2{S}_4-{\displaystyle \frac{{\rho }^4{C}_4}{4}}]\}$$

(5)

$$E_z=E\mathsf{\xi }\{\epsilon C_1+{\epsilon }^3[-{\displaystyle \frac{C_2}{2}}+{\rho }^2{C}_3-{\displaystyle \frac{{\rho }^4{C}_4}{4}}]+{\epsilon }^5[-{\displaystyle \frac{3C_3}{8}}-{\displaystyle \frac{3{\rho }^4{C}_4}{8}}+{\displaystyle \frac{17{\rho }^4{C}_5}{16}}-{\displaystyle \frac{3{\rho }^6{C}_6}{8}}+{\displaystyle \frac{{\rho }^8{C}_7}{32}}]\}$$

(6)

$$B_x=0$$

(7)

$$B_y=E\{S_0+{\epsilon }^2[{\displaystyle \frac{{\rho }_2^{2S}}{2}}-{\displaystyle \frac{{\rho }_3^{4S}}{4}}]+{\epsilon }^4[-{\displaystyle \frac{S_2}{8}}+{\displaystyle \frac{{\rho }_3^{2S}}{4}}+{\displaystyle \frac{5{\rho }^{4S}}{16}}-{\displaystyle \frac{{\rho }_5^{6S}}{4}}+{\displaystyle \frac{{\rho }_6^{8S}}{32}}]\}$$

(8)

$$B_z=E\mathsf{\zeta }\{\mathsf{\epsilon }{C}_1+{\epsilon }^3[{\displaystyle \frac{C_2}{2}}+{\displaystyle \frac{{\rho }^2{C}_3}{2}}-{\displaystyle \frac{{\rho }^4{C}_4}{4}}]+{\epsilon }^5[{\displaystyle \frac{3C_3}{8}}+{\displaystyle \frac{3{\rho }^2{C}_4}{8}}+{\displaystyle \frac{3{\rho }^4{C}_5}{16}}-{\displaystyle \frac{{\rho }^6{C}_6}{8}}+{\displaystyle \frac{{\rho }^8{C}_7}{32}}]\}$$

(9)

Let $\xi =x/{\omega }_0$ and $\zeta =y/{\omega }_0$ denote the normalized coordinates, where $\mathsf{\omega }={\mathsf{\omega }}_0{(1+{(\mathrm{z}/{\mathrm{z}}_{\mathrm{R}})}^2)}^{1/2}$ is the beam waist radius along the z-axis. The radial distance $\mathrm{r}=\sqrt{(x^2+y^2)}$ is normalized as $\mathsf{\rho }=\mathrm{r}/\mathsf{\omega }0$, and the angle of diffraction is characterized by $\epsilon ={\omega }_0/z_r$, with $z_r$ = k ${{\omega }_0}^2$ ($k^{-1}=\lambda /2\pi$) being the Rayleigh length. Based on these parameters, E is given by the following equation:

$$E=E_0{\displaystyle \frac{\omega }{{\omega }_0}}exp[-{\displaystyle \frac{r^2}{{\omega }^2}}]exp[-{(z-ct)}^2/L^2]$$

(10)

where, $E_0$ is the peak amplitude, L is the pulse duration, ${\omega }_0$ is the minimum spot size, $S_n$, $C_n$ are shown below.

$$S_n={({\displaystyle \frac{{\omega }_0}{\omega }})}^n\mathit{sin}(\psi +n\psi )$$

(11)

$$C_n={({\displaystyle \frac{{\omega }_0}{\omega }})}^n\mathit{cos}(\psi +n\psi )$$

(12)

where, $\psi ={\psi }_0+{\psi }_P-{\psi }_R+{\psi }_G$, the initial phase ${\psi }_0$ is constant. ${\mathsf{\psi }}_{\mathrm{p}}=\mathsf{\eta }=\mathsf{\omega }\mathrm{t}-\mathrm{k}\mathrm{z}$ and ${\psi }_G={\mathit{tan}}^{-1}(z/z_R)$ represent the plane-wave phase and the Gouy phase, respectively. The latter is related to the characteristic that the total phase undergoes a change of π during the change of z from $-\infty$ to $+\infty$. ${\psi }_R=k{r}^2/(2R)$ is the phase related to the curvature of the wavefront. ${\mathrm{R}}_{(\mathrm{z})}=\mathrm{z}+{{\mathrm{z}}_{\mathrm{R}}}^2/\mathrm{z}$ denotes the radius of curvature of the wavefront at z intersecting the beam axis. These definitions of phase are important for understanding and analyzing optical phenomena such as the propagation properties of Gaussian beams.

The coupled energy-momentum evolution equations governing the interaction between an ultra-intense laser pulse and a relativistic electron are given by the following:

$${\displaystyle \frac{d\overrightarrow{P}}{dt}}=-e(\overrightarrow{E}+{\displaystyle \frac{\overrightarrow{v}}{c}}\times \overrightarrow{B})$$

(13)

$${\displaystyle \frac{d\gamma }{dt}}=-{\displaystyle \frac{e}{m_0{c}^2}}(\overrightarrow{v}·\overrightarrow{E})$$

(14)

Here, $\overrightarrow{P}=\gamma m\overrightarrow{v}$ is the electron momentum $\mathsf{\gamma }=1/{(1-{\mathrm{v}}^2/{\mathrm{c}}^2)}^2$ is the relativistic factor, $\overrightarrow{E}$ is the laser electric field, and $\overrightarrow{B}$ is the laser magnetic field, and $\overrightarrow{v}$ is the electron velocity.

This work employs the normalized vector potential $\mathsf{\alpha }(={\mathrm{I}{\mathsf{\lambda }}^2/1.37\times {10}^{18})}^{1/2}$ to characterize the relativistic laser intensity, with the laser intensity I (W/cm²) and the laser wavelength $\lambda$ ($\mu m$). The spatial and temporal scales are governed by the normalized wavelength $\lambda$ and laser period T. The laser beam is tightly focused. The volume and intensity gradient of the interaction are controlled by ${\omega }_0$, and at the same time, a significant diffraction phase is triggered when ${\omega }_0<1\mathsf{\mu }\mathrm{m}$. Therefore, we take the laser beam waist spot ${\omega }_0=2\lambda =2\mathsf{\mu }\mathrm{m}$ (corresponding to the laser intensity). When the laser pulse length $L<5T$ (pulse width), the electrons do not complete the acceleration cycle before the end of the pulse, the phase sensitivity is strong, and the energy spectrum will have a discrete structure; therefore, the laser pulse length is taken as $L=1T$. In addition, when the numerical aperture $NA>0.8$, it will lead to phase distortion. Therefore, the numerical aperture (medium focusing) is taken as $NA=\lambda /(\pi {\omega }_0)\approx 0.16<0.5$ (medium focus) when the phase is almost free of aberrations. Electrons with initial kinetic energy 11.4 MeV ($v=0.999c$) are injected from $Z=20\lambda$ along the negative z-direction. In the absence of laser interaction, these electrons would naturally focus at the origin (0, 0, 0), corresponding to the laser beam center.

$${\displaystyle \frac{dP(t^{\prime })}{d\Omega }}={\displaystyle \frac{e^2}{4\pi c}}{\displaystyle \frac{\overrightarrow{n}\times [(\overrightarrow{n}-\overrightarrow{\beta (t})\times \dot{\overrightarrow{\beta (t)}}]}{{(1-\overrightarrow{n}·\overrightarrow{\beta (t}))}^6}}$$

(15)

$${\overrightarrow{E}}_{rad}(\overrightarrow{x},t)=e[{\displaystyle \frac{\overrightarrow{n}-\overrightarrow{\beta }}{{\gamma }^2(1-\overrightarrow{\beta }·\overrightarrow{n})R_e^2}}{]}_{ret}+{\displaystyle \frac{e}{c}}[{\displaystyle \frac{\overrightarrow{n}\times (\overrightarrow{n}-\overrightarrow{\beta })\times \overrightarrow{\beta }}{{(1-\overrightarrow{\beta }·\overrightarrow{n})}^3{R}_e}}{]}_{ret}$$

(16)

Note that the subscript ‘ret’ indicates that all right-hand-side quantities are to be evaluated in the delay time $t^{\prime }=t+R-\overrightarrow{n}·\overrightarrow{r}$, and $R$ denotes the displacement between the observation point and the interaction region. Here, $\overrightarrow{r}$ is the position vector of the electron, the unit vector $\overrightarrow{n}=\mathit{sin}\theta \mathit{cos}\psi \overrightarrow{x}+\mathit{sin}\theta \mathit{sin}\psi \overrightarrow{y}+\mathit{cos}\theta \overrightarrow{z}$ defines the direction of the observation in spherical coordinates, and θ and ψ are the polar and azimuthal angles, respectively. Additionally, there are $\overrightarrow{\beta }=\overrightarrow{v}/c$, $\dot{\overrightarrow{\beta }}=\dot{\overrightarrow{v}}/c$.

The radiated field of a moving charge can be calculated from the Lienard–Wiechert potential, the form of which was first proposed by Lienard (1898) and Wiechert (1901) and refined in modern electrodynamics textbooks [16]. Using the above theory, the temporal evolution of scattered power $P_{(t)}$ and the resultant radiation field at the detector position (defined by a unit vector $\overrightarrow{n}$) are governed by the following:

The spectral angular energy density $d^{2}I/d_{\omega }{d}_{\Omega }$ emitted during electron-laser interactions is given by the following:

$${\displaystyle \frac{d^{2}I}{d\omega d\Omega }}={\left|{\int }_{-\infty }^{+\infty }{\displaystyle \frac{e^2}{4\pi c}}{\displaystyle \frac{(\overrightarrow{n}-\overrightarrow{\beta (t})\times \dot{\overrightarrow{\beta (t)}}}{{(1-\overrightarrow{n}·\overrightarrow{\beta (t}))}^2}}·e^{is(t-\overrightarrow{n}·\overrightarrow{r})}dt\right|}^2$$

(17)

The electron dynamics are numerically integrated via an adaptive Runge–Kutta–Fehlberg algorithm (RKF45), with Equations (9) and (10) governing the time evolution. At each iteration, the particle’s phase-space coordinates (position $r$, velocity $v$, acceleration $a$) are archived. Subsequent radiation analysis derives the spectral-angular distribution from Equations (11) and (12).

3. Results

3.1. Radiation Angular Distribution

In our numerical implementation, since the laser beam is tightly focused, in order to ensure that the intensity of the laser in the interaction region (corresponding to $\alpha =5$), that is $I~{10}^{19} \mathrm{w}/{\mathrm{c}\mathrm{m}}^2$, a linearly polarized laser pulse with a normalized peak intensity $\alpha =5$ was employed, spanning a single optical cycle. Figure 2 displays representative electron trajectories for carrier-envelope phases (CEP) of ${\varphi }_0=0$, ${\varphi }_0=\pi /3$, ${\varphi }_0=\pi /2$, demonstrating clear alignment with the laser’s linear polarization state through their oscillatory patterns. The two-dimensional plots of the motion trajectories show a symmetry about the polarization axis. This symmetry is an important feature of electron dynamics under specific conditions. As shown in the figure below, the electron moves from right to left along the negative direction of the z-axis. We can analyze the physical quantities associated with the radiation field by studying the motion trajectories of electrons under tightly focused linearly polarized laser pulses.

Figure 3 displays the angular distribution of radiation produced by $20\times 0.511 \mathrm{M}\mathrm{e}\mathrm{V}$ electrons in collision with a linearly polarized laser pulse, analyzed for multiple initial phase values (${\varphi }_0=0,\pi /3,\pi /2,2\pi /3,\pi ,4\pi /3,3\pi /2,5\pi /3$). Our simulations reveal that in a head-on collision geometry, where relativistic electrons propagate along the $-\hat{\mathrm{z}}$ direction while interacting with a linearly polarized laser beam advancing in the $+\hat{\mathrm{z}}$ direction, the polar angle ${\theta }_m$ splits at the peak radiation, and the radiant energy appears as a bimodal structure and exhibits a symmetric distribution about the laser polarization axis. The polar angle $\theta$ of radiation is governed by the instantaneous angle between the electron’s velocity vector $\overrightarrow{v}$ and its acceleration vector $\overrightarrow{\beta }$ as prescribed by Liénard–Wiechert potentials. For relativistic electrons (${\gamma }_0\gg 1$), the radiative precession effect brings ${\theta }_m$ the approach to $\pi -c/{\gamma }_0$. With increasing initial phase ${\varphi }_0$, the azimuthal angle $\psi$ remains locked to the laser polarization axis. For a linearly polarized laser with its electric field vector aligned along the direction of the x-axis, the radiation maintains two distinct peaks at $\psi =0,\pi$, corresponding to the direction of the x-axis and the negative direction of the x-axis, respectively, independent of ${\varphi }_0$ variations. Furthermore, the analysis demonstrates that the azimuthal radiation pattern undergoes synchronous rotation with the laser polarization axis, maintaining a fixed angular relationship with the E-field vector orientation. It is worth noting that in Figure 3, we project the 3D bimodal distribution (${\theta }_m=\pm \left(\pi -c/{\gamma }_0\right),\psi =0/\pi$) to the $\theta -\psi$ plane, and if we fix $\psi =0$, then $\theta ={\theta }^{\prime }$, there is no compression; if $\psi \ne 0$, then the polar angle appearing in the projection should be corrected to ${\theta }^{\prime }={\mathit{tan}}^{-1}(\mathit{sin}\theta \mathit{cos}\psi )<\theta$. This will lead to an underestimation of the polar angle.

The effect arises from the regular oscillations of a linearly polarized field, which cause electrons to move alternately along the polarization axis, producing two dominant radiation directions. The initial phase ${\varphi }_0$ only determines the initial amplitude of the laser electric field at the moment t = 0 but does not change the direction of polarization; therefore, the change of ${\varphi }_0$ affects the initial conditions of the electron oscillations, but the radiation bimodal peaks are always along the direction of the electric field and $\psi$ do not change with it.

Numerical simulations were performed for electron initial energies of 20.44, 30.66, 40.88, and 51.1 MeV (corresponding to $\gamma =40,60,80,100$). The angular radiation distributions exhibit a characteristic bifurcation effect; as the initial phase varies, the polar angle $\theta$ develops a double-peak structure at the maximum intensity position, while the azimuthal angle $\psi$ remains constant. It is worth noting that when we use elliptically polarized light, the electrons are in three-dimensional helical motion, and the angular distribution of the radiation is single-peaked (or tilted elliptically distributed).

Building upon these findings, Figure 4 presents the correlation between radiant energy and polar angle $\theta$ for varying initial electron energies ${\gamma }_0$, with the initial phase fixed at ${\varphi }_0=0$. As shown in Figure 4, we find that the bimodal polar angle ${\theta }_m$ decreases with the increase of the initial energy ${\gamma }_0$, but there exists a lower limit (determined by the laser intensity $a$), which ${\theta }_m$ is stabilized at $\pi -c/{\gamma }_0$. This property stems from the planar constraints of the electron motion and the stationarity of the acceleration direction. Therefore, this property of bimodal radiation produced by linearly polarized lasers can be well applied to polarization resolution for a precise measurement of physical parameters or tuning of specific functions. At the same time, the increase of the initial energy of the electron, also causes a superlinear growth of the maximum radiation, which is proportional to ${{\gamma }_0}^4$, and this law provides a key basis for the design of controllable high-intensity radiation sources.

3.2. Temporal Spectrum

The time-domain radiation spectrum was further examined by sweeping the initial phase from 0 to $2\pi$ (steps of $\pi /18$) at an initial electron energy of 20.511 MeV. As shown in Figure 5a–h, we can see that the time spectra at different initial phases are not the same, showing a symmetric bimodal distribution at initial phases 0 and $\pi$, and a single-peak structure at the rest of the phases, while at the same time we observe that the time spectra repeat the same variation at every cycle $\pi$. In reality, during the microscopic transients, the periodic variation of the time spectrum is synchronized with the laser electric field period with a period of $2\pi$, which is due to the half-period inversion symmetry. The indistinguishability of positive and negative half-period radiation leads to macroscopic observations with a time-spectrum period equal to the laser electric field half-period ($\pi$). The time spectrum under tightly focused line polarization is highly sensitive to the initial phase and shows a periodic modulation of the peak high harmonic radiation with a period of $\pi$. This is due to the unidirectional electric field of the line-polarized light, resulting in a periodic energy modulation of the electron motion directly coupled to the phase. The electron motion undergoes a complete acceleration-deceleration process every half laser cycle ($\pi$), corresponding to an increase and then a decrease in the peak energy, which is strictly related to the initial phase. This physical mechanism has a good application for phase control to achieve an active modulation of harmonic intensity. On the contrary, the initial phase under tightly focused circular polarization has a minimal effect on the time spectrum and shows significant similarity across spectra.

As the initial phase changes, the pulse width under linearly polarized light varies, fluctuating in the range of 0.0415–0.0477 as. Figure 6 illustrates the dependence of pulse width on initial phase. Through the images as well as the related data, we can see that the pulse width fluctuates significantly with the change of the initial phase, showing phase sensitivity and nonmonotonicity, and, at the same time, periodicity, with a period of $2\pi$. In addition, we can see that the pulse width reaches a minimum near ${\varphi }_0=\pi /2$; a subminimum near ${\varphi }_0=2\pi /3$; and an extreme maximum near ${\varphi }_0=\pi$. The core physical mechanism is due to the nonlinear coupling effect of the field component. At the same time, the longitudinal field distorts the electron trajectory, and introduces the additional acceleration component, the electron is subjected to longitudinal acceleration, and the motion trajectory is complicated, which widens the radiation pulse. When ${\varphi }_0=0$ ($E_x$ peak, $B_y=0$), a single electric field accelerates and the acceleration lasts for a longer period, creating a moderate pulse width. Thus, in the range 0–$\pi /2$, electric field acceleration dominates, and the pulse width decreases rapidly. When ${\varphi }_0=\pi /2$ ($E_x=0$, $B_y$ peak), the magnetic field deflection dominates, and the Lorentz force induces transient transverse deflection, forming ultrafast transverse oscillations, at which time the pulse is the narrowest; at the same time, in the range of $\pi /2$–$\pi$, the electric field reverses and couples with the longitudinal field at the same time, resulting in a sharp increase in the pulse width. When ${\varphi }_0=\pi$ ($E_x$ negative peak, $B_y=0$), the electrons undergo a two-stage process of deceleration followed by acceleration, and the prolongation of the interaction time leads to the widest pulse width; at the same time, in the range of $\pi$–$3\pi /2$, the reversed electric field carries out acceleration leading to a decrease in the pulse width. When ${\varphi }_0=3\pi /2$, the reverse magnetic field is injected, and again, the reverse Lorentz force induces a transverse deflection, but due to the different direction of the initial velocity being asymmetric concerning that at ${\varphi }_0=\pi /2$, there is a subminimal value of the pulse width; at the same time, in the range of $3\pi /2$–$2\pi$, the electric field starts to recover, while the longitudinal field modulates, causing the pulse width to increase again. Thus, the pulse width has a duration of $2\pi$.

As shown in Figure 7a, when the initial phase is 0, we observe a double-peaked structure at the crest of the linearly polarized laser pulse, and subsequently, by enhancing the laser pulse intensity, the double-peak structure remains prominently observable at the peak intensity of the linearly polarized laser pulse, as evidenced in Figure 7b,c.

In the online polarized light laser field, when the electron oscillatory motion and the laser electric field phase reach a specific matching condition, its radiation electromagnetic wave in space and time will occur in the phase length interference, resulting in the concentration of radiant energy, pulse width compression, this phenomenon is the coherent enhancement effect. As illustrated in Figure 2, the motion of the electron remains in phase with the laser’s electric field, resulting in a driven oscillatory motion under the influence of the field, the electron’s initial phase is zero, then the electron will get the maximum acceleration every half-cycle, and the waveform of the radiation field is synchronized and superimposed. At the same time, the phase difference of electromagnetic waves radiated in adjacent half-cycles is $2\pi$, which is coherently superimposed in the observation direction. When $N$ cycles are coherently superimposed, the radiation intensity is enhanced ($\propto N^2$). In addition, the time-domain interference produces pulse-width compression, resulting in the main peak width $∆\mathsf{\tau }\approx \mathrm{T}/(2{{\mathsf{\gamma }}_0}^2)$ (T for the laser field cycle). It is worth noting that due to the combined effects of time-reversal symmetry, parity conservation, and nonlinear polarization selection, even-order harmonics are suppressed. Only odd-order harmonics satisfy the phase-matching conditions and participate in interference, forming a train of pulses. This mechanism has promising applications in generating isolated attosecond pulses. In contrast, when the electron trajectory is misaligned with the polarization direction or phase of the laser field, radiative energy loss occurs, accompanied by pulse broadening. Experimental measurements reveal a clear dependence of pulse duration on initial electron energy: at 20 × 0.511 MeV, the pulse width is approximately 0.0443 as; at 40 × 0.511 MeV, it decreases to 0.0114 as; and at 60 × 0.511 MeV, further compression to 0.0051 as is observed. As illustrated in Figure 7, a comparative analysis of laser pulse profiles for initial energies of 40 × 0.511 MeV and 60 × 0.511 MeV against the 20 × 0.511 MeV reference demonstrates that higher initial electron energies enhance the peak intensity of the laser pulse while simultaneously reducing its temporal width.

This phenomenon corresponds to the results shown in Figure 4. As the initial energy of the electrons increases, the radiant energy is concentrated in a smaller spatial extent due to the radiation cone angle $\mathsf{\theta }~1$/$\mathsf{\gamma }~$1/$a_0^2$ of the energetic electrons, which manifests itself as pulse narrowing in the time domain.

3.3. Frequency Spectrum

Finally, we also delved into the spectral properties. In our configuration, electrons possessed 20 relativistic mass units (20 × 0.511 MeV) of initial kinetic energy, and by analyzing it, we found that the time domain under linear polarization is correlated with the frequency domain, as shown in Figure 8a–c. When the initial phase ${\varphi }_0=0$, the time-domain bimodal peaks (spaced $0.67T_0$ apart) of the line-polarized radiation lead to the frequency-domain harmonic interference, and this results in sharp harmonic peaks, exclusively of odd-order harmonics, with highly concentrated energy; when the initial phase ${\varphi }_0=\pi /4$, the asymmetric bimodal peaks in the time domain lead to harmonic unfolding, with the harmonic peaks becoming wider, the splitting peaks appearing around them, the continuous spectrum component starts to appear; when the initial phase ${\varphi }_0=\pi /2$, the single-peak broad pulse in the time domain leads to the disappearance of the harmonics, and the continuous spectrum dominates. At the same time, the odd harmonic intensity fluctuates with the initial phase (carrier-envelope phase CEP), realizing a periodic modulation. Analysis reveals that the initial phase plays a crucial role in electron scattering dynamics within tightly focused, linearly polarized laser fields. In line-polarized light, the spectral properties are dynamically tunable through the initial phase, and this mechanism provides a key basis for phase-modulated attosecond pulses. In addition, as shown in Figure 8a, in the generation of high harmonics driven by linearly polarized laser light, we find that the radiation intensity exhibits a stepwise decay with the increase of the harmonic frequency, and the radiation intensity $\propto n^{-3}$ (n is the harmonic order), which embodies the physical mechanism of nonlinear Thomson scattering, and this property provides a theoretical basis for the active modulation of high harmonics. It is worth noting that when we use elliptically polarized light, all harmonics appear in the radiation spectrum, and the harmonics are more densely spaced.

Furthermore, electron energy spectra were analyzed for higher initial energies of 40 × 0.511 MeV (Figure 9b) and 60 × 0.511 MeV (Figure 9c). As can be seen from Figure 9a–c, the maximum radiation intensity corresponds to the values of 60,000, 240,000, and 530,000, respectively, and the threshold frequency (the frequency at which the first significant increase in radiation intensity occurs) corresponds to the values of 2000, 8000, and 18,000, respectively, in the horizontal coordinates. The electron energy increases quadratically from $20\to 40\to 60$ (i.e., to a factor of 2 and 3), while the maximum radiation intensity increases from $I_a\to I_b\to I_c$ satisfies: $I_b/I_a\approx 4,I_c/I_a\approx 9$, and the threshold frequency increases from ${\omega }_{tha}\to {\omega }_{thb}\to {\omega }_{thc}$ satisfies: ${\omega }_{thb}/{\omega }_{tha}\approx 4,{\omega }_{thc}/{\omega }_{tha}\approx 9$, both of which support quadratic increases. Thus, the threshold laser frequency corresponding to the maximum radiation intensity and peak radiation grows quadratically with increasing electron energy. Furthermore, we fit these three plots, and we find that the threshold frequency (vanishing radiation intensity) is linearly proportional to the electron energy.

4. Experimental Considerations

4.1. Experimental Validation

This study examines how varying initial electron energies and phases influence radiation characteristics in tightly focused, linearly polarized laser fields. The findings demonstrate that precise control over the driving laser’s initial phase enables an effective manipulation of (1) the spatial profile of emitted radiation, and (2) key temporal and spectral properties. Notably, initial phase adjustment serves as a powerful tool for pulse structure modulation, enabling selective generation of single- or double-peak temporal spectra through controlled phase variation. Meanwhile, the pulse width will appear to change periodically; synchronized with the laser half-period. In the influence on the spectrum, the phase modulation of the harmonic intensity can be realized, which is maximum at ${\varphi }_0=k\pi$ and almost disappears at ${\varphi }_0=(k+1/2)\pi$, enhancing the odd harmonic selectivity, and, at the same time, the reconstruction of the spectral shape is realized. In the linearly polarized laser field, the modulation of the electron initial energy in the harmonic order, intensity, spatial distribution, and time domain also has a significant effect on the harmonic radiation characteristics. As the electron’s initial energy increases, the harmonic intensity will rise sharply and the time-domain pulse width is compressed; at the same time, the compression of the harmonic spatial distribution leads to a more collimated radiation of the high-energy electrons, which is well applied in high-resolution imaging.

4.2. Experimental Conditions and Limitations

Nonlinear Thomson scattering modulated radiation characteristics research in the experimental aspects, there are still some technical challenges, including CEP stability control, angular resolution accuracy, and electron beam-laser synchronization and other issues. At the same time, there are some limitations in the experimental conditions; first, the experimental environment should simulate a vacuum to avoid the influence of air scattering; second, the laser intensity should be ensured in ${10}^{18}~{10}^{19}\mathrm{W}/{\mathrm{c}\mathrm{m}}^2$. If it is lower than this range, the nonlinear effect is weak, and higher than this range may trigger the interference of QED effect; third, there are some requirements in the quality of the electron beam, the energy dispersion needs to be less than 5%, the transverse size should be smaller than the laser wavelength.

4.3. Future Directions and Research Needs

The study of nonlinear Thomson scattering modulated radiation properties should intervene the QED effect (quantum electrodynamics effect) as well as the multi-electron collective effect in the future, i.e., the correction of the bimodal angular distribution by the quantum radiation backreaction under the hyperintense field and the effect of the electron-beam space charge effect on the radiative bimodal peaks should be specifically considered. Most of the existing studies on attosecond light sources focus on the time-domain control aspect, while our work covers the dual regulation in both time and frequency domains, which makes our study necessary and valuable.

5. Conclusions

This study examines how varying initial electron energies and phases influence radiation characteristics in tightly focused, linearly polarized laser fields. The findings demonstrate that precise control over the driving laser’s initial phase enables an effective manipulation of (1) the spatial profile of emitted radiation, and (2) key temporal and spectral properties. Notably, initial phase adjustment serves as a powerful tool for pulse structure modulation, enabling a selective generation of single- or double-peak temporal spectra through controlled phase variation. Meanwhile, the pulse width will appear to change periodically, synchronized with the laser half-period. In the influence on the spectrum, the phase modulation of the harmonic intensity can be realized, which is maximum at ${\varphi }_0=k\pi$ and almost disappears at ${\varphi }_0=(k+1/2)\pi$, enhancing the odd harmonic selectivity, and, at the same time, the reconstruction of the spectral shape is realized. In the linearly polarized laser field, the modulation of the electron initial energy in the harmonic order, intensity, spatial distribution, and time domain also has a significant effect on the harmonic radiation characteristics. As the electron’s initial energy increases, the harmonic intensity will rise sharply, and the time-domain pulse width is compressed; at the same time, the compression of the harmonic spatial distribution leads to a more collimated radiation of the high-energy electrons, which is well applied in high-resolution imaging.