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 > 10
18 W/cm
2), 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 , with the temporal scale defined via the laser angular frequency (where c is the speed of light, ).
This study examines the propagation dynamics of Laguerre–Gaussian (LG) laser modes in linear dielectric media (
), 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:
where the vector potential A is the relevant solution of the Helmholtz equation, describing the distribution in the space in which it is located.
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
,
. 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.
Let
and
denote the normalized coordinates, where
is the beam waist radius along the z-axis. The radial distance
is normalized as
, and the angle of diffraction is characterized by
, with
= k
(
) being the Rayleigh length. Based on these parameters, E is given by the following equation:
where,
is the peak amplitude,
L is the pulse duration,
is the minimum spot size,
,
are shown below.
where,
, the initial phase
is constant.
and
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
to
.
is the phase related to the curvature of the wavefront.
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:
Here, is the electron momentum is the relativistic factor, is the laser electric field, and is the laser magnetic field, and is the electron velocity.
This work employs the normalized vector potential
to characterize the relativistic laser intensity, with the laser intensity I (W/cm
2) and the laser wavelength
(
). The spatial and temporal scales are governed by the normalized wavelength
and laser period T. The laser beam is tightly focused. The volume and intensity gradient of the interaction are controlled by
, and at the same time, a significant diffraction phase is triggered when
. Therefore, we take the laser beam waist spot
(corresponding to the laser intensity). When the laser pulse length
(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
. In addition, when the numerical aperture
, it will lead to phase distortion. Therefore, the numerical aperture (medium focusing) is taken as
(medium focus) when the phase is almost free of aberrations. Electrons with initial kinetic energy 11.4 MeV (
) are injected from
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.
Note that the subscript ‘ret’ indicates that all right-hand-side quantities are to be evaluated in the delay time , and denotes the displacement between the observation point and the interaction region. Here, is the position vector of the electron, the unit vector defines the direction of the observation in spherical coordinates, and θ and ψ are the polar and azimuthal angles, respectively. Additionally, there are , .
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
and the resultant radiation field at the detector position (defined by a unit vector
) are governed by the following:
The spectral angular energy density
emitted during electron-laser interactions is given by the following:
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 , velocity , acceleration ) are archived. Subsequent radiation analysis derives the spectral-angular distribution from Equations (11) and (12).
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 and almost disappears at , 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 . 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 and almost disappears at , 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.