Abstract
We numerically investigate nonlinear Thomson scattering from an initially stationary electron driven by a finite-waist, linearly polarized Gaussian pulse with quadratic spectral phase. The implemented vector potential includes the transverse component, the finite-waist longitudinal component, and the Gaussian temporal envelope used in the trajectory code. Positive and negative chirp produce similar spatial trajectory envelopes at equal , but they reverse the temporal ordering of the local phase rate and yield different peak powers and spectra. The linearly polarized field produces a two-lobed angular distribution of peak radiation power. For unchirped pulses, a peak-asymmetry metric calculated from decreases from at to at . At every sampled , either chirp sign reduces this imbalance, and all eight chirped cases give values not exceeding . Negative chirp produces the larger peak power and spectral intensity in the sampled parameter range. These results describe trends within the implemented classical single-electron model and are not quantitative predictions for a specific experiment.
1 Introduction
Accurate finite-waist laser fields are required to model relativistic laser–electron interactions beyond the plane-wave approximation []. High-energy scattering of focused pulses and Thomson-backscattered x-ray generation have been studied with relativistic electrons [, ]. Plasma-based and laser-wakefield accelerators provide compact electron sources for these schemes [–7]. These developments form part of the broader strong-field and relativistic-optics landscape [, ].
Laser chirp adds a time-dependent instantaneous frequency to the usual controls of pulse duration, waist, intensity, and polarization. Recent simulations show that chirp can modify nonlinear Thomson radiation intensity, bandwidth, and harmonic structure []. Related calculations have examined circular polarization [], pulse-width effects in tightly focused linearly polarized fields [], pulse-width effects for elliptically polarized fields [, ], and waist-dependent emission [15]. Recent experiments have also measured nonlinear Thomson emission in both hemispheres and exposed the velocity asymmetry of linearly polarized electron motion [16]. All-optical Thomson/Compton platforms now provide tunable radiation energy and polarization [17], while multi-petawatt experiments have reached the strongly nonlinear Compton regime [18].
Building on these chirp and finite-waist studies, we address which observables remain nearly invariant under chirp-sign reversal and which retain sign sensitivity as the pulse width changes. Linear polarization confines the dominant motion to the polarization plane and produces two angular lobes. We therefore compare negative, zero, and positive chirp using identical trajectory and radiation routines. The matched parameter scan separates the weak sign dependence of the large-scale trajectory envelope from the stronger sign dependence of peak temporal power and spectral intensity. It also quantifies how pulse width and chirp reduce the imbalance between the two angular lobes.
2 Theoretical framework
We use the finite-waist Gaussian-pulse construction of Refs. [, 19] and retain its transverse and leading longitudinal components. We take the central wavelength to be and use dimensionless variables and . Here, and , so in the numerical equations. To distinguish the carrier coordinate from the shifted envelope coordinate used in the program, we define the cooordinates as given in Equation 1where is the pulse-width index reported in the figures. The program uses the Gaussian amplitude
Here, is only a shifted envelope coordinate, whereas the carrier coordinate is . This notation reproduces the program exactly: the Gaussian envelope depends on , while the carrier and chirp phase depend on . The displacement is therefore applied only to the envelope and is not an inconsistent phase-coordinate shift. With the Gouy phase and auxiliary parameter defined in Equation 3
The two phases appearing in the code are given by Equations 4, 5.
At fixed position, the local normalized phase rate is
Changing the sign of therefore reverses the ordering of the local phase rate across the pulse.
The field expressions in Equations 2–7c follow the cited focused-pulse model, rewritten in the notation used by the numerical program. The program uses the dimensionless vector potential , whose assigned peak amplitude is . Its components are.
The calculation sets , giving linear polarization with a dominant component and a finite-waist longitudinal component. The program sets and in normalized units. These values correspond to and in physical units. The derived focusing ratio, calculated according to Equation 8, is
This value is derived from the two parameters assigned in the program and is not an independently adjusted input. Throughout this paper, “tightly focused” refers operationally to the implemented finite waist and the retained longitudinal vector-potential component.
The electron trajectory follows the relativistic Lorentz equation, which is expressed in the normalised form of Equation 9 [23].The energy evolution is described by Equation 10.where and the minus sign is the electron charge. Substituting and , then eliminating , gives the component equations integrated numerically.where , , and are the normalized velocity components. The far-field Lienard–Wiechert expression gives the radiated power per unit solid angle [23]:where is normalized by , is the observation direction, and is the emission time. The angular-radiation routine evaluates and at intervals. At each direction, it stores the integrated angular energy defined in Equation 13 and the peak temporal power defined in Equation 14.
Which correspond to the discretely integrated angular energy and the peak temporal power, respectively. Figure 1 uses and normalizes each panel by its own maximum. The direction that maximizes is subsequently used for the temporal and spectral calculations.
FIGURE 1
For the temporal profile, the program places the observer at and uses the retarded-time coordinatewhere is the electron position at emission. Fourier transformation of the same far-field amplitude gives the spectral energy density [23]:
Here, is normalized by , is the radiation frequency, and . The numerical spectrum routine samples and integrates the trajectory with the saved adaptive time steps.
2.1 Numerical parameters and scope
The electron starts at the origin with and , so its initial kinetic energy is zero. The program assigns , , , , and in normalized units. The choice places the interaction in the nonlinear regime while keeping the field amplitude fixed throughout the pulse-width and chirp scan. The chirp values , 0, and form a symmetric control set that isolates chirp-sign effects at fixed . These values define a numerical parameter study and are not fitted to a particular experiment. With , the waist and Rayleigh range are and , respectively. No physical peak intensity is quoted because the program specifies the normalized vector-potential amplitude directly. The stored parameter scan covers –12. Because the intensity envelope is proportional to , the intensity FWHM is . For , –12 corresponds to approximately 7.9–.
The coupled equations are integrated over with MATLAB’s adaptive ode45 solver. The relative tolerance is , and the absolute tolerance is for each of the seven state variables. The parameter set is a numerical scan rather than a fit to a particular experiment. Present Thomson/Compton facilities demonstrate control of laser intensity, collision geometry, radiation energy, and polarization [17, 18], but they generally use electron bunches rather than an isolated stationary electron. Our model therefore tests trends for the precise single-electron initial conditions implemented above.
3 Results and analysis
Figure 2 summarizes the collinear geometry. The pulse propagates along , its electric field is polarized along , and the stationary electron is initially located at the focus.
FIGURE 2
3.1 Evolution of the electron trajectory
Figure 3 shows that the trajectory lies in the polarization plane. At small , its envelope is approximately elliptical. Increasing extends the interaction along and produces a progressively longer trailing structure. This change follows from the slower longitudinal decay of the Gaussian envelope in Equation 2.
FIGURE 3
At equal , the and trajectories have similar large-scale envelopes, although their pointwise paths are not identical. In the program, reversing changes only the sign of the quadratic term while leaving the Gaussian amplitude unchanged. This produces comparable spatial envelopes but different phase ordering. Radiation observables remain sign-sensitive because Equation 12 uses the complete velocity and acceleration histories at retarded time. Similar trajectories in Figure 3 therefore do not require identical peak powers or spectra.
Table 1 states the panel mapping explicitly so that the chirp sign does not have to be inferred from the small labels inside Figure 3.
TABLE 1
| Chirp case | ||||
|---|---|---|---|---|
| Negative chirp | (a) | (b) | (c) | |
| Unchirped | 0 | (d) | (e) | (f) |
| Positive chirp | (g) | (h) | (i) |
Explicit parameter mapping for the trajectory panels in Figure 3.
3.2 Spatial distribution of electron radiation
Figure 1 shows two radiation lobes aligned approximately with the positive and negative observation directions. This orientation follows the trajectory plane in Figure 3. Similar two-hemisphere asymmetry has been traced experimentally to the velocity asymmetry of linearly polarized electron motion [16]. The rows of Figure 1 correspond to , 3, 6, and 9, while the columns correspond to , 0, and . The plotted quantity is the program output from Equation 14. Each primary surface is normalized as and uses the common vertical range 0–1. The inset in each panel is a vertical section through the two dominant lobes using the same normalized data. Figure 1 therefore compares angular shape and relative lobe height rather than absolute peak power between parameter sets.
To quantify the peak-height difference, we define the asymmetry metric A in Equation 17where and are the maxima of in the positive and negative hemispheres. These values were calculated directly from the stored Wsan2.txt arrays and are listed in Table 2. Without chirp, decreases monotonically from at to at . At every sampled pulse width, either chirp sign reduces by more than one order of magnitude relative to the unchirped case. All eight chirped cases satisfy . The insets make this reduction in the relative peak-height difference directly visible. Because is the maximum temporal power at each observation angle, this trend concerns the angular distribution of peak power rather than the integrated angular energy .
TABLE 2
| 0 | ||||
Peak-asymmetry metric calculated directly from the stored angular peak-power arrays used in Figure 1.
3.3 Temporal radiation characteristics
Figure 4 shows the radiated power versus the retarded observation-time coordinate in Equation 15. The observation direction is the direction that maximizes the integrated angular energy . The profiles contain multiple local maxima, and the chirped cases contain more closely spaced features than the unchirped case. This numerical difference accompanies the phase term and should not be interpreted as an independently measured spatiotemporal compression.
FIGURE 4
Figure 5a reports the maximum of each temporal profile rather than the complete waveform. It therefore compares the strongest instantaneous response across the parameter scan.
FIGURE 5
For the unchirped pulse, the stored temporal profiles show a decreasing peak radiated power as increases. The chirped profiles show the opposite trend over the sampled range.
The term in Equation 4 changes the phase accumulated across the envelope but does not change the Gaussian width assigned by Equation 2. Increasing lengthens the interval over which this phase modulation acts. The resulting peak-power trend is a numerical output of the coupled trajectory and radiation calculation rather than an independently isolated chirp mechanism.
Figure 5a also shows a sign dependence that is less apparent in the large-scale trajectory envelope. Equation 6 shows that changing the sign of reverses the temporal ordering of the local phase rate. The complete calculation gives a larger peak power for negative chirp in the sampled range. Because Equation 12 depends on both the acceleration numerator and the beaming denominator, the program output does not attribute this difference to acceleration alone.
Figure 5b shows that the full width at half maximum (FWHM) increases with without chirp but decreases in the chirped cases. The post-processing routine determines the half-maximum crossings around the global maximum by linear interpolation. Negative chirp gives the narrower principal feature in the sampled range.
3.4 Spectral characteristics
Figure 6 shows that increasing narrows and regularizes the unchirped spectrum. The chirped spectra broaden as increases, and the negative-chirp spectra are stronger than the positive-chirp spectra in the plotted cases. These curves are obtained by applying Equation 16 to the saved position, velocity, acceleration, and adaptive time-step arrays. The comparison therefore includes the full sign-dependent trajectory history and is not based on an assumed bandwidth formula.
FIGURE 6
3.5 Scope and limitations
The present work is a classical, deterministic, single-electron parameter study based on the vector potential in Equations 7a–7c. It does not include electron-bunch energy spread, emittance, space charge, focal-volume averaging, alignment jitter, detector response, radiation reaction, or quantum recoil. In plasma-material and electronegative-plasma systems, charge separation can form a sheath whose structure depends on the ion response, electron distribution, collisions, and ionization [20, 22]. Density gradients can further modify collective plasma dynamics [21]. In laser-matter interactions with a target or a background plasma, analogous charge-separation sheaths may therefore affect particle acceleration and subsequent radiation. The present vacuum, initially stationary single-electron model contains neither ions nor a material boundary, so it cannot describe sheath formation or its feedback on the electron trajectory and radiation.
No experiment was performed specifically to validate Figures 2–6. Recent nonlinear Thomson measurements and all-optical scattering platforms support the broader physical framework [16, 17], but they do not validate this stationary-electron configuration. A direct test would require otherwise identical pulses with opposite chirp, fixed pulse energy and waist, and simultaneous measurement of the two angular lobes and emitted spectrum. The numerical trends reported here should therefore be treated as outputs of the specified model rather than experimentally established optimization rules.
4 Summary
This study separates pulse-width and chirp-sign effects in focused nonlinear Thomson scattering from a stationary electron. The principal numerical findings are.
Increasing the pulse-width index causes the electron trajectory to evolve from an elliptical shape into a tadpole-like structure with a trailing tail, while chirp increases the oscillation density.
The angular distribution of peak radiation power has two lobes along the positive and negative directions. Without chirp, decreases from to between and . All eight displayed chirped cases have .
The temporal power profile contains multiple maxima, and chirp increases their density. Negative chirp produces a larger peak and a narrower temporal feature in the sampled range.
Increasing narrows the plotted unchirped spectra, whereas the plotted chirped spectra broaden. Negative chirp produces a larger spectral intensity than positive chirp in the sampled cases.
The main numerical result is that the large-scale trajectory envelope changes little when the chirp sign is reversed, whereas the peak power and spectral intensity remain sign-sensitive. This distinction is specific to the implemented single-electron classical model and provides a testable target for future comparisons.
Statements
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
JW: Writing – review and editing, Data curation, Writing – original draft, Conceptualization. ZW: Writing – review and editing, Supervision. JL: Writing – review and editing, Project administration. YT: Writing – review and editing, Resources.
Funding
The author(s) declared that financial support was received for this work and/or its publication. This work has been supported by the National Natural Sciences Foundation of China under Grant No. 10947170/A05, the Natural Science Foundation of Jiangsu Province under Grant No. BK20240611, the Natural Science Fund for Colleges and Universities in Jiangsu Province under Grant No. 10KJB140006, the Natural Science Foundation of Shanghai under Grant No. 11ZR1441300, and sponsored by the Jiangsu Qing Lan Project and STITP Project under Grant No. 202510293087Z.
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
angular radiation distribution, chirped laser pulse, linearly polarized laser, nonlinear thomson scattering, tight focusing
Citation
Wang J, Wang Z, Li J and Tian Y (2026) The effect of linearly polarized chirped laser pulses on the radiation characteristics of tightly focused nonlinear Thomson scattering. Front. Phys. 14:1847497. doi: 10.3389/fphy.2026.1847497
Received
04 April 2026
Revised
13 July 2026
Accepted
14 July 2026
Published
05 August 2026
Volume
14 - 2026
Edited by
Venugopal Rao Soma, University of Hyderabad, India
Reviewed by
Stefan Karatodorov, Luxembourg Institute of Science and Technology (LIST), Luxembourg
Rajat Dhawan, Maharishi Markandeshwar University, India
Updates
Copyright
© 2026 Wang, Wang, Li and Tian.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Jihong Wang, b23012109@njupt.edu.cn
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.