Over the past few decades, laser technology has advanced rapidly, catalyzing the continuous expansion and deepening of the field of laser-matter interaction [1–3]. Nonlinear inverse Thomson scattering (NITS), as an essential high-quality X-ray source [4, 5], has garnered significant attention from researchers due to its diverse applications ranging from biomedicine to atomic physics [6–8].

NITS devices utilize high-power lasers and relativistic electron beams within a controlled interaction region. Extensive research has been conducted to explore the characteristics of NITS under various parameters, aiming to enhance the modulation of X-rays, as illustrated in Figure 1. Extensive research has explored the characteristics of NITS under various parameters, aiming to enhance the modulation of X-rays [9–11]. Chris Harvey et al. investigated temporal envelope and focusing effects in laser-electron Thomson scattering [12], while S. G. Rykovanov’s team utilized laser chirping to control spectrum broadening for high laser pulse intensities [13].

Previous studies have primarily focused on direct electron collisions with tightly focused laser pulses. However, achieving complete confinement of electron along the laser pulse axis under practical conditions presents significant challenges. Therefore, there is an urgent need to investigate and understand electron-laser interactions under off-axis conditions. Furthermore, there is a lack of research examining the impact of the laser beam waist radius on electron radiation properties, necessitating further investigation.

This paper provides an exploration into how the beam waist radius of a laser pulse influences electron dynamic properties, spatial radiated power characteristics, and spectral attributes during off-axis collisions within a circularly polarized tightly focused laser environment.

The findings demonstrate that the aforementioned characteristics are influenced by the laser beam waist radius, notably revealing distinct properties under ultra-tightly focused (UTF) conditions ( b 0 = 2   λ 0 ), which diminish or disappear as the beam waist radius increases. Under UTF ( b 0 = 2   λ 0 ), the electron experience pronounced asymmetrical forces at the beam waist, leading to considerable axial trajectory deviations, accelerated motion, and energy oscillations.

Observable asymmetries also manifest in the radiation’s spatial distribution, temporal spectrum, and angular distribution. A temporal spectral bias towards the x+axis and spatial spectral disparities around 130° and 230° are observed. Increasing the beam waist radius attenuates these asymmetries and reduces axial trajectory shift, electron acceleration, and energy oscillations along the z-direction. This is attributed to reduced laser intensity attenuation and reduced force disparities at electron positions with increased beam waist radius.

The remaining part of this paper is organized as follows: Section 2 derives analytical expressions encompassing the laser pulse vector potential, electron kinematics, radiation spectrum, and power factors, grounded in classical electrodynamics principles. Section 3 examines the influence of the laser beam waist radius on electron motion dynamics, spatial radiated power distribution, and radiation spectrum characteristics. In Section 4, we consolidate the impact of the laser beam waist radius on electron kinematics, spatial radiated power, and radiation spectrum. Furthermore, we explore methods to generate isolated narrow-second pulses with high signal-to-noise ratios by modulating the laser beam waist radius.

In this paper we introduce a Laguerre–Gaussian (LG) laser pulse propagating along the z-axis at a fluctuation angle σ i n = 0 based on the RNTS (relativistic nonlinear Thomson scattering) model. And the medium is isotropic, homogeneous, nonmagnetic and nonconducting.

Here the wavelength of the laser λ 0 = 1 μ m , c is the speed of light, ϵ is the dielectric constant, and m and e denote the mass and charge of the electron. The first thing to state is that for all the following formula definitions, the spatial and temporal coordinates are normalized by the wave number of the laser k 0 − 1 = λ 0 / 2 π and the frequency of the laser ω 0 − 1 = λ 0 / 2 π c . In a tightly focused Gaussian laser field, the electric field E and magnetic field B can be stated by the following [14]: E = ∇ × Λ (1) B = ϵ   i k ∇ ∇ · Λ + i k Λ (2) the above electric field E and magnetic field B satisfy Maxwell’s system of equations. And it is worth noting that Λ is the solution to the Helmholtz equation: ∇ 2 Λ + k 0 2 Λ = 0 (3)

Circularly polarized lasers and their electromagnetic fields can be decomposed into x-axis and y-axis linearly polarized lasers whose phase difference is π 2 , viz. E = E x p + E y p , B = B x p + B y p . Salamin et al. [15] and Zhang [16] derived Λ , B and E in the x-axis and y-axis linearly polarized laser field from Equations 1–3 respectively. By combining the aforementioned equations, the magnetic field components B = B x , B y , B z are described as Equations 4–6 [17]: B x = A L [ ϖ 0 0 + ε 2 r 2 ϖ 2 0 2 − r 4 ϖ 3 0 4 + ε 4 − ϖ 2 0 8 + r 2 ϖ 3 0 4 + 5 r 4 ϖ 4 0 16 − r 6 ϖ 5 0 4 + r 8 ϖ 6 0 32 (4) B y = A L ϖ 0 1 + ε 2 r 2 ϖ 2 1 2 − r 4 ϖ 3 1 4 + ε 4 − ϖ 2 1 8 + r 2 ϖ 3 1 4 + 5 r 4 ϖ 4 1 16 − r 6 ϖ 5 1 4 + r 8 ϖ 6 1 32 (5) B z = A L β ε ϖ 1 0 + ε 3 ϖ 2 0 2 + r 2 ϖ 3 0 2 − r 4 ϖ 4 0 4 + ε 5 3 ϖ 3 0 8 + 3 r 2 ϖ 4 0 8 + 3 r 4 ϖ 5 0 16 − r 6 ϖ 6 0 4 + r 8 ϖ 7 0 32 − A L σ ε ϖ 1 1 + ε 3 ϖ 2 1 2 + r 2 ϖ 3 1 2 − r 4 ϖ 4 1 4 + ε 5 3 ϖ 3 1 8 + 3 r 2 ϖ 4 1 8 + 3 r 4 ϖ 5 1 16 − r 6 ϖ 6 1 4 + r 8 ϖ 7 1 32 (6) and the electric field component E = E x , E y , E z can be derived as Equations 7–9: E x = A L { ϖ 0 1 + ε 2 ς 2 ϖ 2 1 − r 4 ϖ 3 1 4   + ε 4 ϖ 2 1 8 − r 2 ϖ 3 1 4 − r 2 r 2 − 16 α 2 ϖ 4 1 16 − r 4 r 2 + 2 α 2 ϖ 5 1 8 + r 8 ϖ 6 1 32   + ε 2 ϖ 2 0 + ε 4 r 2 ϖ 4 0 − r 4 ϖ 5 0 4 }   (7) E y = A L { ϖ 0 0 + ε 2 β 2 ϖ 2 0 − r 4 ϖ 3 0 4   + ε 4 ϖ 2 0 8 − r 2 ϖ 3 0 4 − r 2 r 2 − 16 β 2 ϖ 4 0 16 − r 4 r 2 + 2 β 2 ϖ 5 0 8 + r 8 ϖ 6 0 32   + ε 2 ϖ 2 1 + ε 4 r 2 ϖ 4 1 − r 4 ϖ 5 1 4 }   (8) E z = A L σ { ε ϖ 1 0 + ε 3 − ϖ 2 0 2 + r 2 ϖ 3 0 − r 4 ϖ 4 0 4   + ε 5 − 3 ϖ 3 0 8 − 3 r 2 ϖ 4 0 8 + 17 r 4 ϖ 5 0 16 − 3 r 6 ϖ 5 0 16 − 3 r 6 ϖ 6 0 8 + r 8 ϖ 7 0 32 }   − K L β { ε ϖ 1 1 + ε 3 − ϖ 2 1 2 + r 2 ϖ 3 1 − r 4 ϖ 4 1 4   + ε 5 − 3 ϖ 3 1 8 − 3 r 2 ϖ 4 1 8 + 17 r 4 ϖ 5 1 16 − 3 r 6 ϖ 5 1 16 − 3 r 6 ϖ 6 1 8 + r 8 ϖ 7 1 32 }   (9) where fifth-order expansion of the electromagnetic field accurate to the diffraction angle ε = w 0 / z r , and σ = x / w 0 , β = y / w 0 , r = ρ / w 0 , w 0 represents the minimum waist radius. A L can be described as follows: A L = a 0 w 0 w exp − η 2 L 2 exp − ρ 2 w 2 (10) when the laser at z , the waist radius w = w 0 1 + z f 2 z 2 , z r = w 0 2 / 2 indicates the Rayleigh distance, η = z − t , ρ = x 2 + y 2 denotes the perpendicular distance, and a 0 indicates the peak amplitude, whose relationship between laser intensity is a 0 = I λ 0 2 / 1.37 × 10 9 .

The electromagnetic field is fifth-order expanded accurate to the diffraction angle ε = w 0 / z r , among them, ϖ n 0 and ϖ n 1 are shown as Equation 11: ϖ n i = w 0 w n cos φ + n φ G + i π 2   n = 0 , 1 , 2 , … (11) φ = η + φ R − φ G + φ 0 , where φ R = x 2 + y 2 2 R z , R z = z 1 + z f 2 z 2 and φ G = tan − 1 z z f . φ R is the phase associated with the curvature of the wave fronts, and that R z is the radius of a curvature of a wave front intersecting the beam axis at the coordinate z . φ G is the Guoy phase associated with the fact that a Gaussian beam undergoes a total phase change of π as z changes from − ∞ to + ∞ , φ 0 is the initial phase of the laser pulse, which is determined by the pulse. φ 0 is different from the initial phase that electron experiences when it enters the field.

In the course of the calculation, the focal point of the laser pulse was established as the origin, and the electron move from { d 0 , 0 , z 0 } toward the laser, where z 0 represents the enough far away position, and d 0 denotes the initial off-axis position of the electron.

The following Lorentz and energy (Equations 12, 13) can be used to compute the momentum of the electron in an intense laser pulse: d p d t = − e E + u c × B (12) d Γ d t = − e v · E (13)

In this context, Γ = γ m c 2 represents the electron energy, which is defined in terms of the Lorentz factor γ = 1 − v / c 2 − 1 / 2 . The momentum p is given by the relation p = Γ u / c 2 , v is the electron velocity, and u = v / c .

When an electron is in relativistic motion, it emits radiation. The radiated power or energy per unit solid angle is given below [18, 19]: P Ω = d P t d Ω = n × n − u × u 2 1 − n ⋅ u 6 t ′ (14)

Here, the power radiated per unit solid angle P Ω is normalized. The direction of radiation represented by n = sin ⁡ θ ⁡ cos ⁡ ϕ , ⁡ sin ⁡ θ ⁡ sin ⁡ ϕ , ⁡ cos ⁡ θ , is specified in terms of the polar angle θ relative to the laser movement direction. And ϕ denotes the azimuth in the plane perpendicular to the origin. The time at which the electron interacts with the laser pulse t ′ , can be expressed as t = t ′ + R , where  R ∼ R 0 − n · r . R 0 denotes the distance from the origin to the observer, r is the position vector of the electron.

Lee et al. proposed power factor (Equation 15) as a means of characterizing the peak power radiated by the electron in RNTS [20]: F a c t o r = d u x d t 2 + d u y d t 2 1 − u 2 4 (15)

The equation for the radiant energy per unit steradian angle per unit frequency interval during the interaction of electron with a laser pulse can be expressed as follows [21]: d 2 I d ω d Ω = ∫ − ∞ ∞ n × n − u × u 1 − n ⋅ u 2 e i s t − n ⋅ r d t 2 (16)

The normalized expression d 2 I d ω d Ω is derived by employing the constant e 2 4 π 2 c , as well as the harmonic frequency ratio s = ω s ω 0 , ω s is the frequency of the harmonic radiation. Solving Equations 14, 16 gives the full time, space, and spectral characteristics of electron harmonic radiation.

In our observations, we directed our focus towards the radiation emanating from a precisely defined sphere with a radius of 1 m, positioned at the origin of the coordinate system. The peak amplitude of the laser pulse a 0 = 5 , while the pulse duration L is tuned to 6.6 fs. The initialization of the phase φ 0 = 0 . Initially, the electron is endowed with an energy of 5 (equivalent to 2.56 MeV), which is denoted as γ 0 , and its initial off-axis distance d 0 = 0.4 , moving in the opposite direction of the z-axis. The electron can be accelerated to this energy level by a linear accelerator. In view of the characteristics of circularly polarized laser pulses, at the radial off-axis angle α 0 , the effect of the pulse with initial phase φ 0 on the electron is equivalent to the effect of the pulse with initial phase φ 0 + β 0 − α 0 at the radial off-axis angle β 0 , which means that the effect of the off-axis position of the electron on their collision with the off-axis of the laser pulse can be adjusted by the initial phase φ 0 of the laser pulse. Thus, the point of collision of the electron with the laser pulse, namely, the maximum value of the amplitude of the electron motion, is located on the x-axis of the Cartesian coordinate system. In our discussion, the peak radiated power per unit solid angle will be formally designated as d P / d Ω . It should be emphasized that every numerical datum described here has its genesis in the unchanging foundation of physical principles and equations outlined in Section 2.

In this investigation, we carefully explore the effect of laser beam waist radius variations on the radiation properties in nonlinear Thomson scattering, particularly as it pertains to the interaction with off-axis electron. In practical experimental setups, the prevalent scenario involves electron engaging in off-axis collisions, underscoring the profound practical relevance of our study. Under conditions of UTF, denoted by b 0 = 2   λ 0 , electron are exposed to a pronounced asymmetry in the qualitative force along the x+axis and x-axis directions at the laser beam waist. This imbalance leads to a substantial axial deviation in electron trajectory, accompanied by notable instances of electron acceleration and energy oscillations.

Simultaneously, significant asymmetry emerges in the spatial distribution of the electron-radiated power, the temporal spectrum of the electron-radiated pulse, and the angular distribution of the spatial radiation spectrum. Specifically, the radiated pulse power within the temporal spectrum exhibits a marked bias towards the x+axis direction. Moreover, the spatial spectrum’s asymmetry manifests in the disparities observed in spectral peak radiation intensity and harmonic peak frequency at angles around 130°and 230° when Φ coincides with the peak radiated power Φ,and when θ serves as the peak radiant power reference, the spectral distribution showcases evident asymmetry.

With an increase in the beam waist radius b 0 , the aforementioned asymmetries in spatial distribution, temporal spectrum, and angular distributions of spatial radiation spectra, are gradually weakened. At the same time, the magnitude of axial offset in off-axis electron trajectory diminishes, as well as the acceleration of the electron and the energy oscillations along the Z-direction. These phenomena are attributable to the decrease in the laser intensity attenuation after the beam waist radius is increased, coupled with a reduction in the disparity between forces acting along the x+axis and x-axis directions at the specified electron position.

In summary, the manipulation of the laser beam waist radius, particularly when b 0 equals or exceeds 10 λ 0 , is of great significance for the generation of isolated narrow-second pulses boasting remarkable signal-to-noise ratios. This optimization bears greater practical value for investigations within the realms of ultrashort and ultrafast optics. Discerning the laser beam waist radius’s impact on radiation properties in nonlinear Thomson scattering of off-axis electron constitutes a pivotal stride towards utilizing nonlinear inverse Thomson scattering radiation (NITS) as an important source of radiation for both scientific research and practical applications, such as those in cancer therapy. By judiciously fine-tuning the laser beam waist radius, researchers can achieve an optimal balance between electron energy and radiation power for their specific experimental needs.