3.1 X-ray STOV emissions via nonlinear Thomson scattering

The emitted harmonics preserve coherent, mode-resolved STOV field structures, with the vortex charge scaling linearly with the harmonic order. The radiation is tightly collimated (approximately 4 mrad) in the forward direction. To examine the harmonic contents, we perform a Fourier transform of the full radiation field to obtain the spatial spectrum at $\varphi =0$ ; see Figure 2. It should be noted that this spatial spectrum is completely different from spatiospectral optical vortex (SSOV) pulses^[ Reference Martín-Hernández, Gui, Plaja, Kapteyn, Murnane, Liao, Porras and Hernández-García^{45 ]}, which represent an independent category of structured vortex light with intrinsic spatiospectral correlation. In the moderate-intensity regime ( ${a}_0=1$ ), the fundamental-frequency radiation governs the radiation field at a central frequency of ${\omega}^{\prime}\approx 93{\omega}_0\sim 4{\gamma}^2{\omega}_0$ , corresponding to the Doppler-upshifted fundamental-frequency emission (labeled ‘1st’ in Figure 2(a)). Two distinct tilted stripes displayed in the spatial spectrum reveal the STOV spectrum nature of the radiation. Weaker features marked by the dashed red box near ${\omega}^{\prime}\approx 242{\omega}_0$ indicate the presence of third-harmonic emission (labeled ‘3rd’ in Figure 2(a)). The stripe number satisfies $n=l+1$ , where $l$ is the TC of the STOV field. The third-harmonic signal, labeled ‘3rd’ in Figure 2(a), exhibits four tilted stripes (also see Figure 5(c) detailed later), which corresponds to an effective TC of $l=3$ , that is, three times the pump STOV charge ( $l=1$ ). This harmonic-dependent multiplication of the TC is exactly what is expected from angular-momentum conservation in the nonlinear upconversion process. The observed redshifts are attributed to relativistic transverse Doppler effects at ${{a}_0=1}$ ^[ Reference Sakai, Pogorelsky, Williams, O’Shea, Barber, Gadjev, Duris, Musumeci, Fedurin, Korostyshevsky, Malone, Swinson, Stenby, Kusche, Babzien, Montemagno, Jacob, Zhong, Polyanskiy, Yakimenko and Rosenzweig^{70 ]}. Figure 2(b) shows the corresponding one-dimensional spectrum at $\theta =0$ , $\varphi =0$ on a logarithmic scale, confirming the presence of the fundamental, third and fifth (with a central frequency of around 394 ${}{{\omega}_0}$ ) harmonics, with the frequencies extending into the X-ray regime. Notably, no significant harmonic content is observed in the ${E}_x$ and ${E}_z$ components, indicating that the X-ray STOV emissions are linearly polarized along the $y$ -direction. A reversal of the tilted-stripe orientation (i.e., flipping) is both theoretically expected and numerically verified when the TC sign is switched from +1 to –1 (not shown here). This behavior originates from the fact that reversing the TC sign changes the handedness of the STOV phase vortex and thereby flips the sign of the space–frequency correlation in the spatial spectrum, leading to the reversed orientation of the tilted-stripe pattern in the spatiospectral representation.

Figure 2

Full field spectrum. (a) Two-dimensional spatial spectrum of the emitted field ${\mathrm{d}}^2I$ /d $\theta$ d $\omega$ (arb. units) at $\varphi =0$ versus radiation frequency $\omega \kern0.1em \left({\omega}_0\right)$ and polar angle $\theta$ (rad). (b) One-dimensional full spectrum at $\theta =0$ and $\varphi =0$ , shown on a logarithmic scale. ‘1st’, ‘3rd’ and ‘5th’ represent the fundamental, third- and fifth-harmonic radiation, respectively. The central frequencies of the fundamental, third and fifth harmonics are approximately 93 ${\omega}_0$ , 242 ${\omega}_0$ and 394 ${\omega}_0$ , respectively.

We analyze the radiation fields at the fundamental, third and fifth harmonics to characterize their ST field structures and TOAM. For the fundamental-frequency emission (labeled ‘1st’ in Figure 2), the ST amplitude profile in the $t-\theta$ plane exhibits a well-defined fork-shaped pattern with an emission angle of around 4 mrad (Figure 3(a)). Here, ${t}_{\mathrm{s}}$ is the recorded starting time and is set as 3.3356443 $\times {10}^{-9}$ s ( $\approx R/c$ ). We define $t-{t}_{\mathrm{s}}$ as the detection time axis. This distribution results in a null-intensity region with the center temporal position of around 0.85 fs (Figure 3(b)). The spatial spectrum exhibits two tilt interference fringes (Figure 3(c)) due to the interference between the radiation bursts across the singularity. The ST phase profile reveals a single-fork dislocation (Figure 3(d)), which is indicative of an edge-type phase singularity^[ Reference Ge, Liu, Xu, Long, Tian, Liu, Lu and Chen^{71 ]}. After removing the carrier term $\exp \left(i{\omega}^{\prime }t\right)$ (with ${\omega}^{\prime }$ of the central frequency of the target harmonic emission), the phase exhibits a spiral phase structure (Figure 3(e)), analogous to that of a conventional spatial vortex beam. The azimuthal phase variation spans $2\pi$ , indicating a vortex charge of 1. To quantitatively confirm this charge, we perform a TOAM modal decomposition using an optical coordinate transformation implemented via an OAM mode-sorting system within the entire complex radiated field^[ Reference Shangguan and Zheng^{72 ]}. The core principle of this OAM mode sorting involves two key steps: (1) mapping the helical phase of the vortex beam to a transverse phase gradient; (2) focusing the transformed OAM pulses with a lens to OAM mode-related positions. This spatial separation enables different OAM modes to be directed into respective TC channels on the modeled charge-coupled device (CCD) detector. For an STOV field, we utilize an OAM mode-sorting system to retrieve the OAM mode spectrum after removing its carrier term, then the resulting mode spectrum (Figure 3(f)) is obtained, which exhibits a dominant energy peak at $\mathrm{TC}$ of 1, consistent with a unit vortex charge. In addition, we calculate the intrinsic TOAM from ${L}_z={\left(\vec{r}\times \vec{S}\right)}_z/c$ , where the Poynting vector is defined as $\vec{S}=c\left(\vec{E}\times \vec{B}\right)/\left(4\pi \right)$ ^[ Reference Sztul and Alfano^{73 –} Reference Katoh, Fujimoto, Kawaguchi, Tsuchiya, Ohmi, Kaneyasu, Taira, Hosaka, Mochihashi and Takashima^{75 ]}. The resulting intrinsic TOAM value of fundamental-frequency emission is $0.93 \mathrm{\hslash}$ per photon, further validating the STOV pulse.

Figure 3

Fundamental-frequency (1st) STOV radiation: (a) ST amplitude profile with a fork-like dislocation; (b) ST intensity profile with an intensity null; (c) spatial spectrum with two tilted interference fringes; (d) ST phase distribution with the carrier term resembling a single-fork pattern; (e) ST phase distribution without a carrier term presenting a spiral phase structure; (f) TOAM mode spectrum with the dominant energy peak at TC of 1, indicating the vortex charge of 1. Here, ${t}_{\mathrm{s}}$ is the recorded starting time and is set as 3.3356443 $\times {10}^{-9}$ s on the detective time axis.

A pronounced time asymmetry appears around the ST singularity in the intensity map (Figure 3(b)). This time asymmetry originates from the $y$ -dependent carrier-envelope phase (CEP) of the STOV field. In particular, for electrons initialized at negative $y$ -positions, the ST field imposes an additional field oscillation (marked by the red arrow in Figure 4(c)), which induces an abrupt phase transition in their trajectories. This phase transition disrupts the symmetry of the electron’s oscillatory motion and produces a dual-envelope trajectory structure with a corresponding CEP shift (Figure 4(a)). In contrast, electrons at positive $y$ -positions undergo uninterrupted field-driven oscillations without phase discontinuities, retaining a single-envelope trajectory with a constant CEP (Figure 4(b)). This pronounced CEP asymmetry between the two sides about $y=0$ directly creates differential transverse-momentum kicks for off-axis electrons. As illustrated in Figure 4(c), electrons starting at negative $y$ -positions experience the aforementioned extra field oscillation (red arrow); this final kick partly cancels the accumulated transverse momentum, resulting in a diminished net deflection (green/black/yellow curves). Conversely, electrons at positive $y$ -positions interact with the final oscillation, whose phase is consistent with the ongoing oscillatory motion, causing the momentum kicks to add coherently and yielding an enhanced net deflection (purple/blue/red curves). Electrons at different $y$ positions become increasingly inhomogeneous toward the pulse tail due to the nonuniform momentum kick, which in turn degrades the quality of the X-ray STOV generated over the longer time scale. Upon coherent superposition of the Liénard–Wiechert fields, these asymmetric momentum kicks manifest as a time-asymmetric radiation pattern about the singularity (see Figure 3(b)).

Figure 4

Transverse trajectory characteristics of electrons. (a) Typical dual-envelope electron trajectory for an electron initialized at a negative $y$ -position ( ${y}_0=-0.6858\;\mu \mathrm{m}$ ) with the curve shifted upward to starting at the $y$ = 0 axis. (b) Typical single-envelope electron trajectory initialized at a positive $y$ -position ( ${y}_0=+0.6858\;\mu \mathrm{m}$ ) with the curve shifted downward to starting at the $y$ = 0 axis. (c) Transverse trajectories $y(t)$ ( ${\lambda}_0$ ) for electrons launched at distinct off-axis initial $y$ -positions ( ${y}_0$ ) in the STOV field. Color labels for ${y}_0$ : $+0.6875\ \mu \mathrm{m}$ (red), $+0.6858\ \mu \mathrm{m}$ (blue), $+0.6840\ \mu \mathrm{m}$ (purple), $-0.6840\ \mu \mathrm{m}$ (green), $-0.6858\ \mu \mathrm{m}$ (black) and $-0.6875\ \mu \mathrm{m}$ (yellow). The blue–red background indicates the local STOV laser field. Electrons starting at ${y}_0<0$ encounter an additional small field lobe (red arrow) whose sign opposes the preceding motion.

In addition, the asymmetric TOAM modal distribution observed in Figure 3(f) is a direct consequence of the synergistic effect of two distinct factors: the temporal asymmetry of the X-ray STOV field and the intrinsic multimode nature of the STOV-driven radiation. On the one hand, the TOAM of the X-ray STOV field is fundamentally governed by the ST phase winding. The STOV phase (Equation (2)) exhibits a non-ideal linear variance along the azimuthal coordinate, directly leading to an uneven population of distinct TOAM modes. Specifically, the non-ideal ST vortex phase – a direct consequence of temporal asymmetry – disrupts the symmetric distribution of angular momentum across modal orders. On the other hand, the emitted field is not a single pure-OAM mode, due to the broadband and intrinsic multimode character of the STOV-driven radiation from the electron beam with the finite transverse size as well as the nonuniform density distribution. Besides the dominant OAM component, several weaker neighboring satellite modes are naturally generated during the STOV-driven radiation process, reflected in the asymmetry of the OAM modal spectra (see Figures 3(f), 5(d) and 6(d)). Meanwhile, these weaker OAM components further contribute to the unacknowledged electric field dislocations (see the regions outside the black circles in Figures 5(a) and 6(a)).

Figure 5

Third-order (3rd) STOV harmonic emission. (a) ST amplitude profile with three distinct fork-shaped dislocations. The dashed black lines in the black circle highlight the fork dislocations. (b) ST intensity profile with three single-intensity nulls. The red arrows indicate the nulls. (c) Spatial spectrum with four tilted interference fringes. The dashed black lines mark the fringes. (d) TOAM mode spectrum with the dominant energy peak at TC of 3, confirming the vortex charge of 3.

Figure 6

Fifth-order (5th) STOV harmonic emission. (a) ST amplitude profile with five distinct fork-shaped dislocations. The dashed black lines in the black circle highlight the fork-shaped dislocations. (b) ST intensity profile with five single-intensity nulls. The red arrows indicate the nulls. (c) Spatial spectrum with six tilted interference fringes. The dashed black lines mark the fringes. (d) TOAM mode spectrum with the dominant energy peak at TC of 5, confirming the vortex charge of 5.

The fundamental emission exhibits a partial loss of coherence and mode purity (Figure 3(f)). This degradation originates from both ST phase mismatches and spatial asymmetry of the radiation, due to the random spatial distribution of an electron beam ensemble across the on- and off-axis positions. Sampling different phase structures of the driving field, electrons follow distinct dynamics (see Figure 4) and emit radiation with different spatial and phase characteristics^[ Reference Yang, Wang and Tian^{76 ,} Reference Pastor, Álvarez-Estrada, Roso, Guasp and Castejón^{77 ]}. Because the emissions add coherence, the inter-emitter separations during interaction with the STOV pulses modulate the resultant radiation field, leading to asymmetric, non-ideal harmonic wavefronts and reduced mode purity^[ Reference Wu, Meyer-ter Vehn, Fernández and Hegelich^{64 ,} Reference B. e Sá, Sun and Peatross^{78 ]} (see Section 3.2).

These effects are further exacerbated in the higher-order harmonics, exhibiting enhanced coherence loss resulting from the smaller wavelengths. Despite this, the fundamental characteristics of STOV light persist in third- and fifth-order harmonic emissions, which fall in the X-ray spectral region, with the central wavelengths of approximately 4.1 and 2.5 nm, respectively (see Figures 5 and 6). Focusing on the primary intensity region, the ST amplitude profile of the third harmonic reveals three distinct fork-shaped dislocations in the $t$ – $\theta$ plane (Figure 5(a), dashed black lines in the black circle), each corresponding to a null-intensity region in the ST intensity distribution (Figure 5(b), red arrows). In our relativistic scattering configuration, the vortex features of the high-frequency emission are encoded in the ensemble of electron trajectories and their associated Liénard–Wiechert radiation. In the parameter regime considered, this yields a quasi-elliptical angular distribution at high frequencies – an outcome fully consistent with the radiation mechanism of accelerated charges, and one that does not contradict the well-established propagation behavior of freely propagating STOV beams. The spatial spectrum displays four tilted interference fringes (Figure 5(c), dashed black dashed), and the TOAM mode spectrum (Figure 5(d)) shows a dominant energy peak at the TC of 3. These features collectively confirm an ST vortex charge of 3, with a calculated intrinsic TOAM value of approximately $2.11\mathrm{\hslash}$ per photon.

For the fifth-harmonic emission, the ST amplitude profile displays five fork-shaped dislocations (Figure 6(a), dashed black lines in the black circle), corresponding to five null-intensity regions in the ST intensity distribution (Figure 6(b), red arrows) and six tilted interference fringes in the spatial spectrum (Figure 6(c), dashed black lines), all indicating the vortex charge of 5. The transverse OAM mode spectrum (Figure 6(d)) shows the very small energy peak at TC of 5, suggesting a significant coherence loss. The calculated intrinsic TOAM value is approximately $3.57 \mathrm{\hslash}$ per photon. Overall, the TOAM scales approximately linearly with the harmonic order, in agreement with angular-momentum conservation law.

3.2 Impact of the electron beam on coherent STOV radiation

To enable both qualitative and quantitative analysis of ST coherence, we first simulate the radiation fields from two individual electrons located at distinct positions: one on-axis and the other off-axis. Figure 7(a) illustrates the geometry for ST coherence analysis. Here, electron $A$ is positioned on-axis, while electron $B$ is located off-axis with a transverse displacement $h$ . For the radiation emitted toward an observation angle ${\theta}_{\mathrm{coh}}$ , the optical path difference between the two radiation paths is $\Delta =h\sin {\theta}_{\mathrm{coh}}$ . Constructive interference occurs when $\Delta$ becomes comparable to the fundamental-frequency radiation wavelength, approximately ${\lambda}_0/\left(4{\gamma}_0^2\right)$ . This yields an emission angle of ${\theta}_{\mathrm{coh}}\sim {\lambda}_0/\left(4{\gamma}_0^2h\right)$ . For an off-axis displacement of $h={w}_y\sim 2.5\kern0.22em \mu \mathrm{m}$ , this corresponds to an angle of $\theta \approx 4$ mrad, in agreement with simulation results. Figure 7(b) shows the radiation field amplitude from electron $A$ ( $h=0$ ). The field is symmetric with respect to $\theta =0$ , and a null-intensity region appears near 0.86 fs, consistent with the temporal position of the intensity minimum observed in Figure 3 (around 0.85 fs). Figures 7(c) and 7(d) display the radiation fields from electrons $B$ and $C$ located symmetrically at $h=\pm 1.23{\lambda}_0$ . The black dashed rectangles highlight the analysis window with the angular range from –0.05 to 0.05 rad and the detection time from 0.5 to 1.0 fs. Unlike electron $A$ , the off-axis electrons ( $B$ and $C$ ) exhibit asymmetric radiation patterns. For electron $B$ , the peak radiation occurs at approximately 0.59 fs, whereas for electron $C$ , the peak shifts to around 0.68 fs, indicating the radiation fields carry distinct ST phases. When coherently superimposed, the radiation fields achieve temporal and spatial overlap with a specific ST phase matching relation^[ Reference Pausch, Debus, Huebl, Schramm, Steiniger, Widera and Bussmann^{59 ]}. Otherwise, phase mismatches arise. The coherence is determined by the ST phase correlation, the emission angle and the radiation frequency of electron emitters^[ Reference Pausch, Debus, Huebl, Schramm, Steiniger, Widera and Bussmann^{59 ]}. From the highlighted insets in Figures 7(c) and 7(d), a manual count reveals that the radiation field from electron $C$ contains 10 visible cycles, while that from electron $B$ has 9, indicating one additional period for the former. This difference naturally gives rise to a fork-shaped dislocation when the radiation fields interfere coherently^[ Reference Zhang, Ji and Shen^{44 ]}, demonstrating that well-defined STOV structures in the harmonics originate from coherent superposition of radiation fields emitted by individual electrons.

Figure 7

ST coherence analysis. (a) Coherence geometry of the radiation from two individual electrons: electron $A$ is located on-axis and electron $B$ is positioned off-axis; ST amplitude profiles for (b) electron $A$ ( $h$ = 0), (c) electron $B$ ( $h$ = 1.23 ${\lambda}_0$ ) and (d) electron $C$ ( $h$ = –1.23 ${\lambda}_0$ ), respectively. The insets in (c) and (d) are the highlighted analysis windows with the angular range from –0.05 to 0.05 rad and the detection time from 0.5 to 1.0 fs.

Apart from the positions of electron emitters, the transverse envelope of the electron beam, parameterized by the rms widths $\left({\sigma}_{y,z}\right)$ and the effective radius ( $r$ ), exerts a decisive influence on the emitted field. To preserve the STOV phase topology within the interaction volume, ${\sigma}_{y,z}$ must be carefully controlled: when ${\sigma}_{y,z}\lesssim 2\ \mu \mathrm{m}$ , truncation of the STOV wavefront leads to a dipole-like two-lobe pattern for the fundamental-frequency emission intensity. At the opposite extreme, when ${\sigma}_{y,z}\gtrsim 5\ \mu \mathrm{m}$ , the effective collisions with the STOV pulse decrease, resulting in reduced overall brightness and broader angular distribution. This broadening arises from sampling stronger transverse phase gradients (including $y$ -dependent CEP and local incidence), which increases the dispersion of terminal transverse momentum and inter-emitter phase difference (i.e., phase mismatches), thereby widening the ensemble radiation profile.

Furthermore, the temporal coherence of the emitted fields is limited by the longitudinal thickness $\Delta x$ of the relativistic electron beam, which can be modeled as multiple electron sheets. For radiation at wavelength ${\lambda}_m$ (the $m$ th harmonic), electron sheet emitters separated by $\Delta x$ acquire a phase slip:

(11) $$\begin{align}{\Delta \Phi}_m\simeq \frac{2\pi \kern0.1em \Delta x}{\lambda_m}\kern1em \left(\mathrm{at}\;\mathrm{fixed}\;{\omega}_m,\ \theta \right),\end{align}$$

so coherent superposition requires $\Delta x\lesssim \mathcal{O}\left({\lambda}_m\right)$ . To illustrate this, we compare two otherwise identical electron sheets launched at ${x}_{\mathrm{col}}=0$ and ${x}_{\mathrm{col}}=5\ \mathrm{nm}$ . Their field amplitude profiles are nearly identical except for a time translation $\Delta t\approx \Delta x/c\approx 17\ \mathrm{as}$ (Figure 8), which is close to half the fundamental-frequency radiation period ( ${T}_{1\mathrm{st}}\approx {t}_{\mathrm{p}}/93$ ). The on-axis waveforms are nearly $\pi$ out of phase and interfere destructively. For a finite-thickness beam with electrons distributed over $0\le x\le \Delta x$ , the emission phases populate an interval of width ${\Delta \Phi}_m$ ; as $\Delta x$ grows, partial cancellation reduces fringe contrast and degrades coherence. Because ${\Delta \Phi}_m\propto 1/{\lambda}_m$ , higher harmonics (shorter ${\lambda}_m$ ) are more sensitive and demand smaller $\Delta x$ for constructive interference. In our simulations, the fifth harmonic (central wavelength $\sim 2.5\ \mathrm{nm}$ ) is strongly suppressed once $\Delta x\gtrsim 3\ \mathrm{nm}$ , and the third harmonic is nearly extinguished for $\Delta x\gtrsim 5\ \mathrm{nm}$ . The coherence superposition with the increased phase difference (i.e., phase mismatches) at larger $\Delta x$ results in broader angular distribution, as evidenced by the more diffuse third-harmonic amplitude pattern observed for $\Delta x=5$ nm (see the comparison of third-harmonic emissions in Figure 9).

Figure 8

Amplitude profiles for two electron sheet emitters launching at ${x}_{\mathrm{col}}=0$ and 5 nm, respectively. The solid black line and dashed red line are the on-axis waveforms of the two cases, with a time translation of $\Delta t\approx 17$  as.

Figure 9

Comparison of third-order harmonic amplitude fields for the electron beam thickness of (a) 3 nm and (b) 5 nm.

Beyond spatial constraints, the initial electron energy also influences the ST field structure visibility of harmonics. A higher Lorentz factor $\gamma$ yields shorter radiation wavelengths and correspondingly tighter coherence requirements. When $\gamma =6$ , again only the fundamental and third-order harmonics with well-defined ST field structures are observed, with central frequencies of approximately $133\kern0.1em {\omega}_0$ and $350\kern0.1em {\omega}_0$ , respectively. The STOV structures resemble those of $\gamma =5$ , but the associated emission angle narrows to around 3 mrad (not shown here). In addition, increasing the spatial or temporal beam extent promotes multi-band radiation structures^[ Reference Rykovanov, Geddes, Schroeder, Esarey and Leemans^{79 ]}, introducing frequency-dependent phase differences between the pre- and post-singularity bursts and further weakening coherence. Collectively, these effects attenuate interference in the spatial spectrum and reduce the modal purity of high-frequency emissions.

The ST field structures and the TOAM mode analysis show that the visibility of the fork-shaped dislocations and the associated intensity nulls is governed by coherence. In particular, the ensemble-averaged TC serves as a coherence indicator: the higher the coherence, the closer $\left\langle \mathrm{TC}\right\rangle$ approaches the harmonic order $m$ . Figure 10 presents the $\left\langle \mathrm{TC}\right\rangle$ of STOV harmonics versus electron beam thickness for $\gamma =5$ and $\gamma =6$ . For each given harmonic order and each $\gamma$ , increasing the electron beam thickness always reduces the ensemble-average TC and moves it away from the ideal harmonic value, consistent with TC acting as a coherence indicator. Compared with the fundamental-frequency emission, the third- and fifth-harmonic components display weaker coherence, as reflected by their $\left\langle \mathrm{TC}\right\rangle$ values being substantially lower than their respective harmonic orders. The degradation at $\gamma =6$ is particularly pronounced, especially for the fifth-harmonic emission (see the crossing phenomenon between the black and red dashed lines for the 5-nm-thick electron beam). The crossing simply occurs between different harmonics at $\gamma$ = 6: at large thicknesses, the TCs of the third and fifth harmonics are both significantly reduced and become numerically very close, leading to a slight reversal in their ordering. In this regime, all TC values are already far below the ideal harmonic orders; thus, the ensemble-averaged TC should be interpreted as a qualitative, comparative measure of coherence rather than a strictly monotonic metric across all harmonics. This observation confirms that the process with higher $\gamma$ values experiences accelerated coherence loss when the thickness increases, thereby validating the coherence-dependent behavior established in prior analysis.

Figure 10

Relationship between the $\left\langle \mathrm{TC}\right\rangle$ value and electron beam thickness for STOV harmonic radiation when $\gamma =5$ and $\gamma =6$ .

It is important to note that the number of electrons employed in the simulations is significantly lower than that in realistic experimental conditions, which limits the overall radiation efficiency. Nevertheless, the simulations indicate that increasing the electron number promotes the enhancement of ST coherence, as a greater fraction of electrons remains confined within the coherence volume. Consequently, the radiation efficiency exhibits proportional increases accordingly: the field amplitude approximately doubles per additional $2\times {10}^4$ electrons. These results establish quantitative insight into coherence-driven emission scaling and furnish theoretical foundations for experimental validation.

Most existing studies on nonlinear Thomson scattering have focused exclusively on the head-on collision geometry. However, in practical experimental scenarios, electrons always interact with the incident laser at a certain oblique angle rather than in a perfect head-on manner, which deviates from the idealized head-on configuration^[ Reference Ride, Esarey and Baine^{80 –} Reference Liu, Yu, Zou, Xu, Yin and Shao^{82 ]}, and thus necessitates dedicated investigation of oblique collision scenarios. Figure 11 presents a comparative analysis of the radiation properties for different oblique angles ${\zeta}_0$ , defined as the initial incidence angle between the electron momentum and the $-x$ direction. As the incidence angle increases, the spectrum undergoes a noticeable redshift and its overall intensity is reduced; the red arrow marks the redshift of the fundamental-frequency emission. In particular, relative to the head-on case ( ${\zeta}_0=0$ ), the central frequency of the fundamental peak shifts from 93 ${\omega}_0$ to 55 ${\omega}_0$ at ${\zeta}_0=10{}^{\circ}$ . For ${\zeta}_0=10{}^{\circ}$ , the one-dimensional spectrum becomes nearly continuous, making individual harmonics difficult to distinguish; nevertheless, as shown in the inset two-dimensional angular–frequency spectrum, a localized third-harmonic feature with central frequency of around 150 ${\omega}_0$ remains clearly visible.

Figure 11

Comparison of one-dimensional full spectrum at $\theta =0$ and $\varphi =0$ for oblique collision geometry with the initial incidence angle of ${\zeta}_0=0{}^{\circ}$ (head-on), $3{}^{\circ}$ , $7{}^{\circ}$ and $10{}^{\circ}$ , respectively. Inset: two-dimensional spatial spectrum of the emitted field ${\mathrm{d}}^2I/\mathrm{d}\theta \mathrm{d}\omega$ at $\varphi =0$ for ${\zeta}_0=10{}^{\circ}$ .