Zeptosecond-Yoctosecond Pulses Generated by Nonlinear Inverse Thomson Scattering: Modulation and Spatiotemporal Properties

Abstract

Ultrashort light pulses have strong research and application values, while nonlinear inverse Thomson scattering has been considered as a unique source of zepto-yoctosecond pulses. Here, the mechanism of nonlinear inverse Thomson scattering of a high-energy electron colliding with a tightly focused intense laser pulse is investigated through numerical simulation. The time-compression effect was proposed to explain the origin of ultrashort pulses and the nonlinear phenomenon of electron radiation in the time–space joint distribution. It is found that the time scale of electron radiation is orders of magnitude shorter than that of electron motion, and the increases in laser intensity and electron initial energy will result in stronger and shorter pulses. Yoctosecond pulses can be generated by a laser pulse with an intensity of $1.384\times {10}^{20} \mathrm{W}/{\mathrm{c}\mathrm{m}}^2$ and an electron with an initial energy of $51.1 \mathrm{M}\mathrm{e}\mathrm{V}$. These results provide theoretical and numerical basis for generating shorter light pulses.

1. Introduction

Ultrashort pulses are of great significance and application value as a tool in electron-molecular dynamics [1,2,3], surface science [4,5], pump-probe experiments [6] and so on, which need attosecond (as, 10⁻¹⁸ s), zeptosecond (zs, 10⁻²¹ s), and even yoctosecond (ys, 10⁻²⁴ s) pulses. Since Pierre Agostini [7] and Ferenc Krausz [8] who won the newest Nobel Prize produced the first consecutive and isolated single attosecond light pulses, respectively, in the world, the door to ultrafast physics has been opened. The shortest pulse that can be generated experimentally is only 53 as by Goulielmakis’ team [9], but this is almost the limit of current experimental methods. One possible method in theory of producing even shorter (zeptosecond) pulses is injecting an ultra-high power coherent X-ray pulse into an ultra-relativistic flying mirror [10]; however, a very high-intensity laser and novel techniques are required but they are not yet available.

To stably produce shorter pulses with higher energies, nonlinear Thomson/Compton scattering is a method worth considering [5,11,12]. When a single high-energy electron collides with an intense laser pulse, nonlinear inverse Thomson scattering (NITS, Figure 1), or Compton scattering under the Thomson limit [13,14], can be used to explain the motion and radiation of the electron. Lee et al. [15,16] conducted detailed research on attosecond X-ray pulses generated by nonlinear Thomson scattering, pioneered in this field. In our previous research [17,18], the generation of X/γ-ray pulses in the order of zeptosecond through single-electron NITS has been proven theoretically and numerically. However, the potential of NITS goes far beyond that.

In this paper, we investigated and calculated the spatiotemporal properties of NITS pulses generated by low-energy and medium-high-energy single electron collides with a tightly focused short-pulse intense laser field for the first time. The effects of the electron’s initial energy and laser intensity on the spatiotemporal properties of radiation are studied in detail. The temporal distribution of the radiated pulse has a structure of the zepto-yoctosecond and is spatially shown from the perspective of a polar angle (refers to $\theta$ distribution) and also an azimuth angle (refers to $\varphi$ distribution). The time-compression effect was proposed and used to explain the occurrence of ultra-short pulses. Furthermore, when increasing the electron’s initial energy and laser intensity, we discovered a new nonlinear phenomenon of electron radiation properties. This research elucidated the mechanism of generating zeptosecond-yoctosecond pulses through nonlinear Thomson scattering, and also provided the theoretical and numerical basis for experimentally producing shorter light pulses.

2. Materials and Methods

It is necessary to state that the spatial and temporal coordinates in the following equations are normalized by $k_0^{-1}={\lambda }_0/2\pi$ and ${\omega }_0^{-1}={\lambda }_0/2\pi c$, where ${\lambda }_0=1 \mathsf{\mu }\mathrm{m}$ is the wavelength of the laser and $c=2.998\times {10}^8 \mathrm{m}/\mathrm{s}$ is the light speed. The schematic diagram of the NITS process is shown in Figure 1. In a tightly focused Gaussian laser field, the electric field $\mathit{E}$ and magnetic field $\mathit{B}$ which satisfies Maxwell’s equations can be expressed as follows [19]:

$$\mathit{E}=\sqrt{ϵ}\left[\right(i/k)\nabla (\nabla ·\mathit{A})+ik\mathit{A}]$$

(1)

$$\mathit{B}=\nabla \times \mathit{A}$$

(2)

where $ϵ$ is the dielectric constant. In the Lorentz gauge, the vector potential $\mathit{A}$ obeys the following inhomogeneous wave equation:

$${\nabla }^2\mathit{A}+k_0^2\mathit{A}=-\left(4\pi /c\right)\mathit{J}$$

(3)

Considering the electromagnetic field in vacuum, Equation (3) can be reduced to the following:

$${\nabla }^2\mathit{A}+k_0^2\mathit{A}=0$$

(4)

In the Cartesian coordinate, the circularly polarized laser can be decomposed into a pair of linearly polarized lasers with a phase difference of $\pi /2$, whose polarization plane are perpendicular to the y-axis and x-axis, respectively. Thus, the electromagnetic field of the circularly polarized laser can be decomposed as $\mathit{E}={\mathit{E}}_{\mathit{x}\mathit{p}}+{\mathit{E}}_{\mathit{y}\mathit{p}}$ and $\mathit{B}={\mathit{B}}_{\mathit{x}\mathit{p}}+{\mathit{B}}_{\mathit{y}\mathit{p}}$. The Gaussian laser field in this model has the spot size of $2w_0 (\mathsf{\mu }\mathrm{m}$), pulse width of $L \left(\mathsf{\mu }\mathrm{m}\right)$, and the intensity of $I (\mathrm{W}/{\mathrm{c}\mathrm{m}}^2)$, so we can make the definition that

$$E_0=a_0{\displaystyle \frac{w_0}{w}}\exp \left(-{\displaystyle \frac{{\eta }^2}{L^2}}\right)\exp \left(-{\displaystyle \frac{{\rho }^2}{w^2}}\right)$$

(5)

where $a_0=\sqrt{I{\lambda }_0^2/1.384}\times {10}^{-9}$ indicates the peak amplitude, $w=w_0\sqrt{1+z^2/z_0^2}$ is the waist radius at $z$ in which the Rayleigh distance $z_0=k_0{w}_0^2/2$. The longitudinal distance is $\eta =z-t$, and the transverse distance is $\rho =\sqrt{x^2+y^2}$. Under the circumstance of a tightly focused laser [20,21], the expression of the electromagnetic field must undergo high-order expansion, accurate to the diffraction angle $\epsilon =w_0/z_0$. Yousef et al. [20] derived $\mathit{A}$, $\mathit{B}$ and $\mathit{E}$ in the x-axis linearly polarized laser field from Equations (1)–(4). The electromagnetic field expression of the y-axis linearly polarized laser field is proposed by Barton [21] and Zhang [22] using symmetry expression. Combining the wave equation and Gaussian laser field distribution, the fifth-order expansion of the electric field component $\mathit{E}=\{E_x,E_y,E_z\}$ of the Gaussian laser field can be derived as follows:

$$\begin{array}{c}{E}_x=E_0\{S_0+{\epsilon }^2\left[{\alpha }^2{S}_2-{\displaystyle \frac{r^4{S}_3}{4}}\right]\hfill \\ \hspace{1em}\hspace{1em}+{\epsilon }^4\left[{\displaystyle \frac{S_2}{8}}-{\displaystyle \frac{r^2{S}_3}{4}}-{\displaystyle \frac{r^2\left(r^2-16{\alpha }^2\right)S_4}{16}}-{\displaystyle \frac{r^4\left(r^2+2{\alpha }^2\right)S_5}{8}}+{\displaystyle \frac{r^8{S}_6}{32}}\right]\\ \hspace{1em}\hspace{1em}+{\epsilon }^2{C}_2+{\epsilon }^4\left[r^2{C}_4-{\displaystyle \frac{r^4{C}_5}{4}}\right]\}\hfill \end{array}$$

(6)

$$\begin{array}{c}{E}_y=E_0\{C_0+{\epsilon }^2\left[{\beta }^2{C}_2-{\displaystyle \frac{r^4{C}_3}{4}}\right]\hfill \\ \hspace{1em}\hspace{1em}+{\epsilon }^4\left[{\displaystyle \frac{C_2}{8}}-{\displaystyle \frac{r^2{C}_3}{4}}-{\displaystyle \frac{r^2\left(r^2-16{\beta }^2\right)C_4}{16}}-{\displaystyle \frac{r^4\left(r^2+2{\beta }^2\right)C_5}{8}}+{\displaystyle \frac{r^8{C}_6}{32}}\right]\hfill \\ \hspace{1em}\hspace{1em}+{\epsilon }^2{S}_2+{\epsilon }^4\left[r^2{S}_4-{\displaystyle \frac{r^4{S}_5}{4}}\right]\}\hfill \end{array}$$

(7)

$$\begin{array}{c}{E}_z=E_0\alpha \{\epsilon C_1+{\epsilon }^3\left[-{\displaystyle \frac{C_2}{2}}+r^2{C}_3-{\displaystyle \frac{r^4{C}_4}{4}}\right]\hfill \\ \hspace{1em}\hspace{1em}+{\epsilon }^5[-{\displaystyle \frac{3C_3}{8}}-{\displaystyle \frac{3r^2{C}_4}{8}}+{\displaystyle \frac{17r^4{C}_5}{16}}-{\displaystyle \frac{3r^6{C}_5}{16}}-{\displaystyle \frac{3r^6{C}_6}{8}}+{\displaystyle \frac{r^8{C}_7}{32}}]\}\hfill \\ \hspace{1em}\hspace{1em}-E_0\beta \{\epsilon S_1+{\epsilon }^3\left[-{\displaystyle \frac{S_2}{2}}+r^2{S}_3-{\displaystyle \frac{r^4{S}_4}{4}}\right]\hfill \\ \hspace{1em}\hspace{1em}+{\epsilon }^5[-{\displaystyle \frac{3S_3}{8}}-{\displaystyle \frac{3r^2{S}_4}{8}}+{\displaystyle \frac{17r^4{S}_5}{16}}-{\displaystyle \frac{3r^6{S}_5}{16}}-{\displaystyle \frac{3r^6{S}_6}{8}}+{\displaystyle \frac{r^8{S}_7}{32}}]\}\hfill \end{array}$$

(8)

And the magnetic field components are described as follows:

$$B_x=E_0\{C_0+{\epsilon }^2\left[{\displaystyle \frac{r^2{C}_2}{2}}-{\displaystyle \frac{r^4{C}_3}{4}}\right]+{\epsilon }^4\left[-{\displaystyle \frac{C_2}{8}}+{\displaystyle \frac{r^2{C}_3}{4}}+{\displaystyle \frac{5r^4{C}_4}{16}}-{\displaystyle \frac{r^6{C}_5}{4}}+{\displaystyle \frac{r^8{C}_6}{32}}\right]\}$$

(9)

$$B_y=E_0\{S_0+{\epsilon }^2\left[{\displaystyle \frac{r^2{S}_2}{2}}-{\displaystyle \frac{r^4{S}_3}{4}}\right]+{\epsilon }^4\left[-{\displaystyle \frac{S_2}{8}}+{\displaystyle \frac{r^2{S}_3}{4}}+{\displaystyle \frac{5r^4{S}_4}{16}}-{\displaystyle \frac{r^6{S}_5}{4}}+{\displaystyle \frac{r^8{S}_6}{32}}\right]\}$$

(10)

$$\begin{array}{c}{B}_z=E_0\beta \left\{\epsilon C_1+{\epsilon }^3\left[{\displaystyle \frac{C_2}{2}}+{\displaystyle \frac{r_2{C}_3}{2}}-{\displaystyle \frac{r_4{C}_4}{4}}\right]+{\epsilon }^5\left[{\displaystyle \frac{3C_3}{8}}+{\displaystyle \frac{3r^2{C}_4}{8}}+{\displaystyle \frac{3r^4{C}_5}{16}}-{\displaystyle \frac{r^6{C}_6}{4}}+{\displaystyle \frac{r^8{C}_7}{32}}\right]\right\}\hfill \\ \hspace{1em}\hspace{1em}-E_0\alpha \left\{\epsilon S_1+{\epsilon }^3\left[{\displaystyle \frac{S_2}{2}}+{\displaystyle \frac{r_2{S}_3}{2}}-{\displaystyle \frac{r_4{S}_4}{4}}\right]+{\epsilon }^5\left[{\displaystyle \frac{3S_3}{8}}+{\displaystyle \frac{3r^2{S}_4}{8}}+{\displaystyle \frac{3r^4{S}_5}{16}}-{\displaystyle \frac{r^6{S}_6}{4}}+{\displaystyle \frac{r^8{S}_7}{32}}\right]\right\}\hfill \end{array}$$

(11)

where $\alpha =x/w_0$, $\beta =y/w_0$, and $r=\rho /w_0$. In the tightly focused laser ($w_0<4 \mathsf{\mu }\mathrm{m}$), terms expended to the order up to ${\epsilon }^5$ are necessary and sufficient to ensure the accuracy of the description [20]. Among them, $S_m$ and $C_m$ are shown as follows:

$$S_m={\left({\displaystyle \frac{w_0}{w}}\right)}^m\mathit{sin}\left(\psi +m{\psi }_G\right)$$

(12)

$$C_m={\left({\displaystyle \frac{w_0}{w}}\right)}^m\mathit{cos}\left(\psi +m{\psi }_G\right)$$

(13)

where $m=\mathrm{0,1},2,\dots$ The phase $\psi ={\psi }_0+\eta -{\psi }_R+{\psi }_G$, in which the initial phase ${\psi }_0$ is the phase of the plane wave. ${\psi }_R={\rho }^2/2R\left(z\right)$ is the phase related to the curvature of wave fronts where $R\left(z\right)=z+z_0^2/z$ indicates the radius of curvature of a wave front intersecting the beam axis at the coordinate $z$. ${\psi }_G={\tan }^{-1}\left(z/z_0\right)$ is the Gouy phase, which means a total phase transition of $\pi$ will be experienced by the Gaussian beam as $z$ changes from $-\mathrm{\infty }$ to $+\mathrm{\infty }$.

Considering a single electron with a rest mass $m_e=9.1096\times {10}^{-31} \mathrm{k}\mathrm{g}$ and charge $-e=-1.6022\times {10}^{-19}C$, the motion of the electron in an intense laser pulse can be determined by Lorentz and the energy equations given below:

$${\displaystyle \frac{d\mathit{p}}{d{t}_e}}=-e\left(\mathit{E}+\mathit{u}\times \mathit{B}\right)$$

(14)

$${\displaystyle \frac{d\Gamma }{d{t}_e}}-ec\left(\mathit{u}·\mathit{E}\right)$$

(15)

Here, $\Gamma ={\mathsf{\gamma }}_0{m}_e{c}^2$ represents the electron energy defined by equation ${\mathsf{\gamma }}_0^{-1}=\sqrt{1-{\mathit{u}}^2}$ where ${\gamma }_0$ is the Lorentz factor, which represents the normalized initial energy of the electron. The momentum of the electron $\mathit{p}=\Gamma \mathit{v}/c^2$ where $\mathit{v}$ is the electron velocity and $\mathit{u}=\mathit{v}/c$. In the rest frame of the observer, the time the electron moves through is represented by $t_e$. The initial position of electron $(x^{\prime },y^{\prime },z^{\prime })$ is set at $(0,0,14.5{\lambda }_0)$, so the collision center of the NITS process (i.e., the maximum value of the electron motion amplitude) is guaranteed to be at the coordinate origin.

The time-dependent electromagnetic field generated by a moving charged particle can be deduced from the Liénard–Wiechert potentials [23]; thus, the radiated power per unit solid angle can be calculated as follows:

$${\displaystyle \frac{dP\left(t\right)}{d\Omega }}={\left[{\displaystyle \frac{{\left|\mathit{n}\times \left[\left(\mathit{n}-\mathit{u}\right)\times d_{t_e}\mathit{u}\right]\right|}^2}{(1-\mathit{n}\cdot \mathit{u}{)}^6}}\right]}_{t^{\prime }}$$

(16)

where the subscript $t^{\prime }=t_e+R^{\prime }-\mathit{n}\cdot \mathit{j}$ indicates the equation should be calculated at the delay time $t$. $R^{\prime }$ is the distance between the observer and interaction center and $\mathit{j}$ is the electron’s displacement. The power of radiated pulse $dP\left(t\right)/d\Omega$ is normalized by $e^2{\omega }_0^2/4\pi c$, and the radiation direction $\mathit{n}$ is as follows:

$$\mathit{n}=(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$$

(17)

where $\theta$ is the polar angle and $\varphi$ is the azimuth angle shown in Figure 1.

By solving the ordinary differential Equations (14) and (15) with the 8–9 Runge–Kutta–Fehlberg method (RKF89) along with a large-scale GPU parallel solution on MATLAB, the numerical solution of the electron’s position, velocity and acceleration are recorded for each step, and the radiation with different states in time and space can be obtained from Equation (16).

3. Results

In this paper, a tightly focused intense near infrared single laser pulse with the wavelength of ${\lambda }_0=1 \mathsf{\mu }\mathrm{m}$ is used. The rest parameters are as follows: the initial phase ${\psi }_0=0$, the spot size $2w_0=6 \mathsf{\mu }\mathrm{m}$, and the pulse duration $L=10fs$. The normalized peak amplitude of laser $a_0$ can be used to characterize the laser intensity $I$, for $I=1.384\times {10}^{18} \mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ when $a_0=1$. The radiation observation distance $R^{\prime }=1\mathrm{m}$, which is far away enough from the collision center. Figure 2 gives an illustration of several terms used in the following sections. When the polar angle $\theta ={\theta }_0$ is a constant, the distribution of the NITS temporal intensity on the azimuth angle is called the $\varphi$ distribution. Similarly, the distribution of the NITS temporal intensity on the polar angle is called the $\theta$ distribution when the azimuth angle $\varphi ={\varphi }_0$ is a constant. The deviation angle of the NITS radiation is defined as the angle at which peak radiation deviates from the optical axis of the laser.

3.1. Spatial Radiation Features of NITS Radiation

To explore the spatial radiation properties of NITS, the peak power of NITS radiation pulses at each angle in space are recorded and shown in the angular plane, as is shown in Figure 3. A plane projection of the spatial radiation power is added on the top of each figure for clearer observation and analysis. The NITS radiation is almost uniform and continuous with a period of $2\pi$ on the azimuth angle $\varphi$, which can be seen in Figure 3. The angular distribution of NITS radiation power is a shape of a ring, and the polar distribution of the ring-shaped radiation occurs only within a narrow range of polar angles. Although the radiation ring appears to be uniform, there is one single maximum on the ring for each set of parameters, whose power is known as the NITS power peak.

In Figure 4, we further calculated the changes in radiation peak power and deviation angle when the normalized laser intensity $a_0$ increases from $0.1$ to $10$ and the electron’s initial energy increases from $1$ to $100$. As the $a_0$ increases, the radiation peak power increases exponentially [Figure 4a]. In terms of radiation direction, the deviation angle increases linearly [Figure 4b], which is shown in Figure 3 as an increase in the ring-shaped radiation radius. It is worth noting that the radiation vortex gradually becomes significant as $a_0$ increases, which can be seen clearly in Video S1. In fact, when the laser intensity increases, a much greater transverse ponderomotive force is exerted on the electron in average, while the rising edge of the laser also decelerates the electron more violently, so the transverse component of the electron’s average velocity increases at the laser focus. Therefore, the radiation direction changes from backward to sideways, and this kind of change is linear, while the collimation of the radiation decreases at the same time. When the electron is relatively far away from the laser focus, the instantaneous laser intensity is not that high, so the electron still has weak backward radiation, which is also the cause of the radiation vortex in Figure 3.

On the other hand, the peak power of NITS radiation also increases exponentially as ${\gamma }_0$ increases [Figure 4a], which is achieved by adjusting the force exerted by the electron accelerator on the electrons so that the electrons obtain a higher initial velocity when they leave (${\gamma }_0=1/\sqrt{1-v^2/c^2}$). The ultra-intense NITS radiation power reaches $6.4\times {10}^{15}\left(e^2{\omega }_0^2/4\pi c\right)$ when $a_0=10$ (i.e., $I=1.384\times {10}^{20} \mathrm{W}/\mathrm{c}{\mathrm{m}}^2$) and ${\gamma }_0=100$ (i.e., $\Gamma =51.1 \mathrm{M}\mathrm{e}\mathrm{V}$). However, the deviation angle becomes exponentially smaller, and the radiation is concentrated backward with higher collimation [Video S2, Figure 4b]. The initial velocity of the electron is higher owing to the higher initial energy; thus, the time that the electron interacts with the laser is shorter, and the effects of the transverse ponderomotive force on average are less, even if in an ultra-intense laser. The transverse velocity of electrons has little effect compared to the large longitudinal velocity; thus, its average direction is still backward, and the electron radiation still has remarkable backward collimation.

In general, laser intensity and electron initial energy can cooperatively affect radiation intensity, and polar direction, which has significant value for experimentally regulating the spatial scale and intensity of NITS radiation.

3.2. Polar Distribution of NITS Temporal Intensity

To further explore the temporal structure of NITS spatial radiation, the joint distribution of radiation power in time and polar angle is analyzed in in this section. For each set of parament $a_0$ and ${\gamma }_0$, the azimuth angle is fixed in the direction corresponding to the power peak, that is, to explore the temporal intensity of radiation across the radiation ring in Figure 2. The results are illustrated in Figure 5. The analysis of the joint distribution of radiation power in time and azimuth angle will be in the next section.

In Figure 5, we can easily tell that the radiated pulses are clearly symmetrical not only with respect to time, but also the polar angle. From the perspective of time, the NITS radiation pulses are symmetrical about its main radiation peak, whose time of appearance $t=t_{peak}$. Enlarging the cross-section of the main radiation peak in Figure 5g–i, we can obtain Figure 6, and the above time symmetry is more intuitive. The main peak of radiation always has the maximum power and minimum pulse width compared with other peaks, from which a consensus can be clarified that study of the main peak deserves attention.

From the perspective of polar angle distribution, we find the NITS radiation temporal intensity is quasi-axisymmetric about $\theta =180°$. To demonstrate the spatiotemporal structure of the NITS radiation pulse more specifically, we enlarge Figure 5e to Figure 7a, and Figure 7b shows the logarithm projection of the radiation pulses in this case. On one side of $\theta =180°$, we can observe a single main peak with stronger power, while on the other side, two double main peaks with weaker power appear symmetrically. This is actually the impact of the laser’s initial phase ${\psi }_0$, which is clearly shown in Video S3 when the azimuth angle is fixed. However, some more interesting phenomena are shown in Figure 7b. The distribution of one single pulse peak on the polar angle is an extremely thin line, and the power decreases; also, the pulse width increases, as $\theta$ leaves its radiation peak. This is why the middle of the radiation ring in Figure 3 is the brightest. If we connect the radiation peaks on both sides of $\theta =180°$ separately, we will obtain the bracket-shaped dash-line in Figure 7b. The strongest and shortest pulses will appear farthest from $\theta =180°$, and this happens to be the moment when the transverse ponderomotive force exerted on the electron reaches its maximum, and also its transverse velocity. Pulses that precede or follow the main peak of radiation are responsible for the appearance of radiation vortices.

Furthermore, we find the time-compression effect of NITS radiation from Figure 7b that the radiated pules when $\theta =180°$ are tightly compressed in time, and when the polar angle $\theta$ deviates from $180°$, the pulses come earlier than $t=t_{peak}$ will advance, and the pulses come later than $t=t_{peak}$ will lag. And, as $\theta$ moves away from $180°$, this advance–lag extent will increase.

Next, we focus our research on the properties of the main radiation peak. The full-width of half-maximum (FWHM) of the main peak and pulse interval between peaks when $a_0$ from $0.1$ to $10$ and ${\gamma }_0$ from $5$ to $100$ are calculated and shown in Figure 8, respectively. The pulse width becomes shorter with $a_0$ and ${\gamma }_0$ increasing as shown in Figure 8a, which is caused by more severe nonlinear effects. The more intense the laser field is, the greater the ponderomotive force that the electron is subjected to in the laser field. A larger ponderomotive force and a higher electron initial energy will lead to more severe nonlinear effects, causing the electrons to radiate higher order harmonics [24,25]; thus, resulting in a shorter radiation pulse width. In extreme cases when $a_0=10$ and ${\gamma }_0=100$, the FWAM reaches the yoctosecond level [Figure 6c], being ${10}^6$ times shorter than when $a_0=0.1$ and ${\gamma }_0=5$. However, the changes pulse interval of radiated pulses is positively correlated with $a_0$ and negatively correlated with ${\gamma }_0$ according to Figure 8b. It is of great significance for the selection of parameters experimentally obtaining single or continuous ultra-short light pulses.

3.3. Azimuth Distribution of NITS Temporal Intensity

In this section, the joint distribution of NITS radiation in time and azimuth angle is explored, and the results are shown in Figure 9. The joint distribution presents a significant symmetry, and the changes of radiated pulses with time and azimuth angle $\varphi$ is continuous, i.e., the azimuth angle corresponding to the pulse main peak advances continuously as time progressing.

With the increase in laser intensity, the speed of change in the pulse peak power with $\varphi$ becomes faster whether in pulse rise time or fall time, nevertheless, the change with time slows down, causing the increase in pulse interval between the main peak and secondary peaks, and also conspicuous contrast. It is worth mentioning that the subpeak power of the radiated pulse increases with laser field intensity $a_0$, but the increase is far less than that of the main peak, so the subpeak of the radiated pulse is no longer obvious in comparison. Comparing each row in Figure 9, although the number of radiation pulses remains unchanged, the duration of the entire radiation process (note the time scale in Figure 9) is greatly compressed. In fact, this mechanism of compression is also what produces ultra-short pulses. Therefore, we took the same parameters as Figure 5i and analyzed the mechanism of producing ultrashort pulses through NITS in combination with the transverse acceleration of electron in Figure 10.

In the process of nonlinear inverse Thomson scattering, the cause of ultrashort pulses is the nonlinear effect brought about by the ultra-intense laser field and high-energy electron. We know from Section 3.2 that a larger transverse ponderomotive force will bring a more intense pulse, which is why the main peak of pulses in Figure 10a corresponds to the maximum value of the combined transverse acceleration in Figure 10b. Nevertheless, the time scales of the electron motion process [Figure 10b,d] and the pulse radiation process [Figure 10a,c] are not the same because of the time-compression effect.

In fact, when the radiation produced by the electron reaches the observer, the pulse train is not discrete but compressed together for two reasons. One is the relativistic/nonlinear effect caused by the high-energy electron. When the Lorentz factor ${\gamma }_0=100$, the initial velocity of the electron $v_0=0.9995c$; thus, the radiation pulses will be greatly compressed arising from the transformation from the rest frame of the electron to the observer’s frame. The second reason is the time-compression effect of the electron radiation. The pulse generated first has a longer propagation time because it is farther from the observer, while the following pulses have shorter propagation time. The electron moves at a speed close to the light, so the pulses arrive at the observer almost at the same time. Thus, both the time duration of the whole generation of the radiation pulses and the pulse width of the individual pulses are further shortened.

The degree of the time-compression effect is also different under different observation directions. As shown in Figure 7b, the radiation pulses are compressed to different degrees in different polar angles. This is because the electrons have a very high longitudinal velocity, which points in the direction of $\theta =180°$. When the polar angle of the observer deviates from $\theta =180°$, the velocity component of the electron in this direction becomes smaller, and the influence of the above two reasons is both weakened, so the degree of the pulses’ compression is reduced.

All in all, the time-compression effect of electron radiation is the compression of the mutual coupling of high-speed electron radiation processes in time and space under the relativistic effect, which leads to the generation of zepto-yoctosecond pulses and peculiar patterns in the joint distribution of space and time.

4. Conclusions

In conclusion, the mechanism of ultrashort pulses generated by nonlinear inverse Thomson scattering of collision between the high-energy electron and a tightly focused intense laser pulse was studied. Under the framework of classical electrodynamics, this paper explores the effects of laser intensity and electron initial energy on the time–space properties of electron radiation. As the laser intensity and electron initial energy increase, the radiation peak power increases and the pulse width becomes shorter. However, the collimation of radiation is negatively correlated with laser intensity, while positively correlated with the initial intensity of the electron. We further researched the time–space joint distribution of electron radiation and found that the time scale of the radiation is several orders of magnitude shorter than the time scale of electron motion. The time-compression effect is used to explain this phenomenon, which is also the cause of zepto-yoctosecond light pulses. The degree of the time-compression effect is also different in different observation directions. When a laser pulse with intensity $I=1.384\times {10}^{20} \mathrm{W}/\mathrm{c}{\mathrm{m}}^2$ collides with an electron with initial energy $\Gamma =51.1 \mathrm{M}\mathrm{e}\mathrm{V}$, yoctosecond (10–24 s) pulses can be generated. These results enrich the research on ultrafast and nonlinear optics and provide theoretical and numerical supports for the experimental production of zepto-yoctosecond light pulses.