Electron acceleration and emission in a field of a plane and converging dipole wave of relativistic amplitudes with the radiation reaction force taken into account

Let us analyse first the acceleration by the incident semi-bounded plane wave

$$\begin{equation} {\boldsymbol A} = \left\{ {\begin{array}{*{20}c} {a\,{\boldsymbol z}\,\sin \left( {\omega _0 t - ky} \right),} & {\omega t - ky \geqslant 0,} \\ {0,} & {\omega t - ky < 0} \\ \end{array}} \right. \end{equation} \tag{ 2 }$$

of an initially resting electron $\left( {y = 0} \right)$, for which we can obtain exhaustive characteristics of emission.

With the amplitude of the field, much less than $E_{{\text{cr}}}$, we can ignore the effect of radiation reaction and write the corresponding solution of the equation of motion in the parametric form:

$$\begin{equation} \begin{gathered} {\gamma = 1 + \frac{{a^2}} {2}\sin ^2 \varphi ,\,\,p_y = \frac{{a^2}} {2}\sin ^2 \varphi ,\,\,p_z = a\,\sin \varphi ,} \\ {y = \frac{{a^2 c}} {{4\omega _0}}\left( {\varphi - \frac{1} {2}\sin \left( {2\varphi} \right)} \right),\,\,z = \frac{{ac}} {{\omega _0}}\left( {1 - \cos \varphi} \right),} \\ {t = \frac{\varphi} {{\omega _0}} + \frac{{a^2}} {{4\omega _0}}\left( {\varphi - \frac{1} {2}\sin \left( {2\varphi} \right)} \right),} \\ \end{gathered} \end{equation} \tag{ 3 }$$

where $\varphi = \omega _0 t - ky;\,k = \omega _0 /c$; $p_y$ and $p_z$ are the projections of the particle's momentum in relativistic units.

In order to characterise the emission of a particle, we use the expressions for the radiation power $P$, the radiation power per unit solid angle $P_\Omega$ and characteristic radiation frequency $\omega _{\text{c}}$ [4]:

$$\begin{equation} P = \frac{{2e^2}} {{3c^3}}\frac{{{\boldsymbol{w}}^2 - \left[ {{\boldsymbol \upsilon} {\boldsymbol{w}}} \right]^2 /c^2}} {{\left( {1 - \upsilon ^2 /c^2} \right)^3}}, \end{equation} \tag{ 4 }$$

$$\begin{equation} \begin{aligned} P_\Omega & = \frac{{e^2}} {{4\pi c^3}}\left[ {\frac{{2({\boldsymbol n}{\boldsymbol w})({\boldsymbol \upsilon} {\boldsymbol w})}} {{c(1 - ({\boldsymbol \upsilon} {\boldsymbol n})/c)^4}}} \right. + \frac{{w^2}} {{(1 - ({\boldsymbol \upsilon} {\boldsymbol n})/c)^3}} \\ & \left. { - \frac{{({\boldsymbol n}{\boldsymbol w})^2}} {{\gamma ^2 (1 - ({\boldsymbol \upsilon} {\boldsymbol n})/c)^5}}} \right], \\ \end{aligned} \end{equation} \tag{ 5 }$$

$$\begin{equation} \omega _{\text{c}} = \gamma ^3 \frac{{\left| {\left[ {{\boldsymbol \upsilon} {\boldsymbol w}} \right]} \right|}} {{2\upsilon ^2}}, \end{equation} \tag{ 6 }$$

where $\boldsymbol \upsilon$ and $\boldsymbol \omega$ are the velocity and acceleration of a particle; ${\boldsymbol n}$ is the unit vector specifying the direction of the radiation. Equation (6) determines the centre frequency of the radiation when the particle moves along a curved trajectory. Using (3), we can find an expression for $P,\,P_\Omega$ and $\omega _{\text{c}}$:

$$\begin{equation} P = P_0 a^2 \cos ^2 \varphi, \end{equation} \tag{ 7 }$$

$$\begin{eqnarray} P_\Omega &=& \frac{{e^2 \omega _0^2 a^2}} {{8\pi c}} \\ &&\times \frac{{\cos ^2 \varphi \left[ {\cos \theta \left( {2 - a^2 \sin ^2 \varphi} \right) + a\,\sin \varphi \left( {a\,\sin \,\varphi - 2\sin \theta} \right)} \right]^2}} {{\left( {2 + a^2 \sin ^2 \varphi} \right)\left( {1 - a\,\sin \theta \,\sin \varphi + a^2 \,\sin ^2 \left( {\theta /2} \right)\sin ^2 \varphi} \right)^5}}, \end{eqnarray} \tag{ 8 }$$

$$\begin{equation} \omega _{\text{c}} = \frac{{a\omega _0 \left| {\cos \varphi} \right|\left( {2 + a^2 \,\sin ^2 \varphi} \right)^2}} {{4\left( {4 + a^2 \,\sin ^2 \varphi} \right)}}, \end{equation} \tag{ 9 }$$

where $P_0 = 2e^2 \omega _0^2 /3c$ and $\theta$ is the angle between the vector ${\boldsymbol n}$ and the propagation direction of the incident wave in the plane $yz$.

Let us determine the direction and the corresponding moments of time, at which the electron radiates most, as well as the characteristic radiation frequency.

The function $P\left( \varphi \right)$ is maximal when $\varphi = k\pi$, where $k$ is an integer. Then, at $a \gg 1$

$$\begin{equation} \begin{array}{*{20}c} {\frac{P}{{P_0}} = a^2 ,} & {\frac{{P_\Omega}} {{P_{\Omega _0}}} = 2a^2 \cos ^2 \theta} \\ \end{array} , \end{equation} \tag{ 10 }$$

where $P_{\Omega _0} = e^2 \omega _0^2 /\left( {8\pi c} \right)$.

In this case, the direction of the radiation does not depend on the intensity, the electron, because at the moments of time $\varphi - k\pi \ll 1$ its acceleration and velocity are directed almost along the electric field, radiates as a dipole with a characteristic frequency

$$\begin{equation} \omega _{\text{c}} = \frac{{a\omega _0}} {4}. \end{equation} \tag{ 11 }$$

The function $\omega _{\text{c}} \left( \varphi \right)$ is maximum when

$$\begin{equation*} \varphi = \arccos \left[ {\left( {\frac{2} {3} + \frac{{9 - \sqrt {57 + 18a^2 + a^4}}} {{3a^2}}} \right)^{1/2}} \right] + k\pi \end{equation*}$$

and

$$\begin{equation*} \varphi = - \arccos \left[ {\left( {\frac{2} {3} + \frac{{9 - \sqrt {57 + 18a^2 + a^4}}} {{3a^2}}} \right)^{1/2}} \right] + k\pi . \end{equation*}$$

Then, at $a \gg 1$, we have

$$\begin{eqnarray} \omega _{\text{c}} &=& \frac{{a^3 \omega _0}} {{6\sqrt 3}},\,\frac{P} {{P_0}} = \frac{{a^2}} {3}, \\ \frac{{P_\Omega}} {{P_{\Omega _0}}} &=& \frac{{54\left[ {\sqrt 6 \,a\,\sin \theta - 3\,\cos \theta - 2a^2 \,\sin ^2 \left( {\theta /2} \right)} \right]^2}} {{\left[ {3 + 2a^2 \,\sin ^2 \left( {\theta /2} \right) - \sqrt 6 \,a\,\sin \theta} \right]^5}}, \end{eqnarray} \tag{ 12 }$$

the width of the directivity pattern being equal to $3/a^2$, and the direction along which $P_\Omega$ is maximum corresponding to $\theta = \sqrt 6 /a$ (the angle is defined as the ratio of the transverse and longitudinal momenta of the particle, because in the ultrarelativistic case the particles radiate in the direction of their motion in a narrow range of angles around the velocity vector, and the width of the directivity pattern behaves as an inverse function of the gamma factor of the particle).

The function $P_\Omega \left( {\theta ,\varphi} \right)$ is maximal $\left( {a \gg 1} \right)$, when $\theta = 2/\left( {a\,\sin \varphi} \right),\,\varphi = \arccos \left( {1/\sqrt 3} \right) + k{\rm {\pi}}$ and $\varphi = - \arccos \left( {1/\sqrt 3} \right) + k\pi$; this follows from the decomposition of $P_\Omega \left( {\theta ,\varphi} \right)$ in a series in a small angle $\theta$ due to the ultrarelativistic character of motion. In this case, the expression for the power and the characteristic frequency of the radiation coincide with (12), the maximum output power per unit solid angle is given by

$$\begin{equation} \frac{{P_\Omega}} {{P_{\Omega _0}}} = \frac{{16}} {{27}}a^6 , \end{equation} \tag{ 13 }$$

the characteristic width of the directivity pattern is $3/a^2$, and the direction in which $P_\Omega$ is maximum corresponds to $\theta = \sqrt 6 /a$.

Now, using relation (3) for $\varphi$ and $t$, the character of acceleration and emission can be described as follows. Initially, the particle is accelerated by the incident wave and emits into a fairly large solid angle, having the maximum output power (10) at the initial moment of time. Then, on the time interval $0 < t \lesssim 0.12a^2 /\omega _0$, the particle moves at relativistic speeds, the directivity pattern gradually narrows down and is directed along its velocity of motion. The radiation power decreases to a value determined by (12), and the radiation frequency increases to the value of $\omega _{\text{c}}$ from (12). At a time $t = a^2 \pi /\left( {8\omega _0} \right)\,\left( {\varphi = \pi /2} \right)$, the output power is zero because the particle is not subjected to any force. Then, by virtue of the periodic nature of the interaction, the particle starts to slow down, the directivity pattern broadens, and the radiation power increases again to a level (10), corresponding to the stopping time of the particle. Below, the above process of particle acceleration and emission within the approximation neglecting the radiation reaction force is repeated periodically.

Consider now the case when, despite the smallness of the parameter $R_{\text{r}}$, the influence of the reaction force can qualitatively change the character of motion and characteristics of radiation, respectively. For example, Di Piazza et al. [9] presented an example of interaction of an incident wave with a counterpropagating electron and showed that by selecting the wave and the particle parameters, one can change the sign of the longitudinal velocity by taking into account only the radiation reaction force. This is due to the fact that the wave itself under the action of ponderomotive forces can greatly reduce the longitudinal velocity of the electron, and therefore even account for a small radiation reaction force can turn the electron, i.e., make it move in the opposite direction. We show that this situation can be realised for the transverse projection of the velocity; however, in this case the electron at the initial moment shoudl be in the correct phase of the field. This condition can be achieved, for example, during ionisation of atoms by hard photons.

Let the electron be produced with a zero transverse momentum at a time when the field of a plane linearly polarised wave is zero. We use the general solution of the relativistic motion equation, which can be written as [7]

$$\begin{eqnarray} &&{\gamma = \frac{1} {{h\left( \varphi \right)}}\left\{ {\gamma _0 + \frac{1} {{2\rho _0}}\left[ {h^2 \left( \varphi \right) - 1} \right] + \frac{1} {{2\rho _0}}\left[ {a^2 T^2 \left( \varphi \right)} \right]} \right\},} \\ &&{p_{||} = \frac{1} {{h\left( \varphi \right)}}\left\{ {p_{||0} + \frac{1} {{2\rho _0}}\left[ {h^2 \left( \varphi \right) - 1} \right] + \frac{1} {{2\rho _0}}\left[ {a^2 T^2 \left( \varphi \right)} \right]} \right\},} \\ &&{p_ \bot = \frac{{a\,T\left( \varphi \right)}} {{h\left( \varphi \right)}},} \\ &&{h\left( \varphi \right) = 1 + \frac{{R_{\text{r}}}} {2}\left[ {\varphi - \varphi _0 - \sin \,\varphi \,\cos \varphi + \sin \varphi _0 \cos \varphi _0} \right],} \\ &&{T\left( \varphi \right) = h\left( \varphi \right)\cos \varphi - \cos \varphi _0} \\ &&\quad { - R_\text{r} \left[ {\frac{1} {{a^2}} + \frac{1} {3}\left( {\sin ^2 \varphi + \sin ^2 \varphi _0 + \sin \varphi \sin \varphi _0} \right)} \right]\left( {\sin \varphi - \sin \varphi _0} \right).} \\ \end{eqnarray} \tag{ 14 }$$

As is known, in the absence of the radiation reaction force such an electron obtains a maximum constant component of the transverse momentum, so that the transverse momentum is always negative $\left( {p_ \bot < 0} \right)$. Account for the radiation reaction will lead to the fact that at some point in time, the projection of the transverse momentum changes the sign, i.e., the electron will turn around and move in the opposite direction. Assuming $\varphi _0 = 0$, one can easily see this from the solution of (14), taking into account that $p_ \bot \to a\,\cos \,\varphi$ at $\varphi \to \infty$.

We use expression (5) and construct the angular distribution of the radiation power of the electron produced at a zero field. The azimuthal angle is measured from the direction of the electric field at the initial time. As follows from the comparison of the curves in Fig. 1, the effect of the radiation reaction force leads to a qualitative change in the angular distribution of the radiation intensity, which is due to differences in the behaviour of the electron motion.

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Here, we pay particular attention to the fact that the general form of the solution of the relativistic motion equations (14) in the field of a plane travelling wave with the radiation reaction force in the Landau–Lifshitz form allows one to make a fundamentally important conclusion. Indeed, it is easy to see that in the general solution all the trajectories tend to a limit, which corresponds to the trajectory of an electron produced at the zero vector potential of the wave or at the maximum of the electric field. This conclusion not only confirms the theorem on the phase space compressibility of particles interacting with the laser field, taking into account the radiation reaction force [10, 11], but also leads to an even more important statement about the asymptotic synchronisation of the motion of all particles. The trajectory of this limiting motion has an average value of the transverse momentum, equal to zero [$T\left( \varphi \right)/h\left( \varphi \right) \to \cos \varphi$ at $\varphi \to \infty$), and, in particular, can be obtained from equation (14). Assuming $\varphi \gg R_{\text{r}}^{ - 1}$, we obtain

$$\begin{equation} \begin{gathered} {\gamma \approx \frac{{R_{\text{r}} \varphi}} {{4\rho _0}}\left( {1 + a^2 \cos ^2 \varphi} \right),} \\ {p_\parallel \approx \frac{{R_{\text{r}} \varphi}} {{4\rho _0}}\left( {1 + a^2 \cos ^2 \varphi} \right),} \\ {p_ \bot \approx a\,\cos \varphi ,} \\ {t \approx \frac{{R_{\text{r}}^2 \left( {2 + a^2} \right)\varphi ^3}} {{48\rho _0^2 \omega _0}}}. \\ \end{gathered} \end{equation} \tag{ 15 }$$

As is easily seen, the electron in the field of the incident linearly polarised wave is accelerated in the direction of its propagation and its average energy is proportional to $t^{1/3}$, because $t = \omega _0^{ - 1}\int_0^\varphi{h\left( x \right)\gamma \left( x \right)\,{\text{d}}x\sim \varphi ^3}$ and $\gamma \left( x \right)\sim \varphi$ at $\varphi \to \infty$. Note for comparison that in a circularly polarised wave the energy is proportional to $t^{2/3}$ [12].

Let the centre of the converging dipole wave be located at the origin of the Cartesian coordinate system. Then, the expressions for the electric and magnetic fields of the dipole wave can be given in the form [6]

$$\begin{equation} {\boldsymbol{E}} = \frac{{\left[ {{\boldsymbol{n}}\left[ {{\boldsymbol{nd}}_0} \right]} \right]}} {{Rc ^2}}\ddot g_ - \left( {t,\,R} \right) + \frac{{3{\boldsymbol{n}}\left( {{\boldsymbol{nd}}_0} \right) - {\boldsymbol{d}}_0}} {{R^3}}\left[ {\frac{{\boldsymbol{R}}} {c}\dot g_ + \left( {t,\,R} \right) + g_ - \left( {t,\,R} \right)} \right], \end{equation} \tag{ 16 }$$

$$\begin{equation} {\boldsymbol{H}} = - \frac{{\left[ {{\boldsymbol{nd}}_0} \right]}} {{Rc}}\left[ {\frac{{\ddot g_ + \left( {t,\,R} \right)}} {c} + \frac{{\dot g_ - \left( {t,\,R} \right)}} {R}} \right], \end{equation} \tag{ 17 }$$

where ${\boldsymbol{R}}$ is the radius vector from the focusing point to the observation point with $\left| {\boldsymbol{R}} \right| = R$ and ${\boldsymbol n} = {\boldsymbol{R}}/{R}$; $g_ \pm (t,R) = g(t - R/c) \pm g(t + R/c)$; ${\boldsymbol d}_0$ is an arbitrary constant vector, the modulus of which is related to the dipole wave power $P$; and $d_0 = \left( {3Pc^3 /2} \right)^{1/2}$. Consider the trajectory of the particles and compare them with the trajectories in the case of a plane wave. It is useful to bear in mind that the amplitude of the incident dipole wave along its propagation axis is related to the power as follows:

$$\begin{equation*} a_{\text{d}} = \frac{{ed_0}} {{mc^3 R\omega _0}} = \left(\frac{{ {3P} }} {{2c^3}}\right)^{1/2}\frac{e} {{m\omega _0 R}}\left( {R \gtrsim \lambda} \right). \end{equation*}$$

As in Section 2, we assume that the electrons are initially at rest and a semibounded radiation pulse with a sharp switching on of the field is incident on them, so that

$$\begin{equation*} \ddot g\left( {t,\,R} \right) = \cos \varphi ,\,\dot g\left( {t,\,R} \right) = \omega _0^{ - 1} \,\sin \varphi ,\,g\left( {t,\,R} \right) = \omega _0^{ - 2} \cos \varphi, \end{equation*}$$

where $\varphi = \omega _0 t - \omega _0 \,R/c$. For the analysis it is sufficient to consider the behaviour of electrons within the half-cycle of the wave, during which they must lag behind by half of the wavelength from the leading edge of the pulse.

As is known, in the field of a plane wave an initially resting electron acquires a longitudinal momentum $p_{||} = A^2 /2$ along the propagation direction of the wave and shifts in the transverse direction with a transverse momentum ${\boldsymbol p}_ \bot = {\boldsymbol{A}}$ [4], where ${\boldsymbol{A}}$ is the dimensionless vector potential at a point where the particle is located. We take as the amplitude of the vector potential of a plane wave $a_{\text{p}}$ its value corresponding to the amplitude of the incident dipole wave at a point of the initial location of the electron $R_{\text{c}} :a_{\text{p}} = a_{\text{d}} \left( {R = R_{\text{c}}} \right)$. Figure 2 shows the electron trajectories for two different wave powers, $10$ and $100\ \text{PW}$; one can see that they are qualitatively similar, and this is an evidence of the role of small radiation reaction force for the electrons accelerated to the centre. However, the accelerations of the electrons in the propagation directions of the dipole and the plane waves are qualitatively different: the rate of acceleration and achievable values of the longitudinal momentum in a converging dipole wave are much lower than in the plane wave, despite the fact that the converging wave amplitude increases rapidly to the centre. This is due, primarily, to the curvature of the phase front of the dipole wave, and to the transverse nonuniformity of the field distribution. In the converging wave the electron initially located in the azimuthal plane wave $(z = 0)$ first acquires a (transverse) momentum, perpendicular to it, but, deviating from this plane, it also acquires longitudinal acceleration (along the $y$ axis). However, moving in the transverse direction, the electron is affected by the converging wave, the heterogeneous nature of which leads to additional forces, which return it to the side of the azimuthal plane. This limits the motion of the electron in the transverse direction (as seen from Fig. 2, it is pressed to the plane $z = 0$ stronger than in the case of motion in the plane wave), and consequently, the corresponding rate of longitudinal acceleration, typical case of a plane wave, is not achieved. Figure 3 shows the gamma factor of the electron along the trajectory and the vector potential of the converging wave at the point of the electron location; one can see that the acquired longitudinal momentum is much smaller than the corresponding value in the field of a plane wave $\left( {1 + {\boldsymbol{A}}^2 /2} \right)$.

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To determine the number of the electrons accelerated to the centre and the values of the longitudinal momenta acquired by the electrons, we calculate the corresponding distribution of the particles, if they were initially uniformly distributed along the $y$ axis. Figure 4 shows the distribution of the particles inside a sphere of radius $3\lambda /2$ as a function of the longitudinal momentum. One can see that there is a maximum in the vicinity of the highest attainable longitudinal momenta that are about 2.3 times higher than the corresponding values of the vector potential, i.e., $a_{\text{d}} \left( {R = 3\lambda /2} \right) \approx 440$ at $100\ \text{PW}$ and $a_{\text{d}} \left( {R = 3\lambda /2} \right) \approx 140$ at $10\ \text{PW}$. This maximum in the distribution is due to the large time of the interaction of these particles with the wave. Figure 5a shows the dependence of the maximum longitudinal momentum on the power $P$, which agrees with the above assessment, and a longitudinal momentum, acquired in a plane wave with the amplitude corresponding to the field amplitude of the dipole wave at a point of the electron trapping (inset in Fig. 5a). These values are, of course, less than those in the plane of the dipole wave, which is associated with an increase in the amplitude of the dipole wave as it approaches the focus and, consequently, with a large longitudinal momentum acquired. However, it should be noted that in the field of the plane wave the maximum longitudinal momentum $\left( {a_{\text{p}}^2 /2} \right)$ achieved at a distance $a_{\text{p}}^2 \lambda /16$ from the point of initial location of the electron, and it is higher than the maximum value $\left( {\sim 2.3a_{\text{d}}} \right)$ in the dipole configuration (Fig. 5b). Thus, at a power of $200\ \text{PW}$ we have $a_{\text{p}}^2 /2 = 2665$ and $a_{\text{d}} = 620$. Since the distance $a_{\text{p}}^2 \lambda /16$, corresponding to the electron trajectories in the field of the dipole wave, is much greater than the distance to the focal point, we can say that the rate of the delay of the electron from the leading edge of the wave is higher in the dipole wave (Fig. 2).

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For the analysis of realistic scenarios of interaction we consider the incident pulse with a smooth envelope, which may be important for the electrons accelerated to the centre and able to experience the impact of both the incident wave and the wave passed through the centre. First, consider the distribution of electrons over the longitudinal momenta, similar to that shown in Fig. 4. Figure 6 shows the corresponding distribution for a pulse with the envelope $\sin ^2 t$, including 3 and 10 cycles of the field. For a short pulse (Fig. 6a), together with the maximum achievable longitudinal momentum of the order of the maximum field amplitude, additional peaks appear, which correspond to the local maximum in the distribution of the instantaneous pulse intensity. Interesting enough is the fact that for long pulses (Fig. 6b) high-energy peaks are suppressed due to the interaction of electrons with the counter (passed through the centre) wave, which can dramatically reduce the longitudinal momentum due to radiation reaction, as well as complicate the structure of the distribution function due to electron scattering by the counterpropagating wave.

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For more detailed analysis we consider the trajectories of the electrons trapped at the periphery and accelerated toward the centre. Figure 7 shows the trajectories of the electrons moving in the field at two different pulse durations and a power of $200\ \text{PW}$ both including and neglecting the radiation reaction. Each vertex of the trajectories corresponds to the electron delay in the reference frame of the incident wave by $\lambda /2$. One can see that in the field of the plane wave the delay occurs later than in the field of the dipole wave; in addition, in the field of the dipole wave the electron moves along a loop trajectory, which is qualitatively different from the case of the plane wave trajectory. In the field of the plane wave the loop motion is possible when the electron initially moves towards the laser pulse. This fact reduces the maximum attainable Lorentz factor of the accelerated electrons by $\rho _0 = \gamma _0 \left( {1 + \upsilon _0 /c} \right)$ times, where $\gamma_0$ and $\upsilon_0$ are initial values of the gamma factor and the electron velocity. The loop motion in the dipole wave, corresponding to the motion towards the pulse, can be seen in the electron trajectory, shown in Fig. 7b and in Fig. 7d, which shows the function $\cos \alpha$ ($\alpha$ is the angle between the electron velocity vector and electron direction to the centre of the dipole wave). One can see that the angle $\alpha$ in the case of loop motion may be higher than $\pi /2$ and even up to $\sim \pi$.

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Thus, the dipole wave due to the phase front curvature turns the particle and makes it move towards the wave. By analogy with the interaction of an electron with a plane wave, which initially moves towards a wave with the same gamma factor, we can conclude that the effect of the front curvature slows down the acceleration. An important feature of the trajectories shown in Fig. 7 is also their qualitative dependence on the radiation reaction force. As noted above, it can be ignored as long as the particle interacts predominantly with the copropagating wave (corresponding to $y \gtrsim \lambda$ in Fig. 7). Then, the particle is accelerated to the centre, begins to interact with the counterpropagating wave, and the role of the radiation reaction increases significantly; thus, according to (1), the regime of this interaction may become radiation-dominated. For the parameters, corresponding to trajectory (1) in Fig. 7b, $y\sim \lambda ,\,\gamma _{\text{m}} \sim 400,\,a_{\text{d}} \sim 700,\,\rho _0 \sim 2\gamma _{\text{m}}$, where $\gamma_{\text{m}}$ is the gamma factor of the particles at the initial moment of interaction with the counterpropagating wave $R_{\text{r}} \gtrsim 2$. One can see that the radiation reaction can confine the particle in the region of the strong field, while without taking it into account the particle quickly leaves the focal region. From a comparison of the trajectories in Fig. 7b and the field distribution in Fig. 7c one can see that a trajectory can be realised when the electron oscillates around a local maximum of the field for a long time (for a few oscillations of the field) before the ponderomotive force expels it in the transverse direction.

Characteristic features of the electron emission for two pulse durations can be easily obtained through a comparative analysis of the spectral output powers in different parts of the trajectories. For a short pulse the trajectory of motion (1) in Fig. 7a clearly shows that the qualitative change in the radiation behaviour stems from the collision of an electron with the wave passing through the centre (solid curve in Fig. 8a); in this case, the emission occurs mainly in the region $y = \lambda /2$, and its power greatly increases compared to the power at the initial points of the trajectory (in particular, at point $y = \lambda$ shown by the dashed line). Note that the radiation in the region $y = \lambda /2$ occurs at much higher frequencies, which actually corresponds to the emission of a relativistic electron in a counterpropagating wave. In the case of a long pulse the main contribution to the radiation whose spectrum is shown in Fig. 8b, is made by the electrons oscillating in the region of the strong total field of the incident wave and the wave passed through the centre with the gamma-factor corresponding to the local value of the vector potential. The typical trajectory of the electron in the field of the long pulse is shown in Fig. 7b, from which it can be clearly seen that before reaching the centre the electron initially accelerated by the field of the incident wave is sitting in the antinodes of the field at a distance of $3\lambda /4$ from the centre. In this phase of the motion its quasi-periodic trajectory corresponds to a purely oscillatory motion of the electron in the field of the standing wave, which is consistent with the relevant characteristics of the radiated power. For comparison, the dashed lines show the spectral powers emitted by the electron at a previous moment of time, corresponding to the acceleration stage, and therefore interaction only with a copropagating wave, i.e., when it is at point $y = 3\lambda /2$. These curves are an indicative of a much smaller radiative power of the electron before they start interacting with the counterpropagating wave. We also note that the spectral powers represented by the solid curves in Fig. 8 are almost identical, and this, despite the different nature of radiation, is due to the fact that the characteristic values of the longitudinal and transverse momenta in both cases have the same order and are determined by the local values of the vector potential.

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