Parametric interference effect in nonresonant spontaneous bremsstrahlung of an electron in the field of a nucleus and two pulsed laser waves

Nonresonant spontaneous bremsstrahlung of an electron scattered by a nucleus in the field of two moderately strong pulsed waves is studied theoretically. The process is studied in detail within the interference kinematic region. This region is determined by scattering of particles in the same plane at predetermined angles, at which stimulated absorption and emission of photons of external pulsed waves by an electron occur in a correlated manner. It is shown that the probability of the partial process with correlated emission (absorption) by an electron of the equal number of photons of the both waves is of an order of the magnitude greater than the corresponding probability in any other scattering kinematics. The cross section of spontaneous bremsstrahlung in two pulsed waves may be two times greater than the cross section of a free-field process after summation over all stimulated processes of correlated emission and absorption. Obtained results may be experimentally verified, for example, by scientific facilities at sources of pulsed laser radiation (SLAC, FAIR, ELI, XCELS).

Figure 1

Feynman diagrams of spontaneous bremsstrahlung of an electron scattered by a nucleus in the field of two pulsed waves. Dual incoming and outgoing lines correspond to wave functions of an electron in the initial and final states, and the inner line corresponds to the Green's function of an electron in the field of two pulsed waves. Wavy lines correspond to four-momenta of the spontaneous photon and the photon of recoil of a nucleus.

Figure 2

The geometry of ENSB process in the interference region for relativistic electron energies.

Figure 3

The partial probability of emission of a single photon of each of waves $l_{+}=1$ for SB of a relativistic electron $\left(E_i=1.02\mathrm{MeV}\right)$ scattered by a nucleus in the field of two pulsed waves (66), as a function of an outgoing angle ${\theta }_f$. The dotted curves correspond to a partial process within the interference region (68). Black curves (left peak) correspond to the energy of a spontaneous photon ${\omega }^{\prime }=0.5\mathrm{keV}$; blue curves (right peak) correspond to ${\omega }^{\prime }=0.1\mathrm{keV}$. The electron incoming angle ${\theta }_i=\pi /3$, parameters of the external pulsed field are ${\eta }_{01}={\eta }_{02}=0.2,{\omega }_1=2.35\mathrm{eV},{\omega }_2=1\mathrm{eV},{\tau }_1={\tau }_2,\rho =2.$

Figure 4

The partial probability of emission of equal numbers of photons of each of waves for SB of a relativistic electron $\left(E_i=1.02\mathrm{MeV}\right)$ scattered by a nucleus in the field of two pulsed waves (66), as a function of an outgoing angle ${\theta }_f$. The dotted curves correspond to a partial process within the interference region (68). The electron incoming angle ${\theta }_i=\pi /3$; the spontaneous photon energy ${\omega }^{\prime }=0.5\mathrm{keV}$; parameters of the external pulsed field are similar to Fig. 3. Panel(a) corresponds to emission of $l_{+}=3$ photons; panel (b) corresponds to emission of $l_{+}=5$ photons.

Figure 5

The partial probability of emission of three photons $l_{+}=3$ of each of pulsed waves in SB of a nonrelativistic electron $\left(v_i=0.05\right)$ scattered by a nucleus, as a function of an outgoing electron angle ${\theta }_f$. The electron incoming angle ${\theta }_i=\pi /6$. The dotted curves correspond to a partial process within the interference region (77). Black curves (left peak) correspond to the energy of a spontaneous photon ${\omega }^{\prime }=0.2\mathrm{keV}$, blue curves (right peak) correspond to ${\omega }^{\prime }=0.1\mathrm{keV}$. Parameters of the external pulsed field are ${\eta }_{01}={\eta }_{02}=0.2,{\tau }_1={\tau }_2,\rho =2,{\omega }_1=2.35\mathrm{eV},{\omega }_2=1\mathrm{eV}.$

Figure 6

The ratio of the differential cross section of ENSB in the field of two pulsed waves, summed over all the partial processes of stimulated emission and absorption, to the cross section of free-field bremsstrahlung (84), as a function of an interference outgoing angle of an electron ${\theta }_{\mathrm{f}\left(\mathrm{int}\right)}$ (77). Parameters of the external pulsed field are ${\eta }_{01}={\eta }_{02}=0.2,{\omega }_1=2.35\mathrm{eV},{\omega }_2=1\mathrm{eV},{\tau }_1={\tau }_2,\rho =2$, for the initial electron energy $v_i=0.05$ and different values of the energy of a spontaneous photon. The solid curve corresponds to the energy of a spontaneous photon ${\omega }^{\prime }=0.5\mathrm{keV}$; the dotted curve corresponds to ${\omega }^{\prime }=0.3\mathrm{keV}$.