Semiclassical probability of radiation of twisted photons in the ultrarelativistic limit

The semiclassical general formula for the probability of radiation of twisted photons by ultrarelativistic scalar and Dirac particles moving in the electromagnetic field of a general form is derived. This formula is the analog of the Baier-Katkov formula for the probability of radiation of one plane wave photon with the quantum recoil taken into account. The derived formula is used to describe the radiation of twisted photons by charged particles in undulators and laser waves. Thus, the general theory of undulator radiation of twisted photons and radiation of twisted photons in the nonlinear Compton process is developed with account for the quantum recoil. The explicit formulas for the probability to record a twisted photon are obtained in these cases. In particular, we found that the quantum recoil and spin degrees of freedom increase the radiation probability of twisted photons in comparison with the formula for scalar particles without recoil. In the range of applicability of the semiclassical formula, the selection rules for undulator radiation established in the purely classical framework are not violated. The manifestation of the blossoming out rose effect in the nonlinear Compton process in a strong laser wave with circular polarization and in the wiggler radiation is revealed. Several examples are studied: the radiation of mega-electron-volt twisted photons by 180 GeV electrons in the wiggler, the radiation of twisted photons by 256 MeV electrons in strong electromagnetic waves produced by the ${\mathrm{CO}}_2$ and Ti:Sa lasers, and the radiation of MeV twisted photons by 51.1 MeV electrons in the electromagnetic wave generated by the free-electron laser with photon energy 1 keV.

Figure 1

The radiation of twisted photons by 180 GeV electrons in the helical wiggler. The seventh harmonic is presented. The wiggler period is 0.72 cm, the number of periods $N=15$, and the magnetic field strength in the wiggler is 63.3 kG. The applicability conditions (84) are satisfied for ${\sigma }_{\perp }\lesssim {10}^3{m}_e^{-1}$. The photon energy is measured in the electron rest energies. Left panel: The probability of recording a twisted photon produced by one electron in the wiggler against the photon energy. Left inset: The probability distribution over $m$. Right inset: The relative change of radiation probability due to quantum recoil: $\eta ≔(d{P}_{cl}-dP)/d{P}_{cl}$, where $d{P}_{cl}$ is the radiation probability without quantum recoil. Right panel: The distribution over $m$ of the probability per particle to record a twisted photon produced by an incoherent axially symmetric bunch of particles in the wiggler. The radial bunch profile is Gaussian with the width $\sigma =125\mu \mathrm{m}$. The angular momentum per photon is the same as for radiation of one electron. Inset: The asymmetry of distribution over $m$.

Figure 2

The radiation of twisted photons by 51.1 MeV electrons evolving in the circularly polarized electromagnetic wave produced by the FEL with the photon energy 1 keV, intensity $3.38\times {10}^{15}{\mathrm{W}/\mathrm{cm}}^2$, and amplitude envelope (163) with $N=20$. These data correspond to $\mathrm{\Omega }\approx 1.96\times {10}^{-3}$ and $f_0\approx 8.54\times {10}^{-8}$. The applicability conditions (84) are satisfied for ${\sigma }_{\perp }\lesssim {10}^3{m}_e^{-1}$. The photon energy is measured in the electron rest energies. Left panel: The head-on collision. The first harmonic (166) for radiation without recoil (classical), scalar, and Dirac particles is depicted. For the parameters chosen, the quantum recoil halves the energy of radiated photons in comparison with formula without recoil. The probability of radiation of twisted photons by Dirac particles is bigger than the same quantity for scalar particles, which, in turn, is bigger than the probability of radiation of twisted photons without quantum recoil [see the discussion after Eq. (54)]. Right panel: The laser wave is overtaking the electron. The first harmonic (166) is shown. The quantum recoil is negligible in this case. Insets: The distribution over $m$ at the main maximum of the harmonic.

Figure 3

The radiation of twisted photons in head-on collision of 256 MeV electrons with the circularly polarized electromagnetic wave produced by the $\mathrm{Ti}:\mathrm{Sa}$ laser. The parameters are the same as in Fig. 6, but the laser pulse amplitude envelopes are different. Left panel: The amplitude envelope has the form (90). The initial laser wave phase ${\phi }_0=0$. The maximum near $k_0=3{\omega }_{+}$ is shown. Right panel: The amplitude envelope is $f_0{\sin }^4(\mathrm{\Omega }\xi /(2N))$. The third harmonic (166) is depicted. Insets: The distribution over $m$ at the main maximum.

Figure 4

The coordinates $x$, $y$ and “velocities” $r_x$, $r_y$ for head-on collision of an electron with the circularly polarized electromagnetic wave produced by the Ti:Sa laser. The parameters are the same as in Fig. 3. The electron moves initially along the detector axis. The initial laser wave phase ${\phi }_0=0$. The lengths are measured in the Compton wavelengths. a) The amplitude envelope has the form (90). b) The amplitude envelope is given in (163). c) The amplitude envelope is $f_0{\sin }^4(\mathrm{\Omega }\xi /(2N))$.

Figure 5

The radiation of twisted photons in head-on collision of 256 MeV electrons with the circularly polarized electromagnetic wave produced by the ${\mathrm{CO}}_2$ laser with the wavelength $10\mu \mathrm{m}$, intensity $10^{18}{\mathrm{W}/\mathrm{cm}}^2$, and amplitude envelope (163) with $N=20$. These data correspond to $\mathrm{\Omega }\approx 2.8\times {10}^{-7}$ and $f_0\approx 1.47\times {10}^{-6}$. The first three harmonics (166) are depicted. The applicability conditions (84) are satisfied for ${\sigma }_{\perp }\lesssim {10}^3{m}_e^{-1}$. The photon energy is measured in the electron rest energies. Insets: The distribution over $m$ at the main maximum of harmonics.

Figure 6

The radiation of twisted photons in head-on collision of 256 MeV electrons with the circularly polarized electromagnetic wave produced by the Ti:Sa laser with parameters (86) and amplitude envelope (163) with $N=20$. The first three harmonics (166) are depicted. The applicability conditions (84) are satisfied for ${\sigma }_{\perp }\lesssim {10}^3{m}_e^{-1}$. The photon energy is measured in the electron rest energies. Insets: The distribution over $m$ at the main maximum of harmonics.

Figure 7

The imprint of the blossoming out rose effect [9] on the radiation of twisted photons in head-on collision of electrons with circularly polarized electromagnetic wave produced by the ${\mathrm{CO}}_2$ (left panel) and Ti:Sa (right panel) lasers. The parameters are the same as in Figs. 5 and 6. The first harmonic (166) with $s=-1$ dominates. This is just a Lorentz-boosted back lobe of the first harmonic of synchrotron radiation (see Sec. 1.3.4 of Ref. [9]). It is mostly left-handed polarized, and this property is preserved by the Lorentz transformations. This harmonic does not die out even for $n_{\perp }\approx 1$. Upper left insets: The distributions over $m$ at the main maxima of the first two harmonics. Upper right insets: The distribution over $m$ at the main maximum of the third harmonic. Lower insets: The second and third harmonics are separately depicted.