Probability of radiation of twisted photons by classical currents

The general formula for the probability of radiation of a twisted photon by a classical current is derived. The general theory of generation of twisted photons by undulators is developed. It is proved that the probability to record a twisted photon produced by a classical current is equal to the average number of twisted photons in a given state. The general formula for the projection of the total angular momentum of twisted photons given the energy, the longitudinal projection of momentum, and the helicity is obtained. The symmetry property of the average number of twisted photons produced by a charged particle moving along a planar trajectory is found. The explicit formulas for the average number of twisted photons generated by undulators in both the dipole and wiggler regimes are obtained. It is established that, for the forward radiation of an ideal right-handed helical undulator, the harmonic number $n$ of the twisted photon coincides with its projection of the total angular momentum $m$. As for the ideal left-handed helical undulator, we obtain that $m=-n$. It is found that the forward radiation of twisted photons by a planar undulator obeys the selection rule that $n+m$ is an even number. It turns out that the average number of twisted photons produced by the undulator and detected off the undulator axis is a periodic function of $m$ in a certain spectral band of the quantum numbers $m$.

Figure 1

Shown on the left is a pictorial representation of the transition amplitude (36) corresponding to the family (i) of trajectories (see the main text) with the prescribed additional phase factor $e^{im\phi }$. The initial blue trajectory is rotated around ${\mathbf{e}}_3$ by an angle of $\phi$ so that point 1 is shifted to $1^{\prime }$ and the blue trajectory goes to the azure one. The vectors tangent to the trajectories at points 1 and $1^{\prime }$ are the velocity vectors. The vectors lying in the plane normal to the rotation axis are the projections of the velocity vectors onto this plane. Shown on the right is a pictorial representation of the transition amplitude (36) corresponding to the family (ii) of trajectories with the prescribed additional phase factor $e^{-im\phi }$. The initial blue trajectory is rotated around ${\mathbf{e}}_3$ by an angle of $-\phi$, so the point 1 is shifted to $1^{\prime }$ and the blue trajectory goes to the azure one. The vectors tangent to the trajectories at points 1 and $1^{\prime }$ are the velocity vectors. The vectors lying in the plane normal to the rotation axis are the projections of the velocity vectors onto this plane.

Figure 2

Radiation of twisted photons by the planar undulator in the dipole regime at the first harmonic. The trajectory of the electron is taken in the form (73) with $a_z=a_y=0$ and $a_x=K{\lambda }_0/\sqrt{2}\pi \gamma$, where $K=0.03$ is the undulator strength parameter, ${\lambda }_0=1$ cm is the length of the undulator section, and $\gamma ={10}^3$ is the Lorentz factor of the electron. The number of undulator sections $N=40$. The energy of photons is measured in units of the rest energy of the electron, 0.511 MeV. (a) Distribution over $m$, the asymmetry, and the angular momentum projection per one photon. In accordance with (105), the period of oscillations $T_m=4$. (b) Density of the average number of twisted photons against $k_0$ for different observation angles. The position of the peak in the forward radiation and the boundaries of the spectral band in the inset are well described by (91). The dashed vertical line in the inset depicts the photon energy used in (a). (c) Density of the average number of twisted photons against $m$ and the angular momentum projection per one photon at the left, ${\xi }_n=\pi$, and right, ${\xi }_n=0$, peaks appearing in the distribution over the photon energy for $\theta =1/5\gamma$ [see the inset in (b)]. The average number of photons obeys the symmetry relation (60).

Figure 3

Radiation of twisted photons by the helical undulator in the dipole regime at the first harmonic. The trajectory of the electron is taken in the form (73) and (74), where $H_x={10}^{-12}$ and $H_y=1.2\times {10}^{-12}$ in units of the critical field (75), ${\lambda }_0=1$ cm is the length of the undulator section, and $\gamma ={10}^3$ is the Lorentz factor of the electron. These data correspond to $K=4.6\times {10}^{-3}$. The number of undulator sections $N=40$. The energy of photons is measured in units of the rest energy of the electron, 0.511 MeV. (a) Distribution over $m$, the asymmetry, and the angular momentum projection per one photon. In accordance with (105), the period of oscillations $T_m=5$. (b) Density of the average number of twisted photons against $k_0$ for the different observation angles. The position of the peak in the forward radiation and the boundaries of the spectral band in the inset are well described by (91). The dashed vertical line in the inset depicts the photon energy used in (a). (c) Density of the average number of twisted photons against $m$ and the angular momentum projection per one photon at the left, ${\xi }_n=\pi$, and right, ${\xi }_n=0$, peaks appearing in the distribution over the photon energy for $\theta =1/5\gamma$ [see the inset in (b)].

Figure 4

Distribution over $m$ and the angular momentum projection per one photon of the forward radiation of twisted photons produced by the undulators. The energy of photons is measured in units of the rest energy of the electron, 0.511 MeV. The forward radiation of the planar undulator is subject to the symmetry relation (60) and the selection rule that $m+n$ is an even number. The forward radiation of the right-handed helical undulator obeys the rule $m=n$. The undulator forward radiation in the dipole regime at the first harmonic is shown for (a)–(c) the planar undulator and (d) the helical undulator. The trajectories of the electron are the same as in Figs. 2 and 3. The small contribution at $m=-1$ for the helical undulator is a consequence of deviation of the trajectory from an ideal right-handed helix. The density of the average number of twisted photons is well described by (100). The wiggler forward radiation at the fifth harmonic is shown for (${\mathrm{a}}^{\prime })-({\mathrm{c}}^{\prime }$) the planar wiggler and (${\mathrm{d}}^{\prime }$) the helical wiggler. The trajectories of the electron are the same as in Figs. 6 and 7.

Figure 5

Pictorial representation of the radiation amplitude of twisted photons produced by a charged particle moving along a helical trajectory (see the detailed description at the end of Sec. 5a). The red sinusoid depicts the undulator radiation at the first harmonic. The vectors attached to points 1, 2, and $A$ are the velocity vectors. The phase of the electromagnetic wave is $\mathrm{\Phi }(1,2)=d_{1,2}(k_0{\beta }_{\parallel }^{-1}-k_3)$, where $d_{1,2}$ is the distance between points 1 and 2. The resonance occurs when $\mathrm{\Phi }(1,2)=2\pi n$, where $n$ is the harmonic number. The azure trajectory is obtained from the blue one by the rotation by an angle of $\phi$ so that point 1 is shifted to point $1^{\prime }$. This trajectory contributes to the amplitude with the additional phase factor $e^{im\phi }$. Therefore, the radiation from point $A$ adds up constructively to the radiation from points 1 and 2 when $\mathrm{\Phi }(1,2)(1-d_{A,1}/d_{1,2})+m\phi =2\pi n$. Inasmuch as $d_{A,1}/d_{1,2}=\phi /\left(2\pi \right)$, we infer that $(m-n)\phi =0$, i.e., $m=n$. As for the ideal left-handed helical trajectory, one can easily deduce that $m=-n$.

Figure 6

Radiation of twisted photons by the helical wiggler at the fifth harmonic. The trajectory of the electron is taken in the form (114) with the undulator strength parameter $K=4$ and $\omega =2\pi {\beta }_{\parallel }/{\lambda }_0$, where ${\lambda }_0=1$ cm is the length of the undulator section and $\gamma ={10}^3$ is the Lorentz factor of the electron. The number of undulator sections $N=40$. The energy of photons is measured in units of the rest energy of the electron, 0.511 MeV. (a) Distribution over $m$, the asymmetry, and the angular momentum projection per one photon. In accordance with (105), the period of oscillations $T_m=4$. (b) Density of the average number of twisted photons against $k_0$ for different observation angles. The position of the peak in the forward radiation and the boundaries of the spectral band in the inset are well described by (91). The dashed vertical line in the inset depicts the photon energy used in (a). (c) Density of the average number of twisted photons against $m$ and the angular momentum projection per one photon at the left, ${\xi }_n=\pi$, and right, ${\xi }_n=0$, peaks appearing in the distribution over the photon energy for $\theta =1/\left(5\gamma \right)$ [see the inset in (b)].

Figure 7

Radiation of twisted photons by the planar wiggler at the fifth harmonic. The trajectory of the electron is taken in the form (155) with the undulator strength parameter $K=4$ and $\omega =2\pi {\beta }_{\parallel }/{\lambda }_0$, where ${\lambda }_0=1$ cm is the length of the undulator section and $\gamma ={10}^3$ is the Lorentz factor of the electron. The number of undulator sections $N=40$. The energy of photons is measured in units of the rest energy of the electron, 0.511 MeV. (a) Distribution over $m$, the asymmetry, and the angular momentum projection per one photon. In accordance with (105), the period of oscillations $T_m=4$. (b) Density of the average number of twisted photons against $k_0$ for different observation angles. The position of the peak in the forward radiation and the boundaries of the spectral band in the inset are well described by (91). The dashed vertical line in the inset depicts the photon energy used in (a). (c) Density of the average number of twisted photons against $m$ and the angular momentum projection per one photon at the left, ${\xi }_n=\pi$, and right, ${\xi }_n=0$, peaks appearing in the distribution over the photon energy for $\theta =1/5\gamma$ [see the inset in (b)]. The presence of sufficiently high lateral peaks on the upper plots is a consequence of a comparatively large width (179) of the main peak of ${\delta }_N\left(x\right)$. These lateral peaks disappear with increasing $N$. As long as the observation angle $\alpha =0$, the distributions over $m$ have the symmetry property (60).