Generation of hard twisted photons by charged particles in cholesteric liquid crystals

We study the radiation from charged particles crossing a cholesteric plate in the shortwave approximation when the wavelength of photons is much smaller than the pitch of the cholesteric helix, whereas the escaping angle of the photon and the anisotropy of the permittivity tensor can be arbitrary. The radiation of photons is treated in the framework of quantum electrodynamics with classical currents. The radiation of the plane-wave photons and the photons with definite projection of the angular momentum (the twisted photons) produced by charged particles crossing the cholesteric plate and moving rectilinearly and uniformly is considered. The explicit expressions for the average number of radiated photons and their spectra with respect to the energy and the projection of the angular momentum are obtained in this case. It turns out that in the paraxial approximation the projection of the orbital angular momentum, $l$, of radiated twisted photons is related to the harmonic number $n\in \mathbb{Z}$ as $l=2n+1$, i.e., the given system is a pure source of twisted photons as expected. It is shown that in the paraxial shortwave regime the main part of radiated photons is linearly polarized with $l=\pm 1$ at the harmonics $n=\{-1,0\}$. The applicability conditions of the approach developed are discussed. As the examples, we consider the production of 6.3 eV twisted photons from uranium nuclei and the production of x-ray twisted photons from 120 MeV electrons.

Figure 1

The scheme of the experimental setup. The beam of charged particles falls normally onto the CLC plate and then passes through the perforated mirror. The transition radiation created in the CLC plate is reflected by this mirror and is brought out to the detector. The second mirror is placed parallel to the first one so that the spectrum of radiation over $m$ is preserved. Notice that the CLC is depicted only schematically. Actually, CLCs possess neither sublayers nor periodic stricture in these sublayers.

Figure 2

The average number of plane-wave photons, $dP/d{k}_{0}d{k}_{\perp }$, produced in transition radiation by charged particles traversing normally the cholesteric plate. The shortwave approximation is used. The helicity of radiated photons is denoted as $s$ and $n_{\perp }=k_{\perp }/k_0$. The waviness of the plots reflects the presence of resonances in the photon spectrum of the CLC plate. The insets on the plots represent the distribution over $n_{\perp }$ at the maximum of radiation intensity. Upper plots: Transition radiation from the ions ${}^{238}\mathrm{U}^{92+}$ with the Lorentz factor $\gamma =2$ [49]. The plot (a) is for $s=1$ and the plot (b) is for $s=-1$. The width of the CLC plate $L=150\mu \mathrm{m}$, the number of periods $N_u=40$, and the components of the permittivity tensor are ${\varepsilon }_{\perp }=2.1$ and ${\varepsilon }_{\parallel }=2.49$ (see, e.g., Ref. [50]). The parameter of applicability of the shortwave approximation standing on the left-hand side of inequality (19) is 6.6 at the maximum of radiation intensity. Notice that the perturbation theories with respect to $k_{\perp }$ and $\delta \varepsilon$ are not valid for these parameters. The maxima of the corresponding applicability parameters—the expressions on the left-hand sides of (9) and (10)—are 3.5 and 4.2, respectively. Lower plots: Transition radiation from electrons with the Lorentz factor $\gamma =235$. The plot (c) is for $s=1$ and the plot (d) is for $s=-1$. The width of the CLC plate $L=40\mu \mathrm{m}$, the number of periods $N_u=40$, and the components of the permittivity tensor are ${\varepsilon }_{\perp }=1-{\omega }_p^2/\left(3k_0^2\right)$ and ${\varepsilon }_{\parallel }=1-{\omega }_p^2/k_0^2$ with ${\omega }_p=21$ eV (see for details Ref. [51]). The parameter (19) is 0.89 at the maximum of radiation intensity. Therefore, the shortwave approximation describes radiation only qualitatively. The perturbation theory with respect to $\delta \varepsilon$ is not valid for these parameters since the maximum of the applicability parameters Eq. (10) is $2.3\times {10}^3$. The perturbation theory with respect to $n_{\perp }$ works well in this parameter domain. The maximum of the applicability parameters standing on the left-hand side of inequalities (9) is $1.5\times {10}^{-3}$.

Figure 3

The average number of twisted photons, $dP/d{k}_{0}d{k}_{\perp }$, produced in transition radiation from the ions ${}^{238}\mathrm{U}^{92+}$ traversing normally the cholesteric plate. The shortwave approximation is used. The projection of the total angular momentum is denoted as $m$. The insets on the plots represent the distributions over $m$ and $n_{\perp }$ at the maximum of radiation intensity. The parameters are the same as in Fig. 2. The contributions with $s=1$ and $s=-1$ are almost the same indicating that the radiation polarization is linear. Upper plots: Transition radiation from the single ion ${}^{238}\mathrm{U}^{92+}$. The plot (a) is for $s=1$ and the plot (b) is for $s=-1$. The selection rule (64) is fulfilled for $n=-1$. The negative radiation harmonic is realized because $n_{\perp }$ is inside of the Cherenkov cone (63). Lower plots: Transition radiation from the Gaussian beam of $N={10}^5$ ions with transverse and longitudinal dimensions ${\sigma }_{\perp }=1\mu \mathrm{m}$ and ${\sigma }_3=125\mu \mathrm{m}$, respectively. The plot (c) is for $s=1$ and the plot (d) is for $s=-1$. As expected [31, 64], the distribution over $m$ becomes much wider than for the radiation from a single ion since $k_{\perp }{\sigma }_{\perp }\gg 1$. Nevertheless, the projection of the total angular momentum per photon, $\ell$, is the same as for the one-particle radiation.

Figure 4

The average number of twisted photons, $dP/d{k}_{0}d{k}_{\perp }$, produced in transition radiation from electrons traversing normally the cholesteric plate. The shortwave approximation is used. The insets on the plots represent the distributions over $m$ and $n_{\perp }$ at the maximum of radiation intensity. The parameters are the same as in Fig. 2. Upper plots: Transition radiation from the single electron. The plot (a) is for $s=1$ and the plot (b) is for $s=-1$. The selection rule (64) is fulfilled for $n=1$ with good accuracy. The contributions with $s=1$ and $s=-1$ are of the same order of magnitude and so the radiation polarization is close to linear. Lower plots: Transition radiation from the Gaussian beam of $N={10}^5$ electrons with transverse and longitudinal dimensions ${\sigma }_{\perp }=1\mu \mathrm{m}$ and ${\sigma }_3=125\mu \mathrm{m}$, respectively. The plot (c) is for $s=1$ and the plot (d) is for $s=-1$. The distribution over $m$ becomes wider than for the radiation from a single electron since $k_{\perp }{\sigma }_{\perp }\gg 1$. The projection of the total angular momentum per photon, $\ell$, and the projection of the orbital angular momentum per photon are the same as for the one-particle radiation.

Figure 5

The same as on the lower plots in Fig. 2 and on the upper plots in Fig. 4 but the paraxial approximation is employed [29]. The insets on the plots represent the distributions over $m$ and $n_{\perp }$ at the maximum of radiation intensity. For the parameters taken, the paraxial approximation is applicable (the smallness parameter following from inequalities (9) is $2.1\times {10}^{-3}$) whereas the shortwave approximation [the parameter (19) is 1.7] and the perturbation theory with respect to $\delta \varepsilon$ (the smallness parameter following from inequalities (10) is $3.2\times {10}^3$) are not. Despite this fact we see that the shortwave approximation describes the distributions with respect to $m$ and $n_{\perp }$ quite well. The distributions over $k_0$ agree only qualitatively. The selection rule (64) holds for both the approximations. Upper plots: Transition radiation of plane-wave photons. The plot (a) is for $s=1$ and the plot (b) is for $s=-1$. Lower plots: Transition radiation of twisted photons. The plot (c) is for $s=1$ and the plot (d) is for $s=-1$.

Figure 6

The average number of twisted photons, $dP/d{k}_{0}d{k}_{\perp }$, produced in transition radiation from electrons traversing normally the cholesteric plate. The paraxial approximation is used [29]. The insets on the plots represent the distributions over $m$ and $n_{\perp }$ at the maximum of radiation intensity. The width of the CLC plate $L=20\mu \mathrm{m}$, the number of periods $N_u=25$, and the components of the permittivity tensor are ${\varepsilon }_{\perp }=1-{\omega }_p^2/\left(3k_0^2\right)$ and ${\varepsilon }_{\parallel }=1-{\omega }_p^2/k_0^2$ with ${\omega }_p=21$ eV. The parameter (19) is 1.3 at the maximum of radiation intensity and so the shortwave approximation does not work well. In particular, we see that the radiation at the harmonic is not linearly polarized. The perturbation theory with respect to $\delta \varepsilon$ is also not valid for these parameters. The maximum of the applicability parameters (10) is $2.5\times {10}^2$. The maximum of the parameters (9) controlling the applicability of the perturbation theory with respect to $n_{\perp }$ is $2.0\times {10}^{-2}$. Upper plots: Transition radiation from a single electron. The plot (a) is for $s=1$ and the plot (b) is for $s=-1$. The selection rule (64) is fulfilled for $n=1$. Lower plots: Transition radiation from the Gaussian beam of $N={10}^5$ electrons with transverse and longitudinal dimensions ${\sigma }_{\perp }=1\mu \mathrm{m}$ and ${\sigma }_3=125\mu \mathrm{m}$, respectively. The plot (c) is for $s=1$ and the plot (d) is for $s=-1$. The distribution over $m$ becomes wider than for the radiation from a single electron since $k_{\perp }{\sigma }_{\perp }\gg 1$. The projection of the total angular momentum per photon, $\ell$, and the projection of the orbital angular momentum per photon are the same as for the one-particle radiation.