Radiation of twisted photons in elliptical multifrequency undulators

TL;DR

Problem: generate photons in coherent superpositions of twisted modes with definite TAM projection $m$. Approach: develop a theory for radiation of twisted photons by a relativistic charge in an $M$-frequency undulator, derive the one-particle amplitude $\mathcal{A}$, and analyze the energy spectrum and TAM-selection rules, with explicit results for $M=2$ (helical and planar) and for rational frequency ratios via Bézout coefficients. Key contributions: general trajectory formulas, explicit amplitude expressions for arbitrary $M$, TAM-selection rules at fixed harmonic $n$, symmetry under chirality reversal, and a virtual-photon interpretation; the results connect undulator radiation to plane-wave regimes. Significance: provides a tunable, bright source of twisted photons for quantum electrodynamics interference studies and offers a comprehensive framework applicable to plane-wave radiation and to future studies with beams and twisted-electron generation.

Abstract

The theory of radiation of twisted photons in elliptical multifrequency undulators is developed. It is shown that helical multifrequency undulators can be employed as a bright and versatile source of photons in the states that are superpositions of the modes with definite projection of total angular momentum (TAM), amplitude, and relative phase. All these parameters of the state are readily controlled by the undulator design. The explicit expression for the amplitude of radiation of a twisted photon from a charged particle in the multifrequency undulator is derived. The energy spectrum of radiation and the selection rules for the TAM projection of radiated photons are described. The symmetry properties of the spectrum with respect to the TAM projection are established. The interpretation to the energy spectrum and to the selection rules is given in terms of virtual photons mediating between the charged particle and the undulator. The results obtained are also applicable to radiation of twisted photons produced by ultrarelativistic charged particles moving in plane multifrequency electromagnetic waves.

Figures (4)

• Figure 1: The energy and TAM projection spectra of twisted photon radiation from the helical two-frequency undulator. The Lorentz factor of electrons $\gamma=1.566\times 10^4$, the magnetic field strengths in undulator $H_x^i=-H_y^i=1.16\times 10^4$ G, $i=\overline{1,2}$, and the number of undulator sections $N=40$. The frequencies of subundulators are $\omega_1=2.07\times 10^{-5}$ eV, $\omega_2=3.10\times 10^{-5}$ eV and so the undulator strength parameters are $K_1=6.5$, $K_2=4.3$, $K=7.8$, and the frequencies $\tilde{\omega}_1=82.1$ eV, $\tilde{\omega}_2=123.2$ eV. Therefore, $\eta_2=3/2$, $\lambda_1=2$, $\lambda_2=3$, and the respective Bézout coefficients become $n_1^0=2$, $n_2^0=-1$. It is clear from the plots (ii) that the selection rule \ref{['sel_rul2-1']} is fulfilled. The restrictions on the numbers of virtual photons, $|n_i|$, discussed after Eqs. \ref{['n_i_restrictions']}, \ref{['q_ij_restrictions']} determine the positions of the main peaks in the distribution over $m$.
• Figure 2: The energy and TAM projection spectra of twisted photon radiation from the helical two-frequency undulator. The Lorentz factor of electrons $\gamma=1.566\times 10^4$, the magnetic field strengths in undulator $H_x^i=-H_y^i=1.16\times 10^4$ G, $i=\overline{1,2}$, and the number of undulator sections $N=40$. The frequencies of subundulators are $\omega_1=-2.07\times 10^{-5}$ eV, $\omega_2=3.10\times 10^{-5}$ eV and so the undulator strength parameters are $K_1=6.5$, $K_2=4.3$, $K=7.8$, and the frequencies $\tilde{\omega}_1=-82.1$ eV, $\tilde{\omega}_2=123.2$ eV. Therefore, $\eta_2=-3/2$, $\lambda_1=2$, $\lambda_2=-3$, and the respective Bézout coefficients become $n_1^0=-1$, $n_2^0=-1$. It is clear from the plots (ii) that the selection rule \ref{['sel_rul2-1']} is fulfilled. The restrictions on the numbers of virtual photons, $|n_i|$, discussed after Eqs. \ref{['n_i_restrictions']}, \ref{['q_ij_restrictions']} determine the positions of the main peaks in the distribution over $m$. The plot (i) with $m=0$ is depicted only for the photon energies $k_0>27$ eV to cut off the large infrared contribution of edge radiation.
• Figure 3: The energy and TAM projection spectra of twisted photon radiation from the helical three-frequency undulator. The Lorentz factor of electrons $\gamma=1.566\times 10^4$, the magnetic field strengths in undulator $H_x^i=-H_y^i=1.16\times 10^4$ G, $i=\overline{1,2,3}$, and the number of undulator sections $N=40$. The frequencies of subundulators are $\omega_1=2.07\times 10^{-5}$ eV, $\omega_2=-3.09\times 10^{-5}$ eV, $\omega_3=7.23\times 10^{-5}$ eV and so the undulator strength parameters are $K_1=6.5$, $K_2=4.3$, $K_3=1.9$, $K=8.04$, and the frequencies $\tilde{\omega}_1=77.7$ eV, $\tilde{\omega}_2=-116.6$ eV, $\tilde{\omega}_3=272.1$ eV. Therefore, $\eta_2=-3/2$, $\eta_3=7/2$, $\lambda_1=2$, $\lambda_2=-3$, $\lambda_3 = 7$ and the respective Bézout coefficients become $n_1^0=-1$, $n_2^0=-1$, $n_3^0=0$. It is clear from the plots (ii) that the selection rule \ref{['sel_rul_circ']} is fulfilled. The restrictions on the numbers of virtual photons, $|n_i|$, discussed after Eqs. \ref{['n_i_restrictions']}, \ref{['q_ij_restrictions']} determine the positions of the main peaks in the distribution over $m$.
• Figure 4: The energy and TAM projection spectra of twisted photon radiation from the composition of two planar orthogonal undulators. The Lorentz factor of electrons $\gamma=1.566\times 10^4$, the magnetic field strengths in undulator $H_x^2=-H_y^1=1.16\times 10^4$ G, $H_x^1=H_y^2=0$ G, and the number of undulator sections $N=40$. The frequencies of subundulators are $\omega_1=2.07\times 10^{-5}$ eV, $\omega_2=3.44\times 10^{-5}$ eV and so the undulator strength parameters are $K_1=4.6$, $K_2=2.8$, $K=5.4$, and the frequencies $\tilde{\omega}_1=172.8$ eV, $\tilde{\omega}_2=288.0$ eV. Therefore, $\eta_2=5/3$, $\lambda_1=3$, $\lambda_2=5$, and the respective Bézout coefficients become $n_1^0=2$, $n_2^0=-1$. It is clear from the plots (ii) that the selection rule \ref{['sel_rul_2planar1']} is fulfilled. The contribution with $m=0$ at the harmonic $n=1$ that seems to violate the selection rule stems from the neighboring harmonic with $n=2$ and from edge radiation.