Three-frequency helical undulator as a source of photons in composite twisted states

TL;DR

The paper tackles generating photons in composite twisted states with tunable total angular momentum (TAM) projections across a broad energy range. It develops general $M$-frequency undulator radiation theory, then specializes to a three-frequency, rational-ratio case to derive explicit amplitudes $\,\mathcal{A}^*$ and mean numbers for twisted photons, together with TAM-selection rules. By solving linear Diophantine equations, the authors obtain harmonic-specific amplitude expressions and show how the three TAM-carrying modes can be phase-controlled via undulator phases $\\chi_i$, including the phase-locking choice $\\chi_i=\\bar{\\chi}$. Numerical simulations corroborate the theory, revealing resonances and practical tunability of TAM spectra and relative phases across harmonics. Overall, the work establishes three-frequency helical undulators as versatile, bright sources of photons in composite twisted states with controllable properties.

Abstract

The properties of radiation from a three-frequency helical undulator are thoroughly investigated. It is shown that such undulators can be employed for generating photons in the so-called composite twisted states -- the states that are linear superpositions of modes with definite projections of the total angular momentum, amplitudes, relative phases, and polarizations. We find the explicit expressions and the selection rules for these parameters and establish that they can be governed in a predictable way by adjusting the parameters of the multifrequency helical undulator. In particular, the phases of three arbitrary modes admissible by selection rules in the composite state with definite energy can be made arbitrary by tuning the phases of one-frequency undulators comprising the three-frequency one. By solving Diophantine equations, we obtain simple expressions for the complex amplitude of coherent state of photons emitted by a three-frequency helical undulator and for the average number of radiated twisted photons in the case when the ratios of frequencies of the three-frequency undulator are rational numbers. The development of resonances and the control of composite states of radiated photons are studied numerically confirming the theoretical conclusions.

Figures (2)

• Figure 1: The spectrum of the TAM projections of radiation at the first eight harmonics of the three-frequency helical undulator at $n_\perp =K/\gamma$ and different $K_3$ and $s$. The energy of electrons in the undulator is $8$ GeV, the lengths of sections $(l_1,l_2,l_3)=(1,-3/2,7/2)\times 35$ cm. The magnetic field strengths are $(H^1_{x,y},H^2_{x,y},H^3_{x,y})=(0.2,1.16,1.16)\times 10^4$ G for the undulator strength parameters $(K_1,K_2,K_3)=(6.54,8.44,10.85)$. The energy of photons at the fundamental harmonic $k_0=1.87$ eV for these parameters. The undulator phases $\chi_i=0$. Only those values of the TAM projections are shown that give non-negligible contributions. It is seen that the selection rule \ref{['sel_rul3_part']} is fulfilled.
• Figure 2: The spectrum of the TAM projections of radiation at the fourth harmonic of the three-frequency helical undulator at $n_\perp =K/\gamma$ and different $K_3$ and $s$. The energy of electrons in the undulator is $8$ GeV, the lengths of sections $(l_1,l_2,l_3)=(1,-3/2,7/2)\times 35$ cm. The magnetic field strengths are $(H^1_{x,y},H^2_{x,y},H^3_{x,y})=(25,1.16\times 10^4,3.22\times 10^3)$ G for the undulator strength parameters $(K_1,K_2,K_3)=(8.18\times 10^{-2},10.85,8.44)$. The energy of photons at the fourth harmonic $k_0=4.58$ eV for these parameters. The undulator phases $\chi_i=0$. Numerical simulations show that, in changing the undulator phases $\{\chi_i\}$, the relative phase between the modes with $m=3$ and $m=1$ agrees with \ref{['rel_phase_m31']}.