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

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.