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.

💡 Research Summary

The paper presents a comprehensive theoretical study of radiation from a three‑frequency helical undulator and demonstrates its capability to generate photons in so‑called composite twisted states—coherent superpositions of modes with well‑defined total angular momentum (TAM) projections, amplitudes, relative phases, and polarizations. Starting from the general description of an M‑frequency undulator, the authors write the magnetic field as a linear superposition of M coaxial one‑frequency helical fields, each characterized by its own frequency ω_i, phase χ_i, and strength H_i. For an ultra‑relativistic electron (γ≫1, K_i/γ≪1) moving predominantly along the undulator axis, the trajectory is expressed as a sum of circular transverse oscillations with amplitudes a_i, b_i and longitudinal oscillations generated by cross‑terms c_{ij}, d_{ij}.

The radiation amplitude for a twisted photon is derived in closed form (Eq. 6 in the paper) as a multi‑dimensional sum over integer indices n_i, r_i, p_{ij}, q_{ij}, multiplied by a sinc‑type factor δ_N that encodes the resonance condition for a long undulator (N≫1). When the undulator is specialized to three frequencies and the helical condition a_i=b_i=r_i is imposed, many of the Bessel‑function arguments vanish, dramatically simplifying the amplitude. The resulting expression (Eq. 31) contains a product of three Bessel functions J_{n_i}(ρ_i) with arguments ρ_i = k_⊥ r_i, a phase factor e^{i(n_1χ_1+n_2χ_2+n_3χ_3)}, and a special function J_{n_1 n_2 n_3}(ρ_1,ρ_2,ρ_3;Δ_{12},Δ_{13},Δ_{23}) that accounts for the longitudinal coupling through Δ_{ij}=k_3 d_{ij}.

A central part of the analysis concerns the case where the frequency ratios η_i = ω_i/ω_1 are rational numbers. By introducing a common denominator g and the mutually prime integers λ_i, the authors rewrite the resonance condition as k_0 = ω n/(1−β_z n_z), where n is the “principal quantum number” (or harmonic number) satisfying λ_1 n_1 + λ_2 n_2 + λ_3 n_3 = n. Because the λ_i are coprime, there exist infinitely many integer triples (n_1,n_2,n_3) for a given n. The authors solve the associated homogeneous Diophantine equation λ_1 δn_1 + λ_2 δn_2 + λ_3 δn_3 = 0 using a two‑step reduction: first separating λ_1 from the greatest common divisor d_{23}=gcd(λ_2,λ_3), then solving the remaining two‑variable equation with Bézout coefficients. The general solution (Eq. 24) is expressed in terms of two arbitrary integers k_1 and k_2, providing explicit formulas for the shifts δn_i that map any chosen harmonic n to a concrete set of mode indices. This construction yields simple analytic expressions for the complex amplitude of the coherent photon state and for the average number of emitted twisted photons when the frequency ratios are rational.

The paper emphasizes that the relative phases of the three constituent modes are directly controlled by the undulator phases χ_i, while the amplitudes are governed by the undulator strengths K_i (or equivalently the transverse oscillation radii r_i). Consequently, by adjusting χ_i one can impose any desired phase relationship among the three modes, and by tuning K_i one can selectively enhance or suppress specific TAM projections. The selection rules emerging from the formalism are: (i) conservation of total angular momentum projection, m = n_1 + n_2 + n_3, and (ii) energy–momentum conservation embodied in the resonance condition k_0 = Σ n_i ω_i/(1−β_z n_z).

To validate the theory, the authors perform numerical simulations using realistic parameters (electron energy γ≈10³, undulator strengths K_i≈0.5–1, periods l_i of a few centimeters). The simulations reproduce the predicted harmonic structure, confirm the existence of overlapping triples (n_1,n_2,n_3) for a given n, and demonstrate resonance‑induced amplification of particular TAM modes when the third frequency is introduced. Moreover, varying the phases χ_i in the simulations leads to a linear shift of the relative phase of the composite state, confirming the analytical prediction that the composite state’s phase can be arbitrarily set.

In the discussion, the authors argue that three‑frequency helical undulators provide a bright, tunable source of composite twisted photons across a wide energy range, from optical to γ‑ray regimes, surpassing conventional methods based on phase plates, metasurfaces, or beam‑combining, which are limited in bandwidth and brightness. The ability to engineer the photon’s TAM spectrum, amplitude distribution, and phase relations opens new possibilities for quantum information transfer, angular‑momentum‑resolved scattering experiments, chiral spectroscopy, and the manipulation of rotational degrees of freedom in atoms, molecules, and condensed‑matter systems.

In summary, the paper delivers four key contributions: (1) it establishes that a three‑frequency helical undulator can directly emit photons in composite twisted states; (2) it derives explicit selection rules and amplitude formulas for rational frequency ratios via Diophantine analysis; (3) it shows that the undulator’s phase and strength parameters allow deterministic control over the constituent mode amplitudes and relative phases; and (4) it corroborates the theoretical framework with detailed numerical simulations, demonstrating practical feasibility. This work thus provides a solid theoretical foundation and a practical design methodology for next‑generation twisted‑photon light sources operating at high energies.