Helical Multifrequency Undulators

• Helical multifrequency undulators are magnetic structures that superimpose several helical or elliptical field components to generate controlled radiation spectra and twisted photon emission.
• They enable precise tuning of harmonic content and total angular momentum through configurable field amplitudes, periods, and phase offsets, leading to coherent multimode radiation.
• Recent hardware prototypes using NdFeB helices demonstrate compact, high-field implementations while highlighting challenges like off-axis angular flux reduction in practical applications.

Searching arXiv for recent and foundational papers on helical multifrequency undulators, twisted-photon radiation, and helical undulator hardware. A helical multifrequency undulator is a magnetic structure in which the transverse undulator field is formed as a superposition of several helical or elliptical components with different periods, amplitudes, chiralities, and phase offsets. In the modern theoretical formulation, the field is written as

$$H_x=\sum_{i=1}^M H_x^i\sin \tilde{}_i,\qquad H_y=\sum_{i=1}^M H_y^i\cos \tilde{}_i,\qquad H_z=0,$$

with

$\tilde_i=\pm \frac{2\pi z}{l_i}+\chi_i,$

so that each subundulator contributes its own frequency, polarization content, and phase to the overall device (Bogdanov et al., 27 Jul 2025, Bogdanov et al., 8 Feb 2026). Such systems are studied for two closely related reasons. First, helical geometry provides circular or elliptical polarization and strong transverse coupling in both $x$ and $y$. Second, multifrequency composition makes the radiation spectrum, harmonic content, and, in twisted-photon formulations, the projection of total angular momentum (TAM) and the relative phases between modes directly controllable by undulator design (Bogdanov et al., 27 Jul 2025, Bogdanov et al., 8 Feb 2026). At the same time, helical operation imposes nontrivial angular-redistribution constraints on harmonic emission, most notably the suppression of on-axis nonlinear harmonic generation in the ideal helical case (Allaria et al., 2011).

1. Definition and field structure

In the multifrequency theory, the electron is ultrarelativistic and moves mainly along the $z$-axis through a stationary magnetic field built as a superposition of $M$ elliptical undulator sections. The weak-angle and undulator assumptions are

$\gamma\gg 1,\qquad K_i/\gamma\ll 1,$

with

$$K_i^2=\frac{\omega_i^2}{2}(a_i^2+b_i^2),\qquad a_i=\frac{eH_y^i}{m_e\omega_i^2},\qquad b_i=-\frac{eH_x^i}{m_e\omega_i^2},$$

and

$$\beta_3=1-\frac{1+K^2}{2\gamma^2},\qquad K^2=\sum_i K_i^2$$

(Bogdanov et al., 27 Jul 2025). The central physical point is that the charged particle executes a superposition of oscillations at multiple frequencies, and the radiation field therefore acquires a correspondingly rich harmonic structure.

For the three-frequency helical specialization, the field is written as

$H_x=H_x^1\sin\tilde_1+H_x^2\sin\tilde_2+H_x^3\sin\tilde_3,\qquad H_y=-H_x^1\cos\tilde_1-H_x^2\cos\tilde_2-H_x^3\cos\tilde_3,\qquad H_z=0,$

with

$\tilde_i=\pm \frac{2\pi z}{l_i}+\chi_i,$0

so that the electron trajectory becomes

$\tilde_i=\pm \frac{2\pi z}{l_i}+\chi_i,$1

(Bogdanov et al., 8 Feb 2026). This form shows that helical multifrequency operation is not merely a superposition of independent circular motions. Longitudinal mixing terms appear through the coefficients

$\tilde_i=\pm \frac{2\pi z}{l_i}+\chi_i,$2

which encode inter-frequency coupling in the trajectory itself (Bogdanov et al., 8 Feb 2026).

A common misconception is that any helical undulator is automatically a multifrequency undulator. The recent hardware papers on short-period NdFeB helices explicitly do not present experiments on multifrequency tuning, simultaneous multi-harmonic operation, or multiple radiation frequencies; rather, they provide fabrication, magnetization, assembly, and field characterization that are enabling and foundational for more advanced helical multifrequency concepts (Balal et al., 1 Aug 2025, Magory et al., 7 Sep 2025).

2. Radiation spectrum and generalized harmonic conditions

The multifrequency radiation spectrum is governed by an energy-selection condition that generalizes the ordinary undulator harmonic relation. For twisted-photon radiation in an $\tilde_i=\pm \frac{2\pi z}{l_i}+\chi_i,$3-frequency undulator, the main harmonics satisfy

$\tilde_i=\pm \frac{2\pi z}{l_i}+\chi_i,$4

(Bogdanov et al., 27 Jul 2025). In the three-frequency case, this becomes

$\tilde_i=\pm \frac{2\pi z}{l_i}+\chi_i,$5

(Bogdanov et al., 8 Feb 2026). The integers $\tilde_i=\pm \frac{2\pi z}{l_i}+\chi_i,$6 play the role of generalized harmonic indices associated with exchange processes involving the different subundulator frequencies.

When the frequency ratios are rational, the spectrum reduces to an equidistant ladder. If

$\tilde_i=\pm \frac{2\pi z}{l_i}+\chi_i,$7

then one may introduce a common effective base frequency, and the spectrum is rewritten as

$\tilde_i=\pm \frac{2\pi z}{l_i}+\chi_i,$8

(Bogdanov et al., 27 Jul 2025). In the three-frequency construction the reduction is achieved by defining

$\tilde_i=\pm \frac{2\pi z}{l_i}+\chi_i,$9

and then

$x$0

so that the admissible integer triples solve the Diophantine equation

$x$1

(Bogdanov et al., 8 Feb 2026).

This number-theoretic reduction is one of the distinctive features of helical multifrequency theory. It implies that several different tuples $x$2 may contribute to the same emitted energy. The radiated state at fixed energy is therefore naturally a coherent superposition rather than a single harmonic line in the one-frequency sense (Bogdanov et al., 27 Jul 2025, Bogdanov et al., 8 Feb 2026).

A further notable consequence, emphasized for the rational-frequency case, is that the lowest effective frequency is smaller than the smallest subundulator frequency unless all $x$3 are integer multiples of one of them (Bogdanov et al., 27 Jul 2025). This suggests that multifrequency composition can generate spectral structure not obtainable by simply concatenating one-frequency helical sections.

3. Twisted photons, TAM selection rules, and composite states

The most developed contemporary application of helical multifrequency undulators is as a source of twisted photons. In the semiclassical formulation, a twisted photon is labeled by helicity $x$4, TAM projection $x$5, longitudinal and transverse momenta $x$6, and energy $x$7. The general average number of emitted twisted photons is

$x$8

with

$x$9

(Bogdanov et al., 27 Jul 2025).

For trajectories close to the axis, the general multifrequency elliptical selection rule is

$y$0

whereas the helical specialization yields the sharper condition

$y$1

(Bogdanov et al., 27 Jul 2025). In the three-frequency helical case, the near-axis condition is stated as

$y$2

(Bogdanov et al., 8 Feb 2026). The interpretation given in both works is in terms of virtual photons exchanged between the electron and the different helical components of the undulator field; the TAM transferred through those exchanges appears in the emitted twisted photon.

For rational frequencies and fixed harmonic $y$3, the TAM spectrum is constrained by the Diophantine structure. In the general helical multifrequency treatment, one obtains a generalized TAM rule of the form

$y$4

(Bogdanov et al., 27 Jul 2025). In the explicitly three-frequency helical case, one representation is

$y$5

and, when $y$6,

$y$7

(Bogdanov et al., 8 Feb 2026). Thus the allowed $y$8-values at fixed emitted energy form an arithmetic progression.

The three-frequency theory goes beyond selection rules and treats the emitted radiation as a composite twisted state. Its explicit claim is that such undulators can generate photons in linear superpositions of modes with definite projections of the total angular momentum, amplitudes, relative phases, and polarizations, and that these quantities can be governed in a predictable way by adjusting the parameters of the multifrequency helical undulator (Bogdanov et al., 8 Feb 2026). In the two-frequency and general $y$9-frequency formulations, the same idea appears as a coherent superposition of several TAM eigenstates at a fixed harmonic, with amplitudes controlled by $z$0, harmonic content controlled by $z$1, and relative phases controlled by $z$2 (Bogdanov et al., 27 Jul 2025).

This perspective is closely related to earlier work showing that higher harmonics of a one-frequency helical undulator carry orbital angular momentum. That work emphasized that all harmonics with $z$3 carry OAM, whereas the first harmonic is the dominant ordinary helical-undulator line concentrated on axis and does not carry OAM in the same sense (Afanasev et al., 2011). The multifrequency theory extends that single-frequency geometric argument into an explicit mode-engineering framework.

4. Angular distribution, harmonic suppression, and practical flux limits

A central constraint on helical multifrequency operation comes from the angular structure of harmonic radiation. In a helical undulator, unlike the planar case, nonlinear harmonic generation is strongly suppressed on axis and is redistributed off axis (Allaria et al., 2011). The paraxial Maxwell analysis of harmonic emission from a helical undulator gives, for the $z$4-th harmonic,

$z$5

$z$6

with

$z$7

(Allaria et al., 2011). For $z$8, the factor $z$9 forces the emission to vanish on axis in the ideal helical case.

The experimental benchmark at Elettra confirmed this structure. In coherent harmonic generation mode at $M$0 nm, the measured intensity peaks on axis. In nonlinear harmonic generation mode, with the radiator tuned to $M$1 nm and coherent emission observed at $M$2 nm as the second harmonic, the on-axis signal is much weaker, and the off-axis distribution is consistent with the theoretical prediction that the harmonic emission is concentrated away from the axis (Allaria et al., 2011). The comparison of the CHG peak to the residual on-axis NHG signal gave about $M$3 experimentally and about $M$4 in the acceptance-smeared theory, which the authors regarded as satisfactory agreement given alignment uncertainty, beam fluctuations, and model simplifications (Allaria et al., 2011).

This has direct significance for multifrequency helical devices. A plausible implication is that multifrequency design cannot be assessed solely from harmonic resonance conditions or TAM selection rules; it must also account for angular flux redistribution, finite acceptance, and the rapid reduction of usable off-axis harmonic power. The same study explicitly noted a FERMI@Elettra example in which $M$5 on-axis photons/pulse at $M$6 nm would correspond to only about $M$7 photons/pulse at the second harmonic and far fewer at the third harmonic (Allaria et al., 2011). That estimate does not describe a multifrequency device directly, but it quantifies the practical difficulty of using pure helical harmonic radiation as a high-flux operational output.

Related exact work on an extended helical undulator reinforces the importance of angular structure. The exact cylindrical-multipole solution gives the Doppler relation

$M$8

and identifies the fundamental mode $M$9 as the one responsible for radiation exactly on axis, while higher modes contribute near the axis with smaller relative weight (Ivanyan et al., 2024). This exact framework is not a multifrequency theory, but it is directly relevant to controlling harmonic overlap, angular acceptance, and quasi-monochromatic operation in helical devices (Ivanyan et al., 2024).

5. Hardware realizations and enabling technologies

Recent prototype work has established practical fabrication routes for compact helical undulators built from single-piece rare-earth helices. One approach uses wire electrical discharge machining in combination with a flat tool and rotary workpiece motion to cut helical NdFeB magnets directly from a cylinder (Balal et al., 1 Aug 2025). In the reported $\gamma\gg 1,\qquad K_i/\gamma\ll 1,$0 mm-period prototype, an assembly of two oppositely longitudinally magnetized helices with a half-period shift created a measured on-axis field of $\gamma\gg 1,\qquad K_i/\gamma\ll 1,$1 T, which, for a $\gamma\gg 1,\qquad K_i/\gamma\ll 1,$2 mm period, gives $\gamma\gg 1,\qquad K_i/\gamma\ll 1,$3 (Balal et al., 1 Aug 2025). The same paper calculated that Halbach-type helical micro-undulators with periods of $\gamma\gg 1,\qquad K_i/\gamma\ll 1,$4–$\gamma\gg 1,\qquad K_i/\gamma\ll 1,$5 mm built from four helices could provide a field of $\gamma\gg 1,\qquad K_i/\gamma\ll 1,$6 T and $\gamma\gg 1,\qquad K_i/\gamma\ll 1,$7–$\gamma\gg 1,\qquad K_i/\gamma\ll 1,$8 on axis (Balal et al., 1 Aug 2025).

A later implementation study reported two $\gamma\gg 1,\qquad K_i/\gamma\ll 1,$9 mm-period helical microundulators with a $$K_i^2=\frac{\omega_i^2}{2}(a_i^2+b_i^2),\qquad a_i=\frac{eH_y^i}{m_e\omega_i^2},\qquad b_i=-\frac{eH_x^i}{m_e\omega_i^2},$$0 mm bore (Magory et al., 7 Sep 2025). The first, composed of two oppositely magnetized longitudinal helices, produced a transverse on-axis field exceeding $$K_i^2=\frac{\omega_i^2}{2}(a_i^2+b_i^2),\qquad a_i=\frac{eH_y^i}{m_e\omega_i^2},\qquad b_i=-\frac{eH_x^i}{m_e\omega_i^2},$$1 T. The second, a hybrid design using two oppositely longitudinally pre-magnetized rare-earth helices alternating with two unmagnetized steel helices, reached about $$K_i^2=\frac{\omega_i^2}{2}(a_i^2+b_i^2),\qquad a_i=\frac{eH_y^i}{m_e\omega_i^2},\qquad b_i=-\frac{eH_x^i}{m_e\omega_i^2},$$2 T on axis (Magory et al., 7 Sep 2025). In that work, the steel helices function as flux shapers approximating an ideal helical Halbach-type arrangement while avoiding the need for radially magnetized rare-earth helices.

The standard undulator parameter relation used in these hardware studies is

$$K_i^2=\frac{\omega_i^2}{2}(a_i^2+b_i^2),\qquad a_i=\frac{eH_y^i}{m_e\omega_i^2},\qquad b_i=-\frac{eH_x^i}{m_e\omega_i^2},$$3

(Balal et al., 1 Aug 2025), and in the microundulator study

$$K_i^2=\frac{\omega_i^2}{2}(a_i^2+b_i^2),\qquad a_i=\frac{eH_y^i}{m_e\omega_i^2},\qquad b_i=-\frac{eH_x^i}{m_e\omega_i^2},$$4

(Magory et al., 7 Sep 2025). These relations connect field strength and period directly to the compact-FEL parameter regime relevant for multifrequency or harmonic applications.

The current limitation is conceptual rather than mechanical. Both hardware papers explicitly emphasize that they do not directly demonstrate multifrequency emission. Their role is enabling and foundational: they show that precisely fabricated NdFeB helices can achieve strong on-axis fields and useful $$K_i^2=\frac{\omega_i^2}{2}(a_i^2+b_i^2),\qquad a_i=\frac{eH_y^i}{m_e\omega_i^2},\qquad b_i=-\frac{eH_x^i}{m_e\omega_i^2},$$5 values in short-period geometries, which is the kind of platform needed for more advanced helical undulator architectures, including multifrequency variants (Balal et al., 1 Aug 2025, Magory et al., 7 Sep 2025).

A distinct enabling direction is the laser-based helical undulator. A Bessel light beam carrying orbital angular momentum has been proposed as an ultra-short-period helical undulator whose effective period can reach the sub-millimeter range through electron dephasing in a superluminal-phase structure (Jiang et al., 2017). That proposal is not a multifrequency theory, but it broadens the hardware concept of what may serve as a helical undulator.

6. Modeling frameworks, related regimes, and outlook

Three-dimensional and exact modeling frameworks are important because helical multifrequency devices combine polarization control, harmonic structure, transverse mode content, and, in many cases, off-axis emission. A general time-dependent FEL framework capable of modeling planar, helical, and elliptical undulators represents the optical field with Gaussian modes and variable polarization, uses full Newton–Lorentz particle tracking, and includes an analytic APPLE-II model to treat arbitrary elliptical polarizations (Freund et al., 2016). In that treatment, the APPLE-II phase shift $$K_i^2=\frac{\omega_i^2}{2}(a_i^2+b_i^2),\qquad a_i=\frac{eH_y^i}{m_e\omega_i^2},\qquad b_i=-\frac{eH_x^i}{m_e\omega_i^2},$$6 interpolates continuously between planar and helical limits, with

$$K_i^2=\frac{\omega_i^2}{2}(a_i^2+b_i^2),\qquad a_i=\frac{eH_y^i}{m_e\omega_i^2},\qquad b_i=-\frac{eH_x^i}{m_e\omega_i^2},$$7

for $$K_i^2=\frac{\omega_i^2}{2}(a_i^2+b_i^2),\qquad a_i=\frac{eH_y^i}{m_e\omega_i^2},\qquad b_i=-\frac{eH_x^i}{m_e\omega_i^2},$$8, and

$$K_i^2=\frac{\omega_i^2}{2}(a_i^2+b_i^2),\qquad a_i=\frac{eH_y^i}{m_e\omega_i^2},\qquad b_i=-\frac{eH_x^i}{m_e\omega_i^2},$$9

for $$\beta_3=1-\frac{1+K^2}{2\gamma^2},\qquad K^2=\sum_i K_i^2$$0, so that $$\beta_3=1-\frac{1+K^2}{2\gamma^2},\qquad K^2=\sum_i K_i^2$$1 is planar and $$\beta_3=1-\frac{1+K^2}{2\gamma^2},\qquad K^2=\sum_i K_i^2$$2 is helical (Freund et al., 2016). This framework does not demonstrate explicit multifrequency helical output, but it provides the numerical machinery needed to study polarization-resolved harmonic and sideband dynamics.

At long wavelengths, a THz study based on Liénard–Wiechert fields found that helical undulators typically exhibit reduced side lobes and a more focused main radiation peak, whereas planar undulators are more harmonic-rich but show broader angular structure (Choobini et al., 2024). This does not establish a multifrequency helical source in the strict sense, but it clarifies a recurring tradeoff: cleaner directionality and circular polarization in helical geometry versus richer harmonic content in planar geometry.

Twisted-photon work in dispersive media adds another related regime. In undulators filled with a homogeneous dielectric dispersive medium, the helical selection rule

$$\beta_3=1-\frac{1+K^2}{2\gamma^2},\qquad K^2=\sum_i K_i^2$$3

is preserved, plasma media can suppress lower harmonics, and inverse-polarization domains can produce larger orbital angular momentum at a given harmonic (Bogdanov et al., 2020). These results concern one-frequency helical wigglers in a medium rather than vacuum multifrequency devices, but they show that OAM selection structures remain robust under non-vacuum modifications of the spectrum.

Taken together, these results define the present state of the subject. The mature theoretical core lies in the multifrequency twisted-photon treatments, where helical superpositions give explicit control over emitted energies, TAM spectra, amplitudes, and relative phases through $$\beta_3=1-\frac{1+K^2}{2\gamma^2},\qquad K^2=\sum_i K_i^2$$4, $$\beta_3=1-\frac{1+K^2}{2\gamma^2},\qquad K^2=\sum_i K_i^2$$5, and $$\beta_3=1-\frac{1+K^2}{2\gamma^2},\qquad K^2=\sum_i K_i^2$$6 (Bogdanov et al., 27 Jul 2025, Bogdanov et al., 8 Feb 2026). The principal practical constraint is that helical harmonic radiation is strongly structured angularly and may be severely flux-limited on axis (Allaria et al., 2011). The principal engineering advance is the emergence of compact, high-field helical prototypes fabricated from single-piece rare-earth helices, which provide realistic physical platforms for future multifrequency implementations (Balal et al., 1 Aug 2025, Magory et al., 7 Sep 2025). This suggests that the near-term development of helical multifrequency undulators will depend on combining precise field synthesis, exact or full-3D modeling, and application-specific choices about whether the priority is structured twisted states, polarization control, spectral extension, or high-flux operation.