Elliptical Multifrequency Undulators

• Elliptical multifrequency undulators are magnetic devices formed by superposing several one-frequency elliptical subundulators with distinct lengths, frequencies, amplitudes, chiralities, and phases.
• They induce ultrarelativistic electron trajectories whose multifrequency oscillations naturally yield harmonics and allow for precise control of polarization and angular-momentum selection.
• Advanced analysis combines 3D simulations, surrogate modeling, and machine-side studies to optimize radiation output while minimizing impacts on storage-ring dynamics.

Elliptical multifrequency undulators are undulators in which the magnetic field is constructed as a superposition of several elliptical one-frequency subundulators with different section lengths, frequencies, amplitudes, chiralities, and phases. In the direct theoretical formulation, the near-axis field is written as a sum of transverse components $H_x$ and $H_y$, the electron trajectory is an ultrarelativistic multifrequency elliptical motion, and the emitted radiation is naturally resolved into harmonics and, in the twisted-photon basis, into states labeled by helicity $s$ and total angular momentum (TAM) projection $m$. Helical multifrequency undulators are the special case $a_i=b_i$, while planar multifrequency devices are degenerate elliptical limits. In adjacent literatures, however, the same phrase is often used more loosely for APPLE-II elliptically polarizing undulators, quasi-periodic EPUs, and harmonic-resolved FEL models; these are closely related but not equivalent categories (Bogdanov et al., 27 Jul 2025, Bogdanov et al., 8 Feb 2026).

1. Conceptual scope and related device classes

Several technically distinct classes recur under the broader umbrella of elliptical or polarization-variable undulators.

Class	Defining feature	Representative paper
True elliptical multifrequency undulator	Composition of $M$ elliptical one-frequency undulators with different $l_i,\omega_i,\chi_i$	(Bogdanov et al., 27 Jul 2025)
Harmonic-multifrequency elliptical FEL model	One elliptical undulator with resonant odd harmonics $n=1,3,5,\dots$	(Henderson et al., 2016)
Two-period transverse-field EPU mode	Orthogonal field periods satisfy $\lambda_x=1.5\,\lambda_y$	(0802.0237)
Quasi-periodic APPLE-II universal mode	Quasi-periodic APPLE-II EPU used to suppress harmonic contamination	(Sheppard et al., 2022)

In the strict sense developed for twisted-photon radiation, an elliptical multifrequency undulator is a magnetic composition of $M$ coaxial one-frequency undulators. The generic case is elliptical, the helical case is obtained by imposing $H_y$0, and planar constructions arise as limiting cases. This definition is explicit in the multifrequency radiation theory and differs from the more common APPLE-II usage, where one usually has a single fundamental spatial period and varies polarization by phasing the magnet rows rather than by superposing several distinct spatial frequencies (Bogdanov et al., 27 Jul 2025).

A frequent source of confusion is the word “multifrequency.” In the FEL paper on elliptically polarised undulators, “multifrequency” refers to simultaneous coupling to the fundamental and odd resonant harmonics of a single elliptical undulator, not to a magnetic structure with several independent periods. In the PLS-II EPU72 study, the device is an APPLE-II elliptically polarizing undulator whose machine impact changes with polarization phase, but it is not a multifrequency magnetic design. The knot-undulator paper is closer to multifrequency thinking because it deliberately imposes different transverse field periods, while the quasi-periodic APPLE-II work uses non-periodicity chiefly to suppress harmonic contamination rather than to create simultaneous engineered output lines (Henderson et al., 2016, Shin et al., 2016, 0802.0237, Sheppard et al., 2022).

2. Magnetic field representations and electron motion

For an $H_y$1-frequency elliptical undulator, the near-axis stationary magnetic field is modeled as

$H_y$2

with

$H_y$3

Here $H_y$4 is the period parameter of the $H_y$5-th subundulator, $H_y$6 and $H_y$7 are its field amplitudes, and $H_y$8 is its phase. The sign in $H_y$9 fixes chirality, equivalently the sign of the undulator frequency $s$0 (Bogdanov et al., 27 Jul 2025).

The ultrarelativistic trajectory follows from

$s$1

and can be written as

$s$2

$s$3

The total undulator strength is

$s$4

This representation makes the multifrequency character explicit: the transverse motion contains a superposition of oscillations at $s$5, while the longitudinal motion contains sum- and difference-frequency terms through $s$6 and $s$7 (Bogdanov et al., 27 Jul 2025).

A convenient decomposition uses

$s$8

with

$s$9

The parameters $m$0 and $m$1 separate the two circularly rotating components of each elliptical oscillation. In the helical limit, $m$2, so $m$3; many Bessel factors then collapse, and the mode structure becomes much more transparent (Bogdanov et al., 8 Feb 2026).

This general framework should be distinguished from APPLE-II polarization control. In APPLE-II devices, ellipticity is created by longitudinal phasing of orthogonal field components rather than by superposing several independent periods. In the PLS-II EPU72 study, the undulator phase is $m$4, and the on-axis fields are written in the standard form

$m$5

$m$6

That is a single-period variable-polarization architecture, not a true multifrequency superposition (Shin et al., 2016).

3. Radiation spectrum, harmonics, and angular-momentum selection

In the twisted-photon formulation, the average number of emitted photons is expressed as

$m$7

where $m$8 is helicity, $m$9 is the TAM projection, $a_i=b_i$0, and $a_i=b_i$1. The spectral peaks occur at

$a_i=b_i$2

so the emitted energy is a linear combination of the constituent undulator frequencies weighted by integers $a_i=b_i$3 (Bogdanov et al., 27 Jul 2025).

When the frequency ratios are rational, the spectrum can be reduced to an effective fundamental frequency. Writing $a_i=b_i$4, defining $a_i=b_i$5, $a_i=b_i$6, $a_i=b_i$7, and $a_i=b_i$8, one obtains

$a_i=b_i$9

The theory states that $M$0, with equality only if all $M$1. Accordingly, the lowest harmonic of the multifrequency undulator is generally lower than that of any constituent one-frequency undulator unless all frequencies are integer multiples of one of them (Bogdanov et al., 27 Jul 2025).

For generic elliptical multifrequency undulators, the near-axis TAM rule is a parity condition: $M$2 This extends the one-frequency elliptical-undulator selection rule

$M$3

proved for radiation of twisted photons from an electron moving on an elliptical helix (Kazinski et al., 2021). Under chirality reversal,

$M$4

the multifrequency spectrum obeys

$M$5

so the TAM-helicity distribution has a definite reflection symmetry (Bogdanov et al., 27 Jul 2025).

The helical case is sharper. For helical multifrequency undulators,

$M$6

At fixed harmonic $M$7, the allowed $M$8-values are determined by the Diophantine structure of

$M$9

In the fully worked three-frequency helical case, the allowed TAM projections form an equidistant comb,

$l_i,\omega_i,\chi_i$0

and the relative phases of any three admissible modes can be tuned to arbitrary values by adjusting the constituent undulator phases $l_i,\omega_i,\chi_i$1. For $l_i,\omega_i,\chi_i$2, the rule becomes $l_i,\omega_i,\chi_i$3; for $l_i,\omega_i,\chi_i$4, all integer $l_i,\omega_i,\chi_i$5 are allowed at fixed $l_i,\omega_i,\chi_i$6 (Bogdanov et al., 8 Feb 2026).

A different but related harmonic structure appears in FEL theory for a single elliptical undulator. There the averaged 1D model couples only odd harmonics $l_i,\omega_i,\chi_i$7, with ellipticity-dependent coupling $l_i,\omega_i,\chi_i$8. The notable result is that harmonic coupling is not always maximal in the planar limit: for the third harmonic with $l_i,\omega_i,\chi_i$9, $n=1,3,5,\dots$0 is maximized near $n=1,3,5,\dots$1, so ellipticity acts as a harmonic-coupling control parameter as well as a polarization control parameter (Henderson et al., 2016).

4. Polarization-control architectures and representative devices

Representative hardware associated with elliptical multifrequency thinking spans several distinct architectures.

Device or mode	Key structural feature	Representative result
PLS-II EPU72	APPLE-II, $n=1,3,5,\dots$2 period, variable row phasing	Storage-ring optics perturbations remained small (Shin et al., 2016)
Knot undulator	$n=1,3,5,\dots$3, $n=1,3,5,\dots$4	$n=1,3,5,\dots$5 linear polarization at $n=1,3,5,\dots$6 (0802.0237)
QMSC quasi-periodic EPU	Quasi-periodic APPLE-II, $n=1,3,5,\dots$7	Fundamental $n=1,3,5,\dots$8 and $n=1,3,5,\dots$9 modeled over universal mode (Sheppard et al., 2022)

The PLS-II EPU72 is an APPLE-II elliptically polarizing undulator built from four standard Halbach-type magnet arrays, with period length $\lambda_x=1.5\,\lambda_y$0, $\lambda_x=1.5\,\lambda_y$1 maximum peak field, $\lambda_x=1.5\,\lambda_y$2 minimum magnetic gap, and NdFeB permanent magnets. Its polarization is changed by shifting the diagonal arrays relative to fixed arrays. The paper explicitly discusses horizontal mode at $\lambda_x=1.5\,\lambda_y$3, circular mode at $\lambda_x=1.5\,\lambda_y$4, and vertical mode at $\lambda_x=1.5\,\lambda_y$5. Although it is not a multifrequency device in the spectral-engineering sense, it is a canonical case showing how polarization phasing changes both the on-axis field composition and the off-axis 3D field structure (Shin et al., 2016).

The knot undulator is a specialized operating mode for an electromagnetic EPU and is physically closer to a two-frequency transverse-field configuration. In the main example, the vertical field retains period $\lambda_x=1.5\,\lambda_y$6, while the horizontal field is modified to $\lambda_x=1.5\,\lambda_y$7 by inverting the polarization of half the magnetic poles and turning off the poles between one-period right- and left-handed sections to produce a $\lambda_x=1.5\,\lambda_y$8 phase shift. With $\lambda_x=1.5\,\lambda_y$9 and $M$0, the device yields horizontally polarized $M$1 fundamental radiation with $M$2 linear polarization, while the total power in the $M$3 acceptance is $M$4 versus $M$5 for the comparison linear undulator, i.e. $M$6 of the linear-mode value. The same mode lowers the minimum photon energy from $M$7 in linear operation to $M$8 in knot mode (0802.0237).

The QMSC beamline insertion device at the Canadian Light Source is a quasi-periodic APPLE-II EPU with magnetic period $M$9, operated in universal mode with controls $H_y$00. It covers approximately $H_y$01 to $H_y$02 and introduces quasi-periodicity by vertically offsetting certain magnet blocks to reduce contamination of harmonics present in the undulator spectrum. The paper emphasizes that, for this quasi-periodic device, standard periodic-undulator approximations based on effective Fourier fields become inaccurate; one example gives approximately $H_y$03 from the Fourier-derived estimate versus approximately $H_y$04 from SRW, a discrepancy of about $H_y$05 (Sheppard et al., 2022).

5. Simulation formalisms, resonance models, and control surrogates

Modern analysis of elliptical and related multifrequency devices is dominated by 3D, time-dependent, self-consistent simulation. In the polarization-evolving FEL formulation, the essential ingredients are full Newton–Lorentz integration of the particles in analytic 3D magnetostatic fields and simultaneous coupling to two independent optical polarizations. The APPLE-II field is parameterized analytically as the superposition of two orthogonal planar undulators phase-shifted by $H_y$06, and the undulator ellipticity is defined by

$H_y$07

The same work gives the resonance condition

$H_y$08

and the generalized coupling factor

$H_y$09

A crucial result is that magnetic ellipticity and optical polarization are not identical: even when the APPLE-II is configured with $H_y$10, the simulated output is not purely circularly polarized because the APPLE-II field model is not identical to a perfect helical field, especially off axis (Freund et al., 2020).

The earlier 3D formulation with planar, helical, and elliptical undulators supplies the same core numerical ingredients—full Newton–Lorentz particle tracking, analytic APPLE-II magnetostatic fields, Gauss–Laguerre optical modes for elliptical and helical radiation, and multiharmonic time-domain evolution. In that formulation the distance to saturation decreases with increasing ellipticity, consistent with the generalized $H_y$11 trend, and the APPLE-II model is implemented as a near-axis superposition of two orthogonal planar fields with a phase shift $H_y$12 (Freund et al., 2016).

For quasi-periodic APPLE-II control, the QMSC work uses RADIA for magnetic modeling and SRW for radiation calculations, then replaces repeated field/radiation solves by a neural-network surrogate

$H_y$13

The base network, NN4, has architecture

$H_y$14

with ReLU hidden layers, linear output, Adam optimization, batch size $H_y$15, and $H_y$16 epochs. On the $H_y$17-case simulated dataset it achieved average MSE $H_y$18, average MSEV $H_y$19, and on $H_y$20 test cases

$H_y$21

With transfer learning on $H_y$22 measured cases, the calibrated model reached average MSE $H_y$23 and average MSEV $H_y$24, while satisfying the beamline circular-polarization criterion $H_y$25 for $H_y$26 (Sheppard et al., 2022).

These methods collectively indicate that elliptical multifrequency undulators are no longer analyzed solely by on-axis resonance formulas. The dominant practice combines 3D magnetostatics, full-orbit dynamics, harmonic-resolved radiation models, and, where the control manifold is large, surrogate models calibrated to measured data. This suggests that future true multifrequency elliptical devices will likely be designed and operated through the same layered workflow.

6. Storage-ring integration and machine-side constraints

For storage-ring deployment, the decisive issue is not only spectral and polarization performance but also the off-axis 3D magnetic quality. The PLS-II EPU72 study identifies three perturbation classes: orbit distortion from changing gap or row phase; tune shift and coupling from quadrupole and skew-quadrupole components; and intrinsic nonlinear effects from transverse field roll-off (TFR). The paper’s central point is that TFR, combined with the natural oscillatory motion of electrons through the undulator, generates dynamic multipoles that can affect tune, beta-beat, dynamic aperture, injection, and lifetime (Shin et al., 2016).

The machine analysis uses RADIA to generate 3D kick maps, Accelerator Toolbox for $H_y$27D tracking, and Frequency Map Analysis with NAFF over $H_y$28 turns. EPU72 is installed in a long straight section with

$H_y$29

For the three polarization states at minimum gap $H_y$30, the reported tune shifts and rms beta beating are small. In horizontal mode,

$H_y$31

with $H_y$32 horizontal and $H_y$33 vertical rms beta beating. In circular mode,

$H_y$34

with $H_y$35 horizontal and $H_y$36 vertical beta beating. In vertical mode,

$H_y$37

with $H_y$38 horizontal and $H_y$39 vertical beta beating. The dynamic aperture remains up to $H_y$40 horizontally and $H_y$41 vertically, and the study concludes that EPU72 will not reduce lifetime or cause injection problems (Shin et al., 2016).

Several machine-side lessons transfer directly to future elliptical multifrequency undulators. First, polarization tuning or frequency superposition changes the full 3D field, not just the on-axis spectrum. Second, correcting first and second field integrals does not remove the intrinsic nonlinear effects generated by TFR. Third, practical mechanics matter: in EPU72 a horizontal clearance of approximately $H_y$42 between left and right arrays produces a dip in the vertical field at $H_y$43 in zero-phase mode. A plausible implication is that true multifrequency elliptical devices, which introduce richer transverse-field structure, will require the same machine-acceptance tests—kick maps, tune and beta-beat evaluation, $H_y$44D tracking, and frequency-map analysis across all operating phases, gaps, and worst-case field combinations.