---
title: Helical Microundulator Overview
url: https://www.emergentmind.com/topics/helical-microundulator
type: topic
---

# Helical Microundulator Overview

A helical microundulator is an undulator in which the transverse magnetic field rotates along the beam axis with helical symmetry and the magnetic period is reduced to the millimeter or micrometer regime. In the cited literature, this class includes at least two distinct implementations: a ferromagnetic helix embedded in a strong solenoidal guide field, which redistributes part of an initially uniform axial field into a rotating transverse component, and permanent-magnet devices assembled from longitudinally magnetized NdFeB helices machined from a single piece of rare-earth magnet. Across these implementations, the central objective is the same: to obtain short-period, high-field, circularly polarized undulator fields suitable for compact free-electron lasers (FELs), including THz sources, table-top X-ray FELs, and related synchrotron or superradiant systems [1801.05312] [2508.00232] [2509.06186].

## 1. Taxonomy and defining geometry

The literature presents the helical microundulator not as a single construction, but as a family of devices that realize a helical on-axis field with short period. One lineage uses a soft-steel helix with period \(d\), inner radius \(R_1\), outer radius \(R_2\), and axial length of one insertion \(a\), placed coaxially inside a solenoid. The guide field \(B_0\hat z\) magnetizes the steel to saturation along \(\hat z\), while the helical shape converts part of that flux into a rotating transverse field \(B_u(x,y,z)\). Another lineage uses rare-earth helices of NdFeB that are machined first and magnetized afterward, then assembled in either a two-helix or four-helix arrangement to generate the undulator field [1801.05312] [2508.00232].

The permanent-magnet branch itself subdivides into a simple two-helix geometry and a Halbach-like or hybrid geometry. In the two-helix case, two identical helices are uniformly magnetized axially, then assembled with opposite polarity and displaced by half a period. In the four-helix Halbach configuration, or in the hybrid NdFeB-plus-steel configuration, flux is redistributed so that the circularly polarized transverse components are enhanced while the external stray field is reduced. This suggests that the term *helical microundulator* denotes a field topology and scale rather than a unique magnetic architecture [2508.00232] [2509.06186].

| Architecture | Field-generation principle | Representative parameters |
|---|---|---|
| Ferromagnetic helix in solenoid | Redistribution of \(B_0\hat z\) into rotating \(B_u\) | Example: \(d=2.5\,\mathrm{cm}\), \(R_1=0.3\,\mathrm{cm}\), \(b=0.2\,\mathrm{cm}\), \(a=1.2\,\mathrm{cm}\) |
| Two-helix NdFeB device | Opposite axial magnetization with half-period shift | Prototype: \(\lambda_u=20\,\mathrm{mm}\), \(D_i=8\,\mathrm{mm}\), \(B_0=0.53\,\mathrm{T}\) |
| Hybrid NdFeB + steel device | NdFeB flux redirected by steel, approximating Halbach-type field | Prototype: \(d=6\,\mathrm{mm}\), bore \(1\,\mathrm{mm}\), \(B_0\approx1.5\,\mathrm{T}\) |

## 2. Magnetic principle of the helical field

For the ferromagnetic-helix concept, the unperturbed field is the uniform solenoidal field \(B_0\hat z\). The perturbation is the magnetization of the helix, written as \(J(\mathbf r,\phi,z)=J_\infty\hat z\) inside the steel and zero outside. In the thin-shell analytical model, the helix is represented at \(r=R\) with a surface magnetization expanded in harmonics \(e^{in(hz+\phi)}\), where \(h=2\pi/d\) and \(f_n=\sin(nha/2)/(\pi n)\). The resulting scalar potential is solved in Bessel form, and only the \(n=\pm1\) harmonics contribute at the axis. The on-axis field is therefore a rotating transverse mode rather than a mixture of many comparable harmonics [1801.05312].

The transverse on-axis amplitude for that model is
\[
B_u=\frac{1}{\pi}\,h^2Rb\sin(h a/2)\,K_1(hR)\,\mu_0J_\infty,
\]
with the maximum at \(a=d/2\). In the small-argument limit \(hR\ll1\), the modified Bessel function gives \(K_1(hR)\approx1/(hR)\), so
\[
B_u\approx \frac{1}{\pi}\,h\,b\,\mu_0J_\infty
      = \left(\frac{2}{d}\right)b(\mu_0J_\infty),
\]
which shows the basic scaling \(B_u\propto b/d\). The same section of the literature identifies the undulator period as \(\lambda_u=d\) and writes the dimensionless undulator parameter as
\[
K=\frac{eB_ud}{2\pi mc}.
\]
This formulation makes clear that period reduction, magnet thickness, and saturation magnetization are the controlling quantities for the achievable undulator strength [1801.05312].

The permanent-magnet formulation is analytically similar in that a helical geometry produces a rotating field on axis, but the source is remanent magnetization rather than the redistribution of an imposed solenoidal field. For a single, infinitely long, thick helix of rectangular cross section and longitudinal magnetization, the on-axis field is
\[
\mathbf{B}_u^z(r=0,z)
=\frac{\mu_0M_z}{\pi}\,\sin\!\bigl(h\,a/2\bigr)
\int_{hR_i}^{hR_o}\xi\,K_1(\xi)\,d\xi\;
\bigl[\hat x\sin(hz)+\hat y\cos(hz)\bigr],
\]
with \(h=2\pi/d\). The two-helix assembly doubles this field when the second helix has opposite \(M_z\) and axial shift \(h\Delta z=\pi\), so \(B_0^{(2)}=2B_u\) [2508.00232].

## 3. Fabrication routes and magnetic assembly

A central advance in the NdFeB line is the fabrication of helices from a single piece of rare-earth magnet by wire electrical discharge machining (WEDM). In this process, a continuously moving thin conductive wire, specified as brass or stratified copper, acts as the cathode and the NdFeB workpiece as the anode. A dielectric fluid fills the gap until spark discharge vaporizes minute volumes of material, enabling micrometer-scale removal without mechanical stress. For helical cuts, the non-magnetized NdFeB cylinder is mounted on a rotary axis and spun continuously about its symmetry axis while a flat-tool electrode traverses laterally in the \(X\)–\(Y\) plane, tracing a spiral groove [2508.00232].

The reported prototype workpiece had outer radius \(R_o=16\,\mathrm{mm}\), inner radius \(R_i=4\,\mathrm{mm}\), and length \(40\,\mathrm{mm}\). This setup achieves helical periods down to \(1\,\mathrm{mm}\) and likely below, with material damage confined to a micrometer-thick recast layer. For the prototype, two helices of period \(\lambda_u=20\,\mathrm{mm}\) and cross-section radii \(R_i=4\,\mathrm{mm}\), \(R_o=16\,\mathrm{mm}\) were cut simultaneously. Typical dimensional tolerances are on the order of \(\pm 5\,\mu\mathrm{m}\) in period and radius, which the cited work treats as sufficient for high-field undulator performance. After machining, each helix is longitudinally magnetized in a pulsed solenoid with \(>2\,\mathrm{T}\), \(2\,\mathrm{ms}\) pulses, yielding a remanent field \(B_r\approx1.4\,\mathrm{T}\) and magnetization
\[
\mathbf{M}(\mathbf r)=M_z\hat z,\qquad M_z=\frac{B_r}{\mu_0}.
\]
The two-helix prototype then uses one helix magnetized \(+M_z\hat z\), the other \(-M_z\hat z\), with an axial displacement \(\Delta z=d/2\) and inter-screwing on a central rod [2508.00232].

The later \(6\,\mathrm{mm}\)-period prototypes extend this manufacturing route to much smaller apertures. Both devices have a \(1\,\mathrm{mm}\) bore and usable length \(50\,\mathrm{mm}\), approximately \(8\) undulator periods. The simple device uses two identical NdFeB helices with \(R_2=4\,\mathrm{mm}\) and helix width \(a=d/2=3\,\mathrm{mm}\). The hybrid device uses two longitudinally pre-magnetized NdFeB helices with \(R_2=10\,\mathrm{mm}\) alternating with two pre-unmagnetized high-\(\mu\) steel helices with \(R_{2s}=4\,\mathrm{mm}\); the optimized thicknesses are \(a=0.32d\approx1.92\,\mathrm{mm}\) for NdFeB and \(a_s=0.18d\approx1.08\,\mathrm{mm}\) for steel. The reported practical measures include drilling the \(1\,\mathrm{mm}\) bore by EDM to avoid cracking brittle NdFeB, WEDM cutting with pulse current \(\approx12\,\mathrm{A}\), \(10\)–\(40\,\mu\mathrm{s}\), wire tension \(1.5\,\mathrm{N}\), and feed \(1.5\,\mathrm{m/min}\), and pulsed-solenoid magnetization at \(2\)–\(3\,\mathrm{T}\) on the millisecond scale [2509.06186].

## 4. On-axis field amplitude, Halbach enhancement, and \(K\)

The permanent-magnet prototype with \(\lambda_u=20\,\mathrm{mm}\) and \(D_i=8\,\mathrm{mm}\) establishes the basic measured field scale. For a single helix, the on-axis field is \(B_u\approx0.26\,\mathrm{T}\), with analytical, CST, and measurement results agreeing to \(1\%\). The corresponding two-helix assembly gives \(B_0=0.53\,\mathrm{T}\), both measured and simulated. The same work states that an assembly of two oppositely longitudinally magnetized helices with a period of \(20\,\mathrm{mm}\) and a relatively large inner diameter of \(8\,\mathrm{mm}\) creates a field of \(0.53\,\mathrm{T}\) on axis, ensuring an undulator parameter \(K\) close to unity. For four-helix Halbach-type micro-undulators with periods of \(3\)–\(6\,\mathrm{mm}\), the calculation gives a field of \(1\,\mathrm{T}\) and \(K=0.28\)–\(0.6\) at the axis, and the Halbach arrangement exceeds \(2B_u\) by a factor \(\gtrsim1.17\) for identical gap and thickness; elsewhere in the same analysis, the gain from two-helix to four-helix Halbach is stated as \(\sim1.2\)–\(1.4\times\) [2508.00232].

The \(6\,\mathrm{mm}\)-period devices move from prediction to implementation. For the simple \(2\times\)NdFeB prototype, the simulated on-axis field is \(0.95\)–\(0.98\,\mathrm{T}\) and the measured field is \(\ge 0.93\,\mathrm{T}\) with \(\pm2\%\). For the hybrid NdFeB-plus-steel device, the simulated and measured on-axis field is approximately \(1.5\,\mathrm{T}\), with \(\pm5\%\) on the measurement. The simple device was characterized by Hall-probe scans outside at \(r=5\,\mathrm{mm}\) from the axis, comparing CST Microwave Studio™ simulations with Senis 3MTS 3D-Hall-meter data; the reported agreement is better than \(5\%\). Because of the \(1\,\mathrm{mm}\) bore, the hybrid device required a synchronized rotating stage with a miniature 3D-Hall probe inserted through the bore so that the sensor effectively “rides” the helical field maximum while sampling \(B_x\) and \(B_y\) along \(z\) in situ [2509.06186].

The undulator parameter is written in the permanent-magnet studies as
\[
K=\frac{eB_0\lambda_u}{2\pi mc}=0.934\,B_0[\mathrm T]\,\lambda_u[\mathrm{cm}],
\]
and in the \(20\,\mathrm{mm}\) prototype this gives
\[
K_{\rm proto}\approx0.934\times0.53\times2\approx0.99\approx1.
\]
For the predicted four-helix micro-undulator with \(B_0\approx1\,\mathrm{T}\) and \(\lambda_u=3\)–\(6\,\mathrm{mm}\), the estimate is
\[
K_{\rm micro}=0.934\times1\times(0.3\text{--}0.6)\approx0.28\text{--}0.56.
\]
For the \(6\,\mathrm{mm}\), \(1.5\,\mathrm{T}\) helical microundulator in FEL comparison tables, \(K=0.84\), whereas a planar microundulator of the same period and field gives \(K=0.59\) [2508.00232] [2509.06186].

## 5. Scaling laws and microundulator regimes

The scaling analysis in the NdFeB-helical work relates the field to aspect ratios and reduced gap. From the on-axis field expressions, \(B_0\) scales roughly as
\[
B_0\propto \mu_0M\,\frac{\sin(\pi a/d)}{\pi}
\int_{(2\pi/d)R_i}^{(2\pi/d)R_o}\xi\,\bigl[\dots\bigr]\,d\xi.
\]
For fixed aspect ratios \(R_i/d\) and \(R_o/d\), shorter \(d\) produces larger reduced gap \(hR_i\), which boosts the Bessel-\(K\) integrals and increases \(B_0\). The same analysis further states that, for a given gap \(2R_i\), the four-helix Halbach requires a smaller gap to reach \(1\,\mathrm{T}\), or yields a higher field at fixed gap. These combined trends are the basis for the prediction that millimeter-period micro-undulators can reach \(B_0\sim1\,\mathrm{T}\) and \(K\sim0.3\)–\(0.6\) with only four helices of NdFeB [2508.00232].

The ferromagnetic-helix literature gives a complementary miniaturization program. The proposed strategy is to reduce the period to \(d\sim10^2\)–\(10^3\,\mu\mathrm m\) by fabricating a micro-helix, for example by lithographic winding or electroplating around a mandrel, while scaling all dimensions as \(R\sim d/2\), \(b\sim(0.05\text{--}0.2)d\), and \(a\approx d/2\), with \(J_\infty\sim1.5\)–\(2.0\,\mathrm T\). The worked example \(d=300\,\mu\mathrm m\), \(b=30\,\mu\mathrm m\), \(R=150\,\mu\mathrm m\) gives \(B_u\approx0.1\,\mathrm T\), then \(K\approx0.1\), and a fundamental wavelength \(\lambda_r\approx1.5\,\mu\mathrm m\), placing the device in the near-IR regime. In the same framework, a macroscopic THz example with \(d=2.5\,\mathrm{cm}\), \(B_u\approx0.2\,\mathrm T\), and \(\gamma\approx12\) gives \(K\approx0.5\), \(\lambda_r\approx0.092\,\mathrm{mm}\), and \(f_r\approx3.3\,\mathrm{THz}\) [1801.05312].

These scaling statements establish a continuum from centimeter-period helical undulators for THz emission, through millimeter-period permanent microundulators for compact X-ray FELs, down to proposed micrometer-period structures for infrared to soft-X-ray generation. A plausible implication is that the key transition is not the helical field concept itself, but the fabrication technology and magnetic material system used to preserve useful \(B_0\) and \(K\) as \(d\) decreases.

## 6. Radiation formulas, FEL use, and technical considerations

For radiation calculations, the cited work writes the planar-undulator resonance as
\[
\lambda_r=\frac{\lambda_u}{2\gamma^2}\left(1+\frac{K^2}{2}\right),
\]
whereas for a helical undulator the factor \(1/2\) is replaced by \(1\),
\[
\lambda_r^{\rm (helical)}=\frac{\lambda_u}{2\gamma^2}(1+K^2).
\]
In the combined field \(B_0\hat z+B_u^\perp\), an electron beam follows a helical trajectory of period \(d\), and spontaneous coherent radiation is emitted if the bunch length is shorter than \(\lambda_r\). The permanent-magnet studies emphasize that helical microundulators provide strong circularly polarized fields in both transverse directions, higher RMS oscillatory velocities, and two-plane focusing compared to planar devices; the later prototype paper similarly states that, when used in compact FELs from terahertz to X-ray, such devices can provide higher electron oscillation amplitude and radiated power than planar microundulators of similar period [1801.05312] [2508.00232] [2509.06186].

A detailed compact-XFEL case is given for the \(6\,\mathrm{mm}\), \(1.5\,\mathrm{T}\) helical device. The electron beam parameters are \(E_e=1.6\,\mathrm{GeV}\) with \(\gamma\approx3100\), charge \(\approx4.7\,\mathrm{pC}\), r.m.s. length \(140\,\mathrm{nm}\) corresponding to peak current \(I\approx4\,\mathrm{kA}\), relative energy spread \(\Delta E/E\approx0.03\%\), and transverse size \(\sigma_r\approx4.1\,\mu\mathrm m\) under periodic FODO focusing. Using
\[
L_g\approx\frac{\lambda_u}{4\pi\sqrt3\,\rho},
\]
with \(\rho=4.1\times10^{-3}\), the gain length is approximately \(70\,\mathrm{mm}\), about \(12\) periods, and saturation is stated to occur after approximately \((15\text{--}20)L_g\), i.e. \(1\)–\(1.4\,\mathrm m\) of undulator. The predicted SASE output is \(P_{\rm peak}\approx48\,\mathrm{GW}\) at \(\lambda\approx5.3\,\text{\AA}\) for the helical microundulator, compared with \(P_{\rm peak}\approx29\,\mathrm{GW}\) at \(\lambda\approx4.1\,\text{\AA}\) for a planar microundulator under the same beam conditions. The same comparison table gives \(\rho=2.8\times10^{-3}\) for the planar case and \(4.1\times10^{-3}\) for the helical case, and the discussion attributes the power increase to larger \(K\), larger electron oscillation amplitude, and improved microbunching [2509.06186].

The practical limitations are correspondingly specific. In the ferromagnetic-helix experiment, the on-axis transverse field amplitude was approximately \(0.09\,\mathrm T\), axial scans over \(\pm5\,\mathrm{cm}\) showed approximately \(10\%\) variation consistent with solenoid inhomogeneity, the main error source was probe misalignment and mechanical vibration, and induced eddy fields in the helix were negligible at less than \(1\%\) of the magnetization field. In the permanent-magnet devices, the dominant challenges are brittle NdFeB machining, tight alignment to preserve the half-period phase shift, optimization of rare-earth versus steel thickness for flux redirection, and specialized Hall-probe metrology for a \(1\,\mathrm{mm}\) bore. A common misconception is that helical microundulators are synonymous with one specific Halbach geometry; the available literature instead distinguishes redistribution-based ferromagnetic helices, dual-NdFeB helices, and hybrid NdFeB-steel devices, all of which realize the same helical-field objective at reduced period [1801.05312] [2509.06186].

Source: https://www.emergentmind.com/topics/helical-microundulator