Nested Barrel-Shaped Coil Designs

• Nested barrel-shaped coil is a family of configurations characterized by concentric, barrel-like winding surfaces applied in diverse applications such as stellarators, NV-center systems, proton acceleration, and MRI tuning.
• They enable precise control over Fourier content, field homogeneity, and dispersion, with design strategies tailored to add magnetic configurability or enhance field strength.
• Implementations balance trade-offs between manufacturability, performance metrics, and operational complexity to meet application-specific requirements in plasma physics and electromagnetic resonator design.

A nested barrel-shaped coil is a coil architecture in which conductors or winding surfaces are arranged as coaxial, concentric, or otherwise nested barrel-like shells, most often with smooth toroidal, conical, cylindrical, or quasi-cylindrical envelopes. In the cited literature, the term does not denote a single electromagnetic device class; rather, it appears in several technically distinct contexts, including modular stellarator windings, microwave field-forming systems for nitrogen-vacancy ensembles, slow-wave helical structures for target normal sheath acceleration, and tunable ultra-high-field MRI resonators. Across these uses, the common geometric motif is nesting within a barrel-like envelope, while the function of nesting varies from adding controllable Fourier content to increasing central field, flattening dispersion, or widening frequency tunability (Queral, 2016, Biu et al., 12 May 2025, Rezinkin et al., 13 May 2026, Hirsch-Passicos et al., 2023, Ivanov et al., 2020, Rodriguez et al., 14 Apr 2026).

1. Geometric concept and domain-specific meanings

In stellarator research, a barrel-shaped winding surface is a smooth toroidal surface whose poloidal cross-sections are smooth, convex, and have modest indentation or bean features, or none. A nested barrel-shaped configuration then consists of several concentric toroidal winding surfaces obtained as equidistant offsets from a reference surface. In microwave hardware for NV-center control, the barrel shape is realized by two opposing conical windings whose large bases face each other; the nested variant adds an inner coaxial pair. In laser-plasma acceleration, the nested form is a helical coil placed concentrically inside a conducting cylindrical tube. In ultra-high-field MRI, the barrel interpretation is an open, quasi-cylindrical wire-array resonator that can equivalently be realized on a cylindrical former, and is combined with a second coil in one setup (Queral, 2016, Rezinkin et al., 13 May 2026, Hirsch-Passicos et al., 2023, Ivanov et al., 2020).

Domain	Realization	Primary aim
Stellarators	Concentric toroidal winding surfaces	Multiple magnetic configurations
NV-center control	Nested conical windings in parallel	Higher peak $B_1$
TNSA proton optics	Helix inside PEC tube	Dispersion control
UHF MRI	Barrel-like wire-array resonator plus separate $^{1}$H coil	Broad-range X-nuclei tuning

The geometric notion of nesting therefore serves different purposes in different regimes. In stellarators it is a means of decomposing LCFS shaping into independently energizable windings. In microwave and RF devices it is primarily a way to redistribute current paths, inductance, capacitance, or field localization. This suggests that “nested barrel-shaped coil” is best treated as a family of architectures rather than as a single standard topology.

2. Concentric Fourier windings in stellarators

The most explicit nested barrel-shaped stellarator construction in the cited literature is the concentric Fourier windings method. The last closed flux surface is represented in cylindrical coordinates by Fourier series with stellarator symmetry,

$$R(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} R_{mn}\cos\!\big(2\pi(m\,u+n\,v)\big),$$

$$Z(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} Z_{mn}\sin\!\big(2\pi(m\,u+n\,v)\big),$$

with $\theta=2\pi u$ and $\phi=2\pi v/n_p$. Successive winding surfaces are constructed as equidistant offsets of a reference surface,

$$\mathbf r_k(\theta,\phi)=\mathbf r_{\text{ref}}(\theta,\phi)+d_k\,\mathbf n_{\text{ref}}(\theta,\phi),$$

so that each additional winding $W_k$, when added to a base winding $W_b$, introduces exactly one additional Fourier coefficient in the target LCFS. The total field is then assembled by linear superposition of the base field and the fields from the additional windings, with either direct current choices or normalized currents $\lambda_k$ (Queral, 2016).

The case study reported for a 3-period stellarator used up to $^{1}$0 windings, with computational exploration reported for 5 windings. The base winding surface was an external equidistant offset of $^{1}$1 arbitrary units from the reference surface, and additional surfaces followed $^{1}$2. NESCOIL with 7 poloidal and 8 toroidal modes was used to compute each winding, and CASTELL was used to compute Biot–Savart magnetic field grids and assemble linear combinations. For exploration, $^{1}$3 for $^{1}$4, yielding $^{1}$5 configurations. Most magnetic surfaces reproduced the intended LCFS acceptably, with mean errors typically below $^{1}$6 and maximum errors $^{1}$7 for the studied range; the rotational transform varied between $^{1}$8 in the core and $^{1}$9 at the edge depending on the current mix (Queral, 2016).

Three constraints are central in this formulation. First, higher-order coefficients decay faster with distance, so their corresponding windings are placed closer to the plasma. Second, all intended LCFS must lie inside the innermost winding surface; otherwise intersections prevent accurate generation. Third, sharp concavities are difficult to reproduce from distant smooth winding surfaces. The cited formulation states that “concavities in a LCFS are difficult to generate by distant coils if the concavity is not replicated at the winding surface.” This is why low-order, smooth, convex, barrel-shaped surfaces are favored for systematic nesting.

3. Surface-bounded stellarator optimization and equilibrium-based proxies

A distinct stellarator line of work constrains filamentary coils to lie on a prescribed coil-winding surface and parameterizes the coil directly by the surface coordinates. For an axisymmetric circular toroidal winding surface with major radius $$R(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} R_{mn}\cos\!\big(2\pi(m\,u+n\,v)\big),$$0 and minor radius $$R(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} R_{mn}\cos\!\big(2\pi(m\,u+n\,v)\big),$$1, the surface embedding is

$$R(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} R_{mn}\cos\!\big(2\pi(m\,u+n\,v)\big),$$2

and coil curves are written as $$R(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} R_{mn}\cos\!\big(2\pi(m\,u+n\,v)\big),$$3, where $$R(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} R_{mn}\cos\!\big(2\pi(m\,u+n\,v)\big),$$4 and $$R(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} R_{mn}\cos\!\big(2\pi(m\,u+n\,v)\big),$$5 have linear-plus-Fourier form. This keeps the curve exactly on the surface and enables analytic derivatives for coil-shape and winding-surface optimization. In the cited application to an approximately quasisymmetric VMEC equilibrium with $$R(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} R_{mn}\cos\!\big(2\pi(m\,u+n\,v)\big),$$6 m, aspect ratio $$R(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} R_{mn}\cos\!\big(2\pi(m\,u+n\,v)\big),$$7, and $$R(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} R_{mn}\cos\!\big(2\pi(m\,u+n\,v)\big),$$8, the best-performing circular torus winding surface had minor radius $$R(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} R_{mn}\cos\!\big(2\pi(m\,u+n\,v)\big),$$9 m and an optimal plasma-to-CWS distance $$Z(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} Z_{mn}\sin\!\big(2\pi(m\,u+n\,v)\big),$$0 m. After 2000 iterations, the optimized axisymmetric barrel-shaped CWS achieved quadratic flux $$Z(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} Z_{mn}\sin\!\big(2\pi(m\,u+n\,v)\big),$$1 and maximum normal field error $$Z(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} Z_{mn}\sin\!\big(2\pi(m\,u+n\,v)\big),$$2 T for $$Z(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} Z_{mn}\sin\!\big(2\pi(m\,u+n\,v)\big),$$3 T. A non-axisymmetric winding surface rescaled from the plasma boundary performed better, reaching $$Z(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} Z_{mn}\sin\!\big(2\pi(m\,u+n\,v)\big),$$4 and $$Z(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} Z_{mn}\sin\!\big(2\pi(m\,u+n\,v)\big),$$5 T at $$Z(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} Z_{mn}\sin\!\big(2\pi(m\,u+n\,v)\big),$$6 m (Biu et al., 12 May 2025).

These results matter because they delimit what barrel-shaped nesting can and cannot do in fusion coil design. The axisymmetric circular torus simplifies winding fixtures and allows direct deposition or milling, but a winding surface offset from the plasma boundary retains geometric degrees of freedom aligned with the target non-axisymmetry and therefore yields lower error fields. The cited analysis therefore does not treat barrel-shaped surfaces as universally optimal; it treats them as a controlled simplification with an identifiable optimum standoff near $$Z(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} Z_{mn}\sin\!\big(2\pi(m\,u+n\,v)\big),$$7 m for the studied device size (Biu et al., 12 May 2025).

A complementary theoretical framework constructs artificial modular coils directly from an equilibrium by defining a current potential $$Z(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} Z_{mn}\sin\!\big(2\pi(m\,u+n\,v)\big),$$8 on a flux surface. On a smooth toroidal surface $$Z(u,v)=\sum_{m=0}^{m_b}\sum_{n=-n_b}^{n_b} Z_{mn}\sin\!\big(2\pi(m\,u+n\,v)\big),$$9,

$\theta=2\pi u$0

and when the winding surface coincides with a flux surface,

$\theta=2\pi u$1

Filamentary coils are then taken as level sets $\theta=2\pi u$2. Within this framework, coil curvature decomposes into normal and geodesic parts, $\theta=2\pi u$3 and $\theta=2\pi u$4, linking coil complexity directly to local surface curvature and magnetic-field variation. The cited study reports that the equilibrium-based construction provides a lower bound for coil non-planarity and that $\theta=2\pi u$5 is a conservative lower-bound extrapolation for outward-displaced winding surfaces. Precisely quasi-axisymmetric equilibria are therefore especially favorable for barrel-shaped coils because weak toroidal variation of $\theta=2\pi u$6 reduces in-surface bending (Rodriguez et al., 14 Apr 2026).

4. Microwave barrel and nested barrel coils for NV-center control

In microwave control of negatively charged nitrogen-vacancy ensembles in diamond, the nested barrel-shaped coil is a field-forming system designed to generate a spatially uniform microwave magnetic field $\theta=2\pi u$7 around $\theta=2\pi u$8 GHz. The single barrel-shaped coil consists of two identical conical windings, each with three turns, connected in parallel for a total of $\theta=2\pi u$9 turns. The nested barrel-shaped coil adds an inner pair, giving four conical windings in total, all in parallel, for $\phi=2\pi v/n_p$0 turns. The single barrel geometry used in simulation had $\phi=2\pi v/n_p$1 mm, $\phi=2\pi v/n_p$2 mm, and $\phi=2\pi v/n_p$3 mm; the nested variant had $\phi=2\pi v/n_p$4 mm, $\phi=2\pi v/n_p$5 mm, and preserved $\phi=2\pi v/n_p$6 mm. Both used round copper wire of diameter $\phi=2\pi v/n_p$7m and Kapton tape with total thickness $\phi=2\pi v/n_p$8m, so the diameter increment per turn was approximately $\phi=2\pi v/n_p$9m (Rezinkin et al., 13 May 2026).

Uniformity was quantified with

$$\mathbf r_k(\theta,\phi)=\mathbf r_{\text{ref}}(\theta,\phi)+d_k\,\mathbf n_{\text{ref}}(\theta,\phi),$$0

over an inner cylindrical domain. COMSOL simulations reported that the single barrel coil achieved $$\mathbf r_k(\theta,\phi)=\mathbf r_{\text{ref}}(\theta,\phi)+d_k\,\mathbf n_{\text{ref}}(\theta,\phi),$$1 over $$\mathbf r_k(\theta,\phi)=\mathbf r_{\text{ref}}(\theta,\phi)+d_k\,\mathbf n_{\text{ref}}(\theta,\phi),$$2 mm and $$\mathbf r_k(\theta,\phi)=\mathbf r_{\text{ref}}(\theta,\phi)+d_k\,\mathbf n_{\text{ref}}(\theta,\phi),$$3 over $$\mathbf r_k(\theta,\phi)=\mathbf r_{\text{ref}}(\theta,\phi)+d_k\,\mathbf n_{\text{ref}}(\theta,\phi),$$4 mm, with radial nonuniformity at $$\mathbf r_k(\theta,\phi)=\mathbf r_{\text{ref}}(\theta,\phi)+d_k\,\mathbf n_{\text{ref}}(\theta,\phi),$$5 as low as $$\mathbf r_k(\theta,\phi)=\mathbf r_{\text{ref}}(\theta,\phi)+d_k\,\mathbf n_{\text{ref}}(\theta,\phi),$$6 in a small region. The nested barrel coil increased the maximum $$\mathbf r_k(\theta,\phi)=\mathbf r_{\text{ref}}(\theta,\phi)+d_k\,\mathbf n_{\text{ref}}(\theta,\phi),$$7 at the center, but the highly uniform axial region became shorter: $$\mathbf r_k(\theta,\phi)=\mathbf r_{\text{ref}}(\theta,\phi)+d_k\,\mathbf n_{\text{ref}}(\theta,\phi),$$8 only over $$\mathbf r_k(\theta,\phi)=\mathbf r_{\text{ref}}(\theta,\phi)+d_k\,\mathbf n_{\text{ref}}(\theta,\phi),$$9 mm, $W_k$0 over $W_k$1 mm, and $W_k$2 over $W_k$3 mm (Rezinkin et al., 13 May 2026).

Experimental validation used an Element Six diamond with NV density $W_k$4 ppb, cut length $W_k$5m, and nominal thickness $W_k$6m. Rabi oscillations showed that the barrel coil produced $W_k$7–$W_k$8 MHz across positions $W_k$9–$W_b$0 mm, whereas a planar antenna produced $W_b$1–$W_b$2 MHz across $W_b$3–$W_b$4 mm but with rapid decay indicating significant inhomogeneity. The ensemble signal was modeled as $W_b$5, with $W_b$6. The barrel coil exhibited consistently smaller $W_b$7 than the planar antenna, especially away from the coil center, confirming more homogeneous $W_b$8 (Rezinkin et al., 13 May 2026).

A common misconception is that nesting automatically improves all performance metrics. The NV-center study shows the opposite for uniformity: the nested 12-turn variant raised peak $W_b$9, but the 6-turn single barrel geometry performed better when the primary metric was axial homogeneity.

5. Helix-with-tube slow-wave structures for TNSA proton bunching

In target normal sheath acceleration, the nested barrel-shaped architecture takes the form of a helical coil placed concentrically inside a surrounding conducting cylindrical tube. The helix has mean radius $\lambda_k$0, pitch $\lambda_k$1, length $\lambda_k$2, wire diameter $\lambda_k$3, and characteristic speed

$\lambda_k$4

Typical simulated values were $\lambda_k$5–$\lambda_k$6 mm, $\lambda_k$7–$\lambda_k$8 mm, $\lambda_k$9 mm, and $^{1}$00 mm. The surrounding tube had inner radius $^{1}$01 such that $^{1}$02, typical gap $^{1}$03 mm, wall thickness $^{1}$04 mm, and was treated as a perfect electric conductor (Hirsch-Passicos et al., 2023).

The electromagnetic purpose of the outer tube is not merely mechanical enclosure. A bare helix is strongly dispersive: low-frequency components propagate closer to $^{1}$05, while high-frequency components approach $^{1}$06. When the helix is tube-loaded, the slow-wave mode hybridizes with the dispersion-free tube mode, flattening $^{1}$07 and stabilizing $^{1}$08. The reported consequence is a single-polarity current pulse moving at nearly constant speed $^{1}$09, with pulse broadening reduced to a modest factor of approximately $^{1}$10–$^{1}$11 for $^{1}$12 mm helices, and peak amplitude lowered by approximately $^{1}$13 relative to the bare helix (Hirsch-Passicos et al., 2023).

This change in pulse dynamics is directly reflected in the proton spectrum. The cited study reports two narrow-band peaks with characteristic placements

$^{1}$14

where $^{1}$15. For a representative geometry $^{1}$16 mm, $^{1}$17 mm, $^{1}$18 mm, $^{1}$19 mm, the tube-loaded structure produced a single positive current lobe propagating at $^{1}$20, broadened by a factor $^{1}$21–$^{1}$22, and lowered in amplitude by $^{1}$23 versus the bare helix. The proton outcome was two collimated narrow-band beams, with focusing enhanced and divergence reduced by factors $^{1}$24–$^{1}$25 in the high-energy bunch (Hirsch-Passicos et al., 2023).

Here, nesting serves as an electromagnetic loading mechanism. The outer tube adds a coaxial-like capacitance to ground and a return path for magnetic flux, reducing dispersion while preserving the slow-wave interaction needed for synchronism with MeV protons.

6. Barrel-shaped resonators for multiheteronuclear ultra-high-field MRI

In ultra-high-field MRI, a barrel-shaped coil appears as a metamaterial-inspired X-nuclei resonator operating through the fundamental eigenmode of an array of parallel non-magnetic wires. The prototype used five brass telescopic wires with 10 mm center-to-center spacing and adjustable electrical length $^{1}$26–$^{1}$27 mm. Each wire overlapped 4 mm onto metallized patches at both ends on AD1000 ceramic PCBs, and structural capacitance was varied by sliding fully metallized outer AD1000 PCBs so that the overlap changed from 3 to 33 mm. This provided two structural tuning parameters, inductance via wire length and capacitance via plate overlap, with the first-order relation

$^{1}$28

The demonstrated tuning span was 76–203 MHz, covering $^{1}$29H, $^{1}$30C, $^{1}$31Na, $^{1}$32Xe, $^{1}$33B, $^{1}$34Li, and $^{1}$35P at 11.7 T (Ivanov et al., 2020).

The X-nuclei resonator was paired with a separate $^{1}$36H butterfly coil tuned to approximately 500 MHz, forming a dual-coil setup in one assembly. In the simulation and bench geometry, the coil-to-coil separation was 42 mm, both coils were inside a copper RF shield of 82 mm inner diameter, and the $^{1}$37 fields were orthogonal, yielding geometric and field orthogonality for decoupling. Measured or simulated S-parameters gave $^{1}$38 dB for the $^{1}$39H coil, $^{1}$40 dB for the X-nuclei coil at 76, 125.7, and 203 MHz, and inter-coil coupling $^{1}$41 dB at both 500 MHz and X-nuclei frequencies (Ivanov et al., 2020).

Although this device is not a nested barrel in the same sense as the stellarator or NV architectures, the cited description explicitly treats the X-nucleus resonator as a barrel-shaped wire resonator and the overall system as a double-coil setup. Its significance is that barrel-like form can be used to realize a large field of view, broad tunability, and natural decoupling from a second coil tuned to another nucleus. The architecture therefore extends the barrel-shaped concept from field-shaping and confinement into structurally tuned, multimode RF resonator design.

7. Cross-cutting design trade-offs and limitations

Across the cited applications, nested barrel-shaped coil designs are not defined by a single governing figure of merit. In stellarators, nesting increases the number of independently addressable winding families and allows rapid configuration scans, but fidelity decreases for high-order Fourier content, sharp concavities, and large combined coefficients, and all target LCFS must remain inside the innermost winding surface (Queral, 2016). In surface-bounded stellarator optimization, axisymmetric barrel-shaped winding surfaces are manufacturability-oriented simplifications, yet a non-axisymmetric surface offset from the plasma boundary yields better field accuracy for the same coil-curve parameter budget (Biu et al., 12 May 2025). In NV-center control, nesting raises the central $^{1}$42 by adding turns in parallel, but shortens the $^{1}$43 uniform axial region (Rezinkin et al., 13 May 2026). In TNSA, the outer tube suppresses dispersive sign reversals and improves bunching, but reduces peak amplitude by approximately $^{1}$44 relative to the bare helix (Hirsch-Passicos et al., 2023). In MRI, structural tunability is broad, but the resonant frequency is sensitive to the RF shield and sample presence, requiring final tuning in situ (Ivanov et al., 2020).

Several recurrent engineering themes follow from these studies. Spacing and clearance are always central: stellarator windings require intersection avoidance and minimum curvature control; NV coils require balanced parallel branches and unobstructed optical access through $^{1}$45 mm; the TNSA helix-with-tube must avoid electrical contact between helix and tube; and the MRI resonator depends on stable plate overlap, plate parallelism, and feed-loop geometry. A plausible implication is that nested barrel-shaped architectures are most advantageous when the added geometric layer provides a specific additional degree of freedom—spectral control, field-strength enhancement, dispersion flattening, or frequency tuning—without driving the system into excessive complexity.

The literature also indicates that barrel-shaped geometry should not be conflated with universal field uniformity or universal manufacturability. Smooth barrel-like surfaces in stellarators reproduce smooth LCFS features better than sharp indentations, but they are specifically poor at imprinting concave shapes from a distance (Queral, 2016). An axisymmetric barrel-shaped stellarator winding surface is simpler than a plasma-offset surface, but less accurate (Biu et al., 12 May 2025). The single barrel outperforms the nested barrel in NV-field homogeneity (Rezinkin et al., 13 May 2026). These results collectively position the nested barrel-shaped coil as a design strategy of controlled trade-offs rather than an intrinsically optimal form.