Vigorous recent interest in the investigation of electric dipole coils for high field MR has been motivated by two principal observations. On the one hand, it has been asserted that, as wavelength shortens, radiative character – considered a liability at low frequency – becomes useful to transport energy into internal body areas (1). On the other hand, the computation of ideal current patterns has demonstrated that divergence-free (i.e. loop-like) currents alone are insufficient to achieve the ultimate intrinsic SNR (UISNR) at high frequency (2).

The division of ideal surface current patterns into divergence-free and complementary curl-free components is a useful practice (2,3), since all discrete loops fall into the category of divergence-free patterns, and curl-free components are required to represent everything else. Curl-free patterns have commonly been referred to as “electric dipole” currents (2,3), but various efforts to emulate the curl-free components of ideal current patterns using arrays of electric dipole antennae have proven challenging (4). Meanwhile, recent evaluations of relatively straightforward electric dipole array designs have yielded surprising results, including SNR values that exceed the curl-free UISNR limit (5).

In light of this apparent contradiction, we set out to study the fundamental electrodynamic behavior of electric dipole antennae in terms of divergence-free and curl-free components.

At first glance, it is tempting to assume that a simple line-segment electric dipole coil is curl-free, since it is by definition non-closed. That this is not in fact true is relatively straightforward to see by application of Stokes’ Law (drawing a path that branches off into current-free space after following the current filament results in a nonvanishing line integral and corresponding curl). Nor is an electric dipole divergence-free (cf. Gauss’ law for a surface surrounding one end of the dipole.) In a Dyadic Green’s Function (DGF) formalism, which begins with elemental surface current modes and computes associated fields, the electric dipole must therefore be represented by a combination of divergence-free and curl-free basis elements.

Fig. 1 illustrates a sample cylindrical geometry for DGF calculations, indicating how actual and/or ideal current patterns may be displayed by “unwrapping” the cylindrical surface. Fig. 2 expands the current distribution of a z-oriented filamentary electric dipole in a Fourier basis, and summarizes the particular linear combination of divergence-free and curl-free current modes on the cylinder which must be used in order to cancel azimuthal components and to match this current pattern at all spatial positions. The fact that simple linear combinations of divergence-free and curl-free modes result in z- or φ-directed currents also suggests that we might equivalently choose axially- or azimuthally-directed modes to form our elemental current basis. Pursuing this line of thinking, in addition to simulating discrete electric dipoles in our DGF formalism, we also compared ideal current patterns for divergence-free currents only or curl-free currents only with those for z- or φ-directed currents only, to determine which more closely approached the all-mode ultimate SNR.

Particular electric dipole current patterns (with half-cosine standing-wave current distribution along z) and related ideal current patterns were generated and analyzed using a full-wave DGF simulation tool (2) for a 31.5cm cylindrical surface surrounding a uniform dielectric cylinder of 29cm diameter, with conductivity of 0.4 S/m and relative permittivity of 39, at 297.2 MHz (corresponding to our 7T operating frequency). Azimuthal and axial mode orders of n=-40:39 and m=-70:69, respectively, were used to ensure convergence.

Fig. 3 shows the decomposition of a filamentary z-directed electric dipole current pattern into its divergence-free and curl-free components. This decomposition clearly shows the loop-like and the non-loop-like contributions inherent in the electric dipole. Note that the divergence-free component actually has much greater maximum current amplitude than the curl-free component!

Fig 4 shows the ideal current pattern associated with optimal SNR for the center of the cylinder, using divergence-free elements only or curl-free elements only, as compared with the true optimum for both together. Fig 5 shows corresponding ideal current patterns for z-directed current patterns only or phi-directed current patterns only. At the 7T Larmor frequency, z-directed currents are a far better match to the full ideal current pattern, yielding 94% of UISNR as opposed to only 74% or 73% for separately optimized divergence-free or curl-free currents.

As a combination of divergence-free and curl-free current components, electric dipole coils are not bound by limits on the performance of either component (5). Moreover, they appear to represent an alternative basis set well-suited to ideal current patterns at moderate to high field strength (5,6). These observations promise to stimulate new high-performance electric-dipole coil designs, grounded in rigorous electrodynamics.