Synthesis of matched magnetic fields for controlled spin precession

The shaping of magnetic fields is important in many areas of physics, including magnet shimming, electromagnetic traps, magnetic domain switching, and controlled spin precession in nuclear magnetic resonance (NMR). We examine the method of target field matching by orthogonal projection and its application to NMR, whereby the phase of nuclear spins in a strongly inhomogeneous field is corrected through stroboscopic ac irradiation using matching fields. Three-dimensional shaping of static and ac fields can restore the spectral resolution by orders of magnitude using simple linear combinations of a small number of independent sources. Results suggest the possibility of substantially pushing the current limits of high-resolution NMR spectroscopy in weak and inhomogeneous fields. We also discuss conditions under which concomitant gradient effects are important in high magnetic fields and the geometric-phase errors they introduce during precession in ac fields.

Figure 1

(Color online) Effect of concomitant rf gradients on the operator fidelity $F_R$ of the Euler rotation $(\alpha ,\beta ,\gamma )=(10°,20°,30°)$. In (A)–(C) we have $B_{1,x}\left(\mathbf{r}\right)=10+gx$ and $B_{1,y}\left(\mathbf{r}\right)=gy$, while in (D)–(F) we used $B_{1,x}\left(\mathbf{r}\right)=10+g$ $[x(x^2-y^2)-2x{y}^2]$, $B_{1,y}\left(\mathbf{r}\right)=g$ $[-2x^{2}y-y(x^2-y^2)]$ with $(x,y)∊[-0.5,0.5]\times [-0.5,0.5]$. Three values of the gradient strength are shown: (A),(D) $g=0.1$, (B),(E) $g=1.0$, and (C),(F) $g=10.0$.

Figure 2

(Color online) Effect of concomitant rf gradients on echo-planar magnetic-resonance images for (A), (B), (E), (F) a grid pattern and (C), (D), (G), (H) human brain. Images without any concomitant fields are shown in (A) and (C). At the edges of the field of view, the ratio of concomitant field to constant field $\delta B_{1,y}∕B_{1,x}$ was (A) 0, (B) $1∕25$, (F) 1, and (E) $\infty$ for the grid. For the brain image, the ratios are (C) 0, (D) $1∕50$, (H) 1, and (G) $\infty$. A ratio of $\infty$ is obtained, for example, using a Golay or Maxwell pair gradient coil, where the dc component of the $B_1$ field is zero. A surface coil or solenoid, on the other hand, has a nonzero dc component everywhere.

Figure 3

(Color online) Cartesian-grid arrangement for a $4\times 4$ single-sided array of magnetic-field sources on the surface of a one-sided transmitter. These sources are depicted as red circles. The target volume is a rectangular parallelepiped region located near the transmitter surface and is represented as the colored volume raised above the transmitter surface. For simulations, we assume an array surface area of $12\times 12\mathrm{cm}$ and a target volume region with dimensions $6\times 6\mathrm{cm}$ in the $xy$ plane, $2\mathrm{cm}$ thick and positioned $2.5\mathrm{cm}$ away from the sensor surface.

Figure 4

(Color online) $B_z$ component of field maps (estimated vs target) for an $8\times 8$ Cartesian array of current loops calculated using the Gaussian kernel. Field maps are plots of $B_z(x,y,z)$ as $xy$ slices taken at three planes, $z=2.5$, 3.0, and $3.5\mathrm{cm}$ for the following $B_z$ component of target fields: (A) $xy$, (B) $x^2-y^2$, (C) $1+x+y-z+xy+(x^2-y^2)$.

Figure 5

Simulated spectra with the static field inhomogeneity $B_z\propto z$: (a) is the corrected spectrum with an ideal field that perfectly matches the inhomogeneity; (b), (c), and (d) are the corrected spectra at 1, 10, and $50\mathrm{kHz}$; (d) A is the inhomogeneously broadened $50\mathrm{kHz}$, B is the corrected spectrum, and C is B magnified five times. All spectra are drawn to the same vertical scale. The spectral width spans $10\mathrm{kHz}$.

Figure 6

Simulated spectra with the static field inhomogeneity $B_z\propto xy$: (a) is the corrected spectrum with an inhomogeneous broadening measure of $1\mathrm{kHz}$; (b) is for a measure $5\mathrm{kHz}$; (c) for $10\mathrm{kHz}$; (d) A is the uncorrected spectrum at $50\mathrm{kHz}$, B is the corrected spectrum, and C is B magnified five times. All spectra are drawn to the same vertical scale. The spectral width spans $10\mathrm{kHz}$.

Figure 7

Simulated spectra with the static field inhomogeneity $B_z\propto x^2-y^2$: (a) is the shimmed spectrum with an inhomogeneous broadening measure of $3\mathrm{kHz}$; (b) is the unshimmed spectrum; (c) is the shimmed spectrum for $30\mathrm{kHz}$; (d) is the unshimmed spectrum. All spectra are drawn to the same vertical scale. The spectral width spans $10\mathrm{kHz}$.