Signal-to-noise ratio (SNR) in an NMR experiment is linearly proportional to the sensitivity of the RF receiver¹, defined as the B₁-field strength per unit current. The SNR may be increased by reducing the physical size of the coil and increasing the filling factor. However, in many instances, the minimum achievable size of the coil is restricted by technical limitations, or it is not possible to place the active electrical element in close proximity to the sample region of interest (ROI). The purpose of Lenz lenses is to introduce a passive element into the RF transmit/receive chain that remotely focusses the magnetic flux of the transmitter, which, by the principle of reciprocity, also increases the sensitivity of the receiver².

The basic design of a Lenz lens involves two concentric circular conductors of different sizes that are connected in series to form one or more closed loops comprising an annulus³, as depicted in Fig. 1. In the presence of an external field, a current will be induced in the Lenz lens according to Faraday’s law of induction and the current sense will be such that it inductively opposes the incident field, as dictated by Lenz’s law. Since the current sense of the inner loop is opposite to that of the outer loop, the total flux within the inner loop will actually be enhanced. Equivalently, the sensitivity of an NMR system (in receive mode) to fields emanating from within the inner loop will be increased².

The current induced in the Lenz lens, I_L, can be modelled analytically by equating the electromotive force induced by the external field to the electric potential across the equivalent RL-circuit of the lens. For example, for a single lens placed concentrically at the midpoint of a Helmholtz pair (Fig. 1), we obtain:

$$I_{L}=\frac{2i\omega(M_{\mathrm{BH}}-M_{\mathrm{SH}})I_{\mathrm{H}}}{[R_{\mathrm{L}}-i\omega(L_{\mathrm{S}}+L_{\mathrm{B}}-2M_{\mathrm{SB}})]}\qquad\qquad(1)$$

where ω is the frequency, I_H is the current of the Helmholtz pair, R_L is the resistance of the Lenz lens, L_S and L_B are the self-inductances of the inner (S – small) and outer (B – big) loops, and M_BH, M_SH and M_SB are the mutual inductances between different combinations of the various loops (denoted by the subscripts). Note that for high-frequency applications it is important to include the skin-effect in the resistance and self-inductance calculations.

It is relatively straightforward to generalize Eq. (1) to the case of multiple Lenz lenses placed arbitrarily within any external field by introducing additional mutual inductance terms. This can be represented by a matrix equation $$$\mathcal{L}\boldsymbol{I}_{\mathrm{L}}=\boldsymbol{H}$$$, with elements given by:

$$\mathcal{L}_{kk}=R_{\mathrm{L}k}-i\omega(L_{\mathrm{S}k}+L_{\mathrm{B}k}-2M_{\mathrm{S}k\mathrm{B}k})\qquad\qquad$$

$$\mathcal{L}_{kj}=-i\omega(M_{\mathrm{B}k\mathrm{B}j}-M_{\mathrm{S}k\mathrm{B}j}-M_{\mathrm{B}k\mathrm{S}j}+M_{\mathrm{S}k\mathrm{S}j})\qquad\qquad(2)$$

$$\boldsymbol{H}_{k}=i\omega(M_{\mathrm{B}k\mathrm{H1}}+M_{\mathrm{B}k\mathrm{H2}}-M_{\mathrm{S}k\mathrm{H1}}-M_{\mathrm{S}k\mathrm{H2}})I_{\mathrm{H}}\qquad\qquad$$

where k = 1:K, j = 1:K, j ≠ k, and K is the number of lenses. Note, for example, that M_BkSj is the mutual inductance between the outer loop of the k^th lens and the inner loop of the j^th lens (H1 and H2 refer to individual Helmholtz loops for demonstrative purposes only).

Eqs. (1)-(2) were used to optimize the geometry and placement of Lenz lenses for a simple theoretical NMR microscopy example, with the goal of maximizing sensitivity while maintaining a defined level of field homogeneity (i.e. flip angle) within a cylindrical ROI (using the Matlab^® function fmincon). Fig. 2 demonstrates that gains of up to 1.5, 2.2 and 3.0 are possible using one, two and four lenses, respectively, over the use of a Helmholtz pair alone, while maintaining a 1% average field error within the ROI. These gains can be increased further by relaxing the homogeneity constraint.

To demonstrate the effect experimentally, a Lenz lens (not optimized) was patterned by means of gold electro-deposition onto a water-filled, rectangular-shaped capillary, which was embedded within a 10 mm NMR tube containing deuterated agarose gel (Fig. 3a). The tube was aligned carefully within a saddle-shaped coil of a 500 MHz spectrometer (Bruker BioSpin), and images were acquired using a spin echo sequence. Fig. 3 displays an approximately three-fold increase in sensitivity to the water nearby the inner loop of the Lenz lens compared to the water elsewhere in the capillary.

Lenz lenses are simple broadband discrete elements that permit the remote localization of the sensitive volume of an NMR detector, and afford substantial increases in field efficiency.