Reciprocity and gyrotropism in magnetic resonance transduction

Abstract

We give formulas for transduction in magnetic resonance---i.e., the appearance of an emf due to Larmor precession of spins---based upon the modified Lorentz reciprocity principle for gyrotropic (also called ``nonreciprocal'') media, i.e., in which a susceptibility tensor is carried to its transpose by reversal of an external static field [cf., R. F. Harrington and A. T. Villeneuve IRE Trans. Microwave Theory and Technique MTT6, 308 (1958)]. Prior applications of reciprocity to magnetic resonance, despite much success, have ignored the gyrotropism which necessarily arises due to nuclear and/or unpaired electronic spins. For detection with linearly polarized fields, oscillating at the Larmor frequency, the emf is written in terms of a volume integral containing a product of two factors which we define as the antenna patterns, i.e., $({H}_{1x}\ifmmode\pm\else\textpm\fi{}i{H}_{1y})$, where, e.g., for a single transceive antenna, the $H$'s are just the spatially dependent oscillatory magnetic field strengths, per the application of some reference current at the antenna terminals, with the negative sign obtaining for transmission, and the positive for reception. Similar expressions hold for separate transmit and receive antennas; expressions are also given for circular polarization of the fields. We then exhibit a receive-only array antenna of two elements for magnetic resonance imaging of protons, which, due an intensity artifact arising from stray reactive coupling of the elements, produces, despite its own bilateral symmetry, asymmetric proton NMR images of a symmetric cylindrical phantom containing aqueous saline solution [J. Tropp and T. Schirmer, J. Magn. Reson. 151, 146 (2001)]. Modification of this two-port antenna, to function in transmit-receive mode, allows us to demonstrate highly nonreciprocal behavior: that is, to record images (of cylindrical test phantoms containing aqueous saline solution) whose appearance dramatically changes, when the roles of transmission and reception are swapped between the two antenna ports---giving in one instance a signal intensity pattern whose form resembles an umbrella (i.e., with a central column of moderate intensity surmounted by a bright canopy), and in the other, a distorted oval with slight concavities at its horizontal extremes, whose outline suggests that of a cat's eye. The relation between image patterns and drive scheme can be shown to reverse if the static polarizing field is reversed. Electromagnetic and circuit calculations, together with the modified reciprocity principle, allow us to reproduce these pattern changes in numerical simulations, closely and convincingly. Although the imaging experiments are performed at a static field of $3.0\phantom{\rule{0.3em}{0ex}}\mathrm{T}$, and consequently a Larmor frequency of $128\phantom{\rule{0.3em}{0ex}}\mathrm{MHz}$, the nonreciprocal effects are not related to the shortness of the wavelength in aqueous medium, but appear equally in simulations based in either the quasistatic or full electromagnetic regimes. Finally, we show that although antenna patterns for transmission and reception are swapped with reversal of the polarizing field, meaning that the receive pattern equals the transmit pattern with the field reversed, this in no way invalidates the familiar rotating wave model of spin dynamics in magnetic resonance.