A nuclear magnetic resonance well logging tool, where some embodiments comprise two, oppositely oriented magnets separated by a pole piece to guide static magnetic flux into a sensitive volume, and another pole piece serving as a core for several antennas. For some embodiments, the antennas are solenoids. Two of the antennas serve as transmit and receive antennas, where they are driven to generate an elliptically polarized magnetic field, and their antenna responses are combined so that the combined response is sensitive to elliptically polarized magnetic fields, but with zero gradient in the z-direction. A third antenna serves as a receive antenna sensitive to magnetic field vectors having a sinusoidal spatial variation in the z-direction of period equal to the length of the third antenna. A fourth antenna serves as a receive antenna sensitive to sinusoidal magnetic field vectors with the same spatial-frequency as the third antenna, but phase shifted by 90 degrees. A fifth antenna may be utilized, which serves as a receive antenna sensitive to the next higher spatial-frequency component of the received signal. The receive antennas have good cancellation of mutual coupling. Other embodiments are described and claimed.