Square Loop Coil System for Balancing and Calibration of Second-Order SQUID Gradiometers

We have designed and fabricated a uniaxial square- loop coil system to calibrate second-order SQUID gradiometers. The coil system consists of 2 pairs of identical square loops, the first pair which is Helmholtz coils for generation of uniform fields and the second pair for the 2nd-order gradient only in combination with the first pair. We obtained analytical expressions up to the 4th-order of the fields in the midplane and along the coil axis. The coefficient of the normalized second-order field term is calculated to be c(5 - 11gamma² - 18gamma⁴ - 6gamma⁶)/(2 + 3gamma² + gamma⁴)² near the coil center, where c is -1 for the fields along the coil axis and +1/2 for those in the midplane, where gamma is the ratio of the inter-coil distance to the side length of the coil. The Helmholtz condition of a coil pair is gamma = 0.5445. The pure second-order gradient field is generated by subtracting the Helmholtz field from the field of a coil pair off the Helmholtz condition with equal magnitudes of the center fields of the two pairs. The coil system is useful for evaluating second- order SQUID gradiometers.