A new interpolation method for rotation and divergence free fields is presented. It is based on a suitable choice of a tricubic interpolation scheme and reaches an accuracy of third order in grid size (Δx). With the interpolation method it is possible to increase the accuracy with a factor of grid size/distance with respect to the trilinear interpolation method using exactly the same data points. Simulations for several distances of dipoles (r) to the interpolation area show that the maximum relative deviation is approximately 3(Δx/r)³ ppm.