Mapping (or plotting) the magnetic field has a critical importance for the achievement of the homogeneous magnetic field necessary to standard MR experiments. A powerful tool for this purpose is the Spherical Harmonic Expansion (SHE), which provides a simple way to describe the spatial variations of a field in free space. Well-controlled non-zero spatial variations of the field are critical to MRI. The resolution of the image is directly related to the strength of the gradient used to encode space. As a result, it is desirable to have strong variations of the field. In that case, the SHE cannot be used as is, because the field modulus variations are affected by the variations of all components of the field. In this paper, we propose a method based on the SHE to characterize such variations, theoretically and experimentally, in the limit of an axisymmetric magnetic field. Practical applications of this method are proposed through the examples of single-sided magnet design and characterization, along with Stray-Field Imaging (STRAFI).