Cartesian and spherical multipole expansions in anisotropic media

The multipole expansion can be formulated in spherical and Cartesian coordinates. By constructing an explicit map linking both formulations in isotropic media, we discover a lack of equivalence between them in anisotropic media. In isotropic media, the Cartesian multipole tensor can be reduced to a spherical tensor containing fewer independent components. In anisotropic media, however, the loss of propagation symmetry prevents this reduction. Consequently, non-harmonic sources radiate fields that can be projected onto a finite set of Cartesian multipole moments but require potentially infinitely many spherical moments. For harmonic sources, the link between the two approaches provides a systematic way to construct the spherical multipole expansion from the Cartesian one. The lack of equivalence between both approaches results in physically significant effects wherever the field propagation includes the Laplace operator. We demonstrate this issue in an electromagnetic radiation inverse problem in anisotropic media, including an analysis of a large-anisotropy regime and an introduction to vector spherical harmonics. We show that the use of the Cartesian approach increases the efficiency and interpretability of the model. The proposed approach opens the door to a broader application of the multipole expansion in anisotropic media. Anisotropic wave propagation is critical in fields like optical and radio-frequency metamaterials, acoustics, and plasma physics. The authors show the existence of propagation degrees of freedom that cannot be effectively described using the spherical multipole expansion, instead requiring the Cartesian multipole expansion for accurate representation.