On the Degrees of Freedom and Eigenfunctions of Line-of-Sight Holographic MIMO Communications

We consider a line-of-sight communication link between two holographic surfaces (HoloSs), and provide a closed-form expression for the effective degrees of freedom (eDoF), i.e., the number of orthogonal communication modes that can be established between them. The proposed framework can be applied to network deployments beyond the widely studied paraxial setting. This is obtained by partitioning the largest HoloS into sub-HoloSs, and proving that the supports of the Fourier transforms of the kernels of the obtained integral operators are limited and are almost disjoint in the wavenumber domain, provided that the sub-HoloSs are sufficiently small. Using the proposed approach, it is proved that (i) the eDoF correspond to an instance of Landau's second eigenvalue problem; (ii) the eigenvalues polarize asymptotically to multiple values; and (iii) the eDoF depend explicitly on the approximation accuracy according to Kolmogorov's n-width criterion. This result generalizes the analysis in the paraxial setting, in which it is known that the eigenvalues polarize asymptotically to two values. In addition, it is proved that the typical method of analysis utilized in the paraxial setting, which is based on a parabolic approximation of the wavefront in a local coordinates system, is equivalent to a quartic approximation of the wavefront in a general coordinates system. This facilitates the derivation of an explicit formula for the eDoF in terms of key system parameters, including the relative offset between the center-points of the HoloSs, and their relative rotation and tilt. We specialize the framework to canonical network deployments, and provide analytical expressions for the optimal, according to Kolmogorov's n-width criterion, basis functions (communication waveforms) for data encoding and decoding.