Fundamental Limits of Characteristic Mode Slopes

Characteristic Mode analysis is a widely used technique in antenna design, providing insight into the fundamental electromagnetic behavior of radiating structures. In this paper, we establish fundamental bounds on the slope of characteristic mode eigenvalues and angles, demonstrating that their rate of change is subject to fundamental constraints for all possible realizations within a given design region. These bounds are derived using the method of moments and reformulating the slope (frequency derivative) of the eigenvalue quantities as an optimization problem over the current distribution confined to the design region consisting of arbitrary lossless material. The formulation considers the entire geometry as a single structure, and the resulting bounds apply generally. The results reveal a direct analogy between these constraints and classical antenna Q-factors, highlighting the intrinsic limitations on modal evolution and their implications for bandwidth and miniaturization in antenna design. Moreover, by iteratively enforcing orthogonality among the modes the derived bounds can be tightened for higher-order modes, providing deeper insight into the number of simultaneous, usable modes and their associated degrees of freedom. These bounds provide a feasibility criterion for achievable modal behavior by placing limits on the entire eigenvalue range, not only at resonance, and can be interpreted as fundamental modal flow lines of the object, characterizing how the characteristic modes are allowed to evolve over frequency. This offers valuable insights that can guide the design process. Examples are given for various surface PEC structures.