This paper proposes using the curvature of the transmit phase—the second spatial derivative of the phase— as a natural signaling coordinate for continuous, electrically large antenna apertures. The authors first remove the trivial degrees of freedom: an overall phase shift (piston) and a linear tilt. After that quotienting, the curvature field uniquely represents the phase profile and behaves as a well-posed signal space. The work is aimed at systems where the wavefront geometry is measurable and controllable, such as near-field or holographic array surfaces.

The paper builds a mathematical model that shows this curvature description is well behaved. The synthesis operator that maps curvature back to phase is bounded and compact. The authors give an explicit constant (C_L = L^2 / β_1^2) and an exact formula for the modal strengths ρ_m = (L/β_m)^4, where the β_m are the roots of the transcendental equation cosβ coshβ = 1. In plain terms, the natural modes of the curvature coordinate decay with a fourth-power law in index, which controls how many useful modes a finite aperture can support. Under a mild assumption on propagation (a bounded-support square-integrable kernel), the linearized channel is a Hilbert–Schmidt operator and the infinite-dimensional channel capacity can be defined by a Fredholm determinant.

Starting from the exact nonlinear phase-only aperture law, the paper linearizes that law around an operating point using a Fréchet derivative. That produces a coherent “tangent” channel and an explicit bound on the linearization remainder. For large modulations the authors construct a multi-chart atlas, meaning they cover larger excursion ranges by patching multiple local linear models. On the signaling side they derive a new optimal allocation rule: a dual-budget generalized water-filling. This is like the classical water-filling rule in information theory but with two Lagrange multipliers—one that limits curvature power and another that limits phase excursion—to reflect both curvature energy and the nonlinear cost of synthesizing phase.