Beamed-energy propulsion of ultra-light relativistic lightsails confronts two simultaneous electrodynamic instabilities: the continuous reflectivity degradation caused by the relativistic Doppler redshift along the propagation axis, and the transverse radiation-pressure gradient that displaces the sail from the optical axis. Current inverse-design methods address these challenges exclusively through computationally expensive stochastic algorithms with no physical insight into the structure of the solution space. We present a fully deterministic, closed-form analytical framework by decomposing the sail's effective dielectric tensor as (r, z) =_ (r) || (z) within the paraxial scalar approximation. The transverse sector is solved via the inverse Abel transform grounded in the universality theorem for SO (2) -invariant rotating curve families: a quadratic target phase (y) =₀ (1-y^2/R^2) yields the exact closed-form graded-index profile n₁ (r) = (2₀/ R^2) R^2-r^{2} bounded within the Si/SiO₂ range 1. 45, 3. 48 at ₀=1064~nm. The longitudinal sector is governed by the nonlinear Riccati equation for the local reflection coefficient, derived rigorously from the coupled-wave reduction of Maxwell's equations. A formal self-similarity argument under the Lorentz boost subgroup SO (1, 1) is used to prove that a log-periodic layer distribution ₉=₀e^ j is a sufficient condition for broadband Doppler invariance across the target bandwidth ₀e^-₌₀ₗ, ₀. A corrected thin-film Stefan-Boltzmann thermal analysis, using emissivity ₓ₇₈ₑd valid for sub-wavelength films, shows that the log-periodic design maintains equilibrium temperatures below the Si melting point of 1687 K at I₀=1~GW/m^2, whereas the conventional distributed Bragg reflector (DBR) exceeds this limit already at 0. 07. This work provides a rigorous group-theoretic foundation that reduces the broadband inverse-design problem from a stochastic search to a direct algebraic verification.
Ozorio Olea Arnaldo Adrian (Wed,) studied this question.