Borel summability of spectral nonparaxial perturbative series

Fifty years after its introduction, the perturbative approach conceived by Lax <i>et al.</i> [Phys. Rev. A11, 1365 (1975)10.1103/PhysRevA.11.1365] to solve Maxwell's equations remains widely used, despite its inherent divergence in the physical space. Through a vectorial analysis of the LLM algorithm carried out within the spatial frequency domain, it is here shown how each single term of the asymptotic expansion of a free-space propagated coherent light field represents the outcome of a suitable linear filter fed by the boundary field. In this way, since the Fourier transform operator naturally uncouples the roles of the boundary condition and of the propagation process, the decoding mechanism of the LLM scheme can be clearly unveiled. To this end, our main task is the exploration of the convergence features of the so-called <i>spectral series</i>. This is the spectral counterpart of the asymptotic series representation (in the physical space) provided by the LLM algorithm. Through a delicate asymptotic analysis, it is proved that the spectral series is uniformly convergent, within the homogeneous portion of the angular spectrum, to the Fourier transform of the coherent free-space propagator. On employing Borel summation together with analytical continuation, such a fundamental connection is then extended to the remaining evanescent part of the spectrum. In this way, it is rigorously proven that (i) exact free-space solutions of Maxwell's equations are completely encoded within their paraxial approximations and (ii) the Rayleigh-Sommerfeld coherent propagator is the generating function for all nonparaxial corrections iteratively obtained by the LLM algorithm, irrespective of the boundary condition.