  We report on the theoretical study of anomalous phase gradients induced by
Rayleigh anomalies in finite nanoparticle chains. These phase gradients,
defined with respect to the phase of the applied plane wave, cause a deviation
of the diffraction directions from the chain relative to the direction expected
from the grating equation for infinite chains. To study the effect
theoretically, we use an analytical approach based on the discrete dipole
approximation, which reveals the combinatorial nature of the multi-scattering
process that governs the chain dynamics. We find an approximate closed-form
solution to the particles' dipole moments by describing the single reciprocal
system with a successive solution of two non-reciprocal, one-way systems.
Within this framework, we obtain the chain excitation by means of interference
between different scattering paths. Moreover, we show that the dipole moments
along the chain are governed by recursive relations dictated by the generalized
Fibonacci series. The presented results provide a new perspective for
understanding nanoparticle arrays' dynamics. Specifically, the unique approach
for analytically analyzing the spatial excitations of the array inclusions may
shed new light on emerging applications of periodic traveling wave antennas in
the optical regime, such as LIDARs, topological states analysis and arbitrary
beam shaping schemes.
