Method to evaluate the geometrical spin-redirection phase for a nonplanar ray

An evaluation method is suggested for determining the Berry geometrical spin-redirection phase in systems with a set of separated optical elements. The evaluation considers the coordinate transformation, taking into account the directions of the beam. The spin-redirection phase contains information about the nonplanarity of beam propagation. This geometrical phase can be determined even if the ray trajectory is traced as an unclosed topological contour on the unit sphere associated with wave-vector directions.