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or, in matrix notation, 

where ˆ Σ = (ˆ σ$_{1}$, ˆ σ$_{2}$, ˆ σ$_{3}$ ) is the Pauli matrix vector and 

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Equation (5) generalizes the result of [27] for dislocated (amorphous) media. The first two terms are associated with the parallel transport of the beam’s transverse structure [27], while the last describes the parallel transport of the polarization vector along the beam. Note, in particular, the appearance of ˆ σ$_{2}$ which shows the non-Abelian nature of the polarization transport, to be anticipated in view of the anisotropy [39]. The Berry curvature (gauge field strength) associated with this connection is ( ˆ A × ˆ A = 0) 

where D ( θ ) = d ( h cos θ ) /d (cos θ ). 

In the course of propagation, the paraxial beam evolves according to E$_{σlm}$ → e i ˆ $^{Θ}$E$_{σlm}$ , where 

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is the geometric Berry phase. Here C is the beam trajectory in momentum space, Θ$_{0}$ = ∫ C cos θdϕ is the Berry phase accumulated for σ = 1 and m = 0 in the absence of dislocation and I = ∫ $_{C}$h cos θdϕ = (0 , Im ∫ C e iγ cot θ cos θdϕ, Re ∫ C e iγ cot θ cos θdϕ ). The first term yields a rotation (through angle -Θ$_{0}$) of the beam’s transverse structure about the direction of propagation, while the second yields a polarization precession about the direction of I . Such a spin precession is characteristic of anisotropic media [39]. (In the absence of dislocation, the second term in (6) simply yields the phase factor e iσ Θ 0 that leads to the well known Rytov rotation.) 

In view of the polarization evolution, the Berry curvature for a given beam is, therefore, 

where 

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