We present a rigorous vectorial coupled-mode theory (VCMT) for the isotropic waveguide structures with anisotropic disturbance by Maxwell equations and a modified dielectric tensor. The coupling coefficients are related to every element of a disturbed dielectric tensor. In the special case of isotropic disturbance and orthogonal mode coupling, our formulas can be reduced to the previous version of VCMT, in which the coupling coefficient does not include the polarization coupling term appearing in an earlier scalar coupled-mode theory with vector correction (SCMT-VC). The origin of the fundamental error found in the SCMT-VC is analyzed. Finally, we apply our theory to a special coupling between x- and y-polarized modes in a single-mode fiber.
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