An analytic derivation of the true three-dimensional conformal perfectly matched layer (PML) on a general orthogonal curvilinear coordinate system is presented. The conformal PML can be expressed in terms of an anisotropic constitutive tensor depending on the local principal radii of the curvatures of the termination surfaces. The derivation is based on the complex coordinate stretching approach, through a complex stretching of the normal coordinate along the PML. The quasi-PML is shown to be the zeroth order approximation of the anisotropic comformal PML for large radii of curvature. The PML produces an exponential decay of the fields in the normal direction without any reflections in the continuum limit.

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