Without loss of generality, we denote the structural parameters under measurement as an -dimensional vector , where the superscript “” represents the transpose. The vector that consists of the incidence angle and azimuthal angle denotes the measurement configuration. The function is usually applied to estimate the fitting errors between the measured and calculated Mueller matrix elements and , which is defined as Display Formula
(1)where denotes the spectral point from the total number , and indices and show all the Mueller matrix elements except . is the standard deviation associated with . For clarity, the measured Mueller matrix element in Eq. (1) is marked as with the three indices , , and lumped into a single index . The calculated Mueller matrix element is correspondingly marked as . Thus, Eq. (1) can be simply rewritten as Display Formula
(2)where is the weighting factor and is given by and . is an diagonal matrix with diagonal elements . The inverse problem in grating reconstruction is typically formulated as a least square regression problem such that Display Formula
(3)where is the solution of the inverse problem that contains the extracted structural parameters, and is the associated parameter domain. denotes the given value of vector in the parameter extraction.