Without loss of generality, we denote the structural parameters under measurement as an $M$-dimensional vector $x=[x1,x2,\u2026,xM]T$, where the superscript “$T$” represents the transpose. The vector $a=[\theta ,\phi ]T$ that consists of the incidence angle $\theta $ and azimuthal angle $\phi $ denotes the measurement configuration. The $\chi 2$ function is usually applied to estimate the fitting errors between the measured and calculated Mueller matrix elements $mij,kmeas$ and $mij,kcalc(x,a)$, which is defined as Display Formula
$\chi 2=\u2211k=1N\lambda \u2211i,j[mij,kmeas\u2212mij,kcalc(x,a)\sigma (mij,k)]2,$(1)
where $k$ denotes the spectral point from the total number $N\lambda $, and indices $i$ and $j$ show all the Mueller matrix elements except $m11$. $\sigma (mij,k)$ is the standard deviation associated with $mij,k$. For clarity, the measured Mueller matrix element $mij,kmeas$ in Eq. (1) is marked as $yl$ with the three indices $i$, $j$, and $k$ lumped into a single index $l$. The calculated Mueller matrix element $mij,kcalc(x,a)$ is correspondingly marked as $fl(x,a)$. Thus, Eq. (1) can be simply rewritten as Display Formula$\chi 2=\u2211l=1Nwl[yl\u2212fl(x,a)]2=[y\u2212f(x,a)]TW[y\u2212f(x,a)],$(2)
where $wl$ is the weighting factor and is given by $wl=1/\sigma 2(yl)$ and $N=15N\lambda $. $W$ is an $N\xd7N$ diagonal matrix with diagonal elements $wl$. The inverse problem in grating reconstruction is typically formulated as a least square regression problem such that Display Formula$x^=arg\u2009minx\u2208\Omega {[y\u2212f(x,a*)]TW[y\u2212f(x,a*)]},$(3)
where $x^$ is the solution of the inverse problem that contains the extracted structural parameters, and $\Omega $ is the associated parameter domain. $a*$ denotes the given value of vector $a$ in the parameter extraction.