# Chiral Metasurfaces: A Review of the Fundamentals and Research Advances
## 2 Fundamentals of Chiral Metasurfaces
### Jones Matrix and Circular Dichroism  
The incident and scattered field's amplitude are also associated with the Jones matrices, which simplify the characterization of the optical responses of planar materials [92]. Jones matrices are used in experimental setups to depict the transformation of polarization states as light interacts with planar chiral materials, allowing for the prediction and analysis of optical responses like OR and CD. They provide a comprehensive and mathematically rigorous framework to design and understand chiroptical objects, making it simple to manipulate and control polarized light in different applications like imaging, sensing, and communications [98]. \[ \begin{bmatrix}E_{r}^{x}\\ E_{r}^{y}\end{bmatrix} = \begin{bmatrix}r_{xx}&r_{xy}\\ r_{yx}&r_{yy}\end{bmatrix}\begin{bmatrix}E_{i}^{x}\\ E_{i}^{y}\end{bmatrix} = \begin{bmatrix}E_{i}^{x}\\ E_{i}^{y}\end{bmatrix} , \tag{6}\]  
\[ \begin{bmatrix}E_{t}^{x}\\ E_{r}^{y}\end{bmatrix} = \begin{bmatrix}t_{xx}&t_{yy}\\ t_{yx}&t_{yy}\end{bmatrix}\begin{bmatrix}E_{i}^{x}\\ E_{i}^{y}\end{bmatrix} = \begin{bmatrix}E_{i}^{x}\\ E_{i}^{y}\end{bmatrix} , \tag{7}\]  
where \(E_{i}^{x}\), \(E_{r}^{x}\), and \(E_{t}^{x}\) are the incident, reflected, and transmitted polarized electric fields along the x-direction, correspondingly. T and R are the transmission and reflection matrices for linear polarization, respectively. The electric fields polarized in the y-direction are denoted by symbols identical to the superscripts. \[T_{\text{circ}}\ = \begin{bmatrix}t_{++}&t_{+-}\\ t_{-+}&t_{--}\end{bmatrix} =\frac{1}{2} \begin{bmatrix}t_{xx}+t_{yy}+\mathrm{i }\big{(}t_{xy}-t_{yx}\big{)}&t_{xx}-t_{yy}-\mathrm{i}\big{(}t_{xy}+t_{yx}\big{)} \\ t_{xx}-t_{yy}+\mathrm{i}\big{(}t_{xy}+t_{yx}\big{)}&t_{xx}+t_{yy}-\mathrm{i} \big{(}t_{xy}-t_{yx}\big{)}\end{bmatrix} =\mathrm{M}^{-1}\text{TM}, \tag{8}\]  
\[R_{\text{circ}}\ = \begin{bmatrix}r_{++}&r_{+-}\\ r_{-+}&r_{--}\end{bmatrix} =\frac{1}{2} \begin{bmatrix}r_{xx}+r_{yy}+\mathrm{i }\big{(}r_{xy}-r_{yx}\big{)}&r_{xx}-r_{yy}-\mathrm{i}\big{(}r_{xy}+r_{yx}\big{)} \\ r_{xx}-r_{yy}+\mathrm{i}\big{(}r_{xy}+r_{yx}\big{)}&r_{xx}+r_{yy}-\mathrm{i} \big{(}r_{xy}-r_{yx}\big{)}\end{bmatrix} =\mathrm{M}^{-1}\text{RM}. \tag{9}\]  
\[\mathrm{M}=\frac{1}{\sqrt{2}} \begin{pmatrix}1&1\\ \mathrm{i}&-\mathrm{i}\end{pmatrix} \]  
was used to switch the basis from Cartesian to circular. The subscript \"\(--\)\" denotes LCP wave, \"\(+\)\" denotes RCP wave, and likewise \"\(t_{+-}\)\" represents LCP incidence and RCP transmission. \(r_{+-},r_{--},r_{-+}\), and \(r_{++}\) are the reflection coefficients of LCP to RCP, LCP to LCP, RCP to LCP, and RCP to RCP, respectively. Symmetry considerations can notably influence the optical properties of chiral metasurfaces, potentially reducing or enhancing specific chiral effects [83; 99; 100; 101]. For example, C4 symmetry, signifying four-fold rotational symmetry, leads to orientation-dependent chiral effects, essential for designing tailored chiral metasurfaces in applications such as chiral sensing and polarization manipulation. For periodically arranged metamaterials with four-fold (C4) or three-fold (C3) rotational symmetries, the Jones matrices can be expressed as [56]:  
\[\mathrm{T}_{\mathrm{circ}}= \begin{pmatrix}\mathrm{t}_{\mathrm{xx}}+\mathrm{i }_{\mathrm{tyy}}&0\\ 0&\mathrm{t}_{\mathrm{xx}}-\mathrm{i}_{\mathrm{tyy}}\end{pmatrix} \text{, and }\mathrm{R}_{\mathrm{circ}}= \begin{pmatrix}\mathrm{r}_{\mathrm{xx}}&0\\ 0&\mathrm{r}_{\mathrm{xx}}\end{pmatrix} =\mathrm{r}_{\mathrm{xx}}\mathrm{I}, \tag{10}\]  
where I represents the identity matrix. The transmission coefficient matrix can be used in conjunction with the rotation matrix (Dr) to analyze symmetrical circumstances by calculating a transmission matrix Tr. \[\mathrm{Tr}=\mathrm{Dr}^{-1}\mathrm{T}^{\mathrm{f}}\mathrm{Dr}= \begin{bmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}\begin{bmatrix}\mathrm{t}_{\mathrm{xx}}& \mathrm{t}_{\mathrm{yy}}\end{bmatrix}\begin{bmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{bmatrix} .