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Polarization

The orientation of the oscillations of a transverse wave, often explained by the wave nature of light and its interaction with materials.
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The statement of the theorem

Let k∈R3\mathbf{k} \in \mathbb{R}^3 be the unit wave vector defining the direction of propagation, and let ω\omega be the angular frequency. The electric field E(r,t)\mathbf{E}(\mathbf{r}, t) of a monochromatic, transverse electromagnetic wave propagating in a linear, isotropic, and homogeneous medium is given by:\nE(r,t)=E0cos⁡(k⋅r−ωt)+H0sin⁡(k⋅r−ωt)\mathbf{E}(\mathbf{r}, t) = \mathbf{E}_0 \cos(k \cdot \mathbf{r} - \omega t) + \mathbf{H}_0 \sin(k \cdot \mathbf{r} - \omega t) \nwhere E0\mathbf{E}_0 and H0\mathbf{H}_0 are the complex amplitude vectors of the electric and magnetic fields, respectively. Polarization is characterized by the amplitude vector E0\mathbf{E}_0, which must satisfy the transverse condition: E0⋅k=0\mathbf{E}_0 \cdot \mathbf{k} = 0. Furthermore, the polarization state is defined by the ratio of the components of E0\mathbf{E}_0 projected onto an orthonormal basis (e^1,e^2)(\hat{\mathbf{e}}_1, \hat{\mathbf{e}}_2) spanning the plane perpendicular to k\mathbf{k}: E0=E01e^1+E02e^2\mathbf{E}_0 = E_{01} \hat{\mathbf{e}}_1 + E_{02} \hat{\mathbf{e}}_2. The degree of polarization is quantified by the Stokes parameters S=(S0,S1,S2,S3)\mathbf{S} = (S_0, S_1, S_2, S_3), where S0=⟨E2⟩S_0 = \langle E^2 \rangle, S1=⟨Ex2−Ey2⟩S_1 = \langle E_x^2 - E_y^2 \rangle, S2=⟨2ExEy⟩S_2 = \langle 2 E_x E_y \rangle, and S3=⟨2ExEz⟩S_3 = \langle 2 E_x E_z \rangle (assuming a coordinate system where k\mathbf{k} is aligned with zz for simplicity, or generalized components otherwise).
Source: Wikipedia