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Polarization

The orientation of the electric field vector in an electromagnetic wave, describing the wave's transverse nature and influencing its interaction with matter.
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The statement of the theorem

Let k\mathbf{k} be the unit vector defining the direction of propagation. Define the transverse plane P\mathcal{P} such that k⊥P\mathbf{k} \perp \mathcal{P}. Let e^1\hat{\mathbf{e}}_1 and e^2\hat{\mathbf{e}}_2 be two orthonormal basis vectors spanning P\mathcal{P}, forming a basis set {e^1,e^2}\{\hat{\mathbf{e}}_1, \hat{\mathbf{e}}_2\}. The instantaneous electric field E(r,t)\mathbf{E}(\mathbf{r}, t) of the electromagnetic wave is restricted to P\mathcal{P} and can be decomposed as E(r,t)=E1(t)e^1+E2(t)e^2\mathbf{E}(\mathbf{r}, t) = E_1(t) \hat{\mathbf{e}}_1 + E_2(t) \hat{\mathbf{e}}_2. The polarization state is characterized by the time evolution of the complex amplitudes E1(t)E_1(t) and E2(t)E_2(t). Specifically, the polarization vector P(t)\mathbf{P}(t) is defined by the complex components (E1(t),E2(t))(E_1(t), E_2(t)). The state is fully determined by the ratio of the complex amplitudes, ρ=E2(t)/E1(t)\rho = E_2(t)/E_1(t), and the relative phase ϕ=arg⁡(E2(t)/E1(t)\phi = \arg(E_2(t)/E_1(t). For a general polarization state, the field must satisfy the wave equation ∇2E−1c2∂2∂t2E=0\nabla^2 \mathbf{E} - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} \mathbf{E} = 0, with the transverse condition E⋅k=0\mathbf{E} \cdot \mathbf{k} = 0 and B=1c(k^×E)\mathbf{B} = \frac{1}{c}(\hat{\mathbf{k}} \times \mathbf{E}). The polarization state is thus the trajectory of the vector E(t)\mathbf{E}(t) in the plane spanned by e^1\hat{\mathbf{e}}_1 and e^2\hat{\mathbf{e}}_2.