= Polarization
{disambiguate=waves}
{title2=$\mathbf E(t)=(A_x\cos\omega t,A_y\cos(\omega t+\delta))$}
{wiki}
= Polarization of waves
{synonym}
= Wave polarization
{synonym}
<Polarization of waves> describes the direction and time-dependent geometry of a <transverse wave>'s <vector> during its <oscillation> at a fixed position. In three spatial dimensions, a <transverse wave> can have two independent directions of <oscillation>, both perpendicular to its direction of propagation. For <polarization of an electromagnetic wave>, the relevant <vector> is the transverse <electric field>; for a mechanical <transverse wave>, it can be the transverse <displacement>. A purely <longitudinal wave> has no choice between these transverse directions, and the orientation of a static <electric field> is not <polarization of waves>.
Choose transverse axes $x,y$ in a <Cartesian coordinate system>. A <monochromatic wave> with a fixed <phase difference> has the real <electric field>
$$
\mathbf E(t)=\bigl(A_x\cos\omega t,\ A_y\cos(\omega t+\delta)\bigr),\qquad A_x,A_y\geq0,\quad\omega>0.
$$
Here $A_x,A_y$ are <wave amplitudes>, $\omega$ is the <angular frequency>, and $\delta$ is the <phase difference> of the $y$ <oscillation> relative to the $x$ <oscillation>. The <vector>'s tip traces its path during the <wave period> $T=2\pi/\omega$. Both plotted coordinates are real <electric field> components; neither axis represents the <imaginary part> of a <complex amplitude>.
For nonzero <wave amplitudes>, $\delta=0$ or $\pi$ modulo $2\pi$ gives <linear polarization>: the <vector> moves back and forth along a fixed line. Equal nonzero <wave amplitudes> and $\delta=\pm\pi/2$ modulo $2\pi$ give <circular polarization>: the tip traces a <circle> of constant <radius>. Unequal nonzero <wave amplitudes> with $\delta=\pm\pi/2$ give a noncircular <ellipse> whose axes coincide with $x,y$. More generally, nonzero <wave amplitudes> and $\sin\delta\neq0$ give <elliptical polarization>, with <circular polarization> as a special case. If exactly one <wave amplitude> vanishes, the nonzero <wave> has <linear polarization> for every $\delta$; if both vanish, there is no nonzero <wave> to assign a <polarization of waves>. https://www.feynmanlectures.caltech.edu/I_33.html#Ch33-S1[Feynman Lectures, I §33–1: the electric vector of light]{ref}.
For example, take $\omega=2\pi\,\mathrm{s}^{-1}$, so the <wave period> is $T=1\,\mathrm{s}$. The choices $(A_x,A_y,\delta)=(2,1,0)$, $(1,1,\pi/2)$ and $(2,1,\pi/2)$, with <electric field> <wave amplitudes> in $\mathrm{V/m}$, produce respectively the <line segment> $E_y=E_x/2$, the <circle> $E_x^2+E_y^2=1$ and the <ellipse> $E_x^2/4+E_y^2=1$, when coordinates are measured in $\mathrm{V/m}$. At $t=T/4$, their tips are respectively $(0,0)$, $(0,-1)$ and $(0,-1)\,\mathrm{V/m}$. The last two share a point without having the same path.
\Image[/wave-polarization-trajectories.png]
{title=Real transverse <electric field> trajectories for <linear polarization>, <circular polarization> and <elliptical polarization>}
{description=All panels use the same coordinate scale. Orange arrows follow increasing <wave phase> $\phi=\omega t$; the dot marks $\phi=0$. The blue <vector> is the <electric field> at $\phi=\pi/4$. These are paths at one fixed position, rather than spatial drawings of a propagating <wave>.}
Resolving the <electric field> into two transverse components makes <polarization of waves> useful: the same <frequency> and <wave amplitudes> can describe different paths when their <phase difference> changes. An ideal linear <polarizer> selects the component along its pass axis, letting its orientation probe the direction of <linear polarization> through <Malus's law>. A fixed pair of <wave amplitudes> and a fixed <phase difference> describe one stable path; <unpolarized light> requires a statistical description rather than one fixed <ellipse>.
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