= Elliptical polarization
{title2=$E_x^2/A_x^2+E_y^2/A_y^2-2E_xE_y\cos\delta/(A_xA_y)=\sin^2\delta$}
{wiki}
= Elliptically polarized wave
{synonym}
For an <elliptically polarized wave>, the tip of its transverse <vector> traces an <ellipse> at a fixed point during each <wave period>. For <polarization of an electromagnetic wave>, this is the <electric field> <vector>. With the real components in <polarization of waves>, assume $A_x,A_y>0$ and $\sin\delta\neq0$. Eliminating time gives
$$
\frac{E_x^2}{A_x^2}+\frac{E_y^2}{A_y^2}-\frac{2E_xE_y}{A_xA_y}\cos\delta=\sin^2\delta.
$$
A short derivation uses $\phi=\omega t$ and the <cosine> addition formula to obtain $E_y/A_y=(E_x/A_x)\cos\delta-\sin\phi\sin\delta$; squaring and using the <sine> and <cosine> identity $\sin^2\phi=1-\cos^2\phi$ gives the displayed equation. Outside <phase quadrature>, the mixed term gives a tilted <ellipse>; $A_x,A_y$ are component <wave amplitudes> and need not be its <semi-major and semi-minor axes>.
In <phase quadrature>, $\delta=\pm\pi/2$, the mixed term vanishes. For $(A_x,A_y)=(2,1)\,\mathrm{V/m}$, the <ellipse> is $E_x^2/4+E_y^2=1$, its <semi-major and semi-minor axes> have lengths $2$ and $1\,\mathrm{V/m}$, and its <electric field> magnitude varies between $1$ and $2\,\mathrm{V/m}$. At $\phi=\pi/4$ with $\delta=\pi/2$, the <electric field> is $(\sqrt2,-1/\sqrt2)\,\mathrm{V/m}$ and has magnitude $\sqrt{5/2}\,\mathrm{V/m}$. Equal <wave amplitudes> instead give <circular polarization>, a special case of <elliptical polarization> with constant magnitude. As $\delta$ approaches $0$ or $\pi$, the <ellipse> collapses to a <line segment> and gives <linear polarization>; this limit is degenerate rather than a nondegenerate <ellipse>.
Changing $\delta$ from $\pi/2$ to $-\pi/2$ reverses traversal of the same <ellipse>. In the displayed $x,y$ plot, $\delta=\pi/2$ moves from $(A_x,0)$ toward negative $y$ as time increases. A label such as right- or left-handed <circular polarization> additionally requires a stated viewing direction and naming convention: viewing along propagation and viewing against propagation reverse the apparent rotation. https://www.feynmanlectures.caltech.edu/I_33.html#Ch33-S1[Feynman Lectures, I §33–1: polarization conventions]{ref}.
The <ellipse> records both the relative <wave amplitudes> and the <phase difference> of two transverse <oscillations>. This separates a change of <polarization of waves> from a change of overall <wave amplitude>: multiplying both components by the same positive factor enlarges the <ellipse> while preserving its shape and orientation.
A rotating ideal linear <polarizer> can probe this geometry. For the $2:1$ <ellipse> in <phase quadrature>, <orthogonal projection> onto a pass axis at <angle> $\theta$ gives $E_{\mathrm{out}}(t)=2\cos\theta\cos\omega t-\sin\theta\sin\omega t$, in $\mathrm{V/m}$. Averaging its square over a <wave period> gives $(4\cos^2\theta+\sin^2\theta)/2$, while the incoming squared <electric field> magnitude averages to $5/2$. Thus the transmitted fraction of <light intensity> is $(4\cos^2\theta+\sin^2\theta)/5$: it is $4/5$ at $0^\circ$, $1/2$ at $45^\circ$ and $1/5$ at $90^\circ$. There is no single incident line of <linear polarization> to insert into <Malus's law>; projecting the <electric field> and averaging its square is the appropriate generalization. This measurement cannot distinguish $\delta=\pi/2$ from $\delta=-\pi/2$, because reversing the traversal leaves all these squared averages unchanged.
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