The polarization ellipse is a geometric way to describe the state of a monochromatic electromagnetic wave.
If the electric field has two transverse components, one along \(x\) and one along \(y\), then the tip of the electric-field vector traces a curve in the transverse plane as time passes.
In general, that curve is an ellipse. Linear and circular polarization are simply special cases of this general ellipse.
A convenient parametrization is
\[
x(t)=E_x\cos\omega t,
\qquad
y(t)=E_y\cos(\omega t-\delta),
\]
where \(E_x\) and \(E_y\) are the component amplitudes, \(\omega\) is the angular frequency, and \(\delta\) is the phase difference between the two components.
This form is especially useful because it makes the role of the phase difference very clear.
If \(\delta=0^\circ\) or \(180^\circ\), the two components oscillate in step or in opposite phase, so the tip of the field moves back and forth along a straight line.
That is linear polarization.
If the amplitudes are equal and the phase difference is \(\pm90^\circ\), then the field tip rotates with constant magnitude and traces a circle.
That is circular polarization.
When neither of these special situations occurs, the field tip traces a genuine ellipse, so the state is elliptically polarized.
In this calculator, the associated Jones vector is taken as
\[
\mathbf{E}=
\begin{bmatrix}
E_x\\
E_y e^{-i\delta}
\end{bmatrix}.
\]
This matches the time-domain parametrization above when the real field is obtained from the complex representation.
The Jones-vector form is helpful because it connects directly with Stokes parameters and with more advanced polarization tools such as Jones matrices.
The normalized Stokes parameters for this state are
\[
s_1=\frac{E_x^2-E_y^2}{E_x^2+E_y^2},
\qquad
s_2=\frac{2E_xE_y\cos\delta}{E_x^2+E_y^2},
\qquad
s_3=\frac{2E_xE_y\sin\delta}{E_x^2+E_y^2}.
\]
These quantities summarize the polarization state in a compact form.
The parameters \(s_1\) and \(s_2\) encode the linear character and orientation of the ellipse,
while \(s_3\) encodes the sense of rotation, which is interpreted here as handedness.
In this calculator, positive \(s_3\) is labeled right-handed or right-circular, and negative \(s_3\) is labeled left-handed or left-circular.
Different optics texts may adopt different sign conventions, so it is always important to state the convention being used.
The polarization azimuth \(\psi\) and ellipticity angle \(\chi\) are obtained from the Stokes parameters through
\[
\psi=\frac{1}{2}\tan^{-1}\!\left(\frac{s_2}{s_1}\right),
\qquad
\chi=\frac{1}{2}\sin^{-1}(s_3).
\]
The angle \(\psi\) gives the orientation of the major axis of the ellipse, while \(\chi\) gives its shape.
If \(\chi=0^\circ\), the state is linear.
If \(|\chi|=45^\circ\), the state is circular.
Intermediate values correspond to elliptical polarization.
The ratio of minor to major axis is \(\tan|\chi|\), so the closer \(|\chi|\) is to \(45^\circ\), the more circular the ellipse becomes.
The sample input in this calculator is
\(E_x=1\),
\(E_y=1\),
and
\(\delta=90^\circ\).
Substituting into the parametric equations gives
\[
x(t)=\cos\omega t,
\qquad
y(t)=\cos(\omega t-90^\circ)=\sin\omega t.
\]
Since the two components have equal amplitude and differ in phase by \(90^\circ\), the field tip moves on a circle.
Therefore the state is circular.
With the sign convention used in this tool, the corresponding normalized Stokes parameter is
\[
s_3=\frac{2(1)(1)\sin 90^\circ}{1^2+1^2}=1,
\]
so the sample is labeled right-circular polarization.
Another useful set of formulas gives the semi-axis lengths of the ellipse:
\[
a^2=\frac{S_0+\sqrt{(E_x^2-E_y^2)^2+4E_x^2E_y^2\cos^2\delta}}{2},
\]
\[
b^2=\frac{S_0-\sqrt{(E_x^2-E_y^2)^2+4E_x^2E_y^2\cos^2\delta}}{2},
\qquad
S_0=E_x^2+E_y^2.
\]
These expressions let you quantify the geometry of the ellipse directly from the amplitudes and phase difference.
When \(b=0\), the ellipse collapses into a line and the state is linear.
When \(a=b\), the ellipse is a circle and the state is circular.
At a more advanced university level, polarization is often studied using Stokes vectors, Jones matrices, Mueller matrices, and the Poincaré sphere.
But the polarization ellipse remains one of the most intuitive starting points because it turns the abstract phase relation between orthogonal components into a visible geometric object.
\[
x(t)=E_x\cos\omega t,
\qquad
y(t)=E_y\cos(\omega t-\delta),
\qquad
\mathbf{E}=
\begin{bmatrix}
E_x\\
E_y e^{-i\delta}
\end{bmatrix}.
\]
These formulas are enough to decide whether the state is linear, circular, or elliptical, determine the handedness under the chosen convention, and construct the full polarization ellipse.
