Polarization describes how the transverse electric field of light changes in time.
If the field always points along one fixed line, the light is linearly polarized.
If the tip of the electric-field vector rotates with constant magnitude, the light is circularly polarized.
If the tip traces a general ellipse, the state is elliptically polarized.
These three cases are closely related: linear and circular polarization are simply special limiting cases of the general ellipse.
A convenient way to describe a fully polarized state is with two angles:
the azimuth \(\psi\), which sets the orientation of the polarization ellipse,
and the ellipticity angle \(\chi\), which sets its shape.
When \(\chi = 0^\circ\), the ellipse collapses to a line, so the light is linear.
When \(|\chi| = 45^\circ\), the ellipse becomes a circle, so the light is circular.
Intermediate values \(0^\circ < |\chi| < 45^\circ\) give elliptical polarization.
The ratio of minor to major axis is \(\tan|\chi|\), so larger \(|\chi|\) means a “rounder” ellipse.
The normalized Stokes parameters for a fully polarized state are
\[
s_1 = \cos 2\chi \cos 2\psi,\qquad
s_2 = \cos 2\chi \sin 2\psi,\qquad
s_3 = \sin 2\chi.
\]
These quantities summarize the polarization state compactly.
Roughly speaking, \(s_1\) compares horizontal and vertical preference,
\(s_2\) compares \(45^\circ\) and \(135^\circ\) preference,
and \(s_3\) measures the circular component.
If the incident intensity is \(I_0\), then the full Stokes vector is
\((S_0,S_1,S_2,S_3) = (I_0, I_0 s_1, I_0 s_2, I_0 s_3)\).
When polarized light passes through a linear polarizer whose transmission axis makes angle \(\theta\),
the transmitted intensity can be written in a general Stokes-based form:
\[
I(\theta) = \frac{I_0}{2}\left(1 + s_1 \cos 2\theta + s_2 \sin 2\theta\right).
\]
Substituting the expressions for \(s_1\) and \(s_2\) gives a very useful formula directly in terms of \(\psi\) and \(\chi\):
\[
I(\theta) = \frac{I_0}{2}\left[1 + \cos 2\chi \cos 2(\theta-\psi)\right].
\]
This single equation covers linear, circular, and elliptical states.
For linear polarization, \(\chi = 0^\circ\), so \(\cos 2\chi = 1\). Then the formula becomes
\[
I(\theta) = \frac{I_0}{2}\left[1 + \cos 2(\theta-\psi)\right]
= I_0 \cos^2(\theta-\psi),
\]
which is exactly Malus’ law.
This is the standard high-school and introductory-university result:
the transmitted intensity depends on the square of the cosine of the angle difference between the incident linear polarization and the polarizer axis.
For the sample input in this calculator,
linear polarization at \(30^\circ\) means
\(\psi = 30^\circ\) and \(\chi = 0^\circ\),
while the polarizer is set to
\(\theta = 60^\circ\).
Therefore the angle difference is
\[
\theta - \psi = 60^\circ - 30^\circ = 30^\circ.
\]
Malus’ law then gives
\[
\frac{I}{I_0} = \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} = 0.75.
\]
So the correct output for that sample is
\(I = 0.75 I_0\),
not \(0.25 I_0\).
A value of \(0.25 I_0\) would instead correspond to an angle difference of \(60^\circ\), since \(\cos^2 60^\circ = 1/4\).
The same general formula also explains the other special cases.
If the state is circular, then \(|\chi| = 45^\circ\), which makes \(\cos 2\chi = 0\).
The transmission law becomes
\[
I(\theta) = \frac{I_0}{2},
\]
independent of the polarizer angle.
This makes sense physically: circular polarization contains equal components along every linear direction,
so an ideal linear polarizer always passes exactly half the intensity.
If the state is elliptical, then \(0^\circ < |\chi| < 45^\circ\), so \(0 < |\cos 2\chi| < 1\).
In that case the transmitted intensity still oscillates with analyzer angle, but it never reaches perfect extinction.
A single linear polarizer can reduce the beam strongly, but not completely remove it.
That is one of the clearest experimental signatures that the incident light is elliptical rather than purely linear.
In practical optics, this framework helps explain sunglasses cutting glare, polarizers in photography, stress birefringence, liquid-crystal displays, and Stokes-parameter measurements in laboratory optics.
At a more advanced university level one extends the discussion to Jones vectors, partially polarized light, Mueller matrices, wave plates, and full polarization tomography.
Still, for many core problems, the key ideas are the polarization ellipse, Malus’ law for linear states, and the Stokes-based analyzer formula for general states.
\[
s_1 = \cos 2\chi \cos 2\psi,\qquad
s_2 = \cos 2\chi \sin 2\psi,\qquad
s_3 = \sin 2\chi,
\]
\[
I(\theta) = \frac{I_0}{2}\left[1 + \cos 2\chi \cos 2(\theta-\psi)\right],
\qquad
\text{and for } \chi=0,\quad
I = I_0\cos^2(\theta-\psi).
\]
These equations let you identify the polarization type, compute its Stokes parameters, and predict exactly how much light passes through a linear polarizer.
