Polarization: Basis Decomposition

Ask what light can do and the answer is short. An electromagnetic wave coming toward you carries an electric field pointing somewhere in the plane facing you, and as the wave oscillates that arrow sweeps out a shape: a line, a circle, or an ellipse. That is the entire vocabulary of polarization. Why there are only these options, and why each can be written as a blend of two simpler ones, comes down to a single fact about transverse waves with two components.

Two components are a basis, and a basis is a choice

Write the field as a complex Jones vector, dropping the common time dependence and keeping the two amplitudes and their phase difference:

\[ \mathbf{E} \;=\; \begin{pmatrix} a_x \\ a_y\,e^{i\delta} \end{pmatrix}. \]

This two dimensional complex vector can be expanded in any basis. The horizontal and vertical pair is obvious but not special: diagonal and antidiagonal linear states form an equally good basis, and so do the two circular states. Picking a basis just means picking which two reference polarizations everything else is a sum of. The interactive above runs the horizontal and vertical decomposition; the next runs the circular decomposition of the same linear states.

The measurable content of a state is captured by the three normalized Stokes parameters, exactly the coordinates of its point on the Poincaré sphere:

\[ S_1 = a_x^2 - a_y^2, \qquad S_2 = 2 a_x a_y \cos\delta, \qquad S_3 = 2 a_x a_y \sin\delta. \]

Here \( S_1 \) measures how horizontal versus vertical the light is, \( S_2 \) how diagonal, and \( S_3 \) how circular. Each pair of opposite points on the sphere is one basis, so the geometry makes the freedom of choice literal: every axis through the center picks two antipodal states that together span everything.

A linear wave is balanced left and right circular light

The cleanest example of decomposition looks least obvious: take horizontal linear light and split it into equal right and left circular parts:

\[ \begin{pmatrix} 1 \\ 0 \end{pmatrix} \;=\; \tfrac{1}{\sqrt{2}}\,\underbrace{\tfrac{1}{\sqrt 2}\begin{pmatrix} 1 \\ i \end{pmatrix}}_{\text{RCP}} \;+\; \tfrac{1}{\sqrt{2}}\,\underbrace{\tfrac{1}{\sqrt 2}\begin{pmatrix} 1 \\ -i \end{pmatrix}}_{\text{LCP}}. \]

Two arrows of equal length rotating in opposite directions sum to an arrow that never rotates: it just grows and shrinks along a line. The vertical parts always cancel while the horizontal parts reinforce. Slide the relative phase and the cancellation axis tips, so the linear polarization rotates. This is no trick: it is exactly how optical rotation works in sugar solutions and how circular birefringence is engineered, by giving the two circular components slightly different speeds.

Linear = right circular + left circular balanced, opposite handedness

linear orientation 0°

two counter-rotating phasors add to a line

equator point; poles are the RCP / LCP basis

Watch the handedness cancel. The red and blue arrows have the same length and spin in opposite senses, so their resultant (white) is pinned to a single line. On the Poincaré sphere the resultant sits on the equator, halfway between the two poles that represent the circular basis states. Turning the orientation slider feeds a relative phase to the two rotations, and the equator point walks around while the poles stay put.

Birefringence: one crystal, two speeds

Some materials present a different refractive index to the two linear components. A wave entering such a crystal splits along its fast and slow axes into ordinary and extraordinary rays that travel at different speeds and fall out of step. By the time they leave, they have accumulated a phase difference, the retardance:

\[ \Gamma \;=\; \frac{2\pi}{\lambda}\,\bigl(n_e - n_o\bigr)\,L, \]

where \( L \) is the thickness and \( n_e - n_o \) the birefringence. Both components stay perfectly good light throughout; nothing is absorbed or lost. Only their relative phase changes, and that is enough to convert the polarization. Feed in light at 45 degrees, halfway between the axes, and as \( \Gamma \) grows the output sweeps from linear to elliptical to circular and on. On the Poincaré sphere the state rotates about the axis set by the crystal, tracing a circle whose angle is the retardance.

A birefringent slab retards one axis Γ = 0°

retardance Γ 90°

output state of 45° input after the slab

state circles about the S₁ (crystal) axis

Thickness is retardance. The two component waves enter in step. Inside the crystal the slow axis carries more cycles, so by the exit it lags the fast axis by Γ. Press play to let the slab thicken on its own, or drag the slider to scrub. At Γ = 90° a quarter wave of delay turns the 45° input into circular light; at Γ = 180° a half wave flips it to −45°. The output point rides a circle of constant latitude offset around the crystal axis.

The standard toolkit: wave plates and polarizers

Three components do almost all polarization work in a lab, and each is a single move on the Poincaré sphere.

A quarter wave plate is a birefringent slab cut to give exactly \( \Gamma = 90^\circ \). It rotates the sphere by a quarter turn about its fast axis, which is just enough to turn linear light at 45 degrees to its axis into circular light, and circular light back into linear.
A half wave plate gives \( \Gamma = 180^\circ \). It rotates the sphere by a half turn about its fast axis, reflecting any linear state across that axis. A linear input rotates by twice the angle between it and the plate, which makes the half wave plate the standard way to steer a polarization direction.
A linear polarizer is not a rotation at all. It projects the field onto its transmission axis, throwing away the orthogonal part. On the sphere it collapses any input down to one point on the equator, and the surviving intensity follows Malus's law, \( I = I_0 \cos^2\theta \).

Wave plates and polarizers on the sphere QWP

Input state

Element

fast axis 0°

faint = input, bright = output

input, output, and the element's axis

output Right circular

Every element is a move. A wave plate rotates the whole sphere about its axis (the green line) by its retardance, so the input point swings along the green arc to the output point. A quarter wave plate is a 90° turn, a half wave plate a 180° turn. The linear polarizer drops the state straight onto the equator and reports the transmitted intensity from Malus's law. Change the input, the element, or the axis angle and watch where the output lands.

Why the two component picture keeps paying off

Every result above came from one starting point: light has two transverse components, and how you split it between two reference states is free. The polarization ellipse, the Stokes parameters, the Poincaré sphere, and the Jones matrices of wave plates are four views of that one structure. It is the same algebra that governs a spin one half particle and a single qubit, which is why the Poincaré and Bloch spheres are the same sphere with different labels. Learn one and you have read all four.

References

E. Hecht, Optics, 5th ed. Boston, MA, USA: Pearson, 2017.
M. Born and E. Wolf, Principles of Optics, 7th ed. Cambridge, U.K.: Cambridge Univ. Press, 1999.
G. G. Stokes, "On the composition and resolution of streams of polarized light from different sources," Trans. Cambridge Philos. Soc., vol. 9, pp. 399–416, 1852.
R. C. Jones, "A new calculus for the treatment of optical systems. I. Description and discussion of the calculus," J. Opt. Soc. Am., vol. 31, no. 7, pp. 488–493, Jul. 1941, doi: 10.1364/JOSA.31.000488.
D. H. Goldstein, Polarized Light, 3rd ed. Boca Raton, FL, USA: CRC Press, 2011.