Mathematical Description of Linear, Circular, and Elliptical Polarization

In the grand framework of wave optics, light is fundamentally an electromagnetic wave whose state of polarization describes the trajectory traced by the electric field vector in the plane transverse to the direction of propagation. Mastering this mathematical description is essential not only for foundational optical theory but also for driving modern optical communications, laser physics, and advanced photonics engineering.

From classical electrodynamics, a monochromatic plane wave traveling along the positive $z$-axis consists of mutually orthogonal electric ($\vec{E}$) and magnetic ($\vec{B}$) fields. Because the electric vector typically dominates optical interactions with matter, it is designated as the optical vector.

At any arbitrary plane (such as $z = 0$), the instantaneous vibration of $\vec{E}$ can be resolved into two mutually perpendicular components along the $x$ and $y$ axes:

$$E_x(t) = E_{0x} \cos(\omega t - \phi_x)$$
$$E_y(t) = E_{0y} \cos(\omega t - \phi_y)$$

Here, $E_{0x}$ and $E_{0y}$ represent the respective amplitudes, $\omega$ denotes the angular frequency, $t$ is time, and $\phi_x$ and $\phi_y$ are the initial phases. For mathematical convenience, we can set $\phi_x = 0$ and introduce the phase difference $\delta = \phi_y - \phi_x = \phi_y$. The equations then simplify to:

$$E_x(t) = E_{0x} \cos(\omega t)$$
$$E_y(t) = E_{0y} \cos(\omega t - \delta)$$

By eliminating the time parameter $t$ from these coupled equations, we derive the general parametric trajectory of the electric field vector tip in the $xy$-plane:

$$\left(\frac{E_y}{E_{0y}}\right)^2 + \left(\frac{E_x}{E_{0x}}\right)^2 - 2\frac{E_x}{E_{0x}}\frac{E_y}{E_{0y}}\cos\delta = \sin^2\delta$$

This universal expression describes an ellipse. Linear, circular, and elliptical polarizations are simply specialized degeneracies or specific parameter regimes of this overarching elliptical equation.
Linear polarization represents the simplest polarization state, occurring when the electric vector oscillates along a fixed, invariant spatial orientation throughout propagation.

• Mathematical Condition: The phase difference $\delta$ between the orthogonal components satisfies:
  $$\delta = m\pi \quad (m = 0, \pm 1, \pm 2, \dots)$$
• Trajectory Derivation: When $\delta$ is $0$ or an integer multiple of $\pi$, the general equation collapses into a linear relation:
  $$E_y = \pm \frac{E_{0y}}{E_{0x}} E_x$$
• Physical Implication: The positive and negative signs correspond to positive and negative trajectory slopes, respectively. The direction of $\vec{E}$ remains constant in space while its magnitude undergoes simple harmonic motion over time.

Mathematical Formulation of Circular Polarization

In circular polarization, the magnitude of the electric vector remains constant, but its direction rotates uniformly in the transverse plane, tracing out a circle.

• Mathematical Condition: The orthogonal components possess equal amplitudes, and their phase difference is an odd multiple of $\pi/2$:
  $$E_{0x} = E_{0y} = E_0$$
  $$\delta = \pm \frac{\pi}{2} + 2m\pi \quad (m = 0, \pm 1, \pm 2, \dots)$$
• Trajectory Derivation: Substituting these conditions into the general equation yields:
  $$E_x^2 + E_y^2 = E_0^2$$
  which is the standard equation of a circle.
• Handedness:
  • When $\delta = +\pi/2$, the electric vector rotates clockwise (viewed facing the oncoming wave), defining right-handed circular polarization.
  • When $\delta = -\pi/2$, the vector rotates counter-clockwise, defining left-handed circular polarization.

Mathematical Formulation of Elliptical Polarization

Elliptical polarization is the most general state of monochromatic light. Both linear and circular polarizations can be treated as special subsets of this broader category.

• Mathematical Condition: The amplitudes $E_{0x}$ and $E_{0y}$ are unequal, and the phase difference $\delta$ is neither $0$ nor an integer multiple of $\pi$.
• Geometric Parameters: This state is fully characterized not only by the amplitude ratio but also by the orientation of the principal axes (azimuth angle $\psi$) and the ellipticity.
• Jones Vector Representation: In contemporary optics, matrix formalisms offer a streamlined approach. The normalized complex amplitudes are expressed as a column vector:
  $$J = \begin{bmatrix} E_{0x} e^{i\phi_x} \ E_{0y} e^{i\phi_y} \end{bmatrix}$$
  • For instance, a linear polarization state can be written as $\begin{bmatrix} 1 \ 0 \end{bmatrix}$ or $\begin{bmatrix} 1 \ 1 \end{bmatrix}$ (suitably normalized).
  • Circular polarization is represented as $\frac{1}{\sqrt{2}}\begin{bmatrix} 1 \ \pm i \end{bmatrix}$.
  • Elliptical states are represented by more general complex coefficient combinations.

Summary and Analytical Comparison

The defining criteria for the three primary polarization states can be systematically summarized as follows:

In optical engineering, these mathematical models form the theoretical bedrock for designing phase retarders (such as $\lambda/4$ and $\lambda/2$ waveplates) and polarizers. By artificially altering the phase difference $\delta$ using birefringence, engineers can dynamically convert between linear, circular, and elliptical polarization states to suit specific application demands.