Polarization

Circularly polarized electromagnetic wave represented as a rotating electric-field vector. — Circular polarization visualized as a rotating transverse electric-field vector. Image source: Wikimedia Commons, Inductiveload, public domain.

30-second intuitionPolarization describes how the electric-field vector moves at a fixed point in space. If its tip traces a line, the wave is linearly polarized; a circle gives circular polarization; the general case is an ellipse.

Two orthogonal field components

For propagation along $z$,

\[\mathbf E(z,t)=\hat x E_x\cos(\omega t-kz)+\hat y E_y\cos(\omega t-kz+\delta).\]

The amplitude ratio $E_y/E_x$ and phase difference $\delta$ determine polarization.

Linear polarization

If $\delta=0$ or $\pi$, the vector oscillates along a fixed line.

Circular polarization

\[E_x=E_y\]

and

\[\delta=\pm\frac{\pi}{2},\]

the tip traces a circle. The sign convention for right/left handedness depends on viewing convention, so state it explicitly.

Elliptical polarization

Any other nondegenerate combination produces an ellipse. Circular and linear polarization are special cases of elliptical polarization.

Axial ratio

Axial ratio compares major and minor axes:

\[AR=\frac{E_{major}}{E_{minor}}.\]

Circular polarization has ideal $AR=1$ (0 dB); linear polarization corresponds to infinite axial ratio.

Polarization mismatch

For linearly polarized transmit/receive antennas separated by angle $\psi$,

\[\eta_p=\cos^2\psi.\]

At $90^\circ$, an ideal polarization-orthogonal receiver receives zero power; real antennas have finite cross-polarization and scattering often mixes polarization.

Jones-vector representation

A coherent fully polarized field can be represented by

\[\mathbf J= \begin{bmatrix} E_x\\ E_y e^{j\delta} \end{bmatrix}.\]

Jones calculus is convenient for deterministic optical/RF polarization elements. Stokes parameters are better when degree of polarization matters.

Connection to atoms

Relative to a quantization axis, optical/RF polarization decomposes into spherical components:

$\pi$: $\Delta m=0$;
$\sigma^+$: $\Delta m=+1$;
$\sigma^-$: $\Delta m=-1$.

Thus rotating field polarization or quantization axis directly changes atomic transition strengths.

Worked example — 45° mismatch

\[\eta_p=\cos^2 45^\circ=0.5.\]

That is a 3.01 dB polarization mismatch loss.

Measurement

Polarization can be measured by rotating a linearly polarized receive antenna, using orthogonal probes, measuring amplitude/phase of two components, or using polarimetry/Stokes analysis.

Engineering reality

Polarization is a system propertyRadomes, multipath, cables, misalignment, finite cross-polarization, birefringent optics and nearby structures can rotate or mix polarization after it leaves the source.

References

C. A. Balanis, Antenna Theory.
B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics.

Advanced topic — Stokes parameters and polarization coherence

Jones vectors describe fully polarized coherent fields, but partially polarized statistical mixtures require a coherency matrix or Stokes parameters. For transverse complex amplitudes (E_x,E_y), define

\[\mathbf J_c=\left\langle\begin{bmatrix}E_x\\E_y\end{bmatrix}\begin{bmatrix}E_x^*&E_y^*\end{bmatrix}\right\rangle.\]

Under the convention (S_0=J_{xx}+J_{yy}), (S_1=J_{xx}-J_{yy}), (S_2=2\Re J_{xy}), (S_3=-2\Im J_{xy}), the degree of polarization is

\[\mathrm{DoP}=\frac{\sqrt{S_1^2+S_2^2+S_3^2}}{S_0}.\]

The sign assigned to circular polarization depends on propagation direction and the adopted handedness convention. Do not compare RHCP/LHCP labels across references without checking the convention.

Advanced exercise: propagate a partially polarized field through a retarder and polarizer using Mueller matrices, and compare it with the Jones-model result for the fully coherent limit.

Further reading: M. Born and E. Wolf, Principles of Optics.