Linear Polarization

The polarization of an electromagnetic wave describes the orientation and behavior of its electric field vector as the wave propagates. Linear polarization is the simplest and most widely encountered form, where the electric field oscillates along a single fixed direction throughout the wave's travel. It forms the foundation for understanding more complex polarization states such as circular and elliptical polarization.

Core Concept Explanation

A uniform plane wave propagating in the +z direction has its electric field lying entirely in the xy-plane. For the wave to be linearly polarized, the electric field vector must point in the same direction at every point along z at any instant. The field may oscillate back and forth along that direction, but the axis of oscillation does not change with time or position.

A linearly polarized wave can be described as having only one component of the electric field, for example E = Ey ŷ = E₀ cos(ωt − βz) ŷ. This means all the field energy is associated with a single orientation. The magnetic field H is perpendicular both to E and to the direction of propagation, and it is also linearly polarized in the orthogonal transverse direction.

Any linearly polarized wave at an arbitrary angle in the xy-plane can be decomposed into two orthogonal linearly polarized components: a horizontal component Ex and a vertical component Ey. The tilt angle τ of the polarization plane with respect to the x-axis satisfies tan τ = Ey/Ex. If both Ex and Ey are present, are in phase (or exactly 180 degrees out of phase), and have a fixed amplitude ratio, the wave is linearly polarized at angle τ.

The key condition for linear polarization is that the two orthogonal components must have zero relative phase difference (or a phase difference that is an exact multiple of π). Any other phase relationship results in circular or elliptical polarization.

Mathematical Expression

A general linearly polarized wave in the z-direction can be written as:

E(z,t) = (Ex x̂ + Ey ŷ) cos(ωt − βz)

where Ex and Ey are real constants. The total field magnitude is |E| = √(Ex² + Ey²) and the polarization angle is τ = arctan(Ey/Ex). This representation shows that when both components are present with a zero phase difference, the tip of the E vector traces a straight line in the xy-plane as time evolves at a fixed z, hence the name linear polarization.

The Stokes parameters for linearly polarized waves give S1 = I cos(2τ) and S2 = I sin(2τ) with S3 = 0 and S0 = I. The vanishing of S3 is the signature of linear polarization in the Stokes formalism.

Practical Understanding

Most terrestrial broadcast antennas transmit either horizontally or vertically polarized waves. FM radio and television broadcasts in India predominantly use horizontal polarization, whereas mobile phone base station antennas often use dual-slant polarization (±45 degrees to vertical) to serve users with unpredictably oriented handsets while maintaining orthogonality between the two polarization paths.

A receiving antenna must be aligned with the polarization of the incoming wave to capture maximum power. Misalignment by an angle θ results in a received power proportional to cos²θ, a relationship known as Malus's law applied to antennas. A 90-degree misalignment results in zero received power, which is exploited in polarization diversity schemes and in measuring antenna isolation.

In radar systems, horizontal polarization reflects differently from rain droplets compared to vertical polarization. Dual-polarization radar transmits and receives both H and V polarizations simultaneously, which enables estimation of raindrop shape, size, and orientation. This information dramatically improves precipitation estimation accuracy.

Given:
Ex = 3 V/m, Ey = 4 V/m, both in phase (δ = 0), f = 2.4 GHz, free space

Why this formula applies:
Phase difference between components is zero, so wave is linearly polarized.

Formula:
|E| = √(Ex² + Ey²)
τ = arctan(Ey/Ex)
β = ω√(μ₀ε₀) = ω/c

Substitution:
|E| = √(3² + 4²) = √(9 + 16) = √25 = 5 V/m
τ = arctan(4/3) = arctan(1.333) ≈ 53.1°
ω = 2π × 2.4 × 10⁹ rad/s
β = 2π × 2.4 × 10⁹ / (3 × 10⁸) = 50.3 rad/m

Calculation:
E(z,t) = (3x̂ + 4ŷ) cos(ωt − 50.3z)  V/m

Final Answer with units:
Amplitude = 5 V/m
Polarization angle τ = 53.1° with respect to x-axis
Phase constant β = 50.3 rad/m in free space at 2.4 GHz

Exam Tip: Linear polarization requires zero (or π) phase difference between Ex and Ey. If the phase difference is π/2 and |Ex| = |Ey|, the wave is circularly polarized. The polarization angle τ = (1/2) arctan(S2/S1) appears in Stokes parameter problems.

Mechanism Summary

• Linear polarization exists when the phase difference between the two orthogonal E-field components is 0 or π (180 degrees).
• The tip of the E vector traces a straight line in the transverse plane as time evolves at any fixed point along z.
• Any linear polarization at angle τ can be decomposed into horizontal (x̂) and vertical (ŷ) components with a zero phase difference.
• Received power varies as cos²θ with the angular misalignment θ between transmit and receive polarization planes.
• In the Stokes representation, S3 = 0 is the necessary and sufficient condition for linear polarization.

Quick Revision

• Linear polarization: δ = 0 or π between Ex and Ey. E-field oscillates in one fixed plane.
• Polarization tilt angle τ = arctan(Ey/Ex). Total amplitude = √(Ex² + Ey²).
• Power received vs misalignment angle θ: P = P₀ cos²θ (Malus's law for antennas).
• Stokes parameter S3 = 0 for linear polarization. S3 = ±I for circular polarization.
• FM and TV broadcasts typically use horizontal linear polarization.
• Exam trap: If Ex and Ey are present but in phase, the wave is still linearly polarized at an angle, not elliptically polarized.
• Exam trap: A wave with Ex = 0, Ey ≠ 0 is vertically polarized. A wave with Ey = 0, Ex ≠ 0 is horizontally polarized.