Elliptical Polarization

Elliptical polarization is the most general form of polarization for a plane electromagnetic wave. Both linear polarization and circular polarization are special limiting cases of elliptical polarization. In practice, most antennas and propagation environments produce elliptically polarized waves, making this concept central to accurate polarimetric analysis in radar, remote sensing, and satellite link budgeting.

Core Concept Explanation

When a plane wave has two orthogonal electric field components Ex and Ey with arbitrary amplitudes and an arbitrary phase difference δ, the tip of the resultant E vector traces an ellipse in the transverse plane as time progresses at a fixed point in space. This is the general case of elliptical polarization.

The shape of the polarization ellipse is determined by three parameters: the amplitude ratio Ex/Ey, the phase difference δ, and the tilt angle τ of the major axis with respect to the reference x-axis. The semi-major axis OA and semi-minor axis OB of the ellipse depend on all three parameters. As δ approaches 0 or π at any amplitude ratio, the ellipse degenerates into a straight line, giving linear polarization. As Ex approaches Ey and δ approaches ±π/2, the ellipse becomes a circle.

The axial ratio AR is defined as AR = OA/OB, which is always greater than or equal to 1 for physical polarization states. AR = 1 corresponds to circular polarization and AR → ∞ corresponds to linear polarization. In antenna specifications, AR is often expressed in dB as 20 log₁₀(AR). The 3 dB axial ratio criterion (AR ≤ 3 dB, meaning the ellipse ratio ≤ 1.41:1) is the standard engineering acceptance limit for circular polarization performance.

The tilt angle τ of the major axis is the angle between the major axis of the polarization ellipse and the x-axis, defined in the range 0 ≤ τ < π. The tilt angle is related to the wave parameters through the formula tan(2τ) = 2ExEy cos(δ) / (Ex² − Ey²). This result shows that when δ = 90°, the tilt angle expression simplifies, and when Ex = Ey at δ = 90°, the formula becomes indeterminate — consistent with the circular polarization case where the major axis is undefined.

Mathematical Expression

The general elliptically polarized wave propagating in +z direction is written as:

E(z,t) = Ex cos(ωt − βz) x̂ + Ey cos(ωt − βz + δ) ŷ

Eliminating (ωt − βz) from both components gives the equation of the polarization ellipse:

(Ex_inst/Ex)² + (Ey_inst/Ey)² − 2(Ex_inst/Ex)(Ey_inst/Ey)cos(δ) = sin²(δ)

This is the equation of an ellipse in the instantaneous Ex_inst-Ey_inst plane. The semi-major axis OA and semi-minor axis OB are found from the eigenvalues of the associated matrix. The key parameters are:

OA² + OB² = Ex² + Ey² (constant, equals total power)

2OA·OB = 2ExEy |sin(δ)|

AR = OA/OB. The ellipticity angle χ is defined as χ = arctan(1/AR) or equivalently tan(2χ) = 2ExEy sin(δ)/(Ex² + Ey²). This angle χ appears directly in the Poincare sphere representation, which maps all polarization states onto a sphere of radius I (total intensity).

Practical Understanding

Real-world antennas rarely produce perfectly linear or perfectly circular polarization. A dipole antenna illuminated at an oblique angle, a misaligned feed in a reflector antenna, or a circularly polarized antenna operating off-axis will all produce elliptically polarized radiation. Characterizing the axial ratio versus angle is therefore a standard part of antenna measurement procedures.

In polarimetric SAR (Synthetic Aperture Radar), the scattering matrix of a target is a 2×2 complex matrix that maps the incident polarization state to the scattered polarization state. Analysis of this matrix reveals target orientation, shape symmetry, and material properties. Elliptical polarization states produced by scattering from asymmetric targets carry this rich information.

In optical fiber communications, polarization mode dispersion (PMD) arises because real fibers have slight birefringence, causing the two orthogonal linear polarization modes to travel at slightly different phase velocities. An initially linearly polarized pulse becomes elliptically polarized after traveling through such a fiber, broadening the pulse and limiting the data rate.

Given:
Ex = 3 V/m, Ey = 4 V/m, phase difference δ = 60°, propagation in free space

Why this formula applies:
General elliptical polarization formulas apply since δ ≠ 0, π and amplitudes are unequal.

Formula:
OA² + OB² = Ex² + Ey²
2·OA·OB = 2·Ex·Ey·|sin(δ)|
tan(2τ) = 2·Ex·Ey·cos(δ) / (Ex² − Ey²)
AR = OA / OB

Substitution:
Ex² + Ey² = 9 + 16 = 25 V²/m²
Ex·Ey = 3 × 4 = 12 V²/m²
2·OA·OB = 2 × 12 × sin(60°) = 24 × 0.866 = 20.78
OA·OB = 10.39
(OA−OB)² = (OA+OB)² − 4·OA·OB
(OA+OB)² = 25 + 2×10.39 = 45.78,  OA+OB = 6.766
(OA−OB)² = 45.78 − 4×10.39 = 45.78 − 41.56 = 4.22
OA−OB = 2.054

Calculation:
OA = (6.766 + 2.054)/2 = 4.41 V/m
OB = (6.766 − 2.054)/2 = 2.356 V/m
AR = 4.41 / 2.356 = 1.87 (5.43 dB)
tan(2τ) = 2×12×cos60° / (9−16) = 12/(−7) = −1.714
2τ = −59.7° → τ = −29.85° (or 150.15° in 0–180° range)

Final Answer with units:
Semi-major axis OA = 4.41 V/m
Semi-minor axis OB = 2.356 V/m
Axial Ratio AR = 1.87 (5.43 dB) — elliptically polarized
Tilt angle τ ≈ 150.2° from x-axis

Exam Tip: In GATE, AR = 1 (0 dB) means circular polarization. AR → ∞ means linear polarization. The ellipticity angle χ = arctan(OB/OA) = arctan(1/AR). For a left-hand elliptically polarized wave, the sense of rotation is the same as LHCP. Always state both AR and τ to fully specify an elliptical polarization state.

Mechanism Summary

• Elliptical polarization is the most general case. Linear and circular polarization are special cases with AR → ∞ and AR = 1 respectively.
• The polarization ellipse is fully described by three parameters: amplitudes Ex and Ey, phase difference δ, and tilt angle τ.
• Axial ratio AR = OA/OB ≥ 1. In dB, AR = 20 log₁₀(OA/OB). Engineering limit for circular polarization: AR < 3 dB.
• Tilt angle: tan(2τ) = 2ExEy cos(δ)/(Ex² − Ey²). Defined in 0° to 180° range.
• The Poincare sphere provides a geometric map of all polarization states. Ellipticity angle 2χ gives the latitude and tilt angle 2τ gives the longitude on the sphere.

Quick Revision

• Elliptical polarization: general case with arbitrary δ and amplitude ratio. AR > 1.
• Linear polarization: AR = ∞, δ = 0 or π. Circular polarization: AR = 1, δ = ±90°, |Ex|=|Ey|.
• AR = OA/OB. AR in dB = 20 log₁₀(AR). Antenna spec: AR < 3 dB for circular.
• tan(2τ) = 2ExEy cos(δ)/(Ex² − Ey²). tan(2χ) = 2ExEy sin(δ)/(Ex² + Ey²).
• OA² + OB² = Ex² + Ey² (conserved quantity equal to total field power).
• Exam trap: AR is always ≥ 1. Never report AR < 1. If computation gives OB > OA, swap the axes.
• Exam trap: The tilt angle τ is the angle of the major axis, not the instantaneous E vector direction.