Plane Waves & Polarization

Once a changing E-field and H-field detach from an antenna, they form a self-sustaining Electromagnetic Wave. Understanding the geometry and decay of this wave is the basis for all link budgets and coverage planning.

1. The Uniform Plane Wave (TEM)

Far away from the source (Far Field), the spherical wavefront looks essentially flat—like a wall of water moving toward you. We call this a Uniform Plane Wave.

Key Characteristics

1. Transverse Electromagnetic (TEM): $\vec{E}$ and $\vec{H}$ are perpendicular to each other AND perpendicular to the direction of travel ($\vec{k}$).
2. Orthogonal: If the wave moves in Z, E might be in X and H will be in Y.
3. Right-Hand Rule: Point fingers in $\vec{E}$, curl toward $\vec{H}$, thumb points to Propagation Direction ($\vec{k}$).

Intrinsic Impedance ($\eta$)

Just as voltage and current are related by resistance ($R=V/I$), E-fields and H-fields are related by the Intrinsic Impedance of the medium.

$\eta = \frac{E}{H} = \sqrt{\frac{\mu}{\epsilon}}$

• In Free Space ($\eta_0$): $\sqrt{\frac{\mu_0}{\epsilon_0}} \approx 120\pi \approx 377 \Omega$.
• Meaning: For every 1 V/m of Electric field in space, there is necessarily 2.65 mA/m of Magnetic field accompanying it.
• Mismatch: When a wave hits a material with a different $\eta$ (like water, $\eta \approx 40\Omega$), reflections occur—just like an impedance mismatch in a cable.

2. Wave Parameters

• Wavelength ($\lambda$): The physical distance between two wave peaks.
  • $\lambda = \frac{v}{f}$
• Wavenumber ($k$ or $\beta$): Spatial frequency (radians per meter). How much phase shifts per meter traveled.
  • $\beta = \frac{2\pi}{\lambda} = \omega\sqrt{\mu\epsilon}$
• Phase Velocity ($v_p$): How fast a single peak moves.
  • $v_p = \frac{1}{\sqrt{\mu\epsilon}} = \frac{c}{\sqrt{\epsilon_r}}$

3. Propagation in Real (Lossy) Media

Real materials (Air, Water, Soil, PCB FR4) are not perfect vacuum. They have losses.

Loss Tangent ($\tan \delta$)

A measure of how "lossy" a dielectric is. It compares the conduction current (heat) to the displacement current (signal).

$\tan \delta = \frac{\sigma}{\omega\epsilon}$

• Low Loss: Teflon, Air.
• High Loss: Salt water, Carbon.

Attenuation Constant ($\alpha$)

The wave amplitude decays exponentially as it travels: $E(z) = E_0 e^{-\alpha z} e^{-j\beta z}$.

• $\alpha$: Attenuation in Nepers/meter (can convert to dB/m).

Skin Depth ($\delta_s$)

How deep does a wave penetrate into a conductor? This is critical for Shielding and Conductor Loss. Defined as the depth where the field drops to $1/e$ (about 37%) of its surface value.

$\delta_s = \sqrt{\frac{2}{\omega\mu\sigma}}$

• RF Engineering Insight:
  • At 60 Hz, $\delta_s$ in copper is ~8.5 mm.
  • At 1 GHz, $\delta_s$ in copper is ~2.1 $\mu m$.
  • Result: At microwave frequencies, all your current flows in a microscopic skin on the outside of the wire. Surface roughness matters. Gold plating must be thicker than skin depth to be useful.

4. Polarization

Polarization describes the trace of the $\vec{E}$-field vector tip as time passes at a fixed point.

Linear Polarization

The E-field oscillates back and forth on a single line (Vertical or Horizontal).

• Requirement: Both antennas must match (Vertical to Vertical).
• Cross-Pol Isolation: A Horizontal antenna theoretically receives 0 power from a Vertical wave ($\infty$ loss). In practice, you get 20-30 dB isolation.

Circular Polarization (CP)

The E-field vector rotates in a circle while traveling.

• Right-Hand CP (RHCP) vs Left-Hand CP (LHCP).
• Generation: Two linear dipoles fed $90^\circ$ out of phase.
• Why use CP?
  1. Orientation Independence: No need to align polarization (great for drones/satellites).
  2. Rain/Fog Penetration: Better interacts with spherical droplets.
  3. reflection Rejection: When CP bounces off a surface, it flips (RHCP becomes LHCP). An RHCP receiver will reject the reflected LHCP multipath signal!

Axial Ratio (AR)

A measure of "how circular" the polarization is.

• AR = 1 (0 dB): Perfect Circle.
• AR = $\infty$: Pure Linear.
• Typical Spec: An "excellent" CP antenna has AR < 3 dB.

5. Power Density (Poynting Vector)

We don't usually measure E (V/m) or H (A/m) directly. We measure Power. The Poynting Vector ($\vec{S}$) represents directional power flux ($W/m^2$).

$\vec{S}_{avg} = \frac{1}{2} Re \{ \vec{E} \times \vec{H}^* \} = \frac{|E|^2}{2\eta}$

• Inverse Square Law: As the sphere of radiation expands, the surface area grows by $r^2$, so the power density drops by $1/r^2$.