M0. EM Wave Foundations

1. Maxwell Equations and the Wave Equation

In vacuum, the four Maxwell equations in SI units are:

In source-free regions ($\rho=0,\;\mathbf{J}=0$) we take the curl of Faraday's law:

Using the identity $\nabla\times(\nabla\times\mathbf{E})=\nabla(\nabla\cdot\mathbf{E})-\nabla^2\mathbf{E}$and $\nabla\cdot\mathbf{E}=0$ (vacuum) gives the vector wave equation:

An identical equation holds for $\mathbf{B}$. The finite propagation speed $c$is the origin of radar ranging: the two-way travel time $t = 2R/c$ directly encodes range.

2. Plane-Wave Solutions

Seeking solutions of the form $\mathbf{E}(\mathbf{r},t) = \mathbf{E}_0\,e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$substitutes into the wave equation and yields the dispersion relation:

Gauss's law $\nabla\cdot\mathbf{E}=0$ constrains $\mathbf{k}\cdot\mathbf{E}_0=0$: electromagnetic waves are transverse. From Faraday's law we find the magnetic field:

The time-averaged Poynting vector is the power flux in W/m\u00B2:

The impedance of free space $\eta_0$ appears throughout antenna theory: aperture efficiency, mismatch, and gain all trace back to this quantity.

3. Polarization: Linear, Circular, Elliptical

The most general transverse field can be written (for propagation along $\hat{\mathbf{x}}$):

• Linear: $\delta = 0$ or $\pi$. The tip of $\mathbf{E}$ traces a line.
• Right circular (RCP): $E_y=E_z,\;\delta=+\pi/2$. The tip rotates clockwise looking along $\hat{\mathbf{k}}$.
• Left circular (LCP): $\delta=-\pi/2$. Tip rotates counterclockwise.
• Elliptical: general case. Described by Stokes parameters $(I,Q,U,V)$.

The Stokes parameters for a fully-polarized wave are:

Radar systems distinguish HH, VV, HV, VH channels; polarimetric SAR (Module 6) uses the full scattering matrix. Circular polarization is preferred for weather radar because it is insensitive to target orientation.

4. Radiation from Accelerated Charges (Larmor)

A non-relativistic accelerated charge radiates power according to the Larmor formula:

Derivation sketch: start from the Liénard-Wiechert potentials, evaluate the Poynting vector in the far field ($1/r$ terms), and integrate over the sphere. The angular distribution is the familiar $\sin^2\theta$ pattern (dipole):

For a Hertzian dipole of current $I_0\cos\omega t$ and length $\ell\ll\lambda$, the radiated power is $P = (I_0\ell\omega)^2 /(12\pi\varepsilon_0 c^3)$, giving a radiation resistance of $R_{\text{rad}} = 80\pi^2(\ell/\lambda)^2\,\Omega$. This is the starting point for every transmitting antenna.

5. IEEE Radar Frequency Bands

The IEEE 521-2019 standard divides radar spectrum into lettered bands. Each band has a sweet spot balancing range, resolution, atmospheric transmission, and hardware cost.

Why so many bands? Resolution improves with shorter wavelength but absorption and cost rise. $R^4$ losses dominate at low frequencies, hydrometeor clutter at high. Picking a band is already a system-engineering decision.

6. Atmospheric Attenuation

Microwave absorption is dominated by two molecular resonances:

• Water vapor (H\u2082O): rotational lines at 22.235, 183.31, 325.15 GHz. The 22 GHz line limits K-band.
• Molecular oxygen (O\u2082): a complex of ~44 magnetic-dipole lines near 60 GHz plus 118.75 GHz. Gives >10 dB/km at 60 GHz.

The ITU-R Recommendation P.676 models specific attenuation $\gamma$ [dB/km]. For a two-way path of length $R$, total path loss is $L_{atm} = 2\gamma R$. Typical values at sea level, 20 °C, 7.5 g/m\u00B3 humidity:

Rain attenuation adds a term $\gamma_R = a R^b$ (dB/km) where $R$ is the rain rate in mm/hr and the coefficients $(a,b)$ depend on frequency and polarization (ITU-R P.838). Rain absorbs more at higher frequencies — another reason to avoid Ka for long-range weather surveillance.

Simulation: Plane Waves, Polarization & Atmospheric Absorption

The Python code below (i) animates an X-band plane wave, (ii) plots polarization ellipses for linear, circular, and elliptical states, (iii) computes specific atmospheric attenuation via the ITU-R P.676 approximation, and (iv) lays out the IEEE radar band allocation chart.

7. Energy Flow and the Poynting Theorem

The electromagnetic energy density and power flow are defined by:

From Maxwell's equations one derives Poynting's theorem as a local conservation law:

For a transmitted radar pulse, the total energy is $E_t = \int P_t(t)\,dt \approx P_{\text{avg}}\cdot\text{PRI}$; it is this energy (and not peak power) that determines SNR through the matched filter.

8. The Helmholtz Equation and Green's Functions

Taking the Fourier transform of the wave equation in time, we get the Helmholtz equation for each frequency component:

The free-space scalar Green's function satisfying $(\nabla^2+k^2)G = -\delta^3(\mathbf{r}-\mathbf{r}')$ is

which is the key ingredient for every scattering calculation. Integrating over source currents gives the radiated field used in deriving the radar cross section (Module 1) and in physical optics approximations of complex targets.

9. Reflection at Material Boundaries (Fresnel)

Radar returns arise largely from discontinuities in the complex permittivity. For a plane wave incident at angle $\theta_i$ on a plane interface between vacuum and a dielectric of index $n = \sqrt{\varepsilon_r}$, the Fresnel reflection coefficients are

A conductor has $|r|\to 1$ (specular reflection) — this is why metal is the ideal radar target. Sea water has $\varepsilon_r \approx 70 + 40i$ at S-band and reflects strongly; dry soil has $\varepsilon_r \approx 3-5$ and reflects weakly. The Brewster angle $\theta_B = \arctan n$ is where $r_\parallel = 0$; this matters for dual-polarization weather radar.

10. Tropospheric Refraction and the 4/3-Earth Approximation

The index of refraction of air differs from unity by a few parts in 10\u2074. Radio engineers express this as refractivity $N = (n-1)\times 10^6$. A useful approximation at microwave is

where $T$ is temperature (K), $P$ total pressure (hPa), and $e_w$water-vapor partial pressure (hPa). Typical surface refractivity is $N \approx 315$, decreasing exponentially with height with scale $\sim 7\,\text{km}$. The ray curvature is $\kappa = -dn/dh \approx 4\times 10^{-8}\,\text{m}^{-1}$; an effective Earth radius$a_e = 4a/3$ accounts for this so that rays travel in straight lines in the modified geometry. This is the 4/3 Earth radius approximation, used for low-elevation radar range calculations.

Anomalous propagation (ducting) occurs when $dN/dh < -157$ N-units/km over an elevated layer, trapping rays and creating false long-range returns.

11. Hydrometeor and Rain Attenuation

In addition to gas absorption, liquid water in clouds and rain attenuates radar signals. The Mie-theory specific attenuation in rain is well approximated by the ITU-R P.838 power-law:

where $R$ is rainfall rate in mm/hr. Typical coefficients (horizontal pol): at 10 GHz, $k=0.0101,\alpha=1.276$; at 35 GHz, $k=0.242,\alpha=0.997$; at 94 GHz, $k=1.12,\alpha=0.730$. A 25 mm/hr downpour produces ~3 dB/km at X band and ~15 dB/km at Ka band — catastrophic for long paths.

12. Skin Depth in Conductors

In a good conductor of conductivity $\sigma$, the wave decays exponentially over the skin depth

For copper ($\sigma = 5.8\times 10^7\,\text{S/m}$) at 10 GHz, $\delta\approx 0.66\,\mu\text{m}$; this is why the interior of a waveguide must be electroplated smooth. It also explains why thin metallic paint can be used as RAM (radar-absorbent material) only above its cutoff — covered in Module 8 on stealth.