Reflection, Diffraction, and Scattering

When a plane wave hits a smooth planar boundary between two media (permittivities $\varepsilon_1$, $\varepsilon_2$) at incidence angle $\theta_i$, the reflection coefficient depends on polarisation:

Perpendicular (TE) polarisation:

$$\Gamma_\perp = \frac{\sin\theta_i - \sqrt{(\varepsilon_2/\varepsilon_1) - \cos^2\theta_i}} {\sin\theta_i + \sqrt{(\varepsilon_2/\varepsilon_1) - \cos^2\theta_i}}$$

Parallel (TM) polarisation:

$$\Gamma_\parallel = \frac{(\varepsilon_2/\varepsilon_1)\sin\theta_i - \sqrt{(\varepsilon_2/\varepsilon_1) - \cos^2\theta_i}} {(\varepsilon_2/\varepsilon_1)\sin\theta_i + \sqrt{(\varepsilon_2/\varepsilon_1) - \cos^2\theta_i}}$$

At grazing incidence ($\theta_i \to 90°$): $\Gamma \to -1$ for both polarisations.

For a perfect conductor: $\Gamma = -1$ at all angles.

The Brewster angle $\theta_B$ is the incidence angle at which the parallel-polarisation reflection coefficient vanishes ($\Gamma_\parallel = 0$):

$$\theta_B = \arctan\!\left(\sqrt{\varepsilon_2/\varepsilon_1}\right).$$

At Brewster's angle, only perpendicular-polarised waves are reflected. For typical building materials ($\varepsilon_r \approx 5$), $\theta_B \approx 66°$.

The $n$-th Fresnel zone is the locus of points $P$ such that the excess path length via $P$ (compared to the direct path) is $n\lambda/2$:

$$d_1 + d_2 - d = \frac{n\lambda}{2}$$

where $d_1$, $d_2$ are the distances from $P$ to the transmitter and receiver, and $d$ is the direct distance.

The radius of the $n$-th Fresnel zone at a point fraction $d_1/d$ along the path is

$$r_n = \sqrt{\frac{n\lambda d_1 d_2}{d}}.$$

If the first Fresnel zone is at least 60% clear of obstructions, the path loss is approximately equal to free-space.

When a single sharp obstacle of height $h$ above the line of sight blocks the direct path, the Fresnel–Kirchhoff diffraction parameter is

$$\nu = h\,\sqrt{\frac{2(d_1 + d_2)}{\lambda\,d_1\,d_2}}$$

where $d_1$ and $d_2$ are the distances from the obstacle to the transmitter and receiver.

The diffraction loss (dB) relative to free space is approximated by:

$$L_{\text{diff}}(\nu) \approx \begin{cases} 0 & \nu \le -1 \\ 20\log_{10}(0.5 - 0.62\nu) & -1 < \nu \le 0 \\ 20\log_{10}(0.5\,e^{-0.95\nu}) & 0 < \nu \le 1 \\ 20\log_{10}\!\left(\frac{0.4}{\sqrt{0.1216 + (\nu - 0.1)^2}}\right) & 1 < \nu \le 2.4 \\ 20\log_{10}(0.225/\nu) & \nu > 2.4 \end{cases}$$

For $\nu > 1$ (deep shadow): $L_{\text{diff}} \approx 13 + 20\log_{10}(\nu)$ dB.

When the surface roughness is comparable to or larger than the wavelength, the reflected energy is scattered in many directions rather than concentrated in the specular direction.

The Rayleigh roughness criterion determines whether a surface appears smooth or rough:

$$h_{\text{rms}} < \frac{\lambda}{8\sin\theta_i}$$

means the surface is electromagnetically smooth; otherwise it is rough. Scattering from rough surfaces (foliage, building facades) redistributes energy and contributes to both path loss and multipath.