We consider a global coordinate system (GCS) with Cartesian canonical basis $\widehat{𝐱}$, $\widehat{𝐲}$, $\widehat{𝐳}$. The spherical unit vectors are defined as

	$\widehat{𝐫}(\theta ,\phi )$	$=\sin (\theta )\cos (\phi )\widehat{𝐱}+\sin (\theta )\sin (\phi )\widehat{𝐲}+\cos (\theta )\widehat{𝐳}$		(49)
	$\widehat{𝜽}(\theta ,\phi )$	$=\cos (\theta )\cos (\phi )\widehat{𝐱}+\cos (\theta )\sin (\phi )\widehat{𝐲}-\sin (\theta )\widehat{𝐳}$
	$\widehat{𝝋}(\theta ,\phi )$	$=-\sin (\phi )\widehat{𝐱}+\cos (\phi )\widehat{𝐲}.$

For an arbitrary unit norm vector $\widehat{𝐯}=(x,y,z)$, the zenith and azimuth angles $\theta$ and $\phi$ can be computed as

	$\theta$	$={\cos }^{-1}(z)$		(50)
	$\phi$	$=\text{atan2}(y,x)$

where $\text{atan2}(y,x)$ is the two-argument inverse tangent function [20]. As any vector uniquely determines $\theta$ and $\phi$, we sometimes also write $\widehat{𝜽}(\widehat{𝐯})$ and $\widehat{𝝋}(\widehat{𝐯})$ instead of $\widehat{𝜽}(\theta ,\phi )$ and $\widehat{𝝋}(\theta ,\phi )$.

A 3D rotation with yaw, pitch, and roll angles $\alpha$, $\beta$, and $\gamma$, respectively, is expressed by the matrix

$$𝐑(\alpha ,\beta ,\gamma )={𝐑}_z(\alpha ){𝐑}_y(\beta ){𝐑}_x(\gamma )$$

(51)

where ${𝐑}_z(\alpha )$, ${𝐑}_y(\beta )$, and ${𝐑}_x(\gamma )$ are rotation matrices around the $z$, $y$, and $x$ axes, respectively, which are defined as

	${𝐑}_z(\alpha )$			(52)
	${𝐑}_y(\beta )$
	${𝐑}_x(\gamma )$

A closed-form expression for $𝐑(\alpha ,\beta ,\gamma )$ can be found in [17, Sec. 7.1-4]. The inverse rotation is simply defined by ${𝐑}^{-1}(\alpha ,\beta ,\gamma )={𝐑}^{𝖳}(\alpha ,\beta ,\gamma )$. A vector $𝐱$ defined in a first coordinate system is represented in a second coordinate system rotated by $𝐑(\alpha ,\beta ,\gamma )$ with respect to the first one as ${𝐱}^{\prime }={𝐑}^{𝖳}(\alpha ,\beta ,\gamma )𝐱$. If a point in the first coordinate system has spherical angles $(\theta ,\phi )$, the corresponding angles $({\theta }^{\prime },{\phi }^{\prime })$ in the second coordinate system can be found to be

	${\theta }^{\prime }$	$={\cos }^{-1}\left({\widehat{𝐳}}^{𝖳}{𝐑}^{𝖳}(\alpha ,\beta ,\gamma )\widehat{𝐫}(\theta ,\phi )\right)$		(53)
	${\phi }^{\prime }$	$=\arg \left({\left(\widehat{𝐱}+j\widehat{𝐲}\right)}^{𝖳}{𝐑}^{𝖳}(\alpha ,\beta ,\gamma )\widehat{𝐫}(\theta ,\phi )\right).$

For a vector field ${𝐅}^{\prime }({\theta }^{\prime },{\phi }^{\prime })$ expressed in local spherical coordinates

$${𝐅}^{\prime }({\theta }^{\prime },{\phi }^{\prime })=F_{{\theta }^{\prime }}({\theta }^{\prime },{\phi }^{\prime }){\widehat{𝜽}}^{\prime }({\theta }^{\prime },{\phi }^{\prime })+F_{{\phi }^{\prime }}({\theta }^{\prime },{\phi }^{\prime }){\widehat{𝝋}}^{\prime }({\theta }^{\prime },{\phi }^{\prime })$$

(54)

that are rotated by $𝐑=𝐑(\alpha ,\beta ,\gamma )$ with respect to the GCS, the spherical field components in the GCS can be expressed as

(55)

so that

$$𝐅(\theta ,\phi )=F_{\theta }(\theta ,\phi )\widehat{𝜽}(\theta ,\phi )+F_{\phi }(\theta ,\phi )\widehat{𝝋}(\theta ,\phi ).$$

(56)

Sometimes, it is also useful to find the rotation matrix that maps a unit vector $\widehat{𝐚}$ to $\widehat{𝐛}$. This can be achieved with the help of Rodrigues’ rotation formula [21] which defines the matrix

$$𝐑(\widehat{𝐚},\widehat{𝐛})=𝐈+\sin (\theta )𝐊+(1-\cos (\theta )){𝐊}^2$$

(57)

where

	$𝐊$			(58)
	$\widehat{𝐤}$	$={\displaystyle \frac{\widehat{𝐚}\times \widehat{𝐛}}{{\parallel \widehat{𝐚}\times \widehat{𝐛}\parallel }_2}}$		(59)
	$\theta$	$={\widehat{𝐚}}^{𝖳}\widehat{𝐛}$		(60)

such that $𝐑(\widehat{𝐚}$, $\widehat{𝐛})\widehat{𝐚}=\widehat{𝐛}$.

A time-harmonic planar electric wave $𝐄(𝐱,t)\in {ℂ}^3$ traveling in a homogeneous medium with wave vector $𝐤\in {ℂ}^3$ can be described at position $𝐱\in {ℝ}^3$ and time $t$ as

	$𝐄(𝐱,t)$	$={𝐄}_0{e}^{j(\omega t-{𝐤}^{𝖧}𝐱)}$		(61)
		$=𝐄(𝐱)e^{j\omega t}$		(62)

where ${𝐄}_0\in {ℂ}^3$ is the field phasor. The wave vector can be decomposed as $𝐤=k\widehat{𝐤}$, where $\widehat{𝐤}$ is a unit norm vector, $k=\omega \sqrt{\epsilon \mu }$ is the wave number, and $\omega =2\pi f$ is the angular frequency. The permittivity $\epsilon$ and permeability $\mu$ are defined as

	$\epsilon$	$=\eta {\epsilon }_0$		(63)
	$\mu$	$={\mu }_r{\mu }_0$		(64)

where $\eta$ and ${\epsilon }_0$ are the complex relative and vacuum permittivities, ${\mu }_r$ and ${\mu }_0$ are the relative and vacuum permeabilities, and $\sigma$ is the conductivity. The complex relative permittivity $\eta$ is given as

$$\eta ={\epsilon }_r-j\frac{\sigma }{{\epsilon }_0\omega }$$

(65)

where ${\epsilon }_r$ is the real relative permittivity of a non-conducting dielectric.

With these definitions, the speed of light is given as [22, Eq. 4-28d]

$$c=\frac{1}{\sqrt{{\epsilon }_0{\epsilon }_r\mu }}{\left\{\frac{1}{2}\left(\sqrt{1+{\left(\frac{\sigma }{\omega {\epsilon }_0{\epsilon }_r}\right)}^2}+1\right)\right\}}^{-\frac{1}{2}}$$

(66)

where the factor in curly brackets vanishes for non-conducting materials. The speed of light in vacuum is denoted $c_0=\frac{1}{\sqrt{{\epsilon }_0{\mu }_0}}$ and the vacuum wave number $k_0=\frac{\omega }{c_0}$. In conducting materials, the wave number is complex which translates to propagation losses.

The associated magnetic field $𝐇(𝐱,t)\in {ℂ}^3$ is

$$𝐇(𝐱,t)=\frac{\widehat{𝐤}\times 𝐄(𝐱,t)}{Z}=𝐇(𝐱)e^{j\omega t}$$

(67)

where $Z=\sqrt{\mu /\epsilon }$ is the wave impedance. The vacuum impedance is denoted by $Z_0=\sqrt{{\mu }_0/{\epsilon }_0}\approx 376.73\mathrm{\Omega }$.

The time-averaged Poynting vector is defined as

$$𝐒(𝐱)=\frac{1}{2}\mathrm{\Re }\left\{𝐄(𝐱)\times 𝐇(𝐱)\right\}=\frac{1}{2}\mathrm{\Re }\left\{\frac{1}{Z}\right\}{\parallel 𝐄(𝐱)\parallel }_2^2\widehat{𝐤}$$

(68)

which describes the directional energy flux [$\mathrm{W}{\mathrm{m}}^{-2}$], i.e. energy transfer per unit area per unit time. Note that the actual electromagnetic waves are the real parts of $𝐄(𝐱,t)$ and $𝐇(𝐱,t)$.

We assume that the electric far field of an antenna in free space can be described by a spherical wave originating from the center of the antenna:

$$𝐄(r,\theta ,\phi ,t)=𝐄(r,\theta ,\phi )e^{j\omega t}={𝐄}_0(\theta ,\phi )\frac{e^{-j{k}_{0}r}}{r}{e}^{j\omega t}$$

(69)

where ${𝐄}_0(\theta ,\phi )$ is the electric field phasor, $r$ is the distance (or radius), $\theta$ the zenith angle, and $\phi$ the azimuth angle. In contrast to a planar wave, the field strength decays as $1/r$.

The complex antenna field pattern $𝐅(\theta ,\phi )$ is defined as

$𝐅(\theta ,\phi )={\displaystyle \frac{{𝐄}_0(\theta ,\phi )}{{\max }_{\theta ,\phi }{\parallel {𝐄}_0(\theta ,\phi )\parallel }_2}}.$

(70)

The time-averaged Poynting vector for such a spherical wave is

$$𝐒(r,\theta ,\phi )=\frac{1}{2Z_0}{\parallel 𝐄(r,\theta ,\phi )\parallel }_2^2\widehat{𝐫}$$

(71)

where $\widehat{𝐫}$ is the radial unit vector. It simplifies for an ideal isotropic antenna with input power $P_{\text{T}}$ to

$${𝐒}_{\text{iso}}(r,\theta ,\phi )=\frac{P_{\text{T}}}{4\pi r^2}\widehat{𝐫}.$$

(72)

The antenna gain $G$ is the ratio of the maximum radiation power density of the antenna in radial direction and that of an ideal isotropic radiating antenna:

$$G=\frac{{\max }_{\theta ,\phi }{\parallel 𝐒(r,\theta ,\phi )\parallel }_2}{{\parallel {𝐒}_{\text{iso}}(r,\theta ,\phi )\parallel }_2}=\frac{2\pi }{Z_0{P}_{\text{T}}}\underset{\theta ,\phi }{\max }{\parallel {𝐄}_0(\theta ,\phi )\parallel }_2^2.$$

(73)

One can similarly define a gain with directional dependency by ignoring the computation of the maximum the last equation:

$$G(\theta ,\phi )=\frac{2\pi }{Z_0{P}_{\text{T}}}{\parallel {𝐄}_0(\theta ,\phi )\parallel }_2^2=G{\parallel 𝐅(\theta ,\phi )\parallel }_2^2.$$

(74)

If one uses in the last equation the radiated power $P={\eta }_{\text{rad}}{P}_{\text{T}}$, where ${\eta }_{\text{rad}}$ is the radiation efficiency, instead of the input power $P_{\text{T}}$, one obtains the directivity $D(\theta ,\phi )$. Both are related through $G(\theta ,\phi )={\eta }_{\text{rad}}D(\theta ,\phi )$.

Combining $F$ and $G$, we can obtain the following expression of the electric far field

$${𝐄}_{\text{T}}(r,{\theta }_{\text{T}},{\phi }_{\text{T}})=\sqrt{\frac{P_{\text{T}}{G}_{\text{T}}{Z}_0}{2\pi }}\frac{e^{-j{k}_{0}r}}{r}{𝐅}_{\text{T}}({\theta }_{\text{T}},{\phi }_{\text{T}})$$

(77)

where we have added the subscript T to all quantities that are specific to the transmitting antenna.

The input power $P_{\text{T}}$ of an antenna with (conjugate matched) impedance $Z_{\text{T}}$, fed by a voltage source with complex amplitude $V_{\text{T}}$, is given by (see e.g., [23]).

$$P_{\text{T}}=\frac{{|V_{\text{T}}|}^2}{8\mathrm{\Re }\{Z_{\text{T}}\}}.$$

(78)

Although the transmitting antenna radiates a spherical wave ${𝐄}_{\text{T}}(r,{\theta }_{\text{T}},{\phi }_{\text{T}})$, we assume that the receiving antenna observes a planar incoming wave ${𝐄}_{\text{R}}$ that arrives from the angles ${\theta }_{\text{R}}$ and ${\phi }_{\text{R}}$ which are defined in the local spherical coordinates of the receiving antenna. The Poynting vector of the incoming wave ${𝐒}_{\text{R}}$ is hence (71)

$${𝐒}_{\text{R}}=-\frac{1}{2Z_0}{\parallel {𝐄}_{\text{R}}\parallel }_2^2\widehat{𝐫}({\theta }_{\text{R}},{\phi }_{\text{R}})$$

(83)

where $\widehat{𝐫}({\theta }_{\text{R}},{\phi }_{\text{R}})$ is the radial unit vector in the spherical coordinate system of the receiver.

The aperture or effective area $A_{\text{R}}$ of an antenna with gain $G_{\text{R}}$ is defined as the ratio of the available received power $P_{\text{R}}$ at the output of the antenna and the absolute value of the Poynting vector, i.e. the power density:

$$A_{\text{R}}=\frac{P_{\text{R}}}{{\parallel {𝐒}_{\text{R}}\parallel }_2}=G_{\text{R}}\frac{{\lambda }^2}{4\pi }$$

(84)

where $\frac{{\lambda }^2}{4\pi }$ is the aperture of an isotropic antenna. In the definition above, it is assumed that the antenna is ideally directed towards and polarization matched to the incoming wave. For an arbitrary orientation of the antenna (but still assuming polarization matching), we can define a direction dependent effective area

$$A_{\text{R}}({\theta }_{\text{R}},{\phi }_{\text{R}})=G_{\text{R}}({\theta }_{\text{R}},{\phi }_{\text{R}})\frac{{\lambda }^2}{4\pi }.$$

(85)

The available received power at the output of the antenna can be expressed as

$$P_{\text{R}}=\frac{{|V_{\text{R}}|}^2}{8\mathrm{\Re }\{Z_{\text{R}}\}}$$

(86)

where $Z_{\text{R}}$ is the impedance of the receiving antenna and $V_{\text{R}}$ the open circuit voltage. We can now combine (86), (85), and (84) to obtain the following expression for the absolute value of the voltage $|V_{\text{R}}|$ assuming matched polarization:

	$|V_{\text{R}}|$	$=\sqrt{P_{\text{R}}8\mathrm{\Re }\{Z_{\text{R}}\}}$		(87)
		$=\sqrt{{\displaystyle \frac{{\lambda }^2}{4\pi }}{G}_{\text{R}}({\theta }_{\text{R}},{\phi }_{\text{R}}){\displaystyle \frac{8\mathrm{\Re }\{Z_{\text{R}}\}}{2Z_0}}{\parallel {𝐄}_{\text{R}}\parallel }_2^2}$		(88)
		$=\sqrt{{\displaystyle \frac{{\lambda }^2}{4\pi }}{G}_{\text{R}}{\displaystyle \frac{4\mathrm{\Re }\{Z_{\text{R}}\}}{Z_0}}}{\parallel {𝐅}_{\text{R}}({\theta }_{\text{R}},{\phi }_{\text{R}})\parallel }_2{\parallel {𝐄}_{\text{R}}\parallel }_2.$		(89)

By extension of the previous equation, we can obtain an expression for $V_{\text{R}}$ which is valid for arbitrary polarizations of the incoming wave and the receiving antenna:

$$V_{\text{R}}=\sqrt{\frac{{\lambda }^2}{4\pi }{G}_{\text{R}}\frac{4\mathrm{\Re }\{Z_{\text{R}}\}}{Z_0}}{𝐅}_{\text{R}}{({\theta }_{\text{R}},{\phi }_{\text{R}})}^{𝖧}{𝐄}_{\text{R}}.$$

(90)

A single propagation path consists of a sequence of scattering processes, where a scattering process can be anything that prevents the wave from propagating as in free space. This includes specular reflection, refraction, diffraction, and diffuse reflection. For each scattering process, one needs to compute a relationship between the incoming field at the scatter center and the created far field at the next scatter center or the receiving antenna. We can represent this cascade of scattering processes by a single matrix $\widetilde{𝐓}$ that describes the transformation that the radiated field ${𝐄}_{\text{T}}(r,{\theta }_{\text{T}},{\phi }_{\text{T}})$ undergoes until it reaches the receiving antenna:

$${𝐄}_{\text{R}}=\sqrt{\frac{P_{\text{T}}{G}_{\text{T}}{Z}_0}{2\pi }}\widetilde{𝐓}{𝐅}_{\text{T}}({\theta }_{\text{T}},{\phi }_{\text{T}}).$$

(93)

Note that we have obtained this expression by replacing the free space propagation term $\frac{e^{-j{k}_{0}r}}{r}$ in (77) by the matrix $\widetilde{𝐓}$. This requires that all quantities are expressed in the same coordinate system which is also assumed in the following expressions. Further, it is assumed that the matrix $\widetilde{𝐓}$ includes the necessary coordinate transformations. In some cases, e.g., for diffuse reflection (see (123) in Section A.8), the matrix $\widetilde{𝐓}$ depends on the incoming field and is not a linear transformation.

Plugging (93) into (90), we can obtain a general expression for the received voltage of a propagation path:

$$V_{\text{R}}=\sqrt{{\left(\frac{\lambda }{4\pi }\right)}^2{G}_{\text{R}}{G}_{\text{T}}{P}_{\text{T}}8\mathrm{\Re }\{Z_{\text{R}}\}}{𝐅}_{\text{R}}{({\theta }_{\text{R}},{\phi }_{\text{R}})}^{𝖧}\widetilde{𝐓}{𝐅}_{\text{T}}({\theta }_{\text{T}},{\phi }_{\text{T}}).$$

(94)

If the electromagnetic wave arrives at the receiving antenna over $N$ propagation paths, we can simply add the received voltages from all paths to obtain

	$V_{\text{R}}$	$=\sqrt{{\left({\displaystyle \frac{\lambda }{4\pi }}\right)}^2{G}_{\text{R}}{G}_{\text{T}}{P}_{\text{T}}8\mathrm{\Re }\{Z_{\text{R}}\}}{\displaystyle \sum _{n=1}^N}{𝐅}_{\text{R}}{({\theta }_{\text{R},i},{\phi }_{\text{R},i})}^{𝖧}{\widetilde{𝐓}}_i{𝐅}_{\text{T}}({\theta }_{\text{T},i},{\phi }_{\text{T},i})$		(95)
		$=\sqrt{{\left({\displaystyle \frac{\lambda }{4\pi }}\right)}^2{P}_{\text{T}}8\mathrm{\Re }\{Z_{\text{R}}\}}{\displaystyle \sum _{n=1}^N}{𝐂}_{\text{R}}{({\theta }_{\text{R},i},{\phi }_{\text{R},i})}^{𝖧}{\widetilde{𝐓}}_i{𝐂}_{\text{T}}({\theta }_{\text{T},i},{\phi }_{\text{T},i})$		(96)

where all path-dependent quantities carry the subscript $i$. Note that the matrices ${\widetilde{𝐓}}_i$ also ensure appropriate scaling so that the total received power can never be larger than the transmit power.

The channel frequency response $H(f)$ at frequency $f=\frac{c}{\lambda }$ is defined as the ratio between the received voltage and the voltage at the input to the transmitting antenna:

$$H(f)=\frac{V_{\text{R}}}{V_{\text{T}}}=\frac{V_{\text{R}}}{|V_{\text{T}}|}$$

(97)

where it is assumed that the input voltage has zero phase.

It is useful to separate phase shifts due to wave propagation from the transfer matrices ${\widetilde{𝐓}}_i$. If we denote by $r_i$ the total length of path $i$ with average propagation speed $c_i$, the path delay is ${\tau }_i=r_i/c_i$. We can now define the new transfer matrix

$${𝐓}_i={\widetilde{𝐓}}_i{e}^{j2\pi f{\tau }_i}.$$

(98)

Using (78) and (98) in (95) while assuming equal real parts of both antenna impedances, i.e. $\mathrm{\Re }\{Z_{\text{T}}\}=\mathrm{\Re }\{Z_{\text{R}}\}$ (which is typically the case), we obtain the final expression for the channel frequency response:

$$\boxed{H(f)=\sum _{i=1}^N\underset{\triangleq a_i}{\underbrace{\frac{\lambda }{4\pi }{𝐂}_{\text{R}}{({\theta }_{\text{R},i},{\phi }_{\text{R},i})}^{𝖧}{𝐓}_i{𝐂}_{\text{T}}({\theta }_{\text{T},i},{\phi }_{\text{T},i})}}{e}^{-j2\pi f{\tau }_i}}$$

(99)

Taking the inverse Fourier transform, we finally obtain the channel impulse response

$$\boxed{h(\tau )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}H(f)e^{j2\pi f\tau }𝑑f=\sum _{i=1}^N{a}_i\delta (\tau -{\tau }_i)}$$

(100)

The baseband equivalent channel impulse response is then defined as [24, Eq. 2.28]:

$$h_{\text{b}}(\tau )=\sum _{i=1}^N\underset{\triangleq a_i^{\text{b}}}{\underbrace{a_i{e}^{-j2\pi f{\tau }_i}}}\delta (\tau -{\tau }_i).$$

(101)

When a plane wave hits a plane interface which separates two materials, e.g., air and concrete, a part of the wave gets reflected and the other refracted, i.e. it propagates into the other material. We assume in the following description that both materials are uniform non-magnetic dielectrics, i.e. ${\mu }_r=1$, and follow the definitions as in [25]. The incoming wave phasor ${𝐄}_{\text{i}}$ is expressed by two arbitrary orthogonal polarization components, i.e.

$${𝐄}_{\text{i}}=E_{\text{i},s}{\widehat{𝐞}}_{\text{i},s}+E_{\text{i},p}{\widehat{𝐞}}_{\text{i},p}$$

(102)

which are both orthogonal to the incident wave vector, i.e. ${\widehat{𝐞}}_{\text{i},s}^{𝖳}{\widehat{𝐞}}_{\text{i},p}={\widehat{𝐞}}_{\text{i},s}^{𝖳}{\widehat{𝐤}}_{\text{i}}={\widehat{𝐞}}_{\text{i},p}^{𝖳}{\widehat{𝐤}}_{\text{i}}=0$.

Figure 26 shows reflection and refraction of the incoming wave at the plane interface between two materials with relative permittivities ${\eta }_1$ and ${\eta }_2$. The coordinate system is chosen such that the wave vectors of the incoming, reflected, and transmitted waves lie within the plane of incidence, which is chosen to be the x-z plane. The normal vector of the interface $\widehat{𝐧}$ is pointing toward the negative z axis. The incoming wave is must be represented in a different basis, i.e. in the form two different orthogonal polarization components $E_{\text{i},⟂}$ and $E_{\text{i},\parallel }$, i.e.

$${𝐄}_{\text{i}}=E_{\text{i},⟂}{\widehat{𝐞}}_{\text{i},⟂}+E_{\text{i},\parallel }{\widehat{𝐞}}_{\text{i},\parallel }$$

(103)

where the former is orthogonal to the plane of incidence and called transverse electric (TE) polarization (left), and the latter is parallel to the plane of incidence and called transverse magnetic (TM) polarization (right). In the following, we adopt the convention that all transverse components are coming out of the figure (indicated by the $\odot$ symbol). One can easily verify that the following relationships must hold:

	${\widehat{𝐞}}_{\text{i},⟂}$	$={\displaystyle \frac{{\widehat{𝐤}}_{\text{i}}\times \widehat{𝐧}}{{\parallel {\widehat{𝐤}}_{\text{i}}\times \widehat{𝐧}\parallel }_2}}$		(104)
	${\widehat{𝐞}}_{\text{i},\parallel }$	$={\widehat{𝐞}}_{\text{i},⟂}\times {\widehat{𝐤}}_{\text{i}}$
	$\left[\begin{array}{c}{E}_{\text{i},⟂}\\ E_{\text{i},\parallel }\end{array}\right]$

where we have defined the following matrix-valued function

(105)

While the angles of incidence and reflection are both equal to ${\theta }_1$, the angle of the refracted wave ${\theta }_2$ is given by Snell’s law

$$\sin ({\theta }_2)=\sqrt{\frac{{\eta }_1}{{\eta }_2}}\sin ({\theta }_1)$$

(106)

or, equivalently,

$$\cos ({\theta }_2)=\sqrt{1-\frac{{\eta }_1}{{\eta }_2}{\sin }^2({\theta }_1)}.$$

(107)

The reflected and transmitted wave phasors ${𝐄}_{\text{r}}$ and ${𝐄}_{\text{t}}$ are similarly represented as

	${𝐄}_{\text{r}}$	$=E_{\text{r},⟂}{\widehat{𝐞}}_{\text{r},⟂}+E_{\text{r},\parallel }{\widehat{𝐞}}_{\text{r},\parallel }$		(108)
	${𝐄}_{\text{t}}$	$=E_{\text{t},⟂}{\widehat{𝐞}}_{\text{t},⟂}+E_{\text{t},\parallel }{\widehat{𝐞}}_{\text{t},\parallel }$		(109)

where

	${\widehat{𝐞}}_{\text{r},⟂}$	$={\widehat{𝐞}}_{\text{i},⟂}$		(110)
	${\widehat{𝐞}}_{\text{r},\parallel }$	$={\displaystyle \frac{{\widehat{𝐞}}_{\text{r},⟂}\times {\widehat{𝐤}}_{\text{r}}}{{\parallel {\widehat{𝐞}}_{\text{r},⟂}\times {\widehat{𝐤}}_{\text{r}}\parallel }_2}}$
	${\widehat{𝐞}}_{\text{t},⟂}$	$={\widehat{𝐞}}_{\text{i},⟂}$
	${\widehat{𝐞}}_{\text{t},\parallel }$	$={\displaystyle \frac{{\widehat{𝐞}}_{\text{t},⟂}\times {\widehat{𝐤}}_{\text{t}}}{{\parallel {\widehat{𝐞}}_{\text{t},⟂}\times {\widehat{𝐤}}_{\text{t}}\parallel }_2}}$

and

	${\widehat{𝐤}}_{\text{r}}$	$={\widehat{𝐤}}_{\text{i}}-2\left({\widehat{𝐤}}_{\text{i}}^{𝖳}\widehat{𝐧}\right)\widehat{𝐧}$		(111)
	${\widehat{𝐤}}_{\text{t}}$	$=\sqrt{{\displaystyle \frac{{\eta }_1}{{\eta }_2}}}{\widehat{𝐤}}_{\text{i}}+\left(\sqrt{{\displaystyle \frac{{\eta }_1}{{\eta }_2}}}\cos ({\theta }_1)-\cos ({\theta }_2)\right)\widehat{𝐧}.$

The Fresnel equations provide relationships between the incident, reflected, and refracted field components for :

	$r_{⟂}$	$={\displaystyle \frac{E_{\text{r},⟂}}{E_{\text{i},⟂}}}={\displaystyle \frac{\sqrt{{\eta }_1}\cos ({\theta }_1)-\sqrt{{\eta }_2}\cos ({\theta }_2)}{\sqrt{{\eta }_1}\cos ({\theta }_1)+\sqrt{{\eta }_2}\cos ({\theta }_2)}}$		(112)
	$r_{\parallel }$	$={\displaystyle \frac{E_{\text{r},\parallel }}{E_{\text{i},\parallel }}}={\displaystyle \frac{\sqrt{{\eta }_2}\cos ({\theta }_1)-\sqrt{{\eta }_1}\cos ({\theta }_2)}{\sqrt{{\eta }_2}\cos ({\theta }_1)+\sqrt{{\eta }_1}\cos ({\theta }_2)}}$
	$t_{⟂}$	$={\displaystyle \frac{E_{\text{t},⟂}}{E_{\text{i},⟂}}}={\displaystyle \frac{2\sqrt{{\eta }_1}\cos ({\theta }_1)}{\sqrt{{\eta }_1}\cos ({\theta }_1)+\sqrt{{\eta }_2}\cos ({\theta }_2)}}$
	$t_{\parallel }$	$={\displaystyle \frac{E_{\text{t},\parallel }}{E_{\text{i},\parallel }}}={\displaystyle \frac{2\sqrt{{\eta }_1}\cos ({\theta }_1)}{\sqrt{{\eta }_2}\cos ({\theta }_1)+\sqrt{{\eta }_1}\cos ({\theta }_2)}}.$

If $\sqrt{\left|{\eta }_1/{\eta }_2\right|}\sin ({\theta }_1)\ge 1$, we have $r_{⟂}=r_{\parallel }=1$ and $t_{⟂}=t_{\parallel }=0$, i.e. total reflection. For the case of an incident wave in vacuum, i.e. ${\eta }_1=1$, the Fresnel equations (112) simplify to

	$r_{⟂}$	$={\displaystyle \frac{\cos ({\theta }_1)-\sqrt{{\eta }_2-{\sin }^2({\theta }_1)}}{\cos ({\theta }_1)+\sqrt{{\eta }_2-{\sin }^2({\theta }_1)}}}$		(113)
	$r_{\parallel }$	$={\displaystyle \frac{{\eta }_2\cos ({\theta }_1)-\sqrt{{\eta }_2-{\sin }^2({\theta }_1)}}{{\eta }_2\cos ({\theta }_1)+\sqrt{{\eta }_2-{\sin }^2({\theta }_1)}}}$
	$t_{⟂}$	$={\displaystyle \frac{2\cos ({\theta }_1)}{\cos ({\theta }_1)+\sqrt{{\eta }_2-{\sin }^2({\theta }_1)}}}$
	$t_{\parallel }$	$={\displaystyle \frac{2\sqrt{{\eta }_2}\cos ({\theta }_1)}{{\eta }_2\cos ({\theta }_1)+\sqrt{{\eta }_2-{\sin }^2({\theta }_1)}}}.$

Putting everything together, we obtain the following relationships between incident, reflected, and transmitted waves:

	$\left[\begin{array}{c}{E}_{\text{r},⟂}\\ E_{\text{r},\parallel }\end{array}\right]$			(114)
	$\left[\begin{array}{c}{E}_{\text{t},⟂}\\ E_{\text{t},\parallel }\end{array}\right]$			(115)

When an electromagnetic wave impinges on a surface, one part of the energy gets reflected while the other part gets refracted, i.e. it propagates into the surface. We distinguish between two types of reflection, specular and diffuse. The former type is discussed in Section A.7 and we will focus now on the latter type. When a rays hits a diffuse reflection surface, it is not reflected into a single (specular) direction but rather scattered toward many different directions. Since most surfaces give both specular and diffuse reflections, we denote by $S^2$ the fraction of the reflected energy that is diffusely scattered, where $S\in [0,1]$ is the so-called scattering coefficient [26]. Similarly, $R^2$ is the specularly reflected fraction of the reflected energy, where $R\in [0,1]$ is the reflection reduction factor. The following relationship between $R$ and $S$ holds:

$$R=\sqrt{1-S^2}.$$

(119)

Whenever a material has a scattering coefficient $S>0$, the Fresnel reflection coefficients in (112) must be multiplied by (119).

Let us consider an incoming locally planar linearly polarized wave with field phasor ${𝐄}_{\text{i}}(𝐪)$ at the scattering point $𝐪$ on the surface, as shown in Figure 28. We focus on the scattered field of and infinitesimally small surface element $dA$ in the direction ${\widehat{𝐤}}_{\text{s}}$. Note that the surface normal $\widehat{𝐧}$ has an arbitrary orientation with respect to the global coordinate system, whose $(x,y,z)$ axes are shown in green dotted lines. Also, the small surface element is related to the incident ray tube solid angle $d\omega$ through $dA=r^{2}d\omega$, where $r$ is the ray tube length.

The incoming field phasor can be represented by two arbitrary orthogonal polarization components (both orthogonal to the incoming wave vector ${\widehat{𝐤}}_i$):

	${𝐄}_{\text{i}}(𝐪)$	$=E_{\text{i},s}{\widehat{𝐞}}_{\text{i},s}+E_{\text{i},p}{\widehat{𝐞}}_{\text{i},p}$		(120)
		$=E_{\text{i},⟂}{\widehat{𝐞}}_{\text{i},⟂}+E_{\text{i},\parallel }{\widehat{𝐞}}_{\text{i},\parallel }$		(121)
		$=E_{\text{i},\theta }\widehat{𝜽}({\widehat{𝐤}}_{\text{i}})+E_{\text{i},\varphi }\widehat{𝝋}({\widehat{𝐤}}_{\text{i}})$		(122)

where me have omitted the dependence of the field strength on the position $𝐪$ for brevity. The second representation via $(E_{\text{i},⟂},E_{\text{i},\parallel })$ is used for the computation of the specularly reflected field as explained in Section A.7. The third representation decomposes the field into a vertically and horizontally polarized component, where $\widehat{𝜽},\widehat{𝝋}$ are defined in (49).

According to [27, Eq. 9], the diffusely scattered field ${𝐄}_{\text{s}}(𝐫)$ at the observation point $𝐫$ can be modeled as ${𝐄}_{\text{s}}(𝐫)=E_{\text{s},\theta }\widehat{𝜽}({\widehat{𝐤}}_{\text{s}})+E_{\text{s},\phi }\widehat{𝝋}({\widehat{𝐤}}_{\text{s}})$, where the orthogonal field components are computed as

$\left[\begin{array}{c}{E}_{\text{s},\theta }\\ E_{\text{s},\phi }\end{array}\right]$

(123)

Here, ${\mathrm{\Gamma }}^2$ is the percentage of the incoming power that is reflected (specularly and diffuse), which can be computed as

$$\mathrm{\Gamma }=\frac{\sqrt{{|r_{⟂}{E}_{\text{i},⟂}|}^2+{|r_{\parallel }{E}_{\text{i},\parallel }|}^2}}{{\parallel {𝐄}_{\text{i}}(𝐪)\parallel }_2}$$

(124)

where $r_{⟂},r_{\parallel }$ are defined in (112), ${\chi }_1,{\chi }_2\in [0,2\pi ]$ are (optional) independent random phase shifts, and the quantity $K_x\in [0,1]$ is defined by the scattering cross-polarization discrimination

$${\text{XPD}}_{\text{s}}=10{\log }_{10}\left(\frac{{|E_{\text{s},\text{pol}}|}^2}{{|E_{\text{s},\text{xpol}}|}^2}\right)=10{\log }_{10}\left(\frac{1-K_x}{K_x}\right).$$

(125)

This quantity determines how much energy gets transferred from one polarization direction into the other through the scattering process. Lastly, $dA$ is the size of the small area element on the reflecting surface under consideration, and $f_{\text{s}}({\widehat{𝐤}}_i,{\widehat{𝐤}}_s,\widehat{𝐧})$ is the scattering pattern, which has similarities with the BSDF in computer graphics [7, Ch. 4.3.1]. The scattering pattern must be normalized to satisfy the condition

$${\int }_0^{\pi /2}{\int }_0^{2\pi }{f}_{\text{s}}({\widehat{𝐤}}_{\text{i}},{\widehat{𝐤}}_{\text{s}},\widehat{𝐧})\sin ({\theta }_s)𝑑{\varphi }_s𝑑{\theta }_s=1$$

(126)

which ensures the power balance between the incoming, reflected, and refracted fields.