Plane-Wave Representations of Electromagnetic Fields
This chapter describes how the electromagnetic fields of an antenna can be represented by collections of plane-wave components.
Plane-Wave Spectrum Representation of Incident Fields
In a source-free region of space around the origin incident electormagnetic fields can be represented by an integral of the so-called plane-wave spectrum $\bm{P}(\vartheta, \varphi)$ over all propagation directions as
\[\bm{E}(\bm{r}) = \int \limits_{0}^{2\pi} \int \limits_{0}^{\pi} \bm{P}(\vartheta, \varphi)\, \mathrm{e}^{-\mathrm{j} \bm{k}(\vartheta, \varphi) \cdot \bm{r}} \, \sin \vartheta\, \mathrm{d}\vartheta\, \mathrm{d}\varphi\]
and
\[\bm{H}(\bm{r}) = \int \limits_{0}^{2\pi} \int \limits_{0}^{\pi} \dfrac{\bm{k}(\vartheta, \varphi)}{Z_{\mathrm{F}} \lVert\bm{k}(\vartheta, \varphi)\rVert} \times \bm{P}(\vartheta, \varphi)\, \mathrm{e}^{-\mathrm{j} \bm{k}(\vartheta, \varphi) \cdot \bm{r}} \, \sin \vartheta\, \mathrm{d}\vartheta\, \mathrm{d}\varphi\, ,\]
respectively, where the so-called plane-wave propagation vector $\bm{k}$ simultaneously encodes the direction of propagation $\hat{\bm{k}}$ (i.e., the direction of $\bm{k}$) and the wavelength $\lambda$ (from its magnitude $\lVert \bm{k} \rVert =k_0$ via $\lambda= 2\pi / k_0$) of any plane-wave component. The plane-wave spectrum $\bm{P}(\vartheta, \varphi)$ is always perpendicular to the propagation direction $\bm{k}(\vartheta, \varphi)$. We have
\[\bm{k}(\vartheta, \varphi) = k_0 \, \hat{\bm{k}}(\vartheta, \varphi)\]
with
\[k_0 = \lVert \bm{k} \rVert = \frac{2\pi}{\lambda} \]
and
\[\hat{\bm{k}}(\vartheta, \varphi) = \left( \sin \vartheta\, \cos \varphi \,\bm{e}_x + \sin \vartheta\, \sin \varphi \,\bm{e}_y + \cos \vartheta\, \bm{e}_z \right)\]
The above representation of the electric and magnetic field is purely solenoidal, i.e., divergence-free. As such, the representation is only good for representing source-free field solutions of Maxwell's equations. There exists a close relationship between the spherical vector-wave expansion of incident fields which uses only spherical vector-wave modes $\bm{F}_{s \ell m}^{(1)}(r, \vartheta, \varphi)$ of incident type and the above representation of the fields as an integral over propagating plane waves.
Since the propagation vector $\bm{k}$ and the corresponding unit vector into the same direction $\hat{\bm{k}}$ encode the propagation direction of the plane wave, we may use the shorthand notation $\bm{P}(\hat{\bm{k}})$ to represent the slightly longer expression $\bm{P}(\vartheta, \varphi)$ whenever convenient (sometimes we may even mix $\vartheta, \varphi$ with $\hat{\bm{k}}$ in the same expression). This should not lead to any ambiguities because the relation between $\hat{\bm{k}}$ and the tuple $\vartheta, \varphi$ is one-to one, as each encodings uniquely define the same point on the unit sphere.
The plane wave expansion in the form above is sometimes called homogeneous plane wave expansion (because it only contains one type of plane waves, namely non-evanescent, propagating waves) or Whittaker-type expansion.
Far-Field Pattern Representation of Radiated Fields
The radiated field of any current distribution is completely characterized by the radiated far-field pattern
\[\bm{F}(\vartheta, \varphi) = \lim \limits_{r\rightarrow \infty} \dfrac{r}{\mathrm{e}^{-\mathrm{j}k_0 r}}\, \bm{E}(r, \vartheta, \varphi)\,,\]
because it is, in prinicple, possible to reconstruct all radiated spherical vector-wave modes from a spherical expansion of the far fields. Thus, we may represent any radiated field by its far-field pattern.
Do not confuse the spherical vector-wave modes $\bm{F}_{s \ell m}^{(1)}(r, \vartheta, \varphi)$ with the far-field pattern $\bm{F}(\vartheta, \varphi)$!
Despite the fact that the far-field pattern alone contains all relevant information about the radiated field of a source distribution everywhere in space (even in the near field!), evaluating the near-fields of a radiated field given in terms of its far-field pattern $\bm{F}(\vartheta, \varphi)$ is not straightforward. It is more or less necessary to reconstruct the spherical vector-wave expansion of the radiated field from its far-field pattern before one can evaluate the spherical expansion at the desired location $\bm{r}$ (or to perform an equivalent transformation operation).
Translation of a Far-Field Pattern into a Plane Wave Spectrum in Different Coordinate System
Fortunately, there is a convenient way to convert the far-field pattern $\bm{F}(\vartheta, \varphi)$ of a radiated field into an incident plane-wave spectrum $\bm{P}(\vartheta', \varphi')$ in an observation volume around a new coordinate origin which is translated by the vector $\bm{R}$ against the original coordinate system. We use primed coordinates $r', \vartheta', \varphi'$ and primed vectors $\bm{k}'$ to distinguish the translated coordinate system from the original coordinate system. The geometrical situation is depicted in the figure below.
We have
\[\bm{P}(\vartheta', \varphi') = T_L(k_0 \lvert \bm{R} \rvert, \hat{\bm{k}}' \cdot \hat{\bm{R}})\, \bm{F}(\vartheta, \varphi)\bigg\rvert_{\vartheta = \vartheta', \varphi = \varphi`}\, ,\]
where
\[T_L(k_0 \lvert \bm{R} \rvert, \hat{\bm{k}}' \cdot \hat{\bm{R}}) = -\sum \limits_{\ell' =0}^{L}\left(-\mathrm{j}\right)^{\ell'} \, \left(2\ell'+1\right) \, \mathrm{h}_{\ell'}^{(2)}(k_0 \lvert \bm{R} \rvert) \mathrm{P}_\ell'(\hat{\bm{k}}' \cdot \hat{\bm{R}})\]
with $\hat{\bm{R}}$ the unit vector in the direction of the vector from the original coordinate origin to the new coordinate origin. Ideally, the number of sumation terms $L$ should go to $\infty$, but in practice, we can truncate the summation at a carefully chosen finite $L$.
The calculated plane-wave spectrum $\bm{P}(\vartheta', \varphi')$ can be used in the integral expressions above to find the electric and magnetic fields in the observation region.
Plane-Wave Representation of the Received Signal of an Antenna
Let $\hat{\bm{F}}(\vartheta, \varphi)$ be the normalized far field radiated by an antenna such that $\bm{F}(\vartheta, \varphi)= a\,\hat{\bm{F}}(\vartheta, \varphi)$ is the actual radiated far-field when its transmit port is excited by a signal with wave amplitude $a \in \mathbb{C} \, \sqrt{\mathrm{W}}$. To determine the received signal $b \in \mathbb{C} \, \sqrt{\mathrm{W}}$ of this antenna in receive mode under a certain illumination of an incident field, it is convenient to represent the incident field via its plane-wave spectrum $\bm{P}(\vartheta, \varphi)$. In this case, the received signal is given by (remember that $\hat{\bm{k}}$ depends on $\vartheta, \varphi$)
\[b= \int \limits_{0}^{2\pi} \int \limits_{0}^{\pi} \bm{P}(\hat{\bm{k}})\cdot \hat{\bm{F}}\left(- \hat{\bm{k}} \right) \, \sin \vartheta\, \mathrm{d}\vartheta\, \mathrm{d}\varphi\, .\]
Combining this result with the findings from the section above, we can express the $S_{21}$ parameter measured for the transmission between two antennas as
\[S_{21}= \int \limits_{0}^{2\pi} \int \limits_{0}^{\pi} \hat{\bm{F}}_1\left(\hat{\bm{k}} \right)\cdot T_L(k_0 \lvert \bm{R} \rvert, \hat{\bm{k}} \cdot \hat{\bm{R}})\, \hat{\bm{F}}_2\left(- \hat{\bm{k}} \right) \, \sin \vartheta\, \mathrm{d}\vartheta\, \mathrm{d}\varphi\, ,\]
where $\hat{\bm{F}}_1\left(\hat{\bm{k}} \right)$ is the normalized far-field pattern of antenna 1 (the transmit antenna), $\hat{\bm{F}}_2\left( - \hat{\bm{k}} \right)$ is the inverted far-field pattern of antenna 2 (the receive antenna) and $T_L(k_0 \lvert \bm{R} \rvert, \hat{\bm{k}} \cdot \hat{\bm{R}})$ is the translation operator known from previous sections with $\bm{R}$ denoting the vector separating the centers of the two antennas. The geometric situation for the calculation of the $S_{21}$-parameter is depicted below.
Plane-Wave Representation of Upwards Traveling Fields
Apart from the homogeneous plane wave representation of Whittaker type, there exists another common plane wave representation of antenna fields: the inhomogeneous plane wave representation of Wyle-type. Here, the fields are represented by an integral over the cartesian spectral coordinates $\kappa_x\, \kappa_y$ as
\[\bm{E}(\bm{r}) = \int \limits_{-\infty}^{\infty} \int \limits_{-\infty}^{\infty} \bm{\tau}(k_x, k_y)\, \mathrm{e}^{-\mathrm{j} \bm{\kappa}(\kappa_x, \kappa_y) \cdot \bm{r}}\, \mathrm{d}\kappa_x\, \mathrm{d}\kappa_y\, , \]
where
\[\bm{\kappa}({\kappa_x, \kappa_y}) = \kappa_x \bm{e}_x+ \kappa_y \bm{e}_y+\sqrt{k_0^2-\kappa_x^2-\kappa_y^2}\, \bm{e}_z\]
is a wave vector with a $z$-component which is either purely real (if $\kappa_x^2-\kappa_y^2\leq k_0^2$) or purely imaginary (if $\kappa_x^2-\kappa_y^2> k_0^2$). The portion of the spectrum with a purely real $z$-component is called the visible portion of the spectrum and it contains unattenuated propagating plane waves. Conversely, the portion of the spectrum with a purely imaginary $z$-component is called the invisible portion and contains only evanescent waves which are exponentially attenuated in $z$-direction. Since propagating and also evanescent plane waves are involved, this type of expansion is referred to as inhomogeneous plane wave expansion.
It is helpful to think about the spectral domains defined by the wave vectors $\bm{k}$ and $\bm{kappa}$ as two different entities at this point of the discussion, but we will see that there exists a close relationship between the two spectral representations.
Using the coordinate transformations
\[ \kappa_x= k_0\, \sin(\alpha){\cos}({\beta})\]
and
\[ \kappa_y= k_0\,{\sin}({\alpha}){\sin}({\beta}) \,,\]
the expansion can be brought into the form
\[ {\bm{E}}({\bm{r}}) = \int \limits_{0}^{2\pi} \int \limits_{C^\pm} \mathrm{e}^{-\mathrm{j} k_0 \,\hat{\bm{k}}\cdot \bm{r}}\, {\bm{\tau}}({{\kappa_x}({\alpha, \beta}),{\kappa_y}({\alpha, \beta})}) \, k_0^2\, \cos \alpha \, \sin \alpha\, \mathrm{d}\alpha \mathrm{d} \beta \\ = \int \limits_{0}^{2\pi} \int \limits_{\mathcal{C}^\pm} \mathrm{e}^{-\mathrm{j} k_0 \,\hat{\bm{k}}\cdot \bm{r}}\, {\bm{F}}({\hat{\bm{\kappa}}}({\alpha, \beta})) \, \sin \alpha\, \mathrm{d} \alpha \mathrm{d} \beta\, ,\]
where
\[ {\hat{\bm{\kappa}}}({\alpha, \beta}) % = \dfrac{1}{k_0}\vec{k} = {\sin}({\alpha}) {\cos}({\beta})\vec{e}_x + {\sin}({\alpha}) {\sin}({\beta}) \vec{e}_y + {\cos}({\alpha}) {e}_z\, ,\]
and
\[ {\bm{F}}({\hat{\bm{\kappa}}}({\alpha, \beta})) = {\bm{\tau}}({{\kappa_x}({\alpha, \beta}),{\kappa_y}({\alpha, \beta})}) \, k_0^2\, \cos \alpha \,.\]
The integration starts at $\alpha=0$ and ends at $\alpha= \pi/2+ \mathrm{j} \infty$ for $\mathcal{C}^+$ and the integration for $\mathcal{C}^-$ begins at $\alpha= \pi/2- \mathrm{j} \infty$ and ends at $\alpha=\pi$.