Spherical Vector-Wave Expansion of General Time Harmonic Electromangentic Fields

Away from sources, the time harmonic electric and magnetic fields must fulfill the homogeneous curl-curl-equation

\[\nabla \times \nabla \times \bm{E} - k_0^2 \bm{E} =\bm{0}\, .\]

Arbitrary source-free solutions of Maxwell's equations can be expanded into a series of spherical vector-wave functions as (see pp. 9ff of ^[1])

\[\bm{E}(r, \vartheta, \varphi) = k_0 \, \sqrt{Z_{\mathrm{F}}} \sum \limits_{c} \sum \limits_{s=1}^2 \sum \limits_{\ell=1}^{\infty} \sum \limits_{m=-\ell}^{\ell} \alpha_{s\ell m}^{(c)}\, \bm{F}_{s\ell m}^{(c)}(r,\vartheta, \varphi)\]

\[\bm{H}(r, \vartheta, \varphi) =\mathrm{j}\, \dfrac{k_0}{\sqrt{Z_{\mathrm{F}}}} \sum \limits_{c} \sum \limits_{s=1}^2 \sum \limits_{\ell=1}^{\infty} \sum \limits_{m=-\ell}^{\ell} \alpha_{s\ell m}^{(c)}\, \bm{F}_{3-s,\ell m}^{(c)}(r,\vartheta, \varphi)\]

where $Z_{\mathrm{F}}\sqrt{\mu_0/ \varepsilon_{0}}\approx 376.730\,313\,669\, \mathrm{\Omega}$ is the impedance of free space and $\bm{F}_{s\ell m}^{(c)}(r,\vartheta, \varphi)$ are the spherical vector wave mode functions, which are solutions to the homogeneous curl-curl equation (except in the origin of the coordinate system, where some types of vector mode functions become singular). Due to only two of the radial dependencies being linearly independent, it is sufficient to let the sum over $c$ to be an arbitrary pair of two iindices from the set $c \in \{1,2,3,4\}$. Usually one will choose the pair $c=1$ and $c=4$, where only the $c=4$-type modes are needed to describe purely radiated fields and only the $c=1$-type modes are needed to describe purely incident fields.

The vector of coefficients completely defines the fields of a corresponding radiating or incident spherical wave expansion.

Definition of Spherical Vector-Mode Functions

Explicit expressions for the spherical vector-mode functions are (see p. 13 of ^[2])

\[\bm{F}_{1\,\ell m}^{(c)}(r,\vartheta, \varphi) = \dfrac{\mathrm{e}^{\, \mathrm{j}m\varphi}}{\sqrt{2}\pi} \left( \dfrac{z_\ell^{(c)}(k_0r)}{\sqrt{\ell(\ell+1)}} \, \dfrac{\mathrm{j}m \,\overline{\mathrm{P}}_{\ell}^{m}(\cos \vartheta) }{\sin \vartheta} \bm{e}_\vartheta -\dfrac{z_\ell^{(c)}(k_0r)}{\sqrt{\ell(\ell+1)}} \, \dfrac{\mathrm{d}\,\overline{\mathrm{P}}_{\ell}^{m}(\cos \vartheta) }{\mathrm{d} \vartheta} \bm{e}_\varphi \right)\]

\[\bm{F}_{2\,\ell m}^{(c)}(r,\vartheta, \varphi) = \dfrac{\mathrm{e}^{\, \mathrm{j}m\varphi}}{\sqrt{2}\pi} \left( \dfrac{\sqrt{\ell(\ell+1)}}{k_0r}\, z_\ell^{(c)}(k_0r)\, \overline{\mathrm{P}}_{\ell}^{m}(\cos \vartheta) \bm{e}_r \right. \phantom{\dfrac{\mathrm{d}\,\overline{\mathrm{P}}_{\ell}^{m}(\cos \vartheta) }{\mathrm{d} \vartheta}~~~~~} \\ \phantom{\bm{F}_{2\,\ell m}^{(c)}(r,\vartheta, \varphi) = \dfrac{\mathrm{e}^{\, \mathrm{j}m\varphi}}{\sqrt{2}\pi}} + \dfrac{1}{k_0r} \dfrac{\mathrm{d}}{\mathrm{d} k_0r} \left\{k_0 r \dfrac{z_\ell^{(c)}(k_0r)}{\sqrt{\ell(\ell+1)}} \right\}\, \dfrac{\mathrm{d}\,\overline{\mathrm{P}}_{\ell}^{m}(\cos \vartheta) }{\mathrm{d} \vartheta} \bm{e}_\vartheta \phantom{~~~~~} \\ \left. \phantom{\bm{F}_{2\,\ell m}^{(c)}(r,\vartheta, \varphi) = \dfrac{\mathrm{e}^{\, \mathrm{j}m\varphi}}{\sqrt{2}\pi}} + \dfrac{1}{k_0r} \dfrac{\mathrm{d}}{\mathrm{d} k_0r} \left\{k_0 r \dfrac{z_\ell^{(c)}(k_0r)}{\sqrt{\ell(\ell+1)}} \right\}\, \dfrac{\mathrm{j}m \,\overline{\mathrm{P}}_{\ell}^{m}(\cos \vartheta) }{\sin \vartheta} \bm{e}_\varphi\ \right)\, .\]

To ensure that the vector-mode functions $\bm{F}_{s\ell m}^{(c)}(r,\vartheta, \varphi)$ are solutions to the homogeneous curl-curl equation, they are constructed from solutions of the homogeneous scalar Helmholtz equation as (see, e.g., page 393 of ^[3])

\[\bm{M}_{\ell m}^{(c)}= \nabla f_{\ell m}^{(c)} \times \bm{e}_r \overset{!}{=} \bm{F}_{1\,\ell m}^{(c)}\]

and

\[\bm{N}_{\ell m}^{(c)}= \dfrac{1}{k_0} \nabla \times \bm{M}_{\ell m}^{(c)} \overset{!}{=} \bm{F}_{2\,\ell m}^{(c)}\]

where $\bm{e}_r$ is the unit vector in radial direction.

General solution to the scalar Helmholtz equation

\[\Delta f +k_0^2 f=0\, .\]

can be constructed as superpositions of functions from the set

\[f_{s\ell m}^{(c)}(r, \vartheta, \varphi) = \begin{cases} \dfrac{\mathrm{j}_{\ell}(k_0 r)}{\sqrt{\ell(\ell+1)}} \, \mathrm{Y}_{\ell m}(\vartheta, \varphi) & c=1\\[1.5em] \dfrac{\mathrm{\mathrm{n}}_\ell(k_0 r)}{\sqrt{\ell(\ell+1)}} \, \mathrm{Y}_{\ell m}(\vartheta, \varphi) & c=2\\[1.5em] \dfrac{\mathrm{\mathrm{h}}_\ell^{(1)}(k_0 r)}{\sqrt{\ell(\ell+1)}} \, \mathrm{Y}_{\ell m}(\vartheta, \varphi) & c=3\\[1.5em] \dfrac{\mathrm{\mathrm{h}}_\ell^{(2)}(k_0 r)}{\sqrt{\ell(\ell+1)}}\, \mathrm{Y}_{\ell m}(\vartheta, \varphi) & c=4 \end{cases} \]

where

• $\mathrm{j}_{\ell}(k_0 r)$ are spherical Bessel functions which represent the radial behavior standing waves (or incident waves)
• $\mathrm{n}_{\ell}(k_0 r)$ are spherical Neumann functions which usually do not represent a physically meaningful solution
• $\mathrm{h}_\ell^{(1)}(k_0 r) = \mathrm{j}_{\ell}(k_0 r) +\mathrm{j}\, \mathrm{n}_{\ell}(k_0 r)$ are spherical Hankel functions of first kind which represent the radial behavior of waves coming from infinity being absorbed at the coordinate origin
• $\mathrm{h}_\ell^{(2)}(k_0 r) = \mathrm{j}_{\ell}(k_0 r) -\mathrm{j}\, \mathrm{n}_{\ell}(k_0 r)$ are spherical Hankel functions of second kind which represent the radial behavior of waves traveling from the coordinate origin toward infinity
• $\mathrm{Y}_{\ell m}(\vartheta, \varphi)$ are the spherical Harmonics

and $k_0=2\pi/\lambda$ is the wavenumber defined by the frequency of the electromagnetic wave.

The spherical Harmonics are defined as (see §14.30 of ^[1])

\[\mathrm{Y}_{\ell m}(\vartheta, \varphi) = \begin{cases} \dfrac{1}{\sqrt{2\pi}}\, \overline{\mathrm{P}}_{\ell}^{m}(\cos \vartheta) \mathrm{e}^{\, \mathrm{j}m \varphi} & \text{for }|m|\leq \ell\\ 0 & \text{esle} \end{cases}\]

where the normalized associated Legendre functions $\overline{\mathrm{P}}_{\ell}^{m}(x)$ are defined in ordinary Legendre polynomials $\mathrm{P}_{\ell}(x)$ as (compare §14.6 and §14.7 of ^[1])

\[\overline{\mathrm{P}}_{\ell}^{m}(x) = \begin{cases} \sqrt{\dfrac{(2\ell+1)(\ell-m!)}{2(\ell+m)!}} (-1)^m \, (1-x^2)^{m/2} \, \dfrac{\mathrm{d}^m \mathrm{P}_{\ell}(x)}{\mathrm{d}^m} & m\geq 0\\[2em] (-1)^m \overline{\mathrm{P}}_{\ell}^{-m}(x) & m<0\, . \end{cases}\]

Orthogonality Relations of Spherical Vector-Mode Functions

The tangential components of the spherical vector-mode functions are orthogonal on concentric spherical surfaces (centered at the coordinate origin). Concretely, they fulfill the orthogonality properties

\[\int \limits_{0}^{2\pi} \int \limits_{0}^{\pi} \bm{F}_{s\ell m}^{(c)}(r,\vartheta, \varphi) \bigg\rvert_{\mathrm{tan}} \,\cdot\, \bm{F}_{\sigma \lambda \mu}^{(\gamma)}(r,\vartheta, \varphi) \bigg\rvert_{\mathrm{tan}} \, \sin \vartheta\, \mathrm{d}\vartheta \mathrm{d}\varphi = \delta_{s\sigma}\, \delta_{\ell \lambda}\, \delta_{m, -\mu}\, R_{s\ell}^{(c)}(k_0r)\, R_{\sigma\\lambda}^{(\gamma)}(k_0r)\,,\]

where

\[\delta_{\ell \lambda} = \begin{cases} 1 & \text{if } \ell= \lambda \\ 0 & \text{if } \ell\neq \lambda \end{cases}\]

is the Kronecker-Delta and

\[R_{s\ell}^{(c)}(k_0r) = \begin{cases} z_{\ell}^{(c)}(k_0r) & \text{if } s=1 \\[1em] \dfrac{1}{k_0r}\, \dfrac{\mathrm{d}}{\mathrm{d} k_0r} \left\{z_{\ell}^{(c)}(k_0r)\right\} & \text{if } s=2 \end{cases}\]

is a short-hand notation for the radial dependency of the corresponding spherical vector-mode function $\bm{F}_{s\ell m}^{(c)}(r,\vartheta, \varphi)$.

The orthogonality relations are very useful to find the coefficients $\alpha_{s\ell m}^{(c)}$ in the spherical vector-wave expansion of a certain field which is known on a spherical surface.

Far-Field Expressions

Due to the asymptotic behavior of the radial dependencies

\[\lim \limits_{k_0r \rightarrow \infty} \mathrm{h}_\ell^{(2)}(k_0r) = \mathrm{j}^{\ell+1} \dfrac{\mathrm{e}^{-\mathrm{j} k_0r}}{k_0r} \]

\[\lim \limits_{k_0r \rightarrow \infty} \dfrac{1}{k_0r} \dfrac{\mathrm{d}}{\mathrm{d} k_0r} \left\{k_0 r \mathrm{h}_\ell^{(2)}(k_0r) \right\} = \mathrm{j}^{\ell} \dfrac{\mathrm{e}^{-\mathrm{j} k_0r}}{k_0r} \]

for distances approaching infinity, it is convenient to define far-field expressions of the mode functions as

\[\bm{K}_{1\,\ell m}^{(4)}(\vartheta, \varphi) = \lim \limits_{k_0r \rightarrow \infty} \dfrac{k_0r}{\mathrm{e}^{-\mathrm{j} k_0r}} \bm{F}_{1\,\ell m}^{(4)}(r,\vartheta, \varphi)\]

\[\bm{K}_{2\,\ell m}^{(4)}(\vartheta, \varphi) = \lim \limits_{k_0r \rightarrow \infty} \dfrac{k_0r}{\mathrm{e}^{-\mathrm{j} k_0r}} \bm{F}_{2\,\ell m}^{(4)}(r,\vartheta, \varphi)\, .\]

Explicit expressions for the far-field mode functions are given as

\[\bm{K}_{1\,\ell m}^{(4)}(\vartheta, \varphi) = \dfrac{\mathrm{j}^{\ell+1}}{\sqrt{\ell(\ell+1)}} \dfrac{\mathrm{e}^{\, \mathrm{j}m\varphi}}{\sqrt{2\pi}} \left( \dfrac{\mathrm{j}m\, \overline{\mathrm{P}}_{\ell}^m(\cos \vartheta )}{\sin \vartheta} \bm{e}_\vartheta - \dfrac{\mathrm{d} \overline{\mathrm{P}}_{\ell}^m(\cos \vartheta )}{\mathrm{d} \vartheta} \bm{e}_\varphi \right)\]

\[\bm{K}_{2\,\ell m}^{(4)}(\vartheta, \varphi) = \dfrac{\mathrm{j}^{\ell}}{\sqrt{\ell(\ell+1)}} \dfrac{\mathrm{e}^{\, \mathrm{j}m\varphi}}{\sqrt{2\pi}} \left( \dfrac{\mathrm{d} \overline{\mathrm{P}}_{\ell}^m(\cos \vartheta )}{\mathrm{d} \vartheta} \bm{e}_\vartheta + \dfrac{\mathrm{j}m\, \overline{\mathrm{P}}_{\ell}^m(\cos \vartheta )}{\sin \vartheta} \bm{e}_\varphi \right)\, .\]

Naturally, the far field $\bm{F}(\vartheta, \varphi) = \lim \limits_{k_0r \rightarrow \infty} \dfrac{r}{\mathrm{e}^{-\mathrm{j} k_0r}} \bm{E}(r,\vartheta, \varphi)$ can be calculated from a given set of expansion coefficients $\alpha_{s\ell m}^{(4)}$ as

\[\bm{F}(\vartheta, \varphi) = \sqrt{Z_{\mathrm{F}}} \sum \limits_{c} \sum \limits_{s=1}^2 \sum \limits_{\ell=1}^{\infty} \sum \limits_{m=-\ell}^{\ell} \alpha_{s\ell m}^{(4)}\, \bm{K_{s\ell m}^{(c)}}(\vartheta, \varphi)\, .\]

Spherical Expansion Coefficients from a Current Density

The radiating spherical vector-wave expansion for the fields radiated by arbitrary electric and magnetic volume current densities $\bm{J}(\bm{r})$ and $\bm{M}(\bm{r})$ can be calculated by

\[\alpha_{s\ell m}^{(4)}=k_0\, (-1)^{m+1} \iiint \left[ \sqrt{Z_{\mathrm{F}}}\, \bm{F}_{s,\ell,-m}^{(1)} \cdot \bm{J}(\bm{r}) - \dfrac{\mathrm{j}}{\sqrt{Z_{\mathrm{F}}}}(\bm{r})\, \bm{F}_{3-s,\ell,-m}^{(1)}(\bm{r}) \cdot \bm{M}(\bm{r}) \right] \mathrm{d}v \, .\]

The spherical vector-wave functions $\bm{F}_{s\ell m}^{(1)}(\bm{r})$ of incident type which are involved in the above calculation have vanishingly small magnitudes for locations $\bm{r}$ with distance to the coordinate origin $r < \ell /k_0$. Thus, for current densities which are completely enclosed by a minimum sphere of radius $r_{\mathrm{minsphere}}$, one may neglect all coefficients $\alpha_{s\ell m}^{(4)}$ with $\ell < k_0 r_{\mathrm{minsphere}}$ (plus a small buffer depending on the desired accuracy), as long as one is not interested in evaluating the expanded electromagnetic fields extremely close to the original currents.

Spherical Expansion Coefficients by Using Orthogonality

Often, we are interested in finding the spherical vector-wave expansion of a purely incident or a purely radiated field. To be precise, one wants to find the expansion coefficients $\alpha_{s\ell m}^{(c)}$ (with $c=1$ for a purely incident field and $c=4$ for a purely radiated field) to fulfill the equality

\[\bm{E}(r, \vartheta, \varphi) = k_0 \, \sqrt{Z_{\mathrm{F}}} \sum \limits_{s=1}^2 \sum \limits_{\ell=1}^{\infty} \sum \limits_{m=-\ell}^{\ell} \alpha_{s\ell m}^{(c)}\, \bm{F}_{s\ell m}^{(c)}(r,\vartheta, \varphi)\]

for a given field $\bm{E}(r, \vartheta, \varphi)$.

As indicated previously, one may leverage on the orthogonality properties of the spherical vector-wave functions to find the desired expansion coefficients $\alpha_{s\ell m}^{(c)}$. We have

\[\alpha_{s\ell m}^{(c)} = \dfrac{(-1)^{m}}{k_0\sqrt{Z_{\mathrm{F}}}\,R_{s\ell}^{(c)}(k_0r)\, R_{s\ell}^{(\gamma)}(k_0r)} \int \limits_{0}^{2\pi} \int \limits_{0}^{\pi} \bm{E}(r,\vartheta, \varphi) \bigg\rvert_{\mathrm{tan}} \,\cdot\, \bm{F}_{s \ell ,-m}^{(\gamma)}(r,\vartheta, \varphi) \bigg\rvert_{\mathrm{tan}} \, \sin \vartheta\, \mathrm{d}\vartheta \mathrm{d}\varphi\, .\]

The above formula for finding the spherical coefficients $\alpha_{s\ell m}^{(c)}$ is valid at any radial distance $r$ from the origin. In particular, the formula can also be used to find the coefficients $\alpha_{s\ell m}^{(c)}$ from the radiated fields at far-field distance.

Number of Relevant Modes of an Incident Electromagnetic Field

In a source-free region around the coordinate origin, incident fields can be completely described in terms of vector spherical wave functions $\bm{F}_{s\ell m}^{(1)}(\bm{r})$ of incident type by (accordingly for the H-field)

\[\bm{E}(r, \vartheta, \varphi) = k_0 \, \sqrt{Z_{\mathrm{F}}} \sum \limits_{s=1}^2 \sum \limits_{\ell=1}^{\infty} \sum \limits_{m=-\ell}^{\ell} \alpha_{s\ell m}^{(1)}\, \bm{F}_{s\ell m}^{(1)}(r,\vartheta, \varphi)\,.\]

The spherical vector-wave functions $\bm{F}_{s\ell m}^{(1)}(\bm{r})$ of incident type which are involved in the above calculation have vanishingly small magnitudes for locations $\bm{r}$ with distance to the coordinate origin $r < \ell /k_0$. Thus, as long as one is interested only in the fields inside a spherical region around the origin with radius $r_{\mathrm{observation}}$, it is sufficient to consider only modes with $\ell < L = k_0 r_{\mathrm{observation}}$ (plus a small buffer depending on the desired accuracy). We have

\[\bm{E}(r, \vartheta, \varphi)\,\bigg\lvert_{r< \ell/k_0} \approx k_0 \, \sqrt{Z_{\mathrm{F}}} \sum \limits_{s=1}^2 \sum \limits_{\ell=1}^{L} \sum \limits_{m=-\ell}^{\ell} \alpha_{s\ell m}^{(1)}\, \bm{F}_{s\ell m}^{(1)}(r,\vartheta, \varphi) \,.\]

Translation of Radiated Modes into Incident Modes in Different Coordinate System

The radiated electromagnetic fields generated by currents around the coordinate origin may be represented by a superposition of spherical vector-wave functions $\bm{F}_{s\ell m}^{(4)}(r, \vartheta, \varphi)$ of radiated type. However, in a translated coordinate system (denoted by the primed coordinates $r', \vartheta', \varphi'$), the sources are far away from the new coordinate origin and the electromagnetic fields around the new origin may be represented in terms of spherical vector-wave functions $\bm{F}_{s\ell m}^{(1)}(r', \vartheta', \varphi')$ of incident type with respect to the new coordinate origin. The geometrical situation is depicted in the figure below.

The expansion coefficients $\alpha_{\sigma \lambda \mu}^{(1)}$ of the incident field in the new coordinate system can be calculated from the expansion coefficients $\alpha_{s\ell m}^{(4)}$ of the radiated field in the original coordinate system by

\[ \alpha_{\sigma \lambda \mu}^{(1)} = \sum \limits_{s,\ell, m} \mathcal{T}_{s\ell m}^{\sigma \lambda \mu}(\bm{R})\, \alpha_{s\ell m}^{(4)}\]

where the translation coefficient $\mathcal{T}_{s\ell m}^{\sigma \lambda \mu}(\bm{R})$ is given by

\[\mathcal{T}_{s\ell m}^{\sigma \lambda \mu}(\bm{R}) = - \sum \limits_{\ell'=0}^\infty (-1)^{\sigma + \lambda +\mu} \int \limits_{0}^{2\pi} \int \limits_{0}^{\pi} (-\mathrm{j})^{\ell'} \left( 2\ell'+1 \right) {\mathrm{h}_{\ell'}^{(2)}}\left({k_0\lvert{\bm{R}}\rvert}\right)\, {\mathrm{P}_{\ell'}}\left({\hat{\bm{k}}\left(\vartheta, \varphi\right) \cdot\hat{\bm{R}}}\right)\, {\bm{K}_{s\ell m}^{(4)}}\left(\vartheta, \varphi\right) \cdot {\bm{K}_{\sigma, \lambda,- \mu}^{(4)}}\left(\vartheta, \varphi\right) \, \sin \vartheta \mathrm{d}\vartheta\, \mathrm{d}\varphi\, ,\]

where $\hat{\bm{k}}\left(\vartheta, \varphi\right)$ denotes the radial unit vector defined by the angles $\vartheta, \varphi$ and $\hat{\bm{R}}$ denotes the unit vector in the direction of the vector from the original coordinate origin to the new coordinate origin.

Received Signal

Let $\hat{\alpha}_{s\ell m}^{(c)}$ be the normalized expansion coefficients for the fields radiated by an antenna such that $\alpha_{s\ell m}^{(c)}= a\,\hat{\alpha}_{s\ell m}^{(c)}$ are the actual expansion coefficients when the antenna's 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 spherical vector wave expansion

\[\bm{E}(r, \vartheta, \varphi)= k_0 \, \sqrt{Z_{\mathrm{F}}} \sum \limits_{s=1}^2 \sum \limits_{\ell=1}^{L} \sum \limits_{m=-\ell}^{\ell} \alpha_{s\ell m}^{(1)}\, \bm{F}_{s\ell m}^{(1)}(r,\vartheta, \varphi) \,.\]

In this case, the received signal is given by

\[b= \sum \limits_{s=1}^2 \sum \limits_{\ell=1}^{L} \sum \limits_{m=-\ell}^{\ell} \alpha_{s\ell m}^{(1)}\, \hat{\beta}_{s\ell m}, ,\]

where the antenna receive coefficients $\hat{\beta}_{s\ell m}$ are found from the corresponding normalized antenna's transmit coefficients $\hat{\alpha}_{s\ell m}^{(c)}$ via

\[\hat{\beta}_{s\ell m} = \dfrac{(-1)^{m}}{2} \hat{\alpha}_{s\ell,-m}^{(c)}\, .\]

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

\[S_{21} = \sum \limits_{\sigma=1}^2 \sum \limits_{\lambda=1}^{L} \sum \limits_{\mu=-\lambda}^{\lambda} \hat{\beta}_{\sigma \lambda \mu}^{\text{antenna2}} \, \left( \sum \limits_{s=1}^2 \sum \limits_{\ell=1}^{L} \sum \limits_{m=-\ell}^{\ell} \mathcal{T}_{s\ell m}^{\sigma \lambda \mu}(\bm{R})\, \hat{\alpha}_{s\ell m}^{(4),\text{antenna1}} \right)\]

where $\hat{\alpha}_{s\ell m}^{(4),\text{antenna1}}$ are the normalized transmit coefficients of antenna 1 (the transmit antenna), $\hat{\beta}_{\sigma \lambda \mu}^{\text{antenna2}}$ are the receive coefficients of antenna 2 (the receive antenna) and $\mathcal{T}_{s\ell m}^{\sigma \lambda \mu}(\bm{R})$ 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.

References