Metasurface Transformation Theory

Abstract

Metasurfaces constitute a class of thin metamaterials, which are used from microwave to optical frequencies to create new antennas and microwave devices. This chapter describes how to use transformation optics (TO) to create anisotropic modulated-impedance metasurfaces able to transform planar surface waves (SW) into a predefined curved-wavefront surface wave. In fact, the modulated anisotropic impedance imposes a local modification of the dispersion equation and, at constant operating frequency, of the local wavevector. The general effects of metasurface modulation are similar to those obtained by TO in volumetric inhomogeneous metamaterials, namely readdressing the propagation path of an incident wave; however, significant technological simplicity is gained.

Appendices

Appendix A Fundamentals of Transformation Optics

Let us consider Maxwell’s equation in absence of sources in primed space (assumed for simplicity as free space)

$$ \begin{aligned} & \nabla \times {\mathbf{E}}^{\prime} = - j\omega \mu_{0} {\mathbf{H}}^{\prime} \\ & \nabla \times {\mathbf{H}}^{\prime} = j\omega \varepsilon_{0} {\mathbf{E}}^{\prime} \\ \end{aligned} $$

(3.65)

The observation point of this space, that will be denoted in the following as “virtual space” or “transformed space”, will be identified by the vector ${\mathbf{r}}' = x'{\hat{\mathbf{x}}}' + y'{\hat{\mathbf{y}}}' + z'{\hat{\mathbf{z}}}'$. The physical space individuated by ${\mathbf{r}} = x{\hat{\mathbf{x}}} + y{\hat{\mathbf{y}}} + z{\hat{\mathbf{z}}}$ will be instead denoted as “real space”. For simplicity, Cartesian coordinates have been chosen in both spaces. The real space is mapped to the virtual space by a 3D invertible transformation ${\varvec{\Upgamma}}$; conversely, the inverse transformation ${\varvec{\Upgamma}}^{ - 1}$ maps the virtual space to the real space

$$ {\mathbf{r}}^{\prime} = {\varvec{\Upgamma}}\left( {\mathbf{r}} \right):\left\{ {x,y,z} \right\} \to \left\{ {x^{\prime} (x,y,z),y^{\prime} (x,y,z),z^{\prime} (x,y,z)} \right\} $$

(3.66)

$$ {\mathbf{r}} = {\varvec{\Upgamma}}^{ - 1} \left( {{\mathbf{r}}\,^{\prime} } \right):\left\{ {x\,^{\prime} ,y\,^{\prime} ,z\,^{\prime} } \right\} \to \left\{ {x(x\,^{\prime} ,y\,^{\prime} ,z\,^{\prime} ),y(x\,^{\prime} ,y\,^{\prime} ,z\,^{\prime} ),z(x\,^{\prime} ,y\,^{\prime} ,z\,^{\prime} )} \right\} $$

(3.67)

The coordinate transformation ${\varvec{\Upgamma}}$ is the one on which the TO process is based. Note that several authors interchange the definition of transformation and inverse transformation, so that the domain is the virtual space and the co-domain is the real space. On the basis of TO, Maxwell’s equations in the real space are satisfied provided that the real space is filled with an inhomogeneous anisotropic medium characterized by tensors $\underline{\underline{\mu }}$ and $\underline{\underline{{\mathbf{\varepsilon }}}}$ properly related to the coordinate transformation. Hence, Maxwell’s equations in the real space are written as [25, 26]

$$ \begin{aligned} & \nabla \times {\mathbf{E}} = - j\omega \underline{\underline{{\varvec{\mu}}}} \cdot {\mathbf{H}} \\ & \nabla \times {\mathbf{H}} = j\omega \underline{\underline{{\mathbf{\varepsilon }}}} \cdot {\mathbf{E}} \\ \end{aligned} $$

(3.68)

In (3.68), the nabla operator $\nabla \, = \,\frac{\partial }{\partial x}{\hat{\mathbf{x}}} + \frac{\partial }{\partial y}{\hat{\mathbf{y}}} + \frac{\partial }{\partial z}{\hat{\mathbf{z}}}$ acts on the Cartesian coordinates of the real space. The electric and magnetic fields are expressed as

$$ \begin{aligned} & {\mathbf{E}}({\mathbf{r}}) = \underline{\underline{{\mathbf{M}}}}^{T} ({\mathbf{r}}) \cdot {\mathbf{E}}^{\prime} ({\mathbf{r}}^{\prime} ), \\ & {\mathbf{H}}({\mathbf{r}}) = \underline{\underline{{\mathbf{M}}}}^{T} ({\mathbf{r}}) \cdot {\mathbf{H}}^{\prime} ({\mathbf{r}}^{\prime} ) \\ \end{aligned} $$

(3.69)

where ${\mathbf{r}}' = {\varvec{\Upgamma}}\left( {\mathbf{r}} \right)$ and the Jacobian tensor $\underline{\underline{{\mathbf{M}}}}$ is defined as

$$ \begin{aligned} & \underline{\underline{{\mathbf{M}}}} ({\mathbf{r}}) = \left( {\frac{{\partial {\mathbf{r}}^{\prime} }}{{\partial {\mathbf{r}}}}} \right) = \sum\limits_{i,j = 1,3}^{{}} {\frac{{\partial x_{i} ^{\prime} }}{{\partial x_{j} }}{\hat{\mathbf{x}}}_{i} ^{\prime} } {\hat{\mathbf{x}}}_{j} \\ & \underline{\underline{{\mathbf{M}}}}^{ - 1} ({\mathbf{r}}) = \sum\limits_{i,j = 1,3}^{{}} {\frac{{\partial x_{j} }}{{\partial x_{i} ^{\prime} }}{\hat{\mathbf{x}}}_{j} } {\hat{\mathbf{x}}}_{i} ^{\prime} \\ \end{aligned} $$

(3.70)

In (3.70) for simplifying the notation we have redefined $\left\{ {x,y,z} \right\} \equiv \left\{ {x_{1} ,x_{2} ,x_{3} } \right\}$, $\left\{ {x^{\prime} ,y^{\prime} ,z^{\prime} } \right\} \equiv \left\{ {x_{1} ^{\prime} ,x_{2} ^{\prime} ,x_{3} ^{\prime} } \right\}$; ${\hat{\mathbf{x}}},{\hat{\mathbf{y}}},{\hat{\mathbf{z}}} \equiv {\hat{\mathbf{x}}}_{1} ,{\hat{\mathbf{x}}}_{2} ,{\hat{\mathbf{x}}}_{3} ,$ and ${\hat{\mathbf{x}}}^{\prime} ,{\hat{\mathbf{y}}}^{\prime} ,{\hat{\mathbf{z}}}^{\prime} \equiv {\hat{\mathbf{x}}}_{1} ^{\prime} ,{\hat{\mathbf{x}}}_{2} ^{\prime} ,{\hat{\mathbf{x}}}_{3} ^{\prime}$. In the following, we will use this latter indexed notation when needed within summation symbol. The permittivity and permeability TO tensors in (3.68) are related to the Jacobian tensors by the following equations

$$ \frac{1}{{\mu_{0} }}\underline{\underline{{\varvec{\mu}}}} = \frac{1}{{\varepsilon_{0} }} \, \underline{\underline{{\mathbf{\varepsilon }}}} = \underline{\underline{{\varvec{\alpha}}}} \, ; \, \underline{\underline{{\varvec{\alpha}}}} = \det [\underline{\underline{{\mathbf{M}}}} ]\underline{\underline{{\mathbf{M}}}}^{ - 1} \cdot \left( {\underline{\underline{{\mathbf{M}}}}^{T} } \right)^{ - 1} \, $$

(3.71)

The induced electric and magnetic fields are given by

$$ \begin{aligned} {\mathbf{D}}\left( {\mathbf{r}} \right) = \det (\underline{\underline{{\mathbf{M}}}} )\underline{\underline{{\mathbf{M}}}}^{ - 1} \cdot \varepsilon_{0} {\mathbf{E}}^{\prime} \left( {{\mathbf{r}}^{\prime} } \right) \hfill \\ {\mathbf{B}}\left( {\mathbf{r}} \right) = \det (\underline{\underline{{\mathbf{M}}}} )\underline{\underline{{\mathbf{M}}}}^{ - 1} \cdot \mu_{0} {\mathbf{H}}^{\prime} \left( {{\mathbf{r}}^{\prime} } \right) \hfill \\ \end{aligned} $$

(3.72)

Being the real space inhomogeneous, rays propagate there along curvilinear paths; these paths are mapped to straight lines in the virtual space. It is convenient to introduce covariant basis ${\mathbf{g}}_{i} = {{\partial {\mathbf{r}}} \mathord{\left/ {\vphantom {{\partial {\mathbf{r}}} {\partial x_{i} '}}} \right. \kern-0pt} {\partial x_{i} '}} = \sum\nolimits_{j} {{{\partial x_{j} } \mathord{\left/ {\vphantom {{\partial x_{j} } {\partial x_{i} '}}} \right. \kern-0pt} {\partial x_{i} '}}{\hat{\mathbf{x}}}_{j} }$ and contravariant basis ${\mathbf{g}}^{i} = \nabla x_{i} ' = \sum\nolimits_{j} {{{\partial x_{i} '} \mathord{\left/ {\vphantom {{\partial x_{i} '} {\partial x_{j} }}} \right. \kern-0pt} {\partial x_{j} }}{\hat{\mathbf{x}}}_{j} }$ of the coordinate transformation ${\varvec{\Upgamma}}$, as well as covariant basis ${\varvec{\gamma}}_{j} = {{\partial {\mathbf{r}}'} \mathord{\left/ {\vphantom {{\partial {\mathbf{r}}'} {\partial x_{j} }}} \right. \kern-0pt} {\partial x_{j} }} = \sum\nolimits_{i} {{{\partial x_{i} '} \mathord{\left/ {\vphantom {{\partial x_{i} '} {\partial x_{j} }}} \right. \kern-0pt} {\partial x_{j} }}{\hat{\mathbf{x}}}_{i} '}$ and contravariant basis ${\varvec{\gamma}}^{j} = \nabla 'x_{j} = \sum\nolimits_{i} {{{\partial x_{j} } \mathord{\left/ {\vphantom {{\partial x_{j} } {\partial x_{i} }}} \right. \kern-0pt} {\partial x_{i} }}'{\hat{\mathbf{x}}}_{i} '}$ of the inverse coordinate transformation ${\varvec{\Upgamma}}^{ - 1}$. Figure 3.22 illustrates an example. Note that ${\mathbf{g}}_{i} ,{\mathbf{g}}^{i}$ and ${\varvec{\gamma}}_{j} ,{\varvec{\gamma}}^{j}$ are defined in the real space and in the transformed space, respectively. The two introduced bases are bi-orthogonal, namely ${\mathbf{g}}^{i} \cdot {\mathbf{g}}_{j} = \delta_{ij} ,{\varvec{\gamma}}^{i} \cdot {\varvec{\gamma}}_{j} = \delta_{ij}$, and

$$ \begin{array}{*{20}c} {{\mathbf{g}}^{1} \times {\mathbf{g}}^{2} = {\mathbf{g}}_{3} /\sqrt g } \\ {{\mathbf{g}}^{2} \times {\mathbf{g}}^{3} = {\mathbf{g}}_{1} /\sqrt g } \\ {{\mathbf{g}}^{3} \times {\mathbf{g}}^{1} = {\mathbf{g}}_{2} /\sqrt g } \\ \end{array} ;\begin{array}{*{20}c} {{\mathbf{g}}_{1} \times {\mathbf{g}}_{2} = {\mathbf{g}}^{3} \sqrt g } \\ {{\mathbf{g}}_{2} \times {\mathbf{g}}_{3} = {\mathbf{g}}^{1} \sqrt g } \\ {{\mathbf{g}}_{3} \times {\mathbf{g}}_{1} = {\mathbf{g}}^{2} \sqrt g } \\ \end{array} $$

(3.73)

where $\det \underline{\underline{{\mathbf{M}}}} = 1/\sqrt g$. The Jacobian tensor, its inverse, and its determinant can be written in terms of covariant and contravariant basis as

$$ \underline{\underline{{\mathbf{M}}}} = \sum\limits_{i = 1,3}^{{}} {{\hat{\mathbf{x}}}_{i} ^{\prime} } {\mathbf{g}}^{i} = \sum\limits_{j = 1,3}^{{}} {{\varvec{\gamma}}_{j} {\hat{\mathbf{x}}}_{j} } $$

(3.74)

$$ \underline{\underline{{\mathbf{M}}}}^{ - 1} = \sum\limits_{i = 1,3}^{{}} {{\mathbf{g}}_{i} {\hat{\mathbf{x}}}_{i} ^{\prime} } = \sum\limits_{j = 1,3}^{{}} {{\hat{\mathbf{x}}}_{j} } {\varvec{\gamma}}^{j} $$

(3.75)

$$ \det \underline{\underline{{\mathbf{M}}}} = \frac{1}{\sqrt g } = {\mathbf{g}}^{3} \cdot \left( {{\mathbf{g}}^{1} \times {\mathbf{g}}^{2} } \right) = \frac{1}{{{\mathbf{g}}_{3} \cdot \left( {{\mathbf{g}}_{1} \times {\mathbf{g}}_{2} } \right)}} = {\varvec{\gamma}}_{3} \cdot \left( {{\varvec{\gamma}}_{1} \times {\varvec{\gamma}}_{2} } \right) = \frac{1}{{{\varvec{\gamma}}^{3} \cdot \left( {{\varvec{\gamma}}^{1} \times {\varvec{\gamma}}^{2} } \right)}} $$

(3.76)

Fig. 3.22

Coordinate planes and coordinate lines in the real space as transformed from the cartesian planes and coordinates in the virtual space and covariant and contravariant bases

Using (3.74) in (3.69), and observing that the transpose operation in (3.69) swaps the order of the vectors in the dyads in (3.74), leads to

$$ {\mathbf{E}}({\mathbf{r}}) = \sum\limits_{i = 1,3}^{{}} {{\mathbf{g}}^{i} } \left[ {{\hat{\mathbf{x}}}_{i} ^{\prime} \cdot {\mathbf{E}}^{\prime} ({\mathbf{r}}^{\prime} )} \right] $$

(3.77)

$$ {\mathbf{H}}({\mathbf{r}}) = \sum\limits_{i = 1,3}^{{}} {{\mathbf{g}}^{i} } \left[ {{\hat{\mathbf{x}}}_{i} ^{\prime} \cdot {\mathbf{H}}^{\prime} ({\mathbf{r}}^{\prime} )} \right] $$

(3.78)

$$ {\mathbf{D}}\left( {\mathbf{r}} \right) = \frac{1}{\sqrt g }\sum\limits_{i = 1,3}^{{}} {{\mathbf{g}}_{i} } \left[ {{\hat{\mathbf{x}}}_{i} ^{\prime} \cdot {\mathbf{D}}^{\prime} \left( {{\mathbf{r}}^{\prime} } \right)} \right] $$

(3.79)

$$ {\mathbf{B}}\left( {\mathbf{r}} \right) = \frac{1}{\sqrt g }\sum\limits_{i = 1,3}^{{}} {{\mathbf{g}}_{i} } \left[ {{\hat{\mathbf{x}}}_{i} ^{\prime} \cdot {\mathbf{B}}^{\prime} \left( {{\mathbf{r}}^{\prime} } \right)} \right] $$

(3.80)

which identifies the electric and magnetic field Cartesian components in the virtual space with the covariant components of the fields in the real space. On the other hand, the Cartesian components of the electric and magnetic inductions in the virtual space represent the contravariant components of the inductions in the real space.

The normalized tensors are indeed expressed through (3.71) and (3.75–3.76) as

$$ \, \underline{\underline{{\varvec{\alpha}}}} = \frac{1}{\sqrt g }\sum\limits_{i,j = 1,3}^{{}} {{\hat{\mathbf{x}}}_{i} } {\hat{\mathbf{x}}}_{j} \left[ {{\varvec{\gamma}}^{i} \, \cdot {\varvec{\gamma}}^{j} } \right] $$

(3.81)

We observe that for orthogonal transformations covariant vectors coincide with contravariant vectors.

The tensor ${\underline{\underline {{\alpha }}} }$ can be represented in the reference system identified by unit vectors ${\hat{\mathbf{e}}}_{i}$ aligned with its principal axes as follows

$$ \mathop {\varvec{\alpha}}\limits_{ = } = \hat{e}_{1} \hat{e}_{1} \sigma_{1} + \hat{e}_{2} \hat{e}_{2} \sigma_{2} + \hat{e}_{3} \hat{e}_{3} \sigma_{3} $$

(3.82)

where $\sigma_{i}$ are the principal values of ${\underline{\underline {{\alpha }}} }$. Note that the unit vectors ${\hat{\mathbf{e}}}_{i} = {\hat{\mathbf{e}}}_{i} ({\mathbf{r}})$ are in general space dependent. The dispersion equation for local plane waves propagating with ${\mathbf{r}}$-dependent wavevector ${\mathbf{k}} = k_{x} {\hat{\mathbf{x}}} + k_{y} {\hat{\mathbf{y}}} + k_{z} {\hat{\mathbf{z}}} = k_{1} {\hat{\mathbf{e}}}_{1} + k_{2} {\hat{\mathbf{e}}}_{2} + k_{3} {\hat{\mathbf{e}}}_{3}$ ($k_{i} = {\mathbf{k}} \cdot {\hat{\mathbf{e}}}_{i}$) can be written as

$$ k_{1}^{2} \sigma_{1} + k_{2}^{2} \sigma_{2} + k_{3}^{2} \sigma_{3} - k^{2} \sigma_{1} \sigma_{2} \sigma_{3} = 0 $$

(3.83)

where $k = \omega \sqrt {\varepsilon_{o} \mu_{o} }$ is the free-space wavenumber. We note that this equation is valid for both TE and TM modes with respect to one of the three axes; degeneration of TE/TM mode phase velocity in TO media was underlined in Ref. [27] with reference to a spherical cloak. For positive values of $\sigma_{i}$, (3.83) represents in the ${\mathbf{k}}$-space an ellipsoid with axes aligned with the principal axes ${\hat{\mathbf{e}}}_{i}$ and semi-axes given by $k\sqrt {\sigma_{2} \sigma_{3} } ,\,k\sqrt {\sigma_{1} \sigma_{3} } ,\,k\sqrt {\sigma_{1} \sigma_{2} }$ along ${\hat{\mathbf{e}}}_{1} ,\,{\hat{\mathbf{e}}}_{2} ,\,{\hat{\mathbf{e}}}_{3}$ , respectively. The coordinate- free expression of this ellipse is ${\mathbf{k}} \cdot \underline{\underline{{\varvec{\alpha}}}} \cdot {\mathbf{k}} = k^{2} \sigma_{1} \sigma_{2} \sigma_{3} .$ (Fig. 3.23)

Fig. 3.23

Dispersion ellipse for a given constitutive parameter tensor

Appendix B -Dispersion Equation for an Anisotropic Tensor

To solve (3.41) in terms of $\sqrt {{\mathbf{k}}_{t} \cdot {\mathbf{k}}_{t} }$, we project $\underline{\underline{{\mathbf{X}}}}_{S}^{eq}$ along the basis ${\hat{\mathbf{k}}}_{t}$, ${\hat{\mathbf{z}}} \times {\hat{\mathbf{k}}}_{t} = {\hat{\mathbf{k}}}_{t}^{ \bot }$, i.e.,

$$ \begin{aligned} \underline{\underline{{\mathbf{X}}}}_{S}^{eq} & = X^{ee} {\hat{\mathbf{k}}}_{t} {\hat{\mathbf{k}}}_{t} + X^{hh} {\hat{\mathbf{k}}}_{t}^{ \bot } {\hat{\mathbf{k}}}_{t}^{ \bot } + X^{eh} \left( {{\hat{\mathbf{k}}}_{t}^{{}} {\hat{\mathbf{k}}}_{t}^{ \bot } + {\hat{\mathbf{k}}}_{t}^{ \bot } {\hat{\mathbf{k}}}} \right)_{t}^{{}} \hfill \\ X^{ee} & = {\hat{\mathbf{k}}}_{t} \cdot (X_{1} {\hat{\mathbf{e}}}_{1} {\hat{\mathbf{e}}}_{1} + X_{2} {\hat{\mathbf{e}}}_{2} {\hat{\mathbf{e}}}_{2} ) \cdot {\hat{\mathbf{k}}}_{t} = \cos^{2} (\psi - \phi )X_{1} + \sin^{2} (\psi - \phi )X_{2} \hfill \\ X^{hh} & = {\hat{\mathbf{k}}}_{t}^{ \bot } \cdot (X_{1} {\hat{\mathbf{e}}}_{1} {\hat{\mathbf{e}}}_{1} + X_{2} {\hat{\mathbf{e}}}_{2} {\hat{\mathbf{e}}}_{2} ) \cdot {\hat{\mathbf{k}}}_{t}^{ \bot } = \sin^{2} (\psi - \phi )X_{1} + \cos^{2} (\psi - \phi )X_{2} \hfill \\ X^{eh} & = {\hat{\mathbf{k}}}_{t}^{{}} \cdot (X_{1} {\hat{\mathbf{e}}}_{1} {\hat{\mathbf{e}}}_{1} + X_{2} {\hat{\mathbf{e}}}_{2} {\hat{\mathbf{e}}}_{2} ) \cdot {\hat{\mathbf{k}}}_{t}^{ \bot } = \sin (\psi - \phi )\cos (\psi - \phi )(X_{1} - X_{2} ) \hfill \\ \end{aligned} $$

(3.84)

where $\cos \psi = {\hat{\mathbf{e}}}_{1} \cdot {\hat{\mathbf{x}}},\;\sin \psi = {\hat{\mathbf{e}}}_{1} \cdot {\hat{\mathbf{y}}} ; {\text{ cos}}\phi = {\hat{\mathbf{k}}}_{t} \cdot {\hat{\mathbf{x}}},\;\sin \phi = {\hat{\mathbf{k}}}_{t} \cdot {\hat{\mathbf{y}}}$. An explicit expression of $\psi$ is given in (3.22). The dispersion (3.41) can be therefore rewritten as

$$ \det \left[ {\underline{\underline{{\mathbf{X}}}}_{S}^{eq} + \underline{\underline{{\mathbf{X}}}}_{0} } \right] = \left( {X^{ee} + X_{0}^{TM} } \right)\left( {X^{hh} + X_{0}^{TE} } \right) - (X^{eh} )^{2} = 0 $$

(3.85)

The latter equation can be simplified by exploiting the identities $X_{0}^{TE} X_{0}^{TM} = - \zeta^{2}$ and $X_{{}}^{ee} X_{{}}^{hh} - (X_{{}}^{eh} )^{2} = \det \underline{\underline{{\mathbf{X}}}}_{S}^{eq} = X_{1} X_{2}$. This leads to

$$ X^{ee} X_{0}^{TE} + X^{hh} X_{0}^{TM} + X_{1} X_{2} - \zeta^{2} = 0 $$

(3.86)

Solving (3.86) in terms of $\eta = \sqrt {{\mathbf{k}}_{t} \cdot {\mathbf{k}}_{t} /k^{2} - 1} = - X_{TM} /\zeta = \zeta /X_{TE}$ provides

$$ \sqrt {{\mathbf{k}}_{t} \cdot {\mathbf{k}}_{t} /k^{2} - 1} = \frac{1}{{2\zeta X_{hh}^{{}} }}\left\{ {\left( {X_{1}^{{}} X_{2}^{{}} - \zeta^{2} } \right) + \sqrt {\left( {X_{1}^{{}} X_{2}^{{}} - \zeta^{2} } \right)^{2} + 4X_{ee}^{{}} X_{hh}^{{}} \zeta^{2} } } \right\} $$

(3.87)

The above equation can be approximated by

$$ \frac{{{\mathbf{k}}_{t} \cdot {\mathbf{k}}_{t} }}{{k^{2} }} \approx \left( {\frac{{\cos^{2} (\psi - \phi )}}{{1 + \left( {X_{1} /\zeta } \right)^{2} }} + \frac{{\sin^{2} (\psi - \phi )}}{{1 + \left( {X_{2} /\zeta } \right)^{2} }}} \right)^{ - 1} $$

(3.88)

which is an ellipse with the main axes aligned with the eigenvectors ${\hat{\mathbf{e}}}_{1} ,\,{\hat{\mathbf{e}}}_{2}$ and semi-axes equal to $\frac{{k_{i} }}{k} \approx \sqrt {1 + \left( {X_{i} /\zeta } \right)^{2} }$. The maximum deviation of the approximation in (3.88) from (3.87) is exactly given by

$$ \left| {\frac{{{\mathbf{k}}_{t} \cdot {\mathbf{k}}_{t} }}{{k^{2} }}\left[ {\frac{{\cos^{2} (\psi - \phi )}}{{1 + \left( {X_{1} /\zeta } \right)^{2} }} + \frac{{\sin^{2} (\psi - \phi )}}{{1 + \left( {X_{2} /\zeta } \right)^{2} }}} \right] - 1} \right|_{MAX\;\phi } = \left| {f\left( {\frac{{X_{1} }}{\zeta },\frac{{X_{2} }}{\zeta }} \right)} \right| $$

(3.89)

with

$$ f\left( {\frac{{X_{1} }}{\zeta },\frac{{X_{2} }}{\zeta }} \right) = \frac{{\left( {\frac{{X_{1} }}{\zeta } - \frac{{X_{2} }}{\zeta }} \right)^{2} - \left( {\sqrt {\frac{{X_{1} }}{{X_{2} }}} - \sqrt {\frac{{X_{2} }}{{X_{1} }}} } \right)^{2} }}{{4\left( {\frac{{X_{1} }}{\zeta } + \frac{\zeta }{{X_{1} }}} \right)\left( {\frac{{X_{2} }}{\zeta } + \frac{\zeta }{{X_{2} }}} \right)}} $$

(3.90)

This relation has been obtained analytically by imposing a vanishing derivative of the error function with respect to $\psi - \phi$.