# Casimir energy of $N$ magnetodielectric $\delta$-function plates

Venkat Abhignan  Department of Physics, National Institute of Technology,
Tiruchirapalli - 620015, India

###### Abstract

To investigate Casimir electromagnetic interaction in $N$ bodies, we implement
multiple $\delta$-function plates with electric and magnetic properties. We
use their optical properties to study the Casimir energy between the plates by
implementing multiple scattering formalism. We initially solve Green’s
functions for two and three plates configurations to obtain their reflection
coefficients. Further, the coefficients are implemented in multiple scattering
formalism, and a simple method was obtained to depict energy density
distribution in the multiple scattering expansions using diagrammatic loops.
The idea of deriving Casimir energy was extended to $N$ plates by identifying
a systematic pattern.

##  I Introduction

Quantum electrodynamic fluctuations in a vacuum confined between two parallel
perfectly electrically conducting plates were shown to have a force between
them [1]. This force was later generalized for two dielectric slabs [2]. Even
though exact results have been obtained for perfectly conducting boundaries,
it is generally accepted that calculating Casimir energies and their
corresponding forces is a difficult process for dielectric bodies due to the
non-additivity property of the interaction.

Casimir energies can be described in terms of reflection coefficients of the
configurations, for example, using multiple scattering formalism. Using the
optical Fresnel coefficients, Casimir force for parallel multilayered
configurations has been pursued [3, 4]. The central idea revolves around using
the recurrence relations among Fresnel coefficients of neighbourhood media in
an iterative manner and extending the result to $N$ bodies. Primarily, the
Casimir force acting on a particular planar region is calculated in studies of
all the multilayered systems. However, no closed-form expression was solved
for $N$ bodies to our understanding. Beyond this, without using recursive
relations, the Casimir energy for $N$ dielectric cavities separated by plasma
sheets was recently attempted by Allocca et al. [5]. They studied Barton’s
model [6] for plasma sheets to obtain interaction energy for two, three
dielectric cavities, and further inductively, the result was extended to $N$
dielectric cavities.

In this context, we are interested in studying the Casimir interaction of
discrete $N$ plates in a vacuum with magnetodielectric properties. We utilise
$\delta$-function plates to obtain Green’s functions and deduce Casimir energy
using multiple scattering formalism [7, 8, 9, 10, 11]. The physical relevance
of $\delta$-function plates has been discussed previously, and the most
interesting aspect is their nontrivial boundary conditions [12, 13]. We
initially solved for Green’s functions for $N=1,2,3$ $\delta$-function plates
configurations. We extract the reflection coefficients for these
configurations by representing the different regions of the Greens function in
a matrix representation. We obtain Casimir energies for $N=2,3,4,5$
$\delta$-function plates using these reflection coefficients and their
corresponding multiple scattering parameters ($\Delta_{12\cdots N}$’s) from
the multiple scattering formalism. Interpreting these multiple scattering
parameters for $N=3,4,5$ $\delta$-function plates in a diagrammatic manner,
the pattern for $N$ plates seems to be exhibiting.

We theorise that Casimir energy $\Delta E_{(12\cdots N)}$ of $N$ plates
configuration can be represented using the multiple scattering parameter
$\Delta_{12\cdots N}$ such as

$\frac{\Delta E_{(12\cdots
N)}}{A}=\frac{1}{2}\int_{-\infty}^{\infty}\frac{d\zeta}{2\pi}\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\Bigg{[}\ln\Big{[}\Delta_{12\cdots
N}^{H}\Big{]}+\ln\Big{[}\Delta_{12\cdots N}^{E}\Big{]}\Bigg{]},$ (1)

where the integration is over all frequencies and wavenumbers. The parameter
$\Delta_{12\cdots N}$ can be distributed into nearest neighbour scattering
parameter $\Delta_{ij}$ for all $j=i+1$ ($i\in[1,N-1]$ where $i$ and $j$ are
adjacent plates) and next-to-nearest neighbour, next-to-next-to-nearest
neighbour, $\cdots$ scattering parameter $\Delta_{ik}$ for all $k\geq i+2$
($i\in[1,N-2]$ where $i$ and $k$ are not adjacent plates). These parameters
give the different possible ways the paths of propagation contribute to the
energy. They can be obtained in terms of the optical properties of the plates,
reflection coefficients $r_{i}$, transmission coefficients $t_{i}$ and with
exponential dependence on the distance between the plates $l_{ij}$ described
diagrammatically by loops (Expressions derived in Sec. v@). For instance, the
multiple scattering parameter for $N=3$ plates configuration can be obtained
such as

$\Delta_{123}=\Delta_{12}\Delta_{23}+\Delta_{13}$ (2)

and diagrammatic loop distribution of this multiple scattering parameter can
be visualized using Fig. 1. Similarly, for $N=4$ plates configuration the
multiple scattering parameter can be obtained as

$\Delta_{1234}=\Delta_{12}\Delta_{23}\Delta_{34}+\Delta_{12}\Delta_{24}+\Delta_{13}\Delta_{34}+\Delta_{14}$
(3)

and the diagrammatic loop distribution is visualized in Fig. 2.

$l_{12}$$l_{23}$$r_{1}$$r_{2}$$t_{2}$$r_{3}$$\Delta_{123}$$\Delta_{12}\Delta_{23}$$\Delta_{13}$
Figure 1: Visualizing $\Delta_{123}$ with optical properties, $r_{i}$ are
reflection coefficients and $t_{i}$ are transmission coefficients of the
plates. $l_{ij}$ is the distance between the plates where $i$ and $j$ are
adjacent plates.
$l_{12}$$l_{23}$$l_{34}$$r_{1}$$r_{2}$$t_{2}$$r_{3}$$t_{3}$$r_{4}$$\Delta_{1234}$$\Delta_{12}\Delta_{23}\Delta_{34}$$\Delta_{12}\Delta_{24}$$\Delta_{13}\Delta_{34}$$\Delta_{14}$
Figure 2: Visualizing $\Delta_{1234}$ with optical properties, $r_{i}$ are
reflection coefficients and $t_{i}$ are transmission coefficients of the
plates. $l_{ij}$ is the distance between the plates where $i$ and $j$ are
adjacent plates.

In Sec.ii@, we introduce the necessary formalism to handle Maxwell’s equations
in a medium with electric and magnetic properties. Further, electric and
magnetic Green’s functions are defined, representing the TE and TM modes. In
Sec. iii@, Green’s functions are solved for a system of one, two and three
$\delta$-function plates configurations with electric and magnetic properties
in a vacuum. In Sec. iv@, Casimir force is derived between two and three
$\delta$-plates using the stress tensor method in terms of their optical
properties. In Sec.v@, we derive the Casimir energy between $N=2,3,4,5$
$\delta$-plates and generalize it to $N$ plates using multiple scattering
formalism.

##  II Maxwell’s equations with TE and TM modes

The standard Maxwell’s equations for the monochromatic components of electric
and magnetic fields $\mathbf{E(r;\omega)}$ and $\mathbf{H(r;\omega)}$ in
absence of currents and charges, proportional to $\exp{(-\text{i}\omega t)}$
are

$\displaystyle\nabla\times\mathbf{E}=\text{i}\omega\mathbf{B},\ \ \ \ \ \ \ \
\ $ (4a) $\displaystyle-\nabla\times\mathbf{H}=\text{i}\omega\mathbf{(D+P)}.$
(4b)

These equations imply that $\nabla\cdot\mathbf{B}=0$,
$\nabla\cdot(\mathbf{D}+\mathbf{P})=0$ and here $\mathbf{P}$ is the source of
external polarization independent of the polarization in the materials due to
$\mathbf{E}$ and $\mathbf{H}$. Further here, non-linear responses are
neglected in the presence of electric and magnetic materials with boundaries,
and macroscopic fields $\mathbf{D}$ and $\mathbf{B}$ respond linearly to
$\mathbf{E}$ and $\mathbf{H}$ as

$\displaystyle\mathbf{D(r;\omega)}=\bm{\varepsilon}\mathbf{(r;\omega)}\cdot\mathbf{E(r;\omega)},$
(5a)
$\displaystyle\mathbf{B(r;\omega)}=\bm{\mu}\mathbf{(r;\omega)}\cdot\mathbf{H(r;\omega)}.$
(5b)

$\bm{\varepsilon}$ is dielectric permittivity and $\bm{\mu}$ magnetic
permeability of the materials. The Maxwell’s equations decouple and can be
combined to obtain second order differential equations for electric field

$\left[-{\bm{\varepsilon}}({\bf
r};\omega)+\frac{1}{\omega^{2}}{\bm{\nabla}}\times{\bm{\mu}({\bf
r};\omega)}^{-1}\cdot{\bm{\nabla}}\times\right]\cdot{\bf E}({\bf
r},\omega)={\bf P}({\bf r},\omega),$ (6)

and magnetic field in a similar manner. These correspond to transverse
electric (TE) and transverse magnetic (TM) modes rewritten in the form of
Green’s dyadics $\mathbf{\Gamma}$ and $\mathbf{\Phi}$ [14, 12] such as

$\left[-{\bm{\varepsilon}}({\bf
r};\omega)+\frac{1}{\omega^{2}}{\bm{\nabla}}\times{\bm{\mu}({\bf
r};\omega)}^{-1}\cdot{\bm{\nabla}}\times\right]\cdot{\bm{\Gamma}}({\bf r},{\bf
r}^{\prime};\omega)={\bf 1}\delta^{(3)}({\bf r}-{\bf r}^{\prime}),$ (7)

by correlating the the field and source as

$\mathbf{E(r;\omega)}=\int\hbox{d}^{3}r^{\prime}\mathbf{\Gamma(r,r^{\prime},\omega)}\cdot\mathbf{P(r^{\prime},\omega)}.$
(8)

Green’s dyadics $\mathbf{\Gamma}$ and $\mathbf{\Phi}$ can be expressed in form
of scalar Green’s function $g^{E}$ and $g^{H}$, respectively. The scalar
components $\gamma^{ij}$ of dyadic $\bm{\gamma}$ in Fourier space can be
derived and represented as a matrix such as

$\bm{\gamma}(z,z^{\prime};\mathbf{k_{\perp}},\omega)=\left[\begin{array}[]{ccc}\frac{1}{\varepsilon^{\perp}(z)}\frac{\partial}{\partial
z}\frac{1}{\varepsilon^{\perp}(z^{\prime})}\frac{\partial}{\partial
z^{\prime}}g^{H}(z,z^{\prime})&0&\frac{1}{\varepsilon^{\perp}(z)}\frac{\partial}{\partial
z}\frac{ik_{\perp}}{\varepsilon^{||}(z^{\prime})}g^{H}(z,z^{\prime})\\\\[5.69054pt]
0&\omega^{2}g^{E}(z,z^{\prime})&0\\\\[5.69054pt]
-\frac{ik_{\perp}}{\varepsilon^{||}(z)}\frac{1}{\varepsilon^{\perp}(z^{\prime})}\frac{\partial}{\partial
z^{\prime}}g^{H}(z,z^{\prime})&0&-\frac{ik_{\perp}}{\varepsilon^{||}(z)}\frac{ik_{\perp}}{\varepsilon^{||}(z^{\prime})}g^{H}(z,z^{\prime})\end{array}\right]-\delta(z-z^{\prime})\left[\begin{array}[]{llr}\frac{1}{\varepsilon^{\perp}(z)}&0&0\\\
0&0&0\\\ 0&0&\frac{1}{\varepsilon^{||}(z)}\end{array}\right],$ (9)

from using a two-point correlation function

${\bm{\Gamma}}({\bf r},{\bf
r}^{\prime};\omega)=\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\,e^{i{\bf
k}_{\perp}\cdot({\bf r}-{\bf
r}^{\prime})_{\perp}}{\bm{\gamma}}(z,z^{\prime};{\bf k}_{\perp},\omega),$ (10)

where $\bm{\varepsilon}(z)=\varepsilon^{\perp}(z)\,{\bf
1}_{\perp}+\varepsilon^{||}(z)\,\hat{\bf z}\,\hat{\bf z}$ and
$\bm{\mu}(z)=\mu^{\perp}(z)\,{\bf 1}_{\perp}+\mu^{||}(z)\,\hat{\bf
z}\,\hat{\bf z}.$ Electric Green’s function $g^{E}$ referring to TE mode and
$g^{H}$ magnetic Green’s function referring to TM mode can be defined to
satisfy the differential equations such as

$\left[-\frac{\partial}{\partial
z}\frac{1}{\mu^{\perp}(z)}\frac{\partial}{\partial
z}+\frac{k_{\perp}^{2}}{\mu^{||}(z)}-\omega^{2}\varepsilon^{\perp}(z)\right]g^{E}(z,z^{\prime})=\delta(z-z^{\prime})$
(11)

and

$\left[-\frac{\partial}{\partial
z}\frac{1}{\varepsilon^{\perp}(z)}\frac{\partial}{\partial
z}+\frac{k_{\perp}^{2}}{\varepsilon^{||}(z)}-\omega^{2}\mu^{\perp}(z)\right]g^{H}(z,z^{\prime})=\delta(z-z^{\prime}),$
(12)

respectively. As a direct consequence of Eq. (8), the correlations can be
interpreted as [15]

$\frac{\delta{\bf E}({\bf r};\omega)}{\delta{\bf P}({\bf
r}^{\prime};\omega)}={\bm{\Gamma}}({\bf r},{\bf r}^{\prime};\omega).$ (13)

The electric and magnetic field at two distinct points in space can also be
correlated using the Green’s dyadic in form of Green’s functions, and these
correlations are expressed as

$\displaystyle\frac{1}{\tau}\langle{\bf E}({\bf r};-\omega){\bf E}({\bf
r}^{\prime};\omega)\rangle$
$\displaystyle=\frac{1}{\hbox{i}}{\bm{\Gamma}}({\bf r},{\bf
r}^{\prime};\omega),$ (14a) $\displaystyle\frac{1}{\tau}\langle{\bf H}({\bf
r};-\omega){\bf H}({\bf r}^{\prime};\omega)\rangle$
$\displaystyle=\frac{1}{\hbox{i}}{\bm{\Gamma}}({\bf r},{\bf
r}^{\prime};\omega)\Big{|}_{E\leftrightarrow
H,\varepsilon\leftrightarrow\mu},$ (14b)

where $\tau$ is the system’s average observed time.

##  III $\delta$-function plates

In our analysis the interest is in thin $\delta$-function plates [12, 16] with
electric and magnetic properties

$\displaystyle\bm{\varepsilon}(z)=\mathbf{1}+\bm{\lambda}_{ei}(\omega)\delta(z-a_{i}),$
(15a)
$\displaystyle\bm{\mu}(z)=\mathbf{1}+\bm{\lambda}_{gi}(\omega)\delta(z-a_{i}),$
(15b)

at positions $z=a_{i}$ in vacuum up to an infinite extent in $x-y$ axis. Since
we consider disjoint objects the latter term in dyadic $\bm{\gamma}$ in Eq.
(9) containing $\delta(z-z^{\prime})$ can be neglected in this case, since it
does not contribute. The material properties of $\delta$-function plates, with
planar symmetry (homogeneous and isotropic on the plane) are described by the
matrix

$\bm{\lambda}(\omega)=\left[\begin{array}[]{ccc}\lambda^{\perp}(\omega)&0&0\\\
0&\lambda^{\perp}(\omega)&0\\\ 0&0&\lambda^{\parallel}(\omega)\\\
\end{array}\right].$ (16)

The boundary conditions satisfying the planar interface at $z=a_{i}$ are
derived from implementing an Amperian loop integral and a Gaussian surface
integral across the plate. Additional nontrivial contributions to the electric
field boundary conditions are obtained due to $\delta$-functions such as

$\displaystyle\left.E_{2}(\mathbf{k_{\perp}},z;\omega)\right|^{z=a+\delta}_{z=a-\delta}$
$\displaystyle=-\text{i}\omega\lambda^{\perp}_{gi}H_{1}(\mathbf{k_{\perp}},a_{i}),$
(17a)
$\displaystyle\left.E_{1}(\mathbf{k_{\perp}},z;\omega)\right|^{z=a+\delta}_{z=a-\delta}$
$\displaystyle=\text{i}\omega\lambda^{\perp}_{gi}H_{2}(\mathbf{k_{\perp}},a_{i}),$
(17b)
$\displaystyle\left.E_{3}(\mathbf{k_{\perp}},z;\omega)\right|^{z=a+\delta}_{z=a-\delta}$
$\displaystyle=-\text{i}k_{\perp}\lambda^{\perp}_{ei}E_{1}(\mathbf{k_{\perp}},a_{i}).$
(17c)

Similarly, the magnetic field boundary conditions are derived such as

$\displaystyle\left.H_{2}(\mathbf{k_{\perp}},z;\omega)\right|^{z=a+\delta}_{z=a-\delta}$
$\displaystyle=\text{i}\omega\lambda^{\perp}_{ei}E_{1}(\mathbf{k_{\perp}},a_{i}),$
(18a)
$\displaystyle\left.H_{1}(\mathbf{k_{\perp}},z;\omega)\right|^{z=a+\delta}_{z=a-\delta}$
$\displaystyle=-\text{i}\omega\lambda^{\perp}_{ei}E_{2}(\mathbf{k_{\perp}},a_{i}),$
(18b)
$\displaystyle\left.H_{3}(\mathbf{k_{\perp}},z;\omega)\right|^{z=a+\delta}_{z=a-\delta}$
$\displaystyle=-\text{i}k_{\perp}\lambda^{\perp}_{gi}H_{1}(\mathbf{k_{\perp}},a_{i}).$
(18c)

Setting $\lambda=0$ would reduce these equations to the traditional well-known
boundary conditions for electric and magnetic field in vacuum. The presence of
$\delta$-plates with magnetoelectric properties modifies these boundary
conditions on the interface making them nontrivial. And $\delta$-function
plates have boundary constraints

$\displaystyle\lambda^{\parallel}_{ei}E_{3}(\mathbf{k_{\perp}},a_{i};\omega)=0,$
(19a)
$\displaystyle\lambda^{\parallel}_{gi}H_{3}(\mathbf{k_{\perp}},a_{i};\omega)=0.$
(19b)

These reveal that $\lambda^{\parallel}_{ei}=0$ and
$\lambda^{\parallel}_{gi}=0$ unless $E_{3}(\mathbf{k_{\perp}},a_{i};\omega)=0$
and $H_{3}(\mathbf{k_{\perp}},a_{i};\omega)=0$ at $z=a_{i}$. These boundary
conditions on electric and magnetic field govern the boundary conditions on TE
mode and TM mode from Eq.(11) and (12), which in turn dictate the boundary
conditions on the electric Green’s function $g^{E}$ and magnetic Green’s
function $g^{H}$. The boundary conditions on $g^{E}$ are

$\displaystyle g^{E}(z,z^{\prime})\Big{|}^{z=a_{i}+\delta}_{z=a_{i}-\delta}$
$\displaystyle=$
$\displaystyle\frac{\lambda^{\perp}_{gi}}{2}\left[\left\\{\frac{\partial}{\partial
z}g^{E}(z,z^{\prime})\right\\}_{z=a_{i}+\delta}+\left\\{\frac{\partial}{\partial
z}g^{E}(z,z^{\prime})\right\\}_{z=a_{i}-\delta}\right],$ (20a)
$\displaystyle\frac{\partial}{\partial
z}g^{E}(z,z^{\prime})\bigg{|}^{z=a_{i}+\delta}_{z=a_{i}-\delta}$
$\displaystyle=$
$\displaystyle\zeta^{2}\frac{\lambda^{\perp}_{ei}}{2}\big{[}g^{E}(a_{i}+\delta,z^{\prime})+g^{E}(a_{i}-\delta,z^{\prime})\big{]}.$
(20b)

Similarly, the boundary conditions on $g^{H}$ are

$\displaystyle g^{H}(z,z^{\prime})\Big{|}^{z=a_{i}+\delta}_{z=a_{i}-\delta}$
$\displaystyle=$
$\displaystyle\frac{\lambda^{\perp}_{ei}}{2}\left[\left\\{\frac{\partial}{\partial
z}g^{H}(z,z^{\prime})\right\\}_{z=a_{i}+\delta}+\left\\{\frac{\partial}{\partial
z}g^{H}(z,z^{\prime})\right\\}_{z=a_{i}-\delta}\right],$ (21a)
$\displaystyle\frac{\partial}{\partial
z}g^{H}(z,z^{\prime})\bigg{|}^{z=a_{i}+\delta}_{z=a_{i}-\delta}$
$\displaystyle=$
$\displaystyle\zeta^{2}\frac{\lambda^{\perp}_{gi}}{2}\big{[}g^{H}(a_{i}+\delta,z^{\prime})+g^{H}(a_{i}-\delta,z^{\prime})\big{]},$
(21b)

in vacuum where $\omega\rightarrow\text{i}\zeta$ switching to imaginary
frequencies using Euclidean rotation.

$z=z^{\prime}$11212212$z=a_{1}$$z^{\prime}=a_{1}$ (a) Labels for the Green’s
function of one plate ($\lambda^{\perp}_{e1},\lambda^{\perp}_{g1}$ at
$z=a_{1}$).

333111133221222312$z=a_{1}$$z=a_{2}$$z^{\prime}=a_{2}$$z^{\prime}=a_{1}$$z=z^{\prime}$
(b) Labels for the Green’s function of two plates
($\lambda^{\perp}_{e1},\lambda^{\perp}_{g1}$ at $z=a_{1}$ and
$\lambda^{\perp}_{e2},\lambda^{\perp}_{g2}$ at $z=a_{2}$).

Figure 3:  Different labels for the regions in the $z-z^{\prime}$ space for
the Green’s functions of thin magnetodielectric plates.
33311113422441322122231214434434$z=a_{1}$$z=a_{2}$$z=a_{3}$$z^{\prime}=a_{3}$$z^{\prime}=a_{2}$$z^{\prime}=a_{1}$$z=z^{\prime}$
Figure 4: Labels for the Green’s function of three plates
($\lambda^{\perp}_{e1},\lambda^{\perp}_{g1}$ at $z=a_{1}$,
$\lambda^{\perp}_{e2},\lambda^{\perp}_{g2}$ at $z=a_{2}$ and
$\lambda^{\perp}_{e3},\lambda^{\perp}_{g3}$ at $z=a_{3}$).

The magnetic Green’s functions $g^{1,H}$, $g^{2,H}$ and $g^{3,H}$ are solved
using its differential Eq.(12) and boundary conditions in Eq.(21) for the
configuration of one plate with optical properties
$\lambda^{\perp}_{e1},\lambda^{\perp}_{g1}$ at $z=a_{1}$, two plates with
optical properties $\lambda^{\perp}_{e1},\lambda^{\perp}_{g1}$ at $z=a_{1}$,
$\lambda^{\perp}_{e2},\lambda^{\perp}_{g2}$ at $z=a_{2}$ and three plates with
with optical properties $\lambda^{\perp}_{e1},\lambda^{\perp}_{g1}$ at
$z=a_{1}$, $\lambda^{\perp}_{e2},\lambda^{\perp}_{g2}$ at $z=a_{2}$,
$\lambda^{\perp}_{e3},\lambda^{\perp}_{g3}$ at $z=a_{3}$, respectively.
Illustratively, the solution for $g^{H}$ for these configurations in different
regions of $z-z^{\prime}$ space is shown in Fig. (3(a)), (3(b)) and (4). The
solution for $g^{1,H}$ of one plate with the subscript label in Fig. (3(a))
representing the different region in $z-z^{\prime}$ space is

$\displaystyle
g^{1,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{11}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{1}^{H}}{2\kappa}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{1})},$
$\displaystyle z<a_{1}<z^{\prime}$ (22a) $\displaystyle
g^{1,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{12}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{1}{2\kappa}\hbox{e}^{-\kappa|z-z^{\prime}|}+\frac{r_{1}^{H}}{2\kappa}\hbox{e}^{-\kappa(z-a_{1})}\hbox{e}^{-\kappa(z^{\prime}-a_{1})},$
$\displaystyle z,z^{\prime}>a_{1}$ (22b)

and

$\displaystyle
g^{1,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{21}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{1}{2\kappa}\hbox{e}^{-\kappa|z-z^{\prime}|}+\frac{r_{1}^{H}}{2\kappa}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(a_{1}-z^{\prime})},$
$\displaystyle z,z^{\prime}<a_{1},$ (23a) $\displaystyle
g^{1,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{22}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{1}^{H}}{2\kappa}\hbox{e}^{-\kappa(z-a_{1})}\hbox{e}^{-\kappa(a_{1}-z^{\prime})},$
$\displaystyle z^{\prime}<a_{1}<z.$ (23b)

Here, the reflection and transmission coefficients at plate are

$\displaystyle
r_{i}^{H}=-\frac{\lambda^{\perp}_{gi}\zeta^{2}}{\lambda^{\perp}_{gi}\zeta^{2}+2\kappa}+\frac{\lambda^{\perp}_{ei}\kappa}{\lambda^{\perp}_{ei}\kappa+2},\
t_{i}^{H}=1-\frac{\lambda^{\perp}_{gi}\zeta^{2}}{\lambda^{\perp}_{gi}\zeta^{2}+2\kappa}-\frac{\lambda^{\perp}_{ei}\kappa}{\lambda^{\perp}_{ei}\kappa+2}$
(24)

for $i=1$ and $\kappa=\sqrt{k_{\perp}^{2}+\zeta^{2}}$. Perhaps, a better
representation of this Green’s function $g^{1,H}$ is from construction of
matrices $A$, $B^{H}$ and $C$ in relation with regions in Fig. 1a such as

$A=\left[\begin{array}[]{c
c}\hbox{e}^{-\kappa(z^{\prime}-a_{1})}&\hbox{e}^{-\kappa(a_{1}-z^{\prime})}\end{array}\right],B^{H}=\left[\begin{array}[]{c
c}t_{1}^{H}&r_{1}^{H}\\\
r_{1}^{H}&t_{1}^{H}\end{array}\right]\,\hbox{and}\,C=\left[\begin{array}[]{c
c}\hbox{e}^{-\kappa(a_{1}-z)}\\\
\hbox{e}^{-\kappa(z-a_{1})}\end{array}\right].$ (25)

$B^{H}_{ij}$ represents the element of a matrix $B^{H}$ in $i^{th}$ row,
$j^{th}$ column and the regions in Green’s function can be obtained from

$g^{N,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{ij}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}=\begin{cases}\frac{1}{2\kappa}\hbox{e}^{-\kappa|z-z^{\prime}|}+\frac{1}{2\kappa}A_{i}B^{H}_{ij}C_{j}&\text{if}\hskip
5.69054pti+j-2=N,\\\ \frac{1}{2\kappa}A_{i}B^{H}_{ij}C_{j}&\text{if}\hskip
5.69054pti+j-2\neq N,\end{cases}$ (26)

for $i=1,\cdots,N+1$, $j=1,\cdots,N+1$ and $N$ denotes the number of plates.
This representation is helpful in understanding the propagation of multiple
paths possible in different regions of $z-z^{\prime}$ space as can be seen
further in Green’s function of two and three plates. When concerned with one
plate Green’s function the physical interpretation of reflection and
transmission coefficients $r_{1},t_{1}$ can be explained by considering a
point of source and path of propagation. When the point of observation,
$z<z^{\prime}$ the point of source, the Green’s function can be regrouped as

$\displaystyle a_{1}<z<z^{\prime},\hskip 11.38109ptg^{1,H}$
$\displaystyle=\left(\hbox{e}^{-\kappa(a_{1}-z)}+r_{1}^{H}\hbox{e}^{-\kappa(z-a_{1})}\right)\frac{\hbox{e}^{-\kappa(z^{\prime}-a_{1})}}{2\kappa},$
(27a) $\displaystyle z<z^{\prime}<a_{1},\hskip 11.38109ptg^{1,H}$
$\displaystyle=\left(t_{1}^{H}\hbox{e}^{-\kappa(a_{1}-z)}\right)\frac{\hbox{e}^{-\kappa(z^{\prime}-a_{1})}}{2\kappa}.$
(27b)

The terms outside the parenthesis refer to the source and terms in parenthesis
give reflection and transmission amplitudes of the propagator by
$R_{1}^{<,H}\equiv r_{1}^{H}$ and $T_{1}^{<,H}\equiv t_{1}^{H}$ (Superscript
$<$ denoting $z<z^{\prime}$ and subscript $1$ in $R,T$ indicating one plate).
Similarly, when $z>z^{\prime}$ the Green’s function across the planar
interface is

$\displaystyle a_{1}>z>z^{\prime},\hskip 11.38109ptg^{1,H}$
$\displaystyle=\left(\hbox{e}^{-\kappa(z-a_{1})}+r_{1}^{H}\hbox{e}^{-\kappa(a_{1}-z)}\right)\frac{\hbox{e}^{-\kappa(a_{1}-z^{\prime})}}{2\kappa},$
(28a) $\displaystyle z>z^{\prime}>a_{1},\hskip 11.38109ptg^{1,H}$
$\displaystyle=\left(t_{1}^{H}\hbox{e}^{-\kappa(z-a_{1})}\right)\frac{\hbox{e}^{-\kappa(a_{1}-z^{\prime})}}{2\kappa}.$
(28b)

These give the reflection and transmission amplitudes as $R_{1}^{>,H}\equiv
r_{1}^{H}$ and $T_{1}^{>,H}\equiv t_{1}^{H}$. These reflection coefficients
can also be obtained from matrix $B^{H}$, where $R_{1}^{<,H}=B^{H}_{12}$ and
$R_{1}^{>,H}=B^{H}_{21}$. And transmission coefficients can be obtained from
$B^{H}_{11}$ and $B^{H}_{22}$. Transmission amplitudes $T^{<,H}=T^{>,H}$ for
all configurations, but $R_{1}^{<,H}=R_{1}^{>,H}$ only in the case of one
plate which differ for the case of two plate and three plate reflection
amplitudes $R_{12}^{H},R_{123}^{H}$, as can be seen further.
Similarly, the solution for $g^{2,H}$ of two plates with the subscript label
in Fig.(3(b)) representing the different region in $z-z^{\prime}$ space is:

$\displaystyle
g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{11}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{1}^{H}\ \hbox{e}^{-\kappa a}\
t_{2}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{2})},$
$\displaystyle z<a_{1}<a_{2}<z^{\prime},$ (29a) $\displaystyle
g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{12}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{2}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(a_{2}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{2})}+\frac{r_{1}^{H}\
\hbox{e}^{-\kappa a}\
t_{2}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(z-a_{1})}\hbox{e}^{-\kappa(z^{\prime}-a_{2})},$
$\displaystyle a_{1}<z<a_{2}<z^{\prime},$ (29b) $\displaystyle
g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{13}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{1}{2\kappa}\hbox{e}^{-\kappa|z-z^{\prime}|}+\frac{r_{2}^{H}}{2\kappa}\hbox{e}^{-\kappa(z-a_{2})}\hbox{e}^{-\kappa(z^{\prime}-a_{2})}+\frac{t_{2}^{H}\
\hbox{e}^{-\kappa a}\ r_{1}^{H}\ \hbox{e}^{-\kappa a}\
t_{2}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(z-a_{2})}\hbox{e}^{-\kappa(z^{\prime}-a_{2})},$
$\displaystyle a_{1}<a_{2}<z,z^{\prime}$ (29c)

and

$\displaystyle
g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{21}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{1}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{1})}+\frac{t_{1}^{H}\
\hbox{e}^{-\kappa a}\
r_{2}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(a_{2}-z^{\prime})},$
$\displaystyle z<a_{1}<z^{\prime}<a_{2},$ (30a) $\displaystyle
g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{22}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{1}{2\kappa}\hbox{e}^{-\kappa|z-z^{\prime}|}+\frac{r_{1}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(z-a_{1})}\hbox{e}^{-\kappa(z^{\prime}-a_{1})}+\frac{r_{1}^{H}\
\hbox{e}^{-\kappa a}\
r_{2}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(z-a_{1})}\hbox{e}^{-\kappa(a_{2}-z^{\prime})}$
$\displaystyle\hskip
11.38109pt+\frac{r_{2}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(a_{2}-z)}\hbox{e}^{-\kappa(a_{2}-z^{\prime})}+\frac{r_{1}^{H}\
\hbox{e}^{-\kappa a}\
r_{2}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(a_{2}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{1})},$
$\displaystyle a_{1}<z^{\prime},z<a_{2},$ (30b) $\displaystyle
g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{23}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{2}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(z-a_{2})}\hbox{e}^{-\kappa(a_{2}-z^{\prime})}+\frac{r_{1}^{H}\
\hbox{e}^{-\kappa a}\
t_{2}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(z-a_{2})}\hbox{e}^{-\kappa(z^{\prime}-a_{1})},$
$\displaystyle a_{1}<z^{\prime}<a_{2}<z$ (30c)

and

$\displaystyle
g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{31}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{1}{2\kappa}\hbox{e}^{-\kappa|z-z^{\prime}|}+\frac{r_{1}^{H}}{2\kappa}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(a_{1}-z^{\prime})}+\frac{t_{1}^{H}\
\hbox{e}^{-\kappa a}\ r_{2}^{H}\ \hbox{e}^{-\kappa a}\
t_{1}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(a_{1}-z^{\prime})},$
$\displaystyle z,z^{\prime}<a_{1}<a_{2},$ (31a) $\displaystyle
g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{32}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{1}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(z-a_{1})}\hbox{e}^{-\kappa(a_{1}-z^{\prime})}+\frac{r_{2}^{H}\
\hbox{e}^{-\kappa a}\
t_{1}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(a_{2}-z)}\hbox{e}^{-\kappa(a_{1}-z^{\prime})},$
$\displaystyle z^{\prime}<a_{1}<z<a_{2},$ (31b) $\displaystyle
g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{33}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{2}^{H}\ \hbox{e}^{-\kappa a}\
t_{1}^{H}}{2\kappa\Delta_{12}^{H}}\hbox{e}^{-\kappa(z-a_{2})}\hbox{e}^{-\kappa(a_{1}-z^{\prime})},$
$\displaystyle z^{\prime}<a_{1}<a_{2}<z.$ (31c)

The reflection and transmission coefficients at each plate are defined in Eq.
(24) for $i=1,2$. Multiple scattering parameter here is

$\Delta_{12}^{H}=1-r_{1}^{H}\hbox{e}^{-\kappa a}r_{2}^{H}\hbox{e}^{-\kappa
a},$ (32)

with $a=a_{2}-a_{1}$ is the distance between the plates. Similar to matrices
in Eq. (25), this Green’s function $g^{2,H}$ can be represented by
constructing matrices $A$, $B^{H}$ and $C$ such as

$A=\left[\begin{array}[]{c c
c}\hbox{e}^{-\kappa(z^{\prime}-a_{2})}&\left[\begin{array}[]{c
c}\hbox{e}^{-\kappa(z^{\prime}-a_{1})}&\hbox{e}^{-\kappa(a_{2}-z^{\prime})}\end{array}\right]&\hbox{e}^{-\kappa(a_{1}-z^{\prime})}\end{array}\right],\\\
B^{H}=\left[\begin{array}[]{c c c}\frac{t_{1}^{H}\hbox{e}^{-\kappa
a}t_{2}^{H}}{\Delta_{12}^{H}}&\left[\begin{array}[]{c
c}\frac{t_{2}^{H}}{\Delta_{12}^{H}}&\hskip
4.2679pt\frac{r_{1}^{H}\hbox{e}^{-\kappa
a}t_{2}^{H}}{\Delta_{12}^{H}}\end{array}\right]&r_{2}^{H}+\frac{t_{2}^{H}\hbox{e}^{-\kappa
a}r_{1}^{H}\hbox{e}^{-\kappa a}t_{2}^{H}}{\Delta_{12}^{H}}\vspace{3mm}\\\
\left[\begin{array}[]{c c}\frac{t_{1}^{H}}{\Delta_{12}^{H}}\vspace{3mm}\\\
\vspace{3mm}\frac{t_{1}^{H}\hbox{e}^{-\kappa
a}r_{2}^{H}}{\Delta_{12}^{H}}\end{array}\right]&\left[\begin{array}[]{c
c}\frac{r_{1}^{H}\hbox{e}^{-\kappa
a}r_{2}^{H}}{\Delta_{12}^{H}}&\frac{r_{1}^{H}}{\Delta_{12}^{H}}\vspace{3mm}\\\
\vspace{3mm}\frac{r_{2}^{H}}{\Delta_{12}^{H}}&\frac{r_{2}^{H}\hbox{e}^{-\kappa
a}r_{1}^{H}}{\Delta_{12}^{H}}\end{array}\right]&\left[\begin{array}[]{c
c}\frac{r_{1}^{H}\hbox{e}^{-\kappa
a}t_{2}^{H}}{\Delta_{12}^{H}}\vspace{3mm}\\\
\vspace{3mm}\frac{t_{2}^{H}}{\Delta_{12}^{H}}\end{array}\right]\vspace{3mm}\\\
\vspace{3mm}r_{1}^{H}+\frac{t_{1}^{H}\hbox{e}^{-\kappa
a}r_{2}^{H}\hbox{e}^{-\kappa
a}t_{1}^{H}}{\Delta_{12}^{H}}&\left[\begin{array}[]{c
c}\frac{r_{2}^{H}\hbox{e}^{-\kappa
a}t_{1}^{H}}{\Delta_{12}^{H}}&\frac{t_{1}^{H}}{\Delta_{12}^{H}}\end{array}\right]&\frac{t_{2}^{H}\hbox{e}^{-\kappa
a}t_{1}^{H}}{\Delta_{12}^{H}}\end{array}\right]\\\
\hbox{and}\,C=\left[\begin{array}[]{c c c}\hbox{e}^{-\kappa(a_{1}-z)}\\\
\left[\begin{array}[]{c c}\hbox{e}^{-\kappa(a_{2}-z)}\\\
\hbox{e}^{-\kappa(z-a_{1})}\end{array}\right]\\\
\hbox{e}^{-\kappa(z-a_{2})}\end{array}\right].$ (33)

Different regions in Green’s function
$g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{ij}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}$
can be obtained here from Eq. (26) for $N=2$. The reflection coefficients of
two plates configuration can be obtained here from $R_{12}^{<,H}=B^{H}_{13}$
and $R_{12}^{>,H}=B^{H}_{31}$. And transmission coefficients of two plates
configuration are $T_{12}^{<,H}=T_{12}^{>,H}=B^{H}_{11}=B^{H}_{33}$. The
remaining terms describe the multiple possible ways for the path of
propagation depending on $z$ and $z^{\prime}$ with an exponential dependence
on the length of propagation. The reflection coefficients are not the same
when $z<z^{\prime}$ and $z>z^{\prime}$, but the path of propagation defined in
the terms $B^{H}_{11}$, $B^{H}_{12}$ and $B^{H}_{21}$ are same as
$B^{H}_{33}$, $B^{H}_{23}$ and $B^{H}_{32}$, respectively where the matrix is
symmetric around the anti-diagonal terms ${B^{H}_{ij}}$ for $i+j-2=N$.
The solution for $g^{3,H}$ of three plates with the subscript label in Fig.
(4) representing the different region in $z-z^{\prime}$ space is:

$\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{11}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{1}^{H}\ \hbox{e}^{-\kappa a}\ t_{2}^{H}\
\hbox{e}^{-\kappa b}\
t_{3}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{3})},\hskip
5.69054ptz<a_{1}<a_{2}<a_{3}<z^{\prime},$ (34a) $\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{12}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{2}^{H}\ \hbox{e}^{-\kappa b}\
t_{3}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{2}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{3})}+\frac{r_{1}^{H}\
\hbox{e}^{-\kappa a}\ t_{2}^{H}\ \hbox{e}^{-\kappa b}\
t_{3}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{1})}\hbox{e}^{-\kappa(z^{\prime}-a_{3})},\hskip
5.69054pta_{1}<z<a_{2}<a_{3}<z^{\prime},$ (34b) $\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{13}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{3}^{H}\
\Delta_{12}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{3}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{3})}+\frac{r_{2}^{H}\
\hbox{e}^{-\kappa b}\ t_{3}^{H}\
\Delta_{12}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{2})}\hbox{e}^{-\kappa(z^{\prime}-a_{3})}$
$\displaystyle\hskip 11.38109pt+\frac{t_{2}^{H}\ \hbox{e}^{-\kappa a}\
r_{1}^{H}\ \hbox{e}^{-\kappa a}\ t_{2}^{H}\ \hbox{e}^{-\kappa b}\
t_{3}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{2})}\hbox{e}^{-\kappa(z^{\prime}-a_{3})},\hskip
5.69054pta_{1}<a_{2}<z<a_{3}<z^{\prime},$ (34c) $\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{14}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{1}{2\kappa}\hbox{e}^{-\kappa|z-z^{\prime}|}+\frac{r_{3}^{H}}{2\kappa}\hbox{e}^{-\kappa(z-a_{3})}\hbox{e}^{-\kappa(z^{\prime}-a_{3})}+\frac{t_{3}^{H}\
\hbox{e}^{-\kappa b}\ r_{2}^{H}\ \hbox{e}^{-\kappa b}\ t_{3}^{H}\
\Delta_{12}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{3})}\hbox{e}^{-\kappa(z^{\prime}-a_{3})}$
$\displaystyle\hskip 11.38109pt+\frac{t_{3}^{H}\ \hbox{e}^{-\kappa b}\
t_{2}^{H}\ \hbox{e}^{-\kappa a}\ r_{1}^{H}\ \hbox{e}^{-\kappa a}\ t_{2}^{H}\
\hbox{e}^{-\kappa b}\
t_{3}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{3})}\hbox{e}^{-\kappa(z^{\prime}-a_{3})},\hskip
5.69054pta_{1}<a_{2}<a_{3}<z,z^{\prime}.$ (34d)

and

$\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{21}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{1}^{H}\ \hbox{e}^{-\kappa a}\
t_{2}^{H}}{2\kappa\Delta^{H}}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{2})}+\frac{t_{1}^{H}\
\hbox{e}^{-\kappa a}\ t_{2}^{H}\ \hbox{e}^{-\kappa b}\
r_{3}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(a_{3}-z^{\prime})},\hskip
5.69054ptz<a_{1}<a_{2}<z^{\prime}<a_{3},$ (35a) $\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{22}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{2}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{2}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{2})}+\frac{t_{2}^{H}\
\hbox{e}^{-\kappa b}\
r_{3}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{2}-z)}\hbox{e}^{-\kappa(a_{3}-z^{\prime})}$
$\displaystyle\hskip 11.38109pt+\frac{r_{1}^{H}\ \hbox{e}^{-\kappa a}\
t_{2}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{1})}\hbox{e}^{-\kappa(z^{\prime}-a_{2})}+\frac{r_{1}^{H}\
\hbox{e}^{-\kappa a}\ t_{2}^{H}\ \hbox{e}^{-\kappa b}\
r_{3}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{1})}\hbox{e}^{-\kappa(a_{3}-z^{\prime})},\hskip
5.69054pta_{1}<z<a_{2}<z^{\prime}<a_{3},$ (35b) $\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{23}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{1}{2\kappa}\hbox{e}^{-\kappa|z-z^{\prime}|}+\frac{r_{3}^{H}\
\Delta_{12}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{3}-z)}\hbox{e}^{-\kappa(a_{3}-z^{\prime})}$
$\displaystyle\hskip 11.38109pt+\frac{r_{3}^{H}\ \hbox{e}^{-\kappa b}\
r_{2}^{H}\Delta_{12}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{3}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{2})}+\frac{r_{3}^{H}\
\hbox{e}^{-\kappa b}\ t_{2}^{H}\ \hbox{e}^{-\kappa a}\ r_{1}^{H}\
\hbox{e}^{-\kappa a}\
t_{2}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{3}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{2})}$
$\displaystyle\hskip 11.38109pt+\frac{t_{2}^{H}\ \hbox{e}^{-\kappa a}\
r_{1}^{H}\ \hbox{e}^{-\kappa a}\ t_{2}^{H}\ \hbox{e}^{-\kappa b}\
r_{3}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{2})}\hbox{e}^{-\kappa(a_{3}-z^{\prime})}+\frac{r_{2}^{H}\
\hbox{e}^{-\kappa b}\ r_{3}^{H}\
\Delta_{12}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{2})}\hbox{e}^{-\kappa(a_{3}-z^{\prime})}$
$\displaystyle\hskip 11.38109pt+\frac{r_{2}^{H}\
\Delta_{12}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{2})}\hbox{e}^{-\kappa(z^{\prime}-a_{2})}+\frac{t_{2}^{H}\
\hbox{e}^{-\kappa a}\ r_{1}^{H}\ \hbox{e}^{-\kappa a}\
t_{2}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{2})}\hbox{e}^{-\kappa(z^{\prime}-a_{2})},\hskip
5.69054pta_{1}<a_{2}<z,z^{\prime}<a_{3},$ (35c) $\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{24}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{3}^{H}\
\Delta_{12}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{3})}\hbox{e}^{-\kappa(a_{3}-z^{\prime})}+\frac{t_{3}^{H}\
\hbox{e}^{-\kappa b}\ r_{2}^{H}\
\Delta_{12}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{3})}\hbox{e}^{-\kappa(z^{\prime}-a_{2})}$
$\displaystyle\hskip 11.38109pt+\frac{t_{3}^{H}\ \hbox{e}^{-\kappa b}\
t_{2}^{H}\ \hbox{e}^{-\kappa a}\ r_{1}^{H}\ \hbox{e}^{-\kappa a}\
t_{2}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{3})}\hbox{e}^{-\kappa(z^{\prime}-a_{2})},\hskip
5.69054pta_{1}<a_{2}<z^{\prime}<a_{3}<z,$ (35d)

and

$\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{31}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{1}^{H}\
\Delta_{23}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{1})}+\frac{t_{1}^{H}\
\hbox{e}^{-\kappa a}\ r_{2}^{H}\
\Delta_{23}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(a_{2}-z^{\prime})}$
$\displaystyle\hskip 11.38109pt+\frac{t_{1}^{H}\ \hbox{e}^{-\kappa a}\
t_{2}^{H}\ \hbox{e}^{-\kappa b}\ r_{3}^{H}\ \hbox{e}^{-\kappa b}\
t_{2}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(a_{2}-z^{\prime})},\hskip
5.69054ptz<a_{1}<z^{\prime}<a_{2}<a_{3},$ (36a) $\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{32}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{1}{2\kappa}\hbox{e}^{-\kappa|z-z^{\prime}|}+\frac{r_{1}^{H}\
\Delta_{23}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{1})}\hbox{e}^{-\kappa(z^{\prime}-a_{1})}$
$\displaystyle\hskip 11.38109pt+\frac{r_{1}^{H}\ \hbox{e}^{-\kappa a}\
r_{2}^{H}\Delta_{23}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{1})}\hbox{e}^{-\kappa(a_{2}-z^{\prime})}+\frac{r_{1}^{H}\
\hbox{e}^{-\kappa a}\ t_{2}^{H}\ \hbox{e}^{-\kappa b}\ r_{3}^{H}\
\hbox{e}^{-\kappa b}\
t_{2}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{1})}\hbox{e}^{-\kappa(a_{2}-z^{\prime})}$
$\displaystyle\hskip 11.38109pt+\frac{t_{2}^{H}\ \hbox{e}^{-\kappa b}\
r_{3}^{H}\ \hbox{e}^{-\kappa b}\ t_{2}^{H}\ \hbox{e}^{-\kappa a}\
r_{1}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{2}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{1})}+\frac{r_{2}^{H}\
\hbox{e}^{-\kappa a}\ r_{1}^{H}\
\Delta_{23}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{2}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{1})}$
$\displaystyle\hskip 11.38109pt+\frac{r_{2}^{H}\
\Delta_{23}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{2}-z)}\hbox{e}^{-\kappa(a_{2}-z^{\prime})}+\frac{t_{2}^{H}\
\hbox{e}^{-\kappa b}\ r_{3}^{H}\ \hbox{e}^{-\kappa b}\
t_{2}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{2}-z)}\hbox{e}^{-\kappa(a_{2}-z^{\prime})},\hskip
5.69054pta_{1}<z,z^{\prime}<a_{2}<a_{3},$ (36b) $\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{33}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{2}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{2})}\hbox{e}^{-\kappa(a_{2}-z^{\prime})}+\frac{t_{2}^{H}\
\hbox{e}^{-\kappa a}\
r_{1}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{2})}\hbox{e}^{-\kappa(z^{\prime}-a_{1})}$
$\displaystyle\hskip 11.38109pt+\frac{r_{3}^{H}\ \hbox{e}^{-\kappa b}\
t_{2}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{3}-z)}\hbox{e}^{-\kappa(a_{2}-z^{\prime})}+\frac{r_{3}^{H}\
\hbox{e}^{-\kappa b}\ t_{2}^{H}\ \hbox{e}^{-\kappa a}\
r_{1}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{3}-z)}\hbox{e}^{-\kappa(z^{\prime}-a_{1})},\hskip
5.69054pta_{1}<z^{\prime}<a_{2}<z<a_{3},$ (36c) $\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{34}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{3}^{H}\ \hbox{e}^{-\kappa b}\
t_{2}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{3})}\hbox{e}^{-\kappa(a_{2}-z^{\prime})}+\frac{t_{3}^{H}\
\hbox{e}^{-\kappa b}\ t_{2}^{H}\ \hbox{e}^{-\kappa a}\
r_{1}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{3})}\hbox{e}^{-\kappa(z^{\prime}-a_{1})},\hskip
5.69054pta_{1}<z^{\prime}<a_{2}<a_{3}<z,$ (36d)

and

$\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{41}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{1}{2\kappa}\hbox{e}^{-\kappa|z-z^{\prime}|}+\frac{r_{1}^{H}}{2\kappa}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(a_{1}-z^{\prime})}+\frac{t_{1}^{H}\
\hbox{e}^{-\kappa a}\ r_{2}^{H}\ \hbox{e}^{-\kappa a}\ t_{1}^{H}\
\Delta_{23}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(a_{1}-z^{\prime})}$
$\displaystyle\hskip 11.38109pt+\frac{t_{1}^{H}\ \hbox{e}^{-\kappa a}\
t_{2}^{H}\ \hbox{e}^{-\kappa b}\ r_{3}^{H}\ \hbox{e}^{-\kappa b}\ t_{2}^{H}\
\hbox{e}^{-\kappa a}\
t_{1}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{1}-z)}\hbox{e}^{-\kappa(a_{1}-z^{\prime})},\hskip
5.69054ptz,z^{\prime}<a_{1}<a_{2}<a_{3},$ (37a) $\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{42}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{1}^{H}\
\Delta_{23}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{1})}\hbox{e}^{-\kappa(a_{1}-z^{\prime})}+\frac{r_{2}^{H}\
\hbox{e}^{-\kappa a}\ t_{1}^{H}\
\Delta_{23}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{2}-z)}\hbox{e}^{-\kappa(a_{1}-z^{\prime})}$
$\displaystyle\hskip 11.38109pt+\frac{t_{2}^{H}\ \hbox{e}^{-\kappa b}\
r_{3}^{H}\ \hbox{e}^{-\kappa b}\ t_{2}^{H}\ \hbox{e}^{-\kappa a}\
t_{1}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{2}-z)}\hbox{e}^{-\kappa(a_{1}-z^{\prime})},\hskip
5.69054ptz^{\prime}<a_{1}<z<a_{2}<a_{3},$ (37b) $\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{43}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{2}^{H}\ \hbox{e}^{-\kappa a}\
t_{1}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{2})}\hbox{e}^{-\kappa(a_{1}-z^{\prime})}+\frac{r_{3}^{H}\
\hbox{e}^{-\kappa b}\ t_{2}^{H}\ \hbox{e}^{-\kappa a}\
t_{1}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(a_{3}-z)}\hbox{e}^{-\kappa(a_{1}-z^{\prime})},\hskip
5.69054ptz^{\prime}<a_{1}<a_{2}<z<a_{3},$ (37c) $\displaystyle
g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{44}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(z,z^{\prime})$
$\displaystyle=\frac{t_{3}^{H}\ \hbox{e}^{-\kappa b}\ t_{2}^{H}\
\hbox{e}^{-\kappa a}\
t_{1}^{H}}{2\kappa\Delta_{123}^{H}}\hbox{e}^{-\kappa(z-a_{3})}\hbox{e}^{-\kappa(a_{1}-z^{\prime})},\hskip
5.69054ptz^{\prime}<a_{1}<a_{2}<a_{3}<z.$ (37d)

Matrices $A$, $B^{H}$ and $C$ can be defined for this cumbersome expression of
Green’s function as

$A=\left[\begin{array}[]{c c c
c}\hbox{e}^{-\kappa(z^{\prime}-a_{3})}&\left[\begin{array}[]{c
c}\hbox{e}^{-\kappa(z^{\prime}-a_{2})}&\hbox{e}^{-\kappa(a_{3}-z^{\prime})}\end{array}\right]&\left[\begin{array}[]{c
c}\hbox{e}^{-\kappa(z^{\prime}-a_{1})}&\hbox{e}^{-\kappa(a_{2}-z^{\prime})}\end{array}\right]&\hbox{e}^{-\kappa(a_{1}-z^{\prime})}\end{array}\right],\\\
B^{H}=\frac{1}{\Delta_{123}^{H}}\left[\begin{array}[]{cccc}\begin{subarray}{c}t_{1}^{H}\hbox{e}^{-\kappa
a}t_{2}^{H}\hbox{e}^{-\kappa
b}t_{3}^{H}\end{subarray}&B^{\prime}_{12}&B^{\prime}_{13}&\begin{subarray}{c}r_{3}^{H}\Delta_{123}^{H}+t_{3}^{H}\hbox{e}^{-\kappa
b}r_{2}^{H}\hbox{e}^{-\kappa b}t_{3}^{H}\Delta_{12}^{H}\\\
+t_{3}^{H}\hbox{e}^{-\kappa b}t_{2}^{H}\hbox{e}^{-\kappa
a}r_{1}^{H}\hbox{e}^{-\kappa a}t_{2}^{H}\hbox{e}^{-\kappa
b}t_{3}^{H}\end{subarray}\\\
B^{\prime}_{21}&B^{\prime}_{22}&B^{\prime}_{23}&B^{\prime}_{24}\\\
B^{\prime}_{31}&B^{\prime}_{32}&B^{\prime}_{33}&B^{\prime}_{34}\\\
\begin{subarray}{c}r_{1}^{H}\Delta_{123}^{H}+t_{1}^{H}\hbox{e}^{-\kappa
a}r_{2}^{H}\hbox{e}^{-\kappa a}t_{1}^{H}\Delta_{23}^{H}\\\
+t_{1}^{H}\hbox{e}^{-\kappa a}t_{2}^{H}\hbox{e}^{-\kappa
b}r_{3}^{H}\hbox{e}^{-\kappa b}t_{2}^{H}\hbox{e}^{-\kappa
a}t_{1}^{H}\end{subarray}&B^{\prime}_{42}&B^{\prime}_{43}&\begin{subarray}{c}t_{3}^{H}\hbox{e}^{-\kappa
b}t_{2}^{H}\hbox{e}^{-\kappa a}t_{1}^{H}\end{subarray}\end{array}\right]\\\
\hbox{and}\,C=\left[\begin{array}[]{c c c c}\hbox{e}^{-\kappa(a_{1}-z)}\\\
\left[\begin{array}[]{c c}\hbox{e}^{-\kappa(a_{2}-z)}\\\
\hbox{e}^{-\kappa(z-a_{1})}\end{array}\right]\vspace{3mm}\\\
\vspace{3mm}\left[\begin{array}[]{c c}\hbox{e}^{-\kappa(a_{3}-z)}\\\
\hbox{e}^{-\kappa(z-a_{2})}\end{array}\right]\\\
\hbox{e}^{-\kappa(z-a_{3})}\end{array}\right]$ (38)

with

$B^{\prime}_{12}=\left[\begin{array}[]{c
c}\begin{subarray}{c}t_{2}^{H}\hbox{e}^{-\kappa
b}t_{3}^{H}\end{subarray}&\hskip
8.53581pt\begin{subarray}{c}r_{1}^{H}\hbox{e}^{-\kappa
a}t_{2}^{H}\hbox{e}^{-\kappa b}\
t_{3}^{H}\end{subarray}\end{array}\right],\,B^{\prime}_{34}=\left[\begin{array}[]{c
c}\begin{subarray}{c}t_{3}^{H}\hbox{e}^{-\kappa b}t_{2}^{H}\hbox{e}^{-\kappa
a}r_{1}^{H}\end{subarray}\\\ \begin{subarray}{c}t_{3}^{H}\hbox{e}^{-\kappa
b}t_{2}^{H}\end{subarray}\end{array}\right],$ (39)
$B^{\prime}_{13}=\left[\begin{array}[]{c
c}\begin{subarray}{c}t_{3}^{H}\Delta_{12}^{H}\end{subarray}&\hskip
8.53581pt\begin{subarray}{c}r_{2}^{H}\hbox{e}^{-\kappa
b}t_{3}^{H}\Delta_{12}^{H}+t_{2}^{H}\ \hbox{e}^{-\kappa
a}r_{1}^{H}\hbox{e}^{-\kappa a}t_{2}^{H}\hbox{e}^{-\kappa
b}t_{3}^{H}\end{subarray}\end{array}\right],\,B^{\prime}_{24}=\left[\begin{array}[]{c
c}\begin{subarray}{c}t_{3}^{H}\hbox{e}^{-\kappa
b}r_{2}^{H}\Delta_{12}^{H}+t_{3}^{H}\hbox{e}^{-\kappa
b}t_{2}^{H}\hbox{e}^{-\kappa a}r_{1}^{H}\hbox{e}^{-\kappa
a}t_{2}^{H}\end{subarray}\\\
\begin{subarray}{c}t_{3}^{H}\Delta_{12}^{H}\end{subarray}\end{array}\right],$
(40) $B^{\prime}_{21}=\left[\begin{array}[]{c
c}\begin{subarray}{c}t_{1}^{H}\hbox{e}^{-\kappa a}t_{2}^{H}\end{subarray}\\\
\begin{subarray}{c}t_{1}^{H}\hbox{e}^{-\kappa a}t_{2}^{H}\hbox{e}^{-\kappa
b}r_{3}^{H}\end{subarray}\end{array}\right],\,B^{\prime}_{43}=\left[\begin{array}[]{c
c}\begin{subarray}{c}r_{3}^{H}\hbox{e}^{-\kappa b}t_{2}^{H}\hbox{e}^{-\kappa
a}t_{1}^{H}\end{subarray}&\hskip
8.53581pt\begin{subarray}{c}t_{2}^{H}\hbox{e}^{-\kappa
a}t_{1}^{H}\end{subarray}\end{array}\right],$ (41)
$B^{\prime}_{22}=\left[\begin{array}[]{c
c}\begin{subarray}{c}t_{2}^{H}\end{subarray}&\begin{subarray}{c}r_{1}^{H}\hbox{e}^{-\kappa
a}t_{2}^{H}\end{subarray}\\\ \begin{subarray}{c}t_{2}^{H}\hbox{e}^{-\kappa
b}r_{3}^{H}\end{subarray}&\begin{subarray}{c}r_{1}^{H}\hbox{e}^{-\kappa
a}t_{2}^{H}\hbox{e}^{-\kappa
b}r_{3}^{H}\end{subarray}\end{array}\right],\,B^{\prime}_{33}=\left[\begin{array}[]{c
c}\begin{subarray}{c}r_{3}^{H}\hbox{e}^{-\kappa b}t_{2}^{H}\hbox{e}^{-\kappa
a}r_{1}^{H}\end{subarray}&\begin{subarray}{c}t_{2}^{H}\hbox{e}^{-\kappa
a}r_{1}^{H}\end{subarray}\\\ \begin{subarray}{c}r_{3}^{H}\hbox{e}^{-\kappa
b}t_{2}^{H}\end{subarray}&\begin{subarray}{c}t_{2}^{H}\end{subarray}\end{array}\right],$
(42) $B^{\prime}_{31}=\left[\begin{array}[]{c
c}\begin{subarray}{c}t_{1}^{H}\Delta_{23}^{H}\end{subarray}\\\
\begin{subarray}{c}t_{1}^{H}\hbox{e}^{-\kappa
a}r_{2}^{H}\Delta_{23}^{H}+t_{1}^{H}\hbox{e}^{-\kappa
a}t_{2}^{H}\hbox{e}^{-\kappa b}r_{3}^{H}\hbox{e}^{-\kappa
b}t_{2}^{H}\end{subarray}\end{array}\right],\,B^{\prime}_{42}=\left[\begin{array}[]{c
c}\begin{subarray}{c}t_{1}^{H}\Delta_{23}^{H}\end{subarray}&\hskip
8.53581pt\begin{subarray}{c}r_{2}^{H}\hbox{e}^{-\kappa
a}t_{1}^{H}\Delta_{23}^{H}+t_{2}^{H}\hbox{e}^{-\kappa
b}r_{3}^{H}\hbox{e}^{-\kappa b}t_{2}^{H}\hbox{e}^{-\kappa
a}t_{1}^{H}\end{subarray}\end{array}\right],$ (43)
$B^{\prime}_{23}=\left[\begin{array}[]{c
c}\begin{subarray}{c}r_{3}^{H}\hbox{e}^{-\kappa
b}r_{2}^{H}\Delta_{12}^{H}+r_{3}^{H}\hbox{e}^{-\kappa
b}t_{2}^{H}\hbox{e}^{-\kappa a}r_{1}^{H}\hbox{e}^{-\kappa
a}t_{2}^{H}\end{subarray}&\hskip
8.53581pt\begin{subarray}{c}r_{2}^{H}\Delta_{12}^{H}+t_{2}^{H}\hbox{e}^{-\kappa
a}r_{1}^{H}\hbox{e}^{-\kappa a}t_{2}^{H}\end{subarray}\\\
\begin{subarray}{c}r_{3}^{H}\Delta_{12}^{H}\end{subarray}&\begin{subarray}{c}t_{2}^{H}\hbox{e}^{-\kappa
a}r_{1}^{H}\hbox{e}^{-\kappa a}t_{2}^{H}\hbox{e}^{-\kappa
b}r_{3}^{H}+r_{2}^{H}\hbox{e}^{-\kappa
b}r_{3}^{H}\Delta_{12}^{H}\end{subarray}\end{array}\right]$ (44)

and

$B^{\prime}_{32}=\left[\begin{array}[]{c
c}\begin{subarray}{c}t_{2}^{H}\hbox{e}^{-\kappa b}r_{3}^{H}\hbox{e}^{-\kappa
b}t_{2}^{H}\hbox{e}^{-\kappa a}r_{1}^{H}+r_{2}^{H}\hbox{e}^{-\kappa
a}r_{1}^{H}\Delta_{23}^{H}\end{subarray}&\begin{subarray}{c}r_{1}^{H}\Delta_{23}^{H}\end{subarray}\\\
\begin{subarray}{c}r_{2}^{H}\Delta_{23}^{H}+t_{2}^{H}\hbox{e}^{-\kappa
b}r_{3}^{H}\hbox{e}^{-\kappa
b}t_{2}^{H}\end{subarray}&\begin{subarray}{c}r_{1}^{H}\hbox{e}^{-\kappa
a}r_{2}^{H}\Delta_{23}^{H}+r_{1}^{H}\hbox{e}^{-\kappa
a}t_{2}^{H}\hbox{e}^{-\kappa b}r_{3}^{H}\hbox{e}^{-\kappa
b}t_{2}^{H}\end{subarray}\end{array}\right].$ (45)

Different regions in Green’s function
$g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{ij}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}$
can be obtained here from Eq. (26) for $N=3$. Further, multiple scattering
parameters here are

$\Delta_{123}^{H}=(\Delta_{12}^{H}\Delta_{23}^{H}-r_{1}^{H}\hbox{e}^{-\kappa
a}t_{2}^{H}\hbox{e}^{-\kappa b}r_{3}^{H}\hbox{e}^{-\kappa
b}t_{2}^{H}\hbox{e}^{-\kappa
a})\,\hbox{and}\,\Delta_{23}^{H}=1-r_{2}^{H}\hbox{e}^{-\kappa
b}r_{3}^{H}\hbox{e}^{-\kappa b},$ (46)

with $b=a_{3}-a_{2}$ is the distance between the plates. The relevance of
these two plates multiple scattering parameters $\Delta_{12}^{H}$,
$\Delta_{23}^{H}$ and three plates multiple scattering parameter
$\Delta_{123}^{H}$ can be seen further in evaluation of energy (Sec. v@),
where they give all paths of propagation contributing to the energy. The
reflection coefficients of three plates configuration can be obtained here
from $R_{123}^{<,H}=B^{H}_{14}$ and $R_{123}^{>,H}=B^{H}_{41}$. And
transmission coefficients of three plates configuration here are
$T_{123}^{<,H}=T_{123}^{>,H}=B^{H}_{11}=B^{H}_{44}$.
Similarly, $g^{E}$ can be obtained for these configurations from solving
differential equation in Eq. (11) and using boundary conditions in Eq. (20).
However, $g^{E}$ have similar expressions as $g^{H}$, obtained by replacing
superscripts $H\rightarrow E$ and by swapping
$\lambda^{\perp}_{ei}\leftrightarrow\lambda^{\perp}_{gi}$ as can be deduced
from the boundary conditions.

##  IV CASIMIR FORCE FROM STRESS TENSOR METHOD

Here, we are interested in calculating the force on a plate in two plates and
three plates configurations from stress tensor formalism that is based on
conservation of electromagnetic momentum [17]. This is calculated using their
Green’s functions solved in previous section and relating them with their
Green’s dyadic from Eq. (9). From the Maxwell’s equations in Eq. (4) one can
derive the force density ${\bf f}$ across the surface of a volume $V$, which
measures the flux of momentum density of the electromagnetic field when
integrated over volume $V$ from implementing the Gauss law. This stress from
electromagnetic radiation on the surface of volume $V$ is given as

${\bf f}=-{\bm{\nabla}}\cdot{\bf T}.$ (47)

Here $\bf T$ is the stress tensor constituting the force of a plate at
$z=a_{i}$,

${\bf T}={\bf 1}\frac{1}{2}({\bf D}\cdot{\bf E}+{\bf B}\cdot{\bf H})-({\bf
D}{\bf E}+{\bf B}{\bf H})$ (48)

and integration volume $V$ can be represented by a thin film with surfaces
$z=a_{i}-\delta$ and $z=a_{i}+\delta$ enclosing the plate, by taking the limit
$\delta\rightarrow 0$. The force on the surface of plate is given as

${\bf
F}=-\frac{1}{\tau}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\oint_{V}d{\bf
S}\cdot{\bf T}({\bf r},\omega),$ (49)

for $\tau=2T\to\infty$. The fluctuations of quantum vacuum dictate that mean
value of the field $\langle{\bf E}\rangle=0$ do not contribute and only
bilinear description of fields in Eq. (14) contribute to the flux tensor. The
Casimir force, a manifestation of quantum vacuum from flux tensor is obtained
as

${\bf
F}=-\frac{1}{2T}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\oint_{V}d{\bf
S}\cdot\langle{\bf T}({\bf r},\omega)\rangle.$ (50)

The Casimir pressure or force per unit area on the plate at $z=a_{i}$ is

$P=\frac{{\bf F}\cdot\hat{\bf
z}}{A}=\frac{1}{\text{i}}\int_{-\infty}^{\infty}\frac{d\zeta}{2\pi}\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\left[T_{33}(a_{i}+\delta)-T_{33}(a_{i}-\delta)\right],$
(51)

where

$T_{33}(z)=\frac{1}{2}\Big{[}E_{1}^{2}+E_{2}^{2}-E_{3}^{2}\Big{]}\bigg{|}_{z}+\frac{1}{2}\Big{[}H_{1}^{2}+H_{2}^{2}-H_{3}^{2}\Big{]}\bigg{|}_{z}$
(52)

from Eq. (48). Using this expression initially we are interested in measuring
the force between two plates $\bf F^{\text{2-$\delta$}}$ (Superscript
2-$\delta$ refers to force on two plates), using the Green’s function of two
plates in regions of Fig. 1(b). The force on plate at $z=a_{2}$ is derived
from

$P^{\text{2-{\hbox{\delta}}}}|_{z=a_{2}}=\frac{{\bf
F^{\text{2-{\hbox{\delta}}}}}\cdot\hat{\bf
z}}{A}\bigg{|}_{z=a_{2}}=\frac{1}{\text{i}}\int_{-\infty}^{\infty}\frac{d\zeta}{2\pi}\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\left[T_{33}(a_{2}+\delta)-T_{33}(a_{2}-\delta)\right],$
(53)

where

$T_{33}(a_{2}+\delta)=\frac{1}{2\text{i}}\Big{[}\partial\partial^{\prime}g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{13}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{2},a_{2})-\zeta^{2}g^{2,E}_{\rput(2.84526pt,0.0pt){\scriptstyle{13}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{2},a_{2})-k_{\perp}^{2}g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{13}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{2},a_{2})\Big{]}\\\
+\frac{1}{2\text{i}}\Big{[}\partial\partial^{\prime}g^{2,E}_{\rput(2.84526pt,0.0pt){\scriptstyle{13}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{2},a_{2})-\zeta^{2}g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{13}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{2},a_{2})-k_{\perp}^{2}g^{2,E}_{\rput(2.84526pt,0.0pt){\scriptstyle{13}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{2},a_{2})\Big{]}$
(54)

and

$T_{33}(a_{2}-\delta)=\frac{1}{2\text{i}}\Big{[}\partial\partial^{\prime}g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{22}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{2},a_{2})-\zeta^{2}g^{2,E}_{\rput(2.84526pt,0.0pt){\scriptstyle{22}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{2},a_{2})-k_{\perp}^{2}g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{22}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{2},a_{2})\Big{]}\\\
+\frac{1}{2\text{i}}\Big{[}\partial\partial^{\prime}g^{2,E}_{\rput(2.84526pt,0.0pt){\scriptstyle{22}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{2},a_{2})-\zeta^{2}g^{2,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{22}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{2},a_{2})-k_{\perp}^{2}g^{2,E}_{\rput(2.84526pt,0.0pt){\scriptstyle{22}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{2},a_{2})\Big{]}.$
(55)

Implementing two plate Green’s function force between the two plates can be
obtained as

$P^{\text{2-{\hbox{\delta}}}}|_{z=a_{2}}=\frac{{\bf
F^{\text{2-{\hbox{\delta}}}}}\cdot\hat{\bf
z}}{A}\bigg{|}_{z=a_{2}}=-\frac{1}{2\pi^{2}}\int_{0}^{\infty}\kappa^{3}d\kappa\Bigg{[}\frac{r_{1}^{H}r_{2}^{H}\hbox{e}^{-2\kappa
a}}{\Delta_{12}^{H}}+\frac{r_{1}^{E}r_{2}^{E}\text{e}^{-2\kappa
a}}{\Delta_{12}^{E}}\Bigg{]}.$ (56)

Similarly, the force between three plates $\bf F^{\text{3-$\delta$}}$ is
obtained by involving the Green’s function of three plates in regions of Fig.
2. The calculation of force on the plate at $z=a_{3}$ from Eq. (51) involves

$T_{33}(a_{3}+\delta)=\frac{1}{2\text{i}}\Big{[}\partial\partial^{\prime}g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{14}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{3},a_{3})-\zeta^{2}g^{3,E}_{\rput(2.84526pt,0.0pt){\scriptstyle{14}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{3},a_{3})-k_{\perp}^{2}g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{14}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{3},a_{3})\Big{]}\\\
+\frac{1}{2\text{i}}\Big{[}\partial\partial^{\prime}g^{3,E}_{\rput(2.84526pt,0.0pt){\scriptstyle{14}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{3},a_{3})-\zeta^{2}g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{14}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{3},a_{3})-k_{\perp}^{2}g^{3,E}_{\rput(2.84526pt,0.0pt){\scriptstyle{14}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{3},a_{3})\Big{]}$
(57)

and

$T_{33}(a_{3}-\delta)=\frac{1}{2\text{i}}\Big{[}\partial\partial^{\prime}g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{23}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{3},a_{3})-\zeta^{2}g^{3,E}_{\rput(2.84526pt,0.0pt){\scriptstyle{23}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{3},a_{3})-k_{\perp}^{2}g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{23}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{3},a_{3})\Big{]}\\\
+\frac{1}{2\text{i}}\Big{[}\partial\partial^{\prime}g^{3,E}_{\rput(2.84526pt,0.0pt){\scriptstyle{23}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{3},a_{3})-\zeta^{2}g^{3,H}_{\rput(2.84526pt,0.0pt){\scriptstyle{23}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{3},a_{3})-k_{\perp}^{2}g^{3,E}_{\rput(2.84526pt,0.0pt){\scriptstyle{23}}{\psset{linewidth=0.2pt}\pscircle(2.84526pt,0.28452pt){4.2679pt}}}(a_{3},a_{3})\Big{]}.$
(58)

Further, the force between the three plates is evaluated as

$P^{\text{3-$\delta$}}|_{z=a_{3}}=\frac{{\bf
F^{\text{3-$\delta$}}}\cdot\hat{\bf z}}{A}\bigg{|}_{z=a_{3}}=\\\
-\frac{1}{2\pi^{2}}\int_{0}^{\infty}\kappa^{3}d\kappa\Bigg{[}\frac{r_{2}^{H}r_{3}^{H}\hbox{e}^{-2\kappa
b}\Delta_{12}^{H}}{\Delta_{123}^{H}}+\frac{r_{2}^{E}r_{3}^{E}\text{e}^{-2\kappa
b}\Delta_{12}^{E}}{\Delta_{123}^{E}}+\frac{r_{1}^{H}(t_{2}^{H})^{2}r_{3}^{H}\hbox{e}^{-2\kappa
a}\hbox{e}^{-2\kappa
b}}{\Delta_{123}^{H}}+\frac{r_{1}^{E}(t_{2}^{E})^{2}r_{3}^{E}\hbox{e}^{-2\kappa
a}\hbox{e}^{-2\kappa b}}{\Delta_{123}^{E}}\Bigg{]}.$ (59)

##  V CASIMIR ENERGY FROM MULTIPLE SCATTERING FORMALISM

Here we are interested in calculating the Casimir energy between two plates
($\Delta E_{(12)}$), three plates ($\Delta E_{(123)}$), four plates ($\Delta
E_{(1234)}$) and five plates ($\Delta E_{(12345)}$) configurations from
multiple scattering formalism. From the Green’s functions in Sec. iii@, the
reflection coefficients of one plate, two plates and three plates
configurations are used to calculate these energies of multiple plates. The
total energy $E_{(12\cdots i,i+1\cdots N)}$ between two disjoint bodies
$(12\cdots i)$ and $(i+1\cdots N)$ can be decomposed as [18]

$E_{(12\cdots i,i+1\cdots N)}=E_{0}+\Delta E_{(12\cdots i)}+\Delta
E_{(i+1\cdots N)}+\Delta E_{(12\cdots i)(i+1\cdots N)}.$ (60)

Here $E_{0}$ is the energy of the background vacuum which is usually infinite,
$\Delta E_{(12\cdots i)}$, $\Delta E_{(i+1\cdots N)}$ are self-energies of the
bodies and $\Delta E_{(12\cdots i)(i+1\cdots N)}$ is the interaction energy.
Self-energy can be interpreted as energy needed to create the bodies
independently. The finite part of this total energy $E_{(12\cdots i,i+1\cdots
N)}$ after removing the divergent parts ($E_{0}$ and $\Delta E_{i}$,
$i\in[1,N]$) is what we are interested to calculate $\Delta E_{(12\cdots
i,i+1\cdots N)}$, which is responsible for the force and the interaction
energy $\Delta E_{(12\cdots i)(i+1\cdots N)}$ is usually finite. Multiple
scattering formalism predicts this Casimir interaction energy $\Delta
E_{(12\cdots i)(i+1\cdots N)}$ between the two disjoint planar bodies
$(12\cdots i)$ and $(i+1\cdots N)$ with reflection coefficients $R_{(12\cdots
i)}$ and $R_{(i+1\cdots N)}$ (with superscript $H$ denoting TM mode and $E$
for TE mode, subscript $(12\cdots i)$ and $(i+1\cdots N)$ denoting the optical
properties of the two bodies), respectively as

$\frac{\Delta E_{(12\cdots i)(i+1\cdots
N)}}{A}=\frac{1}{2}\int_{-\infty}^{\infty}\frac{d\zeta}{2\pi}\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\Bigg{[}\ln\Big{[}1-R_{(12\cdots
i)}^{<,H}\text{e}^{-\kappa L}R_{(i+1\cdots N)}^{>,H}\text{e}^{-\kappa
L}\Big{]}+\ln\Big{[}1-R_{(12\cdots i)}^{<,E}\text{e}^{-\kappa L}R_{(i+1\cdots
N)}^{>,E}\text{e}^{-\kappa L}\Big{]}\Bigg{]}$ (61)

where $L$ is the distance between the bodies. This assumes that Casimir
interaction energy can be represented only using reflection coefficients
alone.
The Casimir energy between two plates $\Delta E_{(12)}$ with reflection
coefficients $R_{(1)}=r_{1}$, $R_{(2)}=r_{2}$ (Eq. (24) for $i=1,2$) and
distance $l_{12}$ between them, can be easily evaluated as

$\frac{\Delta
E_{(12)}}{A}=\frac{1}{2}\int_{-\infty}^{\infty}\frac{d\zeta}{2\pi}\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\Bigg{[}\ln\Big{[}\Delta_{12}^{H}\Big{]}+\ln\Big{[}\Delta_{12}^{E}\Big{]}\Bigg{]},$
(62)

where $\Delta_{12}$ is multiple scattering parameter of two plates in Eq. (32)
such as

$\Delta_{12}=1-r_{1}\hbox{e}^{-\kappa l_{12}}r_{2}\hbox{e}^{-\kappa l_{12}}.$
(63)

In this case of two plate, $\Delta E_{(12)}$ = $\Delta E_{(1)(2)}$ since self-
energies of plates $\Delta E_{(1)}$ and $\Delta E_{(2)}$ are infinite. This
expression for energy can be checked by finding the force and comparing with
$\bf F^{\text{2-$\delta$}}$ between the two plates (Eq. (56)) where

$P^{\text{2-{\hbox{\delta}}}}|_{z=a_{2}}=-\frac{1}{A}\frac{\partial\Delta
E_{(12)}}{\partial a}$ (64)

with $a=l_{12}$.
The Casimir energy between three plates $\Delta E_{(123)}$ with reflection
coefficients $R_{(1)}=r_{1}$, $R_{(2)}=r_{2}$, $R_{(3)}=r_{3}$ (Eq. (24) for
$i=1,2,3$) and with distances $l_{12}$, $l_{23}$ between them, respectively
can be evaluated from using Eq. (60) such as

$E_{(123)}=E_{0}+\Delta E_{(12)}+\Delta E_{(3)}+\Delta
E_{(12)(3)}=E_{0}+\Delta E_{(1)}+\Delta E_{(23)}+\Delta E_{(1)(23)}.$ (65)

Considering the finite terms in this expression, energy can be solved as

$\frac{\Delta
E_{(123)}}{A}=\frac{1}{2}\int_{-\infty}^{\infty}\frac{d\zeta}{2\pi}\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\Bigg{[}\ln\Big{[}\Delta_{12}^{H}\Big{]}+\ln\Big{[}\Delta_{12}^{E}\Big{]}+\ln\Big{[}1-R_{12}^{<,H}R_{3}^{>,H}\text{e}^{-2\kappa
l_{23}}\Big{]}+\ln\Big{[}1-R_{12}^{<,E}R_{3}^{>,E}\text{e}^{-2\kappa
l_{23}}\Big{]}\Bigg{]}\\\
=\frac{1}{2}\int_{-\infty}^{\infty}\frac{d\zeta}{2\pi}\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\Bigg{[}\ln\Big{[}\Delta_{23}^{H}\Big{]}+\ln\Big{[}\Delta_{23}^{E}\Big{]}+\ln\Big{[}1-R_{1}^{<,H}R_{23}^{>,H}\text{e}^{-2\kappa
l_{12}}\Big{]}+\ln\Big{[}1-R_{1}^{<,E}R_{23}^{>,E}\text{e}^{-2\kappa
l_{12}}\Big{]}\Bigg{]}.$ (66)

Reflection coefficients $R_{12}^{<}$ and $R_{23}^{>}$ can be obtained from
matrix $B$ of two plates Green’s function. The Casimir energy for three plates
is derived as

$\frac{\Delta
E_{(123)}}{A}=\frac{1}{2}\int_{-\infty}^{\infty}\frac{d\zeta}{2\pi}\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\Bigg{[}\ln\Big{[}\Delta_{123}^{H}\Big{]}+\ln\Big{[}\Delta_{123}^{E}\Big{]}\Bigg{]},$
(67)

where $\Delta_{123}$ is multiple scattering parameter of three plates in Eq.
(46) which can be defined as Eq. (2) with

$\Delta_{13}=-r_{1}^{H}\hbox{e}^{-\kappa l_{12}}t_{2}^{H}\hbox{e}^{-\kappa
l_{23}}r_{3}^{H}\hbox{e}^{-\kappa l_{23}}t_{2}^{H}\hbox{e}^{-\kappa l_{12}}$
(68)

and using this we define the generalized multiple scattering parameter such as

$\Delta_{ij}=1-r_{i}\hbox{e}^{-\kappa l_{ij}}r_{j}\hbox{e}^{-\kappa l_{ij}}$
(69)

for all $j=i+1$ (nearest neighbour scattering, $i\in[1,N-1]$ where $i$ and $j$
are adjacent). These terms in $\Delta_{123}$ can be visualized in Fig. 1 where
$\Delta_{12}$ and $\Delta_{23}$ are loops between bodies $(1)-(2)$ and
$(2)-(3)$, respectively. Term $\Delta_{13}$ in Eq.(68) refers to loop between
$(1)-(3)$ with initial reflection with $r_{1}$, propagation with exponential
dependence of length $l_{12}$, transmission with $t_{2}$, propagation of
length $l_{23}$, reflection with $r_{3}$, again propagation of length
$l_{23}$, transmission with $t_{2}$, propagation of length $l_{12}$ and this
propagation continues. Further, these loops can be seen for multiple
scattering parameters $\Delta_{1234}$, $\Delta_{12345}$ of $N=4,5$
configurations. Similarly, this expression of energy for three plates can be
checked by finding the force and comparing with $\bf F^{\text{3-$\delta$}}$
between the three plates (Eq. (59)) where

$P^{\text{3-{\hbox{\delta}}}}|_{z=a_{3}}=-\frac{1}{A}\frac{\partial\Delta
E_{(123)}}{\partial b}.$ (70)

with $a=l_{12}$ and $b=l_{23}$.
In a similar manner, Casimir energy between four plates $\Delta E_{(1234)}$
with reflection coefficients $R_{(1)}=r_{1}$, $R_{(2)}=r_{2}$,
$R_{(3)}=r_{3}$, $R_{(4)}=r_{4}$ (Eq. (24) for $i=1,2,3,4$) and with distances
$l_{12}$, $l_{23}$, $l_{34}$ between them, respectively can be evaluated as

$\frac{\Delta
E_{(1234)}}{A}=\frac{1}{2}\int_{-\infty}^{\infty}\frac{d\zeta}{2\pi}\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}\Bigg{[}\ln\Big{[}\Delta_{1234}^{H}\Big{]}+\ln\Big{[}\Delta_{1234}^{E}\Big{]}\Bigg{]},$
(71)

where multiple scattering parameter of four plates is defined in Eq. (3). Here
$\Delta_{ij}$ for $j=i+1$ is defined in Eq. (69), $\Delta_{13}$ is same as in
Eq.(2),

$\Delta_{24}=-r_{2}\hbox{e}^{-\kappa l_{23}}t_{3}\hbox{e}^{-\kappa
l_{34}}r_{4}\hbox{e}^{-\kappa l_{34}}t_{3}\hbox{e}^{-\kappa l_{23}}$ (72)

and

$\Delta_{14}=-r_{1}\hbox{e}^{-\kappa l_{12}}t_{2}\hbox{e}^{-\kappa
l_{23}}t_{3}\hbox{e}^{-\kappa l_{34}}r_{4}\hbox{e}^{-\kappa
l_{34}}t_{3}\hbox{e}^{-\kappa l_{23}}t_{2}\hbox{e}^{-\kappa l_{12}}.$ (73)

Using this we define

$\Delta_{ik}=-r_{i}\hbox{e}^{-\kappa l_{i,i+1}}t_{i+1}\hbox{e}^{-\kappa
l_{i+1,i+2}}t_{i+2}\cdots\hbox{e}^{-\kappa l_{k-1,k}}r_{k}\hbox{e}^{-\kappa
l_{k-1,k}}\cdots t_{i+1}\hbox{e}^{-\kappa l_{i,i+1}}$ (74)

for all $k\geq i+2$ (next-to-nearest neighbour, next-to-next-to-nearest
neighbour, $\cdots$ scattering, $i\in[1,N-2]$ where $i$ and $k$ are not
adjacent). This Casimir energy between four plates $\Delta E_{(1234)}$ was
evaluated from using Eq. (60) such as

$E_{(1234)}=E_{0}+\Delta E_{(12)}+\Delta E_{(34)}+\Delta
E_{(12)(34)}=E_{0}+\Delta E_{(1)}+\Delta E_{(234)}+\Delta
E_{(1)(234)}=E_{0}+\Delta E_{(123)}+\Delta E_{(4)}+\Delta E_{(123)(4)}.$ (75)

For five plates the multiple scattering parameter can be obtained such as

$\Delta_{12345}=\Delta_{12}\Delta_{23}\Delta_{34}\Delta_{45}+\Delta_{12}\Delta_{24}\Delta_{45}+\Delta_{13}\Delta_{34}\Delta_{45}+\Delta_{12}\Delta_{23}\Delta_{35}+\Delta_{13}\Delta_{35}+\Delta_{12}\Delta_{25}+\Delta_{14}\Delta_{45}+\Delta_{15},$
(76)

where the Casimir energy between five plates $\Delta E_{(12345)}$ was
evaluated from using Eq. (60) such as

$E_{(12345)}=E_{0}+\Delta E_{(123)}+\Delta E_{(45)}+\Delta
E_{(123)(45)}=E_{0}+\Delta E_{(12)}+\Delta E_{(345)}+\Delta E_{(12)(345)}.$
(77)

These multiple scattering parameters can be visualized using a simple
diagrammatic way such as in Figs. 2 and 5 with each term described by loops.
These results for $\Delta$ is similar to the expressions obtained by Allocca
et al. [5] for $N=2,3,4,5$ ($\Delta_{12}$, $\Delta_{123}$, $\Delta_{1234}$,
$\Delta_{12345}$ here referred to as $\Delta_{1}$, $\Delta_{2}$, $\Delta_{3}$,
$\Delta_{4}$, respectively in Ref. [5]). Further, using this similar approach
of separating partitions such as in Eq.s (2), (3) and (76), Casimir energy of
N $\delta$-plates can be generalized such as Eq. (1) where $\Delta_{12\cdots
N}$ can be evaluated from the discussion above.

##  VI Conclusions

The Casimir energy for $N=2,3,4,5$ $\delta-$function plates were derived using
the multiple scattering formalism and by implementing the optical properties
of plates. The pattern seems generic, and the result for $N$ plates was
predicted by extrapolation. Visualization of the distribution using
diagrammatic loops in multiple scattering formalism appears to be an easy way
to obtain $N$ plates result. Green’s functions were derived for two and three
$\delta-$function plates configurations, using which their reflection
coefficients were obtained. Multiple scattering formalism was utilised to
derive the Casimir energy between two, three, four and five plates
configurations. In the case of Casimir energy for two plates configuration,
reflection coefficients of a single plate were used. In the case of Casimir
energy for three plates, reflection coefficients of a single plate and two
plates configurations were used. In the case of Casimir energy for four
plates, reflection coefficients of a single plate and three plates
configurations were used. Similarly, in the case of Casimir energy for five
plates, reflection coefficients of two plates and three plates configurations
were used. For the case of two and three plates configurations, the Casimir
energy was checked with force derived from the stress tensor method. Further,
one can obtain an exact closed-form expression for $N$ plates. It may also be
interesting to expand the matrices defined in Sec. iii@ in order to derive
Green’s functions for $N$ plates in a systematic manner.

$l_{12}$$l_{23}$$l_{34}$$l_{45}$$r_{1}$$r_{2}$$t_{2}$$r_{3}$$t_{3}$$r_{4}$$t_{4}$$r_{5}$$\Delta_{12345}$$\Delta_{12}\Delta_{23}\Delta_{34}\Delta_{45}$$\Delta_{12}\Delta_{24}\Delta_{45}$$\Delta_{13}\Delta_{34}\Delta_{45}$$\Delta_{12}\Delta_{23}\Delta_{35}$$\Delta_{13}\Delta_{35}$$\Delta_{12}\Delta_{25}$$\Delta_{14}\Delta_{45}$$\Delta_{15}$
Figure 5: Visualizing $\Delta_{12345}$ with optical properties, $r_{i}$ are
reflection coefficients and $t_{i}$ are transmission coefficients of the
plates. $l_{ij}$ is the distance between the plates where $i$ and $j$ are
adjacent plates.

## Acknowledgements

We are incredibly grateful to Dr. K. V. Shajesh and Dr. Prachi Parashar for
introducing their work on $\delta$-function plates and for guidance throughout
this work. We thank Dr. R. Sankaranarayanan at NITT for constant support and
making all this happen.

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