**3.1. Statement of the problem**

As the function *L sj*

184 Advanced Electromagnetic Waves

series:

Also the unknown function *zj*

is regular, we can expand it into double Fourier series:

¥ ¥ <sup>+</sup>

q t

<sup>=</sup> å å (15)

and the given potential function are represented by their Fourier

 q

> pp

> > *<sup>n</sup>*=*<sup>∞</sup>* and defining the

*<sup>s</sup>* <sup>=</sup> *<sup>ξ</sup><sup>n</sup> s σn* ,

*<sup>s</sup>* as follows: *ξ*˜ *<sup>n</sup>*

= = å å (16)

( ) (,) . *sj sj in m nm*

( ) , () *j j in in j nj n z eV e* t

¥ ¥ -¥ -¥

 qn

After substitution of all expansions into (9), one can arrive at the system of *N* integral equations:

, ,

*e el e s N*

*n mm n*

æ ö - ç ÷ <sup>=</sup> Î- =

(1 ) , 0, 1, 2...; 1,2,..., .

Following the steps suggested in [33], it can be shown that coefficient matrix in (18) is square


Thus the infinite system (18) is of a second Fredholm kind and can now be effectively solved by a truncation method. The solution of the truncated system monotonically and rapidly converges to the exact solution. The above solution automatically incorporates the reciprocal influence of all charged cylinders, allowing accurate calculation of the line charge densities on the boundaries and then the field potentials at any point of the space between the conductors.

Fourier expansions in (18) are calculated numerically as all functions are regular.

*l n sN*

x


Using orthogonal properties and completeness of the functions {*einφ*}*n*=−*<sup>∞</sup>*

 x s


2 ,

*n mn m*


rescaled unknown Fourier coefficients of charge density function *ξ<sup>n</sup>*

2 [ , ], 1,2,..., .

è ø å åå å å (17)

when *n* ≠0 and *σ*<sup>0</sup> =1, we obtain the following infinite system of linear algebraic

*l vn s N*

å å <¥ = ± ± = (19)


 n q  q

=-¥ =-¥

*n m L l e*

q t

t

*n sn in sj j j sn*

1

*n jn m n*

 q

=-¥ = =-¥ =-¥ =-¥

0 , 1

= =-¥

*<sup>N</sup> sj*

¥

s s

¥

*j m*

*<sup>N</sup> s s sj j n n n mn m m n n*

 ss

¥ ¥¥ ¥

*s N*

*n* x q

x

 d

1

*j m*

= =-¥

*<sup>σ</sup><sup>n</sup>* <sup>=</sup> <sup>|</sup>*<sup>n</sup>* <sup>|</sup> 1/2

equations:

summable:

 x In this section, we consider the scattering problem for the structure which consists of *N* arbitrary profiled perfect electric -conductor cylinders embedded into a homogeneous dielectric medium with relative permittivity *ε*. The main steps of the solution algorithm are similar to those which were carried out to obtain the solution to the Laplace equation, presented in Section 2.2.

The scattered electromagnetic field *U <sup>s</sup>* obeys the Helmholtz equation:

$$\left(\Delta + k^2\right)\mathcal{U}^\*\left(p\right) = 0,\tag{20}$$

where point *p* lies exterior to the structure *S*, *k* =2*π* / *λ* is the wave number and *λ* is the corresponding wave length.

Here we consider incident fields in the form of a plane wave. We focus on a transverse magnetic (TM) wave polarization of the incident field (*U* <sup>0</sup> ), therefore the scattered field should satisfy the Dirichlet boundary condition on metallic surfaces:

$$
\mathcal{L}I^s(p) = -\mathcal{L}I^0(p), \qquad p \in \mathcal{S}.\tag{21}
$$

The field should also satisfy the Sommerfeld radiation condition:

$$\mathcal{U}J^{s}\left(p\right) = O\left(\left|p\right|^{-1/2}\right); \frac{\widehat{\mathcal{O}}\mathcal{U}^{s}\left(p\right)}{\widehat{\mathcal{O}}\left|p\right|} - ik\mathcal{U}^{s}\left(p\right) = O\left(\left|p\right|^{-1/2}\right),\tag{22}$$

As |*p*|→∞ where | *p* | is the radial component of the point *p* in the arbitrary fixed polar coordinate system.

### **3.2. Regularized solution**

Solutions to the Laplace equation can be represented as a single-layer potential at points exterior to the body. Using the superposition principle, we seek the solution as the sum of single-layer potentials contributed by each cylinder:

$$\mathcal{LI}\left(\boldsymbol{q}\right) = \sum\_{n=1}^{N} \int G\left(\left|\boldsymbol{p}\_{n} - \boldsymbol{q}\right|\right) Z\_{n}\left(\boldsymbol{p}\_{n}\right) d\boldsymbol{l}\_{p\_{n}}.\tag{23}$$

Here *G*(| *pn* −*q* |) is the relevant free space Green's function depending on the distance | *pn* −*q* | between the observation point *q* and point *pn* lying on the contour *Sn* : *<sup>G</sup>*(| *<sup>p</sup>* <sup>−</sup>*<sup>q</sup>* |)= <sup>−</sup> *<sup>i</sup>* <sup>4</sup> *<sup>H</sup>*<sup>0</sup> (1) (*k* | *p* −*q* |), and the function *Z*(*p*) is related to the linear current density *J*(*p*) as *J*(*p*)=*ikcZ* / 4*π* (where *k* is a wave number, *c* the light speed).

Applying boundary conditions to (23), we obtain the system of *N* integral equations:

$$\sum\_{j=1}^{N} \int\_{L\_{n}} G\left(\left|p\_{n} - q\right|\right) Z\_{n}\left(p\_{n}\right) dl\_{p\_{n}} = -L\_{n}^{0}\left(q\right), \quad q \in \mathcal{S}\_{m'}, m = 1, \ldots, N. \tag{24}$$

After parameterization of the contours *ηn*(*θ*)=(*xn*(*θ*), *yn*(*θ*)) where *θ* ∈ −*π*, *π* , we use the definition of a line integral and obtain a functional equation in the form:

$$\sum\_{j=1}^{N} \int G\left(R\_{uu}\left(\theta,\tau\right)\right) \mathbf{z}\_{n}\left(\tau\right) d\tau = -\mathsf{L}I^{0}\left(\theta\right). \tag{25}$$

The following notation is used in (25): *zn*(*τ*)=*l <sup>n</sup>*(*τ*)*Zn*(*pn*(*τ*)), where *l <sup>n</sup>* = (*xn*(*τ*) ′ ) <sup>2</sup> <sup>+</sup> (*yn*(*τ*) ′ ) 2 , *<sup>n</sup>* =1, ..., *<sup>N</sup>* ; *Rmn*(*θ*, *<sup>τ</sup>*) is the distance between points *ηm*(*θ*) and *ηn*(*τ*) lying on the *m*-th and *n*-th contours, respectively.

The kernel of the integral equation (25) contains a singularity only in the terms *G*(*Rmn*(*θ*, *τ*)). It is of logarithmic type at the points *θ* =*τ*, and we analytically split the Green's function into a singular and a regular part *H mn*(*θ*, *τ*) similarly to the solution steps for the Laplace equation in Section 2.2:

$$G\left(R\_{nn}\left(\theta,\tau\right)\right) = -\frac{i}{4}H\_0^{(1)}\left(kR\_{nn}\left(\theta,\tau\right)\right) = \begin{cases} \frac{1}{2\pi}\left(\log\left(2\sin\left|\frac{\theta-\tau}{2}\right|\right) + H^{mn}\left(\theta,\tau\right)\right), & m=n\\ \frac{1}{2\pi}H^{mn}\left(\theta,\tau\right), & m\neq n\_\star \end{cases} \tag{26}$$

so that the regular part of Green's function is

$$H^{mn}(\theta,\tau) = \begin{cases} -\frac{i\pi}{2}H\_0^{(1)}\left(kR\_{mn}\right) - \log\left(2\sin\left|\frac{\theta-\tau}{2}\right|\right), & m=n\\ -\frac{i\pi}{2}H\_0^{(1)}\left(kR\_{mn}\right), & m\neq n\_\* \end{cases} \tag{27}$$

The function *H mn*, *m*=*n* is a regular function, defined everywhere except at the points *θ* =*τ* ; the function *H mn*, *m*≠*n* is regular everywhere. An exact expression for *H mm* at the points of singularity is obtained analytically:

$$\left.H^{mm}\left(\theta,\pi\right)\right|\_{\theta=\pi} = -\frac{i\pi}{2} + \gamma + \log\frac{k \cdot l\_w\left(\theta\right)}{2},\tag{28}$$

where *γ* is Euler's constant.

Here *G*(| *pn* −*q* |) is the relevant free space Green's function depending on the distance | *pn* −*q* | between the observation point *q* and point *pn* lying on the contour *Sn* :

Applying boundary conditions to (23), we obtain the system of *N* integral equations:

*n nn p m m*

*G Z p dl U q p q q Sm N*

After parameterization of the contours *ηn*(*θ*)=(*xn*(*θ*), *yn*(*θ*)) where *θ* ∈ −*π*, *π* , we use the

( ( )) ( ) ( ) <sup>0</sup>

The kernel of the integral equation (25) contains a singularity only in the terms *G*(*Rmn*(*θ*, *τ*)). It is of logarithmic type at the points *θ* =*τ*, and we analytically split the Green's function into a singular and a regular part *H mn*(*θ*, *τ*) similarly to the solution steps for the Laplace equation

( )

, , <sup>2</sup>

*<sup>i</sup> H kR m n*

*<sup>i</sup> H kR m n*

<sup>ì</sup> æ ö - ï-- = ç ÷ <sup>ï</sup> <sup>=</sup> è ø <sup>í</sup>


q t

*H mn <sup>i</sup>*

q t

= - (26)

q t

<sup>ì</sup> æ ö æ ö - <sup>ï</sup> ç ÷ ç ÷ + = <sup>ï</sup> <sup>=</sup> è ø è ø <sup>í</sup>

¹ ï

<sup>1</sup> log 2sin , , 2 2 , , <sup>4</sup> <sup>1</sup> , , , <sup>2</sup>

*mn*

 t t

, . *<sup>N</sup>*

*GR z d U*

*J*(*p*) as *J*(*p*)=*ikcZ* / 4*π* (where *k* is a wave number, *c* the light speed).

( ) ( ) ( ) <sup>0</sup>

definition of a line integral and obtain a functional equation in the form:

qt

( ( )) ( ( )) ( )

ï

î

q t

> ( ) ( )

*mm mn*

ï

î

p

p

1 0

( ) ( )

1 0

p

p

log 2sin , 2 2 ,

*mm*

*mn <sup>n</sup> <sup>j</sup>*

The following notation is used in (25): *zn*(*τ*)=*l*

1

lying on the *m*-th and *n*-th contours, respectively.

(1) 0

so that the regular part of Green's function is

( )

q t

*H*

*H k*

*mn mn*

*G R R*

q t = -

p

p

(*k* | *p* −*q* |), and the function *Z*(*p*) is related to the linear current density

, , 1,..., . *<sup>n</sup>*

q

, *<sup>n</sup>* =1, ..., *<sup>N</sup>* ; *Rmn*(*θ*, *<sup>τ</sup>*) is the distance between points *ηm*(*θ*) and *ηn*(*τ*)

*mm*

q t

*H m n*

å ò = - (25)

*<sup>n</sup>*(*τ*)*Zn*(*pn*(*τ*)), where

(27)

å ò - = =- Î (24)

*<sup>G</sup>*(| *<sup>p</sup>* <sup>−</sup>*<sup>q</sup>* |)= <sup>−</sup> *<sup>i</sup>*

186 Advanced Electromagnetic Waves

*l*

*<sup>n</sup>* = (*xn*(*τ*)

in Section 2.2:

′ ) <sup>2</sup> <sup>+</sup> (*yn*(*τ*) ′ ) 2

<sup>4</sup> *<sup>H</sup>*<sup>0</sup> (1)

1

*j L*

=

*N*

*n*

We expand the singular part of Green's functions in the same way as in Section 2.2, and perform the double Fourier series expansion for the regular function *H mm* : *H mn*(*θ*, *τ*)= ∑ *j*,*l*=−*∞ ∞ h jl mn ei*( *<sup>j</sup>θ*+*lτ*) . The unknown function *zj* is also represented by its Fourier series: *zn*(*τ*)= ∑ *p*=−*∞ ∞ ξp neip<sup>τ</sup>* .

After substitution of all expansions into (25), one can arrive at the system of *N* integral equations. Following the regularization steps for the Laplace equation from Section 2.3, we obtain an infinite system of linear algebraic equations of the second kind:

$$
\tilde{\xi}\_l^m(\mathbf{1} - \delta\_{n0}) + \sum\_{j=n-1}^N \sum\_{p=-n}^n \sigma\_l \sigma\_p h\_{l,-p}^{mn} \tilde{\xi}\_p^n = \sigma\_l \mathbf{g}\_l^m \qquad l = 1, 2, \dots, N; \ n = 0, \pm 1, \pm 2, \dots \tag{29}
$$

where the following notations are used:

$$\begin{aligned} \sigma\_p &= \begin{cases} \left| p \right|^{1/2} & \text{when } n \neq 0\\ 1 & \text{when } n = 0, \end{cases} \\ \tilde{\xi}\_p^n &= \sigma\_p^{-1} \tilde{\xi}\_p^n, \qquad -2\mathcal{U}^0 = \sum\_{m = -\alpha}^{\alpha} g\_l^m e^{il\theta}. \end{aligned} \tag{30}$$

The infinite systems (31) can be effectively solved by a truncation method. The solution of the truncated system steadily and rapidly converges to the exact solution [34]. There are no limitations on the number of cylinders with arbitrary smooth cross-sections.
