**3. Integral equations for electric and magnetic currents in thin impedance vibrators and narrow slots**

is omitted) and using the continuity condition for the tangential components of the magnetic field on the holes Σ*n*, we obtain the system of integral equations relative to the density of surface

> → *n <sup>m</sup>*(*r* →

*<sup>n</sup>*) at Σ*n*. The system can be presented

(3)

*<sup>m</sup>*) at *Sm* and equivalent magnetic *J*

rot d

+ = ¢ ¢¢

*q V qm mm m m S*

=

å ò

r rr r r r r

graddiv d

*n*

*V q m Sm m m m m m*

*G rr Z r n J r r*

1

2

r r r rr r r

=+ + ¢ ¢¢ +

<sup>1</sup> <sup>ˆ</sup> () ( ) (, ) ( )

*E r k G rr J r r <sup>i</sup>*

rot d

<sup>r</sup> <sup>r</sup> graddiv <sup>d</sup>

2 1 1 1 2 2 2 1

<sup>1</sup> <sup>ˆ</sup> ) ( ) (,) ()

*p V pn n n n n <sup>N</sup> m m*

*k G rrJ r r <sup>i</sup>*

å ò

+ + ¢ ¢¢

= S

graddiv d

*n*

*m*

*pm mm m*

*rr J r r*

<sup>1</sup> <sup>ˆ</sup> ( ) (,) ()

+ + ¢ ¢¢

*k G rrJ r r <sup>i</sup>*

å ò

= S

*m S*

å ò

=

*n*

2

1

d

<sup>1</sup> <sup>ˆ</sup> ( ) ( , ) ( )[ , ( )]

*k G rrZ r n J r r <sup>i</sup>*

= + - ¢ ¢ ¢¢

1

= S

*n*

ˆ () () (,) ()

å ò

*<sup>k</sup> Z rJr G rrJ r r*

*<sup>N</sup> <sup>e</sup> m m Sq q q q V qn n n n n*

r r r r rr r r

1 1

*<sup>M</sup> m e*

+ ¢ ¢ ¢¢

1 11

1

where *q* =1, 2, ..., *m*, ..., *M* , *p* =1, 2, ..., *n*, ..., *N* , *J*

thin impedance vibrators and narrow slots.

*<sup>M</sup> <sup>e</sup>*

ˆ

*e V*

2 1 1 1

<sup>å</sup> ¢ ¢¢ <sup>ò</sup> <sup>r</sup> rr r r

1

wm

å ò

*m S*

=

0

*H r*

(

wm

we

rot

w


*<sup>k</sup> <sup>G</sup>*

1

*m S*

=

is velocity of light in free space.

*m*

p

0 1

we

w

graddiv d

*m*

1

*<sup>M</sup> e e*

<sup>1</sup> <sup>ˆ</sup> ( , ) ( )[ , ( )] , (a) <sup>4</sup>

*<sup>N</sup> m m*

1

*V pn n n n*

r rr r r

*<sup>M</sup> m e*

*V p m Sm m m m m m*

r rr r r r r

(, ) () , (b)

→ *m <sup>e</sup>* (*r* → *<sup>m</sup>*)= *<sup>c</sup>* <sup>4</sup>*<sup>π</sup> <sup>n</sup>* → *<sup>m</sup>*, *H* →(*r* → *<sup>m</sup>*) , *J* → *n <sup>m</sup>*(*r* → *<sup>n</sup>*)= *<sup>c</sup>* <sup>4</sup>*<sup>π</sup> <sup>n</sup>* → *<sup>n</sup>*, *E* → (*r* → *<sup>n</sup>*) , *c*

Thus, the problem of electromagnetic waves excitation by the impedance bodies of finite dimensions and by the coupling holes between two electrodynamic volumes is formulated as a rigorous boundary value problem of macroscopic electrodynamics, reduced to the system of integral equations for surface currents. Solution of this system is an independent problem, significant in its own right, since it often present considerable mathematical difficulties. If characteristic dimensions of an object are much greater than wavelength (high-frequency region) a solution is usually searched as series expansion in ascending power of inverse wave number. If dimensions of an object are less than wavelength (low-frequency or quasi-static region), representation of the unknown functions as series expansion in wave number powers reduces the problem to a sequence of electrostatic problems. Contrary to asymptotic cases, resonant region, where at least one dimension of an object is comparable with wavelength, is the most complex for analysis, and requires rigorous solution of field equations. It should be noted that, from the practical point of view, the resonant region is of exceptional interest for

r rr r r

currents: electric *J*

152 Advanced Electromagnetic Waves

as

→ *m <sup>e</sup>* (*r* →

> A straightforward solution of the system (3) for the material objects with irregular surface shape and for holes with arbitrary geometry may often be impossible due to the known mathematical difficulties. However, the solution is sufficiently simplified for thin impedance vibrators and narrow slots, i.e. cylinders, which cross-section perimeter is small as compared to their length and the wavelength in the surrounding media and for holes, which one dimension satisfy the analogous conditions [19,20]. The approach used in [19,20] for the analysis of slot-vibrator systems can be generalized for multi-element systems. In addition, the boundary condition (2) can be extended for cylindrical vibrator surfaces with an arbitrary distribution of complex impedance regardless of the exciting field structure and electrophys‐ ical characteristics of vibrator material [4].

> For thin vibrators made of circular cylindrical wire and narrow straight slots the equation system (3) can be easily simplified using inequalities

$$\frac{r\_m}{L\_m} << 1, \frac{r\_m}{\lambda\_{1,2}} << 1, \frac{d\_n}{2L\_n} << 1, \frac{d\_n}{\lambda\_{1,2}} << 1,\tag{4}$$

where *rm* is vibrator radius, *L <sup>m</sup>* is vibrator length, *dn* is slot width, 2*L <sup>n</sup>* is slot length, and *λ*1,2 is wavelength in the corresponding media. The electric current induced on the vibrator surfaces and equivalent magnetic currents in the slots can be presented using the inequalities (4) as

$$
\vec{J}\_{m}^{\epsilon}(\vec{r}\_{m}) = \vec{e}\_{s\_{n}} J\_{m}(\mathbf{s}\_{m}) \boldsymbol{\upmu}\_{m}(\boldsymbol{\uprho}\_{m'} \boldsymbol{\uprho}\_{m}) , \quad \vec{J}\_{n}^{m}(\vec{r}\_{n}) = \vec{e}\_{s\_{n}} J\_{n}(\mathbf{s}\_{n}) \boldsymbol{\upchi}\_{n}(\boldsymbol{\upxi}\_{n}) , \tag{5}
$$

where *e* → *sm* and *e* → *sn* are unit vectors directed along the vibrator and slot axis, respectively; *sm* and *sn* are local coordinates related to the vibrator and slot axes; *ψm*(*ρm*, *φm*) are functions of transverse (⊥*<sup>m</sup>* ) polar coordinates *ρm*, *φm* for the vibrators; *χn*(*ξn*) are functions of transverse coordinates *ξn* for the slots. The functions *ψm*(*ρm*, *φm*) and *χn*(*ξn*) satisfy the normality conditions

$$\int\_{\mathbb{L}\_n} \boldsymbol{\nu}\_m(\boldsymbol{\rho}\_m, \boldsymbol{\rho}\_n) \boldsymbol{\rho}\_n \mathbf{d} \boldsymbol{\rho}\_m \mathbf{d} \boldsymbol{\rho}\_m = 1, \quad \int\_{\mathbb{L}\_n} \boldsymbol{\chi}\_n(\boldsymbol{\xi}\_n) \mathbf{d} \boldsymbol{\xi}\_n = 1,\tag{6}$$

and the unknown currents *Jm*(*sm*) and *Jn*(*sn*) must satisfy the boundary conditions

$$J\_m(\pm L\_m) = 0, \quad J\_n(\pm L\_n) = 0,\tag{7}$$

where upper indexes *e* and *m* are omitted.

Now we take into account that *n* → *<sup>m</sup>*, *J* → *<sup>m</sup>*(*r* → *<sup>m</sup>*) ≪1 according to inequalities (4) and project the equations (3a) and (3b) on the axes of the vibrators and slots, respectively, and arrive at a system of linear integral equations relative to the currents in the vibrators and slots

$$\begin{split} \left(\frac{\mathbf{d}^2}{\mathbf{ds}\_q^2} + k\_1^2\right) \sum\_{n=1}^M \int\_{-L\_n}^{L\_q} f\_n(s\_n') G\_{s\_n}^{V\_1}(s\_q, s\_n') \mathbf{ds}\_n' - i \mathbf{k} \tilde{\varepsilon}\_{s\_q} \operatorname{rot} \sum\_{n=1}^N \int\_{-L\_n} f\_n(s\_n') G\_{s\_n}^{V\_1}(s\_q, s\_n') \mathbf{ds}\_n' \\ = -i \operatorname{oc} \mathbb{E}\_1 \left[ E\_{0s\_q}(s\_q) - z\_{lq}(s\_q) I\_q(s\_q) \right], \\ \frac{1}{\mu\_1} \left( \frac{\mathbf{d}^2}{\mathbf{ds}\_p^2} + k\_1^2 \right) \sum\_{n=1}^N \int\_{-l\_n}^{l\_n} f\_n(s\_n') G\_{s\_n}^{V\_1}(s\_p, s\_n') \mathbf{ds}\_n' + \frac{1}{\mu\_2} \left( \frac{\mathbf{d}^2}{\mathbf{ds}\_p^2} + k\_2^2 \right) \sum\_{n=1}^{l\_n} \int\_{-l\_n}^{l\_n} f\_n(s\_n') G\_{s\_n}^{V\_2}(s\_p, s\_n') \mathbf{ds}\_n' \\ + i k \bar{c}\_{s\_p} \operatorname{rot} \sum\_{n=1}^{M\_n} \int\_{-l\_n}^{l\_n} f\_n(s\_n') G\_{s\_n}^{V\_1}(s\_p, s\_n') \mathbf{ds}\_n' = -i \alpha H\_{0s\_p}(s\_p). \end{split} \tag{8}$$

Here *zim*(*sm*) are internal lineal impedance of the vibrators (*ZSm*(*r* → *<sup>m</sup>*)=2*πrmzim*(*r* → *<sup>m</sup>*)) measured in Ohm/m, *E*0*sm* (*sm*) and *H*0*sn* (*sn*) are projections of extraneous sources on the vibrators and slots axes, *Gsm V*1 (*sm*,*n*, *<sup>s</sup>*′ *<sup>m</sup>*) and *Gsn V*1(2) (*sm*,*n*, *<sup>s</sup>*′ *<sup>n</sup>*) are components of the tensor Green's functions in the volumes *V*1 and *V*2.

For solitary vibrator or slot as well as for the absence of electromagnetic interaction between them, the system (8) splits into two independent equations:

$$\left(\frac{\mathbf{d}^2}{\mathbf{ds}\_v^2} + k\_1^2\right)\Big|\_{-L\_v}^{L\_v} f\_v(\mathbf{s}\_v') \mathbf{G}\_{s\_v}^V(\mathbf{s}\_{v'}, \mathbf{s}\_v') \mathbf{ds}\_v' = -i\alpha \varepsilon\_1 \mathbf{E}\_{0s\_v}(\mathbf{s}\_v) + i\alpha \varepsilon\_1 \mathbf{z}\_l(\mathbf{s}\_v) f\_v(\mathbf{s}\_v) \tag{9}$$

$$\frac{1}{\mu\_1} \left( \frac{\mathbf{d}^2}{\mathrm{ds}\_{\boldsymbol{s}l}^2} + k\_1^2 \right) \Big| \int\_{-l\_d} f\_{\boldsymbol{s}l}(\mathbf{s}\_{\boldsymbol{s}l}') \mathbf{G}\_{\boldsymbol{s}\_d}^{V\_1}(\mathbf{s}\_{\boldsymbol{s}l}, \mathbf{s}\_d') \mathrm{ds}\_d' + \frac{1}{\mu\_2} \left( \frac{\mathbf{d}^2}{\mathrm{ds}\_{\boldsymbol{s}l}^2} + k\_2^2 \right) \Big| \int\_{-l\_d}^{l\_d} f\_{\boldsymbol{s}l}(\mathbf{s}\_{\boldsymbol{s}l}') \mathbf{G}\_{\boldsymbol{s}\_d}^{V\_2}(\mathbf{s}\_{\boldsymbol{s}l}, \mathbf{s}\_d') \mathrm{ds}\_{\boldsymbol{s}l}' = -i\alpha H\_{0s\_d} \{\mathbf{s}\_{\boldsymbol{s}l}\}. \tag{10}$$

Here *e* → *s* ′ *v* and *e* → *s* ′ *sl* are unit vectors of vibrator and slot axes at the sources, and

$$G\_{s\_v}^V(s\_v, s\_v') = \int\_{-\pi}^{\pi} \frac{e^{-ik\_1\sqrt{(s\_v - s\_v')^2 + \left[2r\sin(\phi/2)\right]^2}}}{\sqrt{(s\_v - s\_v')^2 + \left[2r\sin(\phi/2)\right]^2}} \nu(r, \phi) r \, d\phi,\tag{11}$$

d 2 2 1,2 1,2 / 2 ( ) () 2 2 / 2 (,) () . ( ) () *sl sl sl d ik s s V s sl sl <sup>d</sup> sl sl <sup>e</sup> G ss s s* x cx x x - -+ ¢ - ¢ = - + ¢ ò (12)

Solution of the integral equation with the exact kernel expressions (11) and (12) may be very difficult, therefore we will use approximate expressions, the so called "quasi-one-dimension‐ al" kernels [5,15]

$$\mathcal{G}\_{s\_v}^V(\mathbf{s}\_v, \mathbf{s}'\_v) = \frac{e^{-ik\_1 \sqrt{\left(s\_v - s'\_v\right)^2 + r^2}}}{\sqrt{\left(s\_v - s'\_v\right)^2 + r^2}},\tag{13}$$

$$\text{G}\_{s\_d}^{V\_{1,2}}\left(\mathbf{s}\_{sl}, \mathbf{s}\_{sl}'\right) = \frac{e^{-i\mathbf{k}\_{1,2}\sqrt{\left(\mathbf{s}\_{sl} - \mathbf{s}\_{sl}'\right)^2 + \left(d/4\right)^2}}}{\sqrt{\left(\mathbf{s}\_{sl} - \mathbf{s}\_{sl}'\right)^2 + \left(d/4\right)^2}}\tag{14}$$

derived with the assumption that source points belong to the geometric axes of the vibrator and slot while observation points belong to vibrator surface and to slot axis, having coordinates {*ssl* , *<sup>ξ</sup>* / 2}. In that case the functions *Gsv <sup>V</sup>* (*sv*, *<sup>s</sup>*′ *<sup>v</sup>*) and *Gssl V*1,2 (*ssl* , *s*′ *sl*) are everywhere continuous and equations for the currents are simplified significantly.

Since the form of the Green's functions was not specified, the equations (8) are valid for any electrodynamic volumes, provided that the Green's functions for any electrodynamic volumes are known or can be constructed. Although the boundary between the volumes *V*1 and *V*<sup>2</sup> initially was supposed to be of infinitesimal thickness, its actual thickness can be accounted for by introducing into the equations (8) an effective slot width, defined by the formula given in the Section 5.
