**3. Scattering on dielectric wedge placed between metal plates**

Let's consider an incident wave impinging from domain 1 loaded by dielectric with permittivity 3 upon dielectric discontinuity in domain 2, Figure 5. Electromagnetic field of this structure can be described in terms of LM and LE modes represented by *y*-component of electrical *<sup>e</sup>* and magnetic *<sup>m</sup>* Hertz vectors.

**Figure 4.** Imaginary part of normalized transverse wavenumber of fundamental LM mode versus normalized air gap size: (*a*) for certain normalized wavenumbers *kh* while 1=80; (*b*) for various permittivities of dielectric in domain 1 while normalized wavenumber is *kh* = 2.

**Figure 5.** Structure that illustrated scattering problem

284 Dielectric Material

devices.

appreciably less than for the LM mode.

while normalized wavenumber is *kh* = 2.

observed in more complicated tunable structures.

comparable with size of dielectric in domain 1 and quantitatively the alteration is

**Figure 3.** Normalized transverse wavenumber of basic LE mode versus normalized air gap size: (*a*) for certain normalized wavenumbers *kh* while 1 = 80; (*b*) for various permittivities of dielectric in domain 1

(a) (b)

Peculiarity of the LM mode is existence of *<sup>y</sup> E* -component of electrical field which is directed normally to the border of dielectric discontinuity. For the LE mode the component *<sup>y</sup> E* is equal to zero. Therefore to achieve considerable alteration of electromagnetic field a border should be located between dielectric and air to perturb normal component of the electric field. This principle should be applied to all of electromechanically controlled microwave

If the domain 1 contains lossy dielectric characterized by the loss tangent tan, then transverse wavenumber is a complex value and its imaginary part defines dielectric loss. Figure 4 demonstrates dependences of imaginary part of normalized transverse

Negative values of the imaginary part say, that dielectric losses in the structure would be reduced in comparison with homogeneous structure. Moreover, for certain frequency and air gap size the dielectric loss reaches a minimum. This effect is fundamental and is

Rigorous simulation of electromechanically controllable microwave devices requires solving of scattering problem on dielectric wedge placed between metal plates, Figure 5. Solution of

Let's consider an incident wave impinging from domain 1 loaded by dielectric with permittivity 3 upon dielectric discontinuity in domain 2, Figure 5. Electromagnetic field of

wavenumber of domain 1 for fundamental LM mode versus the normalized air gap.

the problem by the boundary element method (BEM) is discussed below.

**3. Scattering on dielectric wedge placed between metal plates** 

An incident wave in domain 1 is described by a sum of partial waves of LM and LE types:

$$\begin{aligned} \Gamma^{e+} &= \sum\_{i=0}^{n\_e} c\_i^e Y\_{1i}^e \left( y \right) X^e \left( \chi \right) e^{-j\mathbb{B}\_{z\_i} z} \\ \Gamma^{m+} &= \sum\_{i=1}^{n\_m} c\_i^m Y\_{1i}^m \left( y \right) X^m \left( \chi \right) e^{-j\mathbb{B}\_{z\_i} z} \end{aligned} \tag{3}$$

where *e m*( ) *<sup>i</sup> c* are amplitudes of partial waves, <sup>1</sup> *<sup>e</sup> Y y <sup>i</sup>* and <sup>1</sup> *<sup>m</sup> Y y <sup>i</sup>* are eigen functions of the domain 1, *<sup>e</sup> X x* and *<sup>m</sup> X x* are solutions of the Helmholtz equation

#### 286 Dielectric Material

 2 ( ) 2 ( ) <sup>2</sup> 0 *e m e m x dX x X x dx* , *x* is a constant, 22 2 *zi* 3 1*yi x k* is the propagation constant in domain 1, *y i* <sup>1</sup> *i h d* is an eigen value of the domain 1, *ne* is the quantity of incident LM modes, *nm* is the quantity of incident LE modes. Eigen functions of the domain 1 are equal to

$$Y\_{1i}^{e} \left( y \right) = \begin{cases} \overline{1\_{h+d}}^{\prime} & i = 0\\ \overline{2\_{h+d}}^{\prime} \cos \left( \mathfrak{h}\_{y\_{1i}} y \right), i \neq 0 \end{cases} \tag{4}$$

Electromechanical Control over Effective Permittivity Used for Microwave Devices 287

*y h*

, (8)

, (9)

*hyhd*

*y h*

*m y i y i i m m*

2 2 4 4 sin sin

*y i y i*

2 2 1 2

sin 2 sin 2

*h d h d*

1 2 1 2

*h d* .

(10)

*m m y i y i m m*

*hyhd*

Eigen functions of the domain 2 are equal to

*e e y i y i e e*

*y i y i*

electrical and magnetic field at the boundary of spatial domains:

*e y*

*e e*

*y k <sup>E</sup> <sup>y</sup> f y Xx Xx*

*i*

*m y*

, 1

, 0 , <sup>2</sup> ,

*<sup>y</sup> y h <sup>i</sup> hyhd*

3 1 2

0

*h d*

0

*h d* ,

sin 2 sin 2

*h d h d*

1 2 1 2 2 2 1122

2

 

2 2 4 4 cos cos

*e y i y i i e e*

where

*N*

where *i*=1,2,

*<sup>e</sup> f y* and *mf y* :

space.

*i*

*Y y*

2

*Y y*

*e y i e e*

1 1

cos

*N h*

1

2 2

*m y i m m*

1

*y*

1

sin

*N d*

2

*y i m m i yi*

sin

*N h*

*hdy*

2

*N*

Using orthogonality property of eigen functions, amplitudes of eigen functions were expressed via indeterminate functions which are proportional to tangential components of

2

*e i i*

0

*m m*

,

,

Equality requirement for another tangential components of electromagnetic filed reduces the scattering problem to set of Fredholm integral equations of the first kind for functions

> 

0

*<sup>y</sup> <sup>k</sup> Z H <sup>y</sup> fy Z Xx Xx*

2

cos

*N d*

2

*y i e e i yi*

sin

cos

sin

*i yi m i m*

*i yi e i e*

cos

*yhd*

, 0

,

, 0

,

2

,

2

*Z* is the characteristic impedance of free

*e em m G y y f y G y y f y dy y j jj j* , (11)

*i*

*m i*

2

0

0

, ( ) , ( ) , 1,2

2

*y*

$$Y\_{1i}^{m}\left(y\right) = \sqrt{\frac{2}{h+d}}\sin\left(\mathfrak{d}\_{y1i}y\right). \tag{5}$$

Reflected wave is represented as series of the 1st domain's eigen functions as

$$\begin{aligned} \Gamma^{\epsilon-} &= \sum\_{i=0}^{\infty} a\_{1i}^{\epsilon} Y\_{1i}^{\epsilon} \left( y \right) \mathbf{X}^{\epsilon} \left( \mathbf{x} \right) e^{j\beta\_{xi} z}, \\ \Gamma^{m-} &= \sum\_{i=1}^{\infty} a\_{1i}^{m} Y\_{1i}^{m} \left( y \right) \mathbf{X}^{m} \left( \mathbf{x} \right) e^{j\beta\_{xi} z}, \end{aligned} \tag{6}$$

where ( ) 1 *e m <sup>i</sup> a* are amplitudes of eigen modes.

Total electromagnetic field in the 1st domain is expressed as a composition of incident (3) and reflected (6) waves.

Electromagnetic field in the 2nd domain is represented as series of domain's eigen functions as

$$\begin{aligned} \Gamma\_2^{\epsilon} &= \sum\_{i=0}^{\infty} a\_{2i}^{\epsilon} \mathfrak{p} \left( \mathfrak{y} \right) Y\_{2i}^{\epsilon} \left( \mathfrak{y} \right) X\_i^{\epsilon} \left( \mathfrak{x} \right) e^{-\beta \mathfrak{k}\_{zi}^{\epsilon} z}, \\ \Gamma\_2^{m} &= \sum\_{k=0}^{\infty} a\_{2i}^{m} Y\_{2i}^{m} \left( \mathfrak{y} \right) X\_i^{m} \left( \mathfrak{x} \right) e^{-\beta \mathfrak{k}\_{zi}^{m} z} \end{aligned} \tag{7}$$

where ( ) 2 *e m <sup>i</sup> a* are amplitude of eigen modes in the domain 2, 1 2 , , *y h y hyhd* is the weight function, 2 2 ( )2 2 2 2 1 2 1 2 *e m e m e m zi x i y i y i k k* is the propagation constant of the domain 2, *k* is the propagation constant in free space, 1(2) *e y i* are *i*-th solutions of the equations (1), 1(2) *m y i* are eigen values of magnetic Hertz vector in the domain 2 computed from the equations (2), ( ) 2 *e m Y y <sup>k</sup>* are eigen functions of the 2nd domain.

Eigen functions of the domain 2 are equal to

286 Dielectric Material

2 ( )

*e m*

1 are equal to

where ( ) 1 *e m*

where ( ) 2 *e m*

equations (1), 1(2)

*m*

from the equations (2), ( )

as

and reflected (6) waves.

*dX x*

constant in domain 1,

2 ( ) <sup>2</sup> 0

*e m x*

*X x*

*dx* , *x* is a constant, 22 2

*h d Y y*

Reflected wave is represented as series of the 1st domain's eigen functions as

*i*

*i*

weight function,

2

of the domain 2, *k* is the propagation constant in free space, 1(2)

22 2 0

*k*

2 22 0

*<sup>i</sup> a* are amplitude of eigen modes in the domain 2,

( )2 2 2 2 1 2 1 2 *e m e m e m zi x i y i y i*

*<sup>i</sup> a* are amplitudes of eigen modes.

*i*

1

*e i*

incident LM modes, *nm* is the quantity of incident LE modes. Eigen functions of the domain

*y i h d*

1 1

*e ee e j z i i*

*aY yX xe*

1 1 0

> 1 1 1

*m mm m j z i i*

Total electromagnetic field in the 1st domain is expressed as a composition of incident (3)

Electromagnetic field in the 2nd domain is represented as series of domain's eigen functions

*a yY yX xe*

*e e ee j z i ii*

2 2

*e m Y y <sup>k</sup>* are eigen functions of the 2nd domain.

*aY yX xe*

*m mm m j z ii i*

*aY yX xe*

*y i*

<sup>2</sup> cos , 0

<sup>1</sup> , 0

1

*i*

 *y i* <sup>1</sup> *i*

*zi* 3 1*yi x k* is the propagation

, (4)

(6)

(7)

*y h*

is the

1 2 , ,

*hyhd*

*y i* are *i*-th solutions of the

*h d* is an eigen value of the domain 1, *ne* is the quantity of

<sup>2</sup> sin *<sup>m</sup> Y y <sup>i</sup> y i<sup>y</sup> h d* . (5)

,

*zi*

*zi*

,

,

*y*

*e*

*e zi*

,

*k k* is the propagation constant

*y i* are eigen values of magnetic Hertz vector in the domain 2 computed

*m zi*

$$Y\_{2i}^{\epsilon} \left( y \right) = \begin{cases} \frac{\cos \left( \mathfrak{k}\_{y1i}^{\epsilon} y \right)}{N\_i^{\epsilon} \varepsilon\_1 \cos \left( \mathfrak{k}\_{y1i}^{\epsilon} h \right)}, & 0 \le y \le h \\\frac{\cos \left( \mathfrak{k}\_{y2i}^{\epsilon} \left( y - h - d \right) \right)}{N\_i^{\epsilon} \varepsilon\_2 \cos \left( \mathfrak{k}\_{y2i}^{\epsilon} d \right)}, h \le y \le h + d \end{cases} \tag{8}$$

$$Y\_{2i}^{m}\left(y\right) = \begin{cases} \frac{\sin\left(\mathfrak{B}\_{y11}^{m}y\right)}{N\_i^{m}\sin\left(\mathfrak{B}\_{y11}^{m}h\right)}, & 0 \le y \le h\\ \frac{\sin\left(\mathfrak{B}\_{y2i}^{m}\left(h+d-y\right)\right)}{N\_i^{m}\sin\left(\mathfrak{B}\_{y2i}^{m}d\right)}, h \le y \le h+d \end{cases} \tag{9}$$

$$\text{where } N\_i^\varepsilon = \sqrt{\frac{\frac{h}{2} + \frac{\sin\left(2\mathfrak{d}\_{y1i}^\varepsilon h\right)}{4\mathfrak{d}\_{y1i}^\varepsilon}}{\varepsilon\_1 \cos^2\left(\mathfrak{d}\_{y1i}^\varepsilon h\right)}} + \frac{\frac{d}{2} + \frac{\sin\left(2\mathfrak{d}\_{y2i}^\varepsilon d\right)}{4\mathfrak{d}\_{y2i}^\varepsilon}}{\varepsilon\_2 \cos^2\left(\mathfrak{d}\_{y2i}^\varepsilon d\right)}}, \text{ } N\_i^\mathfrak{m} = \sqrt{\frac{\frac{h}{2} - \frac{\sin\left(2\mathfrak{d}\_{y1i}^\varepsilon h\right)}{4\mathfrak{d}\_{y1i}^\varepsilon}}{\sin^2\left(\mathfrak{d}\_{y1i}^\varepsilon h\right)}} + \frac{\frac{d}{2} - \frac{\sin\left(2\mathfrak{d}\_{y2i}^\varepsilon d\right)}{4\mathfrak{d}\_{y2i}^\varepsilon}}{\sin^2\left(\mathfrak{d}\_{y2i}^\varepsilon d\right)}.$$

Using orthogonality property of eigen functions, amplitudes of eigen functions were expressed via indeterminate functions which are proportional to tangential components of electrical and magnetic field at the boundary of spatial domains:

$$\begin{aligned} f^{\epsilon} \left( y \right) &= \frac{E\_y}{X^{\epsilon} \left( x \right)} = \frac{\frac{\hat{\partial}^2 \Gamma\_i^{\epsilon}}{\hat{\partial} y^2} + \varepsilon\_i \left( y \right) k^2}{X^{\epsilon} \left( x \right)}, \\\ f^{\epsilon m} \left( y \right) &= \frac{Z\_0 H\_y}{X^m \left( x \right)} = Z\_0 \frac{\frac{\hat{\partial}^2 \Gamma\_i^{\epsilon m}}{\hat{\partial} y^2} + \varepsilon\_i \left( y \right) k^2}{X^m \left( x \right)}, \\\ f^{\epsilon\_{3\epsilon}} \left( \begin{matrix} \varepsilon\_{3\epsilon} & & & \mathbf{i} = \mathbf{1} \\ \varepsilon\_{3\epsilon} & & & \mathbf{Z} \end{matrix} \right) &= 1 \end{aligned} \tag{10}$$

where *i*=1,2, 1 2 , 0 , <sup>2</sup> , *i <sup>y</sup> y h <sup>i</sup> hyhd* , 0 0 0 *Z* is the characteristic impedance of free space.

Equality requirement for another tangential components of electromagnetic filed reduces the scattering problem to set of Fredholm integral equations of the first kind for functions *<sup>e</sup> f y* and *mf y* :

$$\int\_{0}^{h+d} \left( G\_{\slash}^{\epsilon} \begin{pmatrix} y, y' \end{pmatrix} f^{\epsilon} (y) + G\_{\not p}^{m} \begin{pmatrix} y, y' \end{pmatrix} f^{m} (y) \right) dy = \phi\_{\not p} \begin{pmatrix} y \end{pmatrix}, \quad j = 1, 2 \end{pmatrix} \tag{11}$$

where kernels of integral equations , *<sup>e</sup> G yy <sup>j</sup>* and , *<sup>m</sup> G yy <sup>j</sup>* are expressed via eigen functions of domains 1 and 2:

$$G\_{1}^{\epsilon} \left( y, y' \right) = \pm \mathbb{B}\_{x} \sum\_{i=0}^{x} \left| \frac{Y\_{1i}^{\epsilon} \left( y' \right) \frac{dY\_{1i}^{\epsilon} \left( y \right)}{dy}}{\kappa\_{3} k^{2} - \mathfrak{b}\_{y1i}^{2}} - \frac{\rho^{2} \left( y' \right) Y\_{2i}^{\epsilon} \left( y' \right) \frac{dY\_{2i}^{\epsilon} \left( y \right)}{dy}}{\left( \kappa\_{1} k^{2} - \mathfrak{b}\_{y1i}^{\epsilon} \right)^{2}} \right|, \tag{12}$$

Electromechanical Control over Effective Permittivity Used for Microwave Devices 289

*E r* <sup>2</sup> , (18)

, 22 22 <sup>2</sup> 12 3 21 3 31 2 *s* 2( ) ( ) ( ) ,

*dy*

, (19)

, (20)

1 1 **E H**, are

*dy*

*dy*

*dy*

equations was reduced to a system of linear algebraic equations by ordinary Galerkin

For small values of *d*/*h* eigen functions of domains 1 and 2 were selected as a basis of the Galerkin method. However, for large values of *d*/*h* to improve convergence for proper selection of basis it is necessary to take into account that in close proximity to dielectric edge

> 

To satisfy (18) the Gegenbauer polynomials *C y <sup>n</sup>* shall be used as a basis of the Galerkin method. As a consequence, scattered electromagnetic field is calculated from computed

> 

*k kz*

( ) ( )

*e m e m*

**EHe**

\* 1 1

( ) ( )

*e m e m*

**EHe**

\* 1 1

*k kz*

( ) ( )

*e m e m*

**EHe**

\* 2 2

( ) ( )

*e m e m*

**EHe**

\* 1 1

*j jz*

2 2 **E H**, are forward propagated transverse components of electrical and

*j jz*

 

 

 

where indices *je* and *jm* define numbers of incident LM and LE modes, but indexes *ke* and *km* define numbers of scattered LM and LE modes, **e**z is the unit vector of *z*-axis,

transverse components of electrical and magnetic field in the domain 1 forward propagated,

1 1 **E H**, are transverse components of electrical and magnetic field in the domain 1 back

Figure 6 demonstrates a comparison of computed components of scattering matrix by the proposed (BEM) and finite-difference time-domain (FDTD) methods for the structure

1

 <sup>2</sup> 1 arctan 1 <sup>2</sup> ,

electromagnetic field behaves according to the law:

where *r* is the distance to dielectric edge,

Multimode scattering matrix can be computed from the equations:

() ()

() ()

*em em*

*j k*

21

*S*

*em em*

*j k*

11

*S*

*h d*

0

*h d*

0

*h d*

0

*h d*

0

 2 2 2 21 3 12 3 2

solution for functions *<sup>e</sup> f y* and *mf y* .

*s*

procedure.

*tt s*

 <sup>2</sup> 31 2 *t* ( ) .

propagated,

magnetic field in the domain 2.

$$G\_{1}^{m}\left(y,y'\right) = -k\sum\_{i=0}^{\alpha} \left( \frac{\mathfrak{B}\_{zi}Y\_{1i+1}^{m}\left(y'\right)Y\_{1i+1}^{m}\left(y\right)}{\varepsilon\_{3}k^{2} - \mathfrak{B}\_{y1i+1}^{2}} + \frac{\mathfrak{B}\_{zi}^{m}Y\_{2i}^{m}\left(y'\right)Y\_{2i}^{m}\left(y\right)}{\varepsilon\_{1}k^{2} - \mathfrak{B}\_{y1i}^{m}} \right),\tag{13}$$

$$\mathbf{G}\_{2}^{\epsilon} \begin{pmatrix} y, y' \end{pmatrix} = \alpha \varepsilon\_{0} \sum\_{i=0}^{\alpha} \left( \frac{\varepsilon\_{3} \mathfrak{R}\_{zi} \chi\_{1i}^{\epsilon} (y') \chi\_{1i}^{\epsilon} (y)}{\varepsilon\_{3} k^{2} - \mathfrak{R}\_{y1i}^{2}} + \frac{\varepsilon\_{2} \left( y \right) \rho^{2} \left( y' \right) \mathfrak{R}\_{zi}^{\epsilon} \chi\_{2i}^{\epsilon} (y') \chi\_{2i}^{\epsilon} (y)}{\varepsilon\_{1} k^{2} - \mathfrak{R}\_{y1i}^{\epsilon} \end{pmatrix} \right) \tag{14}$$

$$G\_{2}^{m}\left(y, y'\right) = \mp \frac{\beta\_{x}}{Z\_{0}} \sum\_{i=0}^{\infty} \left| \frac{Y\_{1i+1}^{m}\left(y'\right)\frac{dY\_{1i+1}^{m}\left(y\right)}{dy}}{\varepsilon\_{3}k^{2} - \beta\_{y1i+1}^{2}} - \frac{Y\_{2i}^{m}\left(y'\right)\frac{dY\_{2i}^{m}\left(y\right)}{dy}}{\varepsilon\_{1}k^{2} - \beta\_{y1i}^{m}} \right|,\tag{15}$$

where is the circular frequency, 0 is the dielectric constant in vacuum, 0 is the magnetic constant. Sign in (12) depends on relation between signs in *<sup>e</sup> dX x dx* and *<sup>m</sup> X x* . If *<sup>e</sup> m x dX x X x dx* then the sign "+" shall be applied in (12). However if *<sup>e</sup> m x dX x X x dx* then the sign of (12) shall be "—".

Functions *<sup>j</sup> y* are described by incident partial waves:

$$\phi\_1\left(y\right) = -2\operatorname{co}\mu\_0 \sum\_{i=1}^{n\_{1m}} \mathbb{B}\_{zi} c\_{1i}^m Y\_{1i}^m \left(y\right),\tag{16}$$

$$\mathfrak{d}\_2\left(y\right) = 2\cos\varepsilon\_0 \varepsilon\_3 \sum\_{i=0}^{n\_{1c}} \mathfrak{d}\_{zi} c\_{1i}^{\epsilon} Y\_{1i}^{\epsilon} \left(y\right). \tag{17}$$

The set of integral equations (11) was solved by Galerkin method. Functions *<sup>e</sup> f y* and *mf y* were expanded in respect to basis ( ) 0 *e m y* , ( ) 1 *e m y* , … and set of integral equations was reduced to a system of linear algebraic equations by ordinary Galerkin procedure.

For small values of *d*/*h* eigen functions of domains 1 and 2 were selected as a basis of the Galerkin method. However, for large values of *d*/*h* to improve convergence for proper selection of basis it is necessary to take into account that in close proximity to dielectric edge electromagnetic field behaves according to the law:

$$E \sim r^{\frac{\nu-1}{2}},\tag{18}$$

where *r* is the distance to dielectric edge, <sup>2</sup> 1 arctan 1 <sup>2</sup> ,

288 Dielectric Material

functions of domains 1 and 2:

,

,

,

 *<sup>e</sup> m x*

 *<sup>e</sup> m x*

*X x*

*X x*

*dX x*

*dX x*

*dx* then the sign of (12) shall be "—".

Functions *<sup>j</sup> y* are described by incident partial waves:

*mf y* were expanded in respect to basis ( )

,

where kernels of integral equations , *<sup>e</sup> G yy <sup>j</sup>* and , *<sup>m</sup> G yy <sup>j</sup>* are expressed via eigen

*e zi i i zi i i*

<sup>0</sup> <sup>2</sup> <sup>0</sup> 3 11 <sup>1</sup> <sup>1</sup>

*<sup>m</sup> <sup>i</sup> y i y i*

1 1 2

<sup>0</sup> <sup>2</sup> 3 1 <sup>1</sup> <sup>1</sup>

*<sup>e</sup> <sup>i</sup> y i y i Y yY y y y Y yY y G yy <sup>k</sup> <sup>k</sup>*

1 2 2 2

*Y yY y Y yY y G yy k <sup>k</sup> <sup>k</sup>*

*m zi i i zi i i*

11 11 2 2

<sup>0</sup> <sup>2</sup> 3 11 <sup>1</sup> <sup>1</sup>

*<sup>m</sup> <sup>i</sup> y i y i*

1 2 2 2 0 2 3 1

*dY y dY y Y y yY y dy dy G yy <sup>k</sup> <sup>k</sup>*

*i i <sup>e</sup>*

2 0 2 2 2

2 2 2 2

where is the circular frequency, 0 is the dielectric constant in vacuum, 0 is the magnetic

*dx* then the sign "+" shall be applied in (12). However if

 

 

*i*

The set of integral equations (11) was solved by Galerkin method. Functions *<sup>e</sup> f y* and

0

1 2 03 1 1 0

*i*

1 1 0 11 1

*m m zi i i*

*e e zi i i*

*e m y* , ( ) 1

*y cY y* , (16)

*y cY y* . (17)

*e m y* , … and set of integral

<sup>2</sup> *<sup>m</sup> <sup>n</sup>*

<sup>2</sup> *<sup>e</sup> <sup>n</sup>*

*dY y dY y Y y Yy dy dy G yy <sup>Z</sup> <sup>k</sup> <sup>k</sup>*

constant. Sign in (12) depends on relation between signs in *<sup>e</sup> dX x*

*i i <sup>m</sup> <sup>x</sup>*

1 2

*<sup>x</sup> <sup>e</sup> <sup>i</sup> y i y i*

 

1 2 2

*e e e e i i*

1 1

, (12)

, (13)

, (14)

, (15)

and *<sup>m</sup> X x* . If

*dx*

*m m mm m*

*e e ee e*

31 1 2 2 2

2

1 1 2

*m m*

*m m i i*

$$\begin{split} \eta &= \frac{s}{t - \sqrt{t^2 + 2s\left(\mathfrak{e}\_2\left(\mathfrak{e}\_1 + \mathfrak{e}\_3\right)^2 + \mathfrak{e}\_1\left(\mathfrak{e}\_2 + \mathfrak{e}\_3\right)^2\right)}}, \quad s = 2\left(\mathfrak{e}\_1\left(\mathfrak{e}\_2^2 + \mathfrak{e}\_3^2\right) + \mathfrak{e}\_2\left(\mathfrak{e}\_1^2 + \mathfrak{e}\_3^2\right) + \mathfrak{e}\_3\left(\mathfrak{e}\_1 + \mathfrak{e}\_2\right)^2\right), \\ t &= \mathfrak{e}\_3\left(\mathfrak{e}\_1 - \mathfrak{e}\_2\right)^2. \end{split}$$

To satisfy (18) the Gegenbauer polynomials *C y <sup>n</sup>* shall be used as a basis of the Galerkin method. As a consequence, scattered electromagnetic field is calculated from computed solution for functions *<sup>e</sup> f y* and *mf y* .

Multimode scattering matrix can be computed from the equations:

$$\begin{split} \mathbf{S}\_{11}^{j\_{\epsilon(w)}k\_{\epsilon(w)}} &= \sqrt{\frac{\int \mathbf{E}\_{\perp 1k\_{\epsilon(w)}}^{+} \times \mathbf{H}^{\*-}\_{\perp 1k\_{\epsilon(w)}} \cdot \mathbf{e}\_z dy}{\sqrt{\int \mathbf{E}\_{\perp 1j\_{\epsilon(w)}}^{+} \times \mathbf{H}^{\*+}\_{\perp 1j\_{\epsilon(w)}} \cdot \mathbf{e}\_z dy}}, \\\\ \mathbf{S}\_{21}^{j\_{\epsilon(w)}k\_{\epsilon(w)}} &= \sqrt{\frac{\int \mathbf{E}\_{\perp 2k\_{\epsilon(w)}}^{+} \times \mathbf{H}^{\*+}\_{\perp 2k\_{\epsilon(w)}} \cdot \mathbf{e}\_z dy}{\sqrt{\int \mathbf{E}\_{\perp 1j\_{\epsilon(w)}}^{+} \times \mathbf{H}^{\*+}\_{\perp 1j\_{\epsilon(w)}} \cdot \mathbf{e}\_z dy}}, \end{split} \tag{19}$$

where indices *je* and *jm* define numbers of incident LM and LE modes, but indexes *ke* and *km* define numbers of scattered LM and LE modes, **e**z is the unit vector of *z*-axis, 1 1 **E H**, are transverse components of electrical and magnetic field in the domain 1 forward propagated, 1 1 **E H**, are transverse components of electrical and magnetic field in the domain 1 back propagated, 2 2 **E H**, are forward propagated transverse components of electrical and magnetic field in the domain 2.

Figure 6 demonstrates a comparison of computed components of scattering matrix by the proposed (BEM) and finite-difference time-domain (FDTD) methods for the structure characterized by the parameters: 1=10, 2=1, 3=1, *d*/*h*=0.01, / 6 *<sup>x</sup>h* . As it is seen, there is good agreement between proposed and FDTD methods. However computing time for BEM is much less than for FDTD method due to lower order of resulting system of linear algebraic equation.

Electromechanical Control over Effective Permittivity Used for Microwave Devices 291

 2

max 2

0 . At this assumption as it is seen from (1) normalized transverse

*<sup>y</sup>* tends to zero as well. In this case the equations (1) can be solved

*d h k d h*

1

 

Substitution of (23) into (22) gives utmost value of relative alteration of effective

 

2. If medium in domain 2 (Figure 1) is air, then the range of effective permittivity alteration due to displacement of metal plate is from 1 to 1. Such high controllability is not available in

Graphically dependence (24) is presented in Figure 7. This dependence demonstrates utmost controllability of effective permittivity of the dielectric discontinuity by alteration of distance between metal plate and dielectric if medium in the domain 2 is air. For certain

Due to limitation of effective permittivity alteration from 1 to 2 and utmost value of normalized transverse wavenumber equal to /2, to achieve utmost controllability the

lim *eff <sup>k</sup> <sup>h</sup>*

1 2

2 1

*d*

1 2

lim *<sup>e</sup> <sup>y</sup> <sup>k</sup>*

other methods of control including electrical bias control of ferroelectrics.

analogous dependences stand below those presented in Figure 7.

normalized propagation constant shall satisfy to requirement:

permittivity 1 the dependence is an upper asymptote for other values of

0

<sup>1</sup> <sup>0</sup> <sup>2</sup>

*eff <sup>k</sup>* , (22)

*k* maximal alteration of normalized

. (23)

. (24)

*k* . For

*k* 0

0 effective permittivity can be controlled from 1 to

<sup>1</sup> <sup>4</sup> *eff <sup>k</sup>* . Criterion of large and small

where 1 1 *e e*

values of

permittivity 1 and

the assumption of *k*

 1 *e*

It is seen that on the assumption of *k*

wavenumber

analytically:

permittivity:

for large values of normalized propagation constant

*k* will be determined below.

effective permittivity is restricted by the value

*k kh* is normalized propagation constant in free space. Utmost value of the normalized transverse wavenumber is equal to /2 (Figure 2). Therefore

As it follows from (22) relative alteration of effective permittivity is increased while normalized propagation constant is reduced. Utmost range of the alteration can be found on

1 1

*y y h* is normalized transverse wavenumber in domain filled by dielectric with

1 1

 

*e eff y* 2

2

**Figure 6.** Reflection (*S*11) and transmission (*S*21) coefficients computed by boundary element (BEM) and finite-difference time-domain (FDTD) methods. 1=10, 2=1, 3=1, *d*/*h*=0.01, / 6 *<sup>x</sup>h*

### **4. Effective permittivity of one-dimensional dielectric discontinuity**

Transverse wavenumber defines a propagation constant of the structure presented in Figure 1, which contains a discontinuity. Effective permittivity of the structure can be stated as such permittivity of homogeneous structure, which gives numerically the same propagation constant as in inhomogeneous structure. The effective permittivity of basic LM mode can be easily recomputed from transverse wavenumber by the equation:

$$
\varepsilon\_{\rm eff} = \varepsilon\_1 - \frac{\beta\_{y1}^{\epsilon}}{k^2}^2 \,\, . \tag{21}
$$

As it follows from the equation (21), nature of the dependence of effective permittivity on distance between metal plate to dielectric is determined by the function <sup>1</sup> *e <sup>y</sup> d* . Let's consider alteration limit of effective permittivity while displacement of metal plate under the dielectric. It follows from equation (21) that relative alteration of effective permittivity can be derived from the equation:

Electromechanical Control over Effective Permittivity Used for Microwave Devices 291

$$\delta \varepsilon\_{\it eff} = \frac{\varepsilon\_1 - \varepsilon\_{\it eff}}{\varepsilon\_1} = \frac{\tilde{\bf{B}}\_{\it y1}^{\varepsilon}}{\varepsilon\_1 \tilde{k}^2} \; \; \; \tag{22}$$

where 1 1 *e e y y h* is normalized transverse wavenumber in domain filled by dielectric with permittivity 1 and *k kh* is normalized propagation constant in free space.

Utmost value of the normalized transverse wavenumber is equal to /2 (Figure 2). Therefore for large values of normalized propagation constant *k* maximal alteration of normalized effective permittivity is restricted by the value 2 max 2 <sup>1</sup> <sup>4</sup> *eff <sup>k</sup>* . Criterion of large and small

values of *k* will be determined below.

290 Dielectric Material

algebraic equation.

characterized by the parameters: 1=10, 2=1, 3=1, *d*/*h*=0.01, / 6 *<sup>x</sup>h* . As it is seen, there is good agreement between proposed and FDTD methods. However computing time for BEM is much less than for FDTD method due to lower order of resulting system of linear

**Figure 6.** Reflection (*S*11) and transmission (*S*21) coefficients computed by boundary element (BEM) and

Transverse wavenumber defines a propagation constant of the structure presented in Figure 1, which contains a discontinuity. Effective permittivity of the structure can be stated as such permittivity of homogeneous structure, which gives numerically the same propagation constant as in inhomogeneous structure. The effective permittivity of basic LM mode can be

2 1 1 2 *e y*

*eff <sup>k</sup>* . (21)

*e*

*<sup>y</sup> d* . Let's

**4. Effective permittivity of one-dimensional dielectric discontinuity** 

distance between metal plate to dielectric is determined by the function <sup>1</sup>

As it follows from the equation (21), nature of the dependence of effective permittivity on

consider alteration limit of effective permittivity while displacement of metal plate under the dielectric. It follows from equation (21) that relative alteration of effective permittivity

finite-difference time-domain (FDTD) methods. 1=10, 2=1, 3=1, *d*/*h*=0.01, / 6 *<sup>x</sup>h*

easily recomputed from transverse wavenumber by the equation:

can be derived from the equation:

As it follows from (22) relative alteration of effective permittivity is increased while normalized propagation constant is reduced. Utmost range of the alteration can be found on the assumption of *k* 0 . At this assumption as it is seen from (1) normalized transverse wavenumber 1 *e <sup>y</sup>* tends to zero as well. In this case the equations (1) can be solved analytically:

$$\lim\_{\tilde{k}\to 0} \tilde{\mathfrak{F}}\_{y1}^{\varepsilon} = \sqrt{\frac{(\varepsilon\_1 - \varepsilon\_2)\frac{d}{h}}{\frac{\varepsilon\_2}{\varepsilon\_1} + \frac{d}{h}}} \tilde{k} \,. \tag{23}$$

Substitution of (23) into (22) gives utmost value of relative alteration of effective permittivity:

$$\lim\_{\tilde{k}\to 0} \delta \varepsilon\_{\varepsilon \stackrel{\text{eff}}{\ell}} = \frac{\varepsilon\_1 - \varepsilon\_2}{\varepsilon\_2 \frac{h}{d} + \varepsilon\_1} \,. \tag{24}$$

It is seen that on the assumption of *k* 0 effective permittivity can be controlled from 1 to 2. If medium in domain 2 (Figure 1) is air, then the range of effective permittivity alteration due to displacement of metal plate is from 1 to 1. Such high controllability is not available in other methods of control including electrical bias control of ferroelectrics.

Graphically dependence (24) is presented in Figure 7. This dependence demonstrates utmost controllability of effective permittivity of the dielectric discontinuity by alteration of distance between metal plate and dielectric if medium in the domain 2 is air. For certain permittivity 1 the dependence is an upper asymptote for other values of *k* . For *k* 0 analogous dependences stand below those presented in Figure 7.

Due to limitation of effective permittivity alteration from 1 to 2 and utmost value of normalized transverse wavenumber equal to /2, to achieve utmost controllability the normalized propagation constant shall satisfy to requirement:

Electromechanical Control over Effective Permittivity Used for Microwave Devices 293

is reduced at the same condition. The point is that the controllability of effective permittivity depends on ratio of transverse wavenumber to propagation constant rather than transverse

**Figure 8.** Dependence of relative alteration of effective permittivity on distance between metal plate

Requirement (25) can be considered as criteria of smallness of normalized propagation constant. If this requirement is satisfied then the normalized propagation constant may be

Hereby to increase controllability of effective permittivity one should reduce normalized propagation constant. It can be done by two ways. The first method is decreasing of working frequency. However this way has limitation because for many implementations the frequency shall exceed cutoff frequency and cannot be reduced. The second way is to reduce thickness of dielectric *h*. It follows from (25) that efficient controllability of effective

1 2 2

If requirement (26) is not satisfied then the range of effective permittivity alteration is decreased according to the law close to <sup>2</sup> *h* . Moreover required displacement of metal

Effective permittivity model simplifies understanding and simulation of phenomena in controllable microwave devices. This model accurately describes wavelength of fundamental mode in controllable structure. However accuracy of scattering problem description should be investigated. Let's compare scattering matrix derived from effective permittivity model and rigorous solution of scattering problem by the BEM described

*<sup>k</sup>* . (26)

*h*

and dielectric for certain normalized propagation constant *kh* while 1 = 50, 2 = 1.

permittivity the thickness should to satisfy the requirement:

plates for effective permittivity control would be increased.

considered as small, otherwise as large.

wavenumber.

above.

**Figure 7.** Dependence of relative alteration of effective permittivity on distance between metal plate and dielectric on the assumption of *k* 0 .

Neglect of requirement (25) leads to decrease of controllability range and *eff* is limited by the value 2 1 2 <sup>2</sup> <sup>4</sup>*<sup>k</sup>* . The same conclusion can be derived from analysis of the formula (24) on the assumption of *d h* and solution of equations (1).

Figure 8 demonstrates influence of normalized propagation constant on utmost range of effective permittivity alteration. This picture reflects dependence of relative alteration of effective permittivity on normalized distance between metal plate and dielectric with permittivity <sup>1</sup> 50 . For this permittivity of dielectric the requirement (25) is transformed to 0.224 14 *k* . As it is seen in Figure 8 if the last requirement is not satisfied then the range

of effective permittivity alteration is considerably reduced and if *k* 0.6 is only equal to

10% from <sup>1</sup> , and if 0.02 *<sup>d</sup> h* then effective permittivity is almost independent on distance

between metal plate and dielectric. Similar phenomenon in waveguides filled by multilayer dielectric if phase velocity of electromagnetic wave is almost independent on sizes of highpermittivity dielectrics was named as dielectric effect or effect of dielectric waveguide.

As it is seen in Figure 8 dependence of alteration of effective permittivity on normalized propagation constant has opposite trend than analogous dependence of alteration of transverse wavenumber (see Figure 2, *a*): controllability of transverse wavenumber is increased while propagation constant is risen up but controllability of effective permittivity is reduced at the same condition. The point is that the controllability of effective permittivity depends on ratio of transverse wavenumber to propagation constant rather than transverse wavenumber.

292 Dielectric Material

and dielectric on the assumption of *k*

 2

(24) on the assumption of

10% from <sup>1</sup> , and if 0.02 *<sup>d</sup>*

*d*

the value

 0.224 14

*k* . (25)

*k* 0.6 is only equal to

1 2 2

**Figure 7.** Dependence of relative alteration of effective permittivity on distance between metal plate

Neglect of requirement (25) leads to decrease of controllability range and *eff* is limited by

*h* and solution of equations (1).

Figure 8 demonstrates influence of normalized propagation constant on utmost range of effective permittivity alteration. This picture reflects dependence of relative alteration of effective permittivity on normalized distance between metal plate and dielectric with permittivity <sup>1</sup> 50 . For this permittivity of dielectric the requirement (25) is transformed to

*k* . As it is seen in Figure 8 if the last requirement is not satisfied then the range

between metal plate and dielectric. Similar phenomenon in waveguides filled by multilayer dielectric if phase velocity of electromagnetic wave is almost independent on sizes of highpermittivity dielectrics was named as dielectric effect or effect of dielectric waveguide.

As it is seen in Figure 8 dependence of alteration of effective permittivity on normalized propagation constant has opposite trend than analogous dependence of alteration of transverse wavenumber (see Figure 2, *a*): controllability of transverse wavenumber is increased while propagation constant is risen up but controllability of effective permittivity

1 2 <sup>2</sup> <sup>4</sup>*<sup>k</sup>* . The same conclusion can be derived from analysis of the formula

*h* then effective permittivity is almost independent on distance

0 .

of effective permittivity alteration is considerably reduced and if

**Figure 8.** Dependence of relative alteration of effective permittivity on distance between metal plate and dielectric for certain normalized propagation constant *kh* while 1 = 50, 2 = 1.

Requirement (25) can be considered as criteria of smallness of normalized propagation constant. If this requirement is satisfied then the normalized propagation constant may be considered as small, otherwise as large.

Hereby to increase controllability of effective permittivity one should reduce normalized propagation constant. It can be done by two ways. The first method is decreasing of working frequency. However this way has limitation because for many implementations the frequency shall exceed cutoff frequency and cannot be reduced. The second way is to reduce thickness of dielectric *h*. It follows from (25) that efficient controllability of effective permittivity the thickness should to satisfy the requirement:

$$h < \frac{\pi}{2k\sqrt{\varepsilon\_1 - \varepsilon\_2}}.\tag{26}$$

If requirement (26) is not satisfied then the range of effective permittivity alteration is decreased according to the law close to <sup>2</sup> *h* . Moreover required displacement of metal plates for effective permittivity control would be increased.

Effective permittivity model simplifies understanding and simulation of phenomena in controllable microwave devices. This model accurately describes wavelength of fundamental mode in controllable structure. However accuracy of scattering problem description should be investigated. Let's compare scattering matrix derived from effective permittivity model and rigorous solution of scattering problem by the BEM described above.

Scattering matrix for effective permittivity approach can be found from equations:

$$S\_{11} = \frac{\mathfrak{B}\_{z1} - \mathfrak{B}\_{z2}}{\mathfrak{B}\_{z1} + \mathfrak{B}\_{z2}}, \quad S\_{21} = \frac{2\sqrt{\mathfrak{B}\_{z1}\mathfrak{B}\_{z2}}}{\mathfrak{B}\_{z1} + \mathfrak{B}\_{z2}},\tag{27}$$

Electromechanical Control over Effective Permittivity Used for Microwave Devices 295

detached from substrate's surface, Figure 10. Because controllable discontinuity crosses

Conventional methods of microstrip lines analysis, such as Whiller equations [3], Hammerstad equations [4], and their extensions to coplanar lines [5] exploit symmetry of the line with aid of conformal mappings. These methods also introduce effective permittivity to relate quasi-TEM wave propagation characteristics to those of equivalent TEM wave. Transmission lines in Figure 10 still possess symmetry, but their rigorous analysis becomes cumbersome. Thus, numerical techniques could be applied to accurately

Electromagnetic problem can be solved using electric and magnetic scalar *<sup>e</sup>* , *<sup>m</sup>* and

Using these potentials one can introduce electromagnetic filed distribution types with one of components being zero. If electrical vector potential oriented along *z* axis ( *e e* **A e** *<sup>A</sup> <sup>z</sup>* , where *<sup>z</sup>* **<sup>e</sup>** is *<sup>z</sup>* -axis unit vector), then *<sup>e</sup> <sup>A</sup>* and *<sup>e</sup>* functions define E-type field, or TM-mode, for which <sup>0</sup> *Hz* . Similarly, if magnetic vector potential oriented along *z* axis ( *m m* **A e** *<sup>A</sup> <sup>z</sup>* ), then *<sup>m</sup> <sup>A</sup>* and *<sup>m</sup>* functions define Н-type filed, or ТЕ-mode, for which <sup>0</sup> *<sup>z</sup> <sup>E</sup>* . Equation (28) is more convenient in the systems with dielectric only discontinuities, but with uniform

Equations (28), (29) allow ambiguity in relation of vector and scalar potentials with electromagnetic field components. For example, if *<sup>e</sup>* **A** and *<sup>e</sup>* define certain electromagnetic filed distribution, then *<sup>e</sup>* **A** and *<sup>e</sup>* , where is differentiable function, define the

In case of axial symmetry and absence of external currents solution of (30) may be presented

2 2 <sup>2</sup> <sup>0</sup> *e e c*

; .

0

same distribution. This ambiguity is removed applying Lorentz's calibration:

<sup>0</sup> ; ;

*<sup>e</sup> e e <sup>i</sup>* **<sup>A</sup> EA H** (28)

*<sup>m</sup> m m <sup>i</sup>* **<sup>A</sup> E HA** (29)

. (30)

electric field strength lines, it results in higher sensitivity.

calculate electromagnetic field distribution.

vector *<sup>e</sup>* **A** , *<sup>m</sup>* **A** potentials:

permeability.

in the form:

**Figure 10.** Mechanically controllable microstrip (*a*, *b*) and coplanar (*c*) lines

where *<sup>z</sup>*1 is the propagation constant in domain 1 (Figure 5) but *<sup>z</sup>*2 is the propagation constant in domain 2 filed by uniform dielectric with permittivity *eff* .

Comparison of two techniques is demonstrated in Figure 9. Good agreement for reasonable parameters set is observed.

**Figure 9.** Comparison of *S*-parameters computation using boundary element method (BEM) and effective permittivity approach for the structure with parameters: 1 = 10, 2 = 1, 3 = 1, *d/h* = 10-3

Hereby effective permittivity approach is efficient method for investigation of controllable microwave structures. Below this technique is extended for microstrip and coplanar lines.
