ISSN 1064-2269, Journal of Communications Technology and Electronics, 2017, Vol. 62, No. 7, pp. 750–758. © Pleiades Publishing, Inc., 2017. Original Russian Text © B.A. Belyaev, V.V. Tyurnev, 2017, published in Radiotekhnika i Elektronika, 2017, Vol. 62, No. 7, pp. 642–650.

ELECTRODYNAMICS AND WAVE PROPAGATION

Scattering of Electromagnetic Waves by a Metal Lattice Placed at the Interface of Two Media B. A. Belyaeva, b, * and V. V. Tyurneva aKirensky

Institute of Physics, Siberian Branch, Russian Academy of Sciences, Krasnoyarsk, 660036 Russia b Reshetnev Siberian State Aerospace University, Krasnoyarsk, 660014 Russia *e-mail: [email protected] Received June 25, 2016

Abstract—Simple formulas for determination of components of the scattering matrix of a plane electromagnetic wave incident along the normal onto a thin infinite 2D metal lattice with square windows placed at the interface of two media are derived for the first time. Frequency responses (FRs) of this structure calculated with the use of these formulas agree well with responses obtained by the numerical electromagnetic analysis of a 3D model in the case when the lattice period is less than one-half of the wavelength. The capability of efficient control of the FR of the analyzed structure by means of variation in the width of conductors and the lattice period enables application of this lattice for fabrication of a reflector with prescribed reflectivity. Such reflectors at interfaces of dielectric layers are required, for example, in the design of multilayer sufaces radio transparent in a specified frequency band or optical bandpass filters. DOI: 10.1134/S1064226917070026

INTRODUCTION Features of propagation of electromagnetic waves incident onto layered structures composed of dielectric plates with interfaces formed by various periodic structures (2D lattices or grids) made of stripline conductors are intensively discussed in the last few years. Great interest in such structures is caused by the possibility of application of such structures for formation of frequency-selective surfaces serving as band-pass and band-stop filters in frequency ranges from micron [1–3] to decimeter [4–6] wavelength bands. Stripline unit cells forming a 2D periodic structure, for example, metal patches or cells of a metal mesh are usually resonators with properties of parallel or series oscillatory circuits connected in parallel to a transmission line [1]. Therefore, at low frequencies, when the wavelength is larger than the lattice period, they exhibit reactive properties of inductance or capacitance, respectively. In this case, a pair of inductive and capacitive lattices separated by a thin dielectric layer can jointly exhibit properties of a parallel oscillatory circuit in the region of low frequencies. Therefore, such subwavelength multilattice structures can be used for the development of various band-pass filters [6–9]. Note that the wavelength at the center frequency of the passband of such filter is not only substantially larger than the lattice periods but also substantially larger than the thickness of dielectric layers.

There are also designs of band-pass filters on dielectric layers of the half-wavelength thickness separated by nonresonant subwavelength 1D lattices, for example lattices of parallel stripline conductors [10]. In such multilayer structures, lattices serve as reflectors with specified reflectivity ensuring the required coupling between resonators as well as coupling of extreme resonators with space. Varying the lattice reflectivity, for example, by varying the width of conductors at a specified period, one can easily ensure the required width of the filter passband and the permitted value of ripples of the frequency response (FR) in the passband [11]. In the design of frequency-selective surfaces, a combination of the above methods is also possible. For example, a design in which the passband is formed by two pairs of subwavelength 2D lattices of stripline conductors with a sub-wavelength dielectric layer between them was analyzed in [12]. It is important that unloaded Q factors of resonant structures designed on 2D lattices of stripline conductors separated by dielectric layers with thicknesses substantially smaller than the wavelength are substantially lower than the Q factor of half-wavelength dielectric layers separated by subwavelength metal lattices-ref lectors. As a result, frequencyselective surfaces designed on resonant dielectric layers surrounded by ref lectors have lower loss in transparency bands. Therefore, the analysis of properties of subwavelength 2D metal lattices used

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SCATTERING OF ELECTROMAGNETIC WAVES BY A METAL LATTICE

for the design of reflectors with specified ref lectivity is a very important problem. The reflectivity of metal mesh structures is calculated by both numerical and numerical-analytic methods. Numerical methods are most popular, because they are implemented in many commercial universal software packages for electromagnetic simulation of 3D structures. The method of moments using expansion of components of the desired field in terms of basis functions is the most known numerical-analytic method [13]. The method of averaged boundary conditions providing the recipe for finding equivalent boundary conditions for various grid structures is also known [14]. Equivalent boundary conditions for a metal ribbon 1D lattice at the interface of two media were obtained in [15]. Numerical and numerical-analytic calculation methods require rather high computing resources and substantial time of computation. Therefore, researchers often use simple approximate formulas [16, 17] obtained for several particular types of lattices. In the course of derivation of such formulas, metal grids are usually modeled with the use of equivalent circuits [18]. Electric parameters of equivalent circuits are usually calculated with the help of approximate formulas from a reference book [19], which were obtained for 1D gratings of stripline conductors. In addition, the authors of many studies, for example, [20, 21], use the following unproved assumption: a nonresonant metal 2D lattice and a 1D grating of stripline conductors with the same period and width of conductors have identical reflectivity if their spacings are substantially less than the length of the incident wave. In reality, as will be shown below, there is certain difference in reflectivities of a 2D lattice and a 1D grating. Thus, formulas describing reflection and transmission of the electromagnetic wave incident along the normal onto an infinite and infinitely thin perfectly conducting metal mesh with square apertures (Fig. 1) placed at the interface of two media with permittivities ε1 and ε2 are still not derived. This paper is devoted to solution of this problem. It is assumed that the mesh period is substantially less than the wavelength; therefore, components of the electromagnetic field are calculated in the quasi-static approximation. The accuracy of the derived formulas is estimated by comparing the FR calculated with the use of these formulas with the results of the numerical electromagnetic analysis of a 3D model of the considered structure. 1. ELECTROMAGNETIC FIELD OF THE LATTICE Let us consider electromagnetic oscillations near a metal mesh (see Fig. 1) excited by linearly polarized

Conductor

y T/2

IV

III

s/2

I

II

Aperture

751

→

E x

→

H

s/2 T/2

0

Fig. 1. Latticed 2D structure designed from stripline conductors; I, II, III, and IV are the selected parts of its unit cell.

electromagnetic waves incident onto the lattice from both sides along the normal. The waves have the following components: inc

Ex

H yinc

⎧⎪E1inc exp (ik1z − i ω t ), = ⎨ inc ⎪⎩E 2 exp (−ik 2 z − i ω t ), ⎧⎪Z 1E1inc exp (ik1z − i ω t ), =⎨ inc ⎪⎩−Z 2E 2 exp (−ik 2 z − i ω t ),

if z < 0, if z > 0, if z < 0,

(1)

if z > 0,

where

k1,2 = ω ε 0μ 0ε1,2

(2)

are the wave numbers and

Z 1,2 = Z 0

ε1,2 , Z 0 = μ 0 ε 0

(3)

are the characteristic impedances. Here, the reference point of coordinate z is chosen on the lattice surface. Since components (1) of the field of incident waves are   uniform in the lattice plane, components (E latt , H latt ) of the lattice field are generally periodic functions of coordinates x and y with the period equal to the lattice period T. This means that any component of the electromagnetic field can be expanded in a double Fourier series in terms of arguments x and y. In view of the symmetry of the field components relative to the axes

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of symmetry of the lattice, we can write the following expansions in the coordinate system shown in Fig. 1: ∞

E xlatt ( x, y, z) =

∞

∑∑ E

nm x ( z ) cos

( k n x ) cos ( k m y ),

n=0 m =0 ∞ ∞

H y ( x, y, z) = latt

∑∑ H

nm y ( z ) cos

n=0 m =0 ∞ ∞

E zlatt ( x, y, z )

=

∑∑

H zlatt ( x, y, z )

=

( k n x ) cos ( k m y ),

∑∑

nm H z (z) cos

( k n x ) cos ( k m y ),

n=0 m =0 ∞ ∞

( k n x ) sin ( k m y ),

n=0 m =0

where

k n = 2πn T .

We assume that lattice period T is substantially less than the wavelength, i.e., k1,2T ! 2π.

(4) nm E z (z) sin

A. First Partial Solution

(5)

In this case, Helmholtz equations (6) and (7) are simplified and are transformed into the Laplace equations  (11) ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 E latt = 0,

( (∂

2

∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2

E xlatt ( x, y, z) = E x(

1,2)

2

 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 + k1,2 H latt = 0.

)

(7)

1,2)

+

∑∑

(1,2)

where values of undetermined constants E x(

1,2)

Ex

T −s 2 T −s 2

= 12 T

n =1 m =1

H ylatt ( x, y, z) = H y( , 00) exp(ik1,2 z ) 1,2

∞

+

∞

∑∑ H (

1,2) y, n m

(1,2)

Hy (1,2)

cos ( k n x ) cos ( k m y ) exp(−k n m z ),

n =1 m =1

(8)

E zlatt ( x, y, z) ∞

=

∞

∑∑ E (

1,2) z, n m

sin ( k n x ) cos ( k m y ) exp(−k n( m ) z ), 1,2

n =1 m =1

H zlatt ( x, y, z) ∞

=

∞

∑∑ H (

1,2) z, n m

cos ( k n x ) sin ( k m y ) exp(−k n( m ) z ), 1,2

n =1 m =1

where 2 k n( m ) = k n2 + k m2 − k1,2 . 1,2

(9)

Formulas (8) can be considered as expansions of com  ponents E latt and H latt of the lattice field into series in terms of all modes meeting the lattice symmetry and the polarization of the incident wave.

and

are specified by integrals

(1,2)

E x, n m cos ( k n x ) cos ( k m y ) exp(−k n m z ),

(13)

latt

Hy

(1,2)

,

H z ( x, y, z) = 0,

1,2

∞

(12)

latt

E xlatt ( x, y, z) = E x( , 00) exp(ik1,2 z ) ∞

= 0.

E z ( x, y, z) = 0,

(1,2)

In this case, formulas (4) take the following form:

latt

,

H ylatt ( x, y, z) = H y(

)

(∂

)  )H

In sums (8), all terms of the Fourier series, except for principal terms with n = m = 0, rapidly decay with increasing |z|. Therefore, at k1,2T ! k1,2|z| ! 2π, formulas (8) are simplified and, in view of the symmetry relative to x and y, take the following form:

Let us require that each term of the double Fourier series in expressions (4) should satisfy the Helmholtz equations  2 ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 + k1,2 E latt = 0, (6)

(

(10)

= 12 T

∫

∫

latt

E x ( x, y, z)

x =− s 2 y =− s 2

T −s 2

T −s 2

∫

∫

z =0

dxdy,

(14)

H ylatt ( x, y, z ) z = 0 dxdy. (15)

x = −s 2 y = −s 2

Here, s is the width of the lattice aperture. Knowing 1,2 1,2 constants E x( ) and H y( ) , we can calculate elements of scattering matrix S.   Thus, components E latt and H latt of the lattice electromagnetic field must satisfy appropriate boundary conditions on the lattice surface and must be solutions to Eqs. (11) and (12) at k1,2|z| ! 2π. These equations are the differential equations of second order. Therefore, they must have two linearly independent solutions. For definiteness, we require that component H ylatt for one of partial solutions should be an even function of coordinate z and the same component for the second partial solution should be an odd function. As follows from the Maxwell equations, if component H ylatt is an even function, component E xlatt corresponding to this component is an odd function and vice versa.

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One of partial solutions to Eqs. (11) and (12) with even component H ylatt is evident. It has the following form:  E latt ( x, y, z) = 0,

H xlatt ( x, y, z) = 0, H y ( x, y, z) = H 0, latt

(16)

H zlatt ( x, y, z) = 0. This solution does not contain the electric component of the electromagnetic field, including the field on the surfaces of conductors; therefore, it satisfies the boundary condition on the surface of the lattice. B. Second Partial Solution The second solution to Eqs. (11) and (12) sould  have electric component E latt and its tangential component E xlatt should be an even function of coordinate z. At z = 0, it is nonzero only in the region without metallization, i.e., inside the lattice aperture. This region is marked by I in Fig. 1. Normal component E zlatt is nonzero only beyond the aperture, i.e., on the surfaces of conductors. There are three such regions: regions II, III, and IV (see Fig. 1). Magnetic component H ylatt is an odd function of z. It must be zero in region I and nonzero only in regions II–IV. Let us relate electric component E xlatt , which is nonzero only in region I, and magnetic component H ylatt in regions II and IV on the lattice surface. Com-

ponent E xlatt on the lattice surface inside the aperture. Let us find such relation between component E xlatt in region I on the lattice surface and component H ylatt in region IV. Note that, as follows from the charge conservation law, the integral current flowing in any cross section of a stripline conductor in region IV remains the same, i.e., is independent of coordinate x. Since the width of the stripline conductor and the width of the window are fixed, the relation between components E xlatt and H ylatt can be calculated in the approximation of independence of these components of coordinate x. In this case, an exact solution to Laplace equation (12)  for component H latt ( y, z) satisfying the boundary condition

H zlatt ( y, z)

H ylatt

= J x.

(17)

It is directed orthogonally to the edge of metallization. At the edge, rotation of the vector of component H ylatt

H zlatt .

transforms it into component In this case, both components have singularity meeting the Meixner condition at the edge [22]:

H ylatt H zlatt

y = s 2 +ρ

y = s 2 −ρ

∝ρ

∝ρ −1 2

−1 2

= 0,

(19)

latt

Hy

= − ∂ Im Ψ( y + iz), ∂y

(20)

(21) H zlatt = − ∂ Im Ψ( y + iz), ∂z where analytic function Ψ(ζ) of complex argument ζ is defined by the formula

( (πζT ) cos (2πTs )) .

Ψ(ζ) = arccosh cos

(18)

where ρ is the distance from the observation point to the aperture edge. Thus, longitudinal surface current Jx on the conductor in region IV generates a magnetic induction flux penetrating the aperture area. This flux must be accompanied by an electromotive force along the perimeter of the aperture. Therefore, component H ylatt on the surface of metallization in region IV should be directly related to electric com-

(22)

We find from this formula that component H ylatt on the lattice surface is expressed by the formula

( ( ) ( ))

⎡ ⎤ y cos πs ⎥ . (23) = ± Im ⎢ ∂ arccosh cos π ∂ 2 y T T ⎣ ⎦ Here, the plus sign is chosen at z < 0 and the minus sign, at z > 0. latt

Hy

(1,2),IV,III

Let us calculate contributions H y

,

(ρ → 0) ,

y > s 2, z = 0

and, therefore, edge conditions (18) can be obtained with the help of conformal mappings [23]. This solution describes the electromagnetic field of magnetic modes. It has the following form:

H ylatt

ponent in region IV is related to the density of longitudinal current Jx by the equation

753

undetermined constant H y(

1,2)

into

in formula (15) from

component H ylatt in regions IV and III. Substituting expression (23) into (15), we obtain (24) = ∓ π, T where the minus sign corresponds to contribution (1),IV,III and the plus sign corresponds to the conHy (1,2),IV,III

Hy

(2),IV,III

tribution H y

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Let us find the relation between electric compo nent E xlatt in region I and magnetic field H latt expressed by formulas (20)–(22). For this purpose, we turn to the Maxwell equation  (25) (curl E latt ) z = i ωμ 0H zlatt. Integrating over y, we obtain (26) E xlatt = E 0(z) + i ωμ 0 Re Ψ( y + iz), where function E0(z) is an undetermined constant of integration. It must be zero at z = 0 in order to vanish component E xlatt on the metal surface at y > s/2. Note that, as |z| increases, the second term in this equation linearly increases to infinity at |z| > T. Therefore, we require that function E0(z) should be proportional to |z| in order to ensure that component E xlatt gradually ceases to depend on z at |z| > T in accordance with formula (13). As a result, formula (26) takes the form

E xlatt = i ωμ 0 [Re Ψ( y + iz ) − π z T ] .

(27)

(1,2)

Let us now calculate constant E x . Substituting (27) and (22) into formula (14), we obtain

( ( ))

= −i ωμ 0 s ln cos π s . T 2T Here, we took into account that E x(

1,2)

(28)

a

∫

arccosh ( cos ζ cos a ) d ζ = −π ln ( cos a ) . (29)

ζ= − a

Thus, magnetic field H ylatt on the conducting surface in region IV is related to electric field E xlatt char1,2 acterized by constant E x( ) , which is expressed by formula (28). Field H ylatt itself in regions IV and III is (1,2),IV,III

characterized by constant H y expressed by formula (24). Let us now find the relation between electric component E xlatt in region I and magnetic component H ylatt in region II. Note that width s of the aperture, which corresponds to the length of region II, is usually larger than the conductor width T–s. It is also important that electric component E xlatt at boundaries of region I must satisfy the edge condition

E xlatt E xlatt

∝ρ

x = s 2 −ρ

∝ρ

12

y = s 2 −ρ

−1 2

,

(ρ → 0) .

(30)

Therefore, the inequality ∂ 2E xlatt ∂ x 2 @ ∂ 2E xlatt ∂ y 2 should be fulfilled in region II and, consequently,

derivative ∂ 2E xlatt ∂ y 2 in Laplace equation (11) can be neglected. Thus, component E xlatt in region II (at any value of z) and in region I (only on the aperture surface (z = 0) ) can be approximated by the formula

(31) = − A Re Ψ( y) ∂ Re Φ( x + iz ), ∂x where A is an undetermined coefficient that should be determined and analytic function Φ(ζ) of complex argument ζ is defined by the formula latt

Ex

( (πζT ) sin (2πTs )) .

Φ(ζ) = arcsin sin

(32)

Indeed, the coordinate dependence of component specified by formula (31) is a harmonic function. This function satisfies Laplace equation (11) in region II, because its dependence on coordinates x and z is described by analytic function Φ(x + iz) and dependence Ψ(y) can be neglected.

E xlatt

The dependence of component E xlatt on coordinates x and y on the aperture surface (z = 0) in region I meets edge conditions (30) and its dependence on coordinate z can be extended to satisfy Eq. (11). In order to find the value of coefficient A, we calcu1,2 late constant E x( ) once more but, in this case, we start from formula (31). After evaluation of integral (14), we obtain

( ( ))

(33) = A π ln cos π s . T 2T Comparing formulas (33) and (28), we find the value of the coefficient:

E x(

1,2)

A = −i ωμ 0 s π .

(34)

In order to determine component H ylatt on the surface of the metal layer in region II, we turn to the Maxwell equation  (35) (curl H latt ) x = −i ωε 0ε1,2E xlatt . Integrating this equation over z after substitution of formulas (31) and (34), we obtain latt

Hy

= H ( x, y)

(36) ω ε 0ε1,2μ 0 s Re Ψ( y)Im Φ( x + iz), π where function H(x, y) is an undetermined integration “constant”. It should be chosen such that desired component H ylatt is an odd function of argument z. Thus, taking into account formulas (22) and (32), we find that component H ylatt on the conducting surface 2

−

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in region II, i.e., at s/2 < x < T–s/2, –s/2 < y < s/2, and z = 0, is described by the formula

ω ε 0μ 0 s ε1 + ε 2 π 2 π y ⎛ cos ⎞ ⎛ sin ⎜ ⎟ ⎜ T × arccosh ⎜ arccosh ⎜ ⎟ πs ⎜ sin ⎜ cos ⎟ ⎝ ⎝ 2T ⎠ H ylatt = −

(1,2),II

(37)

(2),II

1,2

(1,2)

.

from

stant H y( ) . Substituting expression (37) into double integral (15), we obtain

(38)

where the plus sign corresponds to the contribution (1),II from H y and the minus sign corresponds to the contribution from H y

component H ylatt in region II into undetermined con-

Hy

=±

( ( )) ( ( ))

(πTx ) ⎞⎟ . (2πTs )⎟⎟⎠

Let us calculate the contribution H y

Summing

(1,2),IV,III

contributions

Hy

(1,2),II

( ( )) ( ( ))

1

and the plus sign corresponds to constant H y(

2)

2. FORMULAS FOR THE SCATTERING MATRIX Let us calculate scattering matrix S by considering regions on both side of the lattice as two ports of a twoport network. This matrix relates normalized amplitudes of scattered waves b1 and b2 propagating from the first and second ports with normalized amplitudes of incident waves a1 and a2 by the equation [24]

⎛ b1 ⎞ ⎛ a1 ⎞ (40) ⎜b ⎟ = S ⎜a ⎟. ⎝ 2⎠ ⎝ 2⎠ The normalized amplitudes themselves are determined from the formulas [24] scat Z 1,2 , b1,2 = E1,2

Z 1,2 ,

(41)

inc where E1,2 are the amplitudes of incident waves in forscat mula (1) and E1,2 are the amplitudes of scattered waves. Let us write boundary conditions for electric field Ex and magnetic field Hy at the first port:

(1)

a1 Z 1 + b1 Z 1 = E x

(39)

the order of its period T in order to treat components Ex and Hy as constants E x( ) E 0 and H y( ) E 0 , according to formula (13). Equation (42) takes into account that the wave with amplitude b1 propagates in the direction opposite to the direction of axis z; therefore, the magnetic field of this wave has the opposite sign. Let us write similar equations for the electric and magnetic fields at the second port: 1

.

( 2)

Ex ( 2)

Hy

1

E 0 = b2 Z 2 + a2 Z 2,

E 0 + H 0 = b2

Z 2 − a2

(43)

Z2 .

Note that the wave with amplitude a2 propagates in the direction opposite to the direction of axis z; therefore, the intensity of its magnetic field has the opposite sign, as in the case of the wave with amplitude b1 in formula (42). Solving simultaneously Eqs. (42) and (43) and taking into account dependences of the signs of coeffi1,2 1,2 cients H y( ) , E x( ) on the sign of z, we find amplitudes of scattered waves: 2)

b1 =

E 0,

(42) (1) a1 Z 1 − b1 Z 1 = H y E 0 + H 0, where E0 is an undetermined constant setting the amplitude of the second (odd) partial solution of the system of linear equations (11) and (12). Recall that the amplitude of the first (even) solution is set by undetermined constant H0. We assume here that the reference plane of the first port is separated from the lattice plane by a distance on

and

Hy , which are expressed by formulas (24) and (38), we find the constant

⎡ ⎤ ω2ε 0μ 0 s ε1 + ε 2 ln sin π s ln cos πs ⎥ , = ∓ ⎢π − π 2 2T 2T ⎦ ⎣T

where the minus sign corresponds to constant H y( )

inc a1,2 = E1,2

ω2ε 0μ 0 s ε1 + ε 2 2 π s π π × ln sin ln cos s , 2T 2T (1,2),II

Hy

2

( ) ( )

755

+

ε1 − ε 2 + 2Z 0 H y(

E x(

ε 2 + ε1 − 2Z 0 H y(

E x(

2)

24 ε1ε 2 2 ε 2 + ε1 − 2Z 0 H y( )

24 ε1ε 2 b2 = 2 ε 2 + ε1 − 2Z 0 H y( ) 2)

+

2) 2)

2)

2)

ε 2 + ε1 − 2Z 0 H y(

2)

E x(

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(44)

E x( E x(

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a2 ,

E x(

ε 2 − ε1 + 2Z 0 H y(

a1

a1

2) 2)

a2 .

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BELYAEV, TYURNEV

10

1.0 3 2 |S11|

1

0.6 1

|S21| 0.4

2

6 3

4 2 2

0.2

0

1

8 Δ|S11|/| S11|, %

|S11|, | S21|

0.8

3

5

10

15 f, GHz

20

25

30

Fig. 2. Frequency dependences of transmission coefficients |S21| and reflection coefficients |S11| of the electromagnetic wave incident onto the mesh with period T = 3 mm for three values of s/T: (1) 0.9, (2) 0.75, and (3) 0.5. Solid lines depict the results of calculation by the derived formulas and dots are the results of the numerical analysis of the 3D model of the mesh.

After substitution of expressions (28) and (39) for 2 2 coefficients E x( ) and H y( ) , we obtain the desired formulas:

⎛ ε1 − ε 2 − iZ 0Y ⎜ ε + ε1 + iZ 0Y S=⎜ 2 ⎜ 24 ε1ε 2 ⎜ ε + ε + iZ Y ⎝ 2 1 0 where Y =

⎞ 24 ε1ε 2 ⎟ ε 2 + ε1 + iZ 0Y ⎟ , ε 2 − ε1 − iZ 0Y ⎟ ε 2 + ε1 + iZ 0Y ⎟⎠

(45)

2π

( ( )) ( ( ))

ωμ 0 s ln sec π s (46) 2T ε1 + ε 2 s π ln cosec . − ωε 0T 2T π Note that quantity Y is the susceptance of an unit cell of the lattice. The first term in expression (46) describes the inductive part of susceptance and the second term, the capacitive part of susceptance. As could be expected, elements of matrix S are related by the equalities S12 = S21 and |S11| = |S22|. 3. ANALYSIS OF THE DERIVED FORMULAS In order to estimate the accuracy of the derived formulas, we compare the FRs calculated with the use of these formulas to analogous characteristics calculated with the help of the numerical electromagnetic analy-

0

10

20

30 f, GHz

40

50

Fig. 3. Frequency dependences of the relative difference between values of the reflection coefficient of the electromagnetic wave incident onto the mesh with period T = 3 mm, which were calculated from formula (45) and with the use of numerical electromagnetic analysis of the 3D model for three values of s/T: (1) 0.9, (2) 0.75, and (3) 0.5. The dashed line marks frequency f0 at which λ2/2 = T.

sis of a 3D model of the metal mesh. Frequency dependences of transmission coefficient |S21| and reflection coefficient |S11| for three values of relative aperture width s/T are shown in Fig. 2. Solid lines were plotted with the use of formulas (45) and (46) and dots present the dependences obtained by means of the numerical electromagnetic analysis of the 3D model. The calculation was performed for the case when the structure period T = 3 mm, ε1 = 1, and ε2 = 3. It is seen that, as the relative aperture size decreases, the reflection coefficient increases and the coefficient of transmission of electromagnetic waves through the structure decreases, respectively. Note that, at frequency f0 = 28.8 GHz, the wavelength in the second medium λ2 = 2T. The accuracy of the derived formulas can be estimated from frequency dependences of the relative difference in reflection coefficients of the electromagnetic wave (Fig. 3) calculated with the use of the above formulas and by means of the numerical electromagnetic analysis of the 3D model, which as is well known, is in rather good agreement with experiment. As seen in Fig. 3, the relative error of formulas in the region of low frequencies below f0, which is marked by the dashed line in the figure, is no more than 4%. Frequency dependences of the transmission coefficient for three values of the lattice spacing and a fixed width of stripline conductors, T – s = 0.5 mm, are presented in Fig. 4. The calculation was performed for the case when ε1 = 1 and ε2 = 3. As could be expected, an

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SCATTERING OF ELECTROMAGNETIC WAVES BY A METAL LATTICE

the length of the incident wave. Therefore, it is of interest to compare relative errors of calculation of reflection coefficients of a 2D metal lattice obtained with the use of the derived formulas (45) and (46) and the well-known formula (39) from [23] for a 1D grating of stripline conductors. The results of such a comparison, which was performed at several frequencies for a metal mesh with period T = 3 mm placed at the interface of the media with relative permittivities ε1 = 1 and ε2 = 3, are presented in the table. It is seen that the accuracy of calculation by the derived formulas ( Δ S112D S11 ) is substantially higher than the accuracy of calculation by the well-known formula for the 1D 1D grating ( Δ S11 S11 ).

0.8

|S21|

0.6

3

0.4

2 0.2 1 0

5

10

15 f, GHz

20

757

25

30

Fig. 4. Frequency dependences of the transmission coefficient for T – s = 0.5 mm and three values of the period T = (1) 1, (2) 2, and (3) 3 mm. Solid lines depict the results of calculation by the derived formulas and dots are the results of the numerical electromagnetic analysis of the 3D model of the mesh.

increase in the period of the 2D lattice at a fixed width of stripline conductors results in the corresponding increase in transmission of electromagnetic waves through the structure. As was already noted, it is assumed in many studies that a nonresonant metal 2D mesh lattice and a 1D grating of stripline conductors with the same period and the width of conductors have identical reflectivity if periods of these structures are substantially less than

CONCLUSIONS Thus, simple formulas for calculation of elements of the scattering matrix for a plane electromagnetic wave incident along the normal onto a metal lattice with square windows placed at the interface of two media have been derived for the first time with the use of the quasi-static approximation. Comparison of frequency dependences of transmission and reflection coefficients calculated from the derived formulas and similar dependences calculated by means of the numerical electromagnetic analysis of the 3D model of considered structures has demonstrated rather high accuracy of the derived formulas. Note that rather small relative error (below 4%) is observed in a wide frequency range, up to the frequency at which the lattice spacing is approximately one-half of the wavelength in the medium with larger permittivity. The derived formulas can be used in the design of frequency-selective surfaces on layered dielectric

2D Error of calculation of the reflection coefficient of the 2D lattice by the derived formulas ( Δ S11 S11 ) and by the well-

1D known formulas for the 1D grating ( Δ S11 S11 )

s/T = 0.5

s/T = 0.75

s/T = 0.9

f/ε2 2D Δ S11 S11

1D Δ S11 S11

2D Δ S11 S11

1D Δ S11 S11

2D Δ S11 S11

1D Δ S11 S11

5 GHz/35 mm

7.7 × 10–5

7.0 × 10–4

6.8 × 10–4

2.7 × 10–3

3.2 × 10–4

5.6 × 10–3

10 GHz/17 mm

3.3 × 10–4

2.8 × 10–3

3.0 × 10–3

1.0 × 10–2

1.9 × 10–4

1.8 × 10–2

15 GHz/12 mm

8.4 × 10–4

6.0 × 10–3

7.7 × 10–3

1.9 × 10–2

2.4 × 10–3

2.8 × 10–2

20 GHz/10 mm

1.8 × 10–3

1.0 × 10–2

1.6 × 10–2

2.7 × 10–2

8.8 × 10–3

3.2 × 10–2

25 GHz/7 mm

3.4 × 10–3

1.5 × 10–2

3.0 × 10–2

2.9 × 10–2

2.0 × 10–2

2.7 × 10–2

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BELYAEV, TYURNEV

structures separated by lattices of stripline conductors. In this case, metal lattices, serving as reflectors, ensure the required coupling between resonant dielectric layers as well as coupling of outer layers with space. The reflectivity of the metal mesh can be controlled within broad limits by varying both relative aperture width s/T and the mesh period T. REFERENCES 1. P. A. R. Ade, G. Pisano, C. Tucker, and S. Weaver, Proc. SPIE 6275, 62750 (2006). 2. A. M. Melo, M. A. Kornberg, P. Kaufmann, et al., Appl. Opt. 47 (32), 6064 (2008). 3. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, Rev. Mod. Phys. 82, 729 (2010). 4. B. Munk, Frequency Selective Surfaces: Theory and Design (Wiley, New York, 2000). 5. P. Tomasek, Int. J. Circuits, Syst. Signal Processing 8, 594 (2014). 6. S. Oh, H. Lee, J-H. Jung, and G.-Y. Lee, Int. J. Microwave Sci. Technol. 2014, 857582 (2014). 7. K. Sarabandi and N. Behdad, IEEE Trans. Antennas Propag. 55, 1239 (2007). 8. H. Zhou, S.-B. Qu, J.-F. Wang, et al., Electron. Lett. 48, 11 (2012). 9. M. Salehi and N. Behdad, IEEE Microwave Wireless Comp. Lett. 18, 785 (2008). 10. B. A. Belyaev and V. V. Tyurnev, Opt. Lett. 40, 4333 (2015).

11. B. A. Belyaev and V. V. Tyurnev, Opt. Lett. 41, 536 (2016). 12. Y. Li, Z. Pei, S. Qu, et al., in PIERS Proc., Suzhou, China, Sept. 12–16, 2011 (EMW Publishing, Cambridge, 2011). 13. R. Mittra, C. H. Chan, and T. Cwik, Proc. IEEE 76, 1593 (1988). 14. M. I. Kontorovich, M. I. Astrakhan, V. P. Akimov, and G. A. Fersman, Electrodynamics of Grid Structures (Radio i Svyaz’, Moscow, 1987) [in Russian]. 15. S. E. Bankov and I. V. Levchenko, Radiotekh. Elektron. (Moscow) 33, 2045 (1988). 16. O. Luukkonen, C. Simovski, G. Granet, et al., IEEE Trans. Antennas Propag. 56, 1624 (2008). 17. I. V. Mel’chakova and K. R. Simovskii, J. Commun. Technol. Electron. 53, 874 (2008). 18. C. K. Lee and R. J. Langley, IEE Proc. H 132, 395 (1985). 19. N. Marcuvitz, Waveguide Handbook (McGraw-Hill, New York, 1951). 20. L. B. Whitbourn and R. C. Compton, Appl. Opt. 24, 217 (1985). 21. R. Ulrich, Infrared Phys. 7, 37 (1967). 22. J. Meixner, IEEE Trans. Antennas Propag. 20, 442 (1972). 23. B. A. Belyaev and V. V. Tyurnev, Russian Physics J. 58, 646 (2015). 24. K. C. Gupta, R. Garg, and R. Chadha, Computer-Aided Design of Microwave Circuits (Artech House, Dedham, MA, 1981; Radio i Svyaz’, Moscow, 1987).

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Translated by A. Kondrat’ev

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No. 7

2017