analysis of 2d grating slab

8/6/2019 Analysis of 2D Grating Slab

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Progress In Electromagnetics Research, PIER 74, 195216, 2007

ANALYSIS OF TWO-DIMENSIONALMAGNETO-DIELECTRIC GRATING SLAB

A. M. Attiya

Electronics Research Institute (ERI)El-Tahreer St., Dokki, Giza, 12622, Egypt

A. A. Kishk and A. W. Glisson

Department of Electrical EngineeringCenter of Applied Electromagnetic System Research (CAESR)University of MississippiUniversity, MS 38677, USA

AbstractVectorial modal analysis of a 2-D magneto-dielectricgrating structure is presented. The modal analysis is combined withthe generalized scattering matrix to obtain the transmission andreflection coefficients of multilayered 2-D magneto-dielectric gratingslabs. The results are verified with available commercial codes.Physical interpretation of the grating slab behavior is introduced. Anequivalent homogeneous magneto-dielectric slab is found using a simpleapproach for extracting the equivalent permittivity and permeability.Several examples are presented to find the relation between the physicalparameters of magneto-dielectric grating slabs and their equivalent

parameters. Emphasis on the possibility of designing a metamaterialwith equivalent negative permittivity and/or negative permeability byusing these grating structures is considered.

1. INTRODUCTION

Periodic structures have found great interests in different electromag-netic and optical applications. This was the motivation to developdifferent techniques for studying different configurations of periodicstructures. However, only a few published works have discussed theproperties of magneto-dielectric periodic structures [1, 2]. Recently, ithas been shown that such magneto-dielectric periodic structures canalso be used to design double negative (DNG) metamaterial where both


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196 Attiya, Kishk, and Glisson

the relative permittivity and the relative permeability are negative val-ues [3]. Such metamaterials with negative permittivity and/or negativepermeability have found important applications in focusing the fieldsof low directivity antennas [4, 5]. However, the available study for de-

signing metamaterial based on a periodic magneto-dielectric structureis based on analytical forms for a special case where the structure iscomposed of spherical magneto-dielectric spheres arranged in three-dimensional (3D) cubic cells [3]. For practical applications, it wouldbe more appropriate to study the possibility of realizing such metama-terials by using a 2-D grating slab of finite thickness as shown in Fig. 1.In addition, including the relative permeability as a design parameterrepresents another degree of freedom for designing grating structures.

Figure 1. Geometry of a 2-D Magneto-dielectric grating slab.

From the analytical point of view, the problem of a magneto-dielectric grating slab can be solved by using different techniques; suchas MoM, FDTD, FDFD, FEM, and modal analysis [1, 2, 612]. Thisproblem has been discussed previously by using MoM [1, 2]. However,the main complexity of this method is the large number of expansionfunctions that are required to represent the induced volume electricand magnetic currents. This complexity also increases by increasingthe thickness of the slab. Unlike MoM, modal analysis is based onobtaining the tangential field distributions of the different modes andthe corresponding propagation constants in the longitudinal directions.Then the problem is treated as cascaded sections of guiding structuresby using the generalized scattering method [13]. In this case, thecomplexity of the problem is independent of the thickness of the gratingstructure. Coves et al. [8,9] applied this modal analysis to study


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Progress In Electromagnetics Research, PIER 74, 2007 197

the reflection and transmission of 1-D dielectric grating structures.They showed good agreement with other techniques and experimentalresults. They also showed good convergence and flexibility in theirmodal analysis to include complex dielectric permittivity. Another

advantage of this technique is that the required integrations can becalculated analytically for simple shapes of the implanted rods. Attiyaand Kishk [17] generalized this modal analysis method to study 2-Ddielectric grating slab. This paper presents an additional generalizationto include also the effect of changing the relative permeability on thecharacteristics of a 2-D magneto-dielectric grating slab.

The remaining problem is how to extract the equivalentparameters of the grating slab by using its reflection and transmissioncoefficients. Cheng and Ziolkowski [14] have obtained approximateformulas for such a problem based on the equivalent T-network.However, their approximation is limited to normal incidence and verythin thicknesses. Such an approximation was not found to be suitablefor the structures proposed in this article. Thus, the problem is

formulated as an inverse problem where the equivalent parametersare obtained by a simple optimization process. The objective for thisoptimization is to find the equivalent parameters that minimize thedifference between the actual reflection and transmission coefficientsand the corresponding ones for a homogeneous slab of the samethickness excited by the same incident field.

The following section presents the modal analysis of a 2-Dmagneto-dielectric grating structure and how it can be used to obtainthe transmission and reflection coefficients of multilayered gratingslabs. Then the problem of extracting equivalent parameters isdiscussed in Section 3. Section 4 presents sample results for differentmagneto-dielectric grating slabs and their equivalent parameters. Thissection presents also detailed discussions on the possibility of designing

metamaterials by using magneto-dielectric grating slabs.

2. MODAL ANALYSIS OF MAGNETO-DIELECTRICGRATING

An infinite 2-D grating can be assumed as a guiding structure wherethe total field is a superposition of discrete modes. The transversemagnetic field distribution and propagation constants of these discretemodes can be obtained by solving the following eigenvalue problem:

L[ht] = 2ht (1)

where the longitudinal dependence is assumed to ejz, is thepropagation constant and L[] is a transverse differential operator.


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problem

LTE/TEpq

LTE/TMpq

LTM/TEpq LTM/TMpq

CTE(q)

CTM(q)

= 2

CTE(q)

CTM(q)

(6)

where L/pq =

h(p), L

h(p)

. By solving the above matrix eigenvalue

problem, one can obtain the unknown amplitudes of the expansionfunctions for each mode as the eigenvectors and the propagationconstants as the square roots of the corresponding eigenvalues. For thespecial case of a grating structure composed of rectangular dielectricrods, the matrix elements of Eq. (6) can be obtained in closed formsas shown in Appendix A. Other shapes of implanted rods can also besubdivided to several rectangular rods.

To obtain the reflection and transmission coefficients of thedifferent modes between two adjacent semi-infinite gratings of thesame periodicity, it is required to match the tangential electric andmagnetic field distributions on the transverse interface between twograting structures. The tangential magnetic field distribution of eachmode is obtained directly by using the resultant eigenvectors of Eq. (6).On the other hand, the transverse electric field distribution can beobtained as a function of the transverse magnetic field distribution asfollows:

j0ret =t az

j1

1r tr

ht + t ht

jaz ht

(7)This electric field distribution can also be approximated as a series oforthonormal expansion functions

et = eTEt + e

TMt =

p

TE(p)eTE(p) +

TM(p)

eTM(p) (8)

where the expansion functions of the electric field are orthogonal tothe corresponding expansion functions of the magnetic field such that

eTE(p) = azhTE(p) and

eTM(p) = azhTM(p) . The electric field amplitude

matrix can be obtained by using the bi-orthogonal property of theelectric field modal expansion function as follows:

j0

e, r, et

=

e,

j1C

1r tr

ht+t ht

j Caz

ht

(9)By solving (9), it can be shown that the electric field amplitude matrix is

=1

j0j

1

CQAT

j1

CAAT

+jC

1

(10)


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where A =

ATE

ATM

, Q =

QTE

QTM

, an d =

TE 0

0 TM

.

ATE is a diagonal matrix with the elements ATEpp = (jkx(p) cos inc

jky(p) sin inc) and ATM is a diagonal matrix with the elements ATMpp =

(jkx(p) sin incjky(p) cos inc). TE = TM is a matrix with elementsgiven as

pq =1

DxDy

Dx0

Dy0

r(x, y)ej(kx(p)kx(q))xej(ky(p)ky(q))ydxdy (11)

and the elements of the QTE and QTM matrices are

QTEpq =

Q1x

kx(p) kx(q), ky(p) ky(q)

cos inc

+ Q1y


sin inc

(12a)

QTMpq =Q1x


sin inc

+ Q1y


cos inc

(12b)

where Q1x and Q1y are given by Eqs. (A3e) and (A3f) in Appendix A.By matching the tangential electric and magnetic fields on the

interface between two adjacent semi-infinite 2-D grating structuresof the same periodicity, one can obtain the corresponding scatteringmatrix elements

S11 = I 2

C2C11 + 2

11

1

C2C11 (13a)

S12 = 2

C1C1

2 + 11

21

(13b)

S21 = 2

C2C11 + 2

11

1

(13c)

S22 = I 2

C1C12 + 1

12

1

C1C12 (13d)

where the Ci and i matrices represent the tangential magneticand electric field distributions of the ith grating structure and I isthe unit matrix of the same dimension. By using such scatteringmatrices combined with the corresponding propagation constants ofthe different expansion modes and applying the generalized scatteringmatrix approach [13], one can obtain the reflection and transmissioncoefficients of multilayered grating slabs of finite thicknesses.


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3. EXTRACTION OF EQUIVALENT PARAMETERS

After determining the reflection and transmission coefficients of amultilayered grating slab, it is useful to obtain the parameters of an

equivalent homogenous slab of the same thickness that has the sametransmission and reflection coefficients. These equivalent parameterscan be an appropriate method to understand the characteristics of thedifferent resonances in the grating structures. This also can be a goodtool for designing metamaterials with negative permittivity and/ornegative permeability by using a grating slab. Transmission througha prism represents the most appropriate approach for characterizingDNG metamaterial [15]. However, the present analysis is based ona constant-thickness slab. Cheng and Ziolkowski [14] introduced asimple approach based on an equivalent T network to extract theequivalent parameters in closed forms for a finite-thickness slab fromits scattering parameters. However, their approach is limited to verysmall thickness compared with the operating wavelength and to normal

incidence. To generalize their approach we introduced an equivalentinverse problem, which can be presented as an optimization problemas shown in Fig. 2. The reflection and transmission coefficients of theequivalent homogenous slab are obtained in closed forms [16]. Anysimple optimization technique or minimum-search can be used forobtaining the equivalent parameters that minimize the correspondingerror function.

Figure 2. Block diagram of equivalent parameters extractiontechnique.


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Figure 4. Magnitude of reflection coefficient of a 2-D grating slab dueto normal incident plane wave with Dx = Dy = 20mm, lx1 = 10 mm,ly1 = 10mm, x01 = 10mm, y01 = 10mm, h = 2 mm; the solid linecorresponds to rb = 2, rb = 1, r1 = 7 and r1 = 1, and the dashedline corresponds to the dual problem where rb = 1, rb = 2, r1 = 1and r1 = 7.

Figure 5. Equivalent parameters of the dielectric grating slab of

Fig. 4; rb = 2, rb = 1, r1 = 7 and r1 = 1.


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Figure 6. Magnitude of reflection coefficient of a 2-D grating slabdue to normal incident plane wave. Dx = Dy = 20 mm, lx1 = 10 mm,ly1 = 10mm, x01 = 10mm, y01 = 10 mm and h = 2mm, rb = 2, rb =2, r1 = 3 and r1 = 3.

the reflection coefficients of the above dielectric grating structure (Fig.3) are computed under normal incidence and compared with its dualgrating slab. As expected, the structure and its dual give the samereflection coefficients as shown in Fig. 4. Fig. 5 shows the equivalentmaterial parameters of the dielectric grating structure of Fig. 4. Itis observed that the increase of the reflection coefficient in this caseis effectively due to the increase in the stored electrical energy in thegrating structure, which corresponds to the increase in the equivalentpermittivity. Around the resonance frequency, the dielectric slabis converted from highly positive permittivity to a highly negativepermittivity. Then, the highly negative permittivity is increased tobe positive as the frequency increases farther. On the other hand, therelative permeability is increased as a sharp peak around the centerof the negative permittivity. However, this peak is much less than thepeak of the relative permittivity. In this band, the total stored energyis converted to magnetic energy. At higher frequencies, the equivalentrelative permittivity is converted back gradually to positive value. Thisexplains the reflection minimum that just follows the resonance of thegrating structure in this case. Similarly, one can explain the resonance

behavior of the magnetic grating slab. In the case of magneto-dielectric


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Figure 7. Equivalent relative permittivity and permeability ofthe grating slab of Fig. 6. Shaded region corresponds to DNGmetamaterial.

grating structures with r1 > r1 and rb > rb it is found that suchstructures have quite similar properties to dielectric grating slabs, whilemagneto-dielectric grating structures with r1 > r1 and rb > rbhave quite similar properties to magnetic grating slabs. On the otherhand, a magneto-dielectric grating can be tailored for a special casewhere both the base slab and the implanted rods have the characteristic

impedance of free space and have different propagation constants suchthat r1 = r1 and rb = rb where r1 = rb. In this case thereflection coefficient of such grating structure due to normal incidenceis characterized by a narrow band resonance surrounded by very lowreflection coefficient as shown in Fig. 6 for a 2-D grating slab with thefollowing parameters; Dx = Dy = 20 mm, lx1 = 10 mm, ly1 = 10 mm,x01 = 10mm, y01 = 10mm, h = 2mm, rb = 2, rb = 2, r1 = 3and r1 = 3. The equivalent parameters for such a grating structureare shown in Fig. 7. It can be noticed that the resonance in thiscase is mainly due to stored magnetic energy. It can also be notedthat both the relative permittivity and relative permeability turnsimultaneously into negative values in a very narrow band to forman equivalent DNG slab. Fig. 8 shows the reflection coefficients of

the same magneto-dielectric grating slab for TE and TM obliquely


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Figure 8. Magnitude of TE and TM reflection coefficients of a 2-Dgrating slab due to an obliquely incident plane wave; inc = 30 andinc = 0. The parameters of the grating structure are the same as forFig. 6.

incident waves where inc = 30 and inc = 0. One may noticethat the resonance bandwidth is increased slightly as the interactionbetween the two nearby resonances and the value of the backgroundreflection surrounding the resonance are also increased. Due to thesymmetry of this grating structure and the duality effects of the relativepermittivity and relative permeability on TM and TE waves, the TEand TM responses in this case are identical. The slight difference

shown between the results can be related to the numerical accuracy ofsolving the combined eigenvalue problems. Fig. 9 shows the equivalentparameters in this case for both TM and TE incident waves. Notethat the total resonance is composed of two resonances. For the TMcase these resonances are due to large stored electric energy followedby large stored magnetic energy, while the reverse behavior applies forthe TE case. One may also note that the equivalent parameters do notturn simultaneously into negative values. This means that the DNGproperty, which is found with the normal plane wave incidence in anarrow band, disappears for the oblique incidence case. Fig. 10 showsthe effect of increasing the values of r1 and r1 for the case of normalincidence. A bandwidth increase is noted in this case. The equivalentparameters for Fig. 10 case are shown in Fig. 11 providing a wider

bandwidth for the DNG property. By comparing Fig. 11 with Fig. 7 it


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Figure 10. Magnitude of reflection coefficient of a 2-D gratingslab due to normally incident plane wave. Dx = Dy = 20 mm,lx1 = 10mm, ly1 = 10 mm, x01 = 10mm, y01 = 10 mm and h = 2 mm,rb = 2, rb = 2, r1 = 7 and r1 = 7.

Figure 11. Equivalent relative permittivity and permeability of the

grating slab of Fig. 8 for TE incident wave.


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Figure 12. Magnitude of reflection coefficient of a 2-D gratingslab due to normally incident plane wave. Dx = Dy = 20 mm,lx1 = 5mm, ly1 = 5mm, x01 = 10mm, y01 = 10mm and h = 2mm,rb = 4, rb = 4, r1 = 1 and r1 = 1.

can be noted that the magnitude of the equivalent parameters in theDNG region is decreased by increasing r1 and r1. It is also observedthat the DNG properties disappear as r1 and r1 increase.

In the previous examples the relative permittivity and permeabil-ity of the implanted rods are greater than the corresponding ones ofthe base slab. Fig. 12 shows another example where the implantedrods are free space holes inside a magneto-dielectric slab. The reso-nance behavior in this case due to normal incidence is quite similarto the previous case shown in Fig. 6. However, the equivalent param-eters shown in Fig. 13 for this case present another explanation forthe resonance characteristics, where either the relative permittivity orthe relative permeability becomes negative but not both of them si-multaneously. From these two examples, it can be concluded that theDNG property can be obtained only for implanted rods of relative per-mittivity and permeability greater than the corresponding ones of thesupporting slab.

Figure 14 shows another special case where the values of therelative permittivity and relative permeability of the base slab and theimplanted rods are interchanged such that r1 = rb and rb = r1.

In this case the characteristic impedance of the base and the rods are


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Figure 13. Equivalent relative permittivity and permeability of thegrating slab of Fig. 12.

Figure 14. Magnitude of reflection coefficient of a 2-D grating slabdue to normal incident plane wave. Dx = Dy = 20 mm, lx1 = 10 mm,ly1 = 10mm, x01 = 10mm, y01 = 10 mm and h = 2mm, rb = 2, rb =7, r1 = 7 and r1 = 2.


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Figure 15. Equivalent relative permittivity and permeability of thegrating slab of Fig. 14.

different but the wave propagation constant is the same. It can benoticed that the behavior of the reflection coefficient in this case forthe lower frequency band is similar to the corresponding one of thedielectric or magnetic grating slab shown in Fig. 4. However, overthe higher frequency band multiple closed resonances create a widereflection band. The effective parameters of this grating structure areshown in Fig. 15. It can be noticed that this structure can be used todesign either negative permittivity or negative permeability materialbut it cannot be used to design a DNG metamaterial slab.

5. CONCLUSION

Detailed modal analysis of a 2-D magneto-dielectric grating structurewas discussed. This modal analysis combined with the generalizedscattering matrix approach was used to study the reflection andtransmission coefficients of different multilayered 2-D magneto-dielectric grating slabs. The accuracy of this modal analysis combinedwith the generalized scattering matrix approach was verified bycomparison with MoM and verification of duality between dielectricand magnetic grating slabs. A general approach for extractingthe equivalent parameters of a finite thickness slab based on itstransmission and reflection coefficients for an arbitrary incidence plane

wave was discussed. This approach was used to extract the equivalent


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transverse dimensions of the ith rod, (x0i, y0i) is the transverse centrallocation of the ith rod, and ND is the number of dielectric rods perunit cell. For this case, the matrix elements of Eq. (6) can be obtainedin closed forms as follows

LTE/TEpq = 2ppq + k20R0(kx(p) kx(q), ky(p) ky(q))

+(jkx(q) sin inc jky(q) cos inc)

(R1x(kx(p) kx(q), ky(p) ky(q))sin inc

+R1y(kx(p) kx(q), ky(p) ky(q))cos inc)

(jkx(p) cos inc +jky(p) sin inc)

(Q1x(kx(p) kx(q), ky(p) ky(q))cos inc

+Q1y(kx(p) kx(q), ky(p) ky(q))sin inc) (A2a)

LTM/TMpq = 2ppq + k

20R0(kx(p) kx(q), ky(p) ky(q))

+(jkx(q) cos inc +jky(q) sin inc)

(R1x(kx(p) kx(q), ky(p) ky(q))cos inc

R1y(kx(p) kx(q), ky(p) ky(q))sin inc)

(jkx(p) sin inc jky(p) cos inc)

(Q1x(kx(p) kx(q), ky(p) ky(q))sin inc

+Q1y(kx(p) kx(q), ky(p) ky(q))cos inc) (A2b)

LTE/TMpq = (jkx(q) cos inc jky(q) sin inc)

(R1x(kx(p) kx(q), ky(p) ky(q))sin inc

+R1y(kx(p) kx(q), ky(p) ky(q))cos inc)

(jkx(p) cos inc +jky(p) sin inc)

(Q1x(kx(p) kx(q), ky(p) ky(q))sin inc+Q1y(kx(p) kx(q), ky(p) ky(q))cos inc) (A2c)

LTM/TEpq = (jkx(q) sin inc +jky(q) cos inc)

(R1x(kx(p) kx(q), ky(p) ky(q))cos inc

+R1y(kx(p) kx(q), ky(p) ky(q))sin inc)

(jkx(p) sin inc +jky(p) cos inc)

(Q1x(kx(p) kx(q), ky(p) ky(q))cos inc

+Q1y(kx(p) kx(q), ky(p) ky(q))sin inc) (A2d)

where

p =

rbk20 k2x(p) k2y(p) (A3a)


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R0(kx, ky) =1

DxDy

NDi=1

4(riri rbrb)sin(kxlxi/2)

kx

sin(kylyi/2)

kyej(kxx0i+kyy0i) (A3b)

R1x(kx, ky) =1

DxDy

NDi=1

8j(ri rb)

(ri + rb)sin(kxlxi/2)

sin(kylyi/2)

kyej(kxx0i+kyy0i) (A3c)

R1y(kx, ky) =1

DxDy

NDi=1

8j(ri rb)

(ri + rb)

sin(kxlxi/2)

kx

sin(kylyi/2)ej(kxx0i+kyy0i) (A3d)

Q1x(kx, ky) =1

DxDy

ND

i=1

8j(ri rb)

(ri + rb)sin(kxlxi/2)

sin(kylyi/2)

kyej(kxx0i+kyy0i) (A3e)

Q1y(kx, ky) =1

DxDy

NDi=1

8j(ri rb)

(ri + rb)

sin(kxlxi/2)

kx

sin(kylyi/2)ej(kxx0i+kyy0i) (A3f)

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analysis of 2d grating slab

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