MODAL ANALYSIS OF A TWO-DIMENSIONAL DIELECTRIC GRATING SLAB EXCITED BY AN OBLIQUELY INCIDENT PLANE WAVE

Transcription

1 Prgress In Electrmagnetics Research, PIER 60, , 2006 MODAL ANALYSIS OF A TWO-DIMENSIONAL DIELECTRIC GRATING SLAB EXCITED BY AN OBLIQUELY INCIDENT PLANE WAVE A. M. Attiya Electrnics Research Institute El-Tahreer St., Dkki, Gizza, Egypt A. A. Kishk Department f Electrical Engineering and Center fr Electrmagnetic System Research (CESR University f Mississippi University, MS 38677, USA Abstract A vectrial mdal analysis f tw-dimensinal (2-D dielectric grating is presented. The transmissin and reflectin frm the 2-D dielectric-grating slab are cmputed by cmbining the generalized scattering matrix and the mdal analysis. New insights n the bundary cnditins between tw grating structures are presented t be suitable fr tw-dimensinal gratings. The results btained using the present analysis are verified with published experimental and numerical results fr bth ne- and tw-dimensinal dielectric-grating slabs. The present apprach prvides fast cnvergence and prvides gd agreement with ther numerical techniques. This technique is used t study the effects f different parameters fr designing a wide band FSS cmpsed f multilayered 2-D grating slabs. 1. INTRODUCTION Due t the filtering and diffractive effects f their peridic structures, dielectric gratings have fund different applicatins in cntrlling the prpagatin f electrmagnetic waves. They can be used as frequencyselective surfaces. They can als be used as substrates r superstrates fr micrstrip antennas t suppress the excitatin f surfaces waves r be used in micrstrip circuits t add filtering effects n simple guiding structures like micrstrip lines. These applicatins and thers were the

2 222 Attiya and Kishk mtivatin t intrduce cmprehensive studies f different dielectric grating structures using different techniques during last tw decades [1 12]. These techniques can be classified int tw main categries, numerical techniques such as methd f mment [7] and finite difference time dmain [9] and semi-analytical techniques such as transverse resnance technique [3 5, 10] and mdal analysis [1, 2, 11, 12]. The main advantage f numerical techniques is that they can easily include any perturbatin in finite peridic structure as well as slving infinite peridic structures. On the ther hand, semi-analytical techniques are mainly valid fr infinite peridic structures. Hwever, these techniques require much less cmputatinal effrt t btain the characteristics f the grating structure. Vectrial mdal analysis shwed imprtant advantage cmpared with transverse resnance apprach fr mdeling ne-dimensinal grating structure where the cmplexity f the frmer des nt increase with the number f dielectric slabs present in the unit cell as it is usually happens in the latter apprach [1]. Anther imprtant advantage f vectrial mdal analysis cmpared with the transverse resnance methd is that it can be extended t twdimensinal structure as it is used fr slving a waveguide filled with inhmgeneus materials [13]. In vectrial mdal analysis, the presence f lsses in dielectric slabs can be easily cnsidered by simply intrducing cmplex permittivity [1]. It can be expected that by using semi-analytical techniques t btain the reflectin and transmissin cefficients f grating structures, it may be pssible t frmulate Greens functin f a pint surce abve grating structure. This frmulatin can be quite useful fr simulating an antenna abve grating structure by using integral equatin apprach. Therefre, the present wrk extends the vectrial mdal analysis f ne-dimensinal dielectric grating t the case f twdimensinal dielectric grating. Figure 1 shws a schematic diagram f the prpsed prblem fr a simple case f a single layer-grating slab. It shuld be nted that fr the case f ne dimensinal grating, where the embedded dielectrics are rectangular rds f finite width alng the peridic directin and equal t the dielectric thickness, h. The prblem f tw-dimensinal dielectric grating layered structure can be presented as a multi-sectin guiding structure f finite thicknesses and infinite lateral dimensins. Each guiding structure cnsists f peridic dielectric rds embedded in a base dielectric peridic cell. The fields inside each guiding structure are determined in terms f its mdes. Thus, the vectrial mdal analysis f dielectric grating slabs starts with finding the mdes f each layer. Then the bundary cnditins are btained by matching the tangential fields n the interface between the different layers.

3 Prgress In Electrmagnetics Research, PIER 60, y l x1 ε r1 z TM e D y ε rb x 01 y 01 l y1 x θ inc TE e a r inc h D x φ inc y ε rb x Figure 1. Tw-dimensinal dielectric grating slab excited by bliquely incident plane wave. By enfrcing btaining the apprpriate bundary cnditins, ne can btain the reflectin and transmissin cefficients frm ne grating structure t the ther. Then by using the generalized scattering matrix apprach ne can btain the transmissin and reflectin f multilayered grating structure [14]. The details f the mdal analysis fr twdimensinal dielectric grating structures are discussed in Sectin 2. Then the prblems f the bundary cnditins and the generalized scattering matrix are discussed in Sectin 3. Sample results and discussins are presented in Sectin MODAL ANALYSIS OF TWO-DIMENSIONAL DIELECTRIC GRATING Starting frm Maxwell s equatins, it can be shwn that fr spatial dependent medium the wave equatin f the magnetic field is ( H = k0 2 H (1 ε 1 r Fr the case f a guiding structure, ε r is independent f its lngitudinal directin and the field dependence n the lngitudinal directin is e jβz, assuming that z is the lngitudinal directin. Thus, the transverse field part f (1 can be rewritten as fllws: [ ( ] 2 t + k0ε 2 t ε r r + ( t ht = β 2 ht (2 ε r

4 224 Attiya and Kishk where h t is the field distributin f the transverse magnetic field cmpnents. By slving this eigen value prblem, ne can btain the field distributin f the transverse magnetic field h t and the prpagatin cnstant β. The slutin requires expanding the transverse magnetic field in terms f a set f rthgnal Flquet mde expansin functins. These expansins are btained fr the TE and TM mdes separately, which requires use f bi-rthgnal functins t represent each mde separately. It is imprtant t chse such bi-rthgnal expansin functins. In previus studies f ne-dimensinal dielectric grating, the plane cnsisting f the directin f incidence and the nrmal directin t the interface between the dielectric layers f the same grating structure is cnsidered as the plane f incidence. Then, the TE and TM directins are defined based n this plane. Such definitin simplifies the relatin between the transverse magnetic field btained by slving the abve eigen value prblem and the transverse electric field [1, 4]. Hwever, this apprach cannt be applied fr tw-dimensinal grating because there is n fixed nrmal directin t the interface between the embedded dielectric rds and the surrunding dielectric base medium. Thus, it wuld be preferred t define the plane f incidence as the plane cnsisting f the directin parallel t the embedded rds, which is the z directin, and the directin f incidence. The advantage f this chice is that it wuld be cnsistent with the traditinal definitin f the TE and TM wave fr planar structure. Fr arbitrary incident plane wave, the directin f incidence is a inc = sin θ inc cs φ inc a x + sin θ inc sin φ inc a y cs θ inc a z (3 Thus, the unit vectrs f the TE and TM transverse magnetic fields are a ht E = cs φ inc a x + sin φ inc a y (4a a ht M = sin φ inc a x + cs φ inc a y (4b Based n the abve definitin f TE and TM magnetic field directins, it can be assumed that the bi-rthgnal Flquet mde expansin functins can be defined as hte mn = exp( jk xmx jk yn y (cs φ inc a x + sin φ inc a y (5a Dx D y htm mn = exp( jk xmx jk yn y ( sin φ inc a x + cs φ inc a y (5b Dx D y where k xm = k x0 +2πm/D x (5c

5 Prgress In Electrmagnetics Research, PIER 60, k yn = k y0 +2πn/D y (5d k x0 = k 0 sin θ inc cs φ inc (5e k y0 = k 0 sin θ inc sin φ inc (5f D x and D y are the dimensins f the peridic cell. The bi-rthgnality relatin can be easily verified as fllws D y D x hζ mn, hξ m n = 0 0 hζ mn hξm n dxdy = δ mm δ nn δ ζξ (6 Using these bi-rthgnal expansin functins, the transverse magnetic field cmpnent can be represented as ht = h TE t + h TM t = p C(p TE hte (p + C TM (p htm (p (7 where each (p represents different cmbinatins f mn. Ideally, this series shuld be infinite. Hwever, fr numerical implementatin this series is truncated t be finite. The prblem f btaining the transverse magnetic field distributin is nw cnverted int btaining the amplitudes f such mdal expansin. By inserting (7 int (12, ne can btain L h t = p where C(p TE L hte (p + C(p TM L htm (p = β 2 p C(p TE hte (p + C TM (p htm (p (8a L = L (8b L 0 = 2 t + k0ε 2 rb (8c 1 = k0 2 (ε r (x, y ε rb (8d ( t ε r (x, y 2 = ( t (8e ε r (x, y T btain the unknwn amplitudes f the field expansin functins and the crrespnding prpagatin cnstant f each mde cnvert (8a int an eigen value prblem as fllws [ [ L T E/T E pq L T M/T E pq ] [ ] [ L T E/T M pq L T M/T M pq ] [ ] [ C TE (q C TM (q ] ] = β 2 [ C(q TE [ C(q TM ] ] (9a

6 226 Attiya and Kishk where L T E/T E pq = L T E/T M pq = L T M/T E pq = L T M/T M pq = hte hte (p,l 0 (q hte htm (p, 2 (q htm hte (p, 2 (q htm htm (p,l 0 (q hte + (p, 1 hte (q hte hte + (p, 2 (q (9b (9c (9d htm htm htm htm + (p, 1 (q + (p, 2 (q (9e It shuld be nted that the peratr 2 is the nly part f the peratr L that can cuple TE and TM waves. By btaining the eigen values and the eigen vectrs f (9, ne can btain the field distributin f each mde by using (7 and its crrespnding prpagatin factr. Fr the case f a grating cnsisting f rectangular dielectric rds, the relative dielectric cnstant at each cell can be represented as ND ε r (x, y = ε rb + [(ε ri ε rb (H(x x 0i +l xi /2 H(x x 0i l xi /2 i=1 (H(y y 0i + l yi /2 H(y y 0i l yi /2] (10 where H( is the Heaviside unit step functin, (x 0i,y 0i is the center f the i th rd, l xi and l yi and are the dimensins, and ND is the number f dielectric rds in unit cell. Using (8 and (10 int (9 a clsed frm fr the matrix elements in (9 as L T pq E/T E = β ( pδ 2 pq + k0r 2 0 k x(p k x(q,k y(p k y(q (jk x(q sin φ inc jk y(q cs φ inc L T M/T M + (R 1y (k x(p k x(q,k y(p k y(q cs φ inc ( R 1x k x(p k x(q,k y(p k y(q sin φ inc (11a pq = β ( pδ 2 pq + k0r 2 0 k x(p k x(q,k y(p k y(q + (jk x(q cs φ inc + jk y(q sin φ inc ( R 1y (k x(p k x(q,k y(p k y(q sin φ inc ( R 1x k x(p k x(q,k y(p k y(q cs φ inc (11b

7 Prgress In Electrmagnetics Research, PIER 60, L T pq E/T M = (jk x(q cs φ inc + jk y(q sin φ inc (R 1y (k x(p k x(q,k y(p k y(q cs φ inc ( R 1x k x(p k x(q,k y(p k y(q sin φ inc (11c where L T M/T E pq = (jk x(q sin φ inc jk y(q cs φ inc ( R 1y (k x(p k x(q,k y(p k y(q sin φ inc ( R 1x k x(p k x(q,k y(p k y(q cs φ inc (11d β p = R 0 (k x,k y = R 1x (k x,k y = R 1y (k x,k y = ε rb k 2 0 k2 x(p k2 y(p ND 1 4(ε ri ε rb D x D y i=1 sin(k x l xi /2 sin(k y l yi /2 e j(kxx 0i+k yy 0i k x k y 1 D x D y ND i=1 8j (ε ri ε rb (ε ri + ε rb sin(k x l xi /2 sin(k yl yi /2 k y e j(kxx 0i+k yy 0i 1 D x D y ND i=1 8j (ε ri ε rb (ε ri + ε rb (12a (12b (12c sin(k x l xi /2 sin(k y l yi /2e j(kxx 0i+k yy 0i (12d k x Up t this pint ne can btain the distributin f the transverse magnetic field f each mde and its prpagatin cnstant. 3. BOUNDARY CONDITIONS ON THE INTERFACE BETWEEN TWO ADJACENT DIELECTRIC GRATINGS The tangential magnetic and electric fields shuld be cntinuus n the interface between adjacent dielectric gratings. The cntinuity f the tangential magnetic field can be btained directly by using the mdal representatin btained in the previus sectin as fllws: (I R 11 C 1 = T 21 C 2 (13

8 228Attiya and Kishk where R 11 is the reflectin matrix at the first grating structure, T 21 is the transmissin matrix frm the first grating t the secnd grating, C 1 and C 2 are the transpse matrix cnsisting f the eigen vectrs crrespnding t the mdal analyses f the first and the secnd grating, respectively. The prblem nw is t btain the tangential electric field. It shuld be nted that the transverse electric field distributin culd be btained via anther eigen value prblem similar t (1 f the transverse magnetic field [1]. Hwever, btaining the electric field distributin nly is nt sufficient t btain the secnd bundary cnditin because it des nt present the relatin between the amplitude f the transverse electric field and the transverse magnetic field fr each mde, which is als required t btain the secnd bundary cnditin. Thus, it is required t btain the transverse electric field in terms f the btained transverse magnetic field. Fr the case f ne-dimensinal grating, this prblem is simplified by apprpriate chice f TE and TM definitins as mentined in the previus sectin. In this case, the relatin between the transverse electric and magnetic fields are nly multiplicatin factrs, which represent the TE and the TM characteristic impedances [1, 4]. In this case the TE characteristic impedance is independent f the variatin f the dielectric cnstant. Hwever, in the present case the relatin between the transverse electric and magnetic fields cannt be just multiplicatin factrs. T find this relatin, it wuld be required t btain the transverse electric field in terms f the ttal magnetic field including the lngitudinal magnetic field cmpnent. The lngitudinal magnetic field cmpnent in terms f the transverse magnetic cmpnents can be btained as ( h z = jβ 1 t h t (14 Thus the transverse electric field can be btained in terms f the transverse magnetic field as fllws: jωε 0 ε r e t = [ t a z ( jβ 1 t h t jβ a z ] h t (15 Using the mdal representatin f the transverse magnetic field, (15 can be rewritten in a mdal frm as: ] jωε 0 ε r e t = [( jβ 1 C ( t a z t ht jβc a z ht where ht is a vectr including the TE and TM expansin mdes ht = [[ ] [ ]] T hte htm, β is a diagnal matrix including the prpagatin (p (p

9 Prgress In Electrmagnetics Research, PIER 60, cnstants f the different mdes and C is the transpse matrix cnsisting f the eigen vectrs f the mdal analysis fr the transverse magnetic field. Assuming that the transverse electric field is expanded in terms f transverse mdal functins crrespnding t TE and TM mdes ẽ t = [[ ẽ TE (p ] [ ẽ TE (p ]] T as e t = e TE t + e TM t = p Φ TE (p ẽ TE (p +Φ TM (p ẽ TM (p (16 where ẽ TE (p = a z hte (p and ẽ TM (p = a z htm (p. The electric field amplitude matrix Φ can be btained by using the bi-rthgnal prperty f the electric field mdal expansin functin as fllws: ] jωε 0 ẽ, ε r e t = ẽ, [( jβ 1 C( t a z t ht jβc a z ht (17 By slving (17, it can be shwn that the electric field amplitude matrix Φis Φ= 1 ( jβ 1 CAA T + jβc Ψ 1 (18 jωε 0 [ ] [ ] A TE Ψ TE 0 where A = A TM and Ψ = 0 Ψ TM. A TE is a diagnal matrix whse element is ( jk x(p cs φ inc jk y(p sin φ inc and A TM is a diagnal matrix whse element is (jk x(p sin φ inc jk y(p cs φ inc Ψ TE =Ψ TM is a matrix with element given as ψ pq = 1 D x D y D x 0 D y 0 ε r (x, ye j(k x(p k x(q x e j(k y(p k y(q y dxdy After btaining the transverse electric field amplitude matrix in terms f the previusly btained transverse magnetic field amplitude matrix, ne can btain the equatin f the secnd bundary cnditin as: (I + R 11 Φ 1 = T 21 Φ 2 (19 Slving (13 and (19 simultaneusly, ne can btain the reflectin and transmissin matrices as: ( T 21 = 2 C 2 C1 1 +Φ 2 Φ (20a ( R 11 = I 2 C 2 C1 1 +Φ 2 Φ C2 C1 1 (20b

10 230 Attiya and Kishk The reflectin at the secnd grating structure and the transmissin can be btained frm the secnd t the first grating by exchanging the subscripts 1 and 2 in (20. In practices, tw hmgeneus media at bth sides f the layered structure surrund the layered grating slabs. Hwever, t facilitate the analysis it is assumed that such hmgeneus media are cmpsed f grating structure f the same cell size f the slab gratings. The base dielectric cnstants f these tw semi-infinite gratings are the dielectric cnstants f the tw surrunding media and there are n embedded rds in such media. Fr [ such hmgenus ] medium the C β TE 0 matrix is simply a unit matrix, β = 0 β TM where β TE = β TM are diagnal matrices whse elements are ε r k0 2 k2 x(p k2 y(p. The abve analysis is suitable t btain the reflectin and transmissin matrices between tw semi-infinite grating structures. This analysis can be cmbined with generalized scattering matrix apprach t btain the reflectin and transmissin frm a multilayered medium cmpsed f grating slabs. It shuld be nted that the wrd layer here means a grating slab and shuld nt be cnfused with the structure f the grating itself. As a special case, ne can btain the reflectin and transmissin matrices f a grating slab in free space as fllws: R 11 = R 11 + T 12 (I BR 22 BR 22 1 BR 22 BT 21 (21a T 12 = T 12 (I BR 22 BR 22 1 BT 21 (21b where medium 1 is assumed t be free space and medium 2 is the grating slab, B is a diagnal matrix whse element is exp( jβ i h, β i is the ith mdal prpagatin cnstant f the grating structure, and h is the thickness f the grating slab. 4. RESULTS AND DISCUSSIONS In this sectin, different examples are presented t shw the validity f the abve frmulatin. These examples include bth ne- and tw-dimensinal grating structures. It shuld be nted that nedimensinal grating can be treated as a tw-dimensinal grating whse embedded rds extend in ne directin up t the cell bundaries. Fr the sake f cmparisn with previusly mentined results, it shuld be nted that the TE case crrespnds t electric field nrmal t the plane f incidence and parallel t the interface between the grating layers and TM case crrespnds t electric field parallel t the plane incidence nrmal t that directin, which is cnsistent with the present

11 Prgress In Electrmagnetics Research, PIER 60, Γ TE k 0 h Figure 2. Cmparisn between Transverse Resnance Methd [5] (Dtted pints and Vectrial mdal analysis (Slid line fr calculating TE reflectin cefficient f a ne-dimensinal grating with ε rb =1.44, ε r1 =2.56, D x = D y = 10 mm, l x1 = 10 mm, l y1 = 5 mm, x 01 = 5 mm, y 01 = 7.5 mm and h = mm. φ inc = 90 and θ inc =45. definitin f TE and TM if it is assumed that the directin parallel t the interface between the layers f the ne-dimensinal gratings is the x-directin and φ inc is π/2. Other chices are als pssible. Figure 2 shws the TE reflectin f a ne-dimensinal dielectricgrating slab whse base dielectric cnstant ε rb =1.44. It includes nly ne dielectric rd per cell. The dielectric cnstant f such dielectric rd is ε r1 =2.56. The unit cell is a square f length 10 mm. The dimensin f the embedded dielectric rd is 10 mm 5 mm. The thickness f the substrate is mm. The directin f the incident plane wave is φ inc = π/2 and θ inc = π/4. Excellent agreement is btained between the present results and the previusly published result f Bertni et al. [5, Fig. 3], which is btained by using transverse resnance methd. Figure 3 presents anther example f a ne-dimensinal dielectricgrating slab, where it presents the transmissin cefficients fr bth TE and TM incident waves. The base dielectric is free space and the dielectric cnstant f the embedded rds is ε r1 = The lss tangent f the dielectric rds is tan δ = The unit cell is a square f length 30 mm. The dimensin f the embedded dielectric rd is 30 mm 15 mm. The thickness f the substrate is 8.7 mm. The directin f the incident plane wave is φ inc = π/2 and θ inc = π/180. These results shw gd agreements when cmpared with the crrespnding experimental results f Tibuleac et al. [8]. It shuld be nted that all these results have been btained by using nly

12 db 232 Attiya and Kishk T TE TM T db Frequency (GHz (a Frequency (GHz Figure 3. Cmparisns between measured [8] and calculated TE and TM transmissin cefficients f a ne-dimensinal grating with ε rb =1,ε r1 =2.59, tan δ 1 =0.0067, D x = D y = 30 mm, l x1 = 30 mm, l y1 = 15 mm, x 01 = 15 mm, y 01 =22.5mm and h =8.7mm. φ inc =90 and θ inc =1. Measurements are dtted pints and calculated results are slid lines. (a TE, (b TM. (b 25 Flquet mde expansin functins where 2 m, n 2 in (5. Figure 4 presents an example f the TM reflectin f a twdimensinal dielectric grating slab. The base dielectric is ε rb = 4 and the dielectric cnstant f the embedded rds is ε r1 = 10. The unit cell is a square f length 20 mm and the crss sectin f the embedded rd is als square f length 10 mm. The thickness f the substrate is 2 mm. The φ inc angle is fixed t be zer degree while θ inc

13 Γ TM db db Prgress In Electrmagnetics Research, PIER 60, (a Frequency (GHz (b Γ TM Frequency (GHz (c Figure 4. TM reflectin cefficients f a tw-dimensinal grating with ε rb =4,ε r1 = 10, D x = D y = 20 mm, l x1 = 10 mm, l y1 = 10 mm, x 01 = 10 mm, y 01 = 10 mm and h = 2 mm. φ inc =0 and (a θ inc =0, (b θ inc =15 and (c θ inc =30.

14 234 Attiya and Kishk has three values namely, 0, 15 and 30 degrees. These results shw gd agreement with the crrespnding results btained by Methd f Mment slutin f Yang et al. [7]. Hwever, it is nted that there is a fixed difference in the resnance frequency between their results and the present results. This is examined by increasing the number f the Flquet mde expansin functins. It is fund that the present slutin is highly cnvergent. Therefrethe same prblem is simulated by HFSS fr the case f nrmal incidence. The three results are cmpared in Fig. 4(a. It can be nted that there is an excellent agreement between the results f HFSS simulatin and the results f the vectrial mdal analysis. The main advantage f the present technique is the fast cnvergence as cmpared with the Methd f Mment slutin where the present results are btained by using nly 49 Flquet mde expansin functins, 3 m, n 3, where each mde represent an expansin functin whereas Yang et al. [7] used 1681 Flquet mdes cmbined with 9 expansin functins t btain his results % errr N Figure 5. Percentage errr f TE and TM transmissin and reflectin cefficients as functins f the number f Flquet mde expansin functins N m, n N. The percentage errr is cmpared with the results f m = n = 10. The grating structure is the same as shwn in Fig. 4. θ inc =30 and φ inc =0, f = 10 GHz. T shw the fast cnvergence f the present technique, Fig. 5 shws the percentage errr f TE and TM transmissin and reflectin cefficients as a functin f the number f Flquet mde expansin functins. The grating structure is the same as that f Fig. 4. The angle f incidence is θ inc = 30 and φ inc = 0, and the perating frequency is 10 GHz. The Flquet mde expansin functins are

15 Prgress In Electrmagnetics Research, PIER 60, determined as N m, n N. The errrs are cmputed based n the slutins btained with N = 10 t be the exact ne. It can be nted that, fr N = 2 the maximum percentage errr is less than 4%. The errr decreases mntnically as N increases. This maximum percentage errr decreases t 1.5% fr N = 3, which is the value used t btain the results presented in this paper. Figure 6. TE reflectin cefficients f the same grating structure f Fig. 4. φ inc =0, (a θ inc =0, (b θ inc =15 and (c θ inc =30. Figure 6 shws the TE reflectin f the same grating structures at the same incident angles. It can be nted that due t the symmetry f the embedded dielectric rd with respect t the z-axis, the TE and the TM reflectin cefficients are exactly the same at θ inc =0 and φ inc = 0, thus it has nt been repeated in Fig. 6. By cmparing Figures 5 and 6 fr different θ inc angles fr the same φ inc angle it can be nted that the TM resnance frequencies are slightly affected by changing θ inc. Hwever, the number f the TE resnances increases by increasing the elevatin angle θ inc and the crrespnding resnance frequencies decrease. Figure 7 shws the dependence f the TE and TM reflectin cefficients n the angle φ inc fr fixed tilting where θ inc =30 fr the same grating structure f Figure 4 at 10 GHz. Due t the symmetry f the embedded rd inside the peridic cell, nly π/4 sectin is sufficient t present the dependence f such reflectins cefficients n the angle φ inc. Figure 7 shws π/2 sectin t shw part f this symmetry. It can be nted that, in additin t the frequency selectivity behavir

16 236 Attiya and Kishk Γ Figure 7. Reflectin cefficients at 10 GHz as a functin f φ inc fr the same grating structure f Fig. 4. θ inc =30. φ inc f the dielectric grating structure, the angular selectivity can als be btained fr the same frequency as shwn in Fig. 7. It shuld be nted that all the previus results present nly the specular reflectin and transmissin cefficients. In additin t such specular cmpnents, there are als mde cnversin cmpnents, where the TE incident wave intrduces TM wave and vise versa, and nn-specular cmpnents, which crrespnd t grating lbes. The existence f these cmpnents depends n the dimensins f the peridic cell with respect t the incident wavelength, the cupling effect f the implanted dielectric rds, the directin f incidence and the cupling effect f the embedded rds between the TE and TM waves. Tables 1 and 2 shws samples results fr the grating structure f Fig. 4 where it they intrduces in additin t its specular wave, ther three wave cmpnents including cupling between TE and TM waves. The perating frequency in this case is 15 GHz and the directin f incidence is θ inc =45 and φ inc =30. Pwer cnversin fr bth TE and TM waves can be easily examined by verifying M ( Γ T E/T E i 2 + T T E/T E i 2 + Γ T M/T E i 2 + T T M/T E i 2 =1 (22a i=1 M ( Γ T E/T M i 2 + T T E/T M i 2 + Γ T M/T M i 2 + T T M/T M i 2 =1 (22b i=1

17 Prgress In Electrmagnetics Research, PIER 60, Table 1. Ttal reflectin cefficients at 15 GHz fr the same grating structure f Fig. 4. θ inc =45 and φ inc =30. Directin f diffracted field Mde f diffracted field TE /TE Γ TM /TE Γ TE /TM Γ TM /TM Γ = 45, = 30 ( m = 0, n = = 27.07, = ( m = 0, n = = 58.35, = ( m = 1, n = = 41.08, = ( m = 1, n = Table 2. Ttal transmissin cefficients at 15 GHz fr the same grating structure f Fig. 4. θ inc =45 and φ inc =30. Directin f diffracted field Mde f diffracted field TE /TE T TM /TE T TE /TM T TM /TM T = 45, = 30 ( m = 0, n = = 27.07, = ( m = 0, n = = 58.35, = ( m = 1, n = = 41.08, = ( m = 1, n = where M is the ttal number f the prpagating grating lbes. Figure 8 shws fur-layers 2-D grating slabs as a wide band dielectric FSS. It is cmpsed f fur 2-D grating slabs supprted by three hmgenus dielectric slabs. The unit cell f the 2-D grating slab is a square dielectric ring f ε rb = 12. The dimensins f the unit cell are D x = D y = 10 mm and the dimensins f the air hle inside the dielectric ring are L x = L y = 7 mm. The thickness f the grating slab is h 1 = 2 mm. The supprting hmgenus dielectric slabs have a dielectric cnstant ε r =2.2 and dielectric thickness h 2 = 4 mm. Figure 9 shws the transmissin and reflectin cefficients f this FSS fr the case f nrmally incident plane wave. The results f Fig. 9 are calculated by using bth mdal analysis and HFSS. The cmparisn shws gd agreement between the tw techniques. It can be nted that this FSS structure can be used as a reflecting structure with less than 20 db transmissin in the frequency band frm 13.2 t 15.6 GHz which is nearly 16.6% bandwidth with respect t its central frequency. It can be als used as a transparent surface fr frequencies less than 10 GHz with maximum insertin lss belw 3 db. The supprting dielectric slabs have a significant effect n the reflectin bandwidth. Figure 10 shws the effect f the dielectric

18 238Attiya and Kishk Figure 8. FSS f multilayered dielectric grating slabs. D x = D y = 10 mm, L x = L y = 7 mm, ε rb = 12, h 1 = 2 mm, ε r = 2.2 and h 2 = 4 mm. Figure 9. Reflectin and Transmissin cefficients f multilayered dielectric grating slabs shwn in Fig. 8 fr the case f nrmal incidence. θ inc = 0 and φ inc = 0. The incident field is y-plarized field.

19 Prgress In Electrmagnetics Research, PIER 60, % Reflectin bandwidth % Ref. Bandwidth Center Freq Center frequency (GHz h2 (mm Figure 10. Percentage reflectin bandwidth and its crrespnding center frequency as functins f supprting slab thickness h 2. The remaining parameters are as shwn in Fig. 8. thickness f the supprting dielectric slab n the reflectin bandwidth and the center frequency f the reflectin bandwidth where the dielectric cnstant is ε r eauals 2.2. It can be nticed that the center frequency f the reflectin bandwidth decreases by increasing the thickness f the supprting dielectric slab and the maximum reflectin bandwidth fr such dielectric cnstant is nearly 20% at a dielectric thickness h 2 equals 3 mm. Figure 11 shws the effect f the dielectric cnstant f the supprting dielectric slabs n the reflectin bandwidth. It can be nted that the reflectin band with increases by decreasing the dielectric cnstant f the supprting slabs while the center frequency f the reflectin bandwidth has slightly decreased On the ther hand, the thickness f the grating structure h 1 has a similar effect n the reflectin bandwidth as shwn in Figure 12. Hwever the change f the reflectin bandwidth is mre sensitive t the thickness f the grating structure and the decrease f the center frequency linearly. Finally, Figure 13 shws the effect f base dielectric cnstant f the grating structure ε rb. It can be nted that the reflectin bandwidth increases by increasing ε rb while its center frequency decreases by increasing ε rb. Based n the abve discussins it can be cncluded that t design a wideband FSS cmpsed f 2-D dielectric grating slabs f square

20 240 Attiya and Kishk Figure 11. Percentage reflectin bandwidth and its crrespnding center frequency as functins f supprting slab dielectric cnstant ε r. The remaining parameters are as shwn in Fig. 8. % Reflectin band width % Ref. Bandwidth Center Freq Center frequeny (GHz h1 (mm Figure 12. Percentage reflectin bandwidth and its crrespnding center frequency as functins f grating slab thickness h 1. The remaining parameters are as shwn in Fig. 8.

21 Prgress In Electrmagnetics Research, PIER 60, Figure 13. Percentage reflectin bandwidth and its crrespnding center frequency as functins f ε rb. The remaining parameters are as shwn in Fig. 8. dielectric rings supprted by hmgenus dielectric slabs it is required t increase the base dielectric cnstant f the grating structure and t decrease the dielectric cnstant f the supprting dielectric slab. The thicknesses f the grating slab and the supprt slabs have significant effects n the reflectin bandwidth. Apprpriate chice f these thicknesses is required t btain maximum reflectin bandwidth. 5. CONCLUSION An efficient apprach t study the reflectin and transmissin cefficients f tw-dimensinal dielectric grating structures fr arbitrary incident plane wave was presented. The apprach was based n vectrial mdal analysis f the grating structure t btain the field distributin and the prpagatin cnstant f different mdes in the grating structure. The mdal analysis was cmbined with the generalized scattering matrix. The reflectin and transmissin cefficients f multilayered tw-dimensinal dielectric grating structure were btained. The technique was verified by cmparing the results f the present apprach with previusly published results based n experimental and ther numerical techniques. The present mdal analysis required expanding the field in the transverse directin

22 242 Attiya and Kishk nly while the methd f mments based n vlume integral equatin requires expanding the field in the three-dimensinal space, which requires mre expansin functins especially fr thick grating structures. A wideband FSS cmpsed f multilayered grating slabs supprted by hmgeneus dielectric slabs was designed. It was als fund that this apprach requires less Flquet mdes fr cnvergence as cmpared with the Methd f Mment. REFERENCES 1. Cves, A., B. Gimen, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Bria, Full-wave analysis f dielectric frequency-selective surfaces using vectrial mdal methd, IEEE Trans. Antennas Prpagat., Vl. 52, , Aug Cves, A., B. Gimen, A. A. San Blas, A. Vidal, V. E. Bria, and M. V. Andres, Three-dimensinal scattering f dielectric gratings under plane-wave excitatin, IEEE Antennas and Wireless Prpagatin Letters, Vl. 2, , Peng, S. T., T. Tamir, and H. L. Bertni, Thery f peridic dielectric waveguides, IEEE Trans. Micrwave Thery Techniques, Vl. 23, , Jan Peng, S. T., Rigrus frmulatin f scattering and guidance by dielectric grating waveguides: General case f blique incidence, J. Opt. Sc. Amer. A., Vl. 6, N. 12, , Dec Bertni, H. L., L. H. S. Che, and T. Tamir, Frequency-selective reflectin and transmissin by a peridic dielectric layer, IEEE Trans. Antennas Prpagat., Vl. 37, 78 83, Jan Pai, D. M. and K. A. Awada, Analysis f dielectric gratings f arbitrary prfiles and thicknesses, J. Opt. Sc. Amer. A., Vl. 8, N. 5, , May Yang, H. Y. D., R. Diaz, and N. G. Alexpuls, Reflectin and transmissin f waves frm multilayer structures with planarimplanted peridic material blcks, J. Opt. Sc. Amer. B., Vl. 14, , Oct Tibuleac, S., R. Magnussn, T. A. Maldnad, P. P. Yung, and T. R. Hlzheimer, Dielectric frequency-selective structures incrprating waveguide gratings, IEEE Trans. Micrwave Thery Tech., Vl. 48, , Apr Msallaei, H. and Y. Rahmat-Samii, Peridic bandgap and effective dielectric materials in electrmagnetics: characterizatin and applicatins in nancavities and waveguides, IEEE Trans. Antennas Prpagat., Vl. 51, , Mar

23 Prgress In Electrmagnetics Research, PIER 60, Csta, J. C. W. A. and A. J. Giarla, Electrmagnetic wave prpagatin in multilayer dielectric peridic structures, IEEE Trans. Antennas Prpagat., Vl. 41, , Oct Jarem, J. M. and P. P. Banerjee, Cmputatinal Methds fr Electrmagnetic and Optical Systems, Marcel Dekker, Inc., Mharam, M. G., E. B. Grann, D. A. Pmmet, and T. K. Gaylrd, Frmulatin fr stable and efficient implementatin f the rigrus cupled-wave analysis f binary grating, J. Opt. Sc. Am. A., Vl. 12, , May Silvestre, E., M. A. Abian, B. Gimen, A. Ferrand, M. V. Andres, and V. E. Bria, Analysis f inhmgeneusly filled waveguides using a bi-rthnrmal-basis methd, IEEE Trans. Micrwave Thery Tech., Vl. 48, , Apr Mittra, R., C. H. Chan, and T. Cwik, Techniques fr analyzing frequency selective surfacesa review, Prc. IEEE, Vl. 76, , Dec