Progress In Electromagnetics Research, PIER 51, , 2005

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1 Progress I Electromagetics Research, PIER 51, , 2005 COMPLEX GUIDED WAVE SOLUTIONS OF GROUNDED DIELECTRIC SLAB MADE OF METAMATERIALS C. Li, Q. Sui, ad F. Li Istitute of Electroics Chiese Academy of Sciece Beijig, , P. R. Chia Abstract This paper focuses o the complex guided wave solutios of grouded slab made of metamaterials. Complex solutios of both TE ad TM modes have bee aalyzed. It is foud that they are distributed o the proper Riema sheet. This property differs dramatically from that of a covetioal grouded slab. A umber of other distributio properties are studied aalytically. Some umerical examples are give to verify the aalytical results. It is importat to take the peculiar complex solutios ito accout i the problems of microstrip ad surface wave excitatio. 1 Itroductio 2 Eigevalue Equatios for Grouded Slab Made of MTMs 3 Complex Guided Wave Solutios of TE Modes 4 Complex Guided Wave Solutios of TM Modes 5 Discussios ad Coclusios Ackowledgmet Refereces Also with School of Iformatio, Beijig Broadcastig Istitute, Beijig, , P. R. Chia

2 188 Li, Sui, ad Li 1. INTRODUCTION Theoretical possibility for metamaterials (MTMs) with simultaeously egative permittivity ad permeability was itroduced by Vesalago i 1968 [1]. Recetly, these materials have bee a topic of high iterest after the first experimetal demostratio of left-haded metamaterials was reported [2]. It has bee show that such materials have uique properties such as backward wave (BW) ad egative refractio idex (NRI). So far, a lot of attetio has bee draw to fid the potetial applicatios of such materials. Oe of the pioeerig works was Pedry s suggestio of perfect les made of MTMs [3]. Recet advaces i the theoretical ad experimetal researches of the composite MTMs [4, 5] make further cotributios to the realizatio of these applicatios. The guided surface waves at the iterfaces betwee MTMs ad covetioal materials ca offer ew iterestig pheomea ad may have some importat applicatios [6 9]. So far, a umber of uusual properties have bee foud for the real guided wave solutios of these structures, such as the absece of the fudametal modes ad the existece of slow wave mode [9]. I fact, there may exist lots of complex solutios for the eigevalue equatios. These solutios are importat because their trasitio effects should be icluded i excitatio of surface waves ad microstrip problems, especially whe the cotour i the itegral of the Gree s fuctios is deformed. I this paper, we cosider a slab made of MTMs located o a perfect groud ad place our emphasis o the ivestigatio of the complex guided wave solutios. Based o this aalysis, it was foud that there may exist a ifiite umber of complex TE ad TM surface wave solutios o the proper Riema sheet. This property differs dramatically from that of a covetioal grouded slab. We also study the distributio properties of the complex solutios ad fid some of them i the proper Riema sheet by umerical methods. It is importat to take these peculiar solutios ito accout whe cosiderig excitatio of surface waves ad microstrip problems. 2. EIGENVALUE EQUATIONS FOR GROUNDED SLAB MADE OF MTMS Surface waves of both the TE ad TM type may be guided alog a grouded dielectric slab. Let the thickess of the slab be h, the relative permittivity be ε r, ad relative permeability be µ r, as show i Figure 1. ε r < 1 ad µ r < 1 are assumed ad suppressed i this paper, which meas the refractive idex of the slab is higher tha that

3 Progress I Electromagetics Research, PIER 51, Figure 1. Grouded slab made of MTMs. of vacuum. The eigevalue equatios ca be obtaied by matchig the fields tagetial to the iterface at x = h [10] T e (k) =α 1 h + 1 µ r k 2 h cot(k 2 h) = 0 T m (k) =α 1 h 1 ε r k 2 h ta(k 2 h) = 0 (1a) (1b) (1a) is for TE modes ad (1b) is for TM modes, k is i geeral complex, ad α1 2 = k 2 k0 2 (2a) k2 2 = k0µ 2 r ε r k 2 (2b) It is coveiet to rewrite (2a) ad (2b) as follows: (α 1 h) 2 +(k 2 h) 2 = k0h 2 2 (µ r ε r 1) = a 2 (3) Solutios satisfyig Re(α 1 ) > 0 lie o the proper Riema sheet ad lead to surface wave modes, while solutios with Re(α 1 ) < 0 lie o the improper Riema sheet ad give rise to leaky wave modes. Graphical methods to ivestigate the solutios with real k have bee addressed i [8, 9]. I the preset work, emphasis is placed o aalyzig the solutios with complex k. Complex solutios of TE ad TM modes are aalyzed i Sectios 3 ad 4, respectively. Some distributio properties are preseted i each sectio, with umerical examples give to verify the aalytical results. 3. COMPLEX GUIDED WAVE SOLUTIONS OF TE MODES From Sectio 2, the guided wave solutios of TE modes are determied by a simultaeous roots of (1a) ad (3). I order to fid the complex

4 190 Li, Sui, ad Li solutios, let α 1 h be the complex variable α 1 h = W = u + iv (4) ad let k 2 h be the complex variable Z = x + iy (5) Equatio (1a) becomes W = 1 Z cot(z) (6) µ r or i compoet form u = 1 µ r y tah(y)[1 + ta 2 (x)] + x ta(x)[1 tah 2 (y)] ta 2 (x) + tah 2 (y) v = 1 µ r x tah(y)[1 + ta 2 (x)] y ta(x)[1 tah 2 (y)] ta 2 (x) + tah 2 (y) (7a) (7b) Equatio (3) becomes or i compoet form W 2 + Z 2 = a 2 (8) u 2 v 2 + x 2 y 2 = a 2 (9a) uv + xy = 0 (9b) Elimiatig the variable W i (6) ad (8), we obtai [ ] Z 2 µ 2 r + cot 2 (Z) = µ 2 ra 2 (10) Substitute (5) ito (10), after a rigorous derivatio, the term x ta(x) ca be rewritte i the form x ta(x) = µ2 ra 2 x 2 y 2 P (x 2 + y 2 ) 2 N > 0 (11) where N = [1 + ta 2 (x)]y tah(y)[1 tah 2 (y)] > 0 (12) P = [ta 2 (x) + tah 2 (y)] 2 > 0 (13) From (7a) ad (11), coclusio ca be readily draw that u is positive for all values of x 0 ad y 0. Cosequetly, the real part of α 1

5 Progress I Electromagetics Research, PIER 51, is always positive for the complex solutios. Therefore, the complex solutios of TE modes are located o the proper Riema sheet ad they are all surface wave solutios. O the other had, from (7) ad (9), it s see that, if x 1 + iy 1 is a solutio, the x 1 iy 1, x 1 + iy 1, ad x 1 iy 1 are also solutios. This meas the complex solutios of TE modes are symmetrically distributed i x-y plae. Hece we eed cosider oly positive values of x ad y. Accordig to (11), solutios with positive values of x are located withi the regio π<x<π+ 1 2π, where =0, 1, 2,... Based o the above aalysis, the first three solutios for a = 3, µ r = 1.6, x,y > 0 are foud by umerical method ad listed i Table 1: Table 1. Three complex solutios of TE modes for a =3,µ r = Z = x + iy 1.17 π i 2.02π i 3.01π i W = u + iv i i i 4. COMPLEX GUIDED WAVE SOLUTIONS OF TM MODES I a similar procedure, we fid the complex guided wave solutios for TM modes: Let α 1 h be the complex variable α 1 h = W = u + iv (14) ad let k 2 h be the complex variable Z = x + iy. (15) We ca rewrite equatio (1b) i compoet form u = 1 y tah(y)[1 + cot 2 (x)] x cot(x)[1 tah 2 (y)] ε r cot 2 (x) + tah 2 (16a) (y) v = 1 x tah(y)[1 + cot 2 (x)] + y cot(x)[1 tah 2 (y)] ε r cot 2 (x) + tah 2 (16b) (y) ad rewrite equatio (3) i compoet form u 2 v 2 + x 2 y 2 = a 2 (17a)

6 192 Li, Sui, ad Li uv + xy = 0 (17b) Followig the similar procedure i Sectio 3, the term x cot(x) ca be rewritte i the form x cot(x) = (x2 + y 2 ) 2 N ε 2 ra 2 y 2 Q < 0 (18) where N = [1 + ta 2 (x)]y tah(y)[1 tah 2 (y)] > 0 (19) Q = [1 + ta 2 (x) tah 2 (y)] 2 > 0 (20) As ca be see, u is positive for all values of x 0 ad y 0. Hece, it ca be cocluded that the real part of α 1 is always positive for the complex solutios. Therefore, the complex solutios of TM modes are located o the proper Riema sheet ad they are all surface wave solutios. Similarly, oly the positive values of x ad y eed to be cosidered. From (18), solutios with positive values of x are located withi the regio π + 1 2π<x<π+ π, where =0, 1, 2,... The first three solutios for a =3,ε r = 2.5, x,y>0 are foud by umerical method ad listed i Table 2: Table 2. Three complex solutios of TM modes for a =3,ε r = Z = x + iy 1.52π i 2.51π i 3.501π i W = u + iv i i i 5. DISCUSSIONS AND CONCLUSIONS At this poit, we have aalyzed the complex guided wave solutios for a grouded slab made of MTMs. I the followig, the importace of these peculiar solutios i solvig surface wave excitatio ad microstrip problems will be preseted. It is well kow that the spatialdomai Gree s fuctios i these problems ca be give i the itegral form: F (k, r ) G( r )= dk (21) 0 T e (k)t m (k) The expressios of T e ad T m have bee give i Eq. (1). The zeros of T e ad T m correspod to the TE ad TM poles of the itegrad,

7 Progress I Electromagetics Research, PIER 51, (a) Covetioal grouded slab (b) Grouded slab made of MTMs Figure 2. Surface wave poles i k-plae. respectively. For a covetioal grouded slab, it is kow that TE ad TM surface wave poles oly locate withi the regio k 0 <k<k 0 µr ε r o the Re(k) axis. A ice way to avoid the itegratio ear the poles is to deform the origi cotour C 0 to the cotour C 1 (as show i Figure 2a), which is above the real k axis o the proper Riema sheet. However, for a grouded slab made of MTMs, from Sectios 3 ad 4, we kow that, i additio to the real surface wave poles (this is ot the subject of this paper), there also exists a ifiite umber of complex TE ad TM surface wave poles o the proper Riema sheet. These complex poles will ot directly cotribute to the itegral alog the real axis. However, if the itegratio cotour is deformed above the real axis i the proper Riema sheet (as C 1 i Figure 2b), there is a possibility of capturig a fiite umber of complex surface wave poles, as show i Figure 2b, ad the calculatio of residues at the captured poles is required. I coclusio, we have cosidered a grouded slab made of MTMs ad explored the complex guided wave solutios of both TE ad TM modes. The importat differece betwee the complex solutios of ordiary ad MTMs grouded slab has bee foud aalytically. For MTMs grouded slab, the complex solutios are distributed o the proper Riema sheet ad they are all surface wave solutios. Hece, i the problems of microstrip ad surface wave excitatio, there is a possibility of capturig a fiite umber of complex poles as the itegratio cotour of Grees fuctios is deformed i the proper Riema sheet. I additio, it has bee foud that the complex solutios are symmetrically distributed i x-y plae ad they ca oly exist i specific regios. Some umerical examples of complex solutios have bee give to verify aalytical results of their properties.

8 194 Li, Sui, ad Li ACKNOWLEDGMENT This work was supported by the Natioal Natural Sciece Foudatio of Chia (Grat No ). The authors gratefully ackowledge the reviewers ad editors for several valuable suggestios that led to improvemets i the fial versio of this paper. REFERENCES 1. Veselago, V. G., The electrodyamics of substaces with simultaeously egative values of ε ad µ, Soviet Physics Uspekhi, Vol. 10, , July Shelby, R. A., D. R. Smith, ad S. Schultz, Experimetal verificatio of a egative refractive idex of refractio, Sciece, Vol. 292, 77 99, April Pedry, J. B., Negative refractio makes a perfect les, Phys. Rev. Lett., Vol. 85, , Simovski, C. R. ad S. He, Frequecy rage ad explicit expressios for egative permittivity ad permeability for a isotropic medium formed by a lattice of perfectly coductig Omega-particles, Phys. Lett. A, Vol. 311, 254, Huagfu, J. T., L. X. Ra, ad H. S. Che, et al., Experimetal cofirmatio of egative refractive idex of a metamaterial composed of Ω-like metallic patters, Applied Physics Letters, Vol. 84, No. 9, , Nefedov, I. S. ad S. A. Tretyakov, Theoretical study of waveguidig structures cotaiig backward-wave materials, Proc. of XXXVIIth URSI Geeral Assembly, Maastricht, The Netherlads, August 17 24, Alu, A. ad N. Egheta, Aomalous mode couplig i guided-wave structures cotaiig metamaterials with egative permittivity ad permeability, Proc. IEEE 2002 Naotechology Coferece, Washigto DC, August 26 28, Wu, B.-I., T. M. Grzegorczyk, Y. Zhag, ad J. A. Kog, Guided modes with imagiary traverse wave umber i a slab waveguide with egative permittivity ad permeability, Joural of Applied Physics, Vol. 93, No. 11, , March Ilya V. S., A. A. Sukhorukov, ad Yu. S. Kivshar, Guided modes i egative-refractive-idex waveguides, Physical Review E, Vol. 67, , May Colli, R. E., Field Theory of Guided Waves, 1990.

9 Progress I Electromagetics Research, PIER 51, Chao Li received B.Sc. degree i physics from Qigdao Ocea Uiversity i Chia i 2000, ad M.S. degree i electrical egieerig from Istitute of Electroics, Chiese Academy of Sciece i His research iterests iclude theoretical ad computatioal electromagetics. chaoliew@hotmail.com Qiag Sui received the master s degree from Beijig Broadcastig Istitute (BBI) i Chia i He works as a associate professor at BBI. He is pursuig Ph.D. i the Istitute of Electroics of Chiese Academy of Sciece ow. His research iterests iclude electromagetics theory ad microwave applicatios. suiqiag@bbi.edu.c Fag Li is a Professor of the Istitute of Electroics, Chiese Academy of Sciece. Her research iterests are maily i electromagetic theory ad waves i space plasma. fli@mail.ie.ac.c