Discrimination of Modes of Double- and Single- Negative Grounded Slab

Transcription

1 Wold Acadm of Scinc, Engining and Tchnolog Intnational Jounal of Elctonics and Communication Engining Vol:, No:5, 7 Discimination of Mods of Doubl- and Singl- Ngativ Goundd Slab R. Boghol, T. Aguili Intnational Scinc Indx, Elctonics and Communication Engining Vol:, No:5, 7 wast.og/publication/799 Abstact In this pap, w invstigat thoticall th wavs popagation in a losslss doubl-ngativ goundd slab (DNG). This stud is pfomd b th Tansvs Rsonanc Mthod (TRM). Th pop o impop natu of al and complx mods is obsvd. Th a highl dpndnt on mtamatial paamts, i.. ɛ - ngativ, µ -ngativ, o both. Numical sults povidd that onl th pop complx mods (i.., la mods) xist in DNG slab, and onl th impop complx mods xist in singl-ngativ goundd slab. Kwods Doubl-ngativ goundd slab, al and complx mods, singl-ngativ goundd slab, tansvs sonanc mthod. I. INTRODUCTION HE poduction of atificial dilctics with th pmittivit T and pmabilit abita (mtamatial) has tmndous intst in th communit scintific. Ths atificial dilctics ma giv is to unxpctd and intsting popagation fatu fo wavguiding stuctus [] and th a candidat to duc dg diffaction ffcts and nhanc adiation fficinc fo micostip antnnas []. Th atificial dilctics a usd in man applications such as micowav/millimt wav la wav antnnas [3], wh th ffctiv dilctic constants w btwn zo and unit. Sval goups studid a wavguiding stuctus basd on atificial dilctics mdia. Nfdov and Ttao [4] found that th xist mods whos filds dca xponntiall fom th intfac of th two mdia fo tansvs lctic () and tansvs magntic () polaizations. Ths mods, calld as vanscnt sufac mods, w also found b Alù and Enghta [5]. Ths studis hav no stud on th complx mods. In 6, Shu and Song suggstd th dispsiv popagation of complx mods (la wavs) on mtamatial goundd slab and thi invstigations on th Poting vctos show that ths mods do not tanspot ng in both tansvs and longitudinal dictions [6]. This stud is unfotunatl incomplt. In this pap, w focus to th modal poptis of sufac mods, volum mods, and complx mods suppotd b on o both ngativ paamts of goundd slab using th TRM in od to compltl assss thi popagation and adiation fatus. In Sction II, w psnt a simpl analsis of th dispsion lation of al and complx mods fo and R. Boghol is with th Univsit of Tunis El Mana, National Engining School of Tunis, LR99ES Lab. Ss Com, ENIT Tunis, Tunisia (-mail: bogholiham@gmail.com). T. Aguili is with th Univsit of Tunis El Mana, National Engining School of Tunis, LR99ES Lab. Ss Com, ENIT Tunis, Tunisia. polaizations in ou stuctu. In Sction III, w discuss th numical sults of th chaactistics of ths mods along th considd stuctu. In Sction IV, w conclud ou wo. II. ANALYSIS Th stuctu of intst, as shown in Fig., consists of two mdia, ai and goundd slab. Th infinitl xtnt slab of losslss matial is instd in this stuctu with thicnss d, whil th slab is assumd to b of a losslss isotopic DNG matial with al paamts and at th fqunc of intst. Fig. An infinit goundd slab with ngativ paamts and u To find th popagation constant fo popagation ppndicula to th plan xoz ( diction), w us th wllnown TRM in this diction. Th stuctu can b thn considd as an quivalnt cicuit with mods popagating in th z diction, as shown in Fig.. Ths mods a fd to as z-mods bcaus th popagat along th longitudinal z diction with popagation constant z. This quivalnt cicuit lads to th dispsion quation. (Y Y )E () Fig. Equivalnt cicuit usd to illustat th TRM W consid h that ths mods would b and. Th dispsion quations fo both polaizations and a thn wittn as: Intnational Scholal and Scintific Rsach & Innovation (5) schola.wast.og/37-689/799

2 Wold Acadm of Scinc, Engining and Tchnolog Intnational Jounal of Elctonics and Communication Engining Vol:, No:5, 7 Intnational Scinc Indx, Elctonics and Communication Engining Vol:, No:5, 7 wast.og/publication/799 coth( d) () th( d) Th and chaactistic admittancs in th ai and slab gion a givn b [7]: M M M M (3) (4) j j (5) (6) j j (7) wh and a th tansvs wavnumb. Th analsis pocds b fist assuming that th popagation constant x valu is zo (no vaiation along th x diction). A. Sufac Mod Th sufac mods a chaactizd b xponntiall dcaing filds in th -diction [8]. A -tp sufac mod popagating along th z axis is invstigatd. Th non-vanishing magntic fild componnts which satisf th Maxwll quations in two gions can b dducd as: j Hx C j H x C cosh( ) Th H x componnts fom th two gions a continuous on th sufac =. Fo sufac mods, th sufac impdanc must b inductiv ( al( ) ) in od to povid th fild concntation abov th sufac. Th Hlmholtz wav quation is wittn as: ( ) (9) Th al mods popagat in z diction with a al popagation constant z z. W aim to obtaining an appoximat solution b stting th imagina pats and to zo. Und this assumption, (8) th wav numb is al and, th wav numb is al. Lt j and j. In this cas, () and (8) duc to th( d) ( ) z () Th quation of sstm can b solvd fo th unnowns and using an itativ pocdu to aiv at a final st of solutions fom som initial tial valus. This pocdu is lasd b fsolv in MATLAB. Fo sufac mod, th lctic fild componnts in two gions can b wittn as: j Ex C j Ex C sinh( ) () Assuming th sam appoximation using in mod, w obtain th quation sstm coth( d) ( ) z () To find th dispsion of th sufac mod, w will solv this sstm b th sam pocdu happning in mod. B. Volum Mod Volum mods popagat along th longitudinal z diction with a al popagation constant z z. Th magntic fild in th polaization is psntd b ths quations in th two gions. j j Hx C jz z Hx C cosh( ) (3) Th diffnc btwn th two wav quations allows us to obtain this quation: ( ) (4) W impl that th wav numb of th slab Thfo, w can wit ths two quations: is al. Intnational Scholal and Scintific Rsach & Innovation (5) schola.wast.og/37-689/799

3 Wold Acadm of Scinc, Engining and Tchnolog Intnational Jounal of Elctonics and Communication Engining Vol:, No:5, 7 Intnational Scinc Indx, Elctonics and Communication Engining Vol:, No:5, 7 wast.og/publication/799 th( d) ( ) z (5) In od to find volum mods, w solv this sstm using fsolv in MATLAB. In polaization, th lctic fild is wittn in th two gions b th two quations: j j Ex C j Ex C sinh( ) (6) Volum mod is chaactizd b a al wavnumb and a al wavnumb. It is psntd b th solution of th sstm coth( d) ( ) (7) (7) z C. La Mod Th la mod is chaactizd b two complx wav numbs z,, and. Fo and polaizations, th sstm of quations to solv can b wittn as: th( d) ( ) z coth( d) ( ) z (8) (9) Th solution of th sstm can b ducd to th solution of th singl quation of th two polaizations Z a jz tan(z) () Z a jzcot(z) () wh Z d and a d ( ). Th is no analtical solution to ths quations, thn its solution cannot b don numicall. Th solution of ths quations a fom th complx vaiabl Z, using a Nwton- Raphson algoithm [9]. W chos this mthod bcaus it is simpl and ffctiv. In ou mthod, w chos to ta as initial points, th analtical solutions of () and () fo th paticula cas and. In this cas, ths quations will b wittn as: : jz tan(z) : jz cot(z) Th solution of th two quations is n is an odd intg. Z ( n/)pi wh III. NUMERICAL RESULTS AND DISCUSSION In th numical stud, ou stuctu is mad of a mtamatial mdium with pmabilit and pmittivit qual to: F p ( ) ( ) wh F.56, p GHz and 4 GHz (paamts chosn as in [7]). Ths matial paamts dpnd on fqunc. Fig. 3 Rlativ pmittivit ɛ (solid lin) and lativ pmabilit µ (dashd lin) as a function of fqunc f fo a doubl-ngativs goundd slab In Fig. 3, w illustat th pmabilit and pmittivit dpnding on th fqunc of f 4.8 GHz to f 6. GHz. Th pmabilit bcoms positiv abov f 6 GHz and it bcoms ngativ blow this valu. This fqunc is th citical fqunc. Gaphical psntations of th sufac mod fo and mods suppotd b a goundd slab with th abovpotd paamts and slab hight d 6 mm in a Intnational Scholal and Scintific Rsach & Innovation (5) 7 56 schola.wast.og/37-689/799

4 Wold Acadm of Scinc, Engining and Tchnolog Intnational Jounal of Elctonics and Communication Engining Vol:, No:5, 7 fqunc ang btwn f 4.8 GHz and f 6. GHz a shown in Fig. 4. In polaization, th mod is alwas al and is popl vanscnt abov 5. GHz (d Dash-dot lin) and impop odina blow (d solid lin). But, in polaization, th mod is not vanscnt abov 5. GHz, it is onl impop odina blow. W psnt in Fig. 5 th chaactistics of volum mods popagating along th doubl-ngativs goundd slab. sufac mod odina sufac mod vanscnt sufac mod. Intnational Scinc Indx, Elctonics and Communication Engining Vol:, No:5, 7 wast.og/publication/799 z / x 9 Fig. 4 Dispsion diagams fo and sufac mods of a goundd slab with hight d = 6 mm z / x 9 Fig. 5 Nomalizd al constant of volum mod popagating on a doubl-ngativs goundd slab vsus fqunc fo (solid lin) and (dashd lin) polaizations Th al constants of and volum mods a qual to th constant of f spac β z, abov 5. GHz and th a dcasd blow. At th citical fqunc f c =6 GHz, th nomalizd al constants of mods bcom incasd. But, in mods, th main dcasd. As an illustation, Fig. 6 psnts th nomalizd al constant and th nomalizd attnuation constant of th la mod popagating in doubl-ngativs goundd slab. Two mods fo and a visibl in Fig. 6. and hav two al banchs blow 5. GHz and 5 GHz which to complx banchs abov thos fquncis. and volum mod volum mod hav two al banchs blow 5. GHz and 5.GHz which to complx banchs abov thos fquncis. At fc 6 GHz (whn th mtamatial changs fom doublngativ to -ngativ), th nomalizd al constants of ths two mods fo and bcom zo and tain thi sign. Th nomalizd attnuation constants of mods dcas apidl at f c. Howv, th nomalizd attnuation constants of mods incas at this fqunc. Intnational Scholal and Scintific Rsach & Innovation (5) 7 56 schola.wast.og/37-689/799

5 Wold Acadm of Scinc, Engining and Tchnolog Intnational Jounal of Elctonics and Communication Engining Vol:, No:5, 7.5 (a) z /.5 Intnational Scinc Indx, Elctonics and Communication Engining Vol:, No:5, 7 wast.og/publication/799 - z / (b) x x 9 Fig. 6 Chaactistics of la mods popagating along a doubl-ngativs goundd slab as function fqunc. (a) Nomalizd al constants (b) nomalizd attnuation constant IV. CONCLUSION Th dispsion and adiation poptis of al and complx mods popagating along doubl-ngativ o singl-ngativ goundd slab hav bn invstigatd in this pap. Th gaphical mthod is usd to find th possibl al and complx oots. Evanscnt sufac mod has bn shown to occu onl in mods of DNG cas. W found that onl th impop la mods a obsvd in DNG, and onl th pop la mods a obsvd in SNG. REFERENCES [] P. Baccalli, B. Bughignoli, F. Fzza, A. Galli, P. Lampaillo, G. Lovat and S. Paulotto, Th natu of adiation fom la wavs on singl- and doubl-ngativ mtamatial goundd slabs, IEEE MIT-S Digst, 4. [] Baccalli, P., P. Bughignoli, F. Fzza, A. Galli, P. Lampaillo, G. Lovat, and S. Paulotto, Fundamntal modal poptis of sufac wavs on mtamatial goundd slabs, IEEE Tans. Micowav Tho Tch, Vol. 53, 43 44, Ap. 5. [3] I. J. Bahl and P. Bhatia, La wav antnnas using atificial diltic at millimt wav fquncis, IEEE Tans. Miowav Tho Tch. 8, 5 (98). [4] I. S. Nvdov and S. A. Ttaov, Thotical Stud of Wavguiding Stuctus Containing Bacwad-Wav Matials, [5] A. Alu and N. Enghta, Guidd Mods in a Wavguid Filld with a Pai of Singl-Ngativ (SNG), Doubl-Ngativ (DNG), and/o Doubl-Positiv (DPS) Las, IEEE Tans. Mico. Tho Tch, vol. 5, no., Janua 5. [6] W. Shu and J. M. Song, Wav Popagation in Goundd Dilctic Slabs with Doubl Ngativ Mtamatials, Pogss in Elctomagntics Rsach Smposium 6, Cambidg, USA, Mach 6-9. [7] P. Baccalli, B. Bughignoli, F. Fzza, A. Galli, P. Lampaillo, G. Lovat and S. Paulotto, Effcts of La-Wav Popagation in Mtamatial Goundd Slabs Excits b a Dipol Souc, IEEE Tans. Mico. Tho Tch, vol. 53, no., Janua 5. [8] R. T. Ling, J. D. Scholl, and P. Ya. Ufimtsv, Th Popagation and Excitation of Sufac Wavs in an Absobing La, Pogss In Elctomagntics Rsach, PIER 9, 49-9, 998. [9] O. Balosso, Intoduction ds Onds d sufac dans un Modèl Adapté d Faiscaux Gaussins pou l Taitmnt du Couplag Antnn- Stuctu, thès, univsité Toulous 3 Paul Sabati pp. 49-5, juin 4. Intnational Scholal and Scintific Rsach & Innovation (5) 7 56 schola.wast.og/37-689/799