Study of Electromagnetic Wave Propagation in Periodic Dielectric Structure; MathCAD Analysis

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1 Communictions in Applied Sciences ISSN -737 Volume Nume 3-9 Stud of lectomgnetic Wve Popgtion in Peiodic Dielectic Stuctue; MthCAD Anlsis Ugwu mmnuel.i Ieogu C. nd chi M.I Deptment of Industil phsics oni Stte Univesit Nigei Deptment of phsics Univesit of Agicultue Mudi Nigei Coesponding utho: Ugwu.I Deptment of Industil phsics oni Stte Univesit Nigei Astct. The popgtion of electomgnetic wves in peiodic dielectic stuctue ws nled using plne wve eqution of peiodic electic field in conjunction with the ppliction of the Schodinge's eqution s n opeto with Mthieu's diffeentil equt ion. The eltionship etween electic field () width of dielectic sls peiod tnsvese electic mode nd wve nume wee otined. Gphicll it ws shown tht diffeent vlues of the tio of the width of the dielectic sls to the peiod ffect the wve popgtion. Howeve the ehvious of electomgnetic wves popgtion though dielectic medium depend on the solid stte popeties of the medium. Also how electomgnetic wves popgte s well s its sic popeties ws pesented. Ke wods: lectomgnetic Wves Dielectic Constnt Peiodic Stuctue Dielectic Sls Peiod lectic field Wve nume. Popgtion I. INTRODUCTION In ecent time scientists hve een inteested in the w nd mnne wve ehve s it popgte though mteils of vious solid stte popeties nd s such hve vociousl woing to now the mteils tht offe est conditions to hness the optimum ppliction fo the popgting wve []. lectomgnetic wves popgtion though peiodic stuctue is one of the mn inteesting polems in mthemticl phsics nd hs een of inteest fo mteils fo optoelectonics sol cell etc. Copight 3 the uthos.

2 Communictions in Applied Sciences lectomgnetic wve incident on Cstl t noml o olique on the sufce of cstl l epeiences ttenution s the wve popgte though [3]. The etent of the ttenute suffeed the wve depends on the solid stte popeties of the cstl such s the dielectic function of the cstl which is elted to the efctive inde in which popeties of the solid stte cstl mteils stongl depend on[4]. The stud of the popgtion of wve in the peiodic medium hs comple discete composition which mes it difficult to set up the ound condition fo toms foming the stuctue of the cstl [5]. Though dielectic stuctue does not conduct ut on the ppliction of electic field the dielectic ties to ehve lie tht of non-dielectic ones s the positive nd negtive ions in dielectic stuctue fom polit in the pesence of electic field [6]. Also it hs een shown tht the efctive inde of dielectic cstl is el nd slightl less thn unit giving wves which popgte without ttenution with phse velocit slightl lesse thn the velocit of the light in vcuum [7]. Cstl of diffeent mteils compound e eing gown developed nd chcteied nd studied in ode to identif the ntue of the efctive inde nd dielectic functions. Vious popeties such s efctive inde fequenc limit nd dielectic constnts e pt of the popeties of dielectic cstl when wve e llowed to popgte though it. The nlsis of scl wve ehviou on the ough sufce ws studied [8] Also the popgtion of plne electomgnetic wve though flt sufce is etended to multiple sctteing theo development [9]. The esech on the ehviou of n incident wve ove ough sufce of cstl hs een studied Feineg who otined effective impednce of the intefce coupled with esie studies on sctteing nd multiple sctteing s hve een poposed [].

3 3 Communictions in Applied Sciences In this esech electomgnetic wve popgtion in peiodic dielectic medium ws nled using plne wve eqution of peiodic electic field with the ppliction of Schodinge's eqution s n opeto with Mthieu's diffeentil eqution. The ojective of the stud is to nle the effect of peiodic dielectic stuctue s electomgnetic wve popgtes though it. The esech wo eplins the nlsis of electomgnetic wves popgtion in one-dimensionl peiodic dielectic stuctue. II. Solution of plne wve though one-dimensionl peiodic medi. A numeicl solution to the ect eigenvlue eqution fo the one-dimensionl peiodic polem is otined with the help of MthCAD softwe. Conside one-dimensionl peiodic of dielectic sls with peiod nd dielectic inset width is Fig.; Dielectic Stuctue model with incident wve nomll on it. One-dimensionl lttice of dielectic sls of width in peiodic lttice with peiod.the electic field in peiodic stuctue shown is given ^ e jk jk o e p () whee X is the peiodic electic field tht popgtes onl in the plne i.e. = nd Kois pescied popgtion constnt without loss of genelit. Since the electic field must stisf the wve eqution we ppl the opeto; to in () ove oseving tht the dielectic

4 Communictions in Applied Sciences 4 constntis function of. We hve tht () Fo = dielectic constnt inset Assuming the pllel sls e infinite in the nd Z diections eqution () cn e simplified But (3) Sustituting (3) into () We hve tht (4) (5) But o jk jk p e e (6) i.e. j p e Z ^ (7) Sustituting (7) into (5) nd diffeentite twice with espect to We hve tht (8) (9) Multipling - to oth sides we hve tht () This eqution cn e witten s () O ()

5 5 Communictions in Applied Sciences The ove diffeentil eqution is simil to the cnonicl fom of Mthieu's diffeentil eqution given ; q cos () If we sustitute u = i to the ove eqution then we otin the Mthieu's modified diffeentil eqution q cosh u (3) Accoding to Bloch's theoem fo fied vlues of q Mthieu s eqution dmits comple vlued solution of the fom f ( q ) e p ( i )( q ) (4) Hee µ is comple nume clled the Mthieu eponent nd p is comple vlued function which is peiodic in In eqution ( ) ove we will ppoimte the dielectic peiodic function. cosh which is peiodic with peiod nd Thus we wish to solve the odin diffeentil eqution. We use hpeolic cosine ecuse of eqution (3). cosh (5) Fo the pupose of otining numeicl solution we vectoie the ove eqution s follows; (6) Hence eqution (5) ecomes

6 Communictions in Applied Sciences 6 cosh (7) Fo the numeicl solution we te nd MATCAD SOLUTION D t cosh Deivtive ecto Y...5 Y (8) Define dditionl guments fo the OD solve: t Initil vlue of independent vile t. Teminl vlue of independent vile.5 Y Vecto of initil function vlues N= Nume of solution vlues on t t Solution mti: M Rdpt Y t t N D t M Independent vile vlues M Fist solution function vlues D M Second solution function vlues III. RSULTS AND DISCUSSION When electomgnetic wve impinge on peiodic dielectic medium the eltionship etween lectic field () tnsvese electic mode wve nume width of the dielectic sls nd peiod ws otined fom this eqution.

7 7 Communictions in Applied Sciences cosh On the eltionship some vlues wee otined fo electic field width of the dielectic sls nd peiod fo the wve popgtion fo fo numeicl solution so s to now how electomgnetic wves popgting though peiodic medium with diffeent vlues of the tio of the width of the dielectic sls to the peiod vies with the wve popgtion long diection. Fom Fig. nd fig. it ws oseved tht when the vlues of.3545 nd.5 espectivel fo the wve popgting though the peiodic dielectic medium the oscillting fequenc is low while in cse of the vlue As shown in the Fig.4 the fequenc of the wve ppes highe thn in fist two cses. Fig. : Wve pofile when =.8Fig. 3; Wve pofile when =.3545

8 Communictions in Applied Sciences 8 Fig. 4: Wve pofile when =. IV.CONCLUSION In genel the nlsis of the popgtion of electomgnetic wves in peiodic dielectic stuctue using plne wve eqution of peiodic electic field nd Mthieu's diffeentil eqution hd een cied out numeicll using MthCAD fom whee eltionship etween electic field dielectic constnt wve nume width of dielectic sls nd the peiod of the wve popgtion ws estlished fom the solution. Fom the gphs it ws seen tht the ehviou of electomgnetic wve popgtion though dielectic stuctue depends on the sie of the sl in conjunction with the solid stte popeties of the stuctue such s the dielectic constnt s enshined in the solution lso which s mnifested in the pofile of the popgted wve shown in the gphs fo diffeent vlues of the tio of the width of the dielectic sls Refeences [].I Ugwu nd G.A Ago 7 Wve Popgtion in non-homogeneous thin films of slowl ving efctive inde ; W.K.B solution model PJST Vol.8 No.. pp 46-5 [].I Ugwu nd C. Oee () Itetive Computtionl Scheme of Studing lectomgnetic Wve Popgtion Though Dielectic Thin Film Int Jnl f MultPh.Vol.5 No.3

9 9 Communictions in Applied Sciences pp [3] J.C.W.ACost nd A.J Giol [99] "Popgtion chcteistics in peiodic dielectic stuctues" (in Potuguese) in poceedings of the V. Bsili micowe smposium Bsili DF Bil Jul -4 pp.87-9 [4].I Ugwu A.S Olin nd F.I Olpode9 Anlsis of Wve Popgtion in Homogenous Dielectic Cstl JAS 4() pp6-3. [5] G.S Mtilim[998] Specil issues fo how ging cs sctteing fom ough sufce ntenns popgtion P.S-I998. [6] V.Luichen[ ]Phsics nd Technolog of millimete wve devices nd component (New Yo). [7] A. Lincolnton [] Wve popgtion in cstls Lede Pess New Yo. [8]S.A Ito nd T.D [985] Roc w wve popgtion fo n infinitel conducive one -dimensionl ough sufce I Tns Antenns popgtion. P [9] Bss nd Fus [979] Specil stud of sctteed wve (New Yo). [] J.B. Kelle nd Wtson [983] Reflection sctteing nd Asoption of Acoustic wves ough sufce. Acoustics AM 74 p