Spherical Space Bessel-Legendre-Fourier Mode Solver for Electromagnetic Fields. Mohammed A. Alzahrani, B.Sc., M.A.Sc.

Transcription

1 Sheical Sace Bessel-Legede-Fouie Mode Solve fo Electomagetic Fields. by Mohammed A. Alzahai, B.Sc., M.A.Sc. A Thesis submitted to the Faculty of Gaduate Studies ad eseach i atial fulfilmet of the equiemets fo the degee of Docto of Philosohy i Ottawa-Caleto Istitute fo Electical Egieeig Caleto Uivesity Ottawa, Otaio, Caada August, , Mohammed A. Alzahai

2 Abstact Cotollig ad cofiig light i thee dimesios is of sigificat iteest as it eesets devices i thei etiety with miimal aoximatios. Localized modes i esoato stuctues ae commoly modelled usig the suecell lae wave exasio method, fiite elemet method, o the fiite diffeece time domai. Howeve, these stadad techiques ae exesive i tems of comutatioal esouces whe alied to thee-dimesioal stuctues. The techique eseted i this thesis is a ew method fo solvig Maxwell s vecto wave equatios fo localized modes i thee-dimesioal sheically esoato stuctues as well as its alicatio to seso cofiguatios. The techique equies miimal imlemetatio, ovides omalized esults, woks fo fiite size ad is comutatioally efficiet. The method is such that the stuctues ude test ca be abitay shae, isotoic, aisotoic, lossless, o lossy. Fo the stuctues examied, a modified basis set comosed of sheical Bessel, Legede ad Fouie fuctios (BLF) ae used to exad the electic, magetic, ad ivese elative emittivity. These exasios allow fo Maxwell s wave equatios to be cast as a eigevalue oblem fom which the steady state localized modes (ot oagatig) ca be detemied fom the eige-fequecies ad eigevectos. This wok alies to a umbe of sheically symmetic stuctues. Selectios of stuctues whose esoato oeties ae eoted i the liteatue ae used to comae ad veify the accuacy of the use of the BLF fuctios as a exasio basis. ii

3 Ackowledgemet I would like to begi by thakig Pof. Gauthie fo the geat suot he has ovided thoughout my ost-gaduate studies. I aeciate eveythig he has doe fo me duig my stay hee as iteatioal studet, I feel that I have foud a good fied as well as a sueviso. I would like to thak the Miisty of Highe Educatio i Saudi Aabia ad Taibah Uivesity fo the fiacial suot. To my family ad my wife (Ema), you suot has bee limitless. Thee ae o wods to sufficietly thak you. iii

4 Table of Cotets Abstact... ii 1 Chate: Itoductio Motivatio Cotibutios Thesis ogaizatio Chate: Liteatue eview of calculatio methods Basic methods fo esoace calculatios Advaced umeical techiques Fiite diffeece time domai (FDTD) Fiite elemet method (FEM) Plae wave exasio method (PWEM) ecet techiques Chate: Theoetical develomet of the BLF method Theoy Dielectic ofile ad EM field eesetatio i BLF Buildig the eigevalue matix Solve stes Defiig the dielectic Poulatig the matix Solvig the matix Localized mode ofiles categoies Otimizig the techique Segmetatio of the dielectic ofile Offset-esidue mateial segmetatio Chate: Comutatio examles Sheical cavity High elative emittivity shees Whiseig Galley Modes Sheical Bagg stuctues Aeiodic eflecto esoatos iv

5 4.5 Dielectic disc Chate: Advaced fomulatio ad basis fuctios Bouday coditios Comutatio Examles Aisotoic Mateial Dielectic sheical shell of 6-fold vaiatio i Chate: Sesig alicatio Sheical dielectic shell esults Slot WGMs chael desig Chate: Coclusio Summay Futue wok efeeces Aedix A: Wave Equatio fo H field Aedix B: Deivatives of the Basis Fuctios Aedix C: Develoig the Matix Exessio Aedix D: Matix Exessio Aedix E: Mode Solve MATLAB Pogams v

6 Table of Figues Figue 2.1: Visualizig the comutig method of the udate equatio i time ad sace fo oe dimesio Figue 2.2: Oe dimesioal hotoic stuctue. Black eeset low elative emittivity,, ad white eeset high elative emittivity, Figue 2.3: Bad ga, idicated i gey, i a eidoic mateial with high-dielectic-costat mateial ( h 13) of width 0.2a ad low-dielectic-costat mateial ( h 1) of width 0.8a [45] Figue 3.1: Zeo-value bouday coditio located fa away fom the objective dielectic stuctue Figue 3.2: Segmetatio ocess of ivese of the elative emittivity of the stuctue (to-ight) is show Figue 3.3: Segmetatio of esidue ofile (To Left) ito 4 seaated ofiles, a ofile fo each costituet mateial eset i the esoato stuctue Figue 3.4: efeece, lage etubatio ad small etubatio ofiles Figue 4.1: (Left) Coss sectio of the sheical cavity stuctue. (ight) ecostucted ivese elative emittivity ofile usig 100 Bessel tems. Small deviatios i dielectic value ae uesolved by the esoato states of iteest Figue 4.2: The modal ofile fo E field comoet omalized to a maximum of 1, 110 mode ode, fo k Left (x, y) = (, 90, ) lae. ight (y, z) = (,, 90) lae. The black to white colo eeset the omalized field values fom oe to zeo Figue 4.3: The modal ofile fo E field comoet omalized to a maximum of 1, 230 mode ode, fo ka Left (x, y) = (, 90, ) lae. ight (y, z) = (,, 90) lae. The black to white colo eeset the omalized field values fom oe to zeo Figue 4.4: The covegece fo mode 130. The solid hoizotal lie labels the aalytical solutio [38, 39], the dots eeset the umeical esults obtai usig BLF Figue 4.5:Left Geomety of the solid-uifom, 9, dielectic shee, adius a. The sheical comutatio domai is of adius =2a. ight eal at of the dielectic seies exasio coefficiets lotted vesus sheical Bessel basis fuctio idex. Imagiay ats ae all zeo fo this dielectic ofile Figue 4.6: The (x, y) = (, 90, ) lae modal ofile fo E field comoet (Left), ad the (y, z) = (,, 90) lae modal ofile fo E field comoet (ight). Field ofiles with fo the 110 localized state ae listed i table 2, comuted usig BLF, ka Figue 4.7:The (x, y) = (, 90, ) lae modal ofile fo E field comoet (Left), ad the (y, z) = (,, 90) lae modal ofile fo E field comoet (ight) fo the 150 localized state listed i table 2, comuted usig BLF, ka vi

7 Figue 4.8: The (x, y) = (, 90, ) lae modal ofile fo vii E field comoet (Left), ad the (y, z) = (,, 90) lae modal ofile fo E field comoet (ight), fo 210 localized state listed i table 3, comuted usig BLF, ka The black to white colo eeset the omalized field values fom oe to zeo Figue 4.9: The (x, y) = (, 90, ) lae modal ofile fo E field comoet (Left), ad the (y, z) = (,, 90) laemodal ofile fo E field comoet (ight), fo 250 localized state listed i table 3, comuted usig BLF, ka The black to white colo eeset the omalized field values fom oe to zeo Figue 4.10:The (y, z) = (,, 90) lae modal ofile fo E field comoet fo 210 localized state listed i table 3, comuted usig BLF, ka (Left). The (y, z) = (,, 90) lae modal ofile fo E field comoet fo 250 localized state listed i table 3, comuted usig BLF, ka (ight). The (x, y) = (, 90, ) lae have zeo E field values. The black to white colo eeset the omalized field values fom oe to zeo Figue 4.11: Samle eige-matix showig locatios of zeo ad o-zeo elemets. The white elemets ae the zeo elemets. Each colo eesets diffeet azimuthal ode; the ed, yellow, ad oage cells ae the o-zeo elemets fo the fist, secod, ad thid azimuthal mode odes Figue 4.12: Modal ofile fo the E field comoet of the BLF comuted WGMs of the solid shee i ai. To ai Azimuthal mode ode 20 with 1120 adka Bottom ai Azimuthal mode ode 40 with 1140 ad ka Left (x, y) = (, 90, ) lae. ight (y, z) = (,, 90) lae. The black to white colo eeset the omalized field values fom oe to zeo Figue 4.13: adial ofile of shee coated by dielectic shells. The shells acts as a quate wave Bagg eflecto. The dielectic has eal values oly Figue 4.14: adial dielectic exasio coefficiets fo the sheical Bagg stuctue Figue 4.15: The modal ofile fo H field comoet fo 110, comuted usig BLF. The field ofile ad coesodig eigevalue match those of efeece [20]. Left (x, y) = (, 90, ) lae. ight (y, z) = (,, 90) lae Figue 4.16:The modal ofile fo E field comoet fo 1130, comuted usig BLF, f THz ( ka ).Left (x, y) = (, 90, ) lae. ight (y, z) = (,, 90) lae. 69 Figue 4.17:The modal ofile fo E field comoet fo 1330, comuted usig BLF, f THz ( ka ).Left (x, y) = (, 90, ) lae. ight (y, z) = (,, 90) lae.. 69 Figue 4.18: elative dielectic costat ofile alog the adial diectio fo a shee of ai coated by aeiodic sheical shells Figue 4.19: eal adial ivese elative emittivity exasio coefficiets fo the aeiodic shell stuctue Figue 4.20: Imagiay adial dielectic exasio coefficiets fo the aeiodic shell stuctue.72

8 Figue 4.21: The modal ofile fo E field comoet fo 110, comuted usig BLF. The field ofile ad coesodig eigevalue match those of efeece [53]. Left (x, y) = (, 90, ) lae. ight (y, z) = (, 90, ) lae Figue 4.22: Dielectic disc is show, whee a is the adius of the disc. The stuctue is uifom i the azimuthal diectio Figue 4.23: The exasio coefficiets of the dielectic disc Figue 4.24: Scaled esoace wavelegths ( ) detemied usig BLF ad usig the ay otics a fo the E field Figue 4.25: The waveumbes of modes with the 20 th ode azimuthal mode ode ae show. The field ofiles ae lotted i the (x, y) = (, 90, ) lae (left) ad i the (y, z) = (, 90, ) lae (ight). The dielectic shee adius a is half the solve domai shee. The two highest mode ofiles ae affected by the domai bouday Figue 4.26: The exasio coefficiets fo the zeo idex of the azimuthal diectio q 0, ad 50 idices fo the ola ad the adial diectios Figue 4.27: Exasio coefficiets fo the dielectic disc with azimuthal vaiatio (To) The exasio coefficiets fo the fist idex of the azimuthal diectio q 1ad 1 ad 50 idices fo the ola ad the adial diectios. (Middle) The exasio coefficiets fo the secod idex of the azimuthal diectioq 2 ad 2 ad 50 idices fo the ola ad the adial diectios. (Bottom) The exasio coefficiets fo the thid idex of the azimuthal diectio q 3 ad 3 ad 50 idices fo the ola ad the adial diectios Figue 4.28: The 10 th mode ode i the azimuthal diectio k=4.81 i the disc of azimuthal vaiatio dielectic (left). The 20 th mode ode i the azimuthal diectio k=8.84 i the disc of azimuthal vaiatio dielectic (left) Figue 4.29: The waveumbes of 10 th ode azimuthal mode ae show. The field ofiles ae lotted i the (x, y) = (, 90, ) lae (left) ad i the (y, z) = (, 90, ) lae (ight) Figue 5.1: The mode ofiles ad waveumbes fo a umbe of diffeet localized modes i the ola diectios. Modes (a ad b) have a omalized waveumbe of ka=2.08 ad (c ad d) have omalized waveumbe of ka=2.96. Both ae lotted usig the E field. Modes (e) ka= 3.64 ad (g) ka=4.13 ae lotted usig E field vecto, (f) ka=3.64 ad (h) ka= 4.14 ae lotted usig the E field vecto Figue 5.2: The mode ofiles ad waveumbes fo a umbe of diffeet localized modes i the azimuthal diectio. The field ofiles ae fo the E field vecto Figue 5.3: A coss sectio of a dielectic sheical shell is show, whee a is the adius of the sheical shell. The stuctue is 6-fold symmetic i the azimuthal diectio Figue 5.4: elative emittivity vaiatio i the azimuthal diectio of the sheical shell Figue 5.5: The waveumbes of 3 ed ode azimuthal mode ae show. The field ofiles ae lotted i both the (y, z) lae (left) ad i the (x, y) lae (ight) viii

9 Figue 6.1: esidual dielectic fuctio (es, left) ad offset dielectic fuctio (off, ight) of dielectic sheical shell Figue 6.2: Sesitivity cuves ad mode ofiles fo fist set of modes. The modes ode ae (To): 140, 180, left to ight, (Bottom): 1180, 1300, left to ight Figue 6.3: Sesitivity cuves ad mode ofiles fo secod set of modes. The modes ode ae (To): 1110, 1120, left to ight, (Bottom): 1140, 1160, left to ight Figue 6.4: efeece ad etubatio ofiles of the ivese elative emittivity. The to figue shows the oigial stuctue. The two figues i the bottom show the segmetatios of the oigial stuctue ito efeece (left) ad lage etubatio (ight) Figue 6.5: Segmetatio ito 3 seaated ofiles, a ofile fo each mateial costituet eset i the esoato stuctue Figue 6.6: Sesitivity cuves of ambiet medium ad mode ofiles of E fo selected modes. The mode ofile is show i the (, ) lae. The adius legth of the dielectic sheical shell is omalized Figue 6.7: efeece ad etubatio ivese elative emittivity ofiles. The to figue shows the oigial stuctue. The two figues i the bottom show the segmetatios of the oigial stuctue ito efeece (left) ad small etubatio (ight) Figue 6.8: Side view of the oosed WGMs slot waveguide stuctue. The shee has high elative emittivity value with squaed slot chael filled with ai Figue 6.9: The (x, y) lae modal E field comoet (Left), ad the (y, z) lae modal E field comoet (ight) fo the 110 localized state i the slot chael egio. The filed is high i the slot chael egio. The ba o the left shows the omalized values fo the esoace field ix

10 1 Chate: Itoductio A otical esoato is a oweful otical comoet i hotoic device desig. Geometies ovidig this fuctioality ae vital i may alicatios. The otical esoato is oe of the most commoly used fo the geeatio o cotol of electomagetic adiatio. The esoato is a key comoet i lases, waveguides, sesos, ad filtes devices [1-3]. They ae hotoic stuctues that allow a otio of the field to ciculate i a closed ath withi the device such as a cavity. The esoat fequecies, fequecies that coesod to costuctive itefeece, deed o featues such as the size, geomety of the cavity, o the mateials of the device. Otical esoatos ca be desiged usig two lae mios i ai [4], maco- ad mico-otic ig [3], disk [5], cylidical stuctues [6,7], hotoic cystal cavities [8-10] ad sheical stuctues [11]. Sheical esoatos have gaied iteest as they ovide a eviomet fo cotollig ad cofiig light i thee dimesios ad fom the backdo fo a umbe of moe advaced esoato alicatios [9, 12-17]. The sheical esoatos will be the focus of this wok. Cuet comutatio techiques fo calculatig the esoato oeties i the sheical stuctues ae highly deedet o the hysical size of the device featues elative to the wavelegth of the electomagetic fields ivolved. Fo elatively lage lossless objects a geometical otics aoach ca lead to imotat isights o the modal oeties usig staight fowad equatios [15]. Whe the shee s size, o iteal stuctues, is comaable to the wavelegth the full vecto wave aalysis is equied fo accuate esults. The full vecto wave aoach is eeded to model the diffactio effect which is vital to otical chaacteistics of these devices. I the secific case of a dielectic shee, exact solutios to Maxwell s wave equatio esults i a chaacteistic equatio whose oots ovide access to the esoato wavelegths ad field ofiles [18, 19]. Fo sheical stuctues, cosistig of sigle o multile layes, the 10

11 scatteig matix method has bee used to obtai the esoato oeties [20]. Whe the sheical esoato icludes mateial vaiatios with esect to the agula coodiate, umeical ad aoximate methods ae fequetly equied i detemiig the otical oeties of the esoato. Thee ae 3 domiat methods ecouteed thoughout the liteatue; the lae-wave-method (PWEM) [21, 22], the fiite-diffeece-time-domai method (FDTD) [23, 24] ad the fiiteelemet-method (FEM) [25]. These techiques have advatages ad disadvatages, ad the choice of which is efeed deeds o the alicatios. ecetly, a efficiet umeical mode solve fo cylidically symmetic esoato cofiguatios was develoed by castig Maxwell s wave equatio ito a eigevalue oblem usig the Fouie Bessel basis to exad the fields ad ivese emittivity [26, 27]. That techique is ecofigued hee such that it is alicable to sheical objects of abitay iteal stuctue. Whe solved the eigevalue oblem yields esoato state fequecies o wavelegths (eigevalues) ad field ofiles (ecostucted usig the eigevectos) [28]. The basis selected fo this wok is the Bessel-Legede-Fouie (BLF) basis set whee sheical Bessel fuctios ae used i the adial diectio (), Legede fuctios ae used fo the ola agle ( ) ad the comlex exoetial fuctios (Fouie) ae used fo the azimuthal agle ( ). The BLF techique has bee veified, as the esults of comutatio matches those obtaied usig othe umeical techiques. 1.1 Motivatio Cuet umeical techiques equie lage comute memoy ad log comutatio time whe used to detemie localized modes of 3-D stuctues. The BLF solve is a techique desiged to ovide ifomatio o the localized modes (fequecy, wavelegth, mode ofile) fo stuctues descibable i sheical coodiate. This aoach makes o aoximatios egadig the 11

12 uifomity of the dielectic, iteal featues o loss. The BLF techique solves the full vecto electomagetic wave equatio. I additio, BLF techique ca be otimised to taget a secific mode ode ad ehace the techique futhe. I additio to its alicatios to sheical esoatos, such as lases, sesos, ad filtes devices, a umbe of atually occuig objects ad biological secimes ca be aoximated usig sheically symmetic shaes; liquid dolet [29, 30], ai bubble [31], viuses, biological cells ad bacteia [32]. The otetial exists to exted the techique to moe comlicated stuctues ad i othe aeas of sciece such as the Schödige ad acoustic wave equatios. 1.2 Cotibutios The sigificat cotibutio eseted hee is the use of modified sheical basis fuctios to develo a umeical techique (BLF) to study localized electomagetic waves i thee dimesios systems. The modified sheical basis fuctios geatly simlify the comute imlemetatio. The BLF techique ad its alicatios has bee ublished by the autho (as icile autho) i a ee eview joual ae (chate 3 ad 4) [28] ad thee cofeece cotibutios, 2 oal esetatios (mai theoy of the techique ad segmetatios) [33, 34] ad a oste esetatio (slot chael sesos) [35]. At the time of witig this thesis two aes ae i eaatio, oe about the ew fomulatio i chate 5 ad the othe oe about alicatio of the segmetatios aoach. All calculatios wee efomed usig i-house ogams witte i Matlab by the autho. I additio, wok fo this thesis cotibuted, but ot icluded i the thesis, to two ublished joual aes (as co- autho), which about the cylidical solve FFB ad slot chael desig usig the FFB techique [27, 36] ad oe cofeece esetatio [37]. As a coautho the cotibutios wee to eview the equatios used ad citatio eeded. 12

13 1.3 Thesis ogaizatio Chate 2 is a liteatue eview of commoly used umeical methods fo obtaiig the steady state esoato modes fo the sheically symmetic stuctues selected to be ude study. Chate 3 ovides the theoetical develomet of the BLF mode solve. Chate 4 ovides the umeical esults obtaied usig the BLF techique ad cotasts these esults elative to othe techiques. I chate 5, a simle theoetical aoach to aly the umeical techique ad to model aisotoic stuctue is ovided. I chate 6, the modified umeical techique is alied to sesig cofiguatios. The thesis the cocludes with fial discussios ad suggestios fo futue wok. 13

14 2 Chate: Liteatue eview of calculatio methods While aalytical solutios i closed fom ae aoiate fo a vey esticted umbe of cases the ability to solve Maxwell s equatios is citical to the study of electomagetic waves i comlex stuctues. The advaces i comutig owe have chaged ou ability to solve Maxwell s equatios i a ofoud way. This chate will exloe the umeical ad comutatioal techiques that have allowed eseaches i the field of electomagetics to aalyze ad solve may iteestig oblems icludig heomea such as scatteig, guidig, localizatio, ad oagatio. These iclude the Plae Wave Exasio Method (PWEM), the Fiite Diffeece Time Domai techique (FDTD) ad Fiite Elemet Method (FEM) [21-25]. FDTD ad FEM ae geeal methods that diectly discetize ad solve Maxwell s equatios ad ca be alied to a wide age of dielectic stuctues. Howeve, they equie sigificat amouts of comutatioal esouces i tems of memoy ad time. PWEM is imaily used to calculate the bad stuctue of oagatig wave i hotoic cystals (PhCs). It has symmety estictios ad is imaily limited to stuctues that dislay a taslatioally symmetic lattice. Fo esoace calculatios of PhCs cotaiig defects, a suecell PWEM method that uses lage uit cell is eeded. The suecell PWEM equies iceased comutatioal esouces. 2.1 Basic methods fo esoace calculatios Maxwell s equatios i diffeetial fom ae: B (, t) 0 (2.1) D (, t) (2.2) B(, t) E(, t) t (2.3) 14

15 D(, t) H (, t) J (2.4) t Whee is the emittivity ad is the emeability of the medium ad the elatios ad D E B H ae used. Maxwell s equatios fo liea, lossless, souce fee ( 0, J 0 ) ad omagetic ) mateials ae as follows: ( o H (, t) 0 (2.5) ( ) E (, t) 0 (2.6) H (, t) E(, t) o 0 (2.7) t E(, t) H (, t) ( ) 0 (2.8) t Maxwell s equatios ae commoly solved i Catesia, cylidical, o sheical coodiate systems. The symmety oeties of the stuctue ude test usually dictate the choice of the coodiate system to utilize. The calculatios of the esoace oeties i thee dimesioal systems ae of cosideable iteest, as they ae most suitable whe studyig eal wold esoatos stuctues such as sheical stuctues [8, 9]. Fo a sheical cavity i sheical coodiates the electomagetic wave ca be classified as sheical TE ad TM olaized modes [38, 39]. Fo the sheical TE modes, the field comoets exessios ae: 1 1 F E 0, 1 1 F E, E, (2.9) si 15

16 H k, 2 j 1 1 F H, j F H F, j si (2.10) whee is the agula fequecy, k is the waveumbe, ad F is the solutio that satisfies the scala Helmholtz equatio i sheical coodiates. The sheical TM the field comoets exessios ae: 1 1 F H 0, 1 1 F H, H, (2.11) si E k, 2 j 1 1 F E, E j F F, j si (2.12) The fuctio F has the geeal fom: F ) q iq j ( k ) P (cos e. (2.13) Whee j ( k ) is the sheical Bessel fuctio, P q (cos ) is the Legede fuctios ad iq e is the Fouie fuctio. These exessios ca be used to solve fo esoace mode fequecies ad ofiles fo a homogeous shee bouded by alyig zeo bouday coditio fo the tagetial electic comoets at the shee suface (efect electic coducto o PEC). Fo the TE olaizatio, the E field s comoet is zeo, while the E ad E comoets ae: 1 1 E ) si q iq iq j( k ) P (cos e (2.14) E 1 1 q P (cos ) j( k ) e iq (2.15) 16

17 Fo mooole modes ( q 0, E 0, E 0 ) the TE equatios ca be educed to (2.15) ad a chaacteistic equatio ca be obtaied that ca be solved fo the esoat TE mode olaizatio. Alig the zeo-bouday coditio at the shee adius, 0, esults i a chaacteistic E equatio that ca be solved fo esoace scaled waveumbes of the localized states. The chaacteistic equatio is witte as: j ( k ) 0. (2.16) The esults fo the solutio of (2.16) ae show i table 1.1. Fo each value (Bessel ode) thee will be a umbe of zeos (waveumbe scaled by adius) solutios. The subscit N o ad ae used to idicate the umbe of eaks i the adial ad ola diectios. Table 1.1: Scaled esoace waveumbe i a cavity bouded by a PEC fo the TE olaizatio, obtaied fom efeeces [38, 39]. adial Ode Pola Ode A simila ocedue is alied to the sheical TM olaizatio ovidig the esults show i table 1.2. This solutio method is valid oly fo ideal dielectic shee bouded by the PEC. The solutios fo the esoat modes fo a dielectic shee suouded by ai o immesed i a dielectic mateial will equie alyig bouday coditios fo the tagetial comoets of E o H ad omal comoets of D ad B at the shee suface [18, 19]. 17

18 Table 1.2: Scaled esoace waveumbe i a cavity bouded by PEC obtaied fom efeeces [38, 39], fo the TM olaizatio. adial Ode Pola Ode Fo esoace modes, the field is exected to be oscillatig iside the shee ad decayig outside the dielectic shee. Theefoe, sheical Bessel fuctios of the fist kid, j, ae used to eeset the field iside the dielectic shee ad sheical Bessel fuctios of the thid kid, h, ae used to eeset the field outside the shee: q iq A j ( kd ) P (cos ) e a (,, ), (2.17) q iq B h ( ko ) P (cos ) e a k k. (2.18) d o Alig the cotiuity coditio of (,, ) (,, ) at the bouday a fo the tagetial comoets gives the chaacteistic equatio fo sheical TE mode as: j 1( ka) 1 H 1( kxa) (2.19) j ( ka) H k a x A simila ocedue is used fo calculatig the sheical TM esoace modes [18, 19]. The esoace mode waveumbe is show i tables 1.3 ad

19 Table 1.3: Scaled ka esoace waveumbe i dielectic shee obtaied fom efeeces [18, 19], fo the TE olaizatio. adial Ode Pola Ode Table 1.3: Scaled ka esoace waveumbe i dielectic shee obtaied fom efeeces [18, 19], fo the TM olaizatio. adial Ode Pola Ode Note that this method is oly alicable fo obtaiig esoace states i a uifom dielectic shee. Fo comlex stuctues of abitay cofiguatios such as thee dimesios PhCs with defects to localize light i the stuctues, ehaced umeical solves ae equied. 2.2 Advaced umeical techiques Numeical methods ca be classified as eithe time o fequecy domai methods based o the domai i which they ae used to solve Maxwell s equatios. Oe of the most olific time domai methods is FDTD [24]. It is based o the discetizatio of the oblem i sace ad time makig the techique suitably alicable fo may diffeet stuctues as it makes o assumtios egad the simulatio sace. FDTD icludes the electic ad magetic fields acoss the simulatio sace at vaious time stes makig the calculatios comutatioally exesive. Fequecy domai 19

20 methods ae alied usig the time ideedet Maxwell s equatios ad ca be moe efficiet. FEM ad PWEM ae two well-kow examles of this class of method [21, 25]. FEM equies meshig the oblem i sace ad the alyig aoximate solutio fo each elemet to build a matix ad the solve fo the aoiate bouday coditios. FEM does equie lage comute memoy fo 3D aalysis [40]. PWEM is based o the exasio of the fields ad the stuctue usig basis fuctios ove a uit cell that is assumed to eeat taslatioally though sace. PWEM is maily used to study oagated fields i eiodic stuctues. Theefoe, the PWEM time ad memoy equiemets ae iceased whe used fo studyig localized modes i stuctues of fiite size o disodeed cofiguatio because this equies the use of a suecell [22]. The comutatioal time ad memoy coces of the umeical techiques ae iceased whe alied to 3-D oblems. Whe selectig the umeical techique to use it is citical to coside the stuctue beig ivestigated, the oeties to be detemied, ad the comutig esouces available Fiite diffeece time domai (FDTD) Imlemetatio of the FDTD techique equies discetizatio i sace ad time of Maxwell s cul equatio ad the ceatio of udate equatios that iclude the electic ad magetic fields couled i time ad sace. The udate equatios exess the ukow futue fields i tem of kow ast ad eset fields. Theoetical develomet of the FDTD equatios fo a simle oe-dimesio oblem will be examied. Faaday s ad Amee s law fo a system that has vaiatio oly i the x diectio ad electic field olaized i the z diectio is witte as: H y ( x, t) Ez ( x, t), (2.20) t x 20

21 21 x t x H t t x E y z ), ( ), (. (2.21) The deivatives eed to be exessed as discetized quatities i ode to ceate the udate equatios. The deivative oeatio with esect to time ad sace fo the magetic ad electic field i equatio (2.20) ad (2.21) will be elaced by 1 st ode fiite diffeece exessios i time ad sace. z E ad y H ae as follows: ), ( ), ( t q x m E t x E z z, (2.22) ), ( ), ( t q x m H t x H y y (2.23) whee m ad q ae the ste idices i sace ad time, esectively. x is the satial ste ad t is the temoally ste. The equatios (2.20) ad (2.21) will have the followig fom: x m E m E t m H m H q z q z q y q y (2.24) x m H m H t m E m E q y q y q z q z (2.25) The magetic field ad the electic field ae offset i time ad sace by a half ste i ode to suot a lea fog calculatio alog x. The equatios (2.24) ad (2.25) ae eaaged as follows: m E m E x t m H m H q z q z q y q y (2.26) m H m H x t m E m E q y q y q z q z (2.27)

22 To visualize the udate equatios, figue 2.1 shows the satial offset betwee electic ad magetic fields, the time offset is idicated by the q idices. Figue 2.1: Visualizig the comutig method of the udate equatio i time ad sace fo oe dimesio. Due to the simlicity of usig the techique ad alicability to may stuctues, FDTD is widely used. Howeve, it may equie a sigificat amout of memoy ad log comutatioal time fo some stuctues. The FDTD is efeed fo electomagetic studies i adom o comlex media. The techique has also bee used i [41] to study the elatio betwee adom lasig modes ad localized modes, ad the study of localized modes i hotoic cystals stuctues of egative emittivity elemets [42] Fiite elemet method (FEM) FEM is oe of the most oweful ad flexible umeical techique fo hadlig oblems ivolvig comlex geometies ad ihomogeeous media. The esidual of the equatios is weighted with a abitay fuctio, the the weighted esidual equatio is witte as: L 0 w( x) ( x) dx 0 (2.28) whee w (x) is the weight fuctio ad (x) is the esidual equatios. Equatio 2.28 will be solved to comute the exact solutio ove the iteval fom 0 to L. The fiite elemet aalysis techique begis with the discetizatio of the oblem egio ito a fiite umbe of elemets. Exasio 22

23 fuctios fo a tyical elemet i the solutio egio ae the defied. The exasio fuctios ae the substituted i the weighted esidual equatio ad itegated to fom the weak fom equatio that will be used to oulate the matix elemets. The bouday coditio tems of the weak fom will be secified accodig to the oblem ude study. The fial stes ae the assemblig of all elemets i the solutio egio, ad the solvig the system of equatios that is obtaied. FEM has the caability to be alied to abitay stuctues but it becomes time cosumig ad sesitive to the bouday coditios. FEM ca be used to study the localized modes of esoatos as demostated i [43] usig commecially available softwae. Howeve, it is limited as the azimuthal diectio was esticted to beig uifom to ovide comutatioal seed Plae wave exasio method (PWEM) Plae wave exasio method (PWEM) is a fequecy domai method that solves the time ideedet Maxwell equatios, ad ca be used to solve fo the E-field o the H-field seaately. PWEM is imaily used to calculate the bad stuctue of taslatioally symmetic hotoic cystals (PhCs) [21, 22]. This method is based o the assumtio that the stuctues ude test ae ifiite. PWEM deeds o the use of Fouie seies to eeset the medium ad the Bloch wave theoem to eeset the electomagetic field oagatig though it. These exasios ae iseted i the electomagetic wave equatio to fom a eigevalue matix equatio. The eigevalue matix is the solved to obtai the fequecies (eigevalues) ad ofiles (ecostucted usig the eigevectos) of oagated states. The oe-dimesio hotoic cystal, show i figue 2.2, is the easiest stuctue fo studyig ad udestadig the mai oeties of PWEM. May of the oeties emloyed i oe-dimesioal aalysis ca be alied i two o thee dimesios. 23

24 As such a study of figue 2.2 with would be beeficial. Additioally, PWEM has may similaities to BLF i egads to geeal cocet ad imlemetatio. Figue 2.2: Oe dimesioal hotoic stuctue. Black eeset low elative emittivity,, ad white eeset high elative emittivity,. Fo a oe dimesioal eiodic stuctue i figue2.2 whee the electic field is aallel to the z-diectio ad the layes itefaces the wave equatio is obtaied by combiig the cul equatios (2.7) ad (2.8) ad ca be witte as: 2 1 E 2 ( x) x z 1 2 c E t 2 z 2. (2.29) The elative emittivity of this system is a ositio-deedet fuctio (x). The stuctue is eiodic i x with lattice costat, theefoe (x) must satisfy the elatio: ( x) ( x a). (2.30) The mateial exasio is obtaied by exessig the ivese of the elative emittivity usig the Fouie seies of the fom: 1 ( x) e jg x (2.31) 24

25 25 whee ae the exasio coefficiets ad a G / 2 is the eciocal lattice. The vaiable is a itege idetifyig a secific cell. Similaly, Bloch solutios ae used to eeset the field ad ca be witte as: t j x G k j E z e e t x E E x z ) ( ), ( (2.32) whee the waveumbe 2 / x k ad a G E / 2, is a itege. Note the deivatio assumes the field to be hamoic i time. All ossible omal modes of the matix equatio ca be obtaied by lettig x k vay fom a / to a /, the Billoui zoe fo 1-D eiodic system, ad solvig the matix equatios fo eigevalue ad eigevecto. Isetig equatios (2.31) ad (2.32) ito the oe dimesio electomagetic wave equatio (2.29) ovides the wave equatio of the fom: t j x jk x a j E t j x jk x a a j x E e e e c e e e k a z z ) 2 ( 2 ) 2 2 ( 2 2 (2.33) Multilyig this equatio by the othogoal fuctio x a m j e 2, whee m is a itege ad the itegatig each side ove a uit cell fom esults i: x a m a j a a E m x a m a a j a a x E m dx e c dx e k a z z ) 2 2 ( ) ( (2.34) Usig the othogoality oeties of the basis fuctios, the o-zeo solutios ae obtaied if m fo the left itegal ad m fo the ight itegal. Theefoe, the summatios ca be educed fom two to oe i the ight-had side, ad fom thee to two i the left-had side. E m x E m z z c k a (2.35)

26 The summatio ove ad m ae used to build the matix. As a esult, the exasio idices of the ivese elative emittivity have to be double the exasio idices of the field. To illustate the ocess, the matix fo a small exasio umbe of exasios is show i equatio (2.36). The field exasio idices ad m will cotol the oulatig of the matix elemets i colums ad ows, esectively. Fo each elemet oly oe value of the dielectic exasio will be used, which is the oe that satisfied the coditio m. Q( m, )) Q( m, )) Q( m, )) m2, 2, m1, 2, m0, 2, Q( m, )) m2, 1, Q( m, )) m2, 1, Q( m, )) Q( m, )) m1, 2, m2, 2 E E E E z 2 z 1 z 1 z 2 = c 2 E E z 2 z 1 Ez Ez 1 2 (2.36) whee Q( m, )) m x 2 2 k, solvig the matix fo the eigevalues, the fequecies, that a ae emitted to oagate i the medium at each wave umbe ae lotted to oduce the bad stuctue of the dielectic stuctues ude test. The eigevectos ae obtaied by eisetig idividual eigevalues i the matix ovides the dielectic exasio coefficiets ad the ca be used to defie the oagatig mode ofile. A bad ga is defied as a age of fequecies fo which thee is o coesodig wave vecto ad thus the eegy levels withi the bad ga ae ot available, see figue

27 Figue 2.3: Bad ga, idicated i gey, i a eidoic mateial with high-dielectic-costat mateial ( 13) of width 0.2a ad low-dielectic-costat mateial ( 1) of width 0.8a [45]. h The esece of defects i the hotoic cystal ad itetioal doig esult i the modificatio of the bad stuctue ad ude the ight coditios ca itoduce localized modes withi the oigial bad ga. Fo localized modes aalysis, a suecell PWEM techique is equied. A suecell is a lage uit cell size with the defect i the cete of the suecell. The suecell must be lage eough that it is sufficietly modellig the stuctue ad the defects ae ot omally eeated. This equies moe memoy ad comutatio time. Fom the deivatio, it is aaet that the PWEM equies taslatio symmety i the stuctue, theefoe stuctual assumtios ae imosed whe dealig with o-taslatio symmety o whe geometies ae fiite i size. h 2.3 ecet techiques ecetly, ew aoaches have bee develoed to aalyze stuctues that ossess otatioal symmety [27, 28]. They ae based o takig Maxwell s wave equatio ad exadig the 27

28 electomagetic fields ad the ivese elative emittivity of the dielectic ofile usig Fouie- Bessel (FB) basis fuctios i ola coodiates, ad Fouie-Fouie-Bessel (FFB) basis fuctios i the cylidical coodiates. FB ad FFB wee used to demostate a elatioshi betwee the otatioal ode of the stuctue ad the otatioal ode of the localized modes [27, 28]. A sigificat eductio i the eige-matix ode is ossible usig symmety, which geatly educes the comutatio time. I additio, the solve ca be diected to comute a secific otatioal mode ode. FB aoach equies easoable memoy size ad shot comutatioas time as a mode solve fo 2-D lae PhCs, whee the stuctue is assumed to be ivaiat alog the axial diectio eedicula to the lae. The FFB techique assumes eiodicity i z diectio ad ca be cofigued to solve fo hybid steady states. A atual extesio of the basis fuctio based techiques is to coside sheically symmetic stuctues ovidig a meas of solvig fo esoato states i a 3-D domai. Extedig the solve to 3-D sheical sace will eable the study of actical oblems ot eviously solvable o a deskto PC [28]. The uose of this wok is to develo a techique that is suitable fo aalyzig localized modes i thee dimesioal sheical stuctues. Comutatio time ad memoy equiemets ae issues eseted with may existig techiques whe used to study localized modes fo thee dimesios stuctues. The BLF techique ovides the eseaches with a fast comutatio solve with low memoy equiemets fo thee dimesios sheical domai stuctues. The achievemet of usig basis fuctios ad bouday coditios that ae moe suitable to comute localized modes could oe the ath fo usig diffeet basis fuctios ad bouday coditios fo studyig diffeet oblems. The techique ca be viewed moe clealy whe 28

29 comaed to the PWEM. PWEM was desiged maily to study the oagated waves i eiodic stuctues, while the BLF techique eseted hee is maily desiged to be used fo localized modes comutatios. PWEM use eiodic bouday coditio of uit cells aaged i lattice cofiguatio with the assumtio of ifiite stuctues ad the use of exoetial exasios fuctios i Catesia coodiates. These bouday coditios ad basis fuctios ae moe suitable fo studyig oagatig waves i eiodic stuctues. This makes the PWEM aoiate comutatios method fo the bad stuctues calculatios. The BLF techique is alied to fiite size stuctues with a zeo-field value bouday coditio i all diectios. The exasio fuctios used ae moe suitable to eeset localized modes i sheically symmetic stuctues. I additio, i the BLF thee will be o eed fo taslatio symmety estictios. Uses familia with PWEM will be able to easily imlemet the BLF techique. 29

30 3 Chate: Theoetical develomet of the BLF method I this chate, the details of the theoetical develomet of the BLF techique ae eseted. Fudametally, the modified sheical basis fuctios BLF ae fist itoduced, ad ae used to eeset the ivese elative emittivity ofile ad the vecto fields i Maxwell s H ad E wave equatios i sheical coodiates. The ew exessios with BLF seies eesetatio ae used to fomulate a eigevalue matix. The eigevalue matix ca be solved to obtai the localized mode s oeties, such as agula fequecy (eigevalues), wavelegth, field ofile (eigevectos). 3.1 Theoy The BLF mode solve is develoed fom the vecto wave equatio fo eithe the electic o magetic field i a chage fee, cuet fee, isotoic, omagetic, ad liea medium. Thoughout the comutatio ocess, the time deedece takes the fom of e j t whee is the comlex agula fequecy, j. The geeal statig wave equatios, obtaied fom i combiig Maxwell s cul equatios, ae give i (3.1) ad (3.2) fo the H ad E fields esectively, whee the diffeet dielectic egios ca be eeseted by the satial vaiatios i the elative emittivity,. The oduct of the fee sace costitutive aametes is elaced by the seed of light i vacuum squaed, 2 c. 1 H c 2 H (3.1) 1 E c 2 E (3.2) 30

31 The oeato fom of the equatios i (3.1) ad (3.2) diffe oly i the lacemet of the elative dielectic costat, which is i geeal a fuctio of the coodiates. The oeato fom of the E field equatio ca be obtaied fom the H field equatio whe deivatives of the ae set to zeo. Fo this easo, it is sufficiet to ovide the mathematical stes fo the H equatio. 3.2 Dielectic ofile ad EM field eesetatio i BLF I sheical sace, the fields ad the ivese elative dielectic ofile ae exaded usig basis fuctios of the fom: j ) jq 0 ( ) P (cos e (3.3) whee j0 ( ) is the zeoth ode sheical Bessel fuctio, is the oot of the lowest ode sheical Bessel fuctio, ad is a ositive itege. The oot is scaled by the adius of the comutatioal domai,. The zeoth ode ad th degee associated Legede fuctio, P (cos ), is used fo the deedece i ola agle ( ) ad the comlex exoetial fuctio deedece i azimuthal agle ( ). jq e fo the estictig the adial basis to the lowest ode sheical Bessel fuctios is a deatue fom the stadad aoach i the use of Bessel exasios whee all odes ae used. The lowest ode Bessel fuctio is also emloyed to eeset the decayig field egios. This is made ossible because ay cotiuous adial fuctio f () o the iteval fom 0 to ca be witte as a seies exasio. Theefoe ay highe ode sheical Bessel (o decayig fuctio) ca be accommodated without loss of accuacy usig the exasio: 31

32 f ( ) j0( ) (3.4) whee ae the coefficiets of the exadig fuctio. The beefit to usig oly the lowest ode sheical Bessel fuctio ae two-fold i develoig the eige-matix. The fist is that the deivative alog the adial coodiate will oly ivolve the zeo ad fist ode sheical Bessel fuctios ad theefoe oly two sheical Bessel fuctios ae eeded i buildig the matix elemets. The secod is that i the matix buildig ocess the itegal of the oduct of thee sheical Bessel fuctios (the field, the dielectic ad the othogoal field) is eeded. Usig oly the lowest ode sheical Bessel limits the umbe of diffeet itegads to facilitate the omalizatio ad tabulatio fo eeated use fo all stuctues examied. The sheical Bessel fuctios have the followig othogoality oety [46, 47]: ( ) 0 ( * ) 1( ) * j j d j (3.5), 2 0 Similaly, oly the zeoth ode of the associated Legede fuctios P (cos ) is equied as ay cotiuous ola agula fuctio betwee 0 ad ca be seies eeseted as: f ( ) P (cos ) (3.6) whee ae the coefficiets of the exadig fuctio. The idex of the Legede olyomial is oly limited to ositive itege values ad a seies exasio o these fuctios ca be used to eeset ay ola agle deedet fuctio. This oety geatly simlifies the exessios to be eseted. I the matix buildig ocess the itegal of the oduct of thee Legede olyomials is eset. Usig a sigle uesticted idex limits the umbe of itegad to a 32

33 maageable set that ca be tabulated fo use i the aalysis of all dielectic stuctues. The sigle idex olyomial diffes fom the stadad actice of usig two idices ad tyig oe of the idices to the azimuthal ode though it woks ad is mathematically feasible. Usig the sigle idex Legede olyomial simlifies the matix buildig ocess. The zeoth ode associated Legede fuctio satisfy the othogoality coditios [48]: * * P (cos ) P (cos ) si d (3.7), 0 (2 1) 2 The kow comlex exoetial fuctio jq e elated to the azimuthal agle ( ) ca be used to eeset ay fuctio i the azimuthal agle ( ) ad is othogoal whe itegated ove the iteval [0, 2]. Maxwell s wave equatio makes use of the ivese elative emittivity Ω, ad both the H ad E fields. The ivese elative emittivity ca be exessed usig the BLF fuctios as:,, q j iq 0 ( ) P (cos ) e. (3.8) The ivese elative emittivity exasio coefficiets ca be detemied diectly by the ivese Fouie tasfom of (3.8) oce the details of the dielectic sace ae ovided. See Matlab ogam i aedix E fo details. Each comoet of the magetic field o the electic field is also exaded usig the sheical basis fuctio as: H s,, q s s s s jqs 0( ) P (cos ) e (3.9) j s s 33

34 E s,, q s s s s jqs 0( ) P (cos ) e (3.10) j s s whee s,, is used to distiguish betwee the field comoets ad s ae the exasio coefficiets. Usig these seies exasios, Maxwell s wave equatios ca be cast ito a eigevalue oblem, which ovides the steady state fequecies (eigevalues) ad modal ofiles eigevectos i sheical sace ( exasio coefficiets). 3.3 Buildig the eigevalue matix A eigevalue fomulatio of the wave equatios, suitable fo ovidig ifomatio o the localized state comlex fequecies ad coesodig field ofiles ca be develoed statig fom eithe the magetic (3.1) o electic (3.2) field equatios. Teatig the ivese of the elative emittivity as a scala fuctio of the coodiates, the exessios ca be efomulated as: 0 0 (3.11) I the wave equatio of (3.11), the electic field equatio uses tems with desigatio, the E magetic field equatio uses tems with desigatio H, ad a idicates that the tem is ot 0 icluded. Exessig the wave equatios i sheical coodiates, oeatios left-had side of equatio (3.1) esults i a vecto that ca be defied as:,, o the (3.12) The tems cotaiig deivatives ae collected ito ad ae ovided i aedix A. Fo the E field, the ivese elative emittivity is exteal to the cul, esultig i simle exessios fo 34

35 . The comlicatio fo the H field is that the double cul alicatio i (3.1) esults i fist ad secod deivatives of the basis fuctio with esect to the sheical coodiates fo the field comoets, ad oly fist deivative of the basis fuctio fo the ivese elative emittivity. The ight-had side of (3.1) is also witte i sheical sace as:. (3.13) Usig equatios (3.12) ad (3.13) i (3.1) gives the wave equitatio of the H field i sheical coodiate as: (3.14) Isetig the exasios, (3.8) ad (3.9), ito the exessios fo,, ad o the left-had side, ad ito the ight-had side of the equatios (3.14). Pefomig the deivatios ad alyig the othogoally itegatios ove the sheical domai will esult i the matix oulatig fuctios, Aedices C ad D. The deived exessios that ca be collected ito a eigevalue matix fomulatio, A v eig v (3.15) Fo coveiece, the details of the ocedue used ae show oly fo the ight-had side of (3.1). A simila ocedue is followed fo the left-had side. Fo the ight-had side of equatio (3.14): 35

36 (3.16) Substitutig the exasio (3.9) fo each comoet of field gives: HS comoet 2 2 s iqs ( s) H s s j0( ) P (cos ) e (3.17) s c c,, q s s s Othogoality is used by multilyig each exaded comoet by a exaded othogoal fuctio, ad itegatig ove the volume of the shee domai. 2,,,, s c si. (3.18) Usig the othogoality elatios fo the thee basis fuctios, it ca be oted that o-zeo values ae obtaied oly whe,,,, ) esultig i: HS comoet(s) c 2 2. (3.19) The omalizatio tem Collectig all comoets fo all idices,, ) esults i: 2 ca be moved to the left-had side. HS = I c 2 (3.20) whee I is the idetity matix, ad, Θ, Φ ae the field exasio coefficiets fo the field comoets i the adial diectio, ola agle Θ ad azimuthal agle Φ, esectively. 36

37 37 The left-had side of the exaded wave equatio is obtaied usig simila stes ae alied fo each tem of each comoet of the W equatios, see aedix C. This will esult i thee equatios, a equatio i each of the thee sheical coodiates witte as: W, (3.21) W, (3.22) W. (3.23) These equatios ae fuctios of the thee field comoets, Θ, Φ. The ie fial exessios fo,, s s ad s, ae give i aedix D. This eesets the left-had side of the eigevalue equatios: Av (3.24) Povidig a oveall eigevalue oblem with the followig fial fom: I c 2 (3.25) The eigevalues ae 2 c while the field exasio coefficiets ae the comoets of the eigevecto. The lowe subscit of the elemets i the 3 by 3 matix eesets the coodiate of the equatio. It also seves to idicate the othogoal field comoet that was alied io to comutatio domai itegatio.

38 It is imotat to metio that the satisfactio of the othogoality coditios fo the thee basis fuctios i the ight-had side of the equatio does ot esue that they ae satisfied i the lefthad side. This is due to the existece of the thee basis tems withi each itegal. Theefoe, the sheical Bessel ad Legede othogoality itegals ae calculated ad saved i tables to educe the amout of time comutig as they ae calculated ideedetly fom the stuctue ofile abt 1 ja 0 j j b t 0 * d whee, (3.26) d c c P c P c t P d t s P * d (3.27) c s ab 0 a b whee c cos, s si. Also, i equatio (3.27) the ode idex of the sheical Bessel ad Legede is alteed. This is a cosequece of the deivatives alied o the exaded fields ad the ivese elative dielectic. The adius i the Bessel itegal has bee omalized, Aedix B. Thee ae sigulaities at 0, 0 ad. To umeically deal with these sigulaities, these egios ae excluded fom the itegals. Theefoe, the itegal bouds i (3.26) chaged to be fom to oe, whee is a boud that is close to zeo. Fo the itegals i (3.27) the bouds ae chaged to be fom to 0 whee is a boud that is close to zeo ad 0 is a boud that is close to. The exclusio does ot affect the validity of the solve as esoace fields of iteest have zeo value i these egios. Howeve, these sigulaities ae ot eset whe usig the ew fomulatio fo the techique i chate 5. Fo the exoetial tems jq s e, the othogoal itegal gives Ω 2 Ω, (3.28) 38

39 whe Ω, the itegal will oly esult i a o-zeo value. This gives the advatages of tagetig secific modes i the diectio ad educig the matix ode. The eigevalues fom (3.25), 2 c, ae the waveumbes squaed,. The waveumbes ae omalized by. Settig the dielectic shee stuctue adius to uity, 1, will esult i omalized eigevalues, k omalized,fo the valid localized modes of the stuctue. The localized mode waveumbe values ae give by: komalized k. (3.29) The BLF techique is equied to obtai k omalized, the k values fo shee of diffeet adius ae calculated usig the equatio (3.29). The omalizatio hels to taget ad desig esoatos fo secific wavelegths, simila omalizatio is commo i PWEM [21, 22]. The equatio (3.29) ca be witte as: 2 o k omalized (3.30) Usig equatio (3.30), it is ossible to cofigue ad desig the dielectic shee adius to satisfy ay desied wavelegth. 3.4 Solve stes Defiig the dielectic The fist ste i usig the umeical solve begis by defiig the dielectic stuctue withi the etie comutatio domai, ad the detemiig the exasio coefficiets. Theefoe, the 39

40 coefficiets of the ivese elative dielectic exasio Ω have to be calculated fom the ivese elative dielectic stuctue Ω,, usig ivese Fouie tasfom of (3.8): Ω,,,, Ω,, ρ si (3.31) A zeo-value bouday coditio was used fo the sheical Bessel fuctio omalizatio, the ight had side of equatios (3.5). This will tailo the techique to taget the localized modes, modes that have o electic,, o magetic,, field at the edges of the simulatio domai. The extet of the simulatio domai must be selected to esue that the zeo-value bouday coditio at the edges fo localized modes is satisfied, see figue 3.1. Figue 3.1: Zeo-value bouday coditio located fa away fom the objective dielectic stuctue Poulatig the matix Each of the oeatos i the 3 by 3 matix i equatio (3.25), eesets a squae aay (block) of umbes with a ode equal to umbe of basis fuctios used i the field exasio seies. The idividual umbes fo each oeato ca be geeated usig the exessios give i Aedix D. The geeal fomulatio equies two sets of field elated idices q, 40 * * *, ad, q,. The

41 elemets i a block ae collected fo the field idices costat alog a ow ad the othogoal idices costat dow a colum.. Each idividual matix elemet is oduced though a summatio ove the ivese elative emittivity idices,, q, Solvig the matix The eigevalues ae the set of umbes c 2 etued usig a umeical eigevalue solve such as the eig() fuctio i MATLAB o the matix i (3.25). The eigevectos associated to each eigevalue ae coveietly obtaied usig the eig() fuctio. The eigevecto (exasio coefficiets) s ae used i (3.9) to oduce the satial field ofile Localized mode ofiles categoies Whe used to study dielectic ofiles ossessig sheical symmety, the BLF method ovides fou diffeet categoies of modal fields. Localized states have high field stegths cofied to the cetal egio of the simulatio domai ad egligible field values alog the edge of the comutatioal domai. Edge states o the othe had show stog field comoets cofied mostly alog the comutatioal edge of the simulatio domai. These ae highly deedet o the oeties alog the edge ad ae ot omally states examied usig the BLF techique. Chagig the locatio of the comutatioal edge will affect the edge state eige-fequecies ad field ofiles, while the oeties localized states emai uchaged. Sue-states ae states that ecomass sizeable field values thoughout the etie comutatio domai. These states ae omally ot the states ude ivestigatio whe usig the BLF techique. Iteface states, which ae states that show stog field cofiemet at the iteface betwee dielectic egios. Whe the fields fo these states decay gadually to zeo at the comutatio adial edge, they eeset localized BLF etued states. Such states ca be etued whe seachig fo WGMs, bottle, 41

42 SNAP, ad ig esoato states fo examle [5] ad [13]. I geeal, the mode solve etus oe state fo each basis fuctio used to exess the field comoet seies. Fo istace, usig 100 diffeet basis fuctios to exad the field will etu 300 eigevalues ad eigevectos. The oeties of the states ca be filteed by examiig the domiat field comoet i the eigevecto, the value of the comuted eige-wavelegth comaed to the stuctue size aametes ad the domiat basis fuctio exasio coefficiets. The eal o imagiay atue of the eige-fequecy ad the elative magitude of the eal ad imagiay ats ca seve to filte etued states. A accuate measue of the modes oeties is detemied by ecostuctig the field usig (3.9) ad lottig the field i diffeet laes. 3.5 Otimizig the techique The othogoality itegatio elated to the agle lays a imotat at i educig the eigematix ode ad seves to diect the comutatio ocess to seach fo aticula otatioal odes. The itegal is witte i (3.32) ad laces a estictio o the azimuthal idices combiatios acquied fo o-zeo values of the matix elemets. 2 0 e j qq q* d 2 qq, q* (3.32) Fo examle, if the dielectic ofile shows o vaiatios, the q 0, the ais q, q * must be equal fo the value of the matix elemets to be o-zeo. The ositio of the o-zeo matix elemets ae such that matix blocks comosed of q 0, q* 0 ca be seaated fom q 1, q* 1 ad so o. I this way, a aticula mode tye i the azimuthal lae ca be selected ad solved fo, usig a much-educed matix ode eigevalue system. This featue is aticulaly useful whe seachig fo high ode WGMs i sheical esoatos o fo selectively seachig fo moooles, diole, quadules o othe ode modes. I the table below the q values ae oly show, the othe 42

43 idices ae hidde iside each elemet of the table. This meas that evey elemet i the table eeset a matix of size that is equal to the idices i ad. I additio, this table shows the matix elemets fo oly the azimuthally uifom shee stuctue q 0. Table 3.1: The coditio fo o-zeo matix elemets q q *. This table eeset oe of the 3x3 squae aays i left had side of equatios (3.25). Fo selectig modes, the q, q * eeset the ode of the localized mode ode, theefoe, fo mooole localized mode i the azimuthal diectio oly the matices of the q 0, q* 0 eeds to be calculated ad solved. 3.6 Segmetatio of the dielectic ofile To decease the comutatio time of the techique eve futhe the mateial segmetatio aoach ca be added. A dielectic segmetatio aoach has bee eseted as a suotive comutatio tool fo the FFB techique i [49] ad is alied hee to the BLF techique. The 43

44 aoach is exteded hee to facilitate the segmetatio of the mateial ofile ito two o moe seaate ofiles which whe combied eoduce the oigial iteded stuctue. The esult of this is a system matix that is the sum of the seaate matices fo each mateial segmet Offset-esidue mateial segmetatio I the geeal case, a otical esoato defied i the sheical coodiate system may have mateial vaiatios alog all thee coodiate diectios. These vaiatios defie the stuctues mateial ofile which may be exaded usig (3.8). Figue 3.2 to-left shows the vaiatios i the elative emittivity of a eesetative esoato. The to-ight figue shows the ofile obtaied fom the ivese elative emittivity. The lowe figues show that the ivese elative emittivity ca be divided ito a costat offset value (lowe left, ). The oigial ivese ofile ca be obtaied fom off es. off ) ad a esidue ofile (lowe ight, es 44

45 Figue 3.2: Segmetatio ocess of ivese of the elative emittivity of the stuctue (toight) is show. The wave equatios fo the E ad H fields ca ow be witte as a summatio ove two tems. 2 off E es E c E 2 off H es H H c (3.33) (3.34) Alig the BLF techique deivatios, as i evious sectios, fo (3.33) esults i a matix system that has the fom: es es es es es es c off off off es es es 2 off off off off off off (3.35) The matix elemets of the squae 3 X 3 matices ae oulated usig the exessios ovided i Aedix D. I the case of the offset matix, the ivese emittivity value off is a costat thoughout the comutatio domai ad does ot equie a seies eesetatio. Theefoe, fo the 45

46 ocess of geeatig the offset matix elemets the summatio ove the media is elaced by the offset value. I the case of the esidue matix, the seies exasio fom of (3.8) is equied fo the esidue ofile oly ad used i the matix oulatig exessios. It is easy to show that fo two diffeet media ofiles i which oe ofile, desigated i the equatios as the oigial ofile off es, is the scaled equivalet of the othe ' ' off ' es, the scalig factos elated to the offset ad the esidue ofiles ae give by, ' ' ' K K : off es off off es es K off ' ' ' off off K es off off (3.36) This educes the comutatioal bude i (3.35), because the scalig factos ca be exteal multilies to the matix oulatig ocess ad theefoe the matices geeated fo oe stuctue ca be used (escaled) fo the aalysis of aothe stuctue which has the offset, esidue o both ofiles simly escaled. Figue 3.3 below illustates that the esidue ofile ca be futhe segmeted ito a umbe of ofiles, oe fo each costituet mateial eset i the esoato stuctue. 46

47 Figue 3.3: Segmetatio of esidue ofile (To Left) ito 4 seaated ofiles, a ofile fo each costituet mateial eset i the esoato stuctue. I such a situatio, the oigial ivese emittivity ofile is give by N off i1 i es. Allowig fo the ossibility of escalig oe o moe egios, the system matix obtaied is: 47

48 48 N i i es off c K K i es i es i es i es i es i es i es i es i es off off off off off off off off off 2 1. (3.37) This aticula fom of the system matix is useful whe exloig the stuctues esose whe oe medium is chaged such as i the aalysis of esoat otical sesos [15]. All matices ae built oce ad escaled whe the sesig medium is chaged. The segmetatio of the mateial oeties ito offset ad esidue ofiles ca also be alied i the aalysis of stuctues which udego a etubatio withi oe o moe egios. The offset ofile, which was iitially uifom thoughout the comutatio domai, is elaced by a efeece stuctue which is comosed based o the uetubed stuctue. Figue 3.4 illustates the ocess. The lowe two figues show the uetubed stuctue sevig as the efeece o the left ad o the ight a etubatio alied to the egios chaacteized by a etubatio ofile. The mateial etubatio alied to the efeece stuctue ca be teated as the esidue ofile ad whe the matix is oulated ovides the etubatio matix dm. A chage i the level of the alied etubatio is easily tasfomed ito the escalig of the matix dm K dm et, with et K a costat scalig facto.

49 Figue 3.4: efeece, lage etubatio ad small etubatio ofiles. If the etubatios alied to the efeece stuctue ae small ad the states ae o-degeeate the eigevalues ad eigevectos of the etubed system ca be obtaied diectly though the followig exessios [50]: i o, i o, i dm o, i, (3.38) 49

50 50 j o j i j o i o i o j o i o i dm,,,,,,. (3.39) The matix geeated fo the efeece stuctue ca be solved to yield the uetubed eigevalues o,i ad eigevectos o,i. Whe the etubatios ae lage, the fist ode aalysis leadig to equatios (3.38) ad (3.39) divege fom the exact solutio. I such a case eithe highe ode coectio tems ae added to the exessios o the full stuctue is examied usig a oetubative aoach. The mateial segmetatio ocess eseted does ot ely o the etubatio beig small, a lage etubatio simly esults i a lage scalig facto alied to the etubatio matix. The etie matix ca be aidly oulated usig the followig exessio: c dm M 2 (3.40) e c K e e e e e e e e e ef ef ef ef ef ef ef ef ef 2 (3.41) ad solved fo the eigevalues ad eigevectos. As fo ay umeical techique a umbe of limitatios fo the BLF techique ae eset. Oe of the limitatios is the edge state iclusio i the solutios. Addig a laye of imagiay elative emittivity at the solve domai edge ca hel to filte out those solutios as the imagiay eigevalue of those states will be lage. Aothe limitatio of this techique is the sigulaities i the itegals (3.26) ad (3.27). Dealig with these sigulaities is exlaied i sectio (3.3). I the equatios the ivese of elative emittivity is used, this meas vey high elative emittivity will

51 have vey small ivese elative emittivity, which makes the umeical comutatios hade. I additio, the techique eeds to be imoved to deal with fequecy deedet media. 51

52 4 Chate: Comutatio examles To veify the accuacy of the BLF techique the comutatio esults fo diffeet stuctues ae examied. All comutatios examles show i this chate ae efomed usig the matix system itoduced i chate 3. The elative emeability of all examles efomed i this thesis is costat 1, while the elative emittivity is defied fom the oosed stuctue (,, ). Howeve, the BLF techique ca deal with stuctue ofiles of chagig elative emeability (,, ) ad elative emittivity (,, ). The stuctues ude test ae stuctues whose solutios ae available fom othe techiques. I aticula, the BLF comutatio aoach is alied to 4 secific examles: the sheical cavity with a efectly coductig laye laced o its oute suface; the mooole ad WGM states fo a high elative emittivity shee i ai; esoato states fo eiodic ad aeiodic sheical adial layes; ad a disc of uifom ad o-uifom dielectic ofile i the azimuthal diectio. 4.1 Sheical cavity Figue 4.1 left shows a coss sectio of a esoato comosed of a efect coductig sheical shell of adius filled with homogeeous, ad lossless dielectic mateial such as ai with 1. Due to the uifom atue of the dielectic medium the mooole states decoule ito sheical TE, with comoets H H, E,, ad sheical TM, with comoets E E, H,. I the seies exasio of the ivese elative emittivity ofile usig (3.8), the exasio idices fo ad q fo cotibutios to the basis fuctios ca be set to zeo as the shee is uifom i the ola ad azimuthal diectios. To oely esolve the dielectic adial ofile, 100 sheical Bessel tems wee icluded ( fom 1 to 100). The ecostuctio of the ivese elative emittivity ofile is show i the figue 1-ight, obtaied usig (3.8). 52

53 The matix geeatig exessios i Aedix D would idicate that fo mooole states (q = 0) i a dielectic medium showig o deedece ( q 0) the E comoet is decouled fom the othe comoets. This field comoet ad olaizatio state ae selected as they match u with the esults eseted i [38, 39] that wee comuted diectly fom the solutio of the Maxwell s equatios. Figue 4.1: (Left) Coss sectio of the sheical cavity stuctue. (ight) ecostucted ivese elative emittivity ofile usig 100 Bessel tems. Small deviatios i dielectic value ae uesolved by the esoato states of iteest. To solve fo the E comoets, oly the block is equied i the matix calculatios. The decoulig of fields fo moooles, the fudametal mode i the azimuthal diectio, i the dielectic shee eviomet is simila to the decoulig of the fields obseved whe the out of lae diectio has o dielectic vaiatio as obseved i the lae wave aalysis of hotoic cystals. Poulatio of the matix elemets, fo the E field equatio, usig the geeatig exessios i Aedix D, esults i 4 of the 9 elemets blocks beig zeo added, exessio (4.1), ad eables the E matix block to be solved ideedetly fo its eigevalues ad 53

54 eigevectos. A eductio i the matix ode by a facto of 3, i additio to tagetig a aticula azimuthal mode ode, geatly seeds u the comutatio ocess whe dealig with lage matices c (4.1) To oulate the block i the eige-matix of (4.1), fields wee seies exaded usig (3.10) with 50 sheical Bessel tems ( = 1 to 50) ad 51 Legede tems ( = 0 to 50). The q idex was set to zeo which cofigues the eige-solve fo localized states that eeset the mooole modes. Diectig the BFL to taget a diffeet mode tye with esect to the diectio is also ossible. The matices oduced fo each of the diffeet q field idices ca be seaated ito idividual sub matices (see sectio 3.5). The eigevalues ad eigevectos associated with each mode wee comuted usig the eig() fuctio i MATLAB. A esticted age of the eigevalues wee selected, ad coveted to k fo comaiso with efeeces, fo which the eigevecto is domiated by exasio coefficiet fo the E field comoet ae show i Table 4.1. Table 4.1: Comaiso of the scaled esoace waveumbe obtaied usig BLF ad fom efeeces i [38, 39]. Bold-field ofiles lotted i figues 4.2 ad 4.3 comuted usig the BLF techique with eigevecto domiated by the E field comoet. adial Ode Pola Ode [4.4934] [7.7252] [5.7634] [9.0950] [6.9879] [ ] [8.1825] [ ] [9.3558] [ ]

55 Table 4.1 shows the comaiso of the mooole states i the azimuthal diectio comuted usig BLF ad those quoted i [38, 39] usig aalytical solutios show i squae backets. The comaiso is made by examiig states with simila field oeties, idicatig the umbe of adial zeo cossigs ad the domiat Legede idex odes. Comaed to aalytical solutio fo the same stuctues ad bouday coditios obtaied fom the efeeces, the BLF techique waveumbes comuted match the ublished values. The eigevalues with the bold text i table 4.1, coesod to localized states that ae lotted i figues 4.2 ad 4.3. Whee the field ofiles geeated though equatio (3.10) fo the E field comoet. The fields show a zeo value at the comutatios adial limit due to the imosed of zeo-bouday coditio. The comutig time fo this examle is about 5 miutes, usig a comute of 16 GB AM ad ocesso seed of 2.93 GHz. Figue 4.2: The modal ofile fo E field comoet omalized to a maximum of 1, 110 mode ode, fo k Left (x, y) = (, 90, ) lae. ight (y, z) = (,, 90) lae. The black to white colo eeset the omalized field values fom oe to zeo. 55

56 Figue 4.3: The modal ofile fo E field comoet omalized to a maximum of 1, 230 mode ode, fo ka Left (x, y) = (, 90, ) lae. ight (y, z) = (,, 90) lae. The black to white colo eeset the omalized field values fom oe to zeo. The BLF umeical techique equies the tucatio of the exasio seies; theefoe, covegece of the esults eeds to be addessed. Figue 4.4 shows covegeces of oe of the selected omalized eigevalues (k), obtaied usig BLF techique, the aalytical solutios is the hoizotal lie. The figue shows that usig 40 basis fuctios fo the field exasio ovide well coveged esults. 56

57 Figue 4.4: The covegece fo mode 130. The solid hoizotal lie labels the aalytical solutio [38, 39], the dots eeset the umeical esults obtai usig BLF. The covegece i figue 4.4 is show as the cuve is closely fomig a hoizotal lie. The coveged value of the omalized waveumbe is accuate to thid decimal oit. The techique coveges as the basis umbe used to exad the elative emittivity ad the fields ae iceasig. 4.2 High elative emittivity shees A sigle high dielectic costat shee laced i ai i the secod test fo the accuacy of the BLF mode solvig techique. Published ad well established esults ae available fo comaiso usig aalytical solutios [18, 19]. The dielectic shee is selected to have a adius a ad elative emittivity 9. Fo comutatioal uoses, the ai egio is exteded to a adius 2a with the dielectic shee laced at the cete, as show i figue 4.5-Left. The ai egio extedig to 57

58 2a is sufficiet to isolate the shee localized states fom the comutatioal edge ifluece, a equiemet fo BLF. States of the dielectic shee have effectively a zeo field comoet o the comutatio edge whe well localized. Simila easoig is also emloyed whe seachig fo localized states i sigle defect cotaiig hotoic cystals usig the lae wave exasio techique. Fo the geomety examied hee, the elative dielectic ofiles oly ossess adial vaiatios, with a abut chage at a. The seies eesetatio of the ivese elative emittivity ofile ca be eeseted by settig q ad both to zeo ad summig ove The seies exasio coefficiets of the ivese elative emittivity obtaied usig (3.31) ae lotted i figue 4.5-ight vesus basis fuctio idex.. The exasio coefficiets obtaied cotai oly eal ats as the mediums ivolved ae lossless ad of sheical symmety. The dielectic decomositio is well coveged as demostated by the coefficiets aidly aoachig zeo with iceased odes. A alteative cofimatio is the ocess of ecostuctig the ivese dielectic usig equatio (3.8) ad comaig to the oigial stuctue. To oulate the eige-matix (4.1), the field comoets of the seies i (3.10) wee exessed usig 70 sheical Bessel tems ad 71 Legede tems. Note that the Legede exasio tems ae equied fo the field, eve though the dielectic is uifom, i ode to esolve the elated ofile oeties. The q idex was set to zeo taget mooole like localized states i the (x, y) = (, 90, ) lae. The umbe of basis fuctios was sufficiet to esue coveged localized states fo the dielectic shee i ai. 58

59 Figue 4.5:Left Geomety of the solid-uifom, 9, dielectic shee, adius a. The sheical comutatio domai is of adius =2a. ight eal at of the dielectic seies exasio coefficiets lotted vesus sheical Bessel basis fuctio idex. Imagiay ats ae all zeo fo this dielectic ofile. Whe mooole states ae cosideed i a dielectic medium showig o deedece, the field comoets decoule ito two gous; TE with comoets H, H, E ; ad TM with comoets E, E, H. The eigevalues etued fom (4.1) ad coveted to omalized esoace waveumbes, ka, fo seveal lowe ode moooles ae listed i table 4.2 fo the E (TE) field comoet, alog with coesodig values extacted fom [18, 19]. The esults idicate a excellet ageemet betwee the two diffeet techiques. The bold values coesod to localized states that ae lotted i figues 4.6 ad 4.7. Mode omeclatue efes to the umbe of field maximums obseved alog the uit vecto diectios. Field ofiles also match those lotted i the cited efeece. 59

60 Table 4.2: Scaled esoace waveumbes detemied usig BLF ad fom efeeces i [18, 19] fo the E, TE olaizatio. Bold eties coesod to field ofiles show i figues 4.6 ad 4.7. Pola Ode adial Ode [1.02] [2.06] [1.46] [2.53] [1.90] [2.97] [2.32] [3.44] [2.70] [3.86] 2 Figue 4.6: The (x, y) = (, 90, ) lae modal ofile fo E field comoet (Left), ad the (y, z) = (,, 90) lae modal ofile fo E field comoet (ight). Field ofiles with fo the 110 localized state ae listed i table 2, comuted usig BLF, ka

61 Figue 4.7:The (x, y) = (, 90, ) lae modal ofile fo E field comoet (Left), ad the (y, z) = (,, 90) lae modal ofile fo E field comoet (ight) fo the 150 localized state listed i table 2, comuted usig BLF, ka I a simila fashio the ue left 2 by 2 matix elemet blocks of the matix (4.1) ca be solved fo the TM olaizatio states cotaiig E, E field comoets. The age of scaled waveumbes coesod to the lowe set of localized states listed i table 4.3 alog with those extacted fom [18, 19] that wee comuted usig aalytical solutios. The field ofiles fo the bold eties ae lotted i figues 4.8 to 4.10 fo the vaious field comoets ad ae i excellet ageemet with the featues of the ofiles lotted i the cited efeeces. Table 4.3: Scaled esoace waveumbes detemied usig BLF ad fom efeeces [18, 19] fo the E, E, TM olaizatio. Bold eties coesod to field ofiles show i figues 4.8 to adial Ode Pola Ode [1.46] [2.58] [1.77] [3.03] [2.24] [3.38] [2.68] [3.74] [3.08] [4.17]

62 Figue 4.8: The (x, y) = (, 90, ) lae modal ofile fo E field comoet (Left), ad the (y, z) = (,, 90) lae modal ofile fo E field comoet (ight), fo 210 localized state listed i table 3, comuted usig BLF, ka The black to white colo eeset the omalized field values fom oe to zeo. Figue 4.9: The (x, y) = (, 90, ) lae modal ofile fo E field comoet (Left), ad the (y, z) = (,, 90) laemodal ofile fo E field comoet (ight), fo 250 localized state listed i table 3, comuted usig BLF, ka The black to white colo eeset the omalized field values fom oe to zeo. 62

63 Figue 4.10:The (y, z) = (,, 90) lae modal ofile fo E field comoet fo 210 localized state listed i table 3, comuted usig BLF, ka (Left). The (y, z) = (,, 90) lae modal ofile fo E field comoet fo 250 localized state listed i table 3, comuted usig BLF, ka (ight). The (x, y) = (, 90, ) lae have zeo E field values. The black to white colo eeset the omalized field values fom oe to zeo Whiseig Galley Modes The high dielectic shee i ai is kow to suot high azimuthal ode states efeed to as Whiseig Galley Modes (WGMs). The basic cocet of (WGMs) could be exlaied usig the ay aoach ad the matchig the hase of the wave fo esoace coditios. ays that stike the cocave suface of the shee at a agle lage tha the citical agle will exeiece total iteal eflectio. Fo a wave taed close to the suface the hase matchig coditio is: 2 m (4.2) Whee m is the mode umbe, is the effective efactive idex, ad is the wavelegth. WGMs have may imotat oeties such as small mode volume, high owe desity ad aow sectal lie width. That is due to miimal eflectio losses ad ossibility of usig vey low mateial absotio, which ehace the Q-facto to be vey lage values, ovidig a ootuity to imove seso ad lase alicatios [4, 15]. 63

64 WGMs geeally eset a field ofile highly cofied at the iteface betwee the high dielectic shee ad suoudig ai egio ad a lage umbe of field amlitude cycles withi the azimuthal lae shee s cicumfeece [13]. WGM wee calculated usig BLF by oulatig the eige-matix usig azimuthal ode 20, ad azimuthal mode ode 40. The field comoets wee eeseted usig 60 sheical Bessel fuctios ad 61 fo Legede olyomials. Sice the dielectic exasio has o-zeo exasio coefficiets oly fo q 0, thee is o mixig of the azimuthal mode odes ad they ca be seaately solved, see figue Note the figue is built usig small basis umbes ( = 1 to 2) ad ( = 0 to 2) ad (q = 0 to 2) fo illustatio uoses. The ed, oage, ad yellow cells ae the o-zeo elemets fo the fist, secod, ad thid azimuthal mode odes esectively. Figue 4.11: Samle eige-matix showig locatios of zeo ad o-zeo elemets. The white elemets ae the zeo elemets. Each colo eesets diffeet azimuthal ode; the ed, yellow, ad oage cells ae the o-zeo elemets fo the fist, secod, ad thid azimuthal mode odes. 64

65 The etie oulated eige-matix fo each mode must be solved usig all 9 geeatig exessios. The field ofile a of WGM of ode 20 ad ka , show i figue to, dislays 40 field extemes (20 field amlitude cycles) whe obseved i the (x, y) lae. I figue bottom, the field ofile a WGM of ode 40 ad ka , is show, simila oeties of the mode ofile cofiemet ae eseted but i this case the ofile dislays 80 field extemes (40 field amlitude cycles) whe obseved i the (x, y) lae. The field is highly cofied to the high dielectic side of the shee-ai iteface with decayig field towads the ai side. Figue 4.12: Modal ofile fo the E field comoet of the BLF comuted WGMs of the solid shee i ai. To ai Azimuthal mode ode 20 with 1120 ad ka Bottom ai Azimuthal mode ode 40 with 1140 ad ka Left (x, y) = (, 90, ) lae. ight (y, z) = (,, 90) lae. The black to white colo eeset the omalized field values fom oe to zeo. The comutatio time eeded deeds o may factos such as, the seed of the comute used, ad the basis used to comute the fields ad the dielectic stuctues. The comutatios time fo 65

66 this sectio takes about 2 miutes if the accuacy is about 2 % usig 20 basis fuctios fo the sheical Bessel ad Legede. Fo accuacy of 1% usig 50 basis fuctios fo the sheical Bessel ad Legede, it takes about half a hou usig a comute of 16 GB AM ad ocesso seed of 2.93 GHz. Theefoe, the comutig time ad memoy equied always deeds o the basis umbe used to eeset the field ad the dielectic. I additio, the codes wee witte i a way that makes it flexible to edit but with the cost of lowe seed comutatios. Moe imovemets fo the codes seed ae ossible. 4.3 Sheical Bagg stuctues Whe solid shees cosist of alteatig layes of thi dielectic mateials, seveal of the esoace states ae caused by total/atial iteal eflectio at the shell laye egio. The ability to cotol the esoace oeties i thee dimesios makes the sheical Bagg shell esoato of cosideable iteest ad suitable fo umeous alicatios. Fo a shee coated by Bagg shells light ca be cofied stogly fo a age of wavelegths. I figue 4.13, the adial ofile of a micoshee laced i the cete ad coated with Bagg sheical shell of 24 layes is show, as e the secificatios of [20]. The adius of the micoshee is 1 µm with a elative dielectic costat of The Bagg layes ae fomed fom alteatig egios with µm ( 2. 10) ad µm ( 4. 20). The comutatio domai is exteded to 5 µm beyod the outemost laye esultig i µm. The esults fom [20], wee studied usig the scatteig matix method [51, 52] ad will be comaed to those fom the BLF calculatios. The ivese dielectic was esolved usig 400 sheical Bessel basis tems with ad q set to 0, as the dielectic medium has o o deedece. The adial dielectic exasio coefficiets fo the sheical shell stuctue ae show i figue

67 The field comoets wee eeseted usig 60 sheical Bessels ad 61 Legede olyomials. The BLF comuted state fo the H field comoet was at a fequecy f 188 THz ad show i figue The modal ofile ad comuted fequecy match the cited efeece; Oly a 2% diffeece i fequecy value is obseved. The localized mode is highly cofied aoud the cete, with zeo value at the edge of the comutatio domai. Figue 4.13: adial ofile of shee coated by dielectic shells. The shells acts as a quate wave Bagg eflecto. The dielectic has eal values oly. 67

68 Figue 4.14: adial dielectic exasio coefficiets fo the sheical Bagg stuctue. Figue 4.15: The modal ofile fo H field comoet fo 110, comuted usig BLF. The field ofile ad coesodig eigevalue match those of efeece [20]. Left (x, y) = (, 90, ) lae. ight (y, z) = (,, 90) lae. 68

69 Figue 4.16:The modal ofile fo BLF, (,, 90) lae. f THz ( 8089 E field comoet fo 1130, comuted usig ka 2. ).Left (x, y) = (, 90, ) lae. ight (y, z) = Figue 4.17:The modal ofile fo E field comoet fo 1330, comuted usig BLF, f THz ( ka ).Left (x, y) = (, 90, ) lae. ight (y, z) = (,, 90) lae. esults fo othe moooles of diffeet ode ad field comoet ae also obtaied, while i the efeeced ae the solutio show was just fo oe mode. I figues 4.16 ad 4.17 mode ofiles fo E field comoet fo 1130 at f THz ( ka ) ad 1330, at 69

70 f THz ( ka ) ae show, esectively. The field comoets wee eeseted usig 60 sheical Bessel ad 61 Legede olyomials. 4.4 Aeiodic eflecto esoatos The geomety of the aeiodic eflecto is simila to the Bagg laye stuctue but desiged to have a low dielectic fill facto. The high dielectic egios close to the cete ae thi ad gadually icease i width as thei adius iceases. The outemost laye is desiged to have a thickess that is ¼ wavelegth at the desig value. The aametes fo the stuctue i figue 4.18 ae fom [53] ad cosisted of alteatig sheical layes of ai ad ested alumia. The mode-matchig techique was used fo [53] calculatios. The ested alumia has a comlex dielectic value, 9.8, ad I figue 4.19 ad 4.20, the eal ad imagiay exasio, eal, imagiay sace decomositios ae show. Lage umbes (500 sheical Bessel basis tems) ae used to make accuate eesetatio of the stuctue ( 0 to 500, q 0, ad 0 ). I figue 4.21, the mode ofile fo the E field comoets fo 110 is lotted usig 60 fo the sheical Bessel ad 61 fo the Legede olyomial idices. The localized mode ofile matches the esults obtaied i the efeeced ae, the field is decayig outside the cete shee due to the aeiodic Bagg shells. The fequecy esult also matches the efeeced esult. The esoace fequecy has eal ad imagiay ats. The eal at eesets cofied eegy ad the imagiay at eesets eegy loss due to absotio. I [53] the Q facto was calculated usig a aalytical techique based o the electic fillig facto ad multilyig it by the absotio loss of the thi dielectic. Fo the give stuctue [53] detemied the Q facto to be Usig the BLF method, the losses of the mateial i ae 70

71 icluded ad the Q facto fo a give mode is calculated usig the eal ad imagiay comoets of the fequecy values:. (4.3) The Q facto obtaied usig the BLF is calculated to be I this examle the comaig was with esults obtai usig the mode-matchig techique. The absotio losses ae calculated usig the electic fillig facto fo dielectic egio multilied by the loss taget. The eo ecetage betwee the two umeical calculatios is about 20%. Figue 4.18: elative dielectic costat ofile alog the adial diectio fo a shee of ai coated by aeiodic sheical shells. 71

72 Figue 4.19: eal adial ivese elative emittivity exasio coefficiets fo the aeiodic shell stuctue. Figue 4.20: Imagiay adial dielectic exasio coefficiets fo the aeiodic shell stuctue. 72

73 Figue 4.21: The modal ofile fo E field comoet fo 110, comuted usig BLF. The field ofile ad coesodig eigevalue match those of efeece [53]. Left (x, y) = (, 90, ) lae. ight (y, z) = (, 90, ) lae. While the examles eseted hee have bee limited to stuctues oly showig adial vaiatios this is a limitatio of the available ublish eseach o sheical esoato oeties fo comaiso ad ot a limitatio of the BLF techique. The techique eseted hee ca be eadily alied to othe geometies withi the sheical bouday, cotaiig dielectic vaiatios i ay combiatio of the sheical coodiate diectios. 4.5 Dielectic disc The esoace mode wavelegths ad ofiles fo a dielectic disk ae eseted hee usig the BLF techique. The disc is selected as it has well established esults fom ay otics. Figue 4.22 dislays the stuctue ude test alog with the vaiatio i ola agle ad adial diectio. The dielectic disc is tested to veify the accuacy of the techique as it is the simlest case that has vaiatio i the ola agle lus the adial diectio. The dielectic disc of 9. 0 is assumed to have a width of 0.3a. The domai adius is double the disc adius =2a. 73

74 Figue 4.22: Dielectic disc is show, whee a is the adius of the disc. The stuctue is uifom i the azimuthal diectio. Equatio (4.2) is used to comute the esoace wavelegths based o the ay otics aoach. The use of this aoach equies that the wavelegth should be much smalle tha the stuctue size. This is satisfied by examiig the techique to study high ode modes (fom 50 to 100) fo the E comoets. To cofigue the BLF techique to solve fo the selected mode ode, the coditio 2 0 e j qq q* d 2 qq, q* is alied. The dielectic disc is uifom i the azimuthal agle ( q 0), which additioally meas oly ( q 50, 70, o 100 ) ae eeded fo the field exasio i the azimuthal agle, as the itegal will have zeo value fo othe field s idices. As the mode ode tageted fo this comutatio is high (shot wavelegths), field comoets wee 74

75 eeseted usig 80 basis fuctios fo the sheical Bessel ad 81 basis fuctios fo Legede olyomial idex eve though the stuctue is quite simle. The dielectic exasio of the disc uses 50 basis fuctios fo the sheical Bessel ad 50 basis fuctios fo Legede olyomial idex ae used. The exasio coefficiets ae show i figue It ca be cofimed that the basis umbe used fo the dielectic exasio is sufficiet as the figue 4.23 shows that the coefficiets become egligible fo highe idices. The calculatios of the esoace mode wavelegths ae obtaied omalized to the adius ad togethe with the ay otics esults ae lotted i figue Figue 4.23: The exasio coefficiets of the dielectic disc. 75

76 Figue 4.24: Scaled esoace wavelegths ( ) detemied usig BLF ad usig the ay otics a fo the E field. The BLF calculatios wee doe with a elatively low umbe of basis exasios, but a comaiso of the esults those calculated usig the ay otics aoach shows good ageemet. While the comuted field eigevalue ad eigevecto is thee dimesioal, the selectio ability of the techique educes the field comutatio to what is equivaletly a two dimesios comutatio i tems of the umbe of calculatios. Ulike the ay otics, BLF ca examie modes whose wavelegth is o the same ode as the stuctual featues. Figue 4.25 dislays a lot of the omalized esoace waveumbes fo the 20 th azimuthal mode ode, icludig the mode ofiles fo a umbe of waveumbes, fo the disc show i figue It ca be see that modes that ae affected by the domai edges ae eseted i the solutios. I idetifyig modes these modes ae filteed out, as they do ot eeset a coect localized mode of the dielectic stuctue. 76

77 Eve if the stuctue is ot uifom i the azimuthal diectio, the ability to select modes ad educe the matix emais. If the stuctue has symmetic vaiatio i the azimuthal diectio the oly the idices of dielectic coefficiets that ae o-zeo lus o mis the taget field mode ode ae eeded i buildig the matix. Figue 4.25: The waveumbes of modes with the 20 th ode azimuthal mode ode ae show. The field ofiles ae lotted i the (x, y) = (, 90, ) lae (left) ad i the (y, z) = (, 90, ) lae (ight). The dielectic shee adius a is half the solve domai shee. The two highest mode ofiles ae affected by the domai bouday. 77

78 I the followig sectio a disc that is o-uifom i the azimuthal diectio is studied. This stuctue is thee-dimesioal stuctue i the sheical coodiate. The comutatio time fo this examle was about fou hous, i a comute of 32 GB AM ad ocesso seed of 4 GHz. 4.6 Dielectic disc with azimuthal dielectic chage Fo a dielectic disc with vaiatio i the azimuthal diectio, the BLF ca be used to comute the esoace modes. The dielectic disc has a dielectic vaiatio of the fom of 9.0 cos( ). The exasio coefficiets of this stuctue ae show i figue 4.26 ad It is clea fom the figues that oly low idices of the dielectic exasios ae eeded fo the azimuthal diectios as highe idices have a zeo value. To ut this i cotext the exasio coefficiets have a maximum value of 4.21 whe the zeo idex of the dielectic i the azimuthal diectio is used 6 (see figue 4.26) while fo highe idices the value educes stogly to at the thid azimuthal idex (figue 4.27, bottom). This simlifies the comutatio, as the field calculatio will oly use the fist thee idices i the azimuthal diectio of the dielectic lus o mius the taget field mode ode. 78

79 Figue 4.26: The exasio coefficiets fo the zeo idex of the azimuthal diectio q 0, ad 50 idices fo the ola ad the adial diectios. 79

80 Figue 4.27: Exasio coefficiets fo the dielectic disc with azimuthal vaiatio (To) The exasio coefficiets fo the fist idex of the azimuthal diectio q 1ad 1 ad 50 idices fo the ola ad the adial diectios. (Middle) The exasio coefficiets fo the secod idex of the azimuthal diectio q 2 ad 2 ad 50 idices fo the ola ad the adial diectios. (Bottom) The exasio coefficiets fo the thid idex of the azimuthal diectio q 3 ad 3 ad 50 idices fo the ola ad the adial diectios. 80

81 The othogoality coditio is 2 0 e j qq q* d 2 qq, q*, howeve, sice the azimuthal agle has o-zeo coefficiets oly fo q 0, 1, 2, oly these values ae eeded fo the field exasio i the azimuthal agle. Figue 4.28 below show the esults fo, 10 th ad 20 th, two selected mode odes i the azimuthal diectio. A comaiso of the waveumbe ad mode ofile with the 20 th mode ode i the evious sectio eveals ageemet. The esultig waveumbes ad field ofiles of the disc with cosie vaiatio i the azimuthal diectio ae comaed ad matched to the uifom azimuthal dielectic disc. This esult is exected, as the aveage efactive idices of the two stuctues ae simila. Figue 4.28: The 10 th mode ode i the azimuthal diectio k=4.81 i the disc of azimuthal vaiatio dielectic (left). The 20 th mode ode i the azimuthal diectio k=8.84 i the disc of azimuthal vaiatio dielectic (left). Figue 4.29 dislays othe states fo the 10 th azimuthal mode ode. Simila behavio is obtaied fo the 20 th mode ode. The field is lotted fo the E comoet, ad simila esults ae obtaied fo othe comoets of the field. 81

82 Figue 4.29: The waveumbes of 10 th ode azimuthal mode ae show. The field ofiles ae lotted i the (x, y) = (, 90, ) lae (left) ad i the (y, z) = (, 90, ) lae (ight). Good ageemet has bee show betwee the esults comuted usig BLF ad othe techiques. Moe comlicated stuctue ca be studied simly by esuig the use of eough basis fuctios; this is a basic ule fo ay seies-exasio calculatios. 82

83 5 Chate: Advaced fomulatio ad basis fuctios I this chate, the BLF techique eseted i chate 3 will be e-examied to ovide a moe comact fom. This ste is eeded to facilitate the mathematical wok fo the imlemetatio of the techique fo moe actical stuctues such those that ae Aisotoic. Istead of begiig with the wave equatios the deivatio will begi with Maxwell s time ideedet equatios, 1 E, (5.1) c 1 E. c (5.2) whee jzoh, ad o Z o is the fee sace imedace. The ivese elative emittivity o ad emeability teso matices, esectively, ca be witte as, Ω 0 0 Ω 0 Ω 0, 0 0 Ω Λ 0 0 Λ 0 Λ Λ. (5.3) Usig (5.3), the left-had side of (5.1), becomes: E si, (5.4) Λ si Λ Λ. 83

84 (5.5) Similaly, the ight-had side of (5.1) is witte as, c c. (5.6) Usig equatio (5.5) ad (5.6), equatio (5.1) ca be witte i the followig fom: Λ si Λ Λ. (5.7) Usig the same ocess above, equatio (5.2) becomes, Ω si Ω Ω. (5.8) The ivese elative emeability Λ ca be exaded usig the sheical basis fuctios as: ss,, q ss iq 0 ( ) P (cos ) e. (5.9) j 84

85 85 whee ss is used to label the elemet of the emeability tese matix (5.3), ).,, ( o ss Futhemoe, each comoet of the E ad is also exaded usig the sheical basis fuctios as follows: s s s s s iq q s s e P j E,, 0 s ) (cos ) (, (5.10) s s s s s iq q s s e P j,, 0 s ) (cos ) (. (5.11) I equatio (5.7) the field deivatives ca be elaced with the followig exasios: s s s s s s iq q s s e P j E,, 1 s ) (cos ) (, (5.12) s s s s s s s s s s iq q s s iq q s s s e P j e P j E,, 1 0,, 0 s s ) (cos ) ( si ) (cos ) ( si cos, (5.13) s s s s s iq q s s s e P j iq E,, 0 s ) (cos ) ( ) (. (5.14) Puttig equatios (5.9) (5.11), (5.13) ad (5.14) back ito the comoets of equatio (5.7) esults i the followig exasio, Λ 1 si si

86 86 iq q iq iq q q iq iq q q iq iq q q e P j e P j e P j iq e P j e P j e P j e P j,, 0 0,, 0,, 0,, 1 0,, 0,, 0,, ) (cos ) ( ) (cos ) ( ) (cos ) ( si ) ( ) (cos ) ( ) (cos ) ( si ) (cos ) ( ) (cos ) ( si ) cos cos (. (5.15) Futhemoe, the substitutios (5.9) - (5.14), fo the ad comoet of equatio (5.7) give, Λ 1 1 si iq q iq iq q q iq iq q q iq iq q q e P j e P j e P j e P j e P j e P j e P j iq,, 0 0,, 1,, 0,, 0,, 0,, 0,, ) (cos ) ( ) (cos ) ( ) (cos ) ( ) (cos ) ( ) (cos ) ( 1 ) (cos ) ( ) (cos ) ( si ) (, (5.16) Λ 1

87 ,, q,, q,, q,, q,, q q,,, q q,,, 1 j0( ) P cos j si j si j 1( 0 0 ( (,, q (cos ) e ) P ) P ) P 1 iq (cos ) e j ( 0 (cos ) e (cos ) e ) P j ( 0 iq iq iq j ( 0 j ( j ( 0 0 (cos ) e ) P iq (cos ) e ) P ) P ) P iq (cos ) e (cos ) e (cos ) e iq iq iq. (5.17) At this oit the othogoality elatioshi, as descibed i chate 3, is alied to equatio (5.15). Multilyig the othogoal fuctios, j0( * ) P * (cos ) e equatio (5.15) followed by itegatig ove the sheical domai gives: iq * with the fist tem of,,,, Λ Λ Λ Λ Λ, Λ φ si Λ. Λ Λ Ω (5.15a) The fist tem of equatio (5.15) is witte ow as: Λ 1 2 Λ, Λ 87

88 (5.16) whee ad ae descibed i chate 3. The same othogoality ocess is alied fo the othe tems of equatio (5.15) ad the esults ae witte as follows: 0, (5.18) Λ Λ 2 Λ,, (5.19) Λ 1 Λ 2 Λ, Λ Λ 2 Λ,. (5.20) Futhemoe, the othogoality elatioshi ae also alied to all the tems of equatios (5.16) ad (5.17) givig exessio, Λ 2 Λ, Λ (5.21) 0, (5.22) Λ 2 Λ, Λ 88

89 89 Λ Λ 2 Λ,, (5.23) Λ Λ 2 Λ, Λ Λ 2 Λ, (5.24) Λ Λ 2 Λ,, Λ Λ 2 Λ,, (5.25) 0, (5.26) Icludig the above simlificatios, the geeal matices foms will be E E E c, (5.27) E E E c. (5.28) The geeal matices foms ca be ow used to ucoule (5.1) ad (5.2) as follow,

90 90 H H H H H H c 2 2 (5.29) 5.1 Bouday coditios Fo the adial basis used i this wok, the sheical Bessel fuctios satisfy the othogoality elatio * *, ) ( ) ( d j j (5.30) This geeally hold oly fo ifiite sace, while fo fiite iteval (0, ) the othogoality is satisfied by * *, ) ( ) ( N d j j (5.31) This meas that the bouday coditios at =, eed to be secified to have a set of othogoal basis. Sheical Bessel fuctios with the zeo-bouday coditio have bee used fo all calculatios. The omalized tem that esults whe usig the othogoality itegals as i (5.31) ad (3.19) is witte as: ) ( j N (5.30) whee is the shee adius fo the calculatio domai as eviously stated ad ae the zeos of the Bessel fuctios. This omalizatio imoses zeo bouday coditio, details ae i [46].

91 The use of this zeo-bouday coditio icludes solutios that ae affected by edge of the solve domai esultig i imactical solutios beig icluded, which ceates the issue of the may categoies of mode ofiles that ca be foud as was discussed i the sectio Howeve, usig sheical Bessel fuctio with cotiued bouday coditio eables the solve oly solve fo moe actical solutios. The bouday coditios of the sheical Bessel fuctio ae detemied by alteig the omalized tem whe alyig the othogoality. The modified omalized tem i (5.31) fo cotiued bouday coditio is witte as [46]: N 3 2 j0 ( ). (5.32) 2 Calculatios usig the cotiued bouday coditio is cofimed by matchig esults obtaied usig zeo bouday coditio. This suots the statemet that tue localized modes ae ot affected by the edges of the solve domai. 5.2 Comutatio Examles The develoed BLF techique deived fom Faaday s ad Amee s laws is used i the followig sectios to study localized modes of vaious examles. esults obtaied usig this ew matix fomulatio is validated as it matches those obtaied usig the old matix fomulatio (chate 3) whe used i sectio Aisotoic Mateial The solutios of the electomagetic wave i aisotoic shees have bee studied aalytically ad umeically [54-56]. The umeical aalysis of aisotoic stuctue is alicable usig the comact fomalizatio of the BLF techique fom sectio 5.1. The ew fomalizatio stats fom the cul equatio of Faadays ad Amees law, theefoe has o deivatives o the dielectic 91

92 stuctue. This simlifies dealig with moe comlicated dielectic stuctues, such as teso dielectic stuctues. I the followig examles, the techique is used to study the esoace modes of aisotoic sheical stuctues. I aisotoic medium the electic ad magetic oeties of the stuctue ae deedet o the diectio of the fields vectos. I such a case, the emittivity ad emeability of the medium ae defied by matix tesos. The comutatio examles hee will cove oly diagoal emittivity teso, although the techique is caable of wokig with full aisotoy tesos. Fo diagoal emittivity o diagoal emeability, the teso matix is witte as i (5.3). I the followig comutatio examle, the dielectic will be desiged to have diagoal emittivity teso matix, ad the mateial is assumed to be o-magetic ad lossless. The aisotoic dielectic shee is laced i ai. The emittivity of the shee is set to 9 i the azimuthal agle, 5 i the ola agle, ad 6.5 i the adial diectio. The ivese elative emittivity of the teso ( ) ae exaded usig 100 basis fuctios i the edial diectio ad oly oe basis fo the ola ad azimuthal diectios as the shee is uifom i the ola ad azimuthal diectio. The fields ae exaded usig 20 basis fuctios i adial ad ola diectios. The domai adius is =2a, whee a=1. The omalized waveumbe of comuted azimuthal modes odes is show i figue

93 Figue 5.1: The mode ofiles ad waveumbes fo a umbe of diffeet localized modes i the ola diectios. Modes (a ad b) have a omalized waveumbe of ka=2.08 ad (c ad d) have omalized waveumbe of ka=2.96. Both ae lotted usig the E field. Modes (e) ka= 3.64 ad (g) ka=4.13 ae lotted usig E field vecto, (f) ka=3.64 ad (h) ka= 4.14 ae lotted usig the E field vecto. The esoace modes of the stuctue fo highe mode ode i the azimuthal diectio ae also calculated usig the same ocedues. The fields ae exaded usig 40 basis fuctios, as the uosed mode ode is highe. Figue 5.2 shows the localized waveumbe fo a umbe of selected esoace modes i the azimuthal diectio. 93

94 Figue 5.2: The mode ofiles ad waveumbes fo a umbe of diffeet localized modes i the azimuthal diectio. The field ofiles ae fo the E field vecto. A efectly matched laye (PML) is widely used to tucate the comutatio domai i umeical methods such as FDTD ad FEM. The PML has a absobig laye that is gadually iceased fom zeo. Uiaxial PML (UPML) is a attactive develoed method ad is accomlished by usig aisotoic absobe laye. I equatio (5.3) the dielectic ofile is eeseted by a teso fuctio. To iclude the UPML i the BLF techique the tese fuctio is witte as: Ω Ω 0 0 Ω 0 Ω 0, (5.33) 0 0 Ω I this case, the omal comoets will udego the absotio of the laye ad educe eflectio fom the domai bouday. I sheical 3D domai, the desig of the UPML is much easie tha 94

95 the 3D stuctue i othe coodiates, as it has oly oe comoet that stikes the edges of the domai with a omal agle Dielectic sheical shell of 6-fold vaiatio i The esoace mode wavelegths ad ofiles fo a sheical shell with 6-fold vaiatio i ae studied usig the BLF techique. Figue 5.3 shows a coss sectio of the dielectic sheical shell of 9.0 cos(6 ) i the (, ) lae. The sheical shell is desiged to have a width of a/3, with a comutatio domai adius that is double the shee adius =2a. I figue 5.4 the elative emittivity vaiatio i the azimuthal diectio of the tested sheical shell is show. Figue 5.3: A coss sectio of a dielectic sheical shell is show, whee a is the adius of the sheical shell. The stuctue is 6-fold symmetic i the azimuthal diectio. Fo the dielectic sheical shell 50 sheical Bessel basis seies exasios ae used fo the adial diectio, oe idex fo Legede olyomial idex i the ola diectio, ad 50 Fouie 95

96 basis fuctios i the azimuthal diectios ae used. The ola uses oly oe exasio as the shell is uifom i the ola diectio. The dielectic sheical shell has 6-fold symmety i the azimuthal diectio; theefoe, oly idices of ( q 0, 6, ) ae o-zeo [27]. Fo the azimuthal mode, the techique ca be cofigued to study secific mode ode. Oly the idices of the dielectic i the azimuthal diectio lus o mius the tageted azimuthal mode ode ae eeded fo the field exasio i the azimuthal agle, as the itegal will have zeo value fo othe field s idices [27]. Othe field s comoets wee seies exaded with 20 sheical Bessel tems ad 21 Legede tems. I Figue 5.5 the omalized esoace waveumbes, ad select mode ofiles fo the low ode states havig the 3 ed azimuthal mode ode ae lotted. Figue 5.4: elative emittivity vaiatio i the azimuthal diectio of the sheical shell. 96

97 Figue 5.5: The waveumbes of 3 ed ode azimuthal mode ae show. The field ofiles ae lotted i both the (y, z) lae (left) ad i the (x, y) lae (ight). As show i the figue, the techique is cofimed to study the 3 ed azimuthal mode ode. The mode ode fo the adial is fudametal, while fo the ola seveal diffeet odes ae show. 97

98 6 Chate: Sesig alicatio I ecet yeas, otical devices have gaied i imotace i the field of seso systems. Otical sesos use light fo measuig a wide vaiety of eviomet effects. Comaed to thei electoic couteats, otical sesos ca ovide umeous advatages. Fo examle, they ae immue to electomagetic itefeece, which make them suitable fo oeatio i eviomets with high electic field [57]. They ae small ad light i weight. I geeal, a otical seso cosists of a otical souce, modulatos, ad a detecto. The geeal icile of such devices is the modificatio of the light s aametes (itesity, hase, o wavelegth) i esose to some eviometal effects ude ivestigatio (temeatue, mechaical stai, dislacemets, essue, o othe quatities). The modulato comoet modifies the icomig light by chagig some hysical quatity i esose to the eviomet effect. Sheical esoatos ae of sigificat iteest i sesig alicatios, due to the existece of WGMs with a umbe of imotat of oeties: small mode volume, high owe desity, ad aow sectal lie width. Light ca be couled to a esoato usig a waveguide laced ea the esoato. esoace modes will be stimulated by the evaescet coulig of the waveguide light. The esoace wavelegths of the uloaded esoato ca be detected fom the tasmissio sectum of the waveguide. Whe the quatities ude ivestigatio ae loaded to the esoato the dis i the tasmissio sectum will be shifted i esose to the quatities ude ivestigatio. The shift of the wavelegth is due the chage of the efactive idex of the esoato o the size of the esoato because of the loaded quatities. The BLF techique esults ae used to obtai the esoace wavelegths of the esoatos ad the chage of these wavelegths as the efactive idex o size chage. 98

99 6.1 Sheical dielectic shell The esoace aalysis of a shee dielectic shell esoato is umeically comuted fo two sets of mode odes. The calculatios focus o studyig the localized modes of a umbe of diffeet odes that vay i ola diectios, the fist set, o azimuthal diectios, the secod set. The cuves of wavelegth shift vesus the efactive idex chage of the dielectic shell ae examied ad show esults The segmetatio aoach is used ad the off ad es of a dielectic sheical shell ae show i figue 6.1. The adius of the domai,, is twice the legth of the dielectic sheical shell adius a. Futhe, the dielectic shell thickess is desiged to be equals to a/6. The chages i the shell dielectic value due to the absotio of imosed mateials o temeatue effects ae modeled usig the segmetatio ocedue. The esoace-mode wavelegths ad ofiles ae calculated usig the BLF with both segmetatio ad omalizatio, so as to esue high comutatioal efficiecy. I this calculatio, the omalizatio is alied by settig the comutatio domai adius (=2) ad usig a = 1. Equatios (3.35) is used to comute the omalized esoace wavelegths (0/a) of the dielectic sheical shell as a fuctio of the chages i the dielectic values. The matixes i (3.35) ae oulated oly oce The, by substitutig (3.36) ito (3.37), the chages i the dielectic values ae modeled. This will emit the use to efom faste calculatios as it is ot eeded to oulate the matix evey time the dielectic values chage. The off is desiged to have 1/12 as the efeece value, while off ae chaged i value fom 1/12.25 to 1/ The es also has a efeece ivese elative emittivity of 1-1/12 ad es ae chaged i the ivese elative emittivity fom 1-1/12.25 to 1-1/ The 0/a esults ad 99

100 ofiles fo each dielectic value chage ae the calculated by solvig the eigevalue equatio (3.37). Figue 6.1: esidual dielectic fuctio (es, left) ad offset dielectic fuctio (off, ight) of dielectic sheical shell. I this examle a set of modes icludig the fudametal modes i ad with fou diffeet mode odes i fo the E field is the fist set to be aalyzed. The 0/a behavio ad the sesitivity cuves of the dielectic sheical shell fo the fist set of modes, as fuctios of the chages i the dielectic values, ae show i Fig Aothe set of mode odes ae also comuted usig the BLF techique with the segmetatio ocess. I this secod set, the fudametal mode is i ad, ad thee ae fou diffeet mode odes i. The 0/a behavio ad the sesitivity cuves of the sheical shell fo the secod set of modes, as fuctios of the chages i the dielectic values, ae show i Fig

101 Figue 6.2: Sesitivity cuves ad mode ofiles fo fist set of modes. The modes ode ae (To): 140, 180, left to ight, (Bottom): 1180, 1300, left to ight. The esults show i figues 6.2 ad 6.3 idicate highe sesitivity fo a lowe mode ode i the ola ad azimuthal agles. Note that the sesitivity fo a mode ode geate tha the fudametal i the adial diectio ad the fudametal ode i ad has bee studied i [58]. Usig (3.30), the desige ca set a to taget a secific 0 at a selected mode ode. Fo examle, if the tageted 0 is aoximately 1550 m ad the tageted mode umbe is the 8 th ode, a is calculated to be 1 m usig (3.30). If the value of a is too small fo actical fabicatio, the mode ode ca the be iceased to achieve a desig fo the same 0 age, but with a lage a. Fo the 18 th mode ode, a must be iceased to m i ode to taget the same 1550-m-wavelegth sesig age. 101

102 Figue 6.3: Sesitivity cuves ad mode ofiles fo secod set of modes. The modes ode ae (To): 1110, 1120, left to ight, (Bottom): 1140, 1160, left to ight. The sesitivity is highe fo lowe-ode modes, which is due to the highe cofiemet of the highe-ode modes. Futhe, the sesitivity iceases if a is deceased ad, theefoe, the dielectic shell thickess deceases. We ext coside a shee with a elative dielectic emittivity value 4, which is low coated with a high dielectic laye of 12 high. The substate ad ovelay of the stuctue ca be combied as a efeece matix ad the etubatio of the thid at (the ai egio) is alied by chagig the ivese elative emittivity fom 1 to 0.4, see figue

103 Figue 6.4: efeece ad etubatio ofiles of the ivese elative emittivity. The to figue shows the oigial stuctue. The two figues i the bottom show the segmetatios of the oigial stuctue ito efeece (left) ad lage etubatio (ight). The stuctue ca also be teatig as a sum of seaate multile values of the elative emittivity, see figue 6.5. The wavelegth shift vesus the dielectic chage of the ambiet medium is studied usig (3.37), ad the esoace shifts of two selected modes, 1110 ad 1120, ae show i figue 6.6. The calculatio ocedue used hee is simila to that emloyed i evious sectios. The esultat shifts of 0 ae omalized; theefoe, this aoach ca be alied to desigs fo a lage age of wavelegths. This vesatility is a sigificat advatage fo the develoed BLF techique. 103

104 Figue 6.5: Segmetatio ito 3 seaated ofiles, a ofile fo each mateial costituet eset i the esoato stuctue. Figue 6.6: Sesitivity cuves of ambiet medium ad mode ofiles of E fo selected modes. The mode ofile is show i the (, ) lae. The adius legth of the dielectic sheical shell is omalized. 104

105 Fo small etubatios, equatios (3.38) ad (3.39) ae used to study the chage of the ai egio with etubatio of 0.2 of the elative emittivity, as show i figue 6.7. The esults match the segmetatio aoach with 1% eo. Figue 6.7: efeece ad etubatio ivese elative emittivity ofiles. The to figue shows the oigial stuctue. The two figues i the bottom show the segmetatios of the oigial stuctue ito efeece (left) ad small etubatio (ight). Both segmetatio ad adius omalizatio wee show to be vey effective methods of ehacig the BLF-techique comutatioal ocess. 105

106 6.2 Slot WGMs chael desig Slot chael waveguides ae made by havig low dielectic mateial sadwiched betwee two highe dielectic egios. I this kid of geomety the field will be edomiatly localized i the ai egio, makig this kid of stuctue less affected by the mateial loss. The cotiuity coditio of the omal comoet of the electic flux desity D omal betwee the dielectics iteface ovide a lage electic field i the low dielectic egio. Cotiuity of the omal comoet of D gives D, D (5.1) omal omal, low high E E high omal, low omal, (5.2) high low This demostates a lage discotiuity of the omal comoet of the electic field with times highe electic field i the low dielectic egio tha i the high dielectic egio. Fo Si ai iteface the slot chael will have electic field that is about 12 times lage tha the Si egio. Whe the guided modes i the high dielectic egios have a decayig legth that is lage tha the slot chael width the discotiuity of the field omal comoet will be gaied. The theoetical comutatios fo slot chael waveguide i a cylidical stuctue have eviously bee eseted i usig the FFB techique [36]. high low 106

107 Figue 6.8: Side view of the oosed WGMs slot waveguide stuctue. The shee has high elative emittivity value with squaed slot chael filled with ai. Figue 6.9: The (x, y) lae modal E field comoet (Left), ad the (y, z) lae modal E field comoet (ight) fo the 110 localized state i the slot chael egio. The filed is high i the slot chael egio. The ba o the left shows the omalized values fo the esoace field. I figue 6.8, a shee slot chael cofiguatio is show. The tye of slot ad the size will deed o may factos. Fo lage WGMs ode, the size of the slot chael has to be small comaed to the shee size, figue 6.8. This has the advatage of tagetig small wavelegths modes, such as the 1.55 m i a lage shee size. I figue 6.9, the slot chael localized light is show. The field ae stogly localized i the slot egio. 107