Analysis of TE (Transverse Electric) Modes of Symmetric Slab Waveguide

Size: px

Start display at page:

Download "Analysis of TE (Transverse Electric) Modes of Symmetric Slab Waveguide"

Brittany Alison Lawrence
6 years ago
Views:

1 Adv. Sudis Thor. Phs., Vol. 6,, o. 7, Aalsis of T (Trasvrs lcric Mods of Smmric Slab Wavguid arr Rama SPCTC (Spcrum Tcholog Rsarch Group Dparm of lcrical, lcroic ad Ssms girig Naioal Uivrsi of Malasia 436 UKM-Bagi, Slagor, Malasia hrama@g.um.m Farshad Nasimi SPCTC (Spcrum Tcholog Laboraor Dparm of lcrical, lcroic ad Ssms girig Naioal Uivrsi of Malasia 436 UKM-Bagi, Slagor, Malasia Khairul Auar Isha SPCTC (Spcrum Tcholog Laboraor Dparm of lcrical, lcroic ad Ssms girig Naioal Uivrsi of Malasia 436 UKM-Bagi, Slagor, Malasia Mohammad Suhaimi Ab-Rahma SPCTC (Spcrum Tcholog Laboraor Dparm of lcrical, lcroic ad Ssms girig Naioal Uivrsi of Malasia 436 UKM-Bagi, Slagor, Malasia

$34. Rama, F. Nasimi, K. A. Isha, M. S. Ab-Rahma Absrac Dscripio of igrad mod profil b drmi of κ, γ, δ paramrs as fucios of h propagaio cosa ( ad ffciv rfraciv id ( ff.$

2 34. Rama, F. Nasimi, K. A. Isha, M. S. Ab-Rahma Absrac Dscripio of igrad mod profil b drmi of κ, γ, δ paramrs as fucios of h propagaio cosa ( ad ffciv rfraciv id ( ff. Th profil ca b s from ( formula for ach guid T (Trasvrs lcric mods. Assumpios giv i his slab wavguid is usd for wavlgh (λ.55 μm, h hicss (d of h cor is.9 μm wih a p of smmric sp-id slab wavguid, rfraciv id of is 3.5 ad rfraciv id of is 3, also 3. Th rsuls of aalsis ar prsd i graphical form b combiig T mod, T mod ad T mod.. Kwords: Propagaio cosa, ffciv rfraciv id, slab wavguid, smmric wavguid. Iroducio Th aalsis of T mods ar sard wih h lcric fild polarid alog dircio for a smmric sp id slab wavguid. This calculaio is prformd o drmi h profil mod slab wavguids, ad prov h characrisics of h T mod ha > 3, 3 ad h umbr of frquc ormaliaio. A schmaic diagram of a modl for a smmric slab wavguid is show i Fig. Th rfraciv id idics of h guidig lar, subsra ad covr ar g, s ad c rspcivl. I s assumd ha h rfraciv id of h subsra is grar ha h covr. Fig. A schmaic of a smmric sp-id slab wavguid[]. Dpdig o whhr a oal iral rflcio occurs a h cor-subsra or/ad cor-covr irfacs, hr ar a las hr ps of mods ha ma b suppord b wavguid. Th ar guidd mods, subsra radiaio mods ad suprsra-covr radiaio mods as idicad i Fig blow.

3 Aalsis of T (rasvrs lcric mods 35 Fig. Th ra picur of mod o a-smmric sp id slab wavguid (a φ < θ c o-iral rflcio codiio, (b φ < θ c o-iral rflcio codiio(c. φ > θ c iral rflcio codiio. I Fig (a shows ha h ligh bams comig from subsra lar o guidig lar will occur ligh bams ha cam ou o h covr lar, his icid is ow as radiaio mods []. I Fig (b qual o (a, for his cas, h icid ligh agl vaishigl small, or br ow as la mods. I Fig (c shows ha h ligh bam oal iral rflcio occurs.t of scio. Basic Thor From Fig ad Fig 3, hr ar lcric fild ( ad magic fild (. Two fild of p ca prformd io wo quaios [], i ( ad i ( From Mawll quaio, ha; μ (3 quaio (3 ca b padd io [],

4 36. Rama, F. Nasimi, K. A. Isha, M. S. Ab-Rahma [ ] i i μ h, i i μ μ μ (4 Codiio for slab wavguid is ; hrfor quaio (4 bcoms [3], μ (5 μ (6 μ (7 As plaid i quaio (3, b usig Mawll quaio blow; ε ε (8 from quaio (8 abov, i is padd io, [ ] i i ε i i ε ε ε (9 h, i shows ha, ε (

5 Aalsis of T (rasvrs lcric mods 37 ε ( ε ( 3 T (Trasvrs lcric Mod Fig 3. T mod polariaio [3]. Assum ha is basd o phsical codiio [-4] ( ω (3 ( ω (4 ( ω (5 From quaios (3, (4 ad (5. Th ca b prformd usig diffrial quaio, ω ad ω ad, also ω ad Mai filds ha word i T mod ar,, ad fild. Thrfor, quaio (5, (7 ad ( ca b simplifid io, ωμ (6 ωμ (7 ωε (8 If quaios (6 ad (7 ar subsiud io quaio (8, h ca prformd [3],

6 38. Rama, F. Nasimi, K. A. Isha, M. S. Ab-Rahma ( (9 whr, ε μ ω ε μ ω ω c is fr spac wav umbr. is calld h propagaio cosa. is calld marial rfraciv id. Soluio of diffrial quaio ord- of quaio (9 is ( or ( ( B A cos cos ( i. For ara ff or claddig (suprsra [4-9]: quaio ( ca b chagd o b: from phsical bhavior is ow ha ; for. So, < Soluio of quaio ca b δ ( whr, δ (3 δ is a posiiv ral umbr. ii. For ara ff or guidig (cor [4-9] : quaio ( ca b chagd o b: or si cos B A Soluio of quaio abov o b, κ κ (4 or si( cos( B A κ κ (5

7 Aalsis of T (rasvrs lcric mods 39 whr κ is a ral umbr. (6 κ iii. For ara ff 3 or subsra [4-9]: quaio ( ca chagd o b: (7 from phsical bhavior is ow ha ; for. So, Soluio of quaio o b Whr, γ is a ral umbr. 3 < γ 3 (8 γ 3 (9 4 Calculaio ad Rsuls Th assumpio of his cas is h wavlgh (λ.55 m. Rfraciv id of guidd lar ( is 3.5 ad rfraciv id of subsra lar ( 3 ad covr lar ( ar 3.. I h Fig 4 shows ha h frquc of ormaliaio or V-paramr obaid is [7-] d V π (3 λ V 3.89 Th valu abov is obaid from d.45. I Fig 4 blow, V - valu of is show o h dashd li.

8 33. Rama, F. Nasimi, K. A. Isha, M. S. Ab-Rahma Fig 4. Characrisic quaio diagram T Mods. I h Fig 4 abov showd ha h solid li rprs h graph of h v - T mods ad h dash-do li rprs h graph of h odd - T mods []. Basd o h Fig 4, h firs cofid mod is idifid o b a h valu of κ d.98 whil h scod cofid mod is idifid o b i h rag of < κ d Basicall a h spcific valu of cofid mod ( κ d, h paramrs of h quaio could b dfid b wih h valu blow usig h abov. Tabl. Cofid mod calculaio. κ d V-paramr v T mods odd T mods

9 Aalsis of T (rasvrs lcric mods 33 Fig 5. Mod profil for T, T da T. I h Fig 5 shows h mod profil i h slab wavguid. Profil is obaid from h quaio ( o h ordia ais ad h wavguid lar (subsra, (guidd ad 3 (covr o h abscissa ais. T valus ha mus b m ar: a ( κd V κd whr κd is.3, h h agl of κ d is.98. κd.66. d (3 κd Th rsuls abov will b usd o drmi h propagaio cosas, aml: π κd (3 λ d

10 33. Rama, F. Nasimi, K. A. Isha, M. S. Ab-Rahma π γ (33 λ quaio (3 ad (33 will ild a valu of ad γ From, quaio (34, will g h valu of ffciv rfraciv id ( λ ff, (34 π ff, For T valus ha mus b m ar: co ( κd V κd ff, (35 κd Sam as h abov cas κ d., h h agl of κ d is.347. For h valu 5. 5, 3. 95, 9 γ 5. ad 3. ff, 55. For T valus ha mus b m ar : a ( d V d (36 d for for κ d 3., h h agl of κ d Thrfor 7.53,. 94, 89 γ. ad 3., 8. ff 5 Coclusio W foud ha h mod profils is show b T T ad T. V-paramr or ormalid frquc is Boudar codiio of mod valu o h ach lar ar -.9 < -.45 for subsra lar, for guidd lar ad.45 < 3.9 for covr lar. T T ad T as h mod profil ha was calculad. Simulad quaiaio valu is.. ffciv rfracio id of marial o subsra lar ( ff, is for T, ffciv rfraciv id o guidd lar ( ff, is for T ad ffciv rfraciv id o covr lar is 3. 8 for T. ( ff, Acowldgms This wor is sposord b Rsarch Uivrsi Gra from Uivrsii Kbagsaa Malasia, Bagi, Slagor Darul hsa wih cod umbr UKM-GUP R auhor wish o ha Assoc. Prof. Dr. Ahiruddi Maddu from Dparm Phsic from Bogor Agriculur of Isiu ad Dr. Ar Sahriar DIC from h

11 Aalsis of T (rasvrs lcric mods 333 Idosia Agc for Th Assssm ad Applicaio of Tcholog for hir suppor ad couragm. Rfrcs [] Chri A. A iroducio o opical fibrs: McGraw-ill, 983. [] Kisr G. Opical Fibr Commuicaios: McGraw-ill Compais,. [3] L D L. lcromagic pricipls of igrad opics: Wil, 986. [4] Sdr A W. ad Lov J. Opical wavguid hor. Nw Yor: Sprigr Vrlag, 7. [5] Sms R R A. ad Cos J R. Opical guidd wavs ad dvics: McGraw-ill, 99. [6] Kogli. Thor of dilcric wavguids. i Igrad opics, T. Tamir, d., d Brli: Sprigr, 979. [7] Kasap S O. Opolcroics ad phooics: pricipls ad pracics: Pric all,. [8] Khorasai S. ad Rashidia B,. Guidd ligh propagaio i dilcric slab wavguid wih coducig irfacs. J. Op. A. Pur Appl. Op. 3: pp [9] Yariv A. Opical lcroics i modr commuicaios. Oford: Oford Uivrsi Prss, 997. [] Adams M J. A iroducio o opical wavguids. Nw Yor: Joh Wil & Sos, 98. [] Calvo M L. ad Lashmiaraaa V. Opical wavguids: from hor o applid chologis: CRC Prss, 7. [] Iiua K. lms of Phooics: I fr spac ad spcial mdia: Wil-Irscic,. A.. Slab Wavguid Aalsis Appdi h W assumd ha claddig >> a, wih ad 3 is h widh ad lgh of h slab-wavguid. Th wo codiios for wav o propaga ar :. γ > shows ha wav propaga hrough h cor.. γ < shows ha hr is o wav propagaio hrough h claddig.

12 334. Rama, F. Nasimi, K. A. Isha, M. S. Ab-Rahma Coiui boudar codiio, ˆ ˆ ˆ ˆ (a. ˆ ˆ ˆ ˆ (a. wih h im ad 3 dpdc ( ω 3 (a. 3 Th compo is obaid as soluio of h rduc wav quaio [3] whr, a (a. 4 >, Acos( a B ( a (a. 5 a si a < For T mod ca b wri wav quaio, a, A (a. 6 ( ( [ ( ω ˆ ] 3,,,, (a. 7 3 i dircio of is ulimid uiform valu, is ol var wih ad is prssd as; γ ( (a. 8 hrfor h soluio of ig valu ca b wri as ; cor : ( Acos( γ Bsi( γ, a a (a. 9 claddig: ( ( α c a (a. ( ( α d a (a. wih, cor : γ ω με (a. Claddig: α ω μ ε (a. 3 Boudar codiio a Coiui quaio is ( (, h Acos ( γa B si( γa c ( αa (a. 4 if h γa Boudar codiio a Coiui quaio is (, h ( ( ( (a. 5 si ( γa γb cos( γa α c ( αa (a. 6

13 Aalsis of T (rasvrs lcric mods 335 if A cos ( γa B si( γa d ( αa (a. 7 ( ( h γasi ( γa γb cos( γa α d ( αa (a. 8 Subsiu q (a. 4 ad (a. 7, Acos ( γa B si( γa c ( αa Acos ( γa B si( γa d ( αa Addig abov quaio, Acos( γa ( c d ( α a (a. 9 Subsiu q (a. 6 ad (a. 8, γasi ( γa γb cos( γa α c ( αa γasi ( γa γb cos( γa α d ( αa Subracig abov quaio, γasi( γa α (c d ( αa (a. W ca divid q (a. ad (a.9 ( ( αa γasi γa α (c d A ( a ( c d ( α cos γ a (a. h, α a ( γ a (a. h whr a I compl quaio ca b wri, a h ( ff ff (a. 3 ff usig h umrical mhod of h quaio abov, h ffciv valu of rfraciv id could b drmid. A.. MATLAB Programmig. %%%%%%%%%%%%%%%%%%%%%%%% %Biscio program fucio bisc(f,h,a,b ol.; fa fval (f, h, a; fb fval (f, h, b; if (ol < fprif( ol should b posiiv umbr\ ; rur d %%%%%%%%%%%%%%%%%% γ

14 336. Rama, F. Nasimi, K. A. Isha, M. S. Ab-Rahma if (fa*fb > fprif( Ipu a ad b ar ou of irval\ ; ls whil if (abs (b-a < ol bra d c (ab/; fc fval(f,h,c; %%%%%%%%%%%%%%%%%% If (ca cb fprif ( maimum possibl prcissio achivd\ ; bra d %%%%%%%%%%%%%%%%%% if (fa*fc > a c; fa fc; ls b c; fb fc; d d fprif ( Nffciv valu %8.9f\, b; d %%%%%%%%%%%%%%%%%%%%%%% %cuio program fucio f(h/ a((h*pi/.55*sqr(.468^-.^ sqr(.^-.458^/(sqr(.468^-.^; %%%%%%%%%%%%%%%%%%%%%%%% Rcivd: Spmbr,

Numerical Simulation for the 2-D Heat Equation with Derivative Boundary Conditions

Numerical Simulation for the 2-D Heat Equation with Derivative Boundary Conditions IOSR Joural of Applid Chmisr IOSR-JAC -ISSN: 78-576.Volum 9 Issu 8 Vr. I Aug. 6 PP 4-8 www.iosrjourals.org Numrical Simulaio for h - Ha Equaio wih rivaiv Boudar Codiios Ima. I. Gorial parm of Mahmaics