STRESS ANALYSIS OF POLARIZATION MAINTAINING OPTICAL FIBERS BY THE FINITE ELEMENT METHOD

Transcription

1 IIUM Engineering Journal, Vol. 1, No. 1, Januar STRESS ANALYSIS OF POLARIZATION MAINTAINING OPTICAL FIBERS BY THE FINITE ELEMENT METHOD M. H. Al Facul of Engineering, Universi of Aleandria, Aleandria 1, Egp, A. S. Faraha, M. S. Helmi and M. Farhoud Facul of Science, Universi of Aleandria, Aleandria, Egp Absrac: Sress-induced birefringence in single mode polarizaion mainaining opical fibers has been invesigaed using he finie elemen mehod. The modal birefringence caused b eernal forces in he Panda and he Side Tunnel fibers are calculaed. I is found ha he modal birefringence is direcl proporional o he radial disance from he fiber cener. As epeced, he modal birefringence vanishes wih he variaion in he magniude of he applied eernal loads. Ke Words: Birefringence, Polarizaion, Panda Fiber, Side-Pi Fiber, Finie Elemen Mehod. 1. INTRODUCTION Polarizaion mainaining (PM) opical fibers are of grea imporance in opical coheren communicaions and some opical fiber sensor applicaions [1]. The fibers ha preserve he sae of polarizaion of he guided ligh can be achieved b desroing he circular smmer of he refracive inde disribuion in he core region. However, he birefringence properies of his kind of fiber are much more limied when compared wih wha can be achieved wih sress induced birefringence. Recenl, a varie of srucures has been proposed for he polarizaion mainaining fibers wih sress induced birefringence, mos of which are designed o mainain a differenial sress in he fiber core region [1]. There are man such successful fiber srucure designs, such as Panda, Bow Tie, Side Tunnel and Side Pi fibers [1]. Shih [] described a new fiber manufacuring echnique called preform deformaion. Using his echnique, a new kind of PM fiber wih a circular core, an ellipical sress appling cladding, and an ellipical jacke can be made. This has been sudied eperimenall b D. Marcuse e al. [3]. In our work, full advanage of he finie elemen mehod is aken o develop a numerical sud of he birefringence due o he effecs of eernal sresses in boh Side Pi and he Side Tunnel fibers. The birefringence PM fibers manufacured hrough preform deformaion are analzed. From he calculaed sress disribuion in he fiber, he refracive inde ensor change is obained. The finie elemen mehod is an approimae reamen of phsical problems, defined, for eample, b differenial equaions wih boundar condiions, or b variaion principles. Compared o oher approimae mehods i is successful when he domain of he problem has a complicaed shape or when he funcion involved behaves differenl in differen pars of he domain. The domain is represened approimael b a collecion of finie numbers of conneced subdomains of simple shape (e.g., riangles), called finie elemens. The finie elemen mehod will o be applied in his paper for he deerminaion of he sress disribuion and he modal birefringence, due o he effecs of eernal forces acing on he fiber cross secion, for he Panda and he Side Tunnel pe fibers. The fiber cross secion is firs divided ino elemens. In his research, a riangular elemen is used. A linear ssem of equaions can be esablished according o he principles of he finie elemen mehod, and hen he sress disribuion and, furher, he refracive inde ensor disribuion ma be calculaed in each elemen. The deails of he use of he finie elemen mehod can be found in he lieraure [-].. FORMULATION OF THE PROBLEM BY THE FINITE ELEMENT METHOD.1 Mari Soluion Techniques Solving he waveguide problem b he finie elemen mehod, he ke facor affecing sorage requiremens and compuaional effors is he choice of he algorihm o solve he mari equaion. The advanage of higherorder basis funcions for he fields is ha he give a more accurae soluion. Bu i involves an increased programming effor, paricularl when considering anisoropic maerials, in he finie elemen, and penal funcions [6]. Anoher advanage of he higher order polnomials (for given mari order) is ha i increases he densi of he mari. I should be noed he radeoff opimum choice beween low and high order polnomials depends on he mari algorihm used.. Principle of Minimum Poenial Energ The basic principle of he finie elemen mehod is ha a coninuum (he oal srucure) can be modeled analicall b is subdivision ino regions. The principle of minimum poenial energ can be used following Ref. [1]. Among all displacemens of admissible form, hose ha saisf he equilibrium 7

2 IIUM Engineering Journal, Vol. 1, No. 1, Januar condiions make he poenial energ assume a saionar (minimum) value. The oal poenial energ of a srucure can be epressed as a srain energ, U, plus he poenial energ, V, of he applied loads, i.e., U V (1) The srain energ U of a complee srucure consising of N elemens is simpl he sum of he N elemen srain energies e e e e u K u u e f N N e 1 U U, () T e1 e1 where he superscrip is he operaor o raznspose he vecor or he mari, and {u e }, [K e ], and {f T e } are he vecor of nodal poins displacemens, he siffness mari, and he iniial force vecor in he eh elemen, respecivel. Equaion () is rearranged o consruc he global equaion as: 1 U u K u u f, (3) T where {u}, [K], and {f T } denoe he global nodal poin displacemen vecor, he global siffness mari and he global iniial force vecor, respecivel. The poenial energ, V, due o eernal load is epressed b: V u f, () L where {f L } is he global load vecor acing on each nodal poin. Insering epressions for U and V from Eq. (3) and Eq. () ino Eq. (1), one can ge: 1 u K u u f u T f L, () Appling o his he necessar condiion for minimum energ, i.e., Ku ft fl, (6) u K u ft fl (7) This equaion gives he displacemen a all nodal poins of he fiber under eernal forces. Once he global nodal poin displacemen vecor, {u}, is deermined, sress in each elemen is given b [1] : e D e B e u e o e. () wih e = 1,,3...N, where [D e ], [B e ], and{ o } are he maerial siffness mari, he mari relaing nodal displacemens o srain field, and he iniial srain vecor in he eh elemen, respecivel. 3. DERIVATION OF STRAIN ENERGY As in he case of opical fibers, when he dimension of he srucure in one direcion (he z-direcion) is ver large in comparison wih he oher wo ransverse direcions (- and -) and he applied forces ac in he - plane and do no var in he z-direcion, he problem becomes a plane srain problem [1]. In he plane srain problem, he longiudinal srain z is zero, as are he shear srains z and z. Inroducing he condiion on z ino he relevan srain-sress equaion we ge: 1 1 E T 1 E 1 E E T (9) (1) E, (11) 1 where, and, denoe he normal sresses and srains, respecivel, and, and are he shear sress and srain, respecivel, E is he elasic modulus, is Poisson s raio, is he hermal epansion coefficien, and T is he emperaure change (negaive on cooling). These equaions are epressed in he form: D o, (1) where {}and {} denoe he sress and srain vecors, respecivel, defined as: and o 1, (13), (1) denoes he iniial srain, where: 1 T 1, (1) and [D] is he maerial siffness mari given b [1] : 1,,, E D, 1,,. (16) 1 1 1,, The srain energ in plain srain is given b [1] : L U o dd, (17) S where L is he lengh of he opical fiber and he inegral is carried ou over an region S under consideraion. In

3 IIUM Engineering Journal, Vol. 1, No. 1, Januar his paper a wo-dimensional riangular elemen is used o deermine he variaion of sresses in opical fibers. The formulaion of his elemen is well developed and described in man references [6]. The e h elemen siffness mari and iniial force vecor are: e e e e K L S B D B, (1) e e e e S B D o L T f, (19) L is he fiber lengh and [k e ] and {f T e } denoe he elemen siffness mari and he elemen iniial force vecor, respecivel.. ACCURACY OF THE FINITE ELEMENT ANALYSIS Before conducing he sress analsis of noncircular core fibers, he accurac of he finie elemen analsis iself is firs invesigaed. We can esimae he amoun of error b comparing he finie elemen soluion wih he analical soluions ha show he same behavior. 1. Thermal sress: This is he problem of hermal sress developing in long concenric clinders of differen maerials, which are joined ogeher a some iniial emperaure and heir emperaure is hen changed. This has been reaed b B.M. Azizur e al [7]. Sress disribuions in he clinder are given as: r b a = a b = b b a a b b E T, ( r a) () T b E 1, ( r b) (1) r 1 E T, ( r a) () T b E 1, ( r b) (3) r where a and b are respecivel he inner and ouer radii of he concenric clinder, E,, and T are elasic modules, Poisson s raio and emperaure change in he medium, and 1 and are he hermal epansion coefficiens of he mediums 1 and, respecivel. - Sress b loads: Sress disribuions on he -ais induced b he force W o acing on he clinder along he verical - ais direcion of he clinder are epressed as: Wo b b 1, () b Wo b b 1, () b 3- Sress fields b finie elemen analsis: One-quarer of he enire cross secion is covered b elemens, since he opical fiber under consideraion has wo smmer lines, one along he -ais and he oher along he -ais. Tpicall, we used 1 elemens and 93 nodal poins in his secion. The error in he resuls obained b he finie elemen analsis is less han percen of he analical resuls [1].. RESULTS AND DISCUSSION A compuer program coded in MATLAB is used for he analsis of he birefringence properies of opical fibers using he finie elemen mehod. The srucure of he Side Tunnel and Panda fibers is shown in Figs. 1 and. The fiber parameers used in his sud are shown in Table 1. Taking ino consideraion ha, in he Side Tunnel fiber, d is he disance from he neares edge of SAZ o he core cener and all he refracive indices n 1, n, and n 3 have no effec on he srain. Fig. 1 Side Tunnel fiber d 1 n a n 1 n 3 d Fig. Panda fiber Figure 3 shows he sress disribuion and on he -ais of he Panda pe fiber when force is applied along he -ais. I can be observed from he curve ha sresses d a b n n 3 c b Table I Parameers of he Panda and he Side Tunnel fibers 9

4 IIUM Engineering Journal, Vol. 1, No. 1, Januar Side Tunnel Panda fiber fiber a (m) ~ 1 1 b (m) ( o C -1 ) ( o C -1 ) 3 ( o C -1 ) d/c a/d.1. ~. - - E (kg /mm ) c (m) - 3 d 1 (m) - d (m) - 3 are nearl consan in he core region of he fiber. For he clad region beween he core region and he SAZ region, he firs clad region, sresses decrease graduall. Reaching he SAZ, i is observed ha he sresses nearl equal zero. The behavior of he sresses in he core and he firs clad region can be eplained b he resulan sresses because eernal and inernal sresses eising a hese regions are no equal and he ne sress varies uniforml as we proceed along he fiber cross secion. Afer his region, he sresses are nearl equal resuling in a nearl zero sress. Sress (kg/mm ) ( m) Fig. 3 Sress disribuion on he - ais when force is applied along - ais in he Panda fiber. Figure shows he sress disribuion on he -ais when force is applied along he -ais in he Panda fiber. One can observe he same behavior as in Fig. 3. As we reach he clad region, one observes ha sresses decrease graduall. The decremen of he sresses along he -ais can be eplained b he decremen of he eernal sresses because he area is affeced b he eernal normal force along he ais ha increases when proceeding along he ais. The applied load, W o, in boh Figs. 3 and is. kg/cm. Based on Ref. [] and sresses shown in Fig. 3, he sress-induced modal birefringence in he Panda pe fiber when force is applied along he -ais direcion is shown in Fig.. I is observed from he figure ha he modal birefringence is direcl proporional o he radial disance from he fiber cener. This is due o he effec of he sresses a his region. The value of he birefringence also depends on he magniude of he eernal applied forces. Similarl, he sress-induced modal birefringence when force is applied along he - ais is shown in Fig. 6. The figure shows nearl he same behavior as in Fig.. I is clear ha he birefringence a his applied load equals zero a he core region. Changing he applied load, W o, he sress difference, -, is calculaed when he force is applied in - and - direcions, respecivel. From his difference, he sress-induced modal birefringence is obained. Sress (kg/mm ) Modal Birefringence, B ( m ) Fig. Sress disribuion on he - ais when force is applied along - ais in he Panda fiber (m) Fig. Modal birefringence in he Panda fiber when force is applied along -ais. 1

5 IIUM Engineering Journal, Vol. 1, No. 1, Januar Modal Birefringence, B (m) Fig. 6 Modal birefringence in he Panda fiber when force is applied along -ais direcion. Sress difference, - (kg/mm ) Fig. 9 Sress disribuion in he Panda fiber when force is applied along -ais. The sress difference in he Panda fiber when force is applied along -ais is shown in Fig. 7, from which i is noed ha he sress difference decreases wih he applied loads on he fiber. This means ha he sress along he -ais is greaer han he sress along he - ais. Figure shows he modal birefringence in he Panda fiber when force is applied along he -ais. I is clear ha he modal birefringence decreases wih he applied loads on he fiber. Sress difference- (kg/mm ) Fig. 7 Sress difference in he Panda fiber when force is applied along he -ais. Modal birefringenc, B( 1 ) Fig. Modal birefringence in he Panda fiber when force is applied along -ais. Figure 9 shows he sress disribuion in he Panda fiber when force is applied along he -ais. A direc proporionali beween he applied loads, W o, and he sress difference is noiced. Figure 1 shows he modal birefringence in he Panda fiber when force is applied along he -ais, where an increase is noiced in he modal birefringence wih he applied loads. From hese resuls i is epeced ha we can conrol he magniude of he birefringence in he fiber b conrolling he eernal loads acing on he fiber. Figures 11 and 1 show he sress disribuion for he Side Tunnel fiber on he -ais and he -ais direcions. The wo figures show nearl he same behavior and he same order of magniude as he Panda fiber. This is epeced since he wo fibers are differen onl in he posiion of he SAZs and he parameers This is repeaed for he modal birefringence, Fig. 13 and 1. As he force is applied along he -direcion in he fiber, one observes ha he modal birefringence vanishes nearl a he limi of he fiber. I is also epeced ha he modal birefringence will vanish as he magniude of he eernal applied loads, W o, is varied. Similar o he Panda fiber, he behavior of he Side Tunnel fiber is shown in Figs. 1 o 1. Modal birefringence, B(1 ) Fig. 1 Modal birefringence in he Panda fiber when force is applied along -ais. 11

6 IIUM Engineering Journal, Vol. 1, No. 1, Januar. Sress, (kg/mm ) (m) Fig.11 Sress disribuion for he Side Tunnel fiber on he -ais when force is applied along -ais. Modal Birefringence, B (m) Fig. 1 Modal birefringence in he Side Tunnel fiber when force is applied along he -ais. Sress, (kg/mm ) (m) Fig. 1 Sress disribuion for he Side Tunnel fiber on he -ais when force is applied along -ais. Sress difference, - (kg/mm ) Fig. 1 Sress difference in he Side Tunnel fiber when force is applied along -ais. Modal Birefringence, B (m) Fig. 13 Modal birefringence in he Side Tunnel fiber when force is applied along he -ais. Modal Birefringence, B(1 ) Fig. 16 Modal birefringence in he Side Tunnel fiber when force is applied along - ais. 1

7 IIUM Engineering Journal, Vol. 1, No. 1, Januar Modal birefringence, B(1 ) Sress difference- (kg/mm ) Fig. 17 Sress difference in he Side Tunnel fiber when force is applied along -ais [] S. C. Chao, Eended Gaussian Approimaion for Single-Mode Graded Inde Fibers. IEEE J. Lighwave Technol., Vol. LT-1 (3), pp , 199. [3] D. Marcuse and C. Lin, Low Dispersion Single-Mode Fiber Transmission, IEEE J. Quanum Elecron, Vol. QE-17, No. 6, pp ,191. [] C. X. Sho and U R.Q. Hui, Polarizaion Coupling in Single-Mode Single-Polarizaion Fibers, Op. Le., Vol. 13, No. 1, pp , 19. [] P. K. Bachmann, Dieer Leers, Hermann Wehr and Erich R. Wehrhahn, Dispersion-Flaened Single Mode Fibers prepared wih PCVD: Performance, Limiaions, Design Opimizaion, IEEE J.Lighwave Technol., Vol. LT- (7), pp. -63, 193. [6] K. T. Bahe, Finie Elemen Procedures, Prenice Hall Inc., [7] B. M. A. Rahaman, A. Fernandez, and B. Davies, Review of Finie Elemen Mehods for Microwave and Opical Waveguides, IEEE J. Lighwave Technol., Vol. LT-13, pp. 1-16, [] K. Haaa and M. Koshiba, Sress-induced birefringence of Side-Tunnel Tpe Polarizaion- Mainaining Fiber, IEEE J. Lighwave Technol., Vol. LT-, W o (kg/cm) Fiġ 1 Modal birefringence in he Side Tunnel fiber when force is applied along he -ais. 6. CONCLUSION The finie elemen mehod has been applied for boh he Side Tunnel and Panda fibers o obain variaion in sresses and modal birefringence. I is found ha he modal birefringence is direcl proporional o radial disance from he fiber cener. I is also found ha as he force is applied, he modal birefringence vanishes nearl a he core limis of he fiber. REFERENCES [1] K. Okamoo, T. Hosaka and T. Edahiro, Sress Analsis of Opical Fibers b a Finie Elemen Mehod, IEEE J. Quanum Elecron, Vol. QE-17, No. 1, pp ,

8 IIUM Engineering Journal, Vol. 1, No. 1, Januar BIOGRAPHY Prof. Dr. Mousafa Hussein Al is currenl Professor of Engineering Phsics, Facul of Engineering, Universi of Aleandria, Aleandria, Egp. He was born in Aleandria in 193. He received his B.Sc. in 1976 in Communicaions and Elecrophsics, his M.Sc. in 193 and his Ph.D. in 197 in Engineering Phsics, all from Facul of Engineering, Universi of Aleandria, Egp. He is a member of he Opical Socie of America (OSA) and of he Egpian Socie of Solid Sae (ESSS). His area of ineres is Laser and Fiber Opics where he has abou publicaions. mosal@homail.com Mr. Ashraf M. S. Faraha is currenl a lecurer, Deparmen of Phsics, Facul of Science, Universi of Aleandria, Aleandria, Egp. He was born in Aleandria in 197. He received his B.Sc. in 199 and M.Sc. in 1999, boh in Phsics, Facul of Science, Universi of Aleandria, Egp. Dr. Maher Farhoud is currenl Associae Professor, Deparmen of Phsics, Facul of Science, Universi of Aleandria, Aleandria, Egp. He was born in Aleandria in 3/11/19. He received his B.Sc. in 19 and M.Sc. in 196, boh in Phsics, Facul of Science, Universi of Aleandria, Egp. He received his Ph.D. in Phsics, Adam Mickiwicz Universi, Poland. His area of ineres is Laser and Nonlinear Opics. mfarhoud@ahoo.com. 1