Axial Temperature Distribution in W-Tailored Optical Fibers

Transcription

1 Axial Temperature Distributi i W-Tailred Optical ibers Mhamed I. Shehata (m.ismail34@yah.cm), Mustafa H. Aly(drmsaly@gmail.cm) OSA Member, ad M. B. Saleh (Basheer@aast.edu) Arab Academy fr Sciece, Techlgy ad Maritime Trasprt, Alexadria, Egypt. Abstract- I the preset paper, the thermal effects i W-shaped cre refractive idex ptical fibers are ivestigated. The aim is t miimize its harmful effects the maximum axial temperature iside the fiber cre by selectig suitable values fr the affectig parameters. These iclude the idex tailrig parameters, cre radius, cre absrpti cefficiet ad itesity f light lauched t the fiber cre. Keywrds: Optical ibers, Thermal Effects, W-Tailred Idex ibers, Maximum Axial Temperature. I. INTRODUCTION Whe a laser beam is trasmitted thrugh a medium, a prti f its pwer is absrbed by the medium ad csequetly, heats that medium ad alters the temperature alg the laser path. The ew temperature distributis are resulted frm the chages that ccur i bth directi f prpagati ad rmal t it. The perati f creatig a ew temperature distributi is a trasiet prcess, but geerally, the steady state distributi is ccurred whe a frm f heat trasfer frm the medium balaces the absrbed laser pwer. The idea f sedig eergy thrugh a ptical fiber is t ew. I fact, fibers have bee widely used t guide laser eergy fr idustrial, medical, civil ad military applicatis. Whe a laser beam is trasmitted thrugh a medium, a prti f its pwer is absrbed by the medium ad csequetly, heats that medium ad alters the temperature alg the laser path. The ew temperature distributis are resulted frm the chages that ccur i bth directi f prpagati ad rmal t it. The perati f creatig a ew temperature distributi is a trasiet prcess, but geerally, the steady state distributi is ccurred whe a frm f heat trasfer frm the medium balaces the absrbed laser pwer. The idea f sedig eergy thrugh a ptical fiber is t ew. I fact, fibers have bee widely used t guide laser eergy fr idustrial, medical, civil ad military applicatis. Recet remarkable lss reducti i a ptical fiber imprves the pssibility f usig it i the field f cmmuicatis. Hwever, the eergy trasmissi capacity f a ptical fiber is defied by the iput pwer that raises the temperature f the fiber t the maximum acceptable level. Sigle mde (SM) ptical fibers are the mst suitable fr the high data-rate trasmissi systems due t the absece f the itermdal dispersi. Hwever, the sigle-clad sigle-mde fiber has sme prblems due t the difficulty i the excitati ad splicig because f the small cre radius [1-3]. These prblems ca be slved thrugh utilizig the siglemde graded-cre W-fibers that were firstly prpsed by Kawakami ad Nishida [4]. These W-type fibers have a larger cre tha that i a sigle-clad fiber ad it are als less sesitive t the lauchig agle, which ca icrease the splicig tleraces [3]. I this paper, the thermal effects are ivestigated i W-shaped ptical fibers carryig high pwer as i ptical amplifiers, sesrs ad medical applicatis. Secti 1 is a geeral itrducti, Sec. hadles the basic mdel ad aalysis, while Sec. 3 prcesses the btaied results with a geeral discussi ad Sec. 4 summarizes the majr cclusis. We preset a theretical treatmet, aided by Matlab prgram results, fr the axial temperature distributis i bi-quadratic idex ptical fiber. The axial temperature distributi depeds maily the laser ray trajectry. II. Basic Mdel ad Aalysis The sluti f the ray equati [5] is the rmalized radial psiti with respect t the cre radius [ (r/a)] as a fucti f the rmalized prpagati distace [ (z/a)], the fiber parameters ad the lauchig cditis. 1 N where r is the radial psiti, a is the fiber cre radius, z is the prpagati distace, N is the directi csie f the icidet ray, while is kw as the axial refractive idex at ay icidet pit. I additi, is the refractive idex prfile f the medium that ca be expressed fr W-tailred type as: ( ) α + α (1) 4 1 m () where α ad m are tailrig parameters. 80

2 The use f (m) 1/ ad (α) 1/ /N it Eq. (1) gives: This equati is slved umerically by a Matlab prgram, with iitial cditis (at 0.0): r z 0 C0 m (4) a ad ' d d 0 C1 dr dz z z 0 N m α (5) The maximum axial temperature, T max, is fud by slvig the eergy balace differetial equati [6, 7]: 1 T σi a ( ) ( ) exp( σ N ) k (6) where σ is the absrpti f the medium, T is the abslute temperature at the radial psiti,, k is the thermal cductivity f the fiber material ad is the ht spt parameter. Whe a laser beam is assumed t have a Gaussia distributi, the udistrted itesity ca be defied by: I I (, z) exp( σz ) (7) where I is the utput pwer itesity f laser surce. The udistrted itesity equati is subjected t the fllwig iitial cditis: z0 1.0 ad z (3) 0 (8) Algebraic maipulati f Eqs. (6-8) yields the fllwig differetial equati: d dz 3 z z z [ ] The ht the spt parameter,, ca be cmputed umerically by slvig Eq. (9). It ca als be defied thrugh the light itesity at ay psiti (, ) by: I (, ) I (, ) (9) () The abslute temperature, T, at a radial psiti,, ca be btaied by slvig the eergy balace differetial equati [8]: 1 T σi a ( ) k ( ) exp( σ N ) (11) where k is the thermal cductivity f the fiber material. The heat flux acrss the ht path-cld regi iterface is prprtial t the temperature differece betwee the surface ad the surrudig medium. By defiiti f thermal cductivity, k, ad the cvecti heat trasfer cefficiet, H, e ca write: k a T w H ( T max T ) where T is the ambiet abslute temperature. The use f Eqs. () ad (11) it Eq. (1) yields: T max where: k T + 0.5A (0.5 + ) Ha (1) (13) σia A exp( σan ) (14) k where I is the itesity f light lauched t the fiber cre. III. Results ad Discussi The described mdel is used i MATLAB ver. 7 t perfrm calculatis thrugh this paper. The btaied -, ig. 1, shws a peridic chage ig. 1 The variati f with at α 0.1, m 0.1, a 4 µm ad I 7 W/m. The prcedure is repeated fr ther values f the affectig parameters. T keep the W-shape f the cre refractive idex, the tailrig parameters are take i the rages: 0.1 α 0.9 ad 0.1 m 0.9. The cre radius is take as.0 a 4. µm ad the itesity as 1.0 I 9.0 W/m. The attempts yield the same behavir ad shw that the maximum value f varies accrdig t the values f the affectig parameters. 80

3 The btaied values f are used t get the maximum temperature, T max, Eq. (9), ad the results are displayed i igs. -8 fr differet values f the ctrllig parameters. alpha0.9 a4*^(-6) m^(-6) I7*^() w/m^ The effect f the ctrllig parameter α is first ivestigated ad the results btaied i igs. -6 shw that α has a egligible effect the value f the maximum axial temperature. 5 0 alpha0.1 a4*^(-6) m^(-6) I7*^() w/m^ 0 0 ig. 6 The variati f T max with at α ig. The variati f T max with at α a4*^(-6) m^(-6) I7*^() w/m^ ig. 3 The variati f T max with at α a4*^(-6) m^(-6) I7*^() w/m^ ig. 4 The variati f T max with at α 0.5 alpha0.7 a4*^(-6) m^(-6) I7*^() w/m^ 0 0 ig. 5 The variati f T max with at α 0.7 Thrugh these figures, the maximum temperature is C ad is btaied at , while the miimum temperature is 8.65 C ad is btaied at Bth values are i the allwable rage (belw C) iside the silica fibers [9]. Agai, the prcedure f calculatig the maximum axial temperature is repeated t study the effect f the secd ctrllig parameter, m, i the rage 0.1 m 0.9. Keepig the ther affectig parameters cstat, the btaied results are displayed i igs I5*^() w/m^ ig. 7 The variati f T max with at m 0.1 m0.3 I5*^() w/m^ 0 0 ig. 8 The variati f T max with at m I5*^() w/m^ 0 0 ig. 9 The variati f T max with at m

4 0 m0.7 I5*^() w/m^ 0 0 ig. The variati f T max with at m I3*^() w/m^ ig. 13 The variati f T max with at I 3 W/m m0.9 I5*^() w/m^ I5*^() w/m^ ig. 11 The variati f T max with at m 0.9 It is clear that the ctrllig parameter, m, has a appreciable effect the maximum axial temperature. It is a effective parameter that causes temperature iside ptical fiber t icrease. At m 0.1 t 0.5, the maximum axial temperature lies i the acceptable rage (belw C as metied fr the silica fibers). O the ther had, at m 0.7 t 0.9, the maximum axial temperature exceeds C ad therefre, these values fr the parameter m must be excluded fr the crrect fiber desig. ially, we study the effect f the utput pwer itesity f the light surce, I, which is lauched t the fiber cre. igures 1-16 depict the axial temperature distributi. As expected, the maximum axial temperature is strgly affected by the itesity. Therefre, e ca csider its effect as the mst imprtat ad effective e that raises temperature iside ptical fiber. Greater values f the itesity cause temperature values ver the accepted level ad therefre are rejected I1*^() w/m^ ig. 14 The variati f T max with at I 5 W/m I7*^() w/m^ ig. The variati f T max with at I 7 W/m I9*^() w/m^ ig. 16 The variati f T max with at I 9 W/m ig. 1 The variati f T max with at I 1 W/m 4. Cclusi Maximum axial temperature represets a critical factr i the thermal effect f ptical fibers. A quatitative study fr its value iside a W-shaped cre refractive idex fiber is carried ut. The maximum axial temperature was calculated fr 80

5 differet idepedet variables i differet rages. The affectig parameters iclude the cre radius, the tailrig parameters f the refractive idex ad the itesity f light lauched t the fiber cre. The btaied values f the maximum axial temperature, T max, are summarized i Table 1. m α a(µm) I (W/m ) T max ( C) [5] T.Okshi, Optical iber, Acad. Press Ic., New Yrk, 198. [6] W. Nwacki ad I.N Sedd, Thermdyamics i Slids, Spriger Verlag, New Yrk, [7] D. C. Smith," High-Pwer Laser Prpagati: Thermal Blmig," Prc. IEEE, vl.65, pp , [8] W. Nwacki ad I.N Sedd, Thermdyamics i Slids, Spriger Verlag, New Yrk, [9] S. Nemt ad K. Tetsusaki, "Eergy Trasmissi Capacity f Optical ibers Determied by a Temperature Rise," Appl. Opt., vl. 0,. 8, pp , Table 1 Maximum axial temperature at differet idepedet parameters. Based Ref. [9], the maximum axial temperature iside ptical fiber is t be acceptable belw C. S, the table ca easily be used t get the crrect desig parameters f the cre refractive idex ad cre radius fr the practical use f the W- tailred fiber. Agai, the table shws that the light itesity has the greatest effect the axial temperature. Refereces [1] W. Mu-Shiag, Mei-Hua Lee ad W-Hu Tsai, "Variatial Aalysis f Sigle Mde Graded-Cre W ibers," J. Lightwave Techl., vl.14,.1, pp.11-, [] A. Kumar, R. Chadra, R. A. Sammut ad A. K. Ghatak, "Cutff requecies f a Parablic-Cre W-Type iber," Electr. Lett., vl.14, pp , [3] M. Miyagi ad G. A. Yip,"Excitati ad Splicig f the Step-Idex W-iber," Caadia J. Phys., vl. 57, pp , [4] S. Kawakami ad S. Nishidi, "Amalus Dispersi f New Dubly Clad Optical iber," Electr. Lett., vl., pp.38-,