Numerical Analyses Optical Solitons in Dual

Transcription

1 ppied Mahemaics, 05, 6, Puished Onine Novemer 05 in SciRes. hp:// hp://dx.doi.org/0.436/am Numerica nayses Opica Soions in Dua Core Coupers wih err Law Nonineariy.. Qarni, M.. Banaja, H. O. Bakodah Deparmen of Mahemaics, Facuy of Science for Girs, Bisha Universiy, Bisha, Saudi raia Deparmen of Mahemaics, Facuy of Science for Girs, ing duaziz Universiy, Jeddah, Saudi raia Received 7 May 05; acceped 6 Novemer 05; puished 9 Novemer 05 Copyrigh 05 y auhors and Scienific Research Puishing Inc. This work is icensed under he Creaive Commons riuion Inernaiona License (CC BY). hp://creaivecommons.org/icenses/y/4.0/ srac In his paper, we presen he resus of numerica anaysis of opica soions in dua core coupers. We sudied he opica coupers as an appicaion for he non-inear Schrödinger equaion in he case of err aw for non-inear and carify he exac souion in his case. Then we have provided a numerica sudy of he effec of changing he consans in he form of he hree ypes of soions: righ soion and dark soions and singuar soion. eywords Opica Soion, Dua Core Coupers, err Law Nonineariy. Inroducion The propagaion soions hrough opica fiers have een a major area of sudy given is poenia appicaiiy in opica communicaion sysems. Severa effecs ha are presen in opica fiers and ampifiers imi he performance of opica sysem. Signa propagaion hrough opica fiers can e affeced y group veociy dispersion (GVD), poarizaion mode dispersion (PMD), and noninear effecs. The main noninear effecs ha arise in monomode fiers are Briion scaering, Raman scaering, and he err effec. The err effec of nonineariy is due o he dependence of he fier refracive index on he fied inensiy. The inensiy dependence of he refracive index eads o a arger numer of ineresing noninear effecs. Noae among hem, which have een sudied widey, are sef-phase moduaion (SPM) and cross phase moduaion (XPM). The propagaion soions hrough opica fiers have een we esaished ha his dynamics is descried, o firs approximaion, y he inegra noninear Schrodinger equaion (NLSE) []. The Noninear Schrodinger s Equaion pays a via roe in various areas of physica, ioogica, and engineering sciences. I appears in many appied fieds, incuding fuid dynamics, noninear opics, pasma physics, and proein chemisry. The NLSE ha is going o e sudied is giv- How o cie his paper: Qarni,.., Banaja, M.. and Bakodah, H.O. (05) Numerica nayses Opica Soions in Dua Core Coupers wih err Law Nonineariy. ppied Mahemaics, 6, hp://dx.doi.org/0.436/am

2 .. Qarni e a. en y iq + qxx + F q q = ( 0) () F is a rea-vaued ageraic funcion, q is he dependen variae, x and are he independen variaes. Equaion () is known o suppor soions or soion souions for various kinds of nonineariy. There are various kinds of nonineariies of he funcion F in () ha are known so far. = s, aso known as he kerr aw of nonineariy, is inegrae y he mehod of inverse scaering ransform (IST) []-[5]. The err aw of nonineariy originaes from he fac ha a igh wave in an opica fier faces noninear responses from nonharmonic moion of eecrons ound in moecues, caused y an exerna eecric fied. Even hough he noninear responses are exremey weak, heir effecs appear in various ways over ong disances of propagaion ha are measured in erms of igh waveengh. The specia case where F( s). Dua-Core Coupers Noninear coupers are very usefu devices ha disriue igh from a main fier ino one or more ranch fiers. Coupers aso have appicaions as inensiy dependen swiches and as imiers. Swiching is he process of energy redisriuion eween he cores for a given inpu. The proem of swiching, ahough invoved, can e accompished when he saiiy of soion saes is known. hough here has een a o of work in he area of opica coupers, our approach in his paper is going o presen he specia cases of he kerr-aw nonineariy. so, here has een a o of exac and approximae numerica sudies in he conex of opica coupers. Bu hese numerica echniques coud ge compuaionay inense [6] [7]. For Dua-core coupers, wave propagaion a reaivey high fied inensiies is descried y couped noninear equaions. In he dimensioness form, hey are ( ) ( ) iq + a q + F q q = r () xx ir + a r + F r r = q (3) xx Equaions () and (3) represen a generic mode o sudy he dynamics of opica soions hrough dua-core opica coupers. The firs erm in oh equaions represen inear empora evouion. The coefficiens of a for =, is he group veociy dispersion (GVD) whie represen non-err aw nonineariy, in genera. On he righ hand sides are he couping coefficiens. The dependen variaes q( x, ) and r( x, ) are he compex-vaued wave profies ha propagae hrough hese coupers. In his research, he focus is imied o soion signas. The funciona F represens non-err aw nonineariy, in genera. F s = s. The mode Equaions () and (3) reduces o For err aw nonineariy, ( ) xx iq + aq + q q = r (4) xx ir + ar + r r = q (5) For inegraiy aspecs of his couped equaions y ansaz mehod an assumpion of he foowing form is considered: ( ) P( x) φ(, ) q x, =, e i x (6) ( ) ( ) φ(, ) r x, = P x, e i x (7) where P ( x, ) ( = ; ) represens he ampiude componens of soion whie he phase componen ( x, ) φ( x, ) kx ω θ φ is = + + (8) 958

3 .. Qarni e a. In (6), is he frequency of he souion s whie ω represens he wave numer and θ is he phase consan. Susiuing (6) and (7) ino (4) and (5) and hen decomposing ino rea and imaginary pars give and P a P 0 + ak + F P P P = (9) x ( ω ) ( ) P P ak = 0 x Respecivey. From he imaginary par equaion i is possie o oain he speed (v) of he soion as, Since P( x, ) can e represened as P ( x v ) he ype of nonineariy and v is he speed of he soion. Now, equaing he wo vaues of he soion speed, from () eads o The speed of he soion herefore reduces o (0) v= a (), where he funcion g is he soion wave profie depending a = a () The couped NLSE for dua-core coupers given y (4) and (5) modifies o v = a (3) ( ) ( ) iq + a q + F q q = r (4) xx ir + a r + F r r = q (5) xx where a = a = a Consequeny, he rea par Equaion (9) reduces o P 3 a P ( ω + ak ) + P P = 0 x This equaion wi now e inegraed for hree ypes of soions. They are righ, dark and singuar soion souions. 3. Famiies of Soion Souions 3.. Brigh Soions For righ soions, one assumes [8] [9] where; (6) P P = sech τ; (7) ( v ) τ = B x (8) Here, represens he soion ampiude and B is he inverse widh of he soion. Susiuing (7) ino (6) gives: Baancing principe yieds So ha p ( ω+ ) sech τ + ( + ) 3 3 p P ( sech τ) sech τ 0 ak ap B a p B p + = p sech p+ (9) p + = 3p (0) = for =,. () 959

4 .. Qarni e a. Susiuing () ino (9) we ge: ( ) ( ) ω+ ak ab sechτ + a B sech τ + sechτ = 0 () 3 3 From coefficien sechτ ino () we ge From coefficien which poses he consrain 3 sech τ, we ge ω + + = 0 ak ab ω = a( B k ) a B 0 a = and herefore: and herefore: (3) B = (4) a > 0 for =,. (5) This means ha he GVD and err aw nonineariy mus ear he same sign for righ soions o exis. Nex, equaing he widh of he soions for = ; from (4) impy = nd herefore: which again shows he = (6) > 0 (7) This shows ha he nonineariy of he wo cores mus aso carry he same sign. Then, equaing he wave numers for he wo componens gives he reaion (3) we ge = and herefore: which again shows ha Finay equaing (6) and (8) eads o = (8) > 0 (9) = (30) which is he consrain condiion eween he given coefficiens ha mus hod for righ soions o exis. This eads o he righ -soion souion for dua coupers: ( ) = ( ) ( + ω + θ) q x, sech B x v e i kx (3) (, ) sech (? ) i( kx+ ω + θ) r x = B x v (3) which wi exis for he necessary consrains in pace. 3.. Dark Soions For dark soions, he saring hypohesis is given y [6] P = anh p τ (33) wih he definiion of τ eing he same in (8). However for dark soions he parameers parameers. Susiuing (33) and (8) ino (6) eads o p ( + + ) ( ) ak ap B a B p p ( ) P ω anh τ anh τ p+ 3 3p apb p + anh τ anh τ + anh τ = 0 p and B are free (34) 960

5 .. Qarni e a. Baancing principe yieds So ha Susiuing (36) ino (34) we ge: From coefficien p ( ) ( ) p + = 3p (35) = for =,. (36) ω+ ak ab anhτ + a B anh τ + anhτ = 0 (37) 3 3 anh τ we ge ω + ak ab + = 0 ω = a( B k ) and herefore: (38) From coefficien 3 anh τ we ge a B 0 = and herefore: which poses he consrain a B = (39) a > 0 for =,. (40) This means ha he GVD and err aw nonineariy mus ear he same sign for righ soions o exis. Nex, equaing he widh of he soions for = ; from (39) impied = and herefore: which again shows ha = (4) > 0 (4) This shows ha he nonineariy of he wo cores mus aso carry he same sign. Then, equaing he wave numers for he wo componens gives he reaion (38) we ge = nd herefore: which again shows ha Finay equaing (4) and (43) eads o = (43) > 0 (44) = (45) This gives dark -soion souion for dua-core coupers ( ) = ( ) ( + ω + θ) (46) q x, anh B x v e i kx ( ) = ( ) aong wih heir respecive consrains as indicaed. Noe: These waves known as check waves Singuar Soions ( + ω + θ) (47) r x, anh B x v e i kx For singuar soions, he saring hypohesis is given y [6] 96

6 .. Qarni e a. where τ is he same as in (8) whie he parameers (48) and (6) ino (8) gives Baancing principe yieds So ha ( ) ( ) P = csch P τ (48) and B are again free parameers. Upon susiuing ω+ ak ap B a B p p + csch τ csch τ + csch τ = 0 (49) p p p Susiuing (5) ino (49) we ge: From coefficien cschτ we ge From coefficien which poses he consrain p ( ) ( ) p + = 3p (50) = for =,. (5) ω+ ak ab cschτ + a B csch τ + cschτ = 0 (5) csch τ we ge ω + + = 0 ak ab ω = a( B k ) a B 0 = and herefore: a and herefore: (53) B = (54) a > 0 for =,. (55) This means ha he GVD and err aw nonineariy mus ear he same sign for righ soions o exis. Nex, equaing he widh of he soions for = ; from (54) impied = nd herefore: which again shows he. = (56) > 0 (57) This shows ha he nonineariy of he wo cores mus aso carry he same sign. Then, equaing he wave numers for he wo componens gives he reaion (53) we ge = nd herefore: which again shows ha Finay equaing (56) and (58) eads o. = (58) > 0 (59) = (60) These ead o singuar -soion souions in dua-core opica fiers wih err nonineariy given y ( ) = ( ) ( + ω + θ) (6) q x, csch B x v e i x ( ) = ( ) ( + ω + θ) (6) r x, csch B x v e i x which wi exis for he necessary consrains in pace. 96

7 .. Qarni e a. 4. Resus of Numerica nayses nayic process of he CNLS can ony e found under cerain specia seecions of parameers, i.e. cerain underying physica sysem. For oher cases, numerica anaysis is necessary for seeking he evouion of souions. I is essenia o seec a suiae numerica mehod for soving he equaions [0]. To sudy he effec of parameers on Soions we consider he foowing vaues []: Case : if hey 6 = = = = v = ak,,.003,,, a =, k =, = 6, ω = 3, B =, θ = π a Sudying he effec of changing (non-inear coefficien) in erms of so ake hree cases = 4, =, = 0 Figures -3 shows he effec of his change on Soion ampiude. Case : if hey 6 = = = = v = ak,,.003,,, =, k =, = 6, ω = 3, B =, θ = π 4 a Sudying he effec of changing a (veociy dispersion coefficien) on he wave form and he vaues a =, a =, a = Figure 4-6 shows he effec of his change on he widh Soion. Case 3: if hey 6 =, =, a =, =, v = ak, =, k =, = 6, ω = 3, B =, θ = π 4 a Sudying he effec of changing (Soion ampiude) on he wave form, and ha he vaues of = , =.003, =.006 Figures 7-9 shows he effec of his change on he Soion widh and Soion ampiude. 5. Concusions The sudy coupers dua core is considered one of modern opics of grea imporance in he fied of opica communicaion. In his research, he Schrodinger equaion is inear een sudying hese coupers in he case of kerr aw nonineariy and carifying he exac souion in his case. nd we sudied he effec of changing consans derived under he resricions menioned on Soion form in he case of Soions righ as in he case of a change (non-inear coefficien) impac on soion ampiude. When you changed a (veociy dispersion coefficien), we found ha he effec on Soion widh as we as when changing (Soion ampiude), he impac on he Soion widh and Soion ampiude. Simiary, hese changes have in he case of dark Soions and singuar Soions. The exension of his work can appy differen ypes of non-inear as we as increase he Peruraion erms o 963

8 .. Qarni e a. Figure. Brigh soions. Figure. Dark soions. Figure 3. Singuar soions. 964

9 .. Qarni e a. Figure 4. Brigh soions. Figure 5. Dark soions. Figure 6. Singuar soions. 965

10 .. Qarni e a. Figure 7. Brigh soions. Figure 8. Dark soions. Figure 9. Singuar soions. 966

11 .. Qarni e a. offse he Schrodinger non-inear. s ha finding, hose souions o some numerica mehods provide a o of add-ons in his sudy, and his is wha we hope impemened wih God s hep in he fuure hrough he Maser hesis. References [] Biswas,. (003) Theory of Opica Coupers. Opica and Quanum Eecronics, 35, -35. hp://dx.doi.org/0.03/: [] Zakharov, V.E. and Shaa,.B. (97) Exac Theory of Two-Dimensiona Sef-Focusing and One-Dimensiona Sef- Moduaion of Waves in Non-Linear Media. Journa of Experimena and Theoreica Physics, 34, 6. [3] owiz, M.J. and Segur, H. (98) Soions and H. Segur, Soions and he Inverse Scaering Transform. SIM, Phiadephia. [4] Mihaache, D. and Panoiu, N.C. (99) Exac Souions of Noninear Schrodinger Equaion for Posiive Group Veociy Dispersion. Journa of Mahemaica Physics, 33, 33. hp://dx.doi.org/0.063/ [5] Mihaache, D. and Panoiu, N.C. (993) nayic Mehod for Soving he Noninear Schrodinger Equaion Descriing Puse Propagaion in Dispersive Opic Fiers. Journa of Physics, 6, 679. hp://dx.doi.org/0.088/ /6//06 [6] Ramos, P.M. and Pavia, C.R. (000) Sef-Rouing Swiching of Soion-Like Puses in Muipe-Core Noninear Fier rrays. Journa of he Opica Sociey of merica B, 7, 5. [7] Biswas,. (00) Soions in Noninear Fier rrays. Journa of Eecromagneic Waves and ppicaions, 5, 89. hp://dx.doi.org/0.63/ x00 [8] Biswas,., hen,.r., Rahman,., Yidirim,., Haya, T. and dossary, O.M. (0) Journa of Opoeecronics and dvanced Maerias, 4, 57. [9] Sarescu, M., Bhrawy,.H., saery,.., Hia, E.M., han,.r., Mahmood, M.F. and Biswas,. (04) Opica Soions in Noninear Direciona Coupers wih Spaion-Tempora Dispersion. Journa of Modern Opics, 6, hp://dx.doi.org/0.080/ [0] Banaja, M.., khaee, S.., shaery,.., Hia, E.M., Bhrawy,.H., Moraru, L. and Biswas,. (04) Opica Soions in Dua-Core Coupers. Wufenia Journa,, [] Biswas,., Lo, D.., Suon, B., han,.r. and Mahmood, M.F. (03) Opica Gaussons in Noninear Direciona Coupers. Journa of Eecromagneic Waves and ppicaions, 7,