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IL NUOVO CIMENTO VOL. 112 B, N. 10 Ottobre 1997 NOTE BREVI

Investigation of polarization oscillations of soliton pulses

in lossy low birefringent single-mode optical fibers

M. F. MAHMOOD

Computational Science and Engineering Research Center, Howard University Washington, DC 20059, USA

(ricevuto il 13 Maggio 1997; approvato il 24 Luglio 1997)

Summary. — Nonlinear pulse propagation in a lossy low birefringent single-mode optical fiber is described using a variational technique in terms of an analytical model of coupled nonlinear Schroedinger equations and the frequency of relative oscilla-tions of soliton posioscilla-tions is obtained.

PACS 02.10 – Logic, set theory, and algebra.

There has been recently a great deal of interest in nonlinear pulse propagation in single-mode optical fibers. This is mainly because of the small core size of the fibers allowing strong nonlinear effects with modest input powers, well-characterized transverse profile of the beam, and long optical paths. Theoretical models describing soliton propagation in optical fibers are important both for applications and for fundamental physical interest.

The effect of birefringence on nonlinear codirectional propagation of solitons in single-mode fibers under various conditions has been taken into account and published in a number of papers [1-6] but, so far, no studies have been made accounting for the polarization oscillations and soliton stability in lossy low birefringent optical fibers.

In this note, a variational method is followed to investigate polarization oscillations in a lossy low birefringent optical fiber in terms of a theoretical model of coupled nonlinear Schroedinger equations with oscillating terms into the coupling of the two polarization modes. The polarization instability issues for the solitons will be considered in detail elsewhere.

A soliton propagating in a nonlinear lossy low birefringent optical fiber obeys the following coupled nonlinear Schrödinger equations [ 7 , 8 ]:

i(uz1 dut) 1 1 2utt1 (NuN 2 1 eNvN2) u 1 e 2v 2

u * exp [2iRdz] 42igu ,

(1a) i(vz1 dvt) 1 1 2vtt1 (NvN 2 1 eNuN2) v 1 e 2u 2

v * exp [iRdz] 42igv ,

(1b)

where e 42O3.

M.F.MAHMOOD 1426

u(z , t) and v(z , t) denote slowly varying wave envelopes of the fast and slow linearly

polarized modes, respectively, d is the normalized birefringence, R is the wave vector mismatch due to modal birefringence of the fiber, and g defines losses in the fiber.

The circularly polarized components of the field related to the linearly polarized modes can be conveniently analyzed through the following transformation:

u 4 1 k2 Be 2itz_{(p 1q) , v 4} 1 ik2 Be itz (p 2q) , where t 4 (1O4) Rd.

Thus, one can write eq. (1) into the form

i(pz1 dqt) 1tq1 1 2ptt1 p

g

NpN 2 1 e 12 2 e NqN 2

h

1 igp 4 0 , (2a) i(qz1 dqt) 1tp1 1 2qtt1 q

g

NqN 2 1 e 12 2 e NpN 2

h

1 igq 4 0 . (2b)

Since fiber losses lead to exponential decrease of NLS soliton amplitude, one can use the transformation p Kp 8 e2gz_{and q Kq 8 e}2gz_{to write eqs. (2) in the form}

i(p 8z1 dq 8t ) 1tq 81 1 2p 8tt1 p 8

g

Np 8 N 2 1 e 12 2 e Nq 8 N 2

h

_e22 gz 4 0 , (3a) i(q 8z1 dp 8t ) 1tp 81 1 2q 8tt1 q 8

g

Nq 8 N 2 1 e 12 2 e Np 8 N 2

h

_e22 gz 4 0 . (3b)

The dynamical behavior of localized solution p8 and q8 of eq. (3) using a variational method is based upon self-silimilar–like substitutions of the form [ 2 , 8 , 9 ]

p 84 2 h1

k1 1esech [ 2 h1(t 2z1) ] exp [ 2 iV1(t 2z1) 1iD1] , (4a)

q 84 2 h2

k1 1esech [ 2 h2(t 2z2) ] exp [ 2 iV2(t 2z2) 1iD2] , (4b)

with the evolution parameters hr, zr, Vr, Dr(r 41, 2 corresponds to p8 and q8 solitons, respectively) representing amplitude, central positon, velocity of soliton’s central position as it propagates along the fiber, and phase, respectively, into the Lagrangian

L 4



2Q Q

k

i 2(p 8z p 8*2p 8*z p 8)1 i 2 (q 8zq 8*2q 8*z q 8)1 (5) 1id 2 (p 8t q 8*2q 8*t p 8)1 id 2 (q 8t p 8*2p *8t q 8)1t(p 8* q 81p 8 q 8*)1 1e 12 2 e Np 8 N 2 Nq 8 N2e22 gz 2 1 2 Np 8t N 2 2 1 2Nq 8t N 2 1 1 2Np 8 N 4_e22 gz 1 1 2Nq 8 N 4_e22 gz

l

_{dt ,}

INVESTIGATION OF POLARIZATION OSCILLATIONS ETC. 1427

and in conjunction with Euler-Lagrange equations lead to the evolution of soliton parameters in terms of the following system of coupled ordinary differential equations (ODEs): d dz(hrVr) 4

g

1 1e 8

h

¯L1 ¯zr 1

g

1 1e 8

h

¯L2 ¯zr , (6) dzr dz 4 2 Vr, (7) dhr dz 4 (21 ) r

g

3 e 12 8 e

h

¯L2 dx , (8) dDr dz 4 2 V 2 r2 2 h2r1

g

8 e 3 e 12

h

h 2 r( 1 22gz)1

g

3 e 12 8 e

h

¯L1 ¯hr 1

g

3 e 12 8 e

h

¯L2 ¯hr , (9) where L14 16 e( 3 e 12) ( 3 e 12)2 h1h 2 2



2Q Q sech2_x 1sech2(x11 r) dx1, L24 8 e( 3 e 12) ( 3 e 12) th2cos x



2Q Q sech x1sech (x11 r) dx1, and x 42t(V12 V2) 22(V1z12 V2z2) 2 (D22 D1).

The above integrals are evaluated for nearly equal pulse amplitudes h1`h2`h ,

relative phase f 4D22 D1, and for relative distance between two polarization maxima

r (4 x22 x1).

One may obtain equations for the relative parameters defined for p8 and q8 solitons as follows: writing h124 h12 h2, so that using (8), one can obtain

dh12 dz 4 24 th

g

r sinh r

h

sin

g

V h r 1f

h

, or dh12 dz 4 24 th sin

g

V h r 1f

h

, (10) with rOsinh rB1.

Similarly, for V124 V12 V2and using (6), one can obtain

dV12 dz 4 2 64 15 (e 12) ( 3 e 12)rh 3 e22 gz 2 4 3thr cos

g

V h r 1f

h

. (11)

Also, from r 42h(z12 z2) and using (7), one can obtain

dr

dz 4 4 hV12, (12)

M.F.MAHMOOD 1428

which on differentiating with respect to z and on substituting the value of dV12Odz from

(11) becomes d2r dz2 4 24 h

k

64 15

g

e 12 3 e 12

h

rh 3 e22 gz 1 4 3 trh cos

g

V h r 1f

h

l

. (13)

The two solitons with a phase mismatch form a bound state provided

V

h r 1f4F46p .

Thus eq. (13) can be written as d2r dz2 1 (ge 22 gz 2 h) r(z) 4 0 , (14) with g 4 256 15

g

e 12 3 e 12

h

h 4 _and h 4 16 3 th 2_.

On solving eq. (14), one gets

r(z) 4

g

2 ge gz pkg

h

1 O2 cos

g

kg g

(

1 2gz1o(z 2 )

)

₂ p 2 kh g 2 p 4

h

, (15)

and the frequency, v, of relative oscillations of soliton positions is given by

v2 4 256 15

g

e 12 3 e 12

h

h 4_.

Finally, through an adiabatic approach using a time-averaged variational principle that employs nonlinear Schroedinger solitons with variable parameters used as trial functions we obtained the frequency of relative oscillations of soliton positions in a lossy low birefringent single-mode optical fiber.

* * *

This work was partially sponsored by Army High Performance Computing Research Center (AHPCRC) under the auspices of the Department of Army, Army Research Laboratory cooperative agreement number DAAH 04-95-2-0003Ocontract number DAAH 04-95-C-0008.

R E F E R E N C E S

[1] BLOWK. J., DORANN. J. and WOODD., Opt. Lett., 12 (1987) 202. [2] KIVSHARY. S., J. Opt. Soc. Am. B, 7 (1990) 2204.

[3] WRIGHTE. M., STEGEMANG. I. and WABNITZS., Phys. Rev. A, 40 (1989) 4455. [4] SOTO-CRESPOJ. M., AKHMEDIEVN. and ANKIEWIEZA., Phys. Rev. E, 51 (1995) 3547. [5] MAHMOODM. F., ZACHARYW. W. and GILLT. L., Opt. Fiber Tech., 2 (1996) 343. [6] MAHMOODM. F., ZACHARYW. W. and GILLT. L., Opt. Engin, 35 (1996) 1844. [7] MENYUKC. R., IEEE J. Quant. Electron., 25 (1989) 2674.

[8] MAHMOODM. F., ZACHARYW. W. and GILLT. L., Physica D, 90 (1996) 271. [9] MANAKOVS. V., Sov. Phys. JETP, 38 (1974) 248.