Dark solitons in optical fiber with inhomogeneous effects

Optik 122 (2011) 55–57

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Dark solitons in optical fiber with inhomogeneous effects M. Idrish Miah Department of Physics, University of Chittagong, Chittagong 4331, Bangladesh

a r t i c l e in fo

abstract

Article history: Received 30 April 2009 Accepted 27 September 2009

We study the nonlinear wave propagation in an inhomogeneous optical fiber core in the normal ¨ dispersive regime. In order to include the inhomogeneous physical effects, the nonlinear Schrodinger equation (NLSE), which governs the solitary pulse propagation in optical fiber, is modified by adding terms for phase modulation and power gain or loss. The modified NLSEs are bilinearized and exact dark soliton solutions are obtained. The results are discussed. & 2010 Elsevier GmbH. All rights reserved.

Keywords: Optical material Optical soliton ¨ Nonlinear Schrodinger equation Horota transformation Lax pair

1. Introduction In recent years there has been increased interest from different experimental and theoretical groups in the study of nonlinear optical systems, in particular systems or materials of self-guided optical beams that give rise to energy-localization effects with a relatively long lifetime due to their potential in applications, such as optical communications [1–8]. A typical example for such an effect is called ‘‘optical solitons’’. Solitons are a special breed of optical pulses that can propagate through an optical fiber undistorted over very long distances. The key to soliton formation in optical fiber is the counterbalance of the opposing forces of chromatic dispersion and self-phase modulation [9,10]. The ¨ nonlinear Schrodinger equation (NLSE) which governs this solitary pulse propagation in the fiber is usually given [11,12] by iuz sutt þ 2juj2 u ¼ 0;

ð1Þ

where u(z,t) is the envelope of the axial electric field (subscripts z and t denote partial derivatives with respect to these variables), s= sign(b) = 71 and b is the group velocity dispersion parameter related to the frequency dependence of group velocity and is positive (negative) for normal dispersion (anomalous dispersion). Considering homogeneous fiber core medium, Eq. (1) has widely been studied by numerous authors (e.g. [10–12]). However, in an actual fiber, the core medium is, in general, not uniform or homogeneous [13]. This inhomogeneity arises mainly due to imperfection or defects in fiber core medium and fluctuations in core/cladding radius. Therefore, the study of nonlinear wave propagation in inhomogeneous media is of great interest in the area of fiber optics communications.

E-mail address: m.miah@griffith.edu.au 0030-4026/$ - see front matter & 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.02.003

Recently, depending on the physical situations, several modifications to Eq. (1) has been suggested and the pulse propagation has been discussed (e.g. [3,11,14–16]). Of them, one of the important modifications, for example, was given by Flach and co-workers [3], who discussed resonant light scattering by solitary waves, by considering the process of light scattering by optical solitons in a planar waveguide with homogeneous and inhomogeneous refractive index cores. Considering certain inhomogeneous effect, nonlinear wave propagation in optical fiber has also been discussed in Ref. [17], where the author considered nonlinear compression of chirped solitary waves with and without phase modulation. In case of inhomogeneities in core medium, the NLSE (Eq. (1)) can be modified and be written in the general form iuz sutt þ 2juj2 u ¼ jðzÞt 2 u 7igðzÞu;

ð2Þ

where j is phase modulation and g is the fiber gain (loss) for g4 0 (g o0). Since s= sign(b) the solution of Eq. (2) depends on whether b is positive or negative. In both cases, the NLSE can be solved by the inverse scattering transform method. The pulse-like solutions are found to occur only in the case of anomalous dispersion (b o0), and are called ‘‘bright solitons’’. In the case of normal dispersion (b 40), the solitary wave solutions of Eq. (2) appear as a dip in a constant background, and are called ‘‘dark solitons’’. Eq. (2) contains arbitrary functions of z, so one needs to identify the integrability conditions of the equation through linear eigenvalue problem. The Lax pair assures the complete integrability condition of a nonlinear system of equations and has been used to achieve soliton solutions. In the previous work [18], we studied the wave propagation in a fiber with inhomogeneous core medium in the anomalous dispersive regime (b o0) and showed that the solitary wave propagation is maintained in the core. Here we exactly solve the NLSEs with additional terms for phase modulation and fiber gain or loss, i.e. modified NLSEs (Eq. (2)), in the normal dispersive (b 40) regime to

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M. Idrish Miah / Optik 122 (2011) 55–57

construct dark solitons. In solving the modified NLSEs, the Hirota transformation method has been used. The results show that the product of the depth and width of the soliton in either gain or loss case remains preserved during propagation in the fiber.

The modified NLSE with phase modulation and for b 40 and g4 0 reduces to ð3Þ

Before constructing the Lax pair for Eq. (3), we introduce a variable transformation as uðz; tÞ ¼ Uðz; tÞexpðigt 2 =2Þ:

ð4Þ

Using the above variable transformation in Eq. (3) the resulting equation is i

@U dg 2 U @2 U @U  t  2igUðg 2 jÞUt 2 ¼ 0: þ 2jUj2 U2igt @z dz 2 @t 2 @t

ð5Þ

The Lax pair associated with Eq. (5) is derived as @O ¼ PO @t

@O ¼ QO @z

O ¼ ðO1 O2 ÞT ;

ð6Þ

iD=2 iU 

iU

ð11Þ

c ¼ 1þ sc1 ;

ð12Þ

ðiDz D2t 2igtDt 2ig þ f Þðb0  1Þ ¼ 0;

!

iD=2

2

Q ¼ 2iD

0 B þB @

i=2

0

0

i=2

þ 2D

ijUj2 

@U  þ2igtU  @t

ð13Þ

Eqs. (13) are solved for f and the result gives Rz Rz f ¼ 8j210 expð4 0 g dzÞ, where b0 ¼ 2j10 expf 0 ðif þ 2gÞ dzg. Making use of Eqs. (10)–(12) and then collecting the terms with the same power of s, i.e. the coefficients of s and s2, we obtain the following equations: 9 > ðiDz þD2t 2igtDt Þðb1  1 þ1  c1 Þ ¼ 0 > > > > > 2 2 ðDt kÞð1  c1 þ c1  1Þ þ 4b1 jb0 j ¼ 0 = : ð14Þ 2 > ðiDz þDt 2igtDt Þðb1  c2 Þ ¼ 0 > > > 2 2 > > ðDt kÞð1  c1 þ c  1Þ þ2b1 jb0 j ¼ 0 ; We solve the set of Eqs. (14). In deriving the solutions for b1 and c1 from Eq. (14), we conveniently assume b1 ¼ e4j1 t þ d1 and b1 +c1 =0, where d1 is a constant. After some algebra, Eq. (7) yields

uðz; tÞ ¼ 2j1 ex1 þ igt

and !

f ¼ 2jb0 j2 ;

ð15Þ Uðz; tÞ ¼ 2j1 ex1 tan h g1 ; Rz 2 where x1 ¼ i 0 ð8j1 dz 7 pÞ and g1 ¼ 1=2ð4j1 t þ d1 Þ with R j1 ¼ j10 expð2 0z g dzÞ. Using variable transformation defined earlier (Eq. (4)), we get

where P¼

b ¼ b0 þ sb0 b1

where s is an expansion coefficient. Pluging b ¼ b0 þ sb0 b1 and c ¼ 1þ sc1 in Eq. (10) and collecting the coefficient of s0, we obtain

2. Hirota transformation and soliton solutions

iuz utt þ 2juj2 u ¼ jðzÞt 2 u þigðzÞu:

where f is an arbitrary function. Dark one-soliton solution (1SS) can be obtained by the following perturbation expansions of b and c:

iU

igt iU 

!

2

=2

tan h g1

ð16Þ

Eq. (16) is the exact dark 1SS (of Eq. (3)) with fiber gain and phase modulation, which is plotted in Fig. 1. Now we consider Eq. (2) with phase modulation and fiber loss (g o0) in the normal dispersive regime (b 40). Proceeding as before, it is found that the appropriate equation for the system below is integrable with condition dg=dz2ðjg 2 Þ ¼ 0:

igt 1 @U  2igtU  C  @t C; A ijUj2

iuz utt þ2juj2 u ¼ jðzÞt 2 uigðzÞu

where U* is the complex conjugate of U and D is the variable Rz spectral parameter given by D ¼ D0 expð2 0 gdzÞ. The compatibility condition @P=@z@Q =@t þ ½P; Q  ¼ 0 gives @U @2 U @U  þ 2igU: ð7Þ þ 2jUj2 U ¼ 2igt @z @t 2 @t By comparing Eqs. (5) and (7), we find that Eq. (3) is completely integrable with the integrability condition dg=dz þ2ðjg 2 Þ ¼ 0 and it thus gives exact solutions. Now we introduce the Hirota derivative operators. A symbol Dx is called the Hirota derivative with respect to variable x and defined to act on a pair of functions p(x) and q(x) as follows:

i

ð17Þ

The 1SS of Eq. (17) is obtained by choosing the variable transformation uðz; tÞ ¼ Uðz; tÞ expðigt 2 =2Þ. Following the same procedure, we obtain ð18Þ Uðz; tÞ ¼ 2j2 ex2 tan h g2 ; Rz where x2 ¼ i 0 ð8j22 dz7 pÞ, g2 ¼ 2j2 t þ d2 , d2 is a constant, R j2 ¼ j20 expð2 0z g dzÞ and the eigenvalue problem associated with Eq. (17) is ! iD=2 iU P¼ iD=2 iU 

Dx pðxÞ:qðxÞ ¼ ð@x @x0 ÞpðxÞqðx0 Þjx0 ¼ x ; where qx denotes partial derivative with respect to x. Hirota derivatives are bilinear operators [10]. We define the bilinear operators for our system as m n n 0 0 Dm z Dt bðz; tÞ:cðz; tÞ ¼ ð@x @x0 Þ ð@t @t0 Þ bðz; tÞcðz ; t Þjx0 ¼ x;t 0 ¼ t

ð8Þ

with the transformation in the form U¼

b ; c

ð9Þ

where b(z, t) and c(z, t) are complex and real functions, respectively. Using the above transformation in Eq. (7), we obtain ðiDz D2t 2igtDt 2ig þ f Þðb  cÞ ¼ 0;

D2t ðc  cÞ ¼ 2b2 ;

ð10Þ

Fig. 1. Dark soliton (g 40) propagation. Pulse amplitude (or depth/width) profile: ju(z, t)j as a function of z and t (a 3D plot of Eq. (16) with j10 = 0.5, g = 0.1 z  1 and d1 = 0).

M. Idrish Miah / Optik 122 (2011) 55–57

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3. Conclusions

Fig. 2. Dark soliton (g o0) propagation: ju(z, t)j vs. z and t (Eq. (19) with j20 =0.5, g = 0.1 z  1 and d2 = 0).

2

Q ¼D

i=2

0

0

i=2

0 B þB @

! þD

ijUj2 

@U  @t

2igtU 

igt

iU

!

iU 

igt 1 @U þ 2igtU C  @t C A ijUj2

References

The dark soliton solution with fiber loss is then x2 igt2 =2Þ

uðz; tÞ ¼ 2j2 e

tan h g2

We have considered the more general case of inhomogeneities in fiber core medium. We have modified the NLSE in order to include phase modulation and fiber gain and loss or damping effects. The modified NLSE governs the nonlinear pulse propagation in an inhomogeneous fiber system with fiber gain (loss) where the effects due to fiber gain (loss) and chirping of the pulse exactly balance each other. The complete integrability conditions have been derived for both gain and loss cases and the modified NLSEs in the anomalous dispersive regime have been exactly solved to construct dark soliton solutions. We found that the depth of the soliton is increased (decreased) for gain (loss) as it propagates along the fiber. The pulse width was also found to be compressed (broadened) in propagation with fiber gain (loss) such that the area of the pulse envelope remains conserved, as found for bright solitons in the anomalous dispersive regime.

ð19Þ

We thus have obtained the exact dark 1SS for the wave propagation in the inhomogeneous optical fiber core with phase modulation and fiber loss. Eq. (19) is plotted in Fig. 2. As shown in Fig. 1, the depth of the dark soliton increases as z increases while the width decreases. This is due to the admittance of the terms j1 ðzÞ and b(z)40 in Eq. (16), where the soliton amplitude grows at the same rate as the respective power amplitude with the inclusion of phase modulation and gain. The same modulating term, as in Eq. (16) but with opposite sign, is also appeared in the soliton solution with fiber loss, i.e. in Eq. (19), where the depth of the soliton decreases as z increases while the width increase so that the amplitude decays at the same rate as the respective power amplitude. As a result of chirping, the pulse broadens as it propagates along the fiber, and as a result of damping, the amplitude or depth of the pulse reduces. The depth of the solitary pulse is found to increase or decrease, depending on the sign of g(z), i.e. on fiber gain or loss, with the same amount of broadening in the width during its propagation such that the product of the depth and width, i.e. the area remains constant. With the inclusion of phase modulation and fiber loss, Fig. 2 clearly depicts the damping effect in solitary pulse propagation in a fiber with inhomogeneous core. The results show that the area of the solitary pulse envelope for either gain or damping is preserved during propagation, as found for bright solitons in the anomalous dispersive regime shown earlier [18]. The results are also consistent with those obtained numerically in Ref. [17].

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