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Optical solitons and their shaping in a monomode optical fiber with some inhomogeneous dispersion profiles
Optik - International Journal for Light and Electron Optics 192 (2019) 162906
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Optical solitons and their shaping in a monomode optical fiber with some inhomogeneous dispersion profiles
T
⁎
N. Prathapa, S. Arunprakasha, M.S. Mani Rajanb, , M. Tantawyc a b c
Department of EEE, Anna University, Ramanathapuram 623513, India Department of Physics, Anna University, Ramanathapuram 623513, India Department of Basic Science, October 6 University, 6th of October City, Egypt
A R T IC LE I N F O
ABS TRA CT
Keywords: Optical soliton Exponential profiles Dispersion management Soliton control vc NLS Darboux transformation
In the real fiber optic communication system, the soliton transmission can be signified by nonlinear Schrödinger (NLS) equation with inhomogeneous coefficients. Here, the two-soliton propagation have been investigated by solving the inhomogeneous NLS equation with the process of Darboux transformation. According to obtained soliton solutions, variable coefficients are considered with different forms to study the features of solitons. Since soliton structures are mainly influenced by the controlling parameters, various soliton structures are obtained for specific choices of variable coefficients. Specifically, we adopt various exponential profiles for dispersion coefficient. Through properly manipulating exponential profiles, non-oscillating soliton amplification, dromion-like structure, bounded soliton and parallel soliton transmissions are explored. Results indicate that the characteristics of optical solitons are influenced by variable coefficients. In this work, we report the analytical study on transmission features of optical solitons in inhomogeneous optical fiber under various dispersion profiles. Obtained results are have some theoretical guidance for experimental research on optical amplifier, soliton based optical switches and achievement of parallel transmission of solitons.
1. Introduction In the context of rising traffic in the optical fiber communication links, researches on the production of ultra-short optical soliton have been attracted due to numerous applications [1–4]. Generally, soliton hold some potential values in various areas such as longdistance fiber optic communication systems, fiber lasers and optical switching devices [5–10]. Recently, studies of soliton shaping and management paid attention among researchers owing to potential applications in various nonlinear fiber systems [11–15]. In a real case, GVD, SPM and the gain or loss of fiber medium can be affected by the inhomogeneous nature of the fiber. Thus, nonlinearity, dispersion and gain/loss parameters of transmission line are varying along the propagation distance z. In recent years, the control and management of solitons in inhomogeneous fiber has been acknowledged wide attention due to their potential values in optical technologies [16–21]. Using various cosine and sine profiles, soliton shaping has been studied in inhomogeneous fiber system [22,23]. However, only few articles are reported on soliton structures for various exponential profiles in the inhomogeneous fiber. In this paper, to study the properties of soliton behavior in a real inhomogeneous fiber medium, analytic two soliton solutions are attained by employing Darboux transformation technique. Control parameters for the obtained soliton solutions will be tailoring to shaping the soliton. Here, we study the generalized inhomogeneous NLS equation with inhomogeneous
⁎
Corresponding author. E-mail address: [email protected] (M.S. Mani Rajan).
https://doi.org/10.1016/j.ijleo.2019.06.006 Received 2 June 2019; Accepted 3 June 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 192 (2019) 162906
N. Prathap, et al.
coefficients which governing the soliton transmission in an inhomogeneous fiber medium [24–27],
iqz +
1 i β (z ) qtt + γ (z )|q|2 q − G (z ) q = 0 2 2
(1)
where
W [γ (z ), β (z )] , β (z ) γ (z ) W [γ (z ), β (z )] = γβz − βγz G (z ) =
where q (z, t) is the temporal envelope of optical pulse and z is the propagation axis along the fiber. Inhomogeneous functions β(z) and γ(z) are related to dispersion and nonlinearity management profiles respectively and G(z) is implies the attenuation or amplification of the fiber medium. Moreover, Eq. (1) well describing the soliton propagation and shaping the soliton in inhomogeneous fiber medium. The inhomogeneous effects can be properly tuning which has the major impact on soliton’s shape. Thus, it is necessary to discuss the various soliton structures through soliton shaping in the inhomogeneous optical fiber. By setting control parameters, we examine the controllability of the optical solitons with inhomogeneous dispersion for Eq. (1). To our knowledge, soliton shaping through exponential dispersion profiles for Eq. (1) have not been constructed, which are the main tasks of our paper. 2. Lax pair for Eq. (1) With the aid of Ablowitz–Kaup–Newell–Segur technology [28], the Lax pair associated with Eq. (1) is derived. Through Lax pair, two soliton solutions are generated by employing Darboux transformation scheme. Hence, it is worthful to design the Lax pair for Eq. (1) to analyze the soliton management and shaping. The linear eigenvalue representation of Eq. (1) can be written as given below:
ψt = U ψ ψz = V ψ
(2)
Where ψ = (φ1, φ2 is an eigenfunction for Lax pair (3), φ1 and φ2 are both complex functions. The superscript T represents the transpose and 2 × 2 matrices U and V takes the forms as below.
)T
U = λJ + M V = i β (z ) λ2 J + i β (z ) λ P +
iT 2
β (z ) ⎛ 0 q ⎞ γ (z ) ⎝ q* 0 ⎠
J=
(10 −01), M =
T=
β (z ) γ (z ) qt ⎞ ⎛ γ (z ) q q* ⎜ β (z ) γ (z ) q * − γ (z ) q q* ⎟ t ⎝ ⎠
⎜
⎟
The compatibility condition Uz − Vt + [U , V ] = 0 gives rise to Eq. (1) and λ is known as isospectral parameter which is constant here. In the compatibility condition, brackets imply that the usual matrix commutator. 3. Darboux transformation Once Lax pair is known, via symbolic computation, Darboux Transformation can be applied for Eq. (1) to find soliton solutions. Consequently, Darboux transformation [29] has been employed to many nonlinear systems for obtaining multi-soliton solutions which can be used to understand the physical significance of considered nonlinear system. Also, the Darboux transformation formalism has been showed as an effective method to generate n-soliton solutions for many non-integrable and nonautonomous systems. In order to discuss the soliton shaping and management, we will determine multi soliton solutions for considered Eq. (1) with Darboux Transformation. Here we introduce the linear eigenfunction transformation as given
ψ [1] = (λI − S ) ψ
(3)
We can define S matrix as (4)
S = HΛH−1 Where I is the 2 × 2 identity matrix and S is a nonsingular matrix.
Λ = diag (Λ1 , Λ2 )
(5)
here H is the non-singular matrix, requiring
λ1 0 ⎞ ⎛ ϕ1 − ϕ2* ⎞ H=⎜ , Λ = ⎜⎛ ⎟ ϕ2 ϕ1* ⎟ ⎝ 0 − λ1* ⎠ ⎝ ⎠ 2
Optik - International Journal for Light and Electron Optics 192 (2019) 162906
N. Prathap, et al.
Ψt [1] = U1 Ψ [1] where U1 = λ J + P1 with
0 P1 = ⎜⎛ ∗ q1 ⎝
q1⎞ 0
⎟
⎠
We get the Darboux Transformation for Eq. (1) with following form, (6)
P1 = P + JS − SJ
It is easy to verify that if (φ1, φ2)T and (−φ2*, φ1*)T are solutions of Eq. (2) which are corresponds to the eigenvalue λ1 and − λ1* respectively. Hence the generalized form of Darboux transformation is presented for n-soliton solution through Eq. (6) as given below,
β (z ) γ (z )
qn = q + 2
∑
(λm + λm* ) ϕ1, m (λm) ϕ2,*m (λm) Am
(7)
Where
ϕk, m + 1 (λm + 1) = (λm + 1 + λm* ) ϕk, m (λm + 1) −
Bm (λm + λm* ) ϕk, m (λm) Am
Am = |ϕ1, m (λm)|2 + |ϕ2, m (λm)|2 Bm = ϕ1, m (λm + 1) ϕ1,*m (λm) + ϕ2, m (λm + 1) ϕ2,*m (λm) where k = 1, 2…n, m = 1, 2…n and (ϕ1,1 (λ1), ϕ2,1 (λ1))T is the eigenfunction of Eq. (2) corresponding to λ1. By setting q = 0 in Eq. (7), we can generate one-soliton solution for Eq. (1) by iterating DT once. Using the one-soliton solution as a seed solution in Eq. (7), we can derive the two-soliton solutions to study the interaction characteristics. Thus, one can generate up to n-soliton solution for Eq. (1). 4. Analytical multi-soliton solutions In ideal fiber, soliton cannot involve in the interaction with adjacent one in optical communication system. In real fiber, solitons will undergo interaction due to inhomogeneous nature of the fiber medium. Therefore, this is essential to study the propagation and interaction characteristics of two solitons for considered system (1). Through substituting the value of λ2 = α2+ i β2 and n = 2 in Eq. (7), we can generate the two-soliton solutions for Eq. (1) by repeating the Darboux transformation two times as follows
Q2 =
β (z ) γ (z )
A B
(8)
where
A = a1 Cosh(θ2) Exp (i φ1) + a2 Cosh(θ2) Exp (i φ2) + i a3 (Sinh(θ2) Exp (i φ1) − Sinh(θ1) Exp (i φ2))
B = b1 Cosh (θ1 + θ2) + b2Cosh (θ2 − θ1) + b3Cosh (φ2 − φ1) and z
θ1 = α1 t − α1 β1
∫ β (z ) dz − θ10 0
z
θ2 = α2 t − α2 β2
∫ β (z ) dz − θ20 0
φ1 = α1 t +
φ2 = α2 t +
z
1 ( 2
∫ (α12 − β12) β (z ) dz − φ10)
1 ( 2
∫ (α22 − β22) β (z ) dz − φ20)
0 z
0
α a1 = 1 (α12 − α 22 + (β1 − β2)2) 2 a2 =
α2 2 (α 2 − α12 + (β1 − β2)2) 2
a3 = α1 α2 (β1 − β2) 3
Optik - International Journal for Light and Electron Optics 192 (2019) 162906
N. Prathap, et al.
b1 =
1 ((α1 − α2)2 + (β1 − β2)2) 4
b2 =
1 ((α1 + α2)2 + (β1 − β2)2) 4
b3 = −α1 α2 αn and βn are respectively indicates the amplitude and velocity parameters which are directly related to real and imaginary part of spectral parameter λn (n = 1, 2). Additionally, parameters such as phase and initial position of the solitons are determined by the parameters φ10 and θ10 respectively. 5. Exponentially varying dispersion profiles If the radius of core medium in a fiber can be made to vary while the refractive index profile is invariant, the dispersion coefficient also varies along the propagation axis. If the radius of core is gradually decreasing, the contribution to waveguide dispersion increases while total dispersion decreases and such fibers are popularly known as dispersion decreasing fibers. It can be shown that the dispersion profiles are having significant impact on the soliton shape. Hence, the present work provides much motivational information on soliton shaping in comparison to the existing results in the available literature to investigate the soliton shaping in an inhomogeneous fiber with nonuniform dispersion. (i) Soliton propagation in amplifying medium Soliton amplification [30] where the amplitude is exponentially increasing during propagation in a fiber due to gain medium. To discuss the soliton amplification technique, dispersion coefficient is adopted as exponential profile while nonlinearity as a constant.
β (z ) = Exp (0.05z ) γ (z ) = 1
(9)
As portrayed in Fig. 1(a) & (b), non-oscillating soliton amplification without interaction have been observed along the transmission line which is essential for optical fiber communication link. Besides, the amplified soliton is utterly pedestal free which makes the amplified soliton as extraordinary stable pulse during the propagation. One soliton amplification has been investigated in inhomogeneous waveguides through nonlinear and dispersion management [31]. Additionally, soliton structure is not destroyed while transmitted in the fiber medium which is a key factor for loss less fiber optic-based communication line. From the Fig. 1 (a), we conclude that the amplification takes place without noise, and optical soliton pulses can be amplified without aid of any relay device. (ii) Dromion structure soliton To analyzing soliton property, it may be required to assign the values for variable coefficients with controlling parameters. Here, we assign GVD and nonlinearity coefficients as given below [32].
β (z ) = Exp (−a z 2) γ (z ) = 1
(10)
Recently, dromion structure has been observed for attosecond soliton which can be denoted by sixth order variable coefficient NLS equation [33]. For a = 0.07, multiple dromions have been obtained as depicted in the Fig. 2(a). Distance between dromions can
Fig. 1. (a) Transmission characteristics of solitons with control parameters are α1 = −0.47, α2 = 0.45, β1 = 0.08, β2=−0.06. (b) contour plot. 4
Optik - International Journal for Light and Electron Optics 192 (2019) 162906
N. Prathap, et al.
Fig. 2. (a) Transmission characteristics of solitons with control parameters are α1 = 0.27, α2 = −0.35, β1 = 0.03, β2=−0.07. (b) contour plot.
be controlled by varying the value of “a”. It implies that the controlling parameter “a” has strong influences on interaction between dromions. Therefore, we can conclude that the dromion interaction can be managed by properly tuning the parameter a in dispersion profile β(z). This study may useful to control dromions in an inhomogeneous fiber through dispersion management scheme. (iii) Bounded soliton If we adopt β(z) with the special form, two solitons are propagated in bounded state [34]. In this case, inhomogeneous profiles are adopted as present below.
β (z ) = 1 + Exp (−0.07z 2)
(11)
The two-soliton with periodic interaction has been portrayed in Fig. 3. Furthermore, from the above figures, we infer that the two solitons are attracted and repelled each other alternately, and their central positions oscillate periodically. Meanwhile, it is interestingly to noted that the amplitude of the solitons also oscillates periodical manner. Additionally, through adjusting the angle between two solitons, particle-like behaviors of soliton interaction can be obtained with keeping their basic properties (Fig. 4). (iv) Parallel transmission of breathing soliton Since, parallel soliton transmission is needed for optical communication system, it is essential to discuss here. Through soliton control, one can attained parallel transmission of solitons by considering the inhomogeneous profiles as given in [35].
β (z ) = Exp (−0.07z ) + Sin(z ) γ (z ) = 1
(12)
Fig. 3. (a) Transmission characteristics of solitons with control parameters are α1 = −0.47, α2 = 0.45, β1 = 0.08, β2=−0.06. (b) contour plot. 5
Optik - International Journal for Light and Electron Optics 192 (2019) 162906
N. Prathap, et al.
Fig. 4. (a) Transmission characteristics of solitons with control parameters are α1 = −0.55, α2 = 0.45, β1 = 0.07, β2= −0.09. (b) contour plot.
On the negative z axis, we observed phase shifted breather solitons with more intense. Also, parallel transmission of breathing solitons has been perceived along the positive z direction which is desirable for optical communication. It must be noticed that repulsive and attraction between breathers are balanced along the positive z axis which leads to unbounded breathing behaviors. Recently, breathing solitons and its characteristics has been studied in detail [36,37]. 6. Conclusions Nonlinear Schrödinger system with inhomogeneous functions have been considered which is used to describe the soliton pulse propagation in an inhomogeneous monomode fiber have been discussed in this work. By employing Darboux transformation, two soliton solutions have been obtained through constructed Lax pair. Using obtained soliton solutions, few exponential distributed fiber control systems have been considered and the propagation characteristics of two solitons have been investigated. Specifically, the dynamical properties of the solitons through carefully tailoring dispersion coefficient with specific values of control parameters in the exponential profile which varying along the propagation distance have been studied. Furthermore, we wisely observe the soliton interaction with neighboring soliton through varying control parameters in the variable coefficients. In a dispersion management transmission system of optical fiber, the variable coefficient NLS equation with inhomogeneous functions opens the opportunity for controlling nonlinear properties and suppress specific nonlinear effect in inhomogeneous fiber. The study found that the soliton can be controlled through manipulating dispersion profiles in various nonuniform management systems. While properly tailoring the dispersion profile, the transmission characteristics of solitons are different such as amplification, compression, parallel transmission etc. These results have some significant guidance for soliton control and provide some theoretical analysis for experimental verification. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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Optik - International Journal for Light and Electron Optics 192 (2019) 162906
N. Prathap, et al.
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7
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Optical solitons and their shaping in a monomode optical fiber with some inhomogeneous dispersion profiles
T
⁎
N. Prathapa, S. Arunprakasha, M.S. Mani Rajanb, , M. Tantawyc a b c
Department of EEE, Anna University, Ramanathapuram 623513, India Department of Physics, Anna University, Ramanathapuram 623513, India Department of Basic Science, October 6 University, 6th of October City, Egypt
A R T IC LE I N F O
ABS TRA CT
Keywords: Optical soliton Exponential profiles Dispersion management Soliton control vc NLS Darboux transformation
In the real fiber optic communication system, the soliton transmission can be signified by nonlinear Schrödinger (NLS) equation with inhomogeneous coefficients. Here, the two-soliton propagation have been investigated by solving the inhomogeneous NLS equation with the process of Darboux transformation. According to obtained soliton solutions, variable coefficients are considered with different forms to study the features of solitons. Since soliton structures are mainly influenced by the controlling parameters, various soliton structures are obtained for specific choices of variable coefficients. Specifically, we adopt various exponential profiles for dispersion coefficient. Through properly manipulating exponential profiles, non-oscillating soliton amplification, dromion-like structure, bounded soliton and parallel soliton transmissions are explored. Results indicate that the characteristics of optical solitons are influenced by variable coefficients. In this work, we report the analytical study on transmission features of optical solitons in inhomogeneous optical fiber under various dispersion profiles. Obtained results are have some theoretical guidance for experimental research on optical amplifier, soliton based optical switches and achievement of parallel transmission of solitons.
1. Introduction In the context of rising traffic in the optical fiber communication links, researches on the production of ultra-short optical soliton have been attracted due to numerous applications [1–4]. Generally, soliton hold some potential values in various areas such as longdistance fiber optic communication systems, fiber lasers and optical switching devices [5–10]. Recently, studies of soliton shaping and management paid attention among researchers owing to potential applications in various nonlinear fiber systems [11–15]. In a real case, GVD, SPM and the gain or loss of fiber medium can be affected by the inhomogeneous nature of the fiber. Thus, nonlinearity, dispersion and gain/loss parameters of transmission line are varying along the propagation distance z. In recent years, the control and management of solitons in inhomogeneous fiber has been acknowledged wide attention due to their potential values in optical technologies [16–21]. Using various cosine and sine profiles, soliton shaping has been studied in inhomogeneous fiber system [22,23]. However, only few articles are reported on soliton structures for various exponential profiles in the inhomogeneous fiber. In this paper, to study the properties of soliton behavior in a real inhomogeneous fiber medium, analytic two soliton solutions are attained by employing Darboux transformation technique. Control parameters for the obtained soliton solutions will be tailoring to shaping the soliton. Here, we study the generalized inhomogeneous NLS equation with inhomogeneous
⁎
Corresponding author. E-mail address: [email protected] (M.S. Mani Rajan).
https://doi.org/10.1016/j.ijleo.2019.06.006 Received 2 June 2019; Accepted 3 June 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 192 (2019) 162906
N. Prathap, et al.
coefficients which governing the soliton transmission in an inhomogeneous fiber medium [24–27],
iqz +
1 i β (z ) qtt + γ (z )|q|2 q − G (z ) q = 0 2 2
(1)
where
W [γ (z ), β (z )] , β (z ) γ (z ) W [γ (z ), β (z )] = γβz − βγz G (z ) =
where q (z, t) is the temporal envelope of optical pulse and z is the propagation axis along the fiber. Inhomogeneous functions β(z) and γ(z) are related to dispersion and nonlinearity management profiles respectively and G(z) is implies the attenuation or amplification of the fiber medium. Moreover, Eq. (1) well describing the soliton propagation and shaping the soliton in inhomogeneous fiber medium. The inhomogeneous effects can be properly tuning which has the major impact on soliton’s shape. Thus, it is necessary to discuss the various soliton structures through soliton shaping in the inhomogeneous optical fiber. By setting control parameters, we examine the controllability of the optical solitons with inhomogeneous dispersion for Eq. (1). To our knowledge, soliton shaping through exponential dispersion profiles for Eq. (1) have not been constructed, which are the main tasks of our paper. 2. Lax pair for Eq. (1) With the aid of Ablowitz–Kaup–Newell–Segur technology [28], the Lax pair associated with Eq. (1) is derived. Through Lax pair, two soliton solutions are generated by employing Darboux transformation scheme. Hence, it is worthful to design the Lax pair for Eq. (1) to analyze the soliton management and shaping. The linear eigenvalue representation of Eq. (1) can be written as given below:
ψt = U ψ ψz = V ψ
(2)
Where ψ = (φ1, φ2 is an eigenfunction for Lax pair (3), φ1 and φ2 are both complex functions. The superscript T represents the transpose and 2 × 2 matrices U and V takes the forms as below.
)T
U = λJ + M V = i β (z ) λ2 J + i β (z ) λ P +
iT 2
β (z ) ⎛ 0 q ⎞ γ (z ) ⎝ q* 0 ⎠
J=
(10 −01), M =
T=
β (z ) γ (z ) qt ⎞ ⎛ γ (z ) q q* ⎜ β (z ) γ (z ) q * − γ (z ) q q* ⎟ t ⎝ ⎠
⎜
⎟
The compatibility condition Uz − Vt + [U , V ] = 0 gives rise to Eq. (1) and λ is known as isospectral parameter which is constant here. In the compatibility condition, brackets imply that the usual matrix commutator. 3. Darboux transformation Once Lax pair is known, via symbolic computation, Darboux Transformation can be applied for Eq. (1) to find soliton solutions. Consequently, Darboux transformation [29] has been employed to many nonlinear systems for obtaining multi-soliton solutions which can be used to understand the physical significance of considered nonlinear system. Also, the Darboux transformation formalism has been showed as an effective method to generate n-soliton solutions for many non-integrable and nonautonomous systems. In order to discuss the soliton shaping and management, we will determine multi soliton solutions for considered Eq. (1) with Darboux Transformation. Here we introduce the linear eigenfunction transformation as given
ψ [1] = (λI − S ) ψ
(3)
We can define S matrix as (4)
S = HΛH−1 Where I is the 2 × 2 identity matrix and S is a nonsingular matrix.
Λ = diag (Λ1 , Λ2 )
(5)
here H is the non-singular matrix, requiring
λ1 0 ⎞ ⎛ ϕ1 − ϕ2* ⎞ H=⎜ , Λ = ⎜⎛ ⎟ ϕ2 ϕ1* ⎟ ⎝ 0 − λ1* ⎠ ⎝ ⎠ 2
Optik - International Journal for Light and Electron Optics 192 (2019) 162906
N. Prathap, et al.
Ψt [1] = U1 Ψ [1] where U1 = λ J + P1 with
0 P1 = ⎜⎛ ∗ q1 ⎝
q1⎞ 0
⎟
⎠
We get the Darboux Transformation for Eq. (1) with following form, (6)
P1 = P + JS − SJ
It is easy to verify that if (φ1, φ2)T and (−φ2*, φ1*)T are solutions of Eq. (2) which are corresponds to the eigenvalue λ1 and − λ1* respectively. Hence the generalized form of Darboux transformation is presented for n-soliton solution through Eq. (6) as given below,
β (z ) γ (z )
qn = q + 2
∑
(λm + λm* ) ϕ1, m (λm) ϕ2,*m (λm) Am
(7)
Where
ϕk, m + 1 (λm + 1) = (λm + 1 + λm* ) ϕk, m (λm + 1) −
Bm (λm + λm* ) ϕk, m (λm) Am
Am = |ϕ1, m (λm)|2 + |ϕ2, m (λm)|2 Bm = ϕ1, m (λm + 1) ϕ1,*m (λm) + ϕ2, m (λm + 1) ϕ2,*m (λm) where k = 1, 2…n, m = 1, 2…n and (ϕ1,1 (λ1), ϕ2,1 (λ1))T is the eigenfunction of Eq. (2) corresponding to λ1. By setting q = 0 in Eq. (7), we can generate one-soliton solution for Eq. (1) by iterating DT once. Using the one-soliton solution as a seed solution in Eq. (7), we can derive the two-soliton solutions to study the interaction characteristics. Thus, one can generate up to n-soliton solution for Eq. (1). 4. Analytical multi-soliton solutions In ideal fiber, soliton cannot involve in the interaction with adjacent one in optical communication system. In real fiber, solitons will undergo interaction due to inhomogeneous nature of the fiber medium. Therefore, this is essential to study the propagation and interaction characteristics of two solitons for considered system (1). Through substituting the value of λ2 = α2+ i β2 and n = 2 in Eq. (7), we can generate the two-soliton solutions for Eq. (1) by repeating the Darboux transformation two times as follows
Q2 =
β (z ) γ (z )
A B
(8)
where
A = a1 Cosh(θ2) Exp (i φ1) + a2 Cosh(θ2) Exp (i φ2) + i a3 (Sinh(θ2) Exp (i φ1) − Sinh(θ1) Exp (i φ2))
B = b1 Cosh (θ1 + θ2) + b2Cosh (θ2 − θ1) + b3Cosh (φ2 − φ1) and z
θ1 = α1 t − α1 β1
∫ β (z ) dz − θ10 0
z
θ2 = α2 t − α2 β2
∫ β (z ) dz − θ20 0
φ1 = α1 t +
φ2 = α2 t +
z
1 ( 2
∫ (α12 − β12) β (z ) dz − φ10)
1 ( 2
∫ (α22 − β22) β (z ) dz − φ20)
0 z
0
α a1 = 1 (α12 − α 22 + (β1 − β2)2) 2 a2 =
α2 2 (α 2 − α12 + (β1 − β2)2) 2
a3 = α1 α2 (β1 − β2) 3
Optik - International Journal for Light and Electron Optics 192 (2019) 162906
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b1 =
1 ((α1 − α2)2 + (β1 − β2)2) 4
b2 =
1 ((α1 + α2)2 + (β1 − β2)2) 4
b3 = −α1 α2 αn and βn are respectively indicates the amplitude and velocity parameters which are directly related to real and imaginary part of spectral parameter λn (n = 1, 2). Additionally, parameters such as phase and initial position of the solitons are determined by the parameters φ10 and θ10 respectively. 5. Exponentially varying dispersion profiles If the radius of core medium in a fiber can be made to vary while the refractive index profile is invariant, the dispersion coefficient also varies along the propagation axis. If the radius of core is gradually decreasing, the contribution to waveguide dispersion increases while total dispersion decreases and such fibers are popularly known as dispersion decreasing fibers. It can be shown that the dispersion profiles are having significant impact on the soliton shape. Hence, the present work provides much motivational information on soliton shaping in comparison to the existing results in the available literature to investigate the soliton shaping in an inhomogeneous fiber with nonuniform dispersion. (i) Soliton propagation in amplifying medium Soliton amplification [30] where the amplitude is exponentially increasing during propagation in a fiber due to gain medium. To discuss the soliton amplification technique, dispersion coefficient is adopted as exponential profile while nonlinearity as a constant.
β (z ) = Exp (0.05z ) γ (z ) = 1
(9)
As portrayed in Fig. 1(a) & (b), non-oscillating soliton amplification without interaction have been observed along the transmission line which is essential for optical fiber communication link. Besides, the amplified soliton is utterly pedestal free which makes the amplified soliton as extraordinary stable pulse during the propagation. One soliton amplification has been investigated in inhomogeneous waveguides through nonlinear and dispersion management [31]. Additionally, soliton structure is not destroyed while transmitted in the fiber medium which is a key factor for loss less fiber optic-based communication line. From the Fig. 1 (a), we conclude that the amplification takes place without noise, and optical soliton pulses can be amplified without aid of any relay device. (ii) Dromion structure soliton To analyzing soliton property, it may be required to assign the values for variable coefficients with controlling parameters. Here, we assign GVD and nonlinearity coefficients as given below [32].
β (z ) = Exp (−a z 2) γ (z ) = 1
(10)
Recently, dromion structure has been observed for attosecond soliton which can be denoted by sixth order variable coefficient NLS equation [33]. For a = 0.07, multiple dromions have been obtained as depicted in the Fig. 2(a). Distance between dromions can
Fig. 1. (a) Transmission characteristics of solitons with control parameters are α1 = −0.47, α2 = 0.45, β1 = 0.08, β2=−0.06. (b) contour plot. 4
Optik - International Journal for Light and Electron Optics 192 (2019) 162906
N. Prathap, et al.
Fig. 2. (a) Transmission characteristics of solitons with control parameters are α1 = 0.27, α2 = −0.35, β1 = 0.03, β2=−0.07. (b) contour plot.
be controlled by varying the value of “a”. It implies that the controlling parameter “a” has strong influences on interaction between dromions. Therefore, we can conclude that the dromion interaction can be managed by properly tuning the parameter a in dispersion profile β(z). This study may useful to control dromions in an inhomogeneous fiber through dispersion management scheme. (iii) Bounded soliton If we adopt β(z) with the special form, two solitons are propagated in bounded state [34]. In this case, inhomogeneous profiles are adopted as present below.
β (z ) = 1 + Exp (−0.07z 2)
(11)
The two-soliton with periodic interaction has been portrayed in Fig. 3. Furthermore, from the above figures, we infer that the two solitons are attracted and repelled each other alternately, and their central positions oscillate periodically. Meanwhile, it is interestingly to noted that the amplitude of the solitons also oscillates periodical manner. Additionally, through adjusting the angle between two solitons, particle-like behaviors of soliton interaction can be obtained with keeping their basic properties (Fig. 4). (iv) Parallel transmission of breathing soliton Since, parallel soliton transmission is needed for optical communication system, it is essential to discuss here. Through soliton control, one can attained parallel transmission of solitons by considering the inhomogeneous profiles as given in [35].
β (z ) = Exp (−0.07z ) + Sin(z ) γ (z ) = 1
(12)
Fig. 3. (a) Transmission characteristics of solitons with control parameters are α1 = −0.47, α2 = 0.45, β1 = 0.08, β2=−0.06. (b) contour plot. 5
Optik - International Journal for Light and Electron Optics 192 (2019) 162906
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Fig. 4. (a) Transmission characteristics of solitons with control parameters are α1 = −0.55, α2 = 0.45, β1 = 0.07, β2= −0.09. (b) contour plot.
On the negative z axis, we observed phase shifted breather solitons with more intense. Also, parallel transmission of breathing solitons has been perceived along the positive z direction which is desirable for optical communication. It must be noticed that repulsive and attraction between breathers are balanced along the positive z axis which leads to unbounded breathing behaviors. Recently, breathing solitons and its characteristics has been studied in detail [36,37]. 6. Conclusions Nonlinear Schrödinger system with inhomogeneous functions have been considered which is used to describe the soliton pulse propagation in an inhomogeneous monomode fiber have been discussed in this work. By employing Darboux transformation, two soliton solutions have been obtained through constructed Lax pair. Using obtained soliton solutions, few exponential distributed fiber control systems have been considered and the propagation characteristics of two solitons have been investigated. Specifically, the dynamical properties of the solitons through carefully tailoring dispersion coefficient with specific values of control parameters in the exponential profile which varying along the propagation distance have been studied. Furthermore, we wisely observe the soliton interaction with neighboring soliton through varying control parameters in the variable coefficients. In a dispersion management transmission system of optical fiber, the variable coefficient NLS equation with inhomogeneous functions opens the opportunity for controlling nonlinear properties and suppress specific nonlinear effect in inhomogeneous fiber. The study found that the soliton can be controlled through manipulating dispersion profiles in various nonuniform management systems. While properly tailoring the dispersion profile, the transmission characteristics of solitons are different such as amplification, compression, parallel transmission etc. These results have some significant guidance for soliton control and provide some theoretical analysis for experimental verification. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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