The existence of soliton in the filed of optical fibers has made a wide improvement in various sectors of our life such as communications, medical, aerospace and many others [
Journal of the European Optical Society-Rapid Publications, Volume. 19, Issue 1, 2023025(2023)
Chirped gap solitons in fiber Bragg gratings with polynomial law of nonlinear refractive index
The objective of the present study is to examine the behaviors of chirped optical solitons in fiber Bragg gratings (BGs) with dispersive reflectivity. The form of nonlinear refractive index represents polynomial law nonlinearity. By virtue of phase-matching condition, the discussed model of coupled nonlinear Schrödinger equation is reduced to an integrable form. Consequently, chirped optical solitons having various profiles such as W-shaped, bright, dark, kink and anti-kink solitons are derived. Further to this, the chirp associated with these soliton structures are extracted. The impact of dispersive reflectivity, self-phase modulation and cross-phase modulation on the pulse propagation is investigated and it is induced that the changes of self-phase modulation and cross-phase modulation cause a marked rise in soliton amplitude which is subject to minor variations by dispersive reflectivity. The physical evolutions of chirped optical solitons are described along with the corresponding chirp to pave the way for possible applications in the field of fiber BGs.
1 Introduction
The existence of soliton in the filed of optical fibers has made a wide improvement in various sectors of our life such as communications, medical, aerospace and many others [
During propagation scenario within an optical fiber, soliton is found to be badly affected by the low count of chromatic dispersion (CD) and hence the data transmission is obfuscated. This problem can be handled via applying a new technology, namely, dispersion compensating fiber. One of the perfect candidates for this role is Bragg gratings (BGs) because of their low loss, small footprint, and low optical nonlinearity. Practically, BGs provides induced dispersion to replenish the low count of CD and sufficiently ensure the smooth formation of solitons which are transmitted along fibers for intercontinental distances. Based on the physical nature of fiber BGs medium, the self-phase modulation arises from the nonlinear refractive index structures which has different types such as Kerr law, quadratic-cubic law, parabolic law, polynomial law, parabolic-nonlocal combo law and many others. This diversity of forms of nonlinear refractive index leads to create distinct forms of NLSE.
Various studies in literature are implemented in the area of fiber BGs under the influences of different types of nonlinearity to examine the behaviors of optical solitons by making use of powerful integration schemes. For instance, Biswas et al. [
The recent experimental studies declare that optical solitons with nonlinear chirping have effective role in engineering applications such as the design of fiber-optic amplifiers, spread spectrum communications, photonic and optoelectronic devices. Hence, a lot of intensive theoretical works are directed to the investigation of chirped solitons in fiber-optic media. Many experts have dealt with various mathematical models to derive miscellaneous types of chirped solitons by means of several analytical techniques. Some of obtained soliton structures include kink, dark and bright solitons [
The most substantial purpose of this study is to inquire into the dimensionless form of the coupled NLSE in fiber BGs having polynomial law of nonlinearity given by [
This work is essentially devoted to investigating chirped optical solitons in fiber BG with polynomial law of nonlinear refractive index. The coupled NLSE
2 Mathematical analysis and reduction of governing model
Our aim now is to analyze the system of coupled NLSE
To manipulate the equations obtained above, the following relation is proposed as
From the integrability of equations
To reduce the level of complexity in the system of equations
As a result, the chirp can be written as
Plugging
Integrating the last equations, this leads to
As both equations
Equation
3 Chirped soliton solutions
To derive the chirped soliton solutions, the method of undetermined coefficients with two forms having the hyperbolic secant and tangent functions is employed to equation
3.1 First expression with hyperbolic tangent function
We express that the solution of equation
Set 1
Substituting
Set 2
Set 3
Exploiting
Set 4
Set 5
Employing
3.2 Second expression with hyperbolic secant function
We consider that equation
Set 1
Substituting
Set 2
By virtue of these results with
Set 3
Inserting
Set 4
Set 5
4 Results and remarks
The chirped optical solitons obtained above for the coupled system
In
Figure 1.The intensity profiles of W-shaped, bright, kink, anti-kink and dark solitons given by solutions
Likewise, soliton solutions obtained via the second form of undetermined coefficients method are displayed in
Figure 2.The intensity profiles of W-shaped, bright and dark solitons presented by solutions
Interestingly, the influence level of dispersive reflectivity, self-phase modulation and cross-phase modulation on the pulse propagation is shown in
Figure 3.Effects of dispersive reflectivity, self-phase modulation and cross-phase modulation on the pulse propagation for solution
Conclusion
This study discussed essentially the chirped optical solitons in fiber BGs with dispersive reflectivity having polynomial law of nonlinearity. The model of the coupled NLSE is analyzed under specific conditions in order to be straightforwardly integrable. Then, the soliton solutions are extracted by means of the undetermined coefficients approach which was given in two forms. The created optical solitons have several structures that included W-shaped, bright, dark, kink and anti-kink solitons. The chirping expressions associated with solitons were derived for all obtained solutions as well. The intensities of optical solitons are illustrated in addition to the chirping profiles. Besides, it is found that both of self-phase modulation and cross-phase modulation can highly amplify the soliton amplitude while there is a weak growth of amplitude by reason of dispersive reflectivity. The results obtained are expected to serve the field of optical fibers with BGs.
In the forthcoming work, the current model is studied via the technique of complete discrimination system for polynomial. Due to the intricate form of the coupled NLSE, various implicit solutions are revealed under specific restrictions. Furthermore, an exotic form of the soliton ansatz method having combination of hyperbolic secant and tangent functions is applied to equation
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Khalil S. Al-Ghafri, Mani Sankar, Edamana V. Krishnan, Salam Khan, Anjan Biswas. Chirped gap solitons in fiber Bragg gratings with polynomial law of nonlinear refractive index[J]. Journal of the European Optical Society-Rapid Publications, 2023, 19(1): 2023025
Category: Research Articles
Received: Feb. 18, 2023
Accepted: Apr. 27, 2023
Published Online: Aug. 31, 2023
The Author Email: Al-Ghafri Khalil S. (khalil.ibr@cas.edu.om)