Controllable Nonlinear Waves in Graded-Index Waveguides (GRIN) (SpringerBriefs in Applied Sciences and Technology) 9819704405, 9789819704408

Table of contents :
Preface
Contents
1 Controllable Nonautonomous Waves in Tapered Graded-Index Waveguides
1.1 Introduction
1.2 Model Equation and Similarity Transformation
1.3 Exact Similariton Solutions
1.4 Dynamical Behavior of Self-similar Waves
1.5 Conclusion
References
2 Controllable Optical Similaritons in GRIN by Tailoring of Tapered Profile
2.1 Introduction
2.2 Our Model and Isospectral Deformation
2.3 Riccati Parameterized Waves in s e c h squaredsech2-Type Tapered Fiber
2.4 Dynamics of Similaritons in Parabolic Tapered Profile
2.5 Conclusion
References
3 Butterfly-Shaped and Dromion-Like Waves in GRIN Waveguide
3.1 Introduction
3.2 Our Model and Exact Solutions
3.2.1 Soliton Solution-I
3.2.2 Periodic Solution
3.2.3 Soliton Solution-II
3.3 Numerical Simulations
3.4 Conclusion
References
4 Controlling Optical Similaritons in GRIN Waveguide Under the Action of Cubic–Quintic Nonlinear Media
4.1 Introduction
4.2 Our Model Equation and Similarity Transformations
4.3 Self-similar Solutions
4.3.1 Family of Bright Soliton Solutions
4.3.2 Family of Dark Self-similar Solutions
4.4 The Influence of Cubic Nonlinearity Coefficient
4.5 Computer Simulations
4.6 Conclusion
References
5 Dynamics of Optical Rogue Waves in a GRIN Waveguide with Different Background Beams
5.1 Introduction
5.2 Evolution of Rogons on Background Beams
5.2.1 normal s normal e normal c normal h squaredsech2—Background Beam
5.2.2 normal upper T normal a normal n normal hTanh—Background Beam
5.2.3 Bessel Background Beam
5.2.4 Airy Background Beam
5.3 The Splitting of Rogue Wave
5.4 Conclusion
References
6 Controlling Similaritons in Two-Dimensional GRIN Waveguide Through Riccati Parameter
6.1 Introduction
6.2 The Coupled NLSEs with Distributed Parameters
6.3 Family of Exotic Waves
6.4 The Effect of Dispersion
6.5 Riccati Parametrized Tapering Function and Width
6.6 Conclusion
References
7 Symbiotic Rogons in (2+1)-Dimensional GRIN Waveguide
7.1 Introduction
7.2 Multimode Wave Propagation
7.3 Nonuniform Coupling Coefficients
7.3.1 Bright–Dark Self-similar Waves
7.3.2 Dark Self-similar Waves
7.3.3 Bright Self-similar Waves
7.3.4 Breather-Similaritons and Rogons
7.4 Uniform Coupling Coefficients: Distributed Manakov Model
7.4.1 Semi-rational Rogons
7.5 Conclusion
References
8 Conclusions and Future Perspectives

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SpringerBriefs in Applied Sciences and Technology Thokala Soloman Raju

Controllable Nonlinear Waves in Graded-Index Waveguides (GRIN)

SpringerBriefs in Applied Sciences and Technology

SpringerBriefs present concise summaries of cutting-edge research and practical applications across a wide spectrum of fields. Featuring compact volumes of 50 to 125 pages, the series covers a range of content from professional to academic. Typical publications can be: . A timely report of state-of-the art methods . An introduction to or a manual for the application of mathematical or computer techniques . A bridge between new research results, as published in journal articles . A snapshot of a hot or emerging topic . An in-depth case study . A presentation of core concepts that students must understand in order to make independent contributions SpringerBriefs are characterized by fast, global electronic dissemination, standard publishing contracts, standardized manuscript preparation and formatting guidelines, and expedited production schedules. On the one hand, SpringerBriefs in Applied Sciences and Technology are devoted to the publication of fundamentals and applications within the different classical engineering disciplines as well as in interdisciplinary fields that recently emerged between these areas. On the other hand, as the boundary separating fundamental research and applied technology is more and more dissolving, this series is particularly open to trans-disciplinary topics between fundamental science and engineering. Indexed by EI-Compendex, SCOPUS and Springerlink.

Thokala Soloman Raju

Controllable Nonlinear Waves in Graded-Index Waveguides (GRIN)

Thokala Soloman Raju Department of Physics ICFAI Foundation for Higher Education Dontanapally, Hyderabad, Telangana, India

ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs in Applied Sciences and Technology ISBN 978-981-97-0440-8 ISBN 978-981-97-0441-5 (eBook) https://doi.org/10.1007/978-981-97-0441-5 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Paper in this product is recyclable.

To my Parents: Mrs. and Mr. Thokala Jayamani and Ratna Raju and To my Teacher: Mr. Penmetsa Venkateswara Raju

Preface

By now, it is a proven fact that optical solitons have the potential to become principal information carriers through long-haul telecommunication networks. Due to the delicate balance between nonlinearity and dispersion/diffraction, they have the capability to preserve their shape and velocity during their entire propagation. Although these properties are desirable in long-haul telecommunication networks, they pose certain unwanted difficulties particularly when the solitons encounter any nonuniformity in the waveguide. Suddenly they radiate energy and hence there may be loss of information. To avert such a situation, scientists have devised nonequilibrium waves such as similaritons. Similaritons are nothing but solitons or solitary waves that undergo modulations in their width and amplitude as a function of length of the waveguide. One of the salient features of these waves is that they undergo self-compression or amplification as also self-breathing wherever they encounter nonuniformity in the waveguide. This book has dealt with different types of similaritons and self-similar rogue waves and their dynamics in single-mode graded-index waveguides. It is hoped that these exotic waves will be launched in long-haul telecommunication networks for transmitting high bit rates of data through graded-index waveguides. A great deal of the material presented in this book mainly deals with the solitary wave solutions or similaritons in (1 + 1)-dimensional systems and their dynamical behavior. Novel features such as self-compression and self-breathing of such waves have been exemplified in great detail. In the seventh chapter, we describe symbiotic self-similar rogue waves in (2 + 1)dimensional graded-index waveguide. By suitably lowering the strength of self-phase modulation in comparison to the cross-phase modulation, we could achieve prominence in one of the modes of the waveguide. For equal strengths of these two effects, we could obtain semi-rational self-similar rogue waves. Many of these ideas were conceived while I was working at Karunya University, Coimbatore, in a wonderful collaboration with Prof. C. N. Kumar, Panjab University, Chandigarh, and Prof. P. K. Panigrahi, IISER Kolkata. I profusely thank them. I thank the management of IFHE, Hyderabad, in particular Prof. K. L. Narayana, Director FST, ICFAI Tech, Hyderabad, for providing a conducive atmosphere to finish writing this book. vii

viii

Preface

I owe indebtedness to my parents—Mrs. Thokala Jayamani and Mr. Thokala Ratna Raju—for their love and affection toward me. Despite the fact that they suffered from hunger, they could educate all three children in a magnificent way. I’m forever grateful for their supreme sacrifices. I profusely thank my childhood teacher and mentor— Mr. Penmatsa Venkateswara Raju—for his marvelous teaching and molding me as a good citizen. I thank Dr. Swamy Kumar Polumuri, who mentored me during my school days and nurtured me into a good researcher. I’m indebted to my brothers—Mr. Thokala Mohan Rao and Mr. Thokala Janardhan Rao—and their families for their moral support. Last but not the least, I thank my wife Mrs. P. Swaroopa Rani and two teenage children—Ms. Thokala Nancy Grace and Mr. Thokala Sharon Raju—for understanding why I needed to spend weekends on the book instead of spending time with them. Above all, I extol the name of my Lord and Savior—Jesus Christ—for His everlasting promise recorded in Philippians 1:6. Hyderabad, India December 2023

Thokala Soloman Raju

Contents

1 Controllable Nonautonomous Waves in Tapered Graded-Index Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Model Equation and Similarity Transformation . . . . . . . . . . . . . . . . . . 1.3 Exact Similariton Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Dynamical Behavior of Self-similar Waves . . . . . . . . . . . . . . . . . . . . . . 1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 4 6 7

2 Controllable Optical Similaritons in GRIN by Tailoring of Tapered Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Our Model and Isospectral Deformation . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Riccati Parameterized Waves in sech2 -Type Tapered Fiber . . . . . . . . . 2.4 Dynamics of Similaritons in Parabolic Tapered Profile . . . . . . . . . . . . 2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 11 13 15 17 17

3 Butterfly-Shaped and Dromion-Like Waves in GRIN Waveguide . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Our Model and Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Soliton Solution-I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Periodic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Soliton Solution-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 20 22 23 23 24 28 29

4 Controlling Optical Similaritons in GRIN Waveguide Under the Action of Cubic–Quintic Nonlinear Media . . . . . . . . . . . . . . . . . . . . . . 31 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Our Model Equation and Similarity Transformations . . . . . . . . . . . . . 33

ix

x

Contents

4.3 Self-similar Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Family of Bright Soliton Solutions . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Family of Dark Self-similar Solutions . . . . . . . . . . . . . . . . . . . . 4.4 The Influence of Cubic Nonlinearity Coefficient . . . . . . . . . . . . . . . . . 4.5 Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 36 37 39 39 41 42

5 Dynamics of Optical Rogue Waves in a GRIN Waveguide with Different Background Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Evolution of Rogons on Background Beams . . . . . . . . . . . . . . . . . . . . . 5.2.1 sech2 —Background Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Tanh—Background Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Bessel Background Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Airy Background Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Splitting of Rogue Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 46 47 47 48 49 50 51 51

6 Controlling Similaritons in Two-Dimensional GRIN Waveguide Through Riccati Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 The Coupled NLSEs with Distributed Parameters . . . . . . . . . . . . . . . . 6.3 Family of Exotic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Effect of Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Riccati Parametrized Tapering Function and Width . . . . . . . . . . . . . . . 6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53 53 54 56 58 59 61 61

7 Symbiotic Rogons in (2+1)-Dimensional GRIN Waveguide . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Multimode Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Nonuniform Coupling Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Bright–Dark Self-similar Waves . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Dark Self-similar Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Bright Self-similar Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Breather-Similaritons and Rogons . . . . . . . . . . . . . . . . . . . . . . . 7.4 Uniform Coupling Coefficients: Distributed Manakov Model . . . . . . 7.4.1 Semi-rational Rogons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 64 67 67 68 69 70 72 72 73 73

8 Conclusions and Future Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Chapter 1

Controllable Nonautonomous Waves in Tapered Graded-Index Waveguides

1.1 Introduction In the recent past, tapered graded-index waveguides have become an integral part of various optoelectronic devices. In optical communication systems, we often need efficient coupling techniques for the input and output of optical signals from other fibers or from thin-film waveguides. Tapered graded-index waveguides can be used as coupling and branching devices in optical communication circuits [1, 2], because they help in maximizing light coupled into optical waveguides and integrated optic devices by reducing the reflection losses and mode mismatch [3]. Apart from being used as low loss couplers and connectors between waveguides of different dimensions, they also find applications in Raman lasers [4], supercontinuum generation [5], optical cloaking [6], and fiber optic sensors [7]. Consequently, it is highly desirable to understand the propagation of electromagnetic waves in tapered graded-index waveguide [8]. The complex mathematical structure of evolution equations, due to the presence of tapered geometry, led to the development of various approximation methods to obtain analytic solutions in such media [9, 10]. But, Ponomarenko and Agrawal recently obtained analytically a new class of exact self-similar waves supported by inhomogeneous gain in tapered graded-index waveguide [11]. These optical self-similar waves, also known as similaritons, are of fundamental interest because they maintain their shape but the amplitude and width changes with the modulating system parameters such as dispersion, nonlinearity, and gain. Due to these properties, similaritons have various applications in ultra-fast optics [12–15]. In this chapter, we consider the refractive index distribution inside the waveguide as [16] 2 .n(x, z) = n 0 + n 1 F(z)x + n 2 H (z)I (x, z), (1.1) where the first two terms represent the linear part of refractive index and the last term signifies a Kerr-type nonlinearity. The function . F(z) describes the geometry of tapered waveguide. Such waveguides have applications in image-transmitting © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 T. S. Raju, Controllable Nonlinear Waves in Graded-Index Waveguides (GRIN), SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-97-0441-5_1

1

2

1 Controllable Nonautonomous Waves in Tapered …

processes because they possess a very wide bandwidth. Now, the shape of a taper, i.e. F(z) can be modeled appropriately depending upon its specific application in which it is used. A taper can be made by heating of one or more fibers up to the material softening point and then stretching it until the desired shape has been obtained. Further, the Kerr nonlinearity has the longitudinally varying coefficient. For the wave propagation through the optical fibers, it finds applications in the transverse beam propagation in the layered optical media [17]. The wave propagation in tapered graded-index nonlinear waveguide amplifier can be modeled by the generalized nonlinear Schrödinger equation (GNLSE) with constant dispersion, and distributed nonlinearity, and gain. In Ref. [11], for constant nonlinearity, Ponomarenko and Agrawal obtained similariton solutions for sech.2 -type tapering profile. Extending this work afterward, many authors including the present author studied the dynamics of various similaritons, with source and without source [18–20]. Serkin et al. considered the same problem with all distributed coefficients, in the context of Bose–Einstein condensates (BEC), and obtained nonautonomous soliton solutions using the inverse scattering technique [21]. Control of solitons in BEC has been demonstrated in Ref. [22]. Recently, Dai et al. studied the propagation behaviors of controllable rogue waves, modeled by inhomogeneous GNLSE, under dispersion and nonlinearity management [23]. In this paper, we study the existence of similaritons in the parabolic tapered graded-index waveguide, by reducing the GNLSE to standard NLSE using gauge and similarity transformations [11, 23], which forms an efficient fiber optical coupler, with the Gaussian Kerr coefficient and linear gain coefficient. The intensity profile obtained here is remarkably large.

.

1.2 Model Equation and Similarity Transformation The GNLSE governing the wave propagation through tapered graded-index waveguide is given by i

.

1 ∂ 2u 1 ∂u ig(z) + u + k0 n 1 F(z)x 2 u + k0 n 2 H (z)|u|2 u = 0, − ∂z 2k0 ∂ x 2 2 2

(1.2)

where.g(z) is the gain coefficient,.k0 = 2π n 0 /λ,.λ being the wavelength of the optical source generating the beam. Introducing the normalized variables, . X = x/w0 , Z = √ z/L D , U = k0 |n 2 |L D u, where. L D = k0 w02 is the diffraction length associated with the characteristic transverse scale .w0 = (2k02 n 1 )−1/4 , Eq. (1.2) reduces to dimensionless form given as i

.

1 ∂ 2U i X2 ∂U + U − G(Z )U + σ H (Z )|U |2 U = 0. + F(Z ) 2 ∂Z 2 ∂X 2 2

(1.3)

Here, .G(Z ) = gL D is the effective gain, . F(Z ) = n 1 L 2D F(z) and . H (Z ) = H (z) are the tapering profile and Kerr coefficient respectively in new coordinates. .σ =

1.3 Exact Similariton Solutions

3

sgn(n 2 ) = ±1, with positive (negative) sign corresponding to self-focusing (selfdefocusing) nonlinearity of waveguide. Equation (1.2) reduces to standard NLSE i

.

∂𝚿 1 ∂ 2𝚿 + σ |𝚿|2 𝚿 = 0, + ∂ζ 2 ∂χ 2

(1.4)

by the application of gauge and similarity transformation given as [ U (X, Z ) = A(Z )𝚿

.

] X − X c (Z ) , ζ (Z ) exp(iϕ(X, Z )), W (Z )

(1.5)

where . A(Z ), X c (Z ), ζ (Z ), and .ϕ(X, Z ) are the amplitude, guiding-center position, effective propagation distance, and phase of the self-similar wave. These are related to the width of the similariton wave .W (Z ) as follows: 1 X − X c (Z ) , χ (X, Z ) = , W 2 (Z ) W (Z ) Z dS , ζ (Z ) = ζ0 + 2 0 W (S) ( ) Z dS , X c (Z ) =W (Z ) X 0 + C02 2 0 W (S) Z dS C02 X C2 X 2 dW + − 02 . ϕ(X, Z ) = 2W d Z W 2 0 W 2 (S) .

A(Z ) =

(1.6)

C02 is an integration constant, . X c (0) = X 0 , and .W (0) = 1. Further, the tapering, gain, and width functions are related by the following differential equations:

.

.

.

d 2 W/d Z 2 − F(Z )W = 0,

(1.7)

G(Z ) = −3d[ln W (Z )]/d Z

(1.8)

H (Z ) = W 2 (Z ).

(1.9)

and .

1.3 Exact Similariton Solutions Using the transformation .𝚿(ζ, χ ) = R(ζ )ei(ζ −vχ) in Eq. (1.4), we obtain the following elliptic equation: 1 3 . Rζ ζ + ϵ R + 2σ R = 0. (1.10) 2

4

1 Controllable Nonautonomous Waves in Tapered …

We affirm that all the twelve Jacobian elliptic functions are a solution of the above elliptic equation. However, below we find bright and dark solitary waves as exact solutions under restrictive parameter regime. Furthermore, these solitary waves are physically interesting because of their utility in the long-haul telecommunication networks involving GRINs. Case I: We find that . R(ζ ) = Asech(αζ ) is an exact self-similar bright solitary√wave solution, for the following restrictive conditions: the width of the soliton .α = −2ϵ implying that the detuning parameter .ϵ should always be negative for a physically viable√solution. Also we find that the amplitude of the self-similar wave should be .A = −ϵ/σ . For a physically meaningful soliton solution, the nonlinearity should be always self-focusing. Case II: We find that . R(ζ ) = Btanh(βζ ) is an exact self-similar dark solitary wave √ solution, for the following restrictive conditions: the width of the soliton .β = ϵ implying that the detuning parameter .ϵ should always be positive for a physically viable√solution. Also we find that the amplitude of the self-similar wave should be .B = −ϵ/2σ . For a physically meaningful soliton solution, the nonlinearity should be always self-defocusing.

1.4 Dynamical Behavior of Self-similar Waves In this section, we explicate the mechanism to control the dynamical behavior of the bright self-similar wave and dark self-similar wave for three different varieties of modulations. We observe that Hill’s differential equation (HDE) (1.7) looks similar to the Schrödinger eigenvalue problem. Motivated by this fact, we find three different types of widths and tapering profiles as exact solutions and study the dynamics of bright and dark similaritons. Case I: In this case, we find the solution of the HDE as .

W (z) = e−γ z ;

F(z) = −γ 2 ,

(1.11)

where .γ is a parameter. For these modulations, we depict the bright self-similar wave in Fig. 1.1 and dark self-similar wave in Fig. 1.2 for various parameters specified in the figure captions. In both these cases, we observe that the solitary waves indeed undergo self-compression. Case II: In the second case, we find the solution of the HDE as .

W (z) = 1 − bsin(az);

F(z) =

ba 2 sin(az) , 1 − bsin(az)

(1.12)

where .a and .b are parameters related to the amplitude and width of the periodic modulation. For these modulations, we depict the bright self-similar wave in Fig. 1.3 and dark self-similar wave in Fig. 1.4 for various parameters specified in the figure

1.4 Dynamical Behavior of Self-similar Waves

5

Fig. 1.1 Plot depicting the propagation of bright similariton for parameters . X 0 = 0.3, .γ = 0.02, .ϵ = −0.34, .σ = 0.12, and .v = 0.25

Fig. 1.2 Plot depicting the propagation of bright similariton for parameters . X 0 = 0.3, .γ = 0.02, .ϵ = 0.34, .σ = −0.12, and .v = 0.25

Fig. 1.3 Plot depicting the propagation of bright similariton for parameters . X 0 = 0.3, .a = 0.5, .b = 0.1, .ϵ = −0.34, .σ = 0.12, and .v = 0.25

captions. In both these cases, we observe that the solitary waves get annihilated and then regenerated after some evolution along the length of the waveguide. Case III: In the third case, we find the solution of the HDE as .

W (z) = 1 + bcosh(az);

F(z) =

ba 2 cosh(az) , 1 + bcosh(az)

(1.13)

where .a and .b are parameters related to the amplitude and width of the hyperbolic modulation. For these modulations, we depict the bright self-similar wave in Fig. 1.5 and dark self-similar wave in Fig. 1.6 for various parameters specified in

6

1 Controllable Nonautonomous Waves in Tapered …

Fig. 1.4 Plot depicting the evolution of dark similariton for parameters . X 0 = 0.3, .a = 0.5, .b = 0.1, .ϵ = 0.34, .σ = −0.12, and .v = 0.25

Fig. 1.5 Plot depicting the propagation of bright similariton for parameters . X 0 = 0.3, .a = 0.5, .b = 0.1, .ϵ = −0.34, .σ = 0.12, and .v = 0.25

Fig. 1.6 Plot depicting the propagation of bright similariton for parameters . X 0 = 0.3, .a = 0.5, .b = 0.1, .ϵ = 0.34, .σ = −0.12, and .v = 0.25

the figure captions. In both these cases, we observe that the solitary waves undergo self-compression during the evolution along the length of the waveguide.

1.5 Conclusion In conclusion, we have studied the evolution dynamics of nonautonomous waves in GRIN waveguides for different tapering profiles and widths. Specially, in the case of periodic modulations, we have found that the bright and dark similaritons get

References

7

annihilated and after some time get regenerated along the length of the waveguide during their evolution. In the other two cases, we have observed that the similaritons undergo self-compression. We hope that these intricately modulated waves may be launched in long-haul telecommunication networks involving GRIN waveguides.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

A. Hosseini, J. Covey, D.N. Kwong, R.T. Chen, J. Opt. 12, 075502 (2010) J.R. Salgueiro1, Y.S. Kivshar, Opt. Express 20, 9403 (2012) V.R. Almeida, R.R. Panepucci, M. Lipson, Opt. Lett. 28, 1302 (2003) M. Krause, H. Renner, E. Brinkmeyer, Spectroscopy 21, 26 (2006) T.A. Birks, W.J. Wadsworth, P.St.J. Russell, Opt. Lett. 25, 19, 1415 (2000) I.I. Smolyaninov, V.N. Smolyaninova, A.V. Kildishev, V.M. Shalaev, Phys. Rev. Lett. 102, 213901 (2009) R.K. Verma, A.K. Sharma, B.D. Gupta, Opt. Commun. 281, 1486 (2008) D. Bertilone, A. Ankiewicz, C. Pask, Appl. Opt. 26, 2213 (1987) A.A. Tovar, L.W. Casperson, Appl. Opt. 33, 7733 (1994) C.C. Constantinou, R.C. Jones, IEE Proc.-J. 139, 365 (1992) S.A. Ponomarenko, G.P. Agrawal, Opt. Lett. 32, 1659 (2007) J.M. Dudley, C. Finot, D.J. Richardson, G. Millot, Nat. Phys. 3, 597 (2007) V.I. Kruglov, D. Méchin, J.D. Harvey, J. Opt. Soc. Am. B 24, 833 (2007) C. Finot, G. Millot, Opt. Express 12, 5104 (2004) T. Mansuryan, A. Zeytunyan, M. Kalashyan, G. Yesayan, L. Mouradian, F. Louradour, A. Barthélémy, J. Opt. Soc. Am. B 25, A101 (2008) L. Li, X. Zhao, Z. Xu, Phys. Rev. A 78, 063833 (2008) L. Berg.e, ´ V.K. Mezentsev, J. Juul Rasmussen, P. Leth Christiansen, Yu.B. Gaididei, Opt. Lett. 25, 1037 (2000) T. Soloman Raju, P.K. Panigrahi, Phys. Rev. A 84, 033807 (2011) R. Gupta, S. Loomba, C.N. Kumar, IEEE J. Quantum Electron. 48, 847 (2012) T. Soloman Raju, T. Shreecharan, S.S. Ranjani, IEEE J. Quantum Electron. 58, 6100405 (2022) V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Phys. Rev. Lett. 98, 074102 (2007); V.N. Serkin, A. Hasegawa, T.L. Belyaeva, J. Mod. Opt. 57, 1456 (2010) R. Atre, P.K. Panigrahi, G.S. Agarwal, Phys. Rev. E 73, 056611(5) (2006) C.Q. Dai, Y.Y. Wang, Q. Tian, J.F. Zhang, Ann. Phys. 327, 512 (2012)

Chapter 2

Controllable Optical Similaritons in GRIN by Tailoring of Tapered Profile

2.1 Introduction Much attention has been paid to the study of the design and the properties of different tapered profiles in optical waveguides, both theoretically as well as experimentally [1]. This is because of the potential applications achievable through the tapering effect. Indeed, the tapering effect helps in improving the coupling efficiency between fibers and waveguides by reducing the reflection losses and mode mismatch [2]. Also, it finds applications in highly efficient Raman amplification [3] and extended broadband supercontinuum generation [4]. These days, a lot of work has been done on the study of the propagation of the similaritons in the tapered, graded-index waveguides [5–7]. Similaritons are the waves which maintain their shape, but the amplitude and width change with the modulating system parameters such as dispersion, nonlinearity, and gain. Their temporal and spectral profiles are independent of the input wave profile, but are determined with the input wave energy and amplifier parameters. Owing to these various properties, similaritons are of interest for various applications in ultra-fast optics [8–10]. In this chapter, we consider the refractive index distribution inside the waveguide as 2 2 .n(x, z) = n 0 + n 1 F(z)x + n 2 |u| , (2.1) where the first two terms correspond to the linear part of refractive index and the last term is Kerr-type nonlinearity. The function . F(z) describes the geometry of tapered waveguide, along the waveguide axis. The quadratic variation of the refractive index in transverse direction is a good approximation to the true refractive index distribution, and is often known as a lens-like medium. Such waveguides have applications in image-transmitting processes because they possess a very wide bandwidth. The shape of a taper, i.e. . F(z) can be modeled appropriately depending upon the practical requirements. A taper can be made by heating of one or more fibers up to the material softening point and then stretching it until the desired shape has been obtained. Here, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 T. S. Raju, Controllable Nonlinear Waves in Graded-Index Waveguides (GRIN), SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-97-0441-5_2

9

10

2 Controllable Optical Similaritons in GRIN by Tailoring of Tapered Profile

we study the propagation of similaritons in two types of tapered waveguides—(a) sech2 -type taper, and (b) parabolic taper. In Refs. [5, 6], the authors have considered the lowest-order mode of .sech2 -profile waveguides [11]. It has been shown that parabolic taper, given by . F(z) = F0 (1 + γ z 2 ) [12, 13], is the optimum shape for fiber couplers which can be fabricated by the use of Ag-ion exchange in soda-lime glass [14]. We know that the dynamics of self-similar waves in GRIN can be modeled by generalized nonlinear Schrödinger equation (GNLSE) [15]. By using the inverse scattering technique, Serkin and his collaborators, in a series of papers [16], solved GNLSE arising in the context of Bose–Einstein condensates (BEC) and fiber optics and obtained nonautonomous 1- and 2-soliton solutions. Control of solitons in BEC has been demonstrated in Ref. [17]. Also, utilizing the gauge and group symmetry transformation, Ponomarenko and Agrawal [5] studied the paraxial wave evolution in the tapered, graded-index nonlinear waveguide amplifier and obtained optical similaritons by reducing the GNLSE to standard NLSE. Then, Wu et al. employed lens-type and Husimi transformations to obtain movable exact quasi-soliton similaritons and also studied their interaction dynamics [18]. Later, this equation in the presence of an external source was solved by Soloman Raju and Panigrahi [6] who describe the similariton propagation through asymmetric twin-core fiber amplifiers. More recently, Dai et al. have studied the propagation behavior of controllable rogue waves for this equation through dispersion and nonlinearity management in both (2+1)-dimensions and (1+1)-dimensions [7, 19]. The width of these similaritons obeys the second-order differential equation whose mathematical structure is similar to the Schrödinger equation with tapering profile as potential. In this chapter, we will pay more attention to study the effect of different tapering profiles, on the intensity of optical similaritons in a GRIN. In the subsequent sections, we demonstrate the procedure to tailor .sech2 -type tapering profiles by the crucial observation that the equation governing the tapering profile can be transformed into a Schrödinger-like equation which enables us to introduce the isospectral Hamiltonian approach. It is well-known in supersymmetric quantum mechanics (SUSY QM) that for a given ground state wave function and energy, isospectral deformation of the Hamiltonian can always be done [20]. This identification yields a Riccati differential equation, in which there arises a free parameter called Riccati parameter. This way we can construct one parameter family of potentials and wave functions. Since the newly constructed wave functions are quite different from the original set of wave functions, any wave function-dependent quantity can be tuned through this parameter. Pre Supersymmetric Quantum mechanics era, this aspect was discussed in Ref. [21]. This technique has been proved to be efficacious in obtaining new potentials from a given potential [22–26]. Recently, this approach has been used to control the giant rogue waves in nonlinear fiber optics [27]. We use this aspect to study the evolution of optical similaritons in a graded-index nonlinear waveguide for a class of tapering profiles. The specific form of the tapering profile and width can be adjusted by restricting to a particular value of the Riccati parameter ‘.c’.

.

2.2 Our Model and Isospectral Deformation

11

2.2 Our Model and Isospectral Deformation It is very interesting to study the behavior of solitons and self-similar waves in nonlinear systems exhibiting both spatial inhomogeneity and gain or loss [15] at the same time. A subtle interplay between the linear gain or loss, diffraction, and the inhomogeneity on the one hand, and the nonlinearity of such systems on the other, can result in a rich variety of shape-preserving waves with interesting properties. In this paper, we elucidate dynamical behaviors of .(1 + 1)-D shape-preserving waves in such systems by focusing on a specific example of tapered, graded-index, nonlinear waveguide amplifiers. The paraxial wave evolution corresponding to .n(x, z) in Eq. (2.1) is governed by the inhomogeneous nonlinear Schrödinger equation of the form i

.

1 ∂ 2u 1 i(g(z) − α(z)) ∂u + u + k0 n 1 F(z)x 2 u − 2 ∂z 2k0 ∂ x 2 2 +k0 n 2 |u|2 u = 0,

(2.2)

where .u(x, z) is the complex envelope of the electrical field, .g and .α account for linear gain and loss, respectively,.k0 = 2π n 0 /λ,.λ being the wavelength of the optical source generating the beam, .n 1 is the linear defocusing parameter (.n 1 > 0), and .n 2 represents Kerr-type nonlinearity. The dimensionless profile function . F(z) can be negative or positive, depending on whether the graded-index medium acts as a focusing or defocusing linear lens. Introducing the normalized variables . X = x/w0 , 1/2 . Z = z/L D , . G = [g(z) − α(z)]L D , and .U = (k 0 |n 2 |L D ) u, where . L D = k0 w02 is the diffraction length associated with the characteristic transverse scale .w0 = (k02 n 1 )−1/4 , Eq. (2.2) can be rewritten in a dimensionless form [5], i

.

1 ∂ 2U i X2 ∂U + U − G(Z )U ± |U |2 U = 0. + F(Z ) 2 ∂Z 2 ∂X 2 2

(2.3)

Equation (2.3) has been solved for similariton solutions by transforming it into standard homogeneous NLSE, using the gauge and similarity transformation, together with a generalized scaling with respect to the . Z variable [5, 18, 19], U (X, Z ) =

.

[ ] X − X c (Z ) 1 𝚿 , ζ (Z ) eiϕ(X,Z ) , W (Z ) W (Z )

(2.4)

where .W (Z ) and . X c are the dimensionless width and position of the self-similar wave center. The phase is given by ϕ(X, Z ) = C1 (Z )

.

X2 + C2 (Z )X + C3 (Z ), 2

(2.5)

where .C1 (Z ), C2 (Z ), and .C3 (Z ) are the parameters related to the phase-front curvature, the frequency shift, and the phase offset, respectively, to be determined. Now

12

2 Controllable Optical Similaritons in GRIN by Tailoring of Tapered Profile

substituting Eqs. (2.4) and (2.5) into Eq. (2.3), one obtains the autonomous NLSE given by ∂𝚿 1 ∂ 2𝚿 .i ± |𝚿|2 𝚿 = 0. (2.6) + ∂ζ 2 ∂χ 2 Here the effective propagation distance, similarity variable, guiding-center position, and phase are given respectively as [5] Z

ζ (Z ) = ζ0 +

.

0

dS , W 2 (S) (

.

Z

X c (Z ) = W (Z ) X 0 + C02 0

) dS , W 2 (S)

X 2 dW C02 X C2 + − 02 2W d Z W 2

ϕ(X, Z ) =

.

X − X c (Z ) , W (Z )

χ (X, Z ) =

Z 0

dS , W 2 (S)

(2.7)

where .C2 (0) = C02 , X c (0) = X 0 , and we choose .W (0) = 1. Further, the tapering function, gain, and similariton width .W (Z ) are related as .

d 2 W/d Z 2 − F(Z )W = 0,

(2.8)

G(Z ) = −d[ln W (Z )]/d Z .

(2.9)

.

Interestingly, Eq. (2.8) can be observed as the Schrödinger eigenvalue problem, by identifying . F(Z ) as potential and .W (Z ) as corresponding wave function with zero energy. As Schrödinger equation is exactly solvable for a variety of potentials, so this identification enables us to study Eq. (2.3) for a variety of tapering profiles. Moreover, using the fact that for a given ground state wave function and energy, isospectral deformation of the Hamiltonian can always be done [20], one can write a class of width and tapering profile as [27] √ .

W (Z ) =

c(c + 1) W (Z )

c+

d . F(Z ) = F(Z ) − 2 dZ

Z −∞

W 2 (S)d S

(

,

W 2 (Z ) c+

Z −∞

W 2 (S)d S

(2.10) ) ,

(2.11)

where .c is an integration constant, known as Riccati parameter, which is to be chosen in a way so as to avoid singularities. Hence, the modified gain function becomes .

G(Z ) = −d[ln W (Z )]/d Z .

(2.12)

2.3 Riccati Parameterized Waves in sech2 -Type Tapered Fiber

13

As stated earlier, transformation (2.4) maps Eq. (2.3) to the integrable homogeneous NLSE [5]. Hence, the intensity profile of the bright similariton for Eq. (2.3) is given by a2 . I B (χ , ζ ) = sech2 [a(χ − vζ )], (2.13) W2 where .a and .v are the soliton amplitude and velocity. The intensity profile of the dark similariton is given by I (χ , ζ ) =

. D

u 20 [cos2 (φ) tanh2 ( ) + sin2 (φ)], W2

(2.14)

where .u 0 is the background amplitude and .φ governs the grayness and speed of the dark soliton. The soliton phase . is given as .

(χ , ζ ) = u 0 cos(φ)[χ − u 0 ζ sin(φ)].

(2.15)

Armed with these new functions—. F(Z ), W (Z ), and .G(Z ), we turn our attention to discuss the effect of these new functions on the intensity profiles of the bright and dark similaritons given by Eqs. (2.13) and (2.14), respectively, by taking an explicit example of .sech2 -type tapering in the next section.

2.3 Riccati Parameterized Waves in .sech2 -Type Tapered Fiber For. F(Z ) = n 2 − n(n + 1) sech2 Z ,.n is a positive integer, the width and gain profiles from Eqs. (2.8) and (2.9) are given as W (Z ) = sechn Z ,

(2.16)

G(Z ) = n tanh Z .

(2.17)

.

.

For the lowest order mode, already the dynamics of self-similar waves have been discussed. For the present analysis, we restrict ourselves to cases .n = 2 and .3 and discuss the dynamics of self-similar waves. In particular, we delineate the effect of the controlling-Riccati parameter on the dynamics of the ensuing self-similar waves. Case (i) : For .n = 2, Eqs. (2.16)–(2.17) yield .

F(Z ) = 4 − 6 sech2 Z , W (Z ) = sech2 Z , and G(Z ) = 2 tanh Z .

(2.18)

Substituting .W (Z ) = sech2 Z into Eq. (2.10), a class of .W (Z ) can be obtained as

14

2 Controllable Optical Similaritons in GRIN by Tailoring of Tapered Profile

Fig. 2.1 Plot depicting the propagation of bright similariton for .n = 2 for different tapering; a without generalization, and b, c for generalized tapering for .c = 0.2 and .c = 0.02, respectively. The parameters used in the plots are .C02 = 0.3, . X 0 = 0, .v = 0.3, and .a = 1

Fig. 2.2 Plot depicting the propagation of dark similariton for .n = 2 for different tapering; a without generalization, and b, c for generalized tapering for .c = 0.3 and .c = 0.03, respectively. The parameters used in the plots are .C02 = 0.3, . X 0 = 0, .v = 0.3, and .a = 1

.

W (Z ) =

√ c(c + 1)sech2 (Z ) ( ) . 1 c + 3 2 + sech2 (Z ) tanh(Z )

(2.19)

In much the same way, . F(Z ) and .G(Z ) can also be found from Eqs. (2.11) and (2.12). Now using Eq. (2.19), the intensity of optical bright and dark similaritons for . F(Z ) can be found from Eqs. (2.13) and (2.14). Here we have shown the evolution of bright and dark similaritons (see Figs. 2.1 and 2.2) for .W (Z ) given by Eq. (2.18) and also for .W (Z ) given by Eq. (2.19) for .c = 0.2 and .c = 0.02. For these two values of .c, Figs. 2.1 and 2.2 depict the self-compression of bright and dark similaritons. As the value of c is reduced, the bright similariton as well as the dark similariton undergo nearly 17-fold self-compression as can be seen from Figs. 2.1 and 2.2. Case (ii) : For .n = 3, Eqs. (2.16)–(2.17) result in obtaining .

F(Z ) = 9 − 12 sech2 (Z ), W (Z ) = sech3 (Z ), and G(Z ) = 3 tanh Z .

(2.20)

Now substituting .W (Z ) = sech3 Z into Eq. (2.10), we obtain yet another class of . W (Z ) as

2.4 Dynamics of Similaritons in Parabolic Tapered Profile

15

Fig. 2.3 Plot depicting the propagation of bright similariton for .n = 3 for different tapering; a without generalization, and b, c for generalized tapering for .c = 0.2 and .c = 0.02, respectively. The parameters used in the plots are .C02 = 0.3, . X 0 = 0, .v = 0.3, and .a = 1

Fig. 2.4 Plot depicting the propagation of dark similariton for .n = 3 for different tapering; a without generalization, and b, c for generalized tapering for .c = 0.3 and .c = 0.03, respectively. The parameters used in the plots are .C02 = 0.3, . X 0 = 0, .v = 0.3, and .a = 1

.

W (Z ) =

c+

1 15

[

√ c(c + 1) sech3 Z ]. (8 + 4 sech2 (Z ) + 3 sech4 (Z )) tanh(Z )

(2.21)

Now . F(Z ) and .G(Z ) can also be found from Eqs. (2.11) and (2.12). Now using Eq. (2.21), the intensity of optical bright and dark similaritons for this case can also be found from Eqs. (2.13) and (2.14). Here we have depicted the evolution of both bright and dark similaritons (see Figs. 2.3 and 2.4) for .W (Z ) given by Eq. (2.20) and also for .W (Z ) given by Eq. (2.21) for .c = 0.2 and .c = 0.02. As the value of .c is reduced, we observe that the bright and dark similaritons undergo 13-fold self-compression. Although we have illustrated two interesting cases for which the isospectral deformation technique yielded two different types of .sech2 tapered profiles, we would like to emphasize that the method is quite general and can be applied to generate other 2 .sech -type tapered profiles, as per the requirement of experiment.

2.4 Dynamics of Similaritons in Parabolic Tapered Profile Other type of possible tapering is the parabolic tapering, which finds applications in a number of passive and active optical devices for the propagation of guided wave through them. For instance, parabolic tapering has been proved advantageous for

16

2 Controllable Optical Similaritons in GRIN by Tailoring of Tapered Profile

mode coupling as it reduces the length of multimode interference devices [28–30]. Apart from this, it also finds applications in fiber-based sensors [31, 32]. Here, we study the propagation of similaritons through perimetrically controlled parabolically tapered waveguide and also look into the effects of these parameters on the intensity of similaritons. For parabolic tapered profile, the functional form of . F(Z ) is given as 2 . F(Z ) = −2α1 (1 − 2α1 z ), (2.22) where .α1 is a positive real number. This tapered profile changes its width for different values of .α1 . For example, larger the value of .α1 , the profile becomes narrower. The width and gain profiles from Eqs. (2.8) and (2.9) can be written as .

W (Z ) = exp(−α1 Z 2 ),

(2.23)

G(Z ) = 2α1 Z .

(2.24)

and .

Now using Eq. (2.23), the intensity of optical bright and dark similaritons for parabolic tapering can be found from Eqs. (2.13) and (2.14). Here we have depicted the evolution of bright similariton and dark similariton (see Figs. 2.5 and 2.6) for .α1 = 0.1, 1 and.3, respectively. From Figs. 2.5 and 2.6, it is evident that both the bright

Fig. 2.5 Plot depicting the evolution of bright similariton for parabolic tapering; a .α1 = 0.1, b = 1, and c .α1 = 3. The parameters used in the plots are .C02 = 0.3, . X 0 = 0, .v = 0.3, and .a = 1

.α1

Fig. 2.6 Plot depicting the evolution of dark similariton for parabolic tapering; a .α1 = 0.1, b = 1, and c .α1 = 3. The parameters used in the plots are .C02 = 0.3, . X 0 = 0, .v = 0.3, and .a = 1

.α1

References

17

and dark similaritons undergo 267-fold self-compression, as the taper expansion coefficient—.α1 —increases. Lastly, one can extend the isospectral deformation technique to tailor the parabolic tapering profile also, but due to the complex mathematical structure, it becomes too difficult to handle it analytically.

2.5 Conclusion In conclusion, by utilizing a similarity transformation, we reduced a nonautonomous GNLSE to autonomous NLSE that describes the self-similar propagation through the GRIN amplifier. This reduction yielded a second-order linear differential equation (DE) for the width of the similariton. We have noticed that this DE is similar in mathematical structure to the quantum Schrödinger equation. This in turn enabled us to study the similariton propagation for different types of tapered profiles. By employing the isospectral deformation technique, we have obtained a class of tapering profile through the introduction of Riccati parameter .c. It is found that even small local variations (affected through.c) in the tapering profile lead to rapid self-compression of the respective beam. For illustrative purposes only, .sech2 − and parabolic- type profiles were chosen. It is emphasized that this procedure can be applied for any ‘potential’ . F(Z ). We end this chapter by noting that the chosen ‘wave function’ .W (Z ) in this chapter signifies the ground state wave function of the ‘potential’ . F(Z ), and the excited state wave function will result in singularities of the intensity of the similaritons.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

G.P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 2007) V.R. Almeida, R.R. Panepucci, M. Lipson, Opt. Lett. 28, 1302 (2003) M. Krause, H. Renner, E. Brinkmeyer, Spectroscopy 21, 26 (2006) T.A. Birks, W.J. Wadsworth, P.St.J. Russell, Opt. Lett. 25, 1415 (2000) S.A. Ponomarenko, G.P. Agrawal, Opt. Lett. 32, 1659 (2007) T.S. Raju, P.K. Panigrahi, Phys. Rev. A 84, 033807 (2011) C.Q. Dai, J. Zhang, Opt. Lett. 35, 2651 (2010) V.I. Kruglov, D. M.echin, ´ J.D. Harvey, J. Opt. Soc. Am. B 24, 833 (2007) C. Finot, G. Millot, Opt. Express 12, 5104 (2004) T. Mansuryan, A. Zeytunyan, M. Kalashyan, G. Yesayan, L. Mouradian, F. Louradour, A. Barth.el ´ .emy, ´ J. Opt. Soc. Am. B 25, A101 (2008) G.L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980) W.K. Burns, M. Abebe, C.A. Villarruel, Appl. Opt. 24, 2753 (1985) J. Bures, S. Lacroix, J. Lapierre, Appl. Opt. 22, 1918 (1983) J.C. Campbell, Appl. Opt. 18, 900 (1979) S.A. Ponomarenko, G.P. Agrawal, Phys. Rev. Lett. 97, 013901 (2006) V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Phys. Rev. Lett. 98, 074102 (2007); K. Porsezian, R. Ganapathy, A. Hasegawa, V.N. Serkin, IEEE J. Quantum Electron. 45, 1577 (2009); V.N. Serkin, A. Hasegawa, T.L. Belyaeva, J. Mod. Opt. 57, 1456 (2010)

18

2 Controllable Optical Similaritons in GRIN by Tailoring of Tapered Profile

17. 18. 19. 20.

R. Atre, P.K. Panigrahi, G.S. Agarwal, Phys. Rev. E 73, 056611 (2006) L. Wu, J.F. Zhang, L. Li, C. Finot, K. Porsezian, Phys. Rev. A 78, 053807 (2008) C.Q. Dai, Y.Y. Wang, Q. Tian, J.F. Zhang, Ann. Phys. 327, 512 (2012) F. Cooper, A. Khare, U. Sukhatme, Supersymmetry in Quantum Mechanics (Chap. 6) (World Scientific, Singapore, 2001); and references therein B. Mielnik, J. Math. Phys. 25, 3387 (1984); L. Infeld, T.E. Hull, Rev. Mod. Phys. 23, 21 (1951) A. Khare, C.N. Kumar, Mod. Phys. Lett. A 8, 523 (1993) C.N. Kumar, J. Phys. A: Math. and Gen. 20, 5397 (1987) ; B. Dey, C.N. Kumar, Int. J. Mod. Phys. A 9, 2699 (1994) E.D. Filho, J.R. Ruggiero, Phys. Rev. E 56, 4486 (1997) W. Alka, A. Goyal, C.N. Kumar, Phys. Lett. A 375, 480 (2011) R. Gupta, S. Loomba, C.N. Kumar, IEEE J. Quantum Electron. 48, 847 (2012) C.N. Kumar, R. Gupta, A. Goyal, S. Loomba, T.S. Raju, P.K. Panigrahi, Phys. Rev. A 86, 025802 (2012) W.K. Burns, M. Abebe, C.A. Villarruel, Appl. Opt. 26, 4190 (1987) D.S. Levy, R. Scarmozzino, R.M. Osgood, J.I.E.E.E. Photon, Technol. Lett. 10, 830 (1998) A. Hosseini, J. Covey, D.N. Kwong, R.T. Chen, J. Opt. 12, 075502 (2010) M. Ahmad, L.L. Hench, Biosens. Bioelectron. 20, 1312 (2005) R.K. Verma, A.K. Sharma, B.D. Gupta, Opt. Commun. 281, 1486 (2008)

21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

Chapter 3

Butterfly-Shaped and Dromion-Like Waves in GRIN Waveguide

3.1 Introduction The interplay between nonlinearity and dispersion/diffraction leads to the formation of localized particle-like structures called solitons. They find wider applications in diverse fields of science such as nonlinear optics, condensed matter physics, biology, chemistry, fluid dynamics, and plasmas [1–8]. Indeed, it was Hasegawa and Tappert who, for the first time, predicted the existence of the optical solitons in lossless fibers [9]. Since an optical soliton is a stable pulse and can represent an elementary bit, it has many features that makes it attractive for transmission, storage, and processing of digital information. Solitons can be managed through different choices of dispersion and nonlinearity. One such new kind of soliton pulse called butterfly-shaped pulse was first investigated by Wen et al. [10] in optical fibers with variable group-velocity dispersion (GVD) and constant Kerr nonlinearity. Later, he described interactions between these pulses which affect the pulse propagation qualities [11]. Variant of Nonlinear Schrödinger equation (NLSE) which is a foundational model in the field of nonlinear science is used to study such solitons. Moreover, another type of localized coherent structure called dromion having the property of decaying exponentially in all directions [12] has been obtained for the same model [13]. Recently, great progress has been made to study dromions [14–18]. These dromion structures have been studied for (2+1)-dimensional variable coefficient NLSE (vc-NLSE) with .PT symmetric potential [19]. However, both butterfly pulses and dromions for (1+1)dimensional vc-NLSE having .PT symmetry have been less investigated. The concept of .PT symmetry in quantum mechanics was first introduced by Bender et al. [20] according to which a wide class of non-Hermitian Hamiltonians exhibit entirely real eigenvalue spectra provided they respect parity-time (.PT ) symmetry. In quantum mechanics, .PT symmetry dictates that the potential should satisfy the condition .V (x) = V ∗ (−x) where .∗ sign means complex conjugation which implies that the real component of .PT complex potential must be an even function of position, whereas imaginary component should be an odd function of position [20–22]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 T. S. Raju, Controllable Nonlinear Waves in Graded-Index Waveguides (GRIN), SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-97-0441-5_3

19

20

3 Butterfly-Shaped and Dromion-Like Waves in GRIN Waveguide

However, such a pseudo Hermitian has an interesting feature of spontaneous .PT symmetry breaking [23] which leads to a sudden phase transition (above a critical threshold) due to which the eigenvalue spectra of such Hamiltonian no longer remains real, but it becomes completely or partly complex. .PT symmetry breaking was demonstrated for the first time by Guo et al. [24]. Physical systems exhibiting .PT symmetry have motivated investigations on several fronts in physics [25–28]. Recently,.PT symmetry has received considerable attention in optics due to mathematical similarities between Schrödinger equation in quantum mechanics and nonlinear Schrödinger equation in optics..PT symmetry in optics was first introduced by Musslimani et al. group in 2008 [29], and its experimental observation was reported later on in different optical systems [30, 31]. The proposed .PT symmetry can be implemented by a judicious inclusion of a combination of gain or loss regions and the process of index guiding in optical waveguides. This is modeled by complex refractive index distribution .n(x) = n 0 + n R (x) + in I (x) which plays the role of optical potential. It is suggested that the real part (index waveguiding profile) .n R (x) should be symmetric and the imaginary part .n I (x)(gain(+) or loss(-)) should be antisymmetric in the transverse direction. A large number of novel phenomena such as unidirectional propagation [32], formation of dark solitons, vortices and breathers [33, 34], abrupt phase transitions [35], power oscillations and double refraction [36], and absorption enhanced transmission [24] have been found in optical .PT systems. Nonlinearity in such.PT symmetric systems provides a basis for many effects such as the formation of localized modes, nonlinearly-induced .PT symmetry breaking, and all-optical switching. Nonlinear .PT symmetric systems can serve as powerful building blocks for the development of novel photonic devices targeting an active light control [37]. In a nonlinear domain, the effect of Kerr nonlinearity on beam dynamics in .PT symmetric potentials has been examined in detail leading to the formation of stable .PT solitons [29]. The existence of nonlinear localized modes is reported in .PT symmetric optical media [38, 39]. Stable localized solitons have been obtained analytically in non-Kerr media with .PT potentials [40]. Moreover, Dai et al. studied nonautonomous solitons, their management, and tunneling for .PT symmetric NLSE [41–43]. Khare et al. [44] investigated solitons with competing cubic and quintic nonlinearity in the presence of .PT symmetric potential.

3.2 Our Model and Exact Solutions Waveguides are characterized by a specific variation of refractive index in the transverse direction. To explicate the formation of butterfly pulses and dromions in tapered graded index waveguide, we consider a complex refractive index profile of the form .n(x, z) = n 0 + n R (z, x) + in I (z, x) + n 1 F(z)x 2 + n 2 γ (z)I (z, x) where the first term .n 0 is background refractive index, and .n R and .n I are real and imaginary parts of complex .PT symmetric optical potential which is created due to variation of refractive index from core to cladding. We assume .n 1 > 0 and . F(z) is tapering function. .n 2 is the nonlinear Kerr coefficient which characterizes self-focusing

3.2 Our Model and Exact Solutions

21

or self-defocusing nature of medium, and .γ (z) is a dimensionless variable which describes the inhomogeneity of Kerr nonlinearity. Thus, the propagation of optical beam in the waveguide is governed by the following (1+1)-dimensional distributed coefficient NLSE: iu z +

.

β(z) u x x + γ (z)|u|2 u + F(z)x 2 u + [v1 (x, z) + iw1 (x, z)]u = 0, 2

(3.1)

where .u(z, x) is the complex envelope of electric field. .x and .z are transverse and longitudinal coordinates. Using the following gauge and similarity transformations: u(x, z) = A(z)U (X, Z )eiθ(x,z)

.

(3.2)

where . A(z) is the amplitude and . X = X (x, z), . Z = Z (z). We set X of the form .

x . ω(z)

X=

(3.3)

Here, parameter .ω(z) represents dimensionless width of the beam. The phase and effective propagation distance are given by θ (x, z) = C(z)

.

.

z

Z= 0

x2 2

β(s) ds. Bω2 (s)

(3.4)

(3.5)

Then Eq. (3.1) will reduce to the following NLSE: iU Z +

.

B U X X + G 3 |U |2 U + [V (X ) + i W (X )]U = 0, 2

(3.6)

where . B and .G 3 are constants. Then, the quadratic phase coefficient, amplitude, and tapering related to beam width are given as follows: C(z) =

.

/ .

.

A(z) =

F(z) =

ωz βω(z) βG 3 1 γ B ω(z)

βωzz − βz ωz . β 2 ω(z)

(3.7)

(3.8)

(3.9)

22

3 Butterfly-Shaped and Dromion-Like Waves in GRIN Waveguide

Further constraints on real and imaginary parts of.PT symmetric potentials of Eq. 3.1 are given by .v1 (x, z) = Z z V (X ) (3.10) w1 (x, z) = Z z W (X ).

.

(3.11)

3.2.1 Soliton Solution-I In the following, we obtain the self-similar soliton solution when the complex potential is of Scarf II type V (X ) = V0 sech2 (X ),

(3.12)

W (X ) = W0 sech(X )tanh(X ),

(3.13)

.

and .

and where .V0 and .W0 are arbitrary constants. Then Eq. (3.6) possesses the following soliton solution: / 18B 2 + W02 − 18V0 B sech(X )eiΦ(X,Z ) , (3.14) .U (X, Z ) = 18BG 3 with .Φ(X, Z ) = B Z (z)/2 + W0 /3B tan−1 [sin(x)]. Note that the above solution Eq. (3.14) corresponds to solution in [45] when . B = 2, and .G 3 = 1. Hence from Eqs. (3.12) and (3.13) for the above choices of .PT symmetric potentials, we get β(z) V0 sech2 (X ), Bω2

(3.15)

β(z) W0 sech(X )tanh(X ). Bω2

(3.16)

v (x, z) =

. 1

and w1 (x, z) =

.

Thus the solution of Eq. (3.1) under the transformation Eq. (3.14) reads / u(x, z) =

.

18B 2 + W02 − 18V0 B A(z) sech(X )eiψ(x,z) 18BG 3

(3.17)

where .ψ(x, z) = B Z (z)/2 + W0 /3B tan−1 [sinh(x)] + θ (x, z), and . A(z), . X (x, z), and .θ (x, z) satisfy Eqs. (3.8), (3.3), and (3.4), respectively.

3.2 Our Model and Exact Solutions

23

3.2.2 Periodic Solution Here, we obtain a periodic solution of Eq. (3.1) for the periodic .PT symmetric potential 2 . V (X ) = V0 sin (X ), (3.18) and .

W (X ) =

3BW0 sin(X ), 2

(3.19)

and where .V0 and .W0 are arbitrary constants. Then Eq. (3.6) possesses the following periodic solution: / U (X, Z ) =

.

2V0 + BW02 cos(X )eiΦ(X,Z ) , 2G 3

(3.20)

with.Φ(X, Z ) = (V0 − B/2)Z (z) + W0 sin(X ). We note that when. B = 2 and.G 3 = 1, then the solution (3.20) reduces to the corresponding solution in [45]. Hence from Eqs. (3.18) and (3.19) for the above choices of .PT symmetric potentials β(z) V0 sin2 (X ), Bω2

(3.21)

β(z) 3BW0 sin(X ), Bω2 2

(3.22)

v (x, z) =

. 1

and w1 (x, z) =

.

the solution of Eq. (3.1) under the transformation Eq. (3.20) reads as / u(x, z) =

.

2V0 + BW02 A(z) cos(X )eiψ(x,z) , 2G 3

(3.23)

where .ψ(x, z) = (V0 − B/2)Z (z) + W0 sin(X ) + θ (x, z) and . A(z), . X (x, z), and θ (x, z) satisfy Eqs. (3.8), (3.3) and (3.4), respectively.

.

3.2.3 Soliton Solution-II Now we obtain yet another interesting soliton solution, when the complex potential is like Rosen–Morse Potential, i.e. .

V (X ) = V0 sech2 (X ) + W02 B,

(3.24)

24

3 Butterfly-Shaped and Dromion-Like Waves in GRIN Waveguide

and .

W (X ) =

BW0 tanh(X ), 2

(3.25)

with two arbitrary constants .V0 and .W0 . Then Eq. (3.6) possesses the following soliton solution: /

B − V0 sech(X )eiΦ(X,Z ) , G3

U (X, Z ) =

.

(3.26)

with .Φ(X, Z ) = (7W02 /8 + 1/2)B Z (z) + (W0 /2) X (x, z). Hence from Eqs. (3.24) and (3.25) for the above choices of .PT symmetric potentials: v (x, z) =

. 1

and

β(z) (V0 sech2 (X ) + W02 B), Bω2

β(z) .w1 (x, z) = Bω2

(

) BW0 tanh(X ) , 2

(3.27)

(3.28)

the soliton solution-II of Eq. (3.1) under the transformation Eq. (3.26) reads as / u(x, z) =

.

B − V0 sech(X )eiψ(x,z) , G3

(3.29)

where .ψ(x, z) = (7W02 /8 + 1/2)B Z (z) + (W0 /2) X (x, z) + θ (x, z) and . A(z), . X (x, z), and .θ (x, z) satisfy Eqs. (3.8), (3.3), and (3.4), respectively.

3.3 Numerical Simulations Butterfly-Shaped Pulses In this section, we detail the numerical corroborations that we carried out to obtain the butterfly-shaped pulses for dispersion managed soliton using Eq. (3.1). For the following choices of dispersion and width profiles: β(z) = c + sin(bz)2

(3.30)

1 d + cos(az)

(3.31)

.

where .c > 0 and ω(z) =

.

where .|d| > 1 and

γ (z) = γ0 ,

.

(3.32)

3.3 Numerical Simulations

25

we have obtained both analytically and numerically the butterfly-shaped pulses. Previously, these pulses have been obtained for sine dispersion profile [11] and periodic tapering or width. Here, the constant .b is related to the type of diffraction for different waveguides. Also, Wen et al. [10] proposed this type of pulse by taking cosine dispersion profile of GVD. In order to run the simulations, use was made of the split-step Fourier-transform algorithm. The Fourier domain was composed of . N = 512 grid points, and we have taken the step size of the time integration as .dt = 0.01. In Fig. 3.1a we display the surface plot portraying the analytical butterfly-soliton for the self-similar solution-I given by Eq. (3.14). In Fig. 3.1b we display its contour plot. In Fig. 3.1c we depict the numerically obtained butterfly-soliton solution for the initial condi/ 18B 2 +W 2 −18V B

0 0 A(0) sech(X )eiψ(x,0) . And in Fig. 3.1d we display tion: .u(x, 0) = 18BG 3 its contour plot for various parameters specified in the figure caption. As we can see them, there is an excellent agreement between the analytical and numerical solutions obtained. For the same initial conditions and different set of parameter values, we depict the analytical and numerical butterfly-soliton solutions in Fig. 3.2. Here also, we confirm the excellent agreement between analytical and numerical solutions. In Fig. 3.3, we portray the analytically and numerically obtained periodic butterfly solution. For the numerical simulation, here we have taken the initial condition as

Fig. 3.1 a Surface plot depicting the analytical butterfly-soliton solution for the parameters.a = 0.2, = 0.6, .c = 0.08, and .d = 0.05 b contour plot of the same solution for the parameter values as in (a). c Surface plot depicting the numerical butterfly-soliton solution for the same parameters as in (a) and d depicts its contour plot

.b

26

3 Butterfly-Shaped and Dromion-Like Waves in GRIN Waveguide

Fig. 3.2 a Surface plot depicting the analytical butterfly-soliton solution for the parameters.a = 0.2, = 0.6, .c = 0.08, and .d = 0.05 for longer evolution b contour plot of the same solution for the parameter values as in (a). c Surface plot depicting the numerical butterfly-soliton solution for the same parameters as in (a) and d depicts its contour plot for longer evolution

.b

/ V0 u(x, 0) = 2G A(0) cos(X )eiψ(x,0) . We find that the numerical solution is in con3 formity with the analytical solution. In Fig. 3.4, for the same periodic-type of initial condition, for a different set of various parameters specified in the figure caption, we display analytical and numerical solutions for longer evolution. In this case also, we confirm that both the analytical and numerical solutions complement with each other.

.

Dromion Structures: In general, dromion-like structures represent nonlinear waves that decay exponentially in all directions. In the past, these structures have been investigated in (2+1)dimensional partial differential equations. Here, we report this type of structures in (1+1)-dimensional driven NLSE with .PT symmetric potentials that propagate self-similarly in GRIN waveguide. Dromions appear for Gaussian profile of groupvelocity dispersion (GVD) and hyperbolic profile of Kerr nonlinearity parameters given by β(z) = c1 exp(−c2 z 2 )

.

γ (z) =

.

c3 . 1 + c4 z

(3.33) (3.34)

3.3 Numerical Simulations

27

Fig. 3.3 a Surface plot depicting the analytical periodic butterfly solution for the parameters .a = 0.2, .b = 0.6, .c = 0.08, and .d = 0.05; b contour plot of the same solution for the parameter values as in (a). c Surface plot depicting the numerical periodic butterfly solution for the same parameters as in (a) and d depicts its contour plot

The above choices of .β(z) and .γ (z) are related with dispersion profile functions of dispersion decreasing fibers. Dromion structures can be obtained for negative value of .c1 and positive value of .c2 . Values of .c3 and .c4 could be positive or negative. Amplitude of dromions can be adjusted for different values of .c1 , .c2 , .c3 , and .c4 . Dromion for particular values of these constants has been shown in Fig. 3.5a, when the GVD parameter is perturbed by an amount.0.8. Also, when we reduce the perturbation in GVD parameter to a value.0.08, we observe the vibration of dromion-like structure, and then it transforms into oscillation by the phenomenon of modulational instability of the modified coefficient .c2 , and exhibits interference based on two dromion-like structures. This has been depicted in Fig. 3.5b.

28

3 Butterfly-Shaped and Dromion-Like Waves in GRIN Waveguide

Fig. 3.4 a Surface plot depicting the analytical butterfly-soliton solution for the parameters.a = 0.2, = 0.6, .c = 0.08, and .d = 0.05 for longer evolution b contour plot of the same solution for the parameter values as in (a). c Surface plot depicting the numerical butterfly-soliton solution for the same parameters as in (a) and d depicts its contour plot for longer evolution

.b

Fig. 3.5 Plot depicting the unstable evolution of the dromion-like structure with the parameters a .c1 = −10.0, .c2 = 1.5, .c3 = 5.0, .c4 = 1.7, .a = 0.5, and .d = 0.05 b as in Fig. 3.5a except that .c2 = 0.768

3.4 Conclusion In conclusion, we have obtained, analytically and numerically, the butterfly pulses and dromion-like structures for (1+1)-dimensional inhomogeneous or generalized NLSE in the presence of .PT symmetric potentials. Butterfly-shaped pulses have been attained for different types of.PT potentials, and they appear for sine dispersion profile, periodic tapering, and constant nonlinearity. On the other hand, the dromions

References

29

emerge for Gaussian profile of dispersion and hyperbolic nonlinearity. It has been investigated that nonlinearity influences the amplitude of butterfly pluses. Groupvelocity dispersion (GVD) causes compression of pulses while their period can be controlled through width and dispersion both. Similarly, dromion structures can be managed from a single dromion to two dromions through different variables. Thus, the management of such localized structures might be useful in explaining some phenomenon in optical systems that involve graded-index waveguides.

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3 Butterfly-Shaped and Dromion-Like Waves in GRIN Waveguide

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Chapter 4

Controlling Optical Similaritons in GRIN Waveguide Under the Action of Cubic–Quintic Nonlinear Media

4.1 Introduction Recently, there has been tremendous interest in pursuing similaritons because of their ubiquity in finding applications in various fields such as nonlinear optics, hydrodynamics, and quantum field theory. Their remarkable property of preserving shape by changing their widths and amplitudes in accordance with the system modulation parameters allows us to understand various physical nonlinear phenomena like the pulse propagation in self-written waveguides [1], stimulated Raman scatterin [2], the formation of Cantor set fractals in soliton systems [3], the nonlinear propagation of pulses with parabolic intensity profiles in optical fibers with normal dispersion [4], nonlinear compression of chirped solitary waves [5], and in ultra-fast optics [6–8]. The theoretical and experimental description of similariton propagation in Kerr nonlinear media is governed by the well-known nonlinear Schrödinger equation (NLSE) with a cubic nonlinear term [9]. The higher order terms like third-order dispersion, self-steepening, self-frequency shift, quintic term, etc. need to be added to describe the ultra-short pulse propagation in a nonlinear medium. Various optical solitary wave solutions of NLSE with cubic–quintic nonlinear terms and third-order dispersion have been investigated in the literature [10]. In the present work, we are restricting ourselves to the quintic term, and the resulting equation is known as cubic–quintic NLSE (CQNLSE). The CQNLSE describes many situations of physical interest as it arises in plasma physics [11], condensed matter physics [12], nuclear physics [13], nonlinear optics [14], and Bose–Einstein condensates (BECs) [15]. In the context of nonlinear optical fibers, it has been used to describe the pulse propagation in double doped optical fibers where the proper choice of dopants depends upon the value and sign of cubic and quintic parameters [16]. Additionally, it has been employed to study the propagation of an electromagnetic wave in photorefractive materials. Moreover, many different optical materials such as some semiconductors and doped glasses [17], chalcogenide glasses [18], colloids [19], polydiacetylene para-toluene sulfonate [20], and some transparent organic materials [14] have a refractive index © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 T. S. Raju, Controllable Nonlinear Waves in Graded-Index Waveguides (GRIN), SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-97-0441-5_4

31

32

4 Controlling Optical Similaritons in GRIN Waveguide Under …

that can be well described by a cubic–quintic nonlinearity. Being motivated by the applicability of CQNLSE in the field of nonlinear optics, we are interested to obtain the exact similariton solution of CQNLSE which governs the pulse propagation in a tapered graded-index waveguide whose refractive index is given as n(z, x) = n 0 + n 1 F(z)x 2 + n 2 δ(z)I (x, z) − n 4 v(z)I 2 (x, z)

.

(4.1)

where .x is the spatial coordinate, .z is the propagation distance, and I is the beam intensity. The first two terms represent the linear contribution toward the refractive index, whereas the third intensity dependent term signifies the nonlinear contribution. The fourth term is quintic term which arises due to non-Kerr nonlinearity. Here, we have assumed .n 1 > 0 and the dimensionless tapering function F(z) can be positive or negative depending upon whether the graded-index medium acts as a defocusing or focusing medium. In recent years, a lot of work has been done on the existence of similaritons in nonlinear waveguides. The evolution of bright and dark similaritons is studied in the tapered graded-index waveguide using similarity transformation [21]. Interactions of two identical bright similaritons has been investigated in detail inside a one-dimensional nonlinear graded-index waveguide amplifier using lens type transformation [22]. The dynamics of fundamental and higher order dark solitons on the intense parabolic background in a planar graded-index waveguide with selfdefocusing nonlinearity has been demonstrated in Ref. [24]. Additionally, Dai et al. [23] have obtained the exact bright and dark similariton pair solutions in nonlinear waveguides for the generalized NLSE and studied their interactions for the periodic distributed amplification system. Later, Goyal et al. [25] discussed the control of optical similaritons in a graded-index nonlinear waveguide by tailoring the tapering profile. Recently, the nonlinear tunneling properties of similaritons have been studied in an inhomogeneous tapered graded-index nonlinear waveguide [26]. In this paper, we are presenting the exact analytical solutions for CQNLSE by using similarity transformation in two steps to understand the beam propagation through a tapered graded-index nonlinear waveguide. Although several works have been reported where the exact solutions for the variants of NLSE have been obtained by using similarity transformation [21, 27–34], yet our methodology is different from the known similarity transformation method. In the previous works, the variable coefficient NLSE (vcNLSE) has been reduced to constant coefficient NLSE with the help of similarity transformation under certain conditions of the parameters. But we are employing similarity transformation to reduce vcNLSE to CQNLSE with variable nonlinearity and then applying another set of similarity transformations to reduce the CQNLSE with variable nonlinearity to the .φ 6 field equation. Then, by making use of the known solutions of .φ 6 field equation [35], we have presented the class of similariton solutions for the different parametric conditions.

4.2 Our Model Equation and Similarity Transformations

33

4.2 Our Model Equation and Similarity Transformations The wave propagation through a tapered graded-index waveguide can be modeled by the following CQNLSE [36, 37]: 1 iUz + Ux x + a1 δ(z)|U |2 U + b1 v(z)|U |4 U 2 F(z)x 2 U G(z)U + −ı = 0, 2 2

.

(4.2)

where .z and .x are the normalized propagation distance and the spatial coordinate. δ(z), .v(z), . F(z), and .G(z) represent the cubic nonlinearity, quintic nonlinearity, tapering function, and the gain term, respectively. In the absence of tapering term, Eq. (4.2) has been used to study solitons, Gaussian–Hermite, and the compact solutions [34]. For .v(z) = 0, the model equation has been exploited to understand the dynamics of nonautonomous solitons in an inhomogeneous graded-index waveguide [38]. The corresponding constant coefficient has been explored to reveal the connection between solitons and self-similar waves [39]. In addition to nonlinear optics, Eq. (4.2) has also appeared in the context of Bose–Einstein condensates where the cubic nonlinear term corresponds to atom–atom interaction, while quintic term signifies an effective three-body interaction [40]. To obtain the exact solutions for Eq. (4.2), we use the following similarity transformation [21]: ıφ(x,z) .U (x, z) = A(z)ψ(χ , ξ )e , (4.3)

.

where.ξ is the effective propagation distance. The quadratically chirped phase.φ(x, z) and the similarity variable .χ (x, z) can be given as φ(x, z) = C1 (z)

.

χ=

.

x2 x + B1 (z) + D1 (z), 2 2 x − xc (z) . W (z)

(4.4) (4.5)

The parameters .C1 (z), . B1 (z), and . D1 (z) are associated with the phase-front curvature, the frequency shift, and the phase offset, respectively. .W (z) and .xc (t) represent the dimensionless width and the wave center. Substituting Eq. (4.4) along with Eq. (4.5) into Eq. (4.2), the latter reduces to the following CQNLSE with variable cubic nonlinearity 1 iψξ + ψχχ + a1 δ(z)|ψ|2 ψ + b1 |ψ|4 ψ = 0, 2

.

(4.6)

34

4 Controlling Optical Similaritons in GRIN Waveguide Under …

where the effective propagation distance reads z

ξ=

.

0

1 ds. W 2 (s)

(4.7)

The other parameters are related with the width function .W (z) and can be explicitly expressed as .

A(z) =

1 1 1 ∂W , v(z) = 2 , G(z) = − . W (z) A (z) W (z) ∂z

(4.8)

Wzz . W (z)

(4.9)

Wz B0 , B1 (z) = . W (z) W (z)

(4.10)

.

C1 (z) =

.

.

F(z) =

D1 (z) = −

B02 2

z

1 ds. W 2 (s)

0 z

x (z) = W (z)B0

. c

0

1 ds. W 2 (s)

(4.11)

(4.12)

In order to solve Eq. (4.3), we introduce a new similarity variable .η(χ , ξ ) and choose the following form for .ψ(χ , ξ ) [33]: ψ(χ , ξ ) = ρ(ξ )R(η)eıθ(χ,ξ ) ,

.

with η=

.

χ − χg (ξ ) , W1 (ξ )

(4.13)

(4.14)

where .ρ(ξ ) and .χg (ξ ) stand for the wave amplitude and the wave center in the transformed coordinates. The corresponding quadratic phase becomes θ (χ , ξ ) = C2 (ξ )

.

χ2 χ + B2 (ξ ) + D2 (ξ ). 2 2

(4.15)

On substituting Eq. (4.13) along with Eqs. (4.14)–(4.15) into Eq. (4.3), it reduces to the following .φ 6 field equation: .

Rηη + a R + b R 3 + c R 5 = 0,

(4.16)

4.3 Self-similar Solutions

35

where .a = 2a0 , .b = 2a1 , .c = 2b1 with .a0 as a constant; and the system satisfies the following set of constraint conditions: W1 (ξ ) = 1 + C0 ξ,

(4.17)

1 1 ρ(ξ ) = √ =√ , 1 + C0 ξ W1

(4.18)

.

.

δ(z) =

.

B2 (ξ ) =

(4.19)

C0 1 ∂ W1 = , W1 (ξ ) ∂ξ 1 + C0 ξ

(4.20)

χC2 0 + 2a0 χC0 , D2 (ξ ) = , 1 + C0 ξ 2C0 (1 + C0 ξ )

(4.21)

C2 (ξ ) =

.

.

1 1 = , W1 (ξ ) 1 + C0 ξ

χg (ξ ) = −

.

χC0 + χg0 W1 (ξ ), C0

(4.22)

where.C0 ,. X c0 , and. X g0 are constants. Here,.a0 and.C0 both are free parameters which arise in the solution through phase and cubic nonlinearity coefficients, respectively.

4.3 Self-similar Solutions We obtain the exact solutions for Eq. (4.2) through the solutions of Eq. (4.16) for the width function .W (z) = sech[z]. The chosen form of width yields the explicit analytical forms to the tapering profile, gain, cubic, and quintic nonlinearity coefficients as follows: .

F(z) = 1 − 2sech2 [z],

(4.23)

G(z) = tanh[z].

(4.24)

.

δ(z) =

.

1+

C0 ( 2z

1 , + 41 sinh[2z])

v(z) = sech2 [z].

.

(4.25) (4.26)

In Eq. (4.25), .C0 and the range of .z have to be chosen in such a way so that .δ(z) remains well defined. The considered form of.sech2 [z] tapering can be experimentally realized and has been demonstrated in [41].

36

4 Controlling Optical Similaritons in GRIN Waveguide Under …

4.3.1 Family of Bright Soliton Solutions Here, we are interested to lay out different bright similariton solutions exhibited by Eq. (4.2) depending upon the choices of the constants .a, .b, and .c. Case I For .a < 0, .b = 0, and .c > 0, Eq. (4.16) admits the bright similariton of the form √ . R(η) = A sech[Bη] (4.27) with . B 2 = −4a, .c A4 = 43 B 2 . The corresponding solution of Eq. (4.2) becomes U (x, z) =

.

√ 1 1 A sech[Bη]ei(φ+θ ). √ W (z) 1 + C0 ξ

(4.28)

Here, .ξ(z), .η(ξ, χ ), .φ(x, z), and .θ (χ , ξ ) are given by Eqs. (4.7), (4.14), (4.4), and (4.15), respectively. Since .b is associated with the strength of the cubic nonlinearity, the cubic nonlinearity term vanishes while the other parameters are the same. This type of stable spatial similaritons can be supported by homogeneous conservative optical media demonstrated by Ponomarenko et al. [33]. Case II For .a > 0, .b < 0, and .c > 0, Eq. (4.16) possesses the following bright similariton: √ (4.29) . R(η) = A 1 + sech[Bη], where . B 2 = 4a , . A2 = −8a , .15b2 = 64ac. 5 5b Consequently, the solution of Eq. (4.2) can be given as U (x, z) =

.

√ 1 1 A 1 + sech[Bη]ei(φ+θ ). √ W (z) 1 + C0 ξ

(4.30)

The functions .ξ(z), .η(ξ, χ ), .φ(x, z), and .θ (χ , ξ ) can be obtained by using Eqs. (4.7), (4.14), (4.4), and (4.15), respectively. Case III For.a < 0,.b < 0, and.c > 0, Eq. (4.16) permits another bright similariton of the form A sech[Bη] . R(η) = √ (4.31) 1 − Dtanh2 [Bη] with . B 2 = −a, .b A2 = 2(1 + D)B 2 , and .3b2 /4ac = (1 + D)2 /D for . D < −1. The corresponding solution of Eq. (4.2) reads U (x, z) =

.

1 A sech[Bη] 1 √ eı(φ+θ ). √ W (z) 1 + C0 ξ 1 − Dtanh2 [Bη]

(4.32)

Here, .ξ(z), .η(ξ, χ ), .φ(x, z), and .θ (χ , ξ ) are given by Eqs. (4.7), (4.14), (4.4), and (4.15), respectively.

4.3 Self-similar Solutions

37

Case IV In addition to the above-mentioned bright similaritons, Eq. (4.16) and hence Eq. (4.2) exhibit the following algebraic solution for the same conditions on cubic and quintic nonlinearities (.b < 0 and .c > 0): .

where . B = −b, . A =

−2c , 3b

1 A + Bη2

,

(4.33)

and .a = 0. The solution of Eq. (4.2) can be given as

U (x, z) =

.

R(η) = √

1 1 1 √ ei(φ+θ ), √ W (z) 1 + C0 ξ A + Bη2

(4.34)

where .ξ(z), .η(ξ, χ ), .φ(x, z), and .θ (χ , ξ ) are given by Eqs. (4.7), (4.14), (4.4), and (4.15).

4.3.2 Family of Dark Self-similar Solutions We are presenting the different dark similariton solutions possessed by Eq. (4.2) for the different choices of .a, .b, and .c. Case I For.a > 0,.b < 0, and.c > 0, Eq. (4.16) admits the dark similariton solution of the form Atanh[Bη] (4.35) . R(η) = √ 1 − Dtanh2 [Bη] with .(3D − 2)B 2 = −a, .b A2 = 2(3D − 1)(1 − D)B 2 , and .3b2 /4ac = (3D − 1)2 /D(3D − 2) for . D < 0. The corresponding solution of Eq. (4.2) becomes U (x, z) =

.

1 1 Atanh[Bη] √ ei(φ+θ ), √ W (z) 1 + C0 ξ 1 − Dtanh2 [Bη]

(4.36)

where .ξ(z), .η(ξ, χ ), .φ(x, z), and .θ (χ , ξ ) are given by Eqs. (4.7), (4.14), (4.4), and (4.15), respectively. Case II For .a < 0, .b > 0, and .c < 0, Eq. (4.16) exhibits the following dark similariton: A (4.37) . R(η) = √ 1 − Dtanh2 [Bη] with .(3 − 2D)B 2 = −a D, . Db A2 = 2(3 − D)(1 − D)B 2 , and .3b2 /4ac = (3 − D)2 /(3 − 2D) for .0 < D < 1. Consequently, the solution of Eq. (4.2) reads

38

4 Controlling Optical Similaritons in GRIN Waveguide Under …

U (x, z) =

.

1 1 A √ ei(φ+θ ). √ W (z) 1 + C0 ξ 1 − Dtanh2 [Bη]

(4.38)

Here, .ξ(z), .η(ξ, χ ), .φ(x, z), and .θ (χ , ξ ) are given by Eqs. (4.7), (4.14), (4.4), and (4.15), respectively. The intensity plots of dark similaritons obtained from Eqs. (4.36) and (4.38) have been plotted in Fig. 4.1a and b, for . D = −1. We observe a 2-fold compression of the dark similariton as we increase the value of the detuning parameter. Also in Fig. 4.2a and b we display the dark similariton solution for . D = 0.5. In this case, we notice that the dark similariton undergoes a 1.5-fold compression as we increase the value of the detuning parameter. Besides dark similaritons, Eq. (4.2) admits the kink and the double-kink solutions which are described in detail in the following. Kink Solution For .a < 0, .b > 0, and .c < 0, Eq. (4.16) permits the following kink type solution: .

√ R(η) = A 1 + tanh[Bη]

(4.39)

Fig. 4.1 a Surface √ plot depicting the evolution of dark soliton solution for the parameters. A = 0.2, = 0.2,. B = a/5,.c0 = 0.02, and. D = −1. b Surface plot of the same solution for the parameter values as in (a) excepting that .a = 1.5

.a

Fig. 4.2 a Surface √ plot depicting the evolution of dark soliton solution for the parameters. A = 1.0, = 0.05,. B = 2a,.c0 = 0.02, and. D = 0.5. b Surface plot of the same solution for the parameter values as in (a) excepting that .a = 1.5

.a

4.5 Computer Simulations

39

with . B 2 = −a, . A2 = −(2a/b), and .b2 = (16/3)ac. By using Eq. (4.14) the solution of Eq. (4.2) can be given as U (x, z) =

.

√ 1 1 A 1 + tanh[Bη]ei(φ+θ ). √ W (z) 1 + C0 ξ

(4.40)

Double-Kink Solution For .a /= 0, .b /= 0, and .c /= 0, Eq. (4.16) exhibits the double-kink of the form [42, 43] .

R(η) = √

p sinh[qη]

√ √ −2a(2ϵ − 3)/b(ϵ − 3), .q = (−aϵ)/(ϵ − 3), with .p = (−3q 2 / p 4 )(ϵ − 1)/(ϵ). The solution of Eq. (4.2) becomes U (x, z) =

.

(4.41)

ϵ + sinh2 [qη]

1 1 p sinh[qη] √ ei(φ+θ ). √ W (z) 1 + C0 ξ ϵ + sinh2 [qη]

and

c=

.

(4.42)

In Eqs. (4.40) and (4.42), .ξ(z), .η(ξ, χ ), .φ(x, z), and .θ (χ , ξ ) are given by Eqs. (4.7), (4.14), (4.4), and (4.15), respectively. It is interesting to note that the double-kink solution given by Eq. (4.42) transforms to kink solution for the small value of .ϵ.

4.4 The Influence of Cubic Nonlinearity Coefficient It is investigated that the employment of second similarity transformation, given by Eq. (4.13), on CQNLSE with variable cubic nonlinearity (Eq. (4.3)) introduces a free parameter .C0 through the dimensionless width function .W1 (ξ ) which in turn appears in cubic nonlinearity coefficient .δ(z) in the governing model equation. We observe a 3-fold compression of the bright similariton as evidenced from Fig. 4.3a and b when we change the value of .c0 . The other parameter values are specified in the figure caption. It is worthy to mention here that the tapering profile given by Eq. (4.9) resembles the Schrödinger equation which enables one to introduce the isospectral Hamiltonian approach. This approach provides an efficient mechanism to generate a class of tapering profiles and widths through Riccati parameter [44–48] which is another procedure to amplify the intensity of similaritons.

4.5 Computer Simulations In this section, we check the numerical stability of the exact solutions found in the previous sections. For explicitness, we have taken the solution in Eq. (4.30) as initial wave with perturbation as .U (x, z = 0) = U (x)[1 + ϵ]ei(θ+φ) and checked the sta-

40

4 Controlling Optical Similaritons in GRIN Waveguide Under …

Fig. 4.3 a Surface plot depicting the evolution of bright soliton solution for the parameters.a = 0.2, = 0.6, .c0 = 0.1, and . D = 0.25. b Surface plot of the same solution for the parameter values as in (a) excepting that .c0 = 0.08

.b

Fig. 4.4 a Surface plot depicting the numerical evolution of bright soliton solution for the parameters .a = 0.2, .b = 0.6, .c = 0.08, and .d = 0.05 when the perturbation is turned off (.ϵ = 0); b contour plot of the same solution for the parameter values as in (a). c Surface plot depicting the numerical evolution of bright soliton solution for the parameters .a = 0.2, .b = 0.6, .c = 0.08, and .d = 0.05 when the perturbation is turned on (.ϵ = 0.8) and d depicts its contour plot

bility of the wave when the perturbation is off, i.e. .ϵ = 0 and when the perturbation is turned on, i.e. .ϵ = 0.8. We have evolved the solution and found that the bright soliton under the action of competing cubic–quintic nonlinearities is stable, as depicted in Fig. 4.4a–d. Here the step size has been taken as .d x = 0.3125 and .dt = 0.005. Similarly, we have checked the numerical stability of dark soliton given by Eq. (4.28) when the perturbation is in off state, i.e. .ϵ = 0.0 and on state when .ϵ = 0.8. We

4.6 Conclusion

41

Fig. 4.5 a Surface plot depicting the numerical evolution of dark soliton solution for the parameters = 0.2, .b = 0.6, .c = 0.08, and .d = 0.05 when the perturbation is turned off (.ϵ = 0); b contour plot of the same solution for the parameter values as in (a). c Surface plot depicting the numerical evolution of dark soliton solution for the parameters .a = 0.2, .b = 0.6, .c = 0.08, and .d = 0.05 when the perturbation is turned on (.ϵ = 0.8) and d depicts its contour plot

.a

find that the dark soliton, although is stable for the zero perturbation, it becomes unstable after some evolution, when the perturbation is turned on. We depict these corroborations in Fig. 4.5a–d. Here also the same step size has been taken.

4.6 Conclusion In conclusion, for the variable coefficient CQNLSE, we have obtained the exact similariton solutions and tested the numerical stability of the dark similaritons against finite perturbations. This dynamical system was shown to govern the beam propagation in a tapered graded-index nonlinear waveguide. It has been found that for the considered .sech2 -tapered waveguide, the bright similaritons exist either in the absence or with attractive cubic nonlinearity and repulsive quintic nonlinearity while the dark similaritons have been obtained for the opposite signs of cubic and quintic nonlinearity (attractive or repulsive). Additionally, we have demonstrated the existence of kink and double-kink solutions for the vc-CQNLSE. We have also reported that by controlling detuning parameter and cubic nonlinearity, one can achieve rapid beam compression of different family of similaritons in the waveguide. We expect

42

4 Controlling Optical Similaritons in GRIN Waveguide Under …

that our analytical results will be helpful to understand the beam propagation in a tapered graded-index waveguide experimentally.

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30. A.T. Avelar, D. Bazeia, W.B. Cardoso, Phys. Rev. E 79, (2009) 025602; J. Belmonte-Beitia, J. Cuevas, J. Phys. A: Math. Theor. 42, 165201 (2009) 31. J.R. He, L. Yi, H.M. Li, Phys. Rev. E 90, 013202 (2014) 32. C.Q. Dai, Y.Y. Wang, X.G. Wang, J. Phys. A: Math. Theor. 44, 155203 (2011) 33. S.A. Ponomarenko, S. Haghgoo, Phys. Rev. A 81, 051801 (2010) 34. J.F. Zhang, Q. Tian, Y.Y. Wang, C.Q. Dai, L. Wu, Phys. Rev. A 81, 023832 (2010) 35. A. Khare, A. Saxena, K.J.H. Law, J. Phys. A: Math. Theor. 42, 475404 (2009) 36. K. Senthilnathan, A.M. Abobaker, K. Nakkeeran, Chirped self-similar spatial solitary Waves, in Proceedings of the Progress in Electromagnetics Research Symposium, PIERS (Moscow, Russia, August 18–21) (2009), pp. 1396–1398 37. J.D. He, J.F. Zhang, J. Phys. A: Math. Theor. 44, 205203 (2011) 38. Z.Y. Yang, L.C. Zhao, T. Zhang, Y.H. Li, R.H. Yue, Opt. Commun. 283, 3768 (2010) 39. S.A. Ponomarenko, G.P. Aggrawal, Phys. Rev. Lett. 97, 013901 (2006) 40. P. Das, M. Vyas, P.K. Panigrahi, J. Phys. B At. Mol. Opt. Phys. 42, 245304 (2009) 41. A.W. Snyder, J.D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1973), Table 12-7 and p. 266; G. L. Lamb, Elements of Soliton Theory (Wiley, New York, 1980) 42. N.H. Christ, T.D. Lee, Phys. Rev. D 15, 1606 (1975) 43. A. Alka, R. Goyal, C.N. Gupta, T.S. Kumar Raju. Phys. Rev. A 84, 063830 (2011) 44. C.N. Kumar, R. Gupta, A. Goyal, S. Loomba, T.S. Raju, P.K. Panigrahi, Phys. Rev. A 86, 025802 (2012) 45. A. Goyal, R. Gupta, S. Loomba, C.N. Kumar, Phys. Lett. A 376, 3454 (2012) 46. R. Gupta, S. Loomba, C.N. Kumar, IEEE J. Quantum Electron. 48, 847 (2012) 47. P.K. Panigrahi1, R. Gupta, A. Goyal, C.N. Kumar, Eur. Phys. J. Special Topics 222, 655 (2013) 48. K.K. De, A. Goyal, T.S. Raju, C.N. Kumar, P.K. Panigrahi, Opt. Commun. 341, 15 (2015)

Chapter 5

Dynamics of Optical Rogue Waves in a GRIN Waveguide with Different Background Beams

5.1 Introduction Rogue waves are extreme events that “appear from nowhere and disappear without a trace” [1], and have been studied intensively in different physical contexts [2, 3]. In the field of optics, since the experimental realization of optical rogue waves in microstructured fiber [4], much research activities have been undertaken on this rare event due to its applications in various nonlinear optical processes [5–8]. In recent years, a lot of work has been done on the existence of spatial solitons and rogue waves in graded-index waveguides [9–11]. Various authors have studied the evolution of optical beams in nonlinear waveguides having spatially modulated linear and nonlinear refractive indices along the transverse direction [12–14]. Recently, Liu et al. [15] obtained the exact rogue wave solutions on Gaussian background beam in a graded-index waveguide and discussed its potential applications in nonlinear optics. In this work, we present the more general results to study the evolution of optical rogue waves on any background beam through the spatial modulation of linear and nonlinear refractive index coefficients. We consider the sech.2 -shaped, tanh-shaped, and Bessel and Airy beams as background to study the evolution of first-order rogue waves and rogue wave triplet in graded-index waveguide. The optical beam propagation in graded-index waveguide with a refractive index .n = n 0 + n 1 f (x) + n 2 γ (x)I , where first two terms correspond to the linear part of refractive index and the last intensity-dependent term represents the Kerr nonlinearity, is described by the generalized nonlinear Schrödinger equation (GNLSE) iu z + β(x)u x x + γ (x)|u|2 u + f (x)u = 0,

.

(5.1)

where the beam envelope .u, propagation distance .z, and spatial coordinate .x are the normalized variables. The function .β(x) represents the diffraction coefficient, .γ (x) represents the nonlinearity coefficient, and . f (x) shows the variation of linear refractive index along the spatial direction. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 T. S. Raju, Controllable Nonlinear Waves in Graded-Index Waveguides (GRIN), SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-97-0441-5_5

45

46

5 Dynamics of Optical Rogue Waves in a GRIN Waveguide …

To obtain exact solutions of Eq. (5.1), we introduce the following transformation u(z, x) = A(x) V (z, X ) eikz ,

(5.2)

.

where amplitude . A, variable . X are functions of .x, and .k is a constant. Substituting Eq. (5.2) into Eq. (5.1) reduces this equation into constant coefficient NLSE i Vz + VX X + |V |2 V − kV = 0,

(5.3)

.

where .β(x) = X1 2 , γ (x) = A21(x) , and . X (x) = A21(x) d x + c0 . Further, the linear x refractive index coefficient . f (x) is related to amplitude as .

Ax x f + = 0. A β

(5.4)

It is well known that NLSE is exactly solved and has localized solutions, given by bright/dark solitons and Akhmediev breathers. As a limiting case, AB reduces to the lowest order rational solutions called ‘Peregrine solitons’ or ‘rogue waves’ [23], which themselves are the solutions of NLSE. For rogue wave solutions of Eq. (5.3) for .k = 1, one can write the exact solutions of Eq. (5.1) using transformation Eq. (5.2). The expressions for fundamental rogue wave solutions of Eq. (5.1) are given as u R (z, x) = A(x)

.

( 1−4

1 + 2i z 1 + 4z 2 + 2X 2

) ei z .

(5.5)

Here, choosing a localized profile for . A(x), which in turn modifies the linear refractive index coefficient, one can study the evolution of rogue waves on background beam given by . A(x). Choosing . A(x) as a Gaussian beam, these calculations will lead to the results obtained in the Ref. [15]. Here, we show the optical rogue wave solutions on various background beams. The functional form of . A(x) would be chosen in such a way that diffraction .β(x) and nonlinearity .γ (x) coefficients must be spatially localized.

5.2 Evolution of Rogons on Background Beams In the following, we will study the evolution of rogue wave on different background beams. We find that the amplitude of the rogue wave gets amplified when compared to the respective background beam, in each of these cases.

5.2 Evolution of Rogons on Background Beams

47

a

b

5

30

Amplitude

4

20

3

IR

2

10

5

0 5

1

x

0 4

2

0 x

2

0 z

0

4

5 5

Fig. 5.1 Sech.2 -modulated, a amplitude of the rogue wave (red solid line) and the background beam (blue dashed line), and b intensity profile of .u R . The parameters used in the plots are .c0 = 0, a = 0.5, and .b = 1

5.2.1 .sech2 —Background Beam We first study the evolution of rogue wave on sech.2 -shaped background beam, which can be obtained through spatially modified linear refractive index coefficient . f (x). The choice of . A(x) as 2 . A(x) = a + b sech x (5.6) gives the refractive index coefficient as .

( ) f (x) = − 2b sech2 x 2 − 3 sech2 x ( )3 a + b sech2 x ,

(5.7)

where .a and .b are free parameters, and are chosen in such a way that all spatial coefficients have a localized structure. In Fig. 5.1a, we show the amplitude profile of .u R and sech.2 -shaped background beam. It is seen that the amplitude of first-order rogue wave is three times the maximum value of the background beam. In Fig. 5.1b, we present the corresponding intensity profile, . I R = |u R |2 , of optical rogue wave.

5.2.2 .Tanh—Background Beam In this case, we consider tanh-shaped background beam as .

A(x) =

√ a + b tanhx.

(5.8)

48

5 Dynamics of Optical Rogue Waves in a GRIN Waveguide …

a

b

Amplitude

4

20

3

15 2

IR 10 5 0 5

1 0 10

5

0

5

10

15

x

5 0

0 z x 5

10

5

Fig. 5.2 Tanh-modulated, a amplitude of the rogue wave (red solid line) and the background beam (blue dashed line), and b intensity profile of .u R . The parameters used in the plots are .c0 = 0, a = 1.1, and .b = 0.5

The refractive index coefficient takes the form ) ( 5 2 2 . f (x) = b sech x a tanhx + b sech x . 4

(5.9)

In Fig. 5.2, we show the amplitude and intensity profile of rogue wave on tanh-shaped background beam. Here, it is seen that the center of the rogue wave is shifted along the spatial direction due to the background beam.

5.2.3 Bessel Background Beam Here, we consider the evolution of rogue wave in a Bessel waveguide. For this, we chose background beam as .

1 A(x) = √ , a − Jn2 (b x)

(5.10)

where . Jn (x) is the .n-order Bessel function. The profiles of amplitude and intensity of rogue wave on Bessel background beam is shown in Fig. 5.3. Here also, the rogue wave is localized in both directions and have amplitude three times of the background beam.

5.2 Evolution of Rogons on Background Beams

b

a

4 Amplitude

49

15

3

10

2

IR 1

5

5

0 5

0 4

2

0 x

2

4

x

0 z

0 5 5

Fig. 5.3 Bessel-modulated, a amplitude of the rogue wave (red solid line) and the background beam (blue dashed line), and b intensity profile of .u R . The parameters used in the plots are .c0 = 0, n = 0, a = 1.8, and .b = 2

Amplitude

b

a

4 3

15

2

10 IR

1

5

5

0 5

0 6

4

2

0 x

2

4

6

x

0 z

0 5 5

Fig. 5.4 Airy-modulated, a amplitude of the rogue wave (red solid line) and the background beam (blue dashed line), and b intensity profile of .u R . The parameters used in the plots are .c0 = 0, n = 0, a = 1.8, and .b = 2

5.2.4 Airy Background Beam In the last part, we consider the evolution of Airy-modulated rogue wave by choosing the background beam as .

1 A(x) = / ( 2 ), a − Ai (x) + Ai2 (−x)

(5.11)

where .Ai(x) is the Airy function. In Fig. 5.4, we show the amplitude and intensity profile of rogue wave on Airy background beam.

50

5 Dynamics of Optical Rogue Waves in a GRIN Waveguide …

5.3 The Splitting of Rogue Wave Recently a new phenomenon, known as “splitting of higher order rogue waves into lower order rogue waves”, has been drawing significant attention [16–18]. Here, one observes that a .n th order rogue wave can be decomposed into .n(n + 1)/2 Peregrine breathers, and hence these are also called fissioned higher order rogue waves. To the lowest order, this effect can be observed in the second-order rogue wave, where its splitting into three first-order rogue waves takes place, well known in the literature as rogue wave triplet. The triplet rogue wave comprises a two parameter family [16]. The rogue wave triplet solution of Eq. (5.1) can be written as u RT (z, x) = A(x)

.

where

) ( G + iK ei z , 1+ D

( [ ) = 12 3 − 4X 4 − 12X 2 4z 2 + 1 ] √ −2 2α X − 80z 4 − 72z 2 + 4γ z , [ ( ) √ K = 24 z 15 − 4X 4 + 12X 2 − 2 2α X ( )] ) ( 1 , −8 2X 2 + 1 z 3 − 16z 5 + γ 2z 2 − X 2 − 2 ) ( ( )2 D = 8X 6 + 12X 4 4z 2 + 1 + 6X 2 3 − 4z 2 [ √ + 64z 6 + 432z 4 + 396z 2 + 9 + α α + 2 2X ( )] [ ( )] 12z 2 − 2X 2 + 3 + γ γ + 4z 6X 2 − 4z 2 − 9 .

(5.12)

.G

(5.13)

Here, .α and .γ are the real free parameters, affecting the size and orientation of the wave profile. When these parameters take zero values, the solution gets localized at the origin with the amplitude five times that of an average crest. For arbitrary nonzero values of these parameters, a triplet of rogue wave, distributed on equilateral triangle, is formed [17, 18], where the distance between the peaks depends on their magnitude. The first experimental observation of these rogue wave triplets took place in a water tank [19]. In Fig. 5.5, we present the corresponding intensity profile,. I RT = |u RT |2 , of optical rogue wave triplet assuming the tanh-shaped and Airy as background beams, respectively. Recently, we demonstrate a theoretical approach, based on the isospectral Hamiltonian approach, to control the intensities of similaritons and rogue waves in the tapered nonlinear waveguide [20, 21]. It introduces a free parameter, called Riccati parameter, which enables one to analytically identify a large manifold of allowed tapering profiles with compatible gain functions. This approach is also found useful to manipulate the relative distance between the successive waves of the rogue wave triplet, along both longitudinal and transverse axes, by modulating the tapering of the

References

51 a

b

20 15 IRT 10 5 0 5

15 10 IRT 5 0 x

5

0 z 10

15 5

5 0

5 5 0 x

0 z 5

5

Fig. 5.5 Intensity profile of rogue wave triplet for, a tanh-shaped background beam, and b Airy background beam, respectively. The parameters used in the plots are .α = 150 and .γ = 80

waveguide [22]. Here, Eq. 5.4 governing the amplitude and graded-index profile can be transformed into a Schrödinger-like equation which enables one to introduce the isospectral Hamiltonian approach. This way one can construct one parameter family of graded-index profile which paves the way to produce energetic rogue waves and to manipulate the relative distance between the successive peaks of rogue wave triplet.

5.4 Conclusion In conclusion, in this chapter, we have elaborated the mechanism to generate and control the spatially modulated optical first-order rogue waves and rogue wave triplet for a GNLSE, on background beams such as Airy–Bessel, .sech2 , and .tanh background beams. By this method, we have revealed that the characteristics of RWs are well maintained while the amplitude of the first-order RW gets enhanced three times the maximum value of the Airy–Bessel and sech2 background beams and five times in the case of RW triplet. Thus, we anticipate that the results described in this chapter may be experimentally realizable in situations on small-amplitude background beams for RWs in nonlinear optics, involving graded-index waveguides.

References 1. 2. 3. 4. 5. 6. 7. 8.

N. Akhmediev, A. Ankiewicz, M. Taki, Phys. Lett. A 373, 675 (2009) M. Onorato et al., Phys. Rep. 528, 47 (2013) A. Chabchoub, M. Fink, Phys. Rev. Lett. 112, 124101 (2014) D.R. Solli, C. Ropers, P. Koonath, B. Jalali, Nature 450, 1054 (2007) J.M. Dudley et al., Opt. Exp. 17, 21497 (2009) B. Kibler et al., Nat. Phys. 6, 790 (2010) S. Vergeles, S.K. Turitsyn, Phys. Rev. A 83, 061801 (2011) N. Akhmediev, J.M. Dudley, D.R. Solli, S.K. Turitsyn, J. Opt. 15, 060201 (2013)

52 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

5 Dynamics of Optical Rogue Waves in a GRIN Waveguide …

S.A. Ponomarenko, G.P. Agrawal, Opt. Lett. 32, 1659 (2007) H. Li, F. Song, Opt. Commun. 277, 174 (2007) A. Goyal, R. Gupta, S. Loomba, C.N. Kumar, Phys. Lett. A 376, 3454 (2012) F. Ye, Y.V. Kartashov, B. Hu, L. Torner, Opt. Exp. 17, 11328 (2009) Y. Wang, R. Hao, Opt. Commun. 285, 2171 (2012) W.P. Zhong, M.R. Beli´c, T. Huang, J. Opt. Soc. Am. B 30, 1276 (2013) C. Liu et al., Opt. Lett. 39, 1057 (2014) A. Ankiewicz, D.J. Kedziora, N. Akhmediev, Phys. Lett. A 375, 2782 (2011) P. Gaillard, J. Phys. A: Math. Theor. 44, 45204 (2011) P. Dubard, P. Gaillard, C. Klein, V.B. Matveev, Eur. Phys. J. Spec. Top 185, 247 (2010) A. Chabchoub, N. Akhmediev, Phys. Lett. A 377, 2590 (2013) C.N. Kumar, R. Gupta, A. Goyal, S. Loomba, T.S. Raju, P.K. Panigrahi, Phys. Rev. A 86, 025802 (2012) 21. A. Goyal, R. Gupta, C.N. Kumar, T.S. Raju, P.K. Panigrahi, Opt. Commun. 300, 236 (2013) 22. R. Gupta, C.N. Kumar, V.M. Vyas, P.K. Panigrahi, Phys. Lett. A 379, 314 (2015)

Chapter 6

Controlling Similaritons in Two-Dimensional GRIN Waveguide Through Riccati Parameter

6.1 Introduction It is well-known that vc-NLSE can support the wave propagation in graded-index nonlinear waveguides, in which the interaction between the modes is taken care of by the coupling term known as the cross-phase modulation (XPM) [1]. Practically speaking, vc- NLSE represents an apt model due to the inevitable presence of inhomogeneity in the system. The two-dimensional NLSE with distributed coefficients and sign-alternating Kerr nonlinearity in a layered medium produces stable twodimensional solitons [2, 3]. Few stable localized solutions of the two-dimensional NLSE with distributed coefficients have been obtained by numerical simulations [4–6]. Recently, propagation behavior of two-breather, Kuznetsov–Ma soliton solutions were studied based on (2+1)-dimensional coupled NLSE [7]. Interestingly, Johnson and Drazin [8] and Calogero and Degasperis [9] use the group of transformations under which a given partial differential equation (PDE) is invariant, and they identify the similarity form of the solutions as a standard procedure for reducing PDEs to ODEs. However, in the recent past, scientists are using the direct similarity approach. It is simply assumed, with reference to a single PDE in two independent variables, for example, that the solution is expressible in the form .u(τ, z) = α(τ, z) + β(τ, z) f (η), where .η = η(τ, z) is the similarity variable. This expression is substituted into the given PDE. It is then required that the PDE reduces to an ODE in . f with .η as the independent variable. Over the last few decades, there is a phenomenal growth in the study of self-similar waves in optics for efficient signal propagation through waveguides, as described in the pioneering works [10–15]. In this chapter, we will extract bright soliton, Akhmediev breather, and first- and secondorder rogue waves (rogons) solutions after reducing vc-CNLSE to standard NLSE using gauge and similarity transformation. After that, we will study two important aspects of evolution and control of these waves. Firstly, we will study the effect of dispersion on the intensity of optical similaritons, Akhmediev breathers, and rogons. Secondly, we will identify a connection between the tapering function and width. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 T. S. Raju, Controllable Nonlinear Waves in Graded-Index Waveguides (GRIN), SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-97-0441-5_6

53

54

6 Controlling Similaritons in Two-Dimensional GRIN Waveguide …

The connecting equation is a second-order differential equation whose mathematical structure is similar to the Schrödinger equation of quantum mechanics. By employing the isospectral Hamiltonian approach of quantum mechanics, we will create a series of compatible tapering function and width pair [16, 17]. This process admits a free parameter called Riccati parameter. Afterward, we shall study the effect of different widths and hence tapering on the intensity of the optical similaritons, Akhmediev breathers, and rogons. We illustrate the procedure with .sech2 -type tapering profiles.

6.2 The Coupled NLSEs with Distributed Parameters Wave propagation in inhomogeneous nonlinear waveguide can be described by the vc-CNLSE [18–21]

.i

β ∂u + ∂z 2

( 2 ) ∂ u ∂ 2u 1 + χ (z)(r11 |u|2 + r12 |v|2 )u + f (z)(x 2 + y 2 )u = ig(z)u + 2 ∂x2 ∂ y2

(6.1)

.i

∂v β + ∂z 2

( 2 ) ∂ v ∂ 2v 1 + χ (z)(r21 |u|2 + r22 |v|2 )v + f (z)(x 2 + y 2 )v = ig(z)v. + 2 ∂x2 ∂ y2

(6.2) The refractive index distribution inside the waveguide is taken as n = n 0 + n 1 f (z)(x 2 + y 2 ) + n 2 χ (z)I (x, y, z), . I (x, y, z) being the optical intensity [6]. Here, the first two terms represent the linear part of the refractive index, and the last term describes a Kerr-type nonlinearity of the waveguide. In Eqs. (6.1) and (6.2), .u(x, y, z) and .v(x, y, z) are the two normalized orthogonal components of electric fields. .x, y are the dimensionless transverse variables and z is the dimensionless propagation distance. .χ (z) and .g(z) are the nonlinearity and gain coefficients, respectively. The function . f (z) describes the geometry of tapered waveguide along the waveguide axis. .β is the constant coefficient of dispersion. .r11 and .r22 are the self-phase modulation (SPM) coefficients for .u(x, y, z) and .v(x, y, z), and .r21 and .r 12 are the cross-coupling coefficients which determine the strength of cross-phase modulation (XPM). We employ gauge and similarity transformations [22–26] .

u = A(z)|r22 − r12 | 2 eΩ(z) 𝚿[X (x, y, z), ζ (z)] exp[iϕ(x, y, z)]

(6.3)

v = A(z)|r11 − r21 | 2 eΩ(z) 𝚿[X (x, y, z), ζ (z)] exp[i(ϕ(x, y, z) + δ)]

(6.4)

1

.

1

.

in Eqs. (6.1) and (6.2). Similarity variable and phase are assumed as

6.2 The Coupled NLSEs with Distributed Parameters

.

X (x, y, z) =

55

kx + ly − X c (z) W (z) (

ϕ(x, y, z) = a(z)(x 2 + y 2 ) + b(z)

.

x y + k l

(6.5) ) + c(z)

(6.6)

where .W (z) and . X c (z) are the dimensionless beam width and position coordinate of the self-similar wave center. .a(z), b(z), and .c(z) are the parameters related to phasefront curvature, frequency shift, and phase offset, respectively. When the system satisfies the following constraints: z

Ω(z) =

g(t)dt,

.

(6.7)

0

.

χ (z) =

.

.

A0 , W (z)

(6.8)

β(k 2 + l 2 )e−2Ω(z) , (r11r22 − r12 r21 )A20

(6.9)

A(z) =

z

X c (z) = 2βmW 0 z

ζ (z) =

.

0

1 dt, W 2 (t)

β(k 2 + l 2 ) dt, W 2 (t)

m Wz , b(z) = , a(z) = 2βW W ( ) z 1 β 1 c(z) = −m 2 dt, + 2 2 (t) 2k 2 2l W 0

(6.10)

(6.11)

.

and .

d2W − β f (z)W = 0, dz 2

(6.12)

(6.13)

with .k, l, m being real constants, both Eqs. (6.1) and (6.2) can be reduced into the standard NLSE 1 ∂ 2𝚿 ∂𝚿 + .i + |𝚿|2 𝚿 = 0 (6.14) ∂ζ 2 ∂ X2

56

6 Controlling Similaritons in Two-Dimensional GRIN Waveguide …

6.3 Family of Exotic Waves The nonlinear Schrödinger equation (NLSE), a vital model, arises in various fields of physics, biochemistry, engineering, and finance. In particular, this equation governs the propagation of solitary waves in long-distance optical communication and ultrafast switching devices. Assuming the general form of the bright soliton as [27] .sech[ p(X − vζ )] exp[iq(X − ωζ )], we have 𝚿 B = sech(X − vζ ) eiv(X −

.

v 2 −1 2v ζ )

,

(6.15)

the bright soliton solution of Eq. (6.14) where .v is the velocity of the bright soliton. Equation (6.14) has another localized solution called Akhmediev breather given by [28] √ √ cos( 2X ) + i 2 sinh ζ iζ (6.16) .𝚿 AB = e . √ √ cos( 2X ) − 2 cosh ζ The breathers are nonlinear waves which can carry energy in a localized and oscillatory fashion. In contrast to solitons which are localized in . X , the breathers are localized in .ζ and oscillating in . X . The breathers are not only a mathematical concept but they have also been realized experimentally in different systems such as in BEC [29], dispersion managed optical waveguides and fibers [30], and Josephson arrays [31]. Breather solutions have been obtained for various nonlinear evolution equations like KdV [32], Gardener equation [33], modified-KdV [34], and NLSE. In the context of NLSE, these spatially periodic solutions are termed as Akhmediev breathers. Equation (6.14) has rogue wave solution with the following basic structure [35–39]: ] [ Hν (X, ζ ) + i K ν (X, ζ )) iζ ν e , (6.17) .𝚿 Rν = (−1) + Dν (X, ζ ) where .ν is the order of the solution; . Hν , K ν , Dν are polynomials of different order. For first-order rogue wave solution H1 = 4,

(6.18)

K 1 = 8ζ,

(6.19)

D1 = 1 + 4ζ 2 + 4X 2 .

(6.20)

.

.

.

The second-order rogue wave solution is the .ν = 2 version of Eq. (6.17), with .

H2 = 12[3 − 16X 4 − 24X 2 (4ζ 2 + 1) − 4α X − 80ζ 4 − 72ζ 2 + 4γ ζ ],

(6.21)

6.3 Family of Exotic Waves .K2

= 24[ζ (15 − 16X 4 + 24X 2 − 4α X ) − 8(4X 2 + 1)ζ 3 − 16ζ 5 + γ (2ζ 2 − 2X 2 −

57 1 )], 2

(6.22) and .

D2 = 64X 6 + 48X 4 (4ζ 2 + 1) + 12X 2 (3 − 4ζ 2 )2 + 64ζ 6 + 432ζ 4 + 396ζ 2 + 9 +α[α + 4X (12ζ 2 − 4X 2 + 3)] + γ [γ + 4ζ (12X 2 − 4ζ 2 − 9)], (6.23)

where .α and .γ are arbitrary real constants. The higher order rogue waves have not only been predicted theoretically but up to fifth-order these waves have been observed experimentally [40]. Rogue wave, sometimes known as freak wave or monster wave, is a briefly formed, single, exceptionally large amplitude wave. Rogue waves are localized both in space and time and witness a unique event in which they appear from nowhere and disappear without a trace. We should mention here that, just as solitary waves are known as solitons, the term rogons is coined for rogue waves. The corresponding terminologies oceanic rogons, optical rogons, and matter rogons are used in the field of hydrodynamics, nonlinear optics and BECs, respectively. Because of the constant scaling factor the intensity corresponding to .|v|2 will have same profile as .|u|2 , and hence in the subsequent analysis we shall denote intensity . I = |u|2 . The general expression of the intensity for bright similaritons . I B , Akhmediev breathers . I AB , first-order rogue waves . I R1 , and second-order rogue waves . I R2 are Be2Ω sech2 (X − vζ ), W2 ⎡ ⎤ √ 2 2Ω 2 Be ⎢ cos ( 2X ) + 2 sinh ζ ⎥ . I AB (x, y, z) = ⎣( )2 ⎦ , √ √ W2 cos( 2X ) − 2 cosh ζ [ ] 1 + 4ζ 2 − 4X 2 Be2Ω 1 + 8 , . I R1 (x, y, z) = W2 (1 + 4ζ 2 + 4X 2 )2 [ ] Be2Ω K 22 + (H2 + D2 )2 , . I R2 (x, y, z) = W2 D22 I (x, y, z) =

. B

(6.24)

(6.25)

(6.26) (6.27)

where . B = A20 (r22 − r12 ) and .v is the velocity of bright soliton solution for NLSE. In the subsequent analysis, we shall assume .g(z) = tanh z, r11 = r22 = 2, r12 = r21 = 1, A0 = 1, k = l = 2; m = 1, and . y = 0. Here, we would like to address the interesting issue of interaction of these exact localized self-similar waves, namely similaritons, AB breathers, and rogue waves. Motivated by the pioneering work of Ponomarenko and Agrawal [41] on the interaction of two similaritons, although not presented in the manuscript, we have conducted a collision experiment of two similaritons with differing amplitudes and velocities, and found that they survive mutual collisions just as solitons do, even though their width, amplitude, and chirp

58

6 Controlling Similaritons in Two-Dimensional GRIN Waveguide …

keep changing (undergo self-compression), during and after the collision. Similarly, motivated by the seminal works of Akhmediev et al. [42] and Frisquet et al. [43], we have conducted collision of two Akhmediev breathers and found that if they are moving with finite velocities, then their collision is very similar to the collision of two solitons, except for the direction of localization and the periodicity of each breather. As regards to the collision of rogue waves, we are still in the process of understanding it more deeply as these waves are not propagating waves, and the results will be reported in a future publication.

6.4 The Effect of Dispersion Dispersion plays a vital role in the evolution and dynamics of the self-similar waves. These effects are observed through the study of the intensity profile. In Fig. 6.1a and b we plotted intensity of Akhmediev breathers (AB), and in Fig. 6.1c and d we have displayed first-order rogue waves versus normalized coordinates for different values of .β—the dispersion coefficient. In these plots, we have taken the width function as . W (z) = sech z. Observing these plots we may conclude that the intensity of the AB decreases as the numerical value of dispersion coefficient increases and the intensity of the first-order rogue waves remains the same for different values of .β.

Fig. 6.1 Surface plots depicting intensity profiles of a Akhmediev breather for.β = 2, b Akhmediev breather for .β = 12, c first-order rogue waves for .β = 2, and d first-order rogue waves for .β = 12

6.5 Riccati Parametrized Tapering Function and Width

59

6.5 Riccati Parametrized Tapering Function and Width It is obvious that .W (z) affects the intensity of self-similar waves explicitly as well as implicitly through its appearance in the transformation variables . X and .ζ . For different tapering profiles, .W (z) can be obtained from Eq. (6.13), whose mathematical structure resembles the Schrödinger equation of quantum mechanics, recognizing .β f (z) as potential and .W (z) as corresponding ground state wave function. From supersymmetric quantum mechanics (SUSY QM), it is known that for a given ground state wave function and energy, isospectral deformation of the Hamiltonian can always be done. This isospectral deformation of the Hamiltonian allows one to construct new Hamiltonians with the same eigenvalue spectrum as that of the original Hamiltonian. These new Hamiltonians shall posses different potentials and different wave functions. These array of new potentials and wave functions are related to original potential through the Riccati parameter, and we shall denote it by .c [44]. Based on the results of SUSY QM, we generate a class of tapering function and width as W 2 (z) 2 d (6.28) . f (z) = f (z) − , β dz c + Z W 2 (s)ds −∞

√ .

W (z) =

c(c + 1) W (z)

c+

Z −∞

W 2 (s)ds

,

(6.29)

where .c is an integration constant, known as Riccati parameter and .c ∈ / [−2, 0]. f (z) is called generalized tapering function and .W (z), the generalized width. In the present paper, we work with .sech2 -type tapering commonly used in fiber optics. The general form of tapering function is [23, 45]

.

.

f (z) =

1 2 [n − n(n + 1) sech2 z], β

(6.30)

W (z) = sechn z,

(6.31)

and width profile .

where .n is a positive integer. Equations (6.30) and (6.31) are in conformity with Eqs. (6.13). We shall discuss two cases .n = 1 and .n = 2 for illustration. Case I: For .n = 1, Eqs. (6.30) and (6.31) yield .

f (z) =

1 [1 − 2 sech2 z] and W (z) = sech z. β

(6.32)

It is now straightforward to obtain a class of . f (z) and .W (z) from Eqs. (6.28) and (6.29) as

60

6 Controlling Similaritons in Two-Dimensional GRIN Waveguide …

Fig. 6.2 Surface plots depicting intensity profiles of second-order rogue waves for generalized tapering. The parameter values are .n = 1, .ν = 2, .β = 1, .α = 80, and .γ = 0 and in a .c = 15, b .c = 1.5, and c .c = 0.2

2 d . f (z) = f (z) − β dz

(

sech2 z c + (1 + tanh z)

) (6.33)

√

and .

W (z) =

c(c + 1) sechz . c + (1 + tanhz)

(6.34)

In Fig. 6.2, we have shown the evolution of self-similar second-order rogue waves for .W (z) given by Eq. (6.34). One can observe that, with the reduction in Riccati parameter value, the intensity profiles of self-similar second-order rogue waves undergo self-compression. Case II: For .n = 2, Eqs. (6.30) and (6.31) yield .

f (z) =

1 [4 − 6 sech2 z] and W (z) = sech2 z. β

(6.35)

The corresponding class of . f (z) and .W (z) are given as .

f (z) = f (z) −

2 d β dz

c+ √

and .

W (z) =

c+

1 (2 3

1 (2 3

sech4 z + (2 + sech2 z)tanhz)

c(c + 1) sech2 z . + (2 + sech2 z)tanhz)

(6.36)

(6.37)

The conclusions for both Case I and Case II are similar and hence can be generalized. In general, one can summarize that intensity of similaritons, Akhmediev breathers, and rogons are highly sensitive to change in Riccati parameter. Change in the Riccati parameter brings in a variation in the tapering function. Therefore, a small variation in the tapering function lead to a rapid change in intensity. In fact, waves become more intensive for smaller values of Riccati parameter. This realization may help in producing high-intensity optical pulses from nonlinear optical devices.

References

61

Before closing this section, we would like to append here several noteworthy comments on the Riccati equation (6.13). For example, in the Miura transformation [46]—.vz + v 2 = f (z), if we substitute .v = WWz , then we obtain the Riccati equation (6.13) that helps us to tailor the tapering profile for a known width of the similariton. By utilizing the Cole–Hopf transformation [47]—. f (z) = WWzz , we can manufacture different varieties of tapering profiles that can be experimentally realizable. Furthermore, if the tapering profile takes the form of a linear function of the length of the fiber, then the Riccati equation (6.13) turns out to be an Airy differential equation. In Eq. (6.13), if .β > 0, then the width of the similariton will be an Airy modulated one, and on the other hand if .β < 0, then the width of the similariton will be a Bessel modulated one.

6.6 Conclusion In conclusion, we have achieved the reduction of a (2+1)-dimensional coupled variable coefficient NLSE that appertains to wave propagation through a gradedindex nonlinear waveguide, to integrable homogeneous NLSE. Several conditions were identified. One of the crucial conditions is that the width function satisfies a second-order differential equation whose mathematical structure is similar to a linear Schrödinger equation of quantum mechanics. Then we have employed the isospectral deformation technique and generated a class of tapering profiles through the Riccati parameter. This approach enabled us to study the effects of different tapering profiles on the intensity of solitary waves. It is observed that even a small variation in the tapering profile led to the rapid intensity variation of the solitary waves. Although, for illustrative purposes, we have chosen .sech2 -profiles, we emphasize that the presented results are quite general and can be applied for any form of tapering function. Generally speaking, one can study the effects with other forms of tapering functions also. Additionally, we have studied the effect of dispersion on the evolution of solitary waves and found that the intensity of the solitary waves become less with the increasing value of the dispersion coefficient.

References 1. G.P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 2007) 2. I. Towers, B.A. Malomed, J. Opt. Soc. Am. B 19, 537 (2002) 3. L. Berge, V.K. Mezentsev, J.J. Rasmussen, P.L. Christiansen, Y.B. Gaididei, Opt. Lett. 25, 1037 (2000) 4. A. Alexandrescu, G.D. Montesinos, V.M. Pérez-García, Phys. Rev. E 75, 046609 (2007) 5. S.L. Xu, N.Z. Petrovi´c, M.R. Beli´c, Phys. Scr. 87, 045401 (2013) 6. C.Q. Dai, S.Q. Zhu, L.L. Wang, J.F. Zhang, Europhys. Lett. 92, 24005 (2010) 7. C.Q. Dai, H.P. Zhu, J. Opt. Soc. Am. B 30, 3291 (2013) 8. P.G. Drazin, R.S. Johnson, Solitons: An Introduction (Cambridge University, 1988) 9. F. Calogero, A. Degasperis, Spectral Transform and Solitons (North Holland, 1982)

62

6 Controlling Similaritons in Two-Dimensional GRIN Waveguide …

10. 11. 12. 13. 14. 15. 16.

V.I. Talanov, Radiophys. Quant. Electron. 9, 260–261 (1966) V.I. Talanov, JETP Lett. 11, 199–201 (1970) J.H. Marburger, Prog. Quant. Electr 4, 35–110 (1975) E.G. Lariontsev, V.N. Serkin, Sov. J. Quantum Electron. 5, 796–800 (1975) B.S. Azimov, M.M. Sagatov, A.P. Sukhorukov, Sov. J. Quantum Electron. 21, 785–786 (1991) V.N. Serkin, A. Hasegawa, T.L. Belyaeva, J. Modern Opt. 57, 456–1472 (2010) C.N. Kumar, R. Gupta, A. Goyal, S. Loomba, T.S. Raju, P.K. Panigrahi, Phys. Rev. A 86, 025802 (2012) R. Gupta, S. Loomba, C.N. Kumar, IEEE J. Quantum Electron. 48, 847 (2012) B.A. Malomed, Soliton Manegement in Periodic Systems (Springer, New York, 2006) H.P. Zhu, Z.H. Pan, Laser Phys. 24, 045406 (2014) J. Belmonte-Beitia, G.F. Calvo, Phys. Lett. A 373, 448 (2009) J. Belmonte-Beitia, V.M. Pérez-García, V. Vekslerchik, V.V. Konotop, Phys. Rev. Lett. 100, 164102 (2008) S.A. Ponomarenko, G.P. Agrawal, Phys. Rev. Lett. 97, 013901 (2006) S.A. Ponomarenko, G.P. Agrawal, Opt. Lett. 32, 1659 (2007) A.T. Avelar, D. Bazeia, W.B. Cardoso, Phys. Rev. E 79, 025602 (2009) W.B. Cardoso, A.T. Avelar, D. Bazeia, M.S. Hussein, Phys. Lett. A 374, 2356 (2010) W.B. Cardoso, A.T. Avelar, D. Bazeia, Phys. Lett. A 374, 2640 (2010) M.J. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons (Cambridge University Press, Cambridge, 2011) D.J. Kedziora, A. Ankiewicz, N. Akhmediev, Phy. Rev. E 85, 066601 (2012) A. Trombettoni, A. smerzi, Phy. Rev. Lett. 86, 2353 (2001) J.N. Kutz, S.G. Evangelides, Opt. Lett. 23(9), 685 (1998) E. Trias, J.J. Mazo, T.P. Orlando, Phys. Rev. Lett. 84(4), 741 (2000) K.W. Chow, R.H.J. Grimshaw, E. Ding, Wave Motion 43(2), 158 (2005) A.V. Slyunyaev, J. Exp. Theor. Phys. 92(3), 529 (2001) K.G. Lamb, O. Polukhina, T. Talipova, E. Pelinovsky, W. Xiao, A. Kurkin, Phys. Rev. E 75(4), 046306 (2007) N. Akhmediev, E. Pelinovsky, Eur. Phys. J. Spec. Top. 185, 1 (2010) N. Akhmediev, A. Ankiewicz, M. Taki, Phys. Lett. A 373, 675 (2009) A. Ankiewicz, D.J. Kedziora, N. Akhmediev, Phys. Lett. A 375, 2782 (2011) D.J. Kedziora, A. Ankiewicz, N. Akhmediev, J. Opt. 15, 1 (2013) D.R. Solli, C. Ropers, P. Koonath, B. Jalali, Nature 450, 1054 (2007) A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, N. Akhmediev, Phys. Rev. E 86(5), 056601 (2012) S.A. Ponomarenko, G.P. Agrawal, Opt. Express 15, 2963 (2007) N. Akhmediev, J.M. Soto-Crespo, A. Ankiewicz, Phys. Lett. A 373, 2137 (2009) B. Frisquet, B. Kibler, G. Millot, Phys. Rev. X 3, 041032 (2013) F. Cooper, A. Khare, U. Sukhatme, Supersymmetry in Quantum Mechanics (World Scientific, Singapore, 2001) T.S. Raju, P.K. Panigrahi, Phys. Rev. A 84, 033807 (2011) R. Miura, J. Math. Phys. 9, 1202 (1968) J.D. Cole, Quart. Appl. Math. 9, 225 (1951); E. Hopf, Comm. Pure Appl. Math. 3, 201 (1950)

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.

Chapter 7

Symbiotic Rogons in (2+1)-Dimensional GRIN Waveguide

7.1 Introduction The recent study has shown that the bit rate in optical communication can be enormously increased by employing the multimode wave propagation through the optical fiber [1]. This wave propagation is described by the mathematical model—the coupled nonlinear Schrödinger equation (CNLSE), in which the interaction between the modes is taken care of by the coupling term named as cross-phase modulation (XPM) [2]. The physical meaning of XPM implies that intense waves can modulate the refractive index which in turn imposes an effect on all co-propagating waves. It finds application in Kerr shutter [3], optical switching [4], and optical fiber amplifiers [5, 6]. For an equal contribution of the self-phase modulation (SPM) and XPM, the system is integrable and such a system is called the Manakov system [7]. CNLSE is one of the basic mathematical models for many physical systems. It governs the wave propagation of two overlapped wave packets in hydrodynamics [8], Langmuir and acoustic waves in plasma [9], Rossby waves [10], etc. In the nonlinear fiber optics, apart from multimode propagation, CNLSE also describes wave propagation in twin-core optical fibers [11], birefringent fibers [12], directional couplers [13], etc. Owing to various inhomogeneities present in the systems and their varying boundary conditions, variable coefficients (vc-) NLSE are being studied comparatively more often than their constant counterparts. Another major reason behind their extensive study is the wide range of applications posed by inhomogeneous nonlinear systems. For instance, nonlinear fibers with inhomogeneous dispersion and nonlinearities are useful in pulse compression [14], stimulated modulation instability [15], and amplification of solitons in long-distance communications [16]. The solutions of interest in inhomogeneous systems are self-similar solutions. These have the characteristic property of maintaining their shape though their amplitude and width modulated spatially, as the wave propagates [17]. These solutions have been studied in the context of the single mode inhomogeneous nonlinear optical waveguide and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 T. S. Raju, Controllable Nonlinear Waves in Graded-Index Waveguides (GRIN), SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-97-0441-5_7

63

64

7 Symbiotic Rogons in (2+1)-Dimensional GRIN Waveguide

single component nonautonomous BEC [18, 19]. Here, we consider vc-CNLSE in which dispersion, nonlinearity, gain, and intermodal dispersion have spatial dependence. Kruglov et al. have given self-similar asymptotic solutions for this equation with the constant coefficients [5]. This equation in the absence of intermodal dispersion term and with the equal strength of SPM and XPM has been investigated by Tian et al. [20], and they obtain one- and two-soliton solutions using the Darboux method. A similar kind of equation arises in the context of nonautonomous Bose–Einstein condensates (BEC) in a time-varying trap [21]. Here, using gauge and similarity transformation [18], we find that vc-CNLSE can support bright–dark, bright, and dark similaritons. Apart from similaritons, we also obtain self-similar breathers and rogue waves for this equation. In recent years, a lot of work has been done on the existence of rogue waves in nonlinear fiber optics [19, 22] and BEC [23]. The first experimental observation of optical rogue waves in the nonlinear optical fiber system was reported by Solli et al. [24]. But, as per our knowledge, multimode rogue wave propagation in inhomogeneous optical fibers has not been studied yet. Further, we observe that careful handling of the SPM and XPM coefficients can help to control the relative intensity of these waves. For the equal strengths of XPM and SPM coefficients, we find that vc-CNLSE possesses semi-rational self-similar rogue wave solutions. Earlier, this has been given for the constant coefficient Manakov system [25]. The chapter proceeds as follows. In Sect. 7.2, we give the model equation and similarity transformations which reduce the vc-CNLSE to constant coefficient CNLSE. In Sect. 7.3, various possible self-similar solutions are presented for asymmetric coupling coefficients. Section 7.4 contains semi-rational self-similar rogue wave solutions for distributive Manakov system. Finally, the results of the paper are summarized in Sect. 7.5.

7.2 Multimode Wave Propagation Multimode wave propagation in inhomogeneous nonlinear fiber can be described by the vc-CNLSE ∂V D(Z ) ∂ 2 V + R(Z )(r11 |V |2 + r12 |U |2 )V + ∂Z 2 ∂T 2 1 ∂V + i G(Z )V, = iα(Z ) ∂T 2 ∂U D(Z ) ∂ 2 U .i + R(Z )(r21 |V |2 + r22 |U |2 )U + ∂Z 2 ∂T 2 1 ∂U + i G(Z )U, = iα(Z ) ∂T 2 i

.

(7.1)

(7.2)

where .V (T, Z ) and .U (T, Z ) are the normalized optical fields, . Z and .T are the dimensionless propagation distance and time. Here, . D(Z ), R(Z ), G(Z ), and .α(Z ) are the spatially inhomogeneous coefficients of the dispersion, nonlinearity, gain,

7.2 Multimode Wave Propagation

65

and intermodal dispersion terms respectively. .r11 and .r22 are the SPM coefficients for .V (T, Z ) and .U (T, Z ), and .r21 and .r12 are the cross-coupling coefficients which determine the strength of XPM. Recently, model Eqs. (1)–(2) were considered in Ref. [26] experimentally, for the generation of picosecond pulse train with alternate carrier frequencies using dispersion oscillating fiber. Using gauge and similarity transformations [18] .

V (T, Z ) = A(Z )𝚿1

[ [

T −Tc (Z ) , ζ (Z ) W (Z )

]

T −Tc (Z ) , ζ (Z ) W (Z )

]

eiΦ(T,Z ) , eiΦ(T,Z ) ,

(7.3)

∂ 2 𝚿1 ∂χ 2

+ (r11 |𝚿1 |2 + r12 |𝚿2 |2 )𝚿1 = 0,

(7.4)

∂ 2 𝚿2 ∂χ 2

+ (r21 |𝚿1 |2 + r22 |𝚿2 |2 )𝚿2 = 0.

(7.5)

U (T, Z ) = A(Z )𝚿2 in Eqs. (7.1) and (7.2), we get ∂𝚿1 + ∂ζ ∂𝚿2 + .i ∂ζ i

.

In Eq. (7.3), .W (Z ) and .Tc (Z ) are the dimensionless width and position of the selfsimilar wave center and .Φ(T, Z ), and the quadratically varying chirped phase is chosen to be ( ) T2 + b1 (Z )T + d1 (Z ) , (7.6) .Φ(T, Z ) = c1 (Z ) 2 where .c1 (Z ), b1 (Z ), and .d1 (Z ) are the parameters related to phase-front curvature, c (Z ) is the self-similar variable frequency shift, and phase offset, respectively..χ = T −T W (Z ) 1 1 and .ζ (Z ) = − 2 W (Z ) + ζ0 corresponds to the effective propagation distance. The similariton width is related to the dispersion coefficient as .

W (Z ) =

D(Z )d Z + W0 ,

(7.7)

where .W0 is the width of similariton at . Z = 0. The other ansatz functions can be consistently defined in terms of .W (Z ), G(Z ), and .α(Z ) as follows: G(Z ) 1 1 d Z ; c1 (Z ) = ; exp 2 W (Z ) W (Z ) α(Z )d Z ; R(Z ) = D(Z )/W (Z )2 A(Z )2 ; b1 (Z ) = W (Z ) b12 W Z d Z ; d1 (Z ) = b1 (Z )α(Z )d Z − 2 ) ( W Z b1 α(Z ) Tc = W (Z ) − dZ + dZ . W (Z ) W (Z )

.

A(Z ) = √

(7.8)

66

7 Symbiotic Rogons in (2+1)-Dimensional GRIN Waveguide

Since vc-CNLSE (Eqs. (7.1) and (7.2)) have reduced to standard CNLSE (Eqs. (7.4) and (7.5)) by the similarity transformation, the solution of the former can be obtained by solutions of the latter. This formalism enables us to realize various kinds of selfsimilar solutions governed by vc-CNLSE by the dispersion management. Wave propagation in single mode fiber has been studied for different kinds of dispersion profiles [27, 28]. Here, we specifically restrict to the periodically varying dispersion coefficient . D(Z ) = cos Z , such that dispersion changes sign continuously from positive to negative. In the context of single mode optical fibers, periodic variation of dispersion finds application in enhancing signal-to-noise ratio, reducing Gordon–Hauss time jitter, and also helpful in suppressing the phase matched condition for four-wave mixing [29]. Coupling coefficients .ri j (where i, j = 1, 2) in Eqs. (7.4) and (7.5) are arbitrary yet. Here, we study two cases: one with the asymmetric coefficients, when the SPM and XPM coefficients can take any arbitrary values and are not necessarily equal among themselves. This condition is justified as the difference between the values of SPM and XPM coefficients can be significant in the multimode fibers if the two waves propagate in different fiber modes. Even in single mode fibers, the values of .r 11 , r 12 , r 22 can differ from each other because of the frequency dependence of modal dispersion. But in this case the difference being small can be neglected [2]. In the other choice of parameters, we consider the case when .r11 = r12 = r21 = r22 . We find that the semi-rational rogue wave solutions, first provided for the homogeneous Manakov system [25], can be transcribed here for the inhomogeneous system due to the gauge and similarity transformations. While we discuss the dynamics of these symbiotic solitons in the next sections, here we mention the seminal works done by Serkin and his group [30–32]. At this moment, it is worthwhile to compare our bright–dark, bright, and dark similaritons discovered in this model with the pioneering investigations of nontrivial interaction scenarios of Manakov’s solitons published for the first time in Refs. [33, 34]. In these seminal works, it was discovered that the interaction of equal amplitude solitons manifests itself as a periodic rotation of the soliton polarization without a temporal shift of the soliton; interaction of unequal amplitude solitons is suppressed. Motivated by these works, in the present paper, we study the dynamics of self-similar waves in vc-CNLSE by considering the differing strengths of SPM and XPM coefficients as well as the distributive Manakov system. Also, multicomponent coherently coupled and incoherently coupled solitons and their collisions were considered in Ref. [35]. For a Manakov system, our results are comparable with these results.

7.3 Nonuniform Coupling Coefficients

67

7.3 Nonuniform Coupling Coefficients 7.3.1 Bright–Dark Self-similar Waves Equations (7.4) and (7.5) possess bright–dark soliton solutions given as 𝚿1 = peiδζ sech(ηχ ),

(7.9)

.

𝚿2 =qe

.

iβζ

tanh(ηχ ),

(7.10)

where . p, q, δ, β, and .η are real constant numbers which are related to SPM and XPM coefficients by the following relations: / q2 =

.

β=

.

−r11 +r21 r22 −r12 2

p2 , η =

r22 qp2 , δ =

p2 2

(

r22 r11 −r12 r21 r22 −r12

)

p2 r22 r11 +r12 r21 −2r12 r11 . 2 r22 −r12

,

(7.11) (7.12)

The solutions will be feasible if SPM and XPM terms satisfy constraint .rii > ri j = √ r ji > r j j along with .ri j < rii r j j where, .i, j = 1, 2 and .i /= j. Bright–dark solitons are also called the symbiotic solitons [36] because each of them can exist only due to the interspecies interaction with the other one. Such types of solutions have been observed experimentally in the photorefractive materials [37] and also in the multicomponent BEC in the harmonic trap [38]. It is worthwhile to describe the main physical mechanism that involved in the so-called soliton pairing effect [39– 43]. These symbiotic solitons are formed due to the fact that the existence of each of these waves depends crucially on the nonlinear cross-phase modulation from the other. Therefore, neither of the waves can exist if the other is not there to provide the necessary amount of phase modulation. Solutions for Eqs. (7.1) and (7.2) can be obtained by using transformation (7.3). The solution parameters being functions of space and time, their shape, width, and phase will change along the propagation direction. The expressions of intensity of the bright–dark similariton solutions can be written as |V (T, Z )|2 = A(Z )2 p 2 sech2 (ηχ ),

.

|U (T, Z )|2 = A(Z )2 q 2 tanh2 (ηχ ).

(7.13)

From Eqs. (7.11) and (7.12), it is clear that the amplitude and phase of the propagating wave can be controlled by varying the strength of the SPM and XPM coefficients. The results presented here are true in general for any choices of dispersion, with the ansatz functions satisfying the conditions given in Eqs. (7.7) and (7.8). Here, the intensity profiles for bright–dark similaritons given by Eq. (7.13) are plotted in Fig. 7.1 for . D(Z ) = cos Z . From the figure, it is clear that one of the propagating waves can be made more prominent over the other by increasing the SPM for the later.

68

7 Symbiotic Rogons in (2+1)-Dimensional GRIN Waveguide

Fig. 7.1 Surface plot depicting the intensity of bright and dark similariton solutions for .α(Z ) = cos(Z ) and .G(Z ) = 1 − tanh(Z ) with the values of .r22 = 6, .r11 = 3, .r12 = r21 = −4 and.W0 = 2

7.3.2 Dark Self-similar Waves For .𝚿1 = a𝚿2 , with .a = as i

.

/

r22 −r12 , r11 −r21

Eqs. (7.4) and (7.5) can be equivalently written

∂ 2 𝚿1 ∂𝚿1 + + k|𝚿1 |2 𝚿1 = 0, ∂ζ ∂χ 2

(7.14)

−r12 r21 where .k = r22 rr2211 −r . Equation (7.14) possesses dark soliton solutions for .k < 0 12 which is given as

𝚿1 = a𝚿2 = (i p + q tanh(χ − vζ ))eiωζ ,

.

(7.15)

where . p, q and .ω are real constant numbers and are related to SPM and XPM coefficients by the following relations: q=

.

√

−2/k, p =

√ (ω + 2)/k, v = 2 p/q, ω < −2.

For the solutions to be well behaved .k must be negative and .a should be real, which are in turn related to the .ri' j s, hence certain constraints are imposed on the values of ' .ri j s which are summarized in Table 7.1. Following Eq. (7.3), corresponding expression for the intensities of the solutions of Eqs. (7.1) and (7.2) are given as |V (T, Z )|2 = a 2 |U (T, Z )|2 = A(Z )2 ( p 2 + q 2 tanh2 (χ − vζ )),

.

(7.16)

with the relations between various parameters given by Eq. (7.8). The intensity profiles of dark similaritons are plotted in Fig. 7.2 for. D(Z ) = cos Z . For the periodically varying dispersion, we obtain dark similaritons on the oscillating background. From

7.3 Nonuniform Coupling Coefficients Table 7.1 Dark self-similar waves XPM (.ri j )' s SPM .(rii )' s >0 >0 .rii < 0 .rii < 0 .r 11 > 0, r22 < 0 .r 11 < 0, r22 > 0 .rii

.ri j

.rii

.ri j .ri j .ri j .ri j .ri j

>0 0 0 >0

69

Magnitude of coupling terms Forbidden √ r11 r22 < ri j √ . r 11 r 22 > ri j .rii > ri j or .ri j > rii Forbidden Forbidden Here, .i, j = 1, 2 and .i /= j

.

Fig. 7.2 Surface plot depicting the intensity of dark–dark similariton solutions for .α(Z ) = cos(Z ) and .G(Z ) = 1 − tanh(Z ) with the values of .r22 = 4, .r11 = 3, .r12 = r21 = −4, and .W0 = 2

the figures, it is clear that one of the propagating modes is darker; this is due to its comparatively lower strength of SPM than that of the other.

7.3.3 Bright Self-similar Waves Equation (7.14) also supports bright solitons given as 𝚿1 = a𝚿2 = sech p(χ − vζ )eiq(χ −ωζ ) .

.

(7.17)

Expression for intensity profiles of bright similaritons for the solutions of Eqs. (7.1) and (7.2) is given as |V (T, Z )|2 = a 2 |U (T, Z )|2 = A(Z )2 sech2 p(χ − vζ ),

.

(7.18)

where. p, q, v, and.ω are real constant numbers, related to various equation parameters as

70

7 Symbiotic Rogons in (2+1)-Dimensional GRIN Waveguide

Table 7.2 Bright self-similar waves SPM .(rii )' s XPM (.ri j )' s >0 >0 .rii < 0 .rii < 0 .r 11 > 0, r 22 < 0 .r 11 < 0, r 22 > 0 .rii

.ri j

.rii

.ri j .ri j .ri j .ri j .ri j

>0 0 0 >0

Magnitude of coupling terms > ri j or .ri j > rii √ r11 r22 > ri j √ . r 11 r 22 < ri j Forbidden .rii < ri j .r 22 < ri j Here, .i, j = 1, 2 and .i /= j .rii

.

Fig. 7.3 Surface plot depicting the intensity of bright similariton solutions for .α(Z ) = cos(Z ) and = 1 − tanh(Z ) with the values of .r22 = 4, .r11 = 3, .r12 = r21 = −4, and .W0 = 2

.G(Z )

.

p=

√

k/2, ω = (q 2 − p 2 )/q, v = 2q,

with .q as an arbitrary free parameter. For the solutions to be well behaved, allowed range of .ri' j s is summarized in Table 7.2. The intensity profiles of the bright similaritons for . D(Z ) = cos Z are plotted in Fig. 7.3, where the other ansatz functions take various functional forms following the consistency conditions given in Eq. (7.8). The profiles depict that one wave is more brighter/intense than the other. As in the previous case, here too, comparatively smaller value of SPM for a specific mode makes it more prominent over the other.

7.3.4 Breather-Similaritons and Rogons Apart from dark and bright similariton solutions, Eq. (7.14) also has rational and breather solutions [44]. Breathers are the oscillatory solutions of standard NLSE, and collisions between them leads to the formation of the rogue waves which have amplitude far greater than that of average wave crest.

7.3 Nonuniform Coupling Coefficients

71

Now, following (7.3) it is straightforward to note that Eqs. (7.1) and (7.2) admit self-similar breathers and rogue waves. The conditions for which the equations attain these solutions are the same as the one given in Table 7.2. Expression of intensity of the breather solutions of Eqs. (7.1) and (7.2) is given as |V (T, Z )|2 = a 2 |U (T, Z )|2 ) ( 2sinh2 ζ + cos2 χ A(Z )2 . = √ k ( 2coshζ − cosχ )2

.

(7.19)

In Fig. 7.4 intensity profiles of self-similar breathers are plotted for periodic form of the dispersion profile, i.e. . D(Z ) = cos Z and considering relations between various ansatz functions given by Eqs. (7.7) and (7.8). From the figure, it is clear that breathers have oscillatory behavior in time and also they evolve periodically in space due to the periodic nature of dispersion profile. Expression of intensity of the rogue solutions of Eqs. (7.1) and (7.2) is given as |V (T, Z )|2 = a 2 |U (T, Z )|2 ( ) 32ζ 2 − 16χ 2 + 8 A(Z )2 1+ . = k (1 + 4ζ 2 + 2χ 2 )2

.

(7.20)

The intensity profile given by Eq. (7.20) is plotted in Fig. 7.5 for . D(Z ) = cos Z . Periodic nature of the . D(Z ) results in the trains of rogue waves. This shows the possibility of rogue waves in the multimode fiber. Here, the amplitude of the wave is much higher than the background which occurs due to the presence of the inhomogeneities in the systems. As in the previously discussed cases, here too the intensity of one of the modes can be made comparatively larger by lowering the strength of its SPM relative to the other. This deterministic generation of rogue waves can find various physical applications such as supercontinuum generation [45].

Fig. 7.4 Surface plot depicting the intensity of breather solutions for .α(Z ) = cos(Z ) and .G(Z ) = 1 − tanh(Z ) with the values of .r22 = 4, .r11 = 3, .r12 = r21 = 2, and .W0 = 2

72

7 Symbiotic Rogons in (2+1)-Dimensional GRIN Waveguide

Fig. 7.5 Surface plot depicting the intensity of rogue wave solutions for .α(Z ) = cos(Z ) and = 1 − tanh(Z ) with the values of .r22 = 4, .r11 = 3, .r12 = r21 = 2, and .W0 = 2

.G(Z )

7.4 Uniform Coupling Coefficients: Distributed Manakov Model 7.4.1 Semi-rational Rogons Various distributive Manakov equations, differing from each other depending upon the underlying system, possess dark-bright solitons and one- and two-soliton solutions [20, 46]. Here, with the application of gauge and similarity transformations, we show the existence of semi-rational self-similar rogue waves in inhomogeneous multimode optical fibers. For .ri j = 2, (.i, j = 1, 2) Eqs. (7.4) and (7.5) possess the multiparametric soliton/rogue wave solutions [25], given by ( .

𝚿1 𝚿2

)

[

L = exp(2iωζ ) B

)] ( ) ( M a2 a1 + , a2 B −a1

(7.21)

2 2 2 2 2 with . L = 3/2 − 8ω ζ + 2a χ + | f | exp(2aχ ), . M = 4 f (aχ − 2iωζ − 1/2) exp(aχ + iωζ ) and . B = 1/2 + 8ω2 ζ 2 + 2a 2 χ 2 + | f |2 exp(2aχ )./Here, .a1

and .a2 are arbitrary real parameters, . f is arbitrary complex constant, .a = a12 + a22 , and .ω = a 2 . These solutions being a combination of exponential and rational terms possess the features of both solitons and rogue waves [25]. By varying the values of . f, a1 , and .a2 , interesting features can be observed. For the increase in magnitude of .| f |, the rogue waves and solitons combine together to form one single pulse. For the same choice of XPM and SPM coefficients (.ri j = 2 for .i, j = 1, 2), the solutions of Eqs. (7.1) and (7.2) can be obtained using the transformation (7.3). The corresponding expressions of intensity can be written as ( .

|V (T, Z )|2 |U (T, Z )|2

)

[ = A(Z )

2

|L|2 |B|2

( 2) ( )] |M|2 a22 a1 + . a22 |B|2 a12

(7.22)

References

73

Fig. 7.6 Surface plot depicting the intensity of semi-rational rogue wave solutions for .α(Z ) = cos(Z ) and .G(Z ) = 1 − tanh(Z ) with the values of .r22 = 4, .r11 = 3, .r12 = r21 = 2, and .W0 = 2

The intensity profiles given by Eq. (7.22) are plotted in Fig. 7.6 for. D(Z ) = cos Z . Periodic nature of the . D(Z ) results in the trains of rogue waves.

7.5 Conclusion To conclude this chapter, in a vc-CNLSE dynamical system, we have described the multimode self-similar wave propagation. Analytically, we discovered that this dynamical system possesses various sets of solutions such as dark-bright, dark, bright similaritons, self-similar breathers, and rogue waves. Interestingly, one of the modes can be made more intense over the other by making the strength of self-phase modulation of latter to be larger in comparison to the former, which may find practical applications in the optical phenomena. For equal strengths of SPM and XPM, we proved the existence of semi-rational self-similar rogue wave solutions (rogons), in this dynamical system. We emphasize that the results presented here can be extended to any form of spatial dependence of dispersion, intermodal dispersion, and gain. Finally, it is hoped that the availability of these analytical solutions can provide us a better understanding of the multimode wave propagation in inhomogeneous optical fibers as well as the complicated dynamics of the various other systems modeled by vc-CNLSE.

References 1. 2. 3. 4. 5.

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Chapter 8

Conclusions and Future Perspectives

In this book, we have dealt with the study of exact and numerical self-similar waves that propagate through the GRIN waveguide. We have found bright and dark as well as rogue waves and rogle triplets in diverse media such as competing cubic–quintic nonlinearities and their effect on the ensuing waves. The study of the dynamical behavior of similaritons in graded-index waveguide is very interesting. We have studied nonautonomous wave propagation in the gradedindex (GRIN) waveguide for different tapering profiles. To fulfill this task, we have used a multi-variate similarity transformation, which maps the inhomogeneous generalized nonlinear Schrödinger equation to the standard nonlinear Schrödinger equation. We have shown here that these different tapering profiles can support both bright and dark similaritons with significantly high intensity. Particularly, we have studied the dynamical behavior of bright and dark similaritons for three different sets of tapering profiles and widths. Interestingly, in the case of periodic width and tapering profile, we have observed that the similariton gets annihilated and regenerated after some time during its evolution, along the length of the waveguide. Subsequently, by utilizing the isospectral Hamiltonian approach with the crucial observation that the equation governing the tapering profile can be transformed into a Schrödinger-like equation, we detailed the procedure to tailor .sech2 -type and parabolic tapering profiles of GRIN waveguides. We have demonstrated that this technique yields a wide class of tapering profiles, for the effective control of similariton structure and their dynamical behavior. We exhibited that this novel technique may be profitably exploited to model different tapering profiles that find practical applications.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2024 T. S. Raju, Controllable Nonlinear Waves in Graded-Index Waveguides (GRIN), SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-97-0441-5_8

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8 Conclusions and Future Perspectives

Also, we have reported the discovery of analytical and numerical butterflyshaped and dromion-like optical waves in a tapered graded-index waveguide. Specifically, we obtained these waves in a generalized nonlinear Schrödinger equation (GNLSE), which describes self-similar wave propagation in GRIN waveguide with variable group-velocity dispersion (GVD), nonlinearity, and gain. Despite the fact that dromion-like structures, which have generally been investigated in the (2 + 1) or higher dimensional systems, are reported in the (1+1)-dimensional GNLSE. We have also illustrated the phenomenon of unbreakable .PT symmetry of these exotic nonlinear waves for three specific cases. These days, competing cubic–quintic media in GRIN wavguides is attracting considerable attention. We have obtained a family of bright and dark similaritons for the variable coefficient cubic–quintic nonlinear Schrödinger equation, by employing the similarity reductions in two steps. Further, the existence of kink and double-kink similaritons in various parameter domains is delineated. Then, we discovered optical rogue waves in a nonlinear graded-index waveguide, with spatially modulated dispersion, nonlinearity, and linear refractive index. The evolution of first-order rogue wave and rogue wave triplet on Airy–Bessel, .sech2 , and .tanh background beams has been studied in detail. It is very interesting to realize these RWs in experimentally realizable situations on small-amplitude background beams in nonlinear optics. Further, we have employed an analytical method based on the gauge-similarity transformation technique for mapping a (2+1)-dimensional variable coefficient coupled nonlinear Schrödinger equations (vc-CNLSE) with dispersion, nonlinearity, and gain to standard NLSE. Under certain functional relations, we construct a large family of self-similar waves in the form of bright similaritons, Akhmediev breathers, and rogue waves. Besides this, we illustrated the procedure to amplify the intensity of self-similar waves using the isospectral Hamiltonian approach. The last chapter has been dedicated to study the variable coefficient coupled nonlinear Schrödinger equations (vc-CNLSE) as a model for the description of multimode wave propagation in inhomogeneously coupled nonlinear optical fibers. We have shown that this dynamical system admits bright–dark, bright, dark similaritons, self-similar breathers, and rogue waves as possible localized configurations, under specific conditions on the self- and cross-phase coupling terms. By appropriately lowering the strength of the self-phase modulation in comparison to the cross-phase coupling terms, out of these solutions, one of the propagating modes can be made prominent. For the equal contributions of self- and cross-phase modulations, we have shown that this inhomogeneous system supports semi-rational self-similar rogue waves (rogons). Starting from the different methods and results presented in this book, we foresee a lot of scope for further research that leads to various future perspectives pertaining to self-similar propagation through GRIN waveguides. In the entire book, we have only presented single soliton solutions or self-similar solutions for various dynamical systems. For example, it is very much desirable to find multisoliton solution in the case of variable coefficient coupled nonlinear Schrödinger equations to study switching dynamics. Interaction of two similariton solutions in GRIN media is an interesting issue to be pursued both analytically and numerically.

8 Conclusions and Future Perspectives

77

Furthermore, we have dealt with the dynamics of self-similar waves in single mode GRIN waveguides only. It is exhilarating to study the dynamics of such self-similar waves in multimode GRIN waveguides. We hope to analyze them in the context of the dynamical systems we have dealt with in this book. Finally, we envision that these self-similar light waves may be launched in long-haul telecommunication networks involving GRIN waveguides, for their effective control.