Open access peer-reviewed chapter

Raman Solitons in Nanoscale Optical Waveguides, with Metamaterials, Having Polynomial Law Nonlinearity Using Collective Variables

By Yanan Xu, Jun Ren and Matthew C. Tanzy

Submitted: September 26th 2017Reviewed: February 9th 2018Published: November 5th 2018

DOI: 10.5772/intechopen.75121

Downloaded: 232

Abstract

A mathematical analysis is conducted to illustrate the controllability of the Raman soliton self-frequency shift with polynomial nonlinearity in metamaterials by using collective variable method. The polynomial nonlinearity is due to the expanding nonlinear polarization P NL in a series over the field E up to the seventh order. Gaussian assumption is selected to these pulses on a generalized mode. The numerical simulation of soliton parameter variation is given for the Gaussian pulse parameters.

Keywords

  • Raman solitons
  • polynomial nonlinearity
  • collective variables

1. Introduction

Much attention has been devoted to the understanding of metamaterials [1, 2, 3, 4]. Through its engineered structures, researchers are able to control and manipulate the electromagnetic fields [5]. Using the freedom of design that metamaterials provide, electromagnetic fields can be redirected at will and propose a design strategy [6]. A general recipe for the design of media that create perfect invisibility within the accuracy of geometrical optics is developed. The imperfections of invisibility can be made arbitrarily small to hide objects that are much larger than the wavelength [7].

Especially, mathematical operations can be performed based on suitably designed metamaterials blocks, such as spatial differentiation, integration, or convolution [8]. Soliton pulse can evolve owning to delicate balance between dispersion and nonlinearity. However, it is always a challenge to compensate for the loss when engineering these types of waveguide using metamaterials. The strong perturbation of a soliton envelope caused by the stimulated Raman scattering confines the energy scalability preventing the so-called dissipative soliton resonance [9]. It is important to know the limit we can reach expanding the nonlinear polarization PNLin a series over the field E[10]. The fourth-order nonlinear susceptibility χ4, the fifth-order nonlinearity χ5, and the seventh-order nonlinearity χ7have been measured [11, 12]. The polynomial mode nonlinearity is due to the nonlinear polarization of metamaterials in the power-series expansion form where terms are kept up to the seventh order in the field E[10, 12, 13, 14, 15]. This chapter conducts mathematical analysis to illustrate the controllability of the Raman soliton self-frequency shift with polynomial nonlinearity in metamaterials by using collective variable method.

2. Governing model

The dimensionless form nonlinear Schrödinger’s equation (NLSE) that governs the propagation of Raman soliton through optical metamaterials, with polynomial law nonlinearity, is given by [16, 17, 18, 19, 20, 21, 22, 23, 24].

itΦzt+a2z2Φzt+c1Φzt2+c2Φzt4+c3Φzt6Φzt=zΦzt+zΦzt2Φzt+zΦzt2Φzt.E1

In this model, Φztrepresents the complex valued wave function with the independent variables being zand tthat represent spatial and temporal variables, respectively. The first term represents the temporal evolution of nonlinear wave, while the coefficient ais the group velocity dispersion (GVD). The coefficients of cjfor j=1,2,3correspond to the nonlinear terms. Together, they form polynomial mode nonlinearity. The polynomial mode nonlinearity is due to the nonlinear polarization of metamaterials in the power-series expansion form where terms are kept up to the seventh order in the field E[10, 12, 13, 14, 15]. It must be noted here that when c2=c3=0and c10, the model Eq. (1) collapses to Kerr mode nonlinearity which is due to third-order polarization PNL[15]. However, if c3=0and c10and c20, one arrives at parabolic mode nonlinearity, and it is from the fifth-order polarization PNL[15, 30]. Thus, polynomial mode stands as an extension version to Kerr and parabolic modes. Actually, the Raman effect is not influenced by the properties of the metamaterials; however, the Raman coefficient combines with the dispersive magnetic permeability of the metamaterials leading to additional higher-order nonlinear terms [10, 12, 14]. The group velocity and self-phase modulation term produce the delicate balance dispersion and nonlinearity that accounts for the formation of the stable soliton. On the right hand side, αdescribes intermodel dispersion, λrepresents the self-steepening term in order to avoid the formation of shocks, and νis the complex higher-order nonlinear dispersion coefficient.

3. Mathematical formulation

The pulse may not only be able to translate as a whole entity, but it may also execute more or less complex internal vibrations depending on the type of the perturbations in the system. This particle-like behavior has led to the formulation of the collective variable CVtechniques [25]. The basic idea is that the soliton solution depends on a collective of variables, called CVs, symbolically Zjj=1N, which represent pulse width, amplitude, chirp, frequency, and so on [25, 26, 27, 28]. To this end, the original field is decomposed into two components, say Φztat position zin the metamaterials and at time t, in the following way:

Φzt=fZ1Z2ZNt+qzt,E2

where the first component fconstitutes soliton solution and the second one qrepresents the residual radiation that is known as small amplitude dispersive waves. Introduction of these NCVs increases the phase space of the dynamical system.

In order for the system to remain in the original phase space and best fit for the static solution, the CV method is obtained by configuring the function fZ1Z2ZNtand minimizes residual free energy (RFE) E, where

E=q2dt=Φztf(Z1Z2ZNt)2dt.E3

The approximation of neglecting the residual field is called “bare approximation” in condensed matter physics [27].

Let CVs evolve only in a particular direction to minimize the PFE in the dynamical system with the following simple way:

Cj=EZj=Zjq2dt=qZjq+qZjqdt.E4

The rate of change of Cjwith respect to the normalized distance is defined as.

Ċj=dCjdz=2ddzqZjqdt,E5

where stands for the real part. Here, the weak equality indicates that the constraints Cjneed not be exactly zero [28].

Then, we define a second set of constraints:

dCjdz0.E6

Through Eqs. (2)(6), it leads to the equations of motion:

Ċj=2k=1NfZjfZkdt2fZjZkqdtdZkdt+Rj,E7

where

Rj=2fZjdΦdzdt,E8

for 1jN.

The set of Eqs. (5)(8) is equivalent to the matrix equation:

Ċ=Czż+R,E9

where

z=Z1Z2ZN,E10
R=R1R2RN,E11

while the N×NJacobian is given by

Cz=C1C2CNZ1Z2ZN=CjZkN×N,E12

with

CjZk=2fZjfZkdt2fZjZkqdt,E13

for 1j,kN.

At this stage, through Eq. (6) we can solve Eq. (9) by the following CV equations of motion:

Ẋ=Cz1R.E14

The set of Eqs. (4)(14) represents the complete CV treatment for the generalized NLSE Eq. (1).

4. Computational results

In this part the adiabatic parameter dynamics of solitons in optical metamaterials with polynomial nonlinearity will be obtained by CV method. A Gaussian is given by

fZ1Z2Z3Z4Z5Z6t=Z1exptZ22m/X32+iZ42tZ22+iZ5tZ2+iZ6,E15

where Z1is the soliton amplitude, Z2is the center position of the soliton, Z3is the inverse width of the pulse, Z4is the soliton chirp, Z5is the soliton frequency, and Z6is the soliton phase. Also, mis the Gaussian parameter, where m>0.

In this case, with N=6:

Cz=C1Z1C1Z2C1Z3C1Z5C1Z5C1Z6C2Z1C2Z2C2Z3C2Z5C2Z5C2Z6C3Z1C3Z2C3Z3C3Z5C3Z5C3Z6C4Z1C4Z2C4Z3C4Z5C4Z5C4Z6C5Z1C5Z2C5Z3C5Z5C5Z5C5Z6C6Z1C6Z2C6Z3C6Z5C6Z5C6Z6,E16
z=Z1Z2Z3Z4Z5Z6,E17
R=R1R2R3R4R5R6,E18

where

R1=aZ1Z4+X1Z5Γ12mm2Z3212m2aZ12X4m+12λZ13Z32Γ1m4Z321m2αZ1+4Z13ν,E19
R2=2aZ12m1Z4+Z1Z5+12maZ12Z52m2Z3312m+αZ1216νZ158m4Z3212mΓ12m=Z12Z4mc1Z3241m+c2Z3261m+C3Z3281m4λZ14Z4Z56Z321mΓ1m+αZ12Z5m2Z3212mλZ14Z52m6Z3212mΓ12m+m1λZ14Z323m6Z3212mΓ1m,E20
R3=νZ15Z33aZ13Z32m1Z4mΓ1m2αZ12X33λZ142Z3+c1Z142Z34Z3212maZ12Z4+Z1Z52Z321m+c2Z163Z36Z3212m+c3Z184Z38Z3212mΓ12m2m2,E21
R4=aZ1212m2Z3212mΓ12m4maZ132Z321mλZ14Z54Z321mΓ1mm+αZ12Z422Z322m+2λZ154Z322mΓ2mmZ12c1Z124Z3232m+c2Z146Z3232m+c3Z168Z3232mαZ52Z3232mΓ32mm,E22
R5=2a12mZ12aZ132Z3212mΓ12mm2αZ12Z42X3232mλZ14Z44X3232mΓ32mm2αZ12X52Z321mλZ14Z54X321mΓ1mmaZ13Γ12mm2Z3212m+2a12maZ12,E23
R6=2αZ1Z42Z321m+λZ14Z44Z321mΓ1mm+αZ1Z52Z3212m+λZ14Z54Z3212mΓ12mm+aZ12m1mZ32212mΓ12m2aZ12.E24

5. Numerical simulation

The nonlinear dynamical system discussed in the previous section is plotted to illustrate the collective variables numerically; see Figure 1. The parameter values are as follows: m=1, a=9.9×102, λ=9.8×102, α=1×101, c1=9.9×101, c2=8×102, c3=8×103, and ν=1.01×101[10, 12, 15, 30].

Figure 1.

(a) Surface plot of soliton amplitude, (b) surface plot of soliton center position, (c) surface plot of soliton inverse width of the pulse, (d) surface plot of soliton chirp variation, (e) surface plot of soliton frequency, and (f) surface plot of soliton phase.

This continuous surface plot shows the dynamical relationship between the time and collective variables, in Figure 1. It shows soliton amplitude, center position, inverse width of the pulse, chirp, and phase keeping the original shape as time goes by. This is because group velocity and self-phase modulation term produce the delicate balance dispersion and nonlinearity that accounts for the formation of the stable soliton. It also describes Stokes Raman scattering that is due to transmitted wave at higher frequency and anti-Stokes Raman scattering where transmitted wave is at lower frequency by Figure 1(e) [29]. These results are consistent with Raman soliton scattering effect.

6. Conclusion

This chapter gives Raman soliton solutions in optical metamaterials that is studied with polynomial nonlinearity. The polynomial mode nonlinearity is due to expanding the nonlinear polarization PNLin a series over the field Eup to the seventh order [13, 14, 15]. The polynomial mode nonlinearity is an extension of the Kerr and parabolic mode nonlinearity, which are from third- and fifth-order polarization PNL[15, 30], respectively. The analytical results are supplemented with numerical simulation by collective variables. The continuous surface plot shows the dynamical relationship between the time and collective variables. It shows soliton amplitude, center position, inverse width of the pulse, chirp, and phase keeping the original shape as time goes by since group velocity and self-phase modulation term produce the delicate balance dispersion and nonlinearity. It also describes Stokes Raman scattering that is due to transmitted wave at higher frequency and anti-Stokes Raman scattering where transmitted wave is at lower frequency by Figure 1(e) [8, 29].

In the future, the set of plot with m1will be plotted, and third-order dispersion (TOD) and fourth-order dispersion (FOD) will be included [15]. Nonlinear polarization of medium in the form of a power-series expansion, keeping the terms up to the ninth order, will be explored [10].

Acknowledgments

This work was supported by NSF EAGER grants: 1649173.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Yanan Xu, Jun Ren and Matthew C. Tanzy (November 5th 2018). Raman Solitons in Nanoscale Optical Waveguides, with Metamaterials, Having Polynomial Law Nonlinearity Using Collective Variables, Emerging Waveguide Technology, Kok Yeow You, IntechOpen, DOI: 10.5772/intechopen.75121. Available from:

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