## 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

## 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].

In this model,

## 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

where the first component

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

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:

The rate of change of

where

Then, we define a second set of constraints:

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

where

for

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

where

while the

with

for

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

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

where

In this case, with

where

## 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:

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

In the future, the set of plot with

## Acknowledgments

This work was supported by NSF EAGER grants: 1649173.