Investigation of influence of Kerr and detuning parameters over instability gain in the three wave resonant interaction of Stimulated Raman backscattering

doi:10.21203/rs.3.rs-2445804/v1

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Investigation of influence of Kerr and detuning parameters over instability gain in the three wave resonant interaction of Stimulated Raman backscattering

https://doi.org/10.21203/rs.3.rs-2445804/v1

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published 15 Apr, 2023

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In this article, we study the modulation instability that rules over the evolution of three wave resonant interaction of backward stimulated Raman process. We intend to investigate the influence of Kerr and detuning parameters over the conventional instability gain under both the anomalous and normal dispersion regimes. We also tend to reveal the inertial instability gain arising in the absence of dispersion and explore its dependence over the system parameters.

stimulated Raman backscattering

modulation instability

anomalous and normal dispersion regimes.

In today’s information age, the advent of the mighty fiber optics technology has quenched the thirst for fast, accurate and high capacity communication. The nonlinear interactions of intense light beams with the molecules of the fiber optic transmission medium lead to various fascinating physical effects which have opened up a large field of potential applications in optical communication systems. One such effect is the stimulated Raman scattering, the inelastic process, which involves the interaction of an optical wave with the vibrational level of the material medium resulting in the rapid growth of the Stokes wave at the expense of the pump wave. Such an interaction has the reputation of generating stimulated emission at new wavelengths resulting in self frequency shifts giving rise to generation of short pulses at broader wave- lengths.

Much has been already dealt with forward stimulated Raman scattering which proves to possess the ability of turning optical fibers into broadband Raman amplifiers and tunable lasers [1]. We are interested in the rich dynamics of stimulated Raman back scattering (SRBS). SRBS observed in liquids [2] gives rise to the localized traveling backward scattered Stokes wave packet at the depletion of the forward propagating continuous wave pump. An appropriate model describing the dynamics of the SRBS process in the form of a resonant nonlinear three wave interaction between the pump wave, material wave and the back scattered Stokes wave has been constructed [3]. The solitary solution of the three wave nonlinear partial differential equation governing the resonant SRBS has been derived and studied earlier [4] and an extensive study of the process in the frame of a more general inertial model has also been carried out [5]. While the investigation of instability that arises on the top of Stokes pulse with modulus and phase of pump wave remaining constant is done with the damping of the pump neglected [6], we present in this paper a general detailed investigation of nonlinear modulation instability (MI) of electromagnetic waves originating in the resonant three wave interaction with damping of the pump wave included. Though the resonant three wave interaction problem with bounded envelopes for the pump, Stokes and material wave has been analyzed quiet deeply, the MI analysis with the continuous pump wave poses a non-conservative complex problem and has not been much discussed. The notion behind the approach carried out here is to elucidate the impact of system parameters over the MI-induced dynamics in the semi-infinite media with continuous wave profiles for the three waves with due consideration of the pump, Stokes and material wave damping.

The universal MI was first theoretically analyzed by Hasegawa and Brinkman [7] and first verified experimentally by Tai et al. [8]. This process provides a natural means of breakup of broad optical wave into train of ultra short pulses [9, 10]. Frequency generation, optical switching, laser techniques, such diverse applications of MI has attracted the attention of many researchers recently. Though this MI phenomenon which manifests itself as self-induced breakup of an initially homogeneous wave during its evolution in a nonlinear medium is found as a general feature in very many diverse fields, it is extensively studied in the field of optical fibers. Scalar MI was reported for the single pump wave propagating in a standard non birefringent fiber and in birefringent fiber the cross phase modulation (XPM) between two modes extends the instability do- main to the normal dispersion regime wherein the system involves more than one field component resulting in the vector MI. Various higher order linear and nonlinear effects such as higher order dispersions, self-steepening and time delayed Raman effects have also been found to strongly influence MI in fibers [11,12]. The evolution of sidebands at a frequency separation from the carrier which is proportional to the square root of the optical pump power is the accompanying feature of MI [13]. Long time evolution results in the growth of side bands near the fundamental wave leading to a mutual exchange of energy and hence the detailed study of growth of side bands proves beneficiary. We approach the MI phenomena that rules over the evolution of three wave resonant interaction of backward stimulated Raman process with the key perspective of studying the impact of detuning and nonlinear parameters over the side band evolution. We carry out the analysis exclusively for anomalous and normal regimes to enhance the practical realization of these phenomena in varied domains.

Our paper is structured as follows. In Sec.2, the theoretical formulation of the problem is presented and the MI conditions both for the anomalous and nor- mal dispersion regimes are determined. In Sec.3, the results of the numerical analysis are discussed and conclusions are made in the Sec.4.

The governing coupled mode equation for the resonant case of stimulated Raman scattering where a pump wave and its backscattered Stokes wave is coupled to a material response proportional to the molecular polarization is given by [6]

$$\left({\partial }_{t}+{\partial }_{z}+i\rho {\partial }_{tt}+{\mu }_{p}\right){E}_{p}={-E}_{s}{E}_{a}+\kappa \left({\left|{E}_{p}\right|}^{2}+2{\left|{E}_{s}\right|}^{2}\right){E}_{p},$$

$$\left({\partial }_{t}-{\partial }_{z}+i\rho {\partial }_{tt}+{\mu }_{s}\right){E}_{s}={E}_{p}{{E}_{a}}^{*}+\kappa \left({\left|{E}_{s}\right|}^{2}+2{\left|{E}_{p}\right|}^{2}\right){E}_{s},$$

$$\left(\left(1+2i\alpha {\mu }_{a}\right){\partial }_{t}+{\mu }_{a}\right){E}_{a}={E}_{p}{{E}_{s}}^{*},$$

where Ep, Es, Ea are the complex envelope amplitude of the pump wave, Stokes wave and the material response respectively and κ, α, ρ are the Kerr, detuning and dispersion parameter respectively. For investigating the MI conditions, we assume the steady state continuous wave solutions of the form

$${E}_{p}=\sqrt{{P}_{p}}\text{exp}\left(i{\varphi }_{pi}z\right),$$

$${E}_{s}=\sqrt{{P}_{s}}\text{exp}\left(i{\varphi }_{si}z\right),$$

Now Eq. (1) admits the steady state form Ea as ${E}_{p}{{E}_{s}}^{*}/{\mu }_{a}$, on substituting these steady state forms in Eq. (1) we obtain the phase factors as follows:

$${\varphi }_{pi}=\kappa \left({P}_{p}+2{P}_{s}\right), {\varphi }_{si}=-\kappa \left({2P}_{p}+{P}_{s}\right)$$

With ${\mu }_{s}=\frac{{P}_{p}}{{\mu }_{a}}$ and${\mu }_{p}=-\frac{{P}_{s}}{{\mu }_{a}}$

The stability of the steady state solution is examined by looking into the system in the presence of small amplitude perturbations a_p(z, t), a_s(z, t), a_a(z, t) given by

$${E}_{p}=\left(\sqrt{{P}_{p}}+{a}_{p}[z,t]\right)\text{exp}\left(i{\varphi }_{pi}z\right),$$

$${E}_{s}=\left(\sqrt{{P}_{s}}+{a}_{s}[z,t]\right)\text{exp}\left(i{\varphi }_{si}z\right),$$

$${E}_{a}=\left(\sqrt{{P}_{p}}\sqrt{{P}_{s}}/{\mu }_{a}+{a}_{a}[z,t]\right)\text{exp}\left(-i{\varphi }_{si}z\right)\text{exp}\left(i{\varphi }_{pi}z\right)$$

Substituting Eq. (4) into the three coupled equations and on linearizing in the perturbation aj the following linearized equations are obtained

$$i\frac{\partial {a}_{p}}{\partial t}-{\varphi }_{pi}{a}_{p}+i\frac{\partial {a}_{p}}{\partial z}-\rho \frac{{\partial }^{2}{a}_{p}}{{\partial t}^{2}}+i{{a}_{p}\mu }_{p}+i\sqrt{{P}_{s}}{a}_{a}+\frac{\sqrt{{P}_{p}}\sqrt{{P}_{s}}{a}_{s}}{{\mu }_{a}}+2\kappa {P}_{p}{a}_{p}+\kappa {P}_{p}{{a}_{p}}^{*}+2\kappa {P}_{s}{a}_{p}+2\kappa \sqrt{{P}_{p}}\sqrt{{P}_{s}}{a}_{s}+2\kappa \sqrt{{P}_{p}}\sqrt{{P}_{s}}{{a}_{s}}^{*}=0,$$

$$i\frac{\partial {a}_{s}}{\partial t}-{\varphi }_{si}{a}_{s}-i\frac{\partial {a}_{s}}{\partial z}-\rho \frac{{\partial }^{2}{a}_{s}}{{\partial t}^{2}}+i{{a}_{s}\mu }_{s}-i\sqrt{{P}_{p}}{{a}_{a}}^{*}-\frac{\sqrt{{P}_{p}}\sqrt{{P}_{s}}{a}_{p}}{{\mu }_{a}}+2\kappa {P}_{s}{a}_{s}+\kappa {P}_{s}{{a}_{s}}^{*}+2\kappa {P}_{p}{a}_{s}+2\kappa \sqrt{{P}_{p}}\sqrt{{P}_{s}}{a}_{p}+2\kappa \sqrt{{P}_{p}}\sqrt{{P}_{s}}{{a}_{p}}^{*}=0$$

$$i\left(1+2i\alpha {\mu }_{a}\right)\frac{\partial {a}_{a}}{\partial t}+i{\mu }_{a}{a}_{a}-i\sqrt{{P}_{s}}{a}_{p}-i\sqrt{{P}_{p}}{{a}_{s}}^{*}=0.$$

In order to solve these set of linearized equations, we have assumed a plane wave ansatz constituting of both forward and backward propagation, having the form

$${a}_{j}={a}_{jf}\text{exp}\left[i\left(Kz-\omega t\right)\right]+{a}_{jb}\text{exp}\left[-i\left(Kz-\omega t\right)\right]$$

with f and b denoting forward and backward propagation. On substituting Eq. (8)into the linearized Eqs. (5)-(7) we get a set of 6 homogenous equations. These will have nontrivial solution only when the determinant of 6×6 stability matrix vanishes. ie.,

$$MF=0$$

The elements of non-zero matrix are

$${M}_{11}=-K+\omega -{\varphi }_{pi}+\rho {\omega }^{2}+i{\mu }_{p}+2\kappa {P}_{p}+2\kappa {P}_{s},$$

$${M}_{12}=\kappa {P}_{p},$$

$${M}_{13}=2\kappa \sqrt{{P}_{p}}\sqrt{{P}_{s}}+\frac{i\sqrt{{P}_{p}}\sqrt{{P}_{s}}}{{\mu }_{a}},$$

$${M}_{14}=2\kappa \sqrt{{P}_{p}}\sqrt{{P}_{s}},$$

$${M}_{15}=i\sqrt{{P}_{s}},$$

$${M}_{21}=\kappa {P}_{p},$$

$${M}_{22}=K-\omega -{\varphi }_{pi}+\rho {\omega }^{2}-i{\mu }_{p}+2\kappa {P}_{p}+2\kappa {P}_{s},$$

$${M}_{23}=2\kappa \sqrt{{P}_{p}}\sqrt{{P}_{s}},$$

$${M}_{24}=2\kappa \sqrt{{P}_{p}}\sqrt{{P}_{s}}-\frac{i\sqrt{{P}_{p}}\sqrt{{P}_{s}}}{{\mu }_{a}},$$

$${M}_{26}=-i\sqrt{{P}_{s}}$$

$${M}_{31}=2\kappa \sqrt{{P}_{p}}\sqrt{{P}_{s}}-\frac{i\sqrt{{P}_{p}}\sqrt{{P}_{s}}}{{\mu }_{a}},$$

$${M}_{32}=2\kappa \sqrt{{P}_{p}}\sqrt{{P}_{s}},$$

$${M}_{33}=K+\omega +{\varphi }_{pi}+\rho {\omega }^{2}+i{\mu }_{p}+2\kappa {P}_{p}+2\kappa {P}_{s},$$

$${M}_{34}= \kappa {P}_{s},$$

$${M}_{36}=-i\sqrt{{P}_{p}},$$

$${M}_{41}=2\kappa \sqrt{{P}_{p}}\sqrt{{P}_{s}},$$

$${M}_{42}=2\kappa \sqrt{{P}_{p}}\sqrt{{P}_{s}}-\frac{i\sqrt{{P}_{p}}\sqrt{{P}_{s}}}{{\mu }_{a}},$$

$${M}_{43}= \kappa {P}_{s},$$

$${M}_{44}=-K+\omega -{\varphi }_{pi}+\rho {\omega }^{2}+i{\mu }_{p}+2\kappa {P}_{p}+2\kappa {P}_{s},$$

$${M}_{45}=i\sqrt{{P}_{p}}$$

$${M}_{51}=-i\sqrt{{P}_{s}}$$

$${M}_{54}=-i\sqrt{{P}_{p}}$$

$${M}_{55}=\left(1+2i\alpha {\mu }_{a}\right)\omega +i{\mu }_{a},$$

$${M}_{62}=i\sqrt{{P}_{s}}$$

$${M}_{63}=i\sqrt{{P}_{p}}$$

To have the occurrence of MI there should be an exponential growth in the amplitude of the perturbed wave which implies the existence of a non vanishing imaginary part in the complex eigen value K. The measure of the gain is given by the parameter G = Im (K) where Im K means the imaginary part of K.

The inertial model which we have considered here includes the nonlinear medium response along with the pump and Stokes wave. Apart from the conventional Benjamin-Feir instability, there is a possibility of inertial instability arising even in the absence of group velocity dispersion. We intend to obtain the surface plots of MI gain which would throw light on the various conditions of the MI pertaining to the above said cases. We have obtained the gain spectrum in the following ways. The conventional MI gain spectrum plotted (i) as a function of Stokes power (ii) as a function of Kerr parameter and (iii) as a function of detuning parameter, in normal and anomalous dispersion regimes explicitly. And then the inertial instability gain spectrum plots in the absence of group velocity dispersion, (iv (a)) as a function of Stokes power and (iv (b)) as a function of Kerr parameter. Here Fig. 1(a) and (b) show the gain spectrum against frequency ω and Stokes power with small value of the fixed pump power for the anomalous and normal dispersion regime. While Fig. 1(c) and (d) depict the same at relatively large value of the pump power. As the pump power is increased, the spectral window permitting MI is broadened both in normal and anomalous regimes, with a considerable increase in the optimum gain value too, however the gain and spectral frequency window permitting MI are larger in anomalous regime than that of the normal regime. But the marked difference in normal and anomalous case is that while the gain curves drain down to zero frequency in anomalous case, normal gain curves have flattened edges sustained in a certain level of frequency. Figure 2(a) and (b) portray the gain plots as a function of frequency ω and nonlinearity Kerr parameter κ. While anomalous regime shows MI lobes in the positive Kerr parameter regime, normal plots depict the same scenario at the negative values of Kerr parameter. As we plot the gain as a function of detuning parameter α and frequency ω in Fig. 3(a) and (b), it is found that there is a small peak along the central frequency line with two distinct side peaks on either side in the anomalous regime. But the normal regime plots tend to show a different scenario with diminished side peaks and sharp peak along the central frequency line near the zero detuning regimes. Next as we explore the inertial instability gain as a function of frequency ω and Stokes power Ps in Fig. 4(a), there are sharp narrow lobes of gain found on either side of central frequency line which gets flattened on the edges. Figure 4(b) portrays the inertial gain as a function of frequency ω and Kerr parameter κ. A noteworthy feature in inertial gain plots is that this gain becomes dominant over the modulational instability gain at large detuning value of α. Note that we have arrived at MI gain analysis considering the medium to be very long, semi infinite, it could be said. However, one could consider a finite media, a possible amplifier configuration with proper boundary conditions as in [14]. Considering the backscattering case, with pump field injected at z = 0, and Stokes field at z = L, L being the length of the fiber, the pump and Stokes field amplitude will no longer be constant, instead will be the functions of propagation distance and the stationary solutions will no longer be homogenous. In such a case, the system of linearized ordinary differential equations for the perturbations would involve spatially varying parameters and hence the stability analysis for propagation over a finite distance L should be performed by studying the modulus of the eigen values of the associated Jacobian matrix. Our future work would include this which would be of much practical relevance to amplifier applications.

In this paper, we have obtained the MI conditions for both anomalous and nor- mal dispersion regimes in a resonant case of stimulated Raman backscattering by plotting gain spectrum as function of Stokes power, Kerr parameter and detuning parameter. From these cases discussed above, where gain is plotted as a function of power, a distinct feature of side lobe formation critically depending upon the parameters is depicted. The characteristic feature of occurrence of inertial instability even in the no dispersion regime is portrayed and dependence of inertial gain over the system parameters is clearly depicted. Thus a detailed investigation on modulation instability has been done in the three wave resonant interaction of SRBS process.

Acknowledgements

PM thanks UGC for its financial assistance provided through the RGNF programmed. This paper is a dedication in honor of Prof. K. Porsezian, one of the coauthors who just died away; it was written with his love and support.

Disclosures

The authors declare no conflicts of interest.

Data availability

The data that has been used is confidential.

Author contributions

AJ carried out the concept and design of the article. PM carried out data interpretation for the article as well as draught manuscript. All authors read and approved the final manuscript.

Ethical Approval

Not applicable

Funding

This research received no specific grant from any funding agency in the public, commercial or not for profit sectors.

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No competing interests reported.

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You are reading this latest preprint version

Investigation of influence of Kerr and detuning parameters over instability gain in the three wave resonant interaction of Stimulated Raman backscattering

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1. Introduction

2. Theoretical Formulation Of The Problem

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