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]

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

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

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

1

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

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

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

2

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:

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

3

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 ap(z, t), as(z, t), aa(z, t) given by

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

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

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

4

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

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

5

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

6

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

7

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

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

8

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.,

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.