Impact of self-steepening and intra-pulse Raman scattering on modulation instability in multiple quantum wells

Abstract

This paper presents the impact of self-steepening (SS) and intra-pulse Raman scattering (IRS) on stabilization of modulation instability of a continuous wave probe field in semiconductor double quantum wells under the mechanism of electromagnetically induced transparency, driven by a controlling field. By adjusting the control field and its related parameters, the linear absorption of the probe is comparatively reduced, and simultaneously, the nonlinear coefficient specifically Kerr nonlinearity is enhanced under the optically induced transparency. Due to the presence of SS, the instability gain and frequency bandwidth of gain are dramatically suppressed at higher values of probe peak power $P_0$. Further, the combined effect of SS and IRS on instability gain enhances the number of frequency sidebands with the increase in $P_0$. The probe field can be stabilized against modulation instability, to an extent of desired frequency bandwidth by suitably selecting the SS and IRS, supported by control field Rabi frequency $\Omega _c$. The results of these investigation offer a novel method for the generation of solitonic or ultrashort optical pulses in the context of high-speed data transmission through optical media.

Appendix

The analytical expression of the right-hand side of Eqs. (15)–(17) is given by

$$\begin{aligned} M^{(1)}= & {} N^{(1)}=R^{(1)}=0, \end{aligned}$$

(56)

$$\begin{aligned} M^{(2)}= & {} -i\frac{\partial a_2^{(1)}}{\partial t_1}, N^{(2)}=-i\frac{\partial a_3^{(1)}}{\partial t_1}, \end{aligned}$$

(57)

$$\begin{aligned} R^{(2)}= & {} -i\left( \frac{\partial }{\partial z_1}+\frac{1}{c}\frac{\partial }{\partial t_1}\right) \Omega _p^{(1)}, \end{aligned}$$

(58)

$$\begin{aligned} M^{(3)}= & {} -i\frac{\partial a_2^{(2)}}{\partial t_1}, N^{(3)}=-i\frac{\partial a_3^{(2)}}{\partial t_1}-\Omega _p^{(1)}a_1^{(2)}, \end{aligned}$$

(59)

$$\begin{aligned} R^{(3)}= & {} -i\left( \frac{\partial \Omega _p^{(2)}}{\partial z_1}+\frac{\partial \Omega _p^{(1)}}{\partial z_2}+\frac{1}{c}\frac{\partial \Omega _p^{(2)}}{\partial t_1}\right) -\kappa a_3^{(1)}a_1^{(2)^*}, \end{aligned}$$

(60)

$$\begin{aligned} M^{(4)}= & {} -i\frac{\partial a_2^{(3)}}{\partial t_1}, N^{(4)}=-i\frac{\partial a_3^{(3)}}{\partial t_1}-\Omega _p^{(1)}a_1^{(3)}-\Omega _p^{(2)}a_1^{(2)},\end{aligned}$$

(61)

$$\begin{aligned} R^{(4)}= & {} -i\left( \frac{\partial \Omega _p^{(3)}}{\partial z_1}+\frac{\partial \Omega _p^{(2)}}{\partial z_2}+\frac{\partial \Omega _p^{(1)}}{\partial z_3}+\frac{1}{c}\frac{\partial \Omega _p^{(3)}}{\partial t_1}\right) -\kappa (a_3^{(1)}a_1^{(3)^*}+a_3^{(2)}a_1^{(2)^*}). \end{aligned}$$

(62)

The elements of the matrix associated with MI are given by

$$\begin{aligned}{} & {} T_{11}(\Omega )=\left\{ -\frac{\beta _2}{2}\Omega ^2-\frac{\beta _3}{6}\Omega ^3+\gamma P_0+P_0 \Omega (2s-\tau _R)\right\} , \end{aligned}$$

(63)

$$\begin{aligned}{} & {} T_{22} (\Omega )=\left\{ \frac{\beta _2}{2}\Omega ^2-\frac{\beta _3}{6}\Omega ^3-\gamma P_0+P_0 \Omega (2s-\tau _R)\right\} , \end{aligned}$$

(64)

$$\begin{aligned}{} & {} T_{12} (\Omega )=\gamma P_0+P_0 \Omega (s-\tau _R), \end{aligned}$$

(65)

$$\begin{aligned}{} & {} T_{21} (\Omega )=-\gamma P_0+P_0 \Omega (s-\tau _R). \end{aligned}$$

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