Excitation and propagation of surface polaritonic rogue waves and breathers

Excitation and propagation of the surface polaritonic rogue waves and breathers are investigated by proposing a coupler free optical waveguide that consists of a transparent layer, middle negative index metamaterial layer, and bottom layer of the cold four level atomic medium. In this planar optical waveguide, a giant controllable Kerr nonlinearity is achieved by sufficient field concentration and a proper set of intensities and detunings of the driven laser fields. As a result, various kinds of temporal surface polaritonic solitons, rogue waves, and breathers can be propagated in the narrow window for electromagnetically induced transparency. We find that the giant intensity and extreme concentration of surface polaritons with low generation power can be achieved by excitation of the first- and second-order peregrine rogue waves. Furthermore, the first- and higher-order surface polaritonic Akhmediev breathers can be propagated at the slow light level due to modulation instability in the proposed optical waveguide. We demonstrate that surface-polariton propagation length can be significantly enhanced by Kuznetsov-Ma breather dynamics.

Figure 1

Free space coupled planar waveguide: the surface polaritons can be excited and propagated through the $x$ axis in the NIMM–quantum emitter interface [39]. The system consists of transparent medium (the optical properties of this medium ${\varepsilon }_t$ and ${\mu }_t$), middle layer of NIMM (with electrical permittivity ${\varepsilon }_N$ and magnetic permeability ${\mu }_N$), and coherently driven cold four-level $N$ type atomic medium is set as a quantum emitter in the bottom layer (the white dots). The inset of the figure denotes the cold atomic medium with energy-level diagram. ${\omega }_1$; $\text{l}\in \left\{\text{c,p,s}\right\}$ are the center frequencies of the couple, probe, and signal laser fields, $|l\rangle$ is the bare atomic state, and ${\mathrm{\Delta }}_1$ are corresponded detunings (see the text for details).

Figure 2

Plot of (a) linear absorption and (b) the real part of the dielectric constant of atomic medium as a function of the frequency deviation ($\omega$). The ultranarrow EIT window is observed due to the existence of the signal and couple laser fields and the resultant interference effects. Moreover, the optical properties of the atomic medium reach to the three level $\mathrm{\Lambda }$ type atomic system when the signal laser switched off ${\mathrm{\Omega }}_{\text{s}}=0$. The parameters used for these figures are ${\mathrm{\Omega }}_{\text{c}}=34.56\text{MHz}$, ${\mathrm{\Omega }}_{\text{s}}=30.79\text{MHz}$, ${\mathrm{\Delta }}_{\text{s}}=15.71\text{MHz}$, and ${\mathrm{\Delta }}_{\text{p}}=11.31\text{MHz}$.

Figure 3

Enhancement factor and the reflection coefficient is plotted as a function of (a) frequency deviation ($\omega$) and (b) angle of incident ($\theta$). In the ultranarrow EIT windows, giant field factor is observed with negligible reflection. The giant field factor (up to $|E_{\text{nd}}{|/E_0|}^2$) in the EIT position promises the efficient excitation of SP modes in the coupler free scheme (see the text for details).

Figure 4

Group velocity of the excited SP modes as a function of $\omega$. The group velocity can be switched from subluminal to superluminal with reduced atomic absorption. The blue solid line represents the SP group velocity for negative small coupling detunings, while the red dashed line denotes the behavior of ${\tilde{V}}_g$ for positive ${\mathrm{\Delta }}_{\text{c}}$ and negligible absorption (see the text for details).

Figure 5

Giant Kerr nonlinearity and sufficiently enhanced group-velocity dispersion. The Kerr nonlinearity ($\text{Re}\left[{\chi }_{\text{pp}}^{\left(3\right)}(\mathit{r},\omega )\right]$) in panel (a) represents a giant self-focusing and self-defocusing optical Kerr in the EIT position. Panel (b) denotes the various types of (${\mathrm{\Delta }}_{\text{c}}$) assisted normal and anomalous GVD as a function of frequency deviation. Balanced GVD-optical Kerr nonlinearity couples are achieved for both self-focusing and self-defocusing regimes. The parameters are the same as Fig. 2.

Figure 6

Formation of bright and dark surface polaritonic solitons: the spatiotemporal dynamics of bright nonlinear SP's in the EIT window is depicted in (a) which demonstrates the existence of the bright mode surface polaritonic soliton. The stability of the two mode subluminal bright surface polaritonic solitons is investigated in (b) by plotting the collision of the corresponding SP solitons. The propagation of superluminal dark nonlinear SP's with negligible linear absorption is depicted in (c). Sufficient stability of these dark SP's [panel (d)] represents the fairly long distance propagation of the superluminal dark SP soliton.

Figure 7

Excitation of the weak light first-order surface polaritonic peregrine waves: 3D representation of the formation and evolution of the first-order surface polaritonic peregrine wave in the NIMM-quantum emitter interface is represented in panel (a). Panel (b) represents the contour map of panel (a), while the effect of phase engineering on the localization and the modulation of its contour map is represented in panel (c).

Figure 8

Observation of the second-order surface polariton peregrine breathers. The giant SP amplitude enhancement $|{\mathrm{\Omega }}_{\text{PP}}^{\left(\text{B}\right)}|\approx 4.58U_0$ with sufficient localized distribution is achieved.

Figure 9

Dynamics of the SP Akhemediev breathers. The dynamics of the probe field intensity $|{\mathrm{\Omega }}_{\text{pp,AB}}/U_0{|}^2$ as a function of normalized length $x/L_{\text{Non}}$ and normalized time $(t-z/{\tilde{V}}_g)/{\tau }_0$ is represented in panel (a). The parameters used are $\mathrm{\Omega }=1.2$, $a=0.32$, and $b=0.96$. Panel (b) represents the corresponding contour map of the panel (a) when the modulation frequency is set as real values. This plot shows the zero velocity of the SP Akhmedieve breather propagation. (c) The nonzero velocity of the SP Akhmediev breathers: this breather propagation can be achieved by considering the complex frequency modulation. The values are the same as panel (a) with ${\mathrm{\Omega }}_{\text{i}}=0.3$.

Figure 10

Dynamics of the higher-order NLSE solution: the collision of the SP Akhmediev breathers is plotted as a function of normalized time ($\tau /{\tau }_0$) and normalized length ($z/L_{\text{Non}}$). The picture shows the significant time and space localizations of the SP breather. The parameters used are $a_2=0.30$, ${\mathrm{\Omega }}_{\text{i}}=0.3$, $l_1=0.6$, and $l_2=-0.6$. Other parameters are the same as Fig. 9.

Figure 11

Observation of the SP Kuznetsov-Ma breathers. This solution of the standard NLSE is localized in both in $x$ and $\tau$ axes and propagates along the interface of the NIMM–atomic medium interface. The probe field intensity $|{\mathrm{\Omega }}_{\text{p}}/U_0{|}^2$ has maximas along the propagation direction with giant SP intensity peaks. The propagation length can be greatly enhanced compared to that of the temporal soliton. The parameter used for this simulation is $\mathrm{\Omega }=1.26$, $b=1.5$, and $T_z\approx 1.3\pi$.