Phonon-Polaritons in Nonlinear Dielectric Medium

Abstract

We discuss the properties of polaritons and obtain theoretically the phonon-polariton spectrum in nonlinear dielectric medium with the third order Kerr-type nonlinearity. We investigate the dependence of number of the polariton spectrum branches on the intensity of electromagnetic field and demonstrate that the appearance of new branches located in the polariton spectrum gap is caused by the dispersion of the third order dielectric susceptibility at the intensive electromagnetic field in the medium. The modulation instability of new spectrum branch waves leads to the appearance of the spatial solitons or cnoidal waves. Also we theoretically investigate the properties of scalar and vector phonon-polariton spatial solitons and cnoidal waves propagating in boundless dielectric medium. These new nonlinear waves one can use for designing the optical devices such as the optical converter, controllable filter, all-optical logic gates, etc.

Appendices

Appendix A

The set of (1.18) can be simplified for linearly polarized wave by turning of the coordinate axes at the angle $\pi /4$,

$$- \frac{\partial }{\partial y}\left( {\frac{\partial e}{\partial x} - \frac{\partial e}{\partial y}} \right) + \left( {\alpha_{1} + \alpha_{3} \left| e \right|^{2} } \right)\,e = 0,\quad \frac{\partial }{\partial x}\left( {\frac{\partial e}{\partial x} - \frac{\partial e}{\partial y}} \right) + \left( {\alpha_{1} + \alpha_{3} \left| e \right|^{2} } \right)\,e = 0,$$

where $e = \sqrt 2 e_{x} = \sqrt 2 e_{y}$. Introducing the “light-cone” coordinates $\xi = \left( {x + y} \right)/2$ and $\eta = \left( {x - y} \right)/2$ we obtain two uncoupled scalar equations

$$\frac{{\partial^{2} e}}{{\partial \eta^{2} }} + 2\left( {\alpha_{1} + \alpha_{3} \left| e \right|^{2} } \right)\,e = 0,\quad \quad \frac{{\partial^{2} e}}{\partial \xi \eta } = 0.$$

The first equation of equations has the cnoidal or soliton solutions $e\left( \eta \right)$, the second equation has the solution $e\left( \eta \right) = e\left( {x,y} \right)$ describing a one-dimensional wave varying in a single direction $\eta$.

Appendix B

The first integral of the equation $d^{2} e/dx^{2} - \bar{\alpha }_{1} e + \alpha_{3} e^{3} = 0$ looks like $\left( {de/dx} \right)^{2} = \bar{\alpha }_{1} e^{2} - \alpha_{3} e^{4} /2 + C$, where $C$ is an integration constant.

The boundary conditions for soliton $e \to 0$, $de/dx \to 0$ at $\left| x \right| \to \infty$ allow to define the integration constant as $C = 0$ [41]. The boundary conditions $e\left( 0 \right) = const$ and $de\left( 0 \right)/dx = 0$ for soliton centre $x = 0$ allow to define the phase shift as $q = \left[ {\alpha_{3} e^{2} \left( 0 \right)/2 + c^{ - 2} \omega^{2} \left( {1 + 4\pi \chi_{1} } \right) - k^{2} } \right]/2k$ in case without perturbation. The second integral of the equations for bright soliton looks like $\sqrt {\alpha_{3} /2} x = \int {e^{ - 1} \left( {\,\alpha '\, - \,e^{2} } \right)}^{ - 1/2} de$, where $\alpha^{\prime} = 2\bar{\alpha }_{1} \alpha_{3}^{ - 1}$, and after its integration we obtain $e\left( x \right) = \left| {\sqrt {\alpha '} } \right|sch\left( {sch^{ - 1} \left| {e\left( 0 \right)/\sqrt {\alpha '} } \right| - \sqrt {\bar{\alpha }_{1} } x} \right)$.

If we choose the boundary conditions as $e = const$, $de/dx = 0$ at $\left| x \right| \to \infty$, i.e. the integration constant is not equal zero $C = \alpha_{3} e_{\infty }^{4} /2 - \bar{\alpha }_{1} e_{\infty }^{2}$, we obtain the cnoidal wave . In this case the phase shift at the boundary conditions $e\left( 0 \right) = const$ and $de\left( 0 \right)/dx = 0$ is equal $q = \left[ {\alpha_{3} e^{2} \left( 0 \right)/2 + c^{ - 2} \omega^{2} \left( {1 + 4\pi \chi_{1} } \right) - k^{2} - Ce^{ - 2} \left( 0 \right)} \right]/2k$. Then the second integral of the equations for cnoidal wave looks like $\sqrt {\alpha_{3} /2} x = \int\nolimits_{0}^{e} {\left( {\,C^{\prime} + \alpha^{\prime}\,e^{2} - \,e^{4} } \right)^{ - 1/2} de}$, where $C^{\prime} = 2C\alpha_{3}^{ - 1}$.