Vector dissipative light bullets in optical laser beam

Abstract

The dynamics of light bullets propagating in nonlinear media with linear/nonlinear, gain/loss and coupling described by the (2+1)-dimensional vectorial cubic–quintic complex Ginzburg–Landau (CGL) equations is considered. The evolution and the stability of the vector dissipative optical light bullets, generated from an asymmetric input with respect to two transverse coordinates x and y, are studied. We use the variational method to find a set of differential equations characterizing the variation of the light bullet parameters in the laser cavity. This approach allows us to analyze the influence of various physical parameters on the dynamics of the propagating beam and its relevant parameters. Then, we solve the original coupled (2+1)D cubic–quintic CGL equation using the split-step Fourier method. Numerical results and analytical predictions are confronted, and a good agreement between the two approaches is obtained.

Appendix

The effective potential obtained for the stability analysis is given by

$$\begin{aligned} \begin{aligned} U =&\left( \left( \left( 5 - \frac{3}{{2{Y^2}}}\right) {X^2} + \frac{{55}}{{336}} + \frac{9}{{224X_m^2}} + \frac{{43}}{{252}}X_m^2\right. \right. \\&\left. \left. - \frac{{\frac{{4X_m^4}}{9} + \frac{{17X_m^2}}{{36}} + \frac{1}{8}}}{{{X^2}}}\right) {\delta _i}{\gamma _i}{A^6} - \frac{1}{2}{A^2}\ln (X)\right) {e^{ - \frac{{14X_m^2}}{{3{X^2}}}}}\\&+ \left( \frac{1}{8}\left( \left( \frac{7}{2} - \frac{1}{{{Y^2}}}\right) {X^2} + \frac{3}{{16X_m^2}} + \frac{7}{9}\right) \right. \\&\left. +\frac{{8X_m^2}}{{81}}- \frac{{\frac{1}{{16}} + \frac{{16}}{{81}}X_m^4 + \frac{{2X_m^2}}{9}}}{{{X^2}}}\right) \delta _i^2{A^8}{e^{ - \frac{{16X_m^2}}{{{X^2}}}}} \\&+ \left( \left( \left( 4 - \frac{1}{{2{Y^2}}}\right) {X^2}+ \frac{{8X_m^2}}{9} + \frac{1}{2}\right) \frac{{\mu {\delta _i}{A^8}}}{9}\right. \\&+ \left( \left( 7 - \frac{1}{{2{Y^2}}}\right) \frac{{{X^2}}}{8} + \frac{{X_m^2}}{9}+ \frac{1}{{16}}\right) \varepsilon {\delta _i}{A^6}\\&+ \left( \left( \frac{\delta }{2} - S + \left( {C^2} - \frac{1}{{2{Y^2}}} + \frac{{16{C^2}X_m^2}}{9}\right. \right. \right. \\&\left. \left. + \frac{4}{3}C\right) \beta \right) {\delta _i}{X^2} + {\delta _i}\left( \frac{2}{3} - \frac{{16CX_m^2}}{9} - C\right) \\&\left. \left. + \beta {\delta _i}\left( \frac{7}{{12}}+\frac{3}{{16X_m^2}} - {C^2}{X^4} - \frac{{\frac{1}{4} + \frac{{4X_m^2}}{9}}}{{{X^2}}}\right) \right) {A^4}\right) {e^{ - \frac{{8X_m^2}}{{3{X^2}}}}}\\&+ \left( \left( \left( \left( 3 - \frac{1}{{{Y^2}}}\right) {X^2} + 1 + X_m^2 + \frac{1}{{4X_m^2}}\right) \right. \right. \\&\left. \left. - \frac{{\frac{1}{4} + X_m^2 + X_m^4}}{{2{X^2}}}\right) \gamma _i^2{A^4}+ \frac{{\beta {\gamma _i}{A^2}}}{2}\right) {e^{ - \frac{{4X_m^2}}{{{X^2}}}}}\\&+ \left( \left( \frac{{11}}{2} - \frac{1}{{{Y^2}}}\right) \frac{{{X^2}}}{{18}} + \frac{{X_m^2}}{9}+ \frac{1}{{18}}\right) \mu {\gamma _i}{A^6}\\&+ \left( \left( 1 - \frac{1}{{32{Y^2}}}\right) \frac{{{X^2}}}{4} + \frac{{X_m^2}}{8} + \frac{1}{{16}}\right) \varepsilon {\gamma _i}{A^4} \\&+ \left( \left( - {C^2}{X^4} + \frac{1}{{8X_m^2}} - \frac{{1 + 2X_m^2}}{{4{X^2}}}\right) \beta {\gamma _i}\right. \\&- {\gamma _i}C(1 +2X_m^2)\\&+ \frac{{{\gamma _r}}}{2} + \left( {\gamma _i}\frac{{\delta - 2S - 2C}}{4}+ \beta {\gamma _i}\left( \frac{C}{2} + 2{C^2}X_m^2\right. \right. \\&\left. \left. \left. + {C^2} + \frac{1}{{4{Y^2}}}\right) \right) {A^2} + \frac{1}{{{X^2}}}\right) {e^{ - \frac{{2X_m^2}}{{{X^2}}}}}\\&+ \left( {X^2}+ {Y^2}\right) \left( \frac{{{\mu ^2}{A^8}}}{9} + \frac{{5\varepsilon \mu {A^6}}}{{24}}\right) \\&+ \left( - \frac{4}{9}\left\{ (\nu + \beta \mu )(\ln (X)+ \ln (Y))+ \beta \mu \left( {C^2}{X^4} + {S^2}{Y^4}\right) \right\} \right. \\&\left. + \frac{{5{\varepsilon ^2}}}{{64}}({X^2} + {Y^2})+ \left( 4\beta C\nu + 2\mu \delta - 4\mu S - \frac{{2\beta \mu }}{{{Y^2}}}\right) \left( \frac{{{X^2}}}{9}\right) \right. \\&\left. + \left( 4\beta S\nu + 2\mu \delta - 4\mu C - \frac{{2\beta \mu }}{{{X^2}}}\right) \left( \frac{{{Y^2}}}{9}\right) \right) {A^4}\\&+ \left( - \frac{{\beta \varepsilon }}{2}({C^2}{X^4}+ {S^2}{Y^4})- (\ln (X) + \ln (Y))\left( \frac{{2 + \beta \varepsilon }}{4}\right) \right. \\&+ \left( - \frac{{\beta \varepsilon }}{{{Y^2}}}+ 2\varepsilon C + 4\delta \varepsilon - 2\varepsilon S\right) \frac{{{X^2}}}{8}\\&+ \left( 4\beta S + 2\varepsilon S - 2\varepsilon C + \delta \varepsilon \right. \\&\left. \left. - \frac{{\beta \varepsilon }}{{{X^2}}}\right) {Y^2}\right) {A^2}\\&- 4{\beta ^2}({C^4}{X^6} + {S^4}{Y^6}) + 4\beta ({S^3}{Y^4} + {C^3}{Y^4}) \\&- 6{\beta ^2}({C^2}{X^2} + {S^2}{Y^2})+ 12\beta \left( S\ln (Y)\right. \\&\left. + C\ln (X)\right) - \frac{{{\beta ^2}}}{4}\left( \frac{1}{{{X^2}}} + \frac{1}{{{Y^2}}}\right) \end{aligned} \end{aligned}$$