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Propagation of dissipative simple vortex-, necklace- and azimuthon-shaped beams in Kerr and non-Kerr negative-refractive-index materials beyond the slowly varying envelope approximation

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Abstract

We present an explicit derivation of a (3+1)-dimensional [(3+1)D] cubic- quintic-septic complex Ginzburg–Landau (CQS-CGL) equation, including diffraction, linear dispersions up to the seventh order, loss, gain, cubic-quintic-septic nonlinearities, as well as cubic-quintic-septic first-order self-steepening effects. The new model equation, derived from Maxwell equations beyond the slowly varying envelope approximation, describes the dynamics of dissipative light bullets in nonlinear metamaterials (MMs). Using direct numerical simulations of the whole (3+1)D CQS-CGL equation, we present the evolution of various dissipative optical bullets in MMs characterized by different topological charges, namely, the fundamental vortex, necklace, and azimuthons. The bullet amplitudes and phase distributions support the emergence of new propagating modes under parameter values that promote their instability. However, with the right choice of higher-order parameters, especially the cubic, quintic and septic self-steepening coefficients, the numerical simulations are capable of achieving the stability of the studied. Under unstable conditions, even multipole vortices are found to converge in the rotating frame, the fundamental spherical light bullet, while their amplitude drops drastically. The results suggest that the presence of higher-order nonlinear effects, balanced by the higher-order dispersive terms, prevent the light bullets, with different topological charges, from collapsing, with rotation direction specific to negative-index MMs.

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The simulation data related to the current study are not publicly available due to but can be obtained from the corresponding author, CBT, on reasonable request.

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Funding

CBT thanks the Kavli Institute for Theoretical Physics (KITP), University of California Santa Barbara (USA), where this work was supported in part by the National Science Foundation Grant no.NSF PHY-1748958, NIH Grant no.R25GM067110, and the Gordon and Betty Moore Foundation Grant no.2919.01.

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Authors and Affiliations

  1. Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon

    L. Tiam Megne & J. A. Ambassa Otsobo

  2. Department of Physics and Astronomy, Botswana International University of Science and Technology, Private Bag 16, Palapye, Botswana

    C. B. Tabi, C. M. Muiva & T. C. Kofané

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  1. L. Tiam Megne
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Appendices

Appendix A: Coefficients \( W_{2}\), \( W_{3}\),\( W_{4}\),\( W_{5}\), \( W_{6}\) of Eq.(11)

$$\begin{aligned} W_{2}= & {} -c^{2}v^{-2}_{g}+\frac{1}{2}\omega _{0}(\mu \alpha _{2}+\varepsilon \beta _{2})+ \alpha _{1}\beta _{1},\nonumber \\ W_{3}= & {} \frac{\omega _{0}}{6}(\mu \alpha _{3}+\varepsilon \beta _{3})+\frac{1}{2}(\beta _{1}\alpha _{2}+\beta _{2}\alpha _{1}),\nonumber \\ W_{4}= & {} \frac{\omega _{0}}{24}(\mu \alpha _{4}{+}\varepsilon \beta _{4})+\frac{1}{6}(\beta _{1}\alpha _{3}+\beta _{3}\alpha _{1})+\frac{1}{4}(\alpha _{2}\beta _{2}),\nonumber \\ W_{5}= & {} \frac{\omega _{0}}{120}(\mu \alpha _{5}+\varepsilon \beta _{5})+\frac{1}{24}(\beta _{1}\alpha _{4}+\beta _{4}\alpha _{1}) \nonumber \\{} & {} +\frac{1}{12}(\alpha _{3}\beta _{2}+\alpha _{2}\beta _{3}),\end{aligned}$$
(22)
$$\begin{aligned} W_{6}= & {} \frac{\omega _{0}}{720}(\mu \alpha _{6}+\varepsilon \beta _{6})+\frac{1}{120}(\beta _{1}\alpha _{5}+\beta _{5}\alpha _{1})\nonumber \\{} & {} +\frac{1}{48}(\alpha _{4}\beta _{2}+\alpha _{2}\beta _{4})+\frac{1}{36}(\alpha _{3}\beta _{3}). \end{aligned}$$
(23)

Appendix B: Coefficients of the (3+1)D cubic-quintic-septic CGL Eq.(14)

$$\begin{aligned} \sigma _{\bot }= & {} \textrm{sgn}(n)-2i\frac{k_{i}}{k^{2}_{r}},\;k_{2r}=\textrm{sgn}\left( -\frac{\textrm{Re}[W_{2}]}{n\varpi }\right) ,\nonumber \\ k_{2i}= & {} \frac{\textrm{Im}[W_{2}]}{n\varpi }\left| \frac{n\varpi }{\textrm{Re}[W_{2}]}\right| ,\nonumber \\ k_{3r}= & {} \frac{3}{6}\left( \left| \frac{n\varpi }{\textrm{Re}[W_{2}]}\right| \right) ^{\frac{3}{2}}\left( \omega _{p}\frac{\textrm{Re}[W_{3}]}{n\varpi }-\frac{\textrm{Re}[W_{2}]c}{n^{2}\varpi ^{2}v_{g}}\right) ,\nonumber \\ k_{3i}= & {} \frac{3}{6}\left( \left| \frac{n\varpi }{\textrm{Re}[W_{2}]}\right| \right) ^{\frac{3}{2}}\left( \omega _{p}\frac{\textrm{Im}[W_{3}]}{n\varpi }-\frac{\textrm{Im}[W_{2}]c}{n^{2}\varpi ^{2}v_{g}}\right) ,\nonumber \\ k_{4r}= & {} \frac{12}{24}\left( \left| \frac{n\varpi }{\textrm{Re}[W_{2}]}\right| \right) ^{2}\nonumber \\{} & {} \times \left( \omega ^{2}_{p}\frac{\textrm{Re}[W_{4}]}{n\varpi }-\frac{(\textrm{Re}[W_{2}])^{2}-(\textrm{Im}[W_{2}])^{2}}{4n^{2}\varpi ^{2}}\right. \nonumber \\{} & {} \left. -\omega _{p}\frac{\textrm{Re}[W_{3}]}{n^{2}\varpi ^{2}v_{g}}\right) ,\nonumber \\ k_{4i}= & {} \frac{12}{24}\left( \left| \frac{n\varpi }{\textrm{Re}[W_{2}]}\right| \right) ^{2}\nonumber \\{} & {} \times \left( \omega ^{2}_{p}\frac{\textrm{Im}[W_{4}]}{n\varpi }-\frac{\textrm{Re}[W_{2}]\textrm{Im}[W_{2}]}{2n^{2}\varpi ^{2}}\right. \nonumber \\{} & {} \left. -\omega _{p}\frac{\textrm{Im}[W_{3}]}{n^{2}\varpi ^{2}v_{g}}\right) ,\nonumber \\ k_{5r}= & {} \frac{60}{120}\left( \left| \frac{n\varpi }{\textrm{Re}[W_{2}]}\right| \right) ^{\frac{5}{2}}\nonumber \\{} & {} \times \Big (\omega ^{3}_{p}\frac{\textrm{Re}[W_{5}]}{n\varpi } -\omega _{p}c\frac{\textrm{Re}[W_{4}]}{n^{2}\varpi ^{2}v_{g}}\nonumber \\{} & {} -\omega _{p}\frac{(\textrm{Re}[W_{2}]\textrm{Im}[W_{3}]+\textrm{Im}[W_{2}]\textrm{Re}[W_{3}])}{n^{3}\varpi ^{3}}\Big ),\nonumber \\ k_{5i}= & {} \frac{60}{120}\left( \left| \frac{n\varpi }{\textrm{Re}[W_{2}]}\right| \right) ^{\frac{5}{2}}\Big (\omega ^{3}_{p}\frac{\textrm{Im}[W_{5}]}{n\varpi }{-}\omega _{p}c\frac{\textrm{Im}[W_{4}]}{n^{2}\varpi ^{2}v_{g}}\nonumber \\{} & {} -\omega _{p}\frac{(\textrm{Re}[W_{2}]\textrm{Re}[W_{3}]-\textrm{Im}[W_{2}]\textrm{Im}[W_{3}])}{n^{3}\varpi ^{3}}\Big ),\nonumber \\ k_{6r}= & {} \frac{360}{720}\left( \left| \frac{n\varpi }{\textrm{Re}[W_{2}]}\right| \right) ^{3}\Big (-\omega ^{4}_{p}\frac{\textrm{Re}[W_{6}]}{n\varpi }\nonumber \\{} & {} +\omega ^{2}_{p}\left( \frac{\textrm{Re}[W_{2}]\textrm{Re}[W_{4}]-\textrm{Im}[W_{2}]\textrm{Im}[W_{4}]}{4n^{3}\varpi ^{3}}\right) \nonumber \\{} & {} +\omega ^{2}_{p}\frac{\textrm{Re}[W_{3}]}{4n^{3}\varpi ^{3}}+\omega ^{3}_{p}c\frac{\textrm{Re}[W_{5}]}{n^{2}\varpi ^{2}v_{g}}\Big ),\nonumber \\ k_{6i}= & {} \frac{360}{720}\left( \left| \frac{n\varpi }{\textrm{Re}[W_{2}]}\right| \right) ^{3}\Big (-\omega ^{4}_{p}\frac{\textrm{Im}[W_{6}]}{n\varpi }\nonumber \\{} & {} +\omega ^{2}_{p}\left( \frac{\textrm{Re}[W_{2}]\textrm{Im}[W_{4}]+\textrm{Im}[W_{2}]\textrm{Re}[W_{4}]}{4n^{3}\varpi ^{3}}\right) \nonumber \\{} & {} +\omega ^{2}_{p}\frac{\textrm{Im}[W_{3}]}{4n^{3}\varpi ^{3}}-\omega ^{3}_{p}c\frac{\textrm{Im}[W_{5}]}{n^{2}\varpi ^{2}v_{g}}\Big ),\nonumber \\ \delta= & {} i\frac{\varpi }{2n}\textrm{Im}[\varepsilon \mu ],\;\;N_{3}=\textrm{sgn}(\chi ^{(3)}_{r})+i\frac{\chi ^{(3)}_{i}}{|\chi ^{(3)}_{r}|}, \nonumber \\ N_{5}= & {} \frac{-\chi ^{(5)}_{r}}{|\chi ^{(3)}_{r}|}-\frac{\chi ^{(3)}_{r}\chi ^{(3)}_{r}-\chi ^{(3)}_{i}\chi ^{(3)}_{i}}{{4n|\chi ^{(3)}_{r}|}}\nonumber \\{} & {} -i\left( \frac{\chi ^{(5)}_{i}}{|\chi ^{(3)}_{r}|}+\frac{2\chi ^{(3)}_{r}\chi ^{(3)}_{i}}{{4n|\chi ^{(3)}_{r}|}}\right) ,\nonumber \\ N_{7}= & {} \frac{\chi ^{(7)}_{r}}{|\chi ^{(3)}_{r}|}+\frac{\chi ^{(3)}_{r}\chi ^{(5)}_{r}-\chi ^{(3)}_{i}\chi ^{(5)}_{i}}{{4n|\chi ^{(3)}_{r}|}}\nonumber \\{} & {} +i\left( \frac{\chi ^{(7)}_{i}}{|\chi ^{(3)}_{r}|}+\frac{\chi ^{(3)}_{r}\chi ^{(5)}_{i}+\chi ^{(5)}_{r}\chi ^{(3)}_{i}}{{4n|\chi ^{(3)}_{r}|}}\right) \nonumber \\ SS_{3}= & {} -\frac{1}{\varpi }\left( 1+\frac{1}{2nv_{g}}\right) \frac{\chi ^{(3)}_{i}}{\left| \left( -\frac{\chi ^{(3)}_{r}\textrm{Re}[W]}{n\varpi }\right) \right| }\nonumber \\{} & {} +i\left( \frac{1}{\varpi }\left( 1+\frac{1}{2nv_{g}}\right) \frac{\textrm{sgn}\left( \chi ^{(3)}_{r}\right) }{\left| \left( -\frac{\textrm{Re}[W]}{n\varpi }\right) \right| }\right) ,\nonumber \\ SS_{5}= & {} -\frac{1}{\varpi }\left( 1+\frac{1}{2nv_{g}}\right) \frac{\chi ^{(5)}_{i}}{\left| \left( -\frac{\textrm{Re}[W]}{n\varpi }\right) \right| |\chi ^{(3)}_{r}|}\nonumber \\{} & {} +i\left( \frac{1}{\varpi }\left( 1+\frac{1}{2nv_{g}}\right) \frac{\chi ^{(5)}_{r}}{\left| \left( -\frac{\textrm{Re}[W]}{n\varpi }\right) \right| |\chi ^{(3)}_{r}|}\right) ,\nonumber \\ SS_{7}= & {} -\frac{1}{\varpi }\left( 1+\frac{1}{2nv_{g}}\right) \frac{\chi ^{(7)}_{i}}{\left| \left( -\frac{\textrm{Re}[W]}{n\varpi }\right) \right| |\chi ^{(3)}_{r}|}\nonumber \\{} & {} +i\left( \frac{1}{\varpi }\left( 1+\frac{1}{2nv_{g}}\right) \frac{\chi ^{(7)}_{r}}{\left| \left( -\frac{\textrm{Re}[W]}{n\varpi }\right) \right| |\chi ^{(3)}_{r}|}\right) \nonumber \\ \end{aligned}$$
(24)

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Megne, L.T., Tabi, C.B., Otsobo, J.A.A. et al. Propagation of dissipative simple vortex-, necklace- and azimuthon-shaped beams in Kerr and non-Kerr negative-refractive-index materials beyond the slowly varying envelope approximation. Nonlinear Dyn 111, 20289–20309 (2023). https://doi.org/10.1007/s11071-023-08939-0

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