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Propagation dynamics of rotating high-order cosine-Gaussian array beams induced by initial cross phase

Authors:
Zhuo-Yue Sun
Zhuo-Yue Sun
Zhuo-Yue Sun
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Jia-Ji Wu
Jia-Ji Wu
Jia-Ji Wu
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Zhen-Jun Yang
Zhen-Jun Yang
Zhen-Jun Yang
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Zhao-Guang Pang
Zhao-Guang Pang
Zhao-Guang Pang
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Abstract and Figures

The propagation dynamics of the complex variable cosine-Gaussian cross-phase (CVCGCP) array beams in strongly nonlocal nonlinear media are researched based on the nonlocal nonlinear Schrödinger equation. Under the effect of cross-phase, the transverse mode of CVCGCP array beams changes periodically and rotates during propagation. Compared with higher-order temporal solitons in nonlinear optical fibers, CVCGCP array beams can be considered as a new form of higher-order spatial solitons. The expression of the optical field distribution for the propagation evolution of CVCGCP array beams is presented. According to the different parameters, three cases are studied in detail. The light intensity pattern, phase and statistical width of CVCGCP array beams are discussed and analyzed. The results show that CVCGCP array beams have rich transmission characteristics and can form a linear shape distribution, which has the practical application value. By selecting the parameters, the light intensity patterns can be repeated and controlled, so as to achieve the purpose of controlling the light intensity patterns. The results of this paper enrich the types of higher-order spatial solitons, and also provide theoretical references for beam control and information transmission, etc.
The effect of parameter b on the CVCGCP array beams propagation characteristics. a–c: The transverse light intensity patterns at different positions in the transmission process, where the curve illustrates the two-dimensional distribution of light intensity in the ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} direction. d–f: The evolution of on-axis light intensity during transmission. g–i: The statistical beam width evolution of array beams in the transmission process. The solid line represents the spot size of the beams in the ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} direction, while the dashed line represents the spot size of the beams in the η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} direction. Parameter b: b=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=2$$\end{document} for (a), (d) and (g); b=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=4$$\end{document} for (b), (e) and (h); b=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=6$$\end{document} for (c), (f) and (i). Other parameters: a=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=0$$\end{document}, c=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=1$$\end{document}, ωξ0=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\xi 0}=2$$\end{document}, ωη0=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\eta 0}=1$$\end{document}, P0=4Pgc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{0}=4P_{gc}$$\end{document} for all cases
The effect of parameter b on the CVCGCP array beams propagation characteristics. a–c: The transverse light intensity patterns at different positions in the transmission process, where the curve illustrates the two-dimensional distribution of light intensity in the ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} direction. d–f: The evolution of on-axis light intensity during transmission. g–i: The statistical beam width evolution of array beams in the transmission process. The solid line represents the spot size of the beams in the ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} direction, while the dashed line represents the spot size of the beams in the η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} direction. Parameter b: b=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=2$$\end{document} for (a), (d) and (g); b=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=4$$\end{document} for (b), (e) and (h); b=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=6$$\end{document} for (c), (f) and (i). Other parameters: a=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=0$$\end{document}, c=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=1$$\end{document}, ωξ0=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\xi 0}=2$$\end{document}, ωη0=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\eta 0}=1$$\end{document}, P0=4Pgc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{0}=4P_{gc}$$\end{document} for all cases
… 
a: The transverse light intensity evolution of the linear shape CVCGCP array beams during the transmission process. b and c: The evolution of CVCGCP array beams in the diagonal direction. The parameters we use in this figure are: a=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=0$$\end{document}, b=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=6$$\end{document}, c=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=1$$\end{document}, ωξ0=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\xi 0}=2$$\end{document}, ωη0=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\eta 0}=1$$\end{document}, P0=4Pgc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{0}=4P_{gc}$$\end{document}
a: The transverse light intensity evolution of the linear shape CVCGCP array beams during the transmission process. b and c: The evolution of CVCGCP array beams in the diagonal direction. The parameters we use in this figure are: a=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=0$$\end{document}, b=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=6$$\end{document}, c=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=1$$\end{document}, ωξ0=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\xi 0}=2$$\end{document}, ωη0=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\eta 0}=1$$\end{document}, P0=4Pgc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{0}=4P_{gc}$$\end{document}
… 
The light intensity and phase evolution of CVCGCP array beams during transmission. Parameters: a=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=0$$\end{document}, b=8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=8$$\end{document}, c=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=2$$\end{document}, ωξ0=ωη0=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\xi 0}=\omega _{\eta 0}=1$$\end{document}, P0=4Pgc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{0}=4P_{gc}$$\end{document}
The light intensity and phase evolution of CVCGCP array beams during transmission. Parameters: a=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=0$$\end{document}, b=8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=8$$\end{document}, c=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=2$$\end{document}, ωξ0=ωη0=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\xi 0}=\omega _{\eta 0}=1$$\end{document}, P0=4Pgc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{0}=4P_{gc}$$\end{document}
… 
a–c: Evolution of light intensity of CVCGCP array beams during transmission. d–f: Beam width evolution of the CVCGCP array beams during transmission. Parameters a: a=8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=8$$\end{document} for (a) and (d); a=9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=9$$\end{document} for (b) and (e); a=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=10$$\end{document} for (c) and (f). Other parameters: b=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=0$$\end{document}, c=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=2$$\end{document}, ωξ0=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\xi 0}=2$$\end{document}, ωη0=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\eta 0}=1$$\end{document}, P0=20Pgc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{0}=20P_{gc}$$\end{document}
a–c: Evolution of light intensity of CVCGCP array beams during transmission. d–f: Beam width evolution of the CVCGCP array beams during transmission. Parameters a: a=8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=8$$\end{document} for (a) and (d); a=9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=9$$\end{document} for (b) and (e); a=10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=10$$\end{document} for (c) and (f). Other parameters: b=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=0$$\end{document}, c=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=2$$\end{document}, ωξ0=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\xi 0}=2$$\end{document}, ωη0=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\eta 0}=1$$\end{document}, P0=20Pgc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{0}=20P_{gc}$$\end{document}
… 
Lines 1–3: The transverse light intensity distribution of CVCGCP array beams during transmission when parameter c is zero. Line 4: The evolution of the statistical beam width of CVCGCP array beams with different parameters. The ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} direction width for the solid line and the η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} direction width for the dashed line. Parameters a and b: a=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=1$$\end{document}, b=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=2$$\end{document} for (a) and (d); a=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=2$$\end{document}, b=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=3$$\end{document} for (b) and (e); a=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=3$$\end{document}, b=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=4$$\end{document} for (c) and (f). Other parameters: c=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0$$\end{document}, ωξ0=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\xi 0}=2$$\end{document}, ωη0=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\eta 0}=3$$\end{document}, P0=0.5Pgc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{0}=0.5P_{gc}$$\end{document}
+5
Lines 1–3: The transverse light intensity distribution of CVCGCP array beams during transmission when parameter c is zero. Line 4: The evolution of the statistical beam width of CVCGCP array beams with different parameters. The ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} direction width for the solid line and the η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document} direction width for the dashed line. Parameters a and b: a=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=1$$\end{document}, b=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=2$$\end{document} for (a) and (d); a=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=2$$\end{document}, b=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=3$$\end{document} for (b) and (e); a=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a=3$$\end{document}, b=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=4$$\end{document} for (c) and (f). Other parameters: c=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0$$\end{document}, ωξ0=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\xi 0}=2$$\end{document}, ωη0=3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{\eta 0}=3$$\end{document}, P0=0.5Pgc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{0}=0.5P_{gc}$$\end{document}
… 
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Nonlinear Dyn (2024) 112:2893–2908
https://doi.org/10.1007/s11071-023-09226-8
ORIGINAL PAPER
Propagation dynamics of rotating high-order
cosine-Gaussian array beams induced by initial cross phase
Zhuo-Yue Sun ·Jia-Ji Wu ·Zhen-Jun Yang ·
Zhao-Guang Pang ·Hui Wang
Received: 20 April 2023 / Accepted: 13 December 2023 / Published online: 5 January 2024
© The Author(s), under exclusive licence to Springer Nature B.V. 2024
Abstract The propagation dynamics of the com-
plex variable cosine-Gaussian cross-phase (CVCGCP)
array beams in strongly nonlocal nonlinear media are
researched based on the nonlocal nonlinear Schrödinger
equation. Under the effect of cross-phase, the trans-
verse mode of CVCGCP array beams changes peri-
odically and rotates during propagation. Compared
with higher-order temporal solitons in nonlinear optical
fibers, CVCGCP array beams can be considered as a
new form of higher-order spatial solitons. The expres-
sion of the optical field distribution for the propaga-
tion evolution of CVCGCP array beams is presented.
According to the different parameters, three cases are
studied in detail. The light intensity pattern, phase and
statistical width of CVCGCP array beams are discussed
and analyzed. The results show that CVCGCP array
beams have rich transmission characteristics and can
form a linear shape distribution, which has the practi-
cal application value. By selecting the parameters, the
light intensity patterns can be repeated and controlled,
so as to achieve the purpose of controlling the light
intensity patterns. The results of this paper enrich the
types of higher-order spatial solitons, and also provide
theoretical references for beam control and information
transmission, etc.
Z.-Y. Sun ·J.-J. Wu ·Z.-J. Yang (B)·Z.-G. Pang ·H. Wang
Hebei Key Laboratory of Photophysics Research and Applica-
tion, College of Physics, Hebei Normal University, Shijiazhuang
050024, China
e-mail: zjyang@vip.163.com
Keywords Nonlocal nonlinear media ·Array beams ·
Cross phase ·Beam transmission
1 Introduction
Since the concept of strongly nonlocal nonlinearity was
proposed [1], the transmission characteristics of laser
beams in strongly nonlocal nonlinear media (SNNM)
have been one of the key subjects of research. The
strongly nonlocal nonlinear media have many unique
properties that can suppress the instability of the laser
beams during transmission, and thus can support many
laser beams to form new nonlocal beam forms, such as
high-order Gaussian beams [2,3], vortex solitons [4–
6], high-order multipole solitons [7–9], dark and anti-
dark solitons [10,11], elliptic solitons [12–14], com-
plex variable solitons [15–18], soliton clusters [19–
22], etc. At the same time, nonlocal optical beams
exhibit unique dynamic transmission properties, such
as mutual attraction between antiphase solitons [1,23],
self-induced mode transformation [24], optical field
self-reconstruction [25], three-dimensional scale trans-
formation [26], and self-induced fractional Fourier
transformation [27], etc. At present, it has been found
that many optical media have some nonlocal properties,
and strongly nonlocal nonlinearity can exist in nematic
liquid crystals [28,29], lead glasses [30,31], atomic
vapors [32], and other media. Based on these nonlo-
cal optical beams and their characteristics, some novel
all-optical controllers have been realized [33–36].
123
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  • May 2024
  • OPT QUANT ELECTRON
The evolution of a controllable vortex anomalous hollow beam (CVAHB) in free space is studied theoretically and numerically. Based on the Collins integral, the analytical formula of its propagation under free space paraxial approximation is derived. The corresponding beams with different parameters are simulated numerically, and the influence of each beam parameter on the beam type and intensity distribution during beam propagation is studied. Through the parameter control, the CVAHB can present a variety of intensity modes, such as hollow, three-segment, boat, etc. It is obvious that the evolution of CVAHB is different from that of the controllable anomalous hollow beam (CAHB). By changing the value of the topological charge (TC), the role of the orbital angular momentum (OAM) in the transverse distribution and propagation of light intensity is studied in detail.
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Nonlinear propagation dynamics of lossy tripolar breathers in nonlocal nonlinear media
Article
Full-text available
  • Jul 2022
  • NONLINEAR DYNAM
In this work, the propagation characteristics of tripolar breather trial solution in nonlocal nonlinear media with loss are studied theoretically. The approximate equations of parameters of the tripolar breather are obtained analytically by the variational method. The analytical solution was verified by numerical simulation. Tripolar loss soliton and tripolar loss breather can be formed under suitable incident conditions. By analogy with Newton’s laws of motion in classical mechanics, we regard the evolution of the triple breather as a particle with a mass equal to 1. By studying the evolution law of the equivalent force and the equivalent potential energy, the in-depth physical reasons for the periodic evolution of the tripolar breather are analyzed.
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Controllable propagation paths of gap solitons
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This paper numerically investigates the evolution of solitons in an optical lattice with gradual longitudinal manipulation. We find that the stationary solutions (with added noise to the amplitude) keep their width, profile, and intensity very well, although the propagation path is continuously changing during the modulated propagation. Discontinuities in the modulation functions cause the scattering of the beam that may end the stable propagation. Our results reveal a method to control the trajectory of solitons by designed variation of the optical lattice waveguides. Interesting examples presented include the snakelike and spiraling solitons that both can be adaptively induced in sinusoidally and helically shaped optical lattices. The controlled propagation paths provide an excellent opportunity for various applications, including optical switches and signal transmission, among others.
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Spatiotemporal control of laser intensity through cross-phase modulation
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Spatiotemporal pulse shaping provides control over the trajectory and range of an intensity peak. While this control can enhance laser-based applications, the optical configurations required for shaping the pulse can constrain the transverse or temporal profile, duration, or orbital angular momentum (OAM). Here we present a novel technique for spatiotemporal control that mitigates these constraints by using a "stencil" pulse to spatiotemporally structure a second, primary pulse through cross-phase modulation (XPM) in a Kerr lens. The temporally shaped stencil pulse induces a time-dependent focusing phase within the primary pulse. This technique, the "flying focus X," allows the primary pulse to have any profile or OAM, expanding the flexibility of spatiotemporal pulse shaping for laser-based applications. As an example, simulations show that the flying focus X can deliver an arbitrary-velocity, variable-duration intensity peak with OAM over distances much longer than a Rayleigh range.
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Stabilization of higher-order vortex solitons by means of nonlocal nonlinearity
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Full-text available
  • Jan 2022
We report the experimental observation of the instability suppression of higher-order vortex solitons in cylindrical lead glass with thermal nonlocal nonlinearity. A scalar higher-order vortex soliton with a topological charge l=10, which is vulnerable to azimuthal breakup when it is perturbed by an initial noise, is stabilized by a coaxially propagating, mutually incoherent Gaussian beam, forming a Gauss-vortex vector soliton (GVVS). The vortical annular profile and topological charge in the GVVS, and the vortical annular profile in the Gauss-vortex vector breather (GVVB), can be preserved during the propagation. Numerical simulations for unperturbed and perturbed scalar vortex solitons, GVVS, and GVVB demonstrate close agreement with experimental results.
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Periodic collision theory of multiple cosine-Hermite-Gaussian solitons in Schrödinger equation with nonlocal nonlinearity
Article
  • Jun 2023
  • APPL MATH LETT
In this paper, the dynamic characteristics of periodic collisions of multiple cosine-Hermite-Gaussian (CHG) solitons in Schrödinger equation with nonlocal nonlinearity is investigated in detail, which can be used to describe the soliton propagation in novel nonlocal nonlinear optical materials. The analytic expression of the interaction of multiple CHG solitons is derived. Taking the interaction of two CHG solitons and four CHG solitons as examples, a series of analytical results of the propagation trajectory, motion velocity, acceleration, light intensity evolution of solitons are given. The typical characteristics of periodic collisions of multiple CHG solitons are illustrated by numerical simulations.
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Propagation characteristics of cosine-Gaussian cross-phase beams in strongly nonlocal nonlinear media
Article
  • Sep 2022
  • OPTIK
In this paper, the propagation expression of cosine-Gaussian cross-phase beams (CGCPBs) is derived based on the nonlocal nonlinear Schrödinger equation and the propagation characteristics of CGCPBs in strongly nonlocal nonlinear media (SNNM) are researched. It is worth noting that the mode of CGCPBs can be controlled by modifying the parameters of cosine term and cross phase term. CGCPBs can form generalized high-order breathers, which signifies that the multi-peak light intensity pattern and the beam width evolve periodically during the transmission process. Apart from that, the impact of the parameters on the light intensity mode, the phase and the beam width is discussed by changing the parameters in the initial incident transmission expression. The research content of this paper provides reference value for controlling beams.
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Parameters controlling of vortex solitons in nonlocal nonlinear medium with gradually characteristic length
Article
  • Aug 2022
  • CHAOS SOLITON FRACT
This paper discusses the dynamics of vortex beam in nematic liquid crystals with the tunable characteristic length (CL) of response function. It is found that the transition of the soliton states with different widths during propagations can be controlled by changing the CL under the condition that the nonlocal degree is constant. Furthermore, the transitions propagation of different soliton state is stable and can resist noise. In addition, the numerical results prove that the parameters of vortex solitons, such as rotating periods of angles, orbital angular momentum (OAM), gradient force upon micro-particles all can be controlled conveniently at the desired positions. The controllable density of OAM and gradient force have important academic value and application prospect in the field of optical tweezers and spanner.
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Evolution characteristics of generalized Hermite-cosine-Gaussian solitons in optical nonlinear medium with spatial nonlocality
Article
  • Apr 2022
The evolution characteristics of Hermite-cosine-Gaussian beams in optical nonlinear medium with spatial nonlocality are investigated theoretically in detail. We obtain a series of analytical expressions to describe the evolution characteristics of Hermite-cosine-Gaussian beams in nonlocal medium, such as the expressions of the evolutionary propagation, the beam spot size, the statistical wavefront curvature, the critical energy etc. The results show that the statistical spot size of Hermite-cosine-Gaussian beam can remain unchanged or change periodically according to different initial incident energy. However, the transverse intensity mode of Hermite-cosine-Gaussian beam always presents periodic transformation. This is consistent with high-order temporal solitons, so the evolution of Hermite-cosine-Gaussian beams can also be called generalized solitons or breathers. The influences of initial incident energy and cosine parameters on the evolution characteristics of Hermite-cosine-Gaussian beams are analyzed in detail. Through numerical simulation, some typical evolution characteristics are shown graphically.
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Transformation of multipole and vortex solitons in the nonlocal nonlinear fractional Schrödinger equation by means of Lévy-index management
Article
  • Apr 2022
  • CHAOS SOLITON FRACT
The structure and stability of multipole and vortex solitons in the nonlocal nonlinear fractional Schrödinger equation with a gradually decreasing Lévy index, α, are numerically studied. It is found that the solitons adiabatically compress with the decrease of Lévy index, and new species of stable ones are produced by means of this technique. It is known that, under the action of the normal diffraction (α = 2), the nonlocal cubic self-trapping can support, at most, quadrupole solitons and vortex ones with winding number m = 2 as stable modes in the one- and two-dimensional space, respectively. In contrast to that, we find that the application of the Lévy index management (the gradual decrease of α) leads to the formation of stable five-poles and sextupoles in one-dimensional, and vortices with m = 3 in two-dimensional. Weak dissipation does not essentially affect the observed results.
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