Flexible rotation of transverse optical field for 2D self-accelerating beams with a designated trajectory

Lei Zhu; Xuesong Zhao; Chen Liu; Songnian Fu; Yuncai Wang; Yuwen Qin

doi:10.29026/oea.2021.200021

Journals >Opto-Electronic Advances >Volume 4 >Issue 3 >Page 200021-1 > Article

Opto-Electronic Advances
Vol. 4, Issue 3, 200021-1 (2021)

Flexible rotation of transverse optical field for 2D self-accelerating beams with a designated trajectory

Lei Zhu¹, Xuesong Zhao¹, Chen Liu¹, Songnian Fu^2、*, Yuncai Wang², and Yuwen Qin²

Author Affiliations

¹Wuhan National Laboratory for Optoelectronics, and School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan 430074, China

²School of Information Engineering, Guangdong University of Technology, and Guangdong Provincial Key Laboratory of Photonics Information Technology, Guangzhou 510006, China.

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DOI: 10.29026/oea.2021.200021 Cite this Article

Lei Zhu, Xuesong Zhao, Chen Liu, Songnian Fu, Yuncai Wang, Yuwen Qin. Flexible rotation of transverse optical field for 2D self-accelerating beams with a designated trajectory[J]. Opto-Electronic Advances, 2021, 4(3): 200021-1 Copy Citation Text

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Abstract

Self-accelerating beams have the unusual ability to remain diffraction-free while undergo the transverse shift during the free-space propagation. We theoretically identify that the transverse optical field distribution of 2D self-accelerating beam is determined by the selection of the transverse Cartesian coordinates, when the caustic method is utilized for its trajectory design. Based on the coordinate-rotation method, we experimentally demonstrate a scheme to flexibly manipulate the rotation of transverse optical field for 2D self-accelerating beams under the condition of a designated trajectory. With this scheme, the transverse optical field can be rotated within a range of 90 degrees, especially when the trajectory of 2D self-accelerating beams needs to be maintained for free-space photonic interconnection.

Keywords

caustic optical field self-accelerating beams

$ \begin{split} & E\left( {X,Z} \right) \\ & =\frac{1}{{2{\rm{π}} }}\int {A\left( {{k_x}} \right){\rm{exp}} \left\{ {{\rm{i}}\left[ {{k_x}X + \sqrt {{k^2} - {k_x}^2} Z + \varphi \left( {{k_x}} \right)} \right]} \right\}} {\rm{d}}{k_x} \\ & =\frac{1}{{2{\rm{π}} }}\int A \left( {{k_x}} \right){\rm{exp}} \left[ {{\rm{i}}\psi \left( {{k_x}} \right)} \right]{\rm{d}}{k_x}\;,\\[-13pt]\end{split}$

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$\frac{{{\rm{d}}\psi \left( {{k_x}} \right)}}{{{\rm{d}}{k_x}}} = X - \frac{{{k_x}}}{{\sqrt {{k^2} - {k_x}^2} }}Z + \phi '\left( {{k_x}} \right) = 0\;.$

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$\left[ {\begin{array}{*{20}{c}} {{X_\theta }} \\ {{Y_\theta }} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\cos \theta }&{\sin \theta } \\ { - \sin \theta }&{\cos \theta } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {f\left( Z \right)} \\ {g\left( Z \right)} \end{array}} \right]\;.$

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$\left[ {\begin{array}{*{20}{c}} {{x_{0,}}_\theta } \\ {{y_{0,\theta }}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\cos \theta }&{\sin \theta } \\ { - \sin \theta }&{\cos \theta } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {f\left( Z \right) - Zf\,'\left( Z \right)} \\ {g\left( Z \right) - Zg\,'\left( Z \right)} \end{array}} \right]\;,$

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$\left[ {\begin{array}{*{20}{c}} {{k_{0,x,\theta }}} \\ {{k_{0,y,\theta }}} \end{array}} \right] = k\left[ {\begin{array}{*{20}{c}} {\cos \theta }&{\sin \theta } \\ { - \sin \theta }&{\cos \theta } \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {f\,'\left( Z \right)} \\ {g\,'\left( Z \right)} \end{array}} \right]\;.$

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$\begin{split} &{W_\theta }\left( {{x_\theta },{y_\theta },{k_{x,\theta }},{k_{y,\theta }}} \right)\\ & =\! \int {\delta \left[ {{x_\theta } \!-\! {x_{0,\theta }},{y_\theta } \!-\! {y_{0,\theta }}} \right]} \delta \left[ {{k_{x,\theta }} \!-\! {k_{0,x,\theta }},{k_{y,\theta }} \!-\! {k_{0,y,\theta }}} \right]{\rm{d}}Z\;,\end{split}$

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${\varphi _\theta } = \int { - \frac{{\int {{x_\theta }{y_\theta }{W_\theta }{\rm{d}}{x_\theta }{\rm{d}}{y_\theta }} }}{{\int {{W_\theta }{\rm{d}}{x_\theta }{\rm{d}}{y_\theta }} }}{\rm{d}}{k_{x,\theta }}{\rm{d}}{k_{y,\theta }}} \;,$

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$\begin{split} {W_{\theta + {{90}^ \circ }}}\!\!= & \int {\delta \left( {{x_{\theta + {{90}^ \circ }}} - {x_{0,\theta + {{90}^ \circ }}},{y_{\theta + {{90}^ \circ }}} - {y_{0,\theta + {{90}^ \circ }}}} \right)} \\ & \times\! \delta \left( {{k_{x,\theta \! +\! {{90}^ \circ }}} \!-\! {k_{0,x,\theta + {{90}^ \circ }}},{k_{y,\theta + {{90}^ \circ }}} \!\!-\! {k_{0,y,\theta + {{90}^ \circ }}}} \right){\rm{d}}Z \\{\rm{ = }} & \int {\delta \left( {{y_\theta } - {y_{0,\theta }}, - {x_\theta } + {x_{0,\theta }}} \right)} \\& \times \delta \left( {{k_{y,\theta }} - {k_{0,y,\theta }}, - {k_{x,\theta }} + {k_{0,x,\theta }}} \right){\rm{d}}Z = {W_\theta }\;. \\[-11pt] \end{split} $

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$\left\{ {\begin{aligned} X = & 7.7896 \times {{10}^{ {\text{-}} 3}}{Z^2} - 3.9470 \times {{10}^{ {\text{-}} 3}}Z \\ & + 0.8750 \times {{10}^{ {\text{-}} 3}}/\rm{m} \\ Y = & 5.5445 \times {{10}^{ {\text{-}}3}}{Z^2} - 5.2862 \times {{10}^{ {\text{-}} 3}}Z \\& + 1.6800 \times {{10}^{ {\text{-}} 3}}/\rm{m} \end{aligned}} \right.\;.$

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