Multifunctional metalens generation using bilayer all-dielectric metasurfaces: erratum

Li Chen; Li Chen; Yuan Hao; Yuan Hao; Lin Zhao; Ruihuan Wu; Ruihuan Wu; Yue Liu; Yue Liu; Zhongchao Wei; Ning Xu; Zhaotang Li; Hongzhan Liu; Hongzhan Liu

doi:10.1364/OE.431809

Abstract

We provide corrected equations for our previous publication [Opt. Express 29, 9332(2021) [CrossRef] ].

1. Introduction

In our previous publication [1], Eqs. (8), (9), (13), and (14) accidentally omitted a square term.

Equation (8) in [1] is

(1)$$\left\{ \begin{array}{l} {\varphi _1}({x_i},{y_i},{\lambda _1},{f_1}) = 2({\theta _2} - {\theta _1}) = \frac{{2\pi }}{{{\lambda _1}}}(\sqrt {{x_i}^2 + {y_i}^2 + {f_1}^2} - {f_1}) \\ {\varphi _2}({x_i},{y_i},{\lambda _1},{f_2}) ={-} 2{\theta _2} = \frac{{2\pi }}{{{\lambda _1}}}(\sqrt {{x_i}^2 + {y_i}^2 + {f_2}^2} - {f_2}) \end{array} \right.$$

Equation (9) in [1] is

(2)$${\varphi _{3}}({x_i},{y_i},{\lambda _1},{f_1},{f_2}) = - 2{\theta _{1}} = \frac{{2\pi }}{{{\lambda _1}}}(\sqrt {{x_i}^2 + {y_i}^2 + {f_2}^2} - {f_2} + \sqrt {{x_i}^2 + {y_i}^2 + {f_1}^2} - {f_1})$$

Equation (13) in [1] is

(3)$${\varphi _{3}}({x_i},{y_i},{\lambda _2},{f_3}) = \frac{{2\pi }}{{{\lambda _2}}}(\sqrt {{x_i}^2 + {y_i}^2 + {f_3}^2} - {f_3})$$

Equation (14) in [1] is

(4)$$\frac{1}{{{\lambda _2}}}(\sqrt {{x_i}^2 + {y_i}^2 + {f_3}^2} - {f_3}) = \frac{1}{{{\lambda _1}}}(\sqrt {{x_i}^2 + {y_i}^2 + {f_2}^2} - {f_2} + \sqrt {{x_i}^2 + {y_i}^2 + {f_1}^2} - {f_1})$$

It should be mentioned that all the simulated results presented in the published paper [1] are not affected.

Funding

Science and Technology Program of Guangzhou (2019050001); National Natural Science Foundation of China (61475049, 61774062, 61875057).

References

1. L. Chen, Y. Hao, L. Zhao, R. Wu, Y. Liu, Z. Wei, N. Xu, Z. Li, and H. Liu, “Multifunctional metalens generation using bilayer all-dielectric metasurfaces,” Opt. Express 29(6), 9332–9345 (2021). [CrossRef]

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Equations (4)

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{\begin{cases} φ_{1} (x_{i}, y_{i}, λ_{1}, f_{1}) = 2 (θ_{2} - θ_{1}) = \frac{2 π}{λ_{1}} (\sqrt{{x_{i}}^{2} + {y_{i}}^{2} + {f_{1}}^{2}} - f_{1}) \\ φ_{2} (x_{i}, y_{i}, λ_{1}, f_{2}) = - 2 θ_{2} = \frac{2 π}{λ_{1}} (\sqrt{{x_{i}}^{2} + {y_{i}}^{2} + {f_{2}}^{2}} - f_{2}) \end{cases}

φ_{3} (x_{i}, y_{i}, λ_{1}, f_{1}, f_{2}) = - 2 θ_{1} = \frac{2 π}{λ_{1}} (\sqrt{{x_{i}}^{2} + {y_{i}}^{2} + {f_{2}}^{2}} - f_{2} + \sqrt{{x_{i}}^{2} + {y_{i}}^{2} + {f_{1}}^{2}} - f_{1})

φ_{3} (x_{i}, y_{i}, λ_{2}, f_{3}) = \frac{2 π}{λ_{2}} (\sqrt{{x_{i}}^{2} + {y_{i}}^{2} + {f_{3}}^{2}} - f_{3})

\frac{1}{λ_{2}} (\sqrt{{x_{i}}^{2} + {y_{i}}^{2} + {f_{3}}^{2}} - f_{3}) = \frac{1}{λ_{1}} (\sqrt{{x_{i}}^{2} + {y_{i}}^{2} + {f_{2}}^{2}} - f_{2} + \sqrt{{x_{i}}^{2} + {y_{i}}^{2} + {f_{1}}^{2}} - f_{1})