Microscopic analysis of the energy, momentum and spin distributions in a surface plasmon-polariton wave: erratum

A. Y. Bekshaev; A. Y. Bekshaev; O. V. Angelsky; O. V. Angelsky; J. Zheng; J. Zheng; S. G. Hanson; C. Yu. Zenkova

doi:10.1364/OME.437630

Abstract

We correct inessential mistakes in equations of the recently published paper [Opt. Mater. Express 11, 2165 (2021) [CrossRef] ]. The mistakes do not affect the numerical results and conclusions of the paper.

Four misprints are detected in equations of our paper [1]:

(1) The 1^st Eq. (39) (Section 3.2, p. 2178) erroneously contains the coefficient “g” which should be replaced by “1/8π” so that the correct Eq. (39) reads: $(39)$${\textbf{p}^F} = \textbf{z}{p^F} = \frac{1}{{8\pi c}}\textrm{Re} ({\textbf{E}_{}^\ast{\times} \textbf{H}} ),\,\,\,{p^F} = \frac{1}{{8\pi c}}E_x^\ast {H_y}. $$$
(2) In the unnumbered equation at the top of p. 2181 (Section 3.3), also “g” should be replaced by “1/8π” so that the correct form is as follows: $\mathbf{z} p_{e}^{M}=\frac{1}{4} \mathbf{z} \operatorname{Im}\left(\alpha_{x x}^{\prime} E_{x}^{*} \frac{\partial E_{x}}{\partial z}+\alpha_{z z}^{\prime} E_{z}^{*} \frac{\partial E_{z}}{\partial z}\right) \simeq \frac{1}{16 \pi} \frac{d \varepsilon_{2}}{d \omega} \operatorname{Im}\left[\mathbf{E}^{*} \cdot(\nabla) \mathbf{E}\right]$
(3) In Eq. (47) (Section 3.2, p. 2179) the “–” sign before the last summand in the brackets should be replaced by “+” so the correct form of Eq. (47) is $(47)$$s_R^M ={-} \frac{{2g}}{\omega }\frac{\eta }{{\varepsilon _2^2}}\frac{{{\kappa _2}}}{{{k_s}}}\left[ {{e^{2{\kappa_2}x}} - \left( {1 + \frac{{k_s^2}}{{\gamma {\kappa_2}}}} \right){e^{({{\kappa_2} + \gamma } )x}} + \frac{{k_s^2}}{{\gamma {\kappa_2}}}{e^{2\gamma x}}} \right]. $$$
(4) In Eq. (71) (Section 3.4, p. 2185), the coefficient “η” was erroneously omitted in the last summand in brackets; the correct form of this equation reads: $(71)$$s = {s^F} + s_m^M + s_R^M ={-} \frac{g}{\omega }\frac{{{\kappa _2}}}{{\varepsilon _2^2{k_s}}}[{({1 + \eta } ){e^{2{\kappa_2}x}} - 2\eta {e^{({{\kappa_2} + \gamma } )x}}} ]. $$$

These corrections do not influence the paper’s conclusions and the numerical data presented in figures.

References

1. A. Y. Bekshaev, O. V. Angelsky, J. Zheng, S. G. Hanson, and C. Y. Zenkova, “Microscopic analysis of the energy, momentum and spin distributions in a surface plasmon-polariton wave,” Opt. Mater. Express 11(7), 2165–2191 (2021). [CrossRef]

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Equations (4)

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p^{F} = z p^{F} = \frac{1}{8 π c} Re (E_{}^{*} \times H), p^{F} = \frac{1}{8 π c} E_{x}^{*} H_{y} .

z p_{e}^{M} = \frac{1}{4} z Im (α_{x x}^{'} E_{x}^{*} \frac{\partial E_{x}}{\partial z} + α_{z z}^{'} E_{z}^{*} \frac{\partial E_{z}}{\partial z}) ≃ \frac{1}{16 π} \frac{d ε_{2}}{d ω} Im [E^{*} \cdot (\nabla) E]

s_{R}^{M} = - \frac{2 g}{ω} \frac{η}{ε_{2}^{2}} \frac{κ_{2}}{k_{s}} [e^{2 κ_{2} x} - (1 + \frac{k_{s}^{2}}{γ κ_{2}}) e^{(κ_{2} + γ) x} + \frac{k_{s}^{2}}{γ κ_{2}} e^{2 γ x}] .

s = s^{F} + s_{m}^{M} + s_{R}^{M} = - \frac{g}{ω} \frac{κ_{2}}{ε_{2}^{2} k_{s}} [(1 + η) e^{2 κ_{2} x} - 2 η e^{(κ_{2} + γ) x}] .

Abstract

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Equations (4)

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