Analytical design of quasi-closed subwavelength electromagnetic rectangular resonators using stacks of dielectric&#x2013;plasmonic bilayers: errata

Hongyuan Zhao; Guanghao Zhu

doi:10.1364/JOSAB.30.002928

Abstract

We correct several clerical errors of a previously published paper and provide a material loss $Q$ factor formula with improved accuracy.

We have found several clerical errors for the material loss $Q$ factor derived in Appendix B of a previously published paper [1]. Equations (B3 )–(B9) should read as

δ [\frac{1}{ε_{x}}] = - \frac{2 (1 - η)}{ε_{0} {(1 - ω_{p}^{2} / ω^{2})}^{2}} \frac{ω_{p}^{2}}{ω^{2}} \frac{δ ω - i γ_{d} / 2}{ω} = - \frac{Γ_{x}}{ε_{0}} (\frac{δ ω}{ω} - \frac{i γ_{d}}{2 ω}),

δ [\frac{1}{ε_{y}}] = - \frac{2 (1 - η) ε_{0}}{{(η ε_{1} + (1 - η) ε_{0} (1 - ω_{p}^{2} / ω^{2}))}^{2}} \frac{ω_{p}^{2}}{ω^{2}} \frac{δ ω - i γ_{d} / 2}{ω} = - \frac{Γ_{y}}{ε_{0}} (\frac{δ ω}{ω} - \frac{i γ_{d}}{2 ω}),

Γ_{x} = \frac{2 (1 - η)}{{(1 - ω_{p}^{2} / ω^{2})}^{2}} \frac{ω_{p}^{2}}{ω^{2}},

Γ_{y} = \frac{2 (1 - η)}{{(η ε_{1} / ε_{0} + (1 - η) (1 - ω_{p}^{2} / ω^{2}))}^{2}} \frac{ω_{p}^{2}}{ω^{2}},

2 ω δ ω = \frac{π^{2}}{μ_{0}} (- \frac{Γ_{y}}{ε_{0}} \frac{1}{L_{x}^{2}} - \frac{Γ_{x}}{ε_{0}} \frac{1}{L_{y}^{2}}) (\frac{δ ω}{ω} - \frac{i γ_{d}}{2 ω}),

δ ω = {(1 + \frac{2 ω^{2}}{\frac{π^{2}}{μ_{0}} (\frac{Γ_{y}}{ε_{0}} \frac{1}{L_{x}^{2}} + \frac{Γ_{x}}{ε_{0}} \frac{1}{L_{y}^{2}})})}^{- 1} \cdot \frac{i γ_{d}}{2} \approx {(1 + \frac{4}{\frac{λ_{p}^{2}}{L_{x}^{2}} \frac{1}{1 - η} + \frac{λ_{p}^{2}}{L_{y}^{2}} (1 - η)})}^{- 1} \cdot \frac{i γ_{d}}{2},

and

Q_{mat} \approx (1 + \frac{4}{\frac{λ_{p}^{2}}{L_{x}^{2}} \frac{1}{1 - η} + \frac{λ_{p}^{2}}{L_{y}^{2}} (1 - η)}) \frac{ω}{γ_{p}},

where, in the last step of the derivation of Eq. (B8),

Γ_{x} \approx 2 (1 - η) ω^{2} / ω_{p}^{2}

and

Γ_{y} \approx 2 {(1 - η)}^{- 1} ω^{2} / ω_{p}^{2}

have been used. Note that, when

η ≪ 1

, Eq. (B9) presented here reduces to Eq. (B9) of the previously published paper [1].

We emphasize that the formulas provided in this errata improve the accuracy of the analytic calculation of the material loss $Q$ factor. To demonstrate this point, we numerically evaluate the material loss $Q$ factor of a dielectric–plasmonic bilayer resonator (consisting of 10 pairs) using COMSOL Multiphysics and compare the results with those obtained using the two analytic formulas. For this purpose, we fix the resonator width, the resonator height, the dielectric permittivity, and the plasmonic damping rate as $L_{x} = L_{y} = λ_{p}$ , $ε_{1} = 12.96 ε_{0}$ , and $γ = 0.002 ω_{p}$ , and sweep the dielectric filling ratio $η$ from 0.01 to 0.5. The obtained results of the material loss $Q$ factor are plotted in Fig. 1, where the squares correspond to the numerical values from COMSOL Multiphysics, the solid line corresponds to the analytic value from Eq. (B9) of the previously published paper [1], and the dashed line corresponds to the analytic value from Eq. (B9) of this errata. From Fig. 1, the accuracy improvement of the material loss $Q$ factor formula of this errata is confirmed.

Fig. 1. Material loss $Q$ factor as a function of the dielectric filling ratio. Solid line: analytic result from Eq. (B9) of the previously published paper [1]. Dashed line: analytic result from Eq. (B9) of this errata. Squares: numerical results from COMSOL Multiphysics.

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REFERENCE

1. G. Zhu, “Analytical design of quasi-closed subwavelength electromagnetic rectangular resonators using stacks of dielectric–plasmonic bilayers,” J. Opt. Soc. Am. B 29, 2575–2580 (2012). [CrossRef]

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Figures (1)

Fig. 1.

Fig. 1. Material loss

Q

factor as a function of the dielectric filling ratio. Solid line: analytic result from Eq. (B9) of the previously published paper [1]. Dashed line: analytic result from Eq. (B9) of this errata. Squares: numerical results from COMSOL Multiphysics.

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Equations (7)

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δ [\frac{1}{ε_{x}}] = - \frac{2 (1 - η)}{ε_{0} {(1 - ω_{p}^{2} / ω^{2})}^{2}} \frac{ω_{p}^{2}}{ω^{2}} \frac{δ ω - i γ_{d} / 2}{ω} = - \frac{Γ_{x}}{ε_{0}} (\frac{δ ω}{ω} - \frac{i γ_{d}}{2 ω}),

δ [\frac{1}{ε_{y}}] = - \frac{2 (1 - η) ε_{0}}{{(η ε_{1} + (1 - η) ε_{0} (1 - ω_{p}^{2} / ω^{2}))}^{2}} \frac{ω_{p}^{2}}{ω^{2}} \frac{δ ω - i γ_{d} / 2}{ω} = - \frac{Γ_{y}}{ε_{0}} (\frac{δ ω}{ω} - \frac{i γ_{d}}{2 ω}),

Γ_{x} = \frac{2 (1 - η)}{{(1 - ω_{p}^{2} / ω^{2})}^{2}} \frac{ω_{p}^{2}}{ω^{2}},

Γ_{y} = \frac{2 (1 - η)}{{(η ε_{1} / ε_{0} + (1 - η) (1 - ω_{p}^{2} / ω^{2}))}^{2}} \frac{ω_{p}^{2}}{ω^{2}},

2 ω δ ω = \frac{π^{2}}{μ_{0}} (- \frac{Γ_{y}}{ε_{0}} \frac{1}{L_{x}^{2}} - \frac{Γ_{x}}{ε_{0}} \frac{1}{L_{y}^{2}}) (\frac{δ ω}{ω} - \frac{i γ_{d}}{2 ω}),

δ ω = {(1 + \frac{2 ω^{2}}{\frac{π^{2}}{μ_{0}} (\frac{Γ_{y}}{ε_{0}} \frac{1}{L_{x}^{2}} + \frac{Γ_{x}}{ε_{0}} \frac{1}{L_{y}^{2}})})}^{- 1} \cdot \frac{i γ_{d}}{2} \approx {(1 + \frac{4}{\frac{λ_{p}^{2}}{L_{x}^{2}} \frac{1}{1 - η} + \frac{λ_{p}^{2}}{L_{y}^{2}} (1 - η)})}^{- 1} \cdot \frac{i γ_{d}}{2},

Q_{mat} \approx (1 + \frac{4}{\frac{λ_{p}^{2}}{L_{x}^{2}} \frac{1}{1 - η} + \frac{λ_{p}^{2}}{L_{y}^{2}} (1 - η)}) \frac{ω}{γ_{p}},

Analytical design of quasi-closed subwavelength electromagnetic rectangular resonators using stacks of dielectric–plasmonic bilayers: errata

Abstract

REFERENCE

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Figures (1)

Equations (7)

Journal of the Optical Society of America B