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# 金、銀、銅の反射シミュレーション

## Abstract

The reflection spectra of visible to near-infrared light for perpendicular incidence of gold, silver and copper smooth surfaces were derived. Gold, silver, and copper have plasma frequencies near the visible light band, and the physical properties near the plasma frequencies can be approximated by a combination of Drude and Lorentz dispersion models. The reflection spectra were derived by FDTD method and compared with those calculated from the measured complex refractive index. The simulated and measured reflectance of both metals agreed well overall, confirming the validity of the dispersion parameters used in the analysis and the analysis of KeyFDTD for multipolar dispersible materials.

## 2．解析条件

Fig.1にシミュレーションモデルを示す。金と銅の複素誘電率はDrude-Lorentz-Lorentzモデル(eq.1)、銀はDrude-Lorentzモデル(eq.2)で近似した。 ωは入射光の角周波数である。金、銀、銅の式中の各係数をTable1.に示す。

Fig.1 Simulation model

$$\varepsilon_r(\omega) = \varepsilon_\infty + \frac{\omega^2_{p1}}{\omega(j\nu_c-\omega)}+\Delta\varepsilon_2 \frac{\omega^2_{p2}} {\omega^2_{p2}+j\omega\delta_{p2}-\omega^2}+\Delta\varepsilon_3 \frac{\omega^2_{p3}} {\omega^2_{p3}+j\omega\delta_{p3}-\omega^2} ・・・(eq.1)$$　 $$\varepsilon_r(\omega) = \varepsilon_\infty + \frac{\omega^2_{p1}}{\omega(j\nu_c-\omega)}+\Delta\varepsilon_2 \frac{\omega^2_{p2}} {\omega^2_{p2}+j\omega\delta_{p2}-\omega^2} ・・・(eq.2)$$
Table.1 Dispersion parameters
Parameter Gold Silver Copper
$\varepsilon_\infty$ $6.00$ $2.00$ $6.00$
$\omega_{p1}$[rad/sec] $1.40\times10^{16}$ $1.30\times10^{16}$ $1.37\times10^{16}$
$\omega_{p2}$[rad/sec] $5.50\times10^{15}$ $7.00\times10^{15}$ $4.50\times10^{15}$
$\omega_{p3}$[rad/sec] $4.30\times10^{15}$   $3.60\times10^{15}$
$\nu_c$[rad/sec] $9.80\times10^{13}$ $7.36\times10^{13}$ $9.59\times10^{13}$
$\Delta\varepsilon_2$ $2.00$ $1.00$ $2.00$
$\Delta\varepsilon_3$ $0.600$   $0.300$
$\delta_{p2}$[rad/sec] $2.20\times10^{15}$ $2.10\times10^{15}$ $2.80\times10^{15}$
$\delta_{p3}$[rad/sec] $8.17\times10^{14}$   $5.40\times10^{14}$

Table.2の解析条件で、電磁波解析ソフトKeyFDTDTRを用いて各金属の反射スペクトルを導出した。反射スペクトルは、ガウシアンパルスの入射波形と反射波形のフーリエ変換後のエネルギー比から計算した。

Table.2 Analysis condition
Incident pulse $Gaussian\ pulse: \tau_0 = 2\times10^{-15}[sec]$ $x,y: PERIODIC, z:MUR1$ $10\times10\times2000[nm]$ $2.5[nm]=λ/160～480$ $4.18×10^{-18}[sec]$

## 3．解析結果

Fig.2 Reflectance spectra of gold

Fig.3 Reflectance spectra of silver

Fig.4 Reflectance spectra of copper

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