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Notes on Metals in meep


Material Dispersion

Dielectric materials in meep are implemented in terms of a frequency and position dependent $ \mu$ and $ \epsilon$. Specifically, the electric field as a function of angular frequency $ \epsilon(\omega)$ is defined according to

$\displaystyle \epsilon(\omega)= \epsilon_\infty+\sum_n \frac{\sigma_n \omega_n^2} {\omega_n^2-\omega^2-{\mathrm{i}}\omega\Gamma_n}$ (1)

where $ \epsilon_\infty$ is the instantaneous dielectric response (DC), $ \sigma_D$ is the electric conductivity, $ \omega_n$ and $ \Gamma_n$ are constants, and $ \sigma_n$ is a function of position specifying the strength of the $ n^$th resonance.1

In the literature however, metals are commonly specified according to the Lorentz-Drude (LD) model.

$\displaystyle \epsilon_{LD}=\epsilon_D+\epsilon_L$ (2)

where $ \epsilon_D$ is contribution from the Drude model, representing the free electron effects

$\displaystyle \epsilon_D=1-\frac{f_1 \omega_p^{\prime 2}}{\omega(\omega - {\mathrm{i}}\Gamma_1^\prime)}$ (3)

and $ \epsilon_L$ is the Lorentz contribution, representing the bound electron effects

$\displaystyle \epsilon_L =\sum_{n}\frac{f_n\omega_p^{\prime 2}}{\omega_n^{\prime 2}-\omega^2+{\mathrm{i}}\omega\Gamma_n^\prime}$ (4)

This particular model and coefficients are obtained from [1]. In this paper, the coefficients are in electron volts. Since meep operates in dimensionless units (e.g. $ c=\hbar=1$), the units in [1] must be converted. First a length scale $ a$ is chosen. The frequencies are then expressed in $ c/a$, where $ c$ is the speed of light. The conversion from joules to electron volts is given by Thus to convert from $ \si{\eV}$ to angular frequency, simply divide the result by $ \hbar$. Likewise to convert this angular frequency to dimensionless units, the result must be divided by $ 2\pi c/a$. Putting this together, given a value $ X_{\si{\eV}}$ in $ {\si{\eV}}$, the corresponding value in dimensionless units $ X_0$ is obtained by

$\displaystyle X_0$ $\displaystyle =\left. \frac{X_{\si{\eV}}}{\hbar} \middle/ (2\pi c/a)\right.$ (5)
  $\displaystyle =X_{\si{\eV}}\frac{1}{2\pi\hbar c /a}$ (6)
  $\displaystyle =X_{\si{\eV}}\frac{a}{h c}$ (7)

Once normalized, the LD coefficients can be imported into meep with the transformations2

$\displaystyle \omega_1$ $\displaystyle = \num{1e-20}$ (8)
$\displaystyle \sigma_n$ $\displaystyle = \frac{f_n \omega_p^{\prime 2}}{\omega_n^2}$ (9)
$\displaystyle \epsilon_\infty$ $\displaystyle = 1$ (10)
$\displaystyle \Gamma_n$ $\displaystyle = \Gamma_n^\prime$ (11)

To check these results, a $ 1\times1\times1$ pixel region of dielectric was simulated and $ \epsilon$ was exported with the command meep-fields-analytic-chi1. The results, which follow, were then compared to the complex $ \epsilon$ predicted by the LD model (via LD.m courtesy of Bora Ung of Ecole Polytechnique de Montreal)3.

meep Code for Metals

The following is meep code for selected metals with $ a=\SI{1}{\micro\meter}$.

Silver


\begin{picture}(9600,7680)(0,0)%
{\catcode\lq \%=14\relax\special {''
%!PS-Adobe-2...
...,0)[r]{\strut{}4}}%
\put(1128,1015){\makebox(0,0)[r]{\strut{}2}}%
\end{picture}
(define myAg (make dielectric (epsilon 1)
(polarizations
 (make polarizability
(omega 1e-20) (gamma 0.038715) (sigma 4.4625e+41))
(make polarizability
(omega 0.65815) (gamma 3.1343) (sigma 7.9247))
(make polarizability
(omega 3.6142) (gamma 0.36456) (sigma 0.50133))
(make polarizability
(omega 6.6017) (gamma 0.052426) (sigma 0.013329))
(make polarizability
(omega 7.3259) (gamma 0.7388) (sigma 0.82655))
(make polarizability
(omega 16.365) (gamma 1.9511) (sigma 1.1133))
)))
;Additional Information
;Normalization length=1e-06 in meter
;Material_used_is_Agfrom Rakic et al.,Applied Optics (1998)
;Plasma Angular Frequency (and plasma wave vector,kp) in normalized units=6.6802

Aluminum


\begin{picture}(9600,7680)(0,0)%
{%
\catcode\lq \%=14\relax\special {''
%!PS-Adob...
...0,0)[r]{\strut{}6}}%
\put(1128,968){\makebox(0,0)[r]{\strut{}4}}%
\end{picture}
(define myAl (make dielectric (epsilon 1)
(polarizations
 (make polarizability
(omega 1e-20) (gamma 0.037908) (sigma 7.6347e+41))
(make polarizability
(omega 0.13066) (gamma 0.26858) (sigma 1941))
(make polarizability
(omega 1.2453) (gamma 0.25165) (sigma 4.7065))
(make polarizability
(omega 1.4583) (gamma 1.0897) (sigma 11.396))
(make polarizability
(omega 2.8012) (gamma 2.7278) (sigma 0.55813))
)))
;Additional Information
;Normalization length=1e-06 in meter
;Material_used_is_Alfrom Rakic et al.,Applied Optics (1998)
;Plasma Angular Frequency (and plasma wave vector,kp) in normalized units=8.7377

Gold


\begin{picture}(9600,7680)(0,0)%
{%
\catcode\lq \%=14\relax\special {''
%!PS-Adob...
...0,0)[r]{\strut{}4}}%
\put(1128,966){\makebox(0,0)[r]{\strut{}2}}%
\end{picture}
(define myAu (make dielectric (epsilon 1)
(polarizations
 (make polarizability
(omega 1e-20) (gamma 0.042747) (sigma 4.0314e+41))
(make polarizability
(omega 0.33472) (gamma 0.19438) (sigma 11.363))
(make polarizability
(omega 0.66944) (gamma 0.27826) (sigma 1.1836))
(make polarizability
(omega 2.3947) (gamma 0.7017) (sigma 0.65677))
(make polarizability
(omega 3.4714) (gamma 2.0115) (sigma 2.6455))
(make polarizability
(omega 10.743) (gamma 1.7857) (sigma 2.0148))
)))
;Additional Information
;Normalization length=1e-06 in meter
;Material_used_is_Aufrom Rakic et al.,Applied Optics (1998)
;Plasma Angular Frequency (and plasma wave vector,kp) in normalized units=6.3493

Beryllium


\begin{picture}(9600,7680)(0,0)%
{%
\catcode\lq \%=14\relax\special {''
%!PS-Adob...
...0)[r]{\strut{}3.5}}%
\put(1128,932){\makebox(0,0)[r]{\strut{}3}}%
\end{picture}
(define myBe (make dielectric (epsilon 1)
(polarizations
 (make polarizability
(omega 1e-20) (gamma 0.028229) (sigma 1.8722e+41))
(make polarizability
(omega 0.080655) (gamma 1.3421) (sigma 1062.1))
(make polarizability
(omega 0.83236) (gamma 2.7383) (sigma 45.038))
(make polarizability
(omega 2.5673) (gamma 3.5924) (sigma 17.923))
(make polarizability
(omega 3.7134) (gamma 1.4534) (sigma 2.1013))
)))
;Additional Information
;Normalization length=1e-06 in meter
;Material_used_is_Befrom Rakic et al.,Applied Optics (1998)
;Plasma Angular Frequency (and plasma wave vector,kp) in normalized units=4.3269

Chromium


\begin{picture}(9600,7680)(0,0)%
{%
\catcode\lq \%=14\relax\special {''
%!PS-Adob...
...0)[r]{\strut{}2}}%
\put(1128,886){\makebox(0,0)[r]{\strut{}1.5}}%
\end{picture}

(define myCr (make dielectric (epsilon 1)
(polarizations
 (make polarizability
(omega 1e-20) (gamma 0.037908) (sigma 1.263e+41))
(make polarizability
(omega 0.097593) (gamma 2.5608) (sigma 1191.9))
(make polarizability
(omega 0.43796) (gamma 1.0526) (sigma 58.791))
(make polarizability
(omega 1.5889) (gamma 2.1583) (sigma 34.214))
(make polarizability
(omega 7.0775) (gamma 1.0768) (sigma 1.2382))
)))
;Additional Information
;Normalization length=1e-06 in meter
;Material_used_is_Crfrom Rakic et al.,Applied Optics (1998)
;Plasma Angular Frequency (and plasma wave vector,kp) in normalized units=3.5538

Copper


\begin{picture}(9600,7680)(0,0)%
{%
\catcode\lq \%=14\relax\special {''
%!PS-Adob...
...0,0)[r]{\strut{}4}}%
\put(1128,883){\makebox(0,0)[r]{\strut{}2}}%
\end{picture}
(define myCu (make dielectric (epsilon 1)
(polarizations
 (make polarizability
(omega 1e-20) (gamma 0.024197) (sigma 4.3873e+41))
(make polarizability
(omega 0.23471) (gamma 0.30488) (sigma 84.489))
(make polarizability
(omega 2.385) (gamma 0.85172) (sigma 1.395))
(make polarizability
(omega 4.2747) (gamma 2.5915) (sigma 3.0189))
(make polarizability
(omega 9.0173) (gamma 3.4722) (sigma 0.59868))
)))
;Additional Information
;Normalization length=1e-06 in meter
;Material_used_is_Cufrom Rakic et al.,Applied Optics (1998)
;Plasma Angular Frequency (and plasma wave vector,kp) in normalized units=6.6236

Nickel


\begin{picture}(9600,7680)(0,0)%
{%
\catcode\lq \%=14\relax\special {''
%!PS-Adob...
...0,0)[r]{\strut{}3}}%
\put(1128,863){\makebox(0,0)[r]{\strut{}2}}%
\end{picture}
(define myNi (make dielectric (epsilon 1)
(polarizations
 (make polarizability
(omega 1e-20) (gamma 0.038715) (sigma 1.5828e+41))
(make polarizability
(omega 0.14034) (gamma 3.6384) (sigma 837.12))
(make polarizability
(omega 0.46941) (gamma 1.0759) (sigma 101.01))
(make polarizability
(omega 1.2881) (gamma 1.7567) (sigma 10.534))
(make polarizability
(omega 4.9111) (gamma 5.0748) (sigma 4.9834))
)))
;Additional Information
;Normalization length=1e-06 in meter
;Material_used_is_Nifrom Rakic et al.,Applied Optics (1998)
;Plasma Angular Frequency (and plasma wave vector,kp) in normalized units=3.9784

Palladium


\begin{picture}(9600,7680)(0,0)%
{%
\catcode\lq \%=14\relax\special {''
%!PS-Adob...
...0,0)[r]{\strut{}3}}%
\put(1128,853){\makebox(0,0)[r]{\strut{}2}}%
\end{picture}
(define myPd (make dielectric (epsilon 1)
(polarizations
 (make polarizability
(omega 1e-20) (gamma 0.0064524) (sigma 2.0282e+41))
(make polarizability
(omega 0.271) (gamma 2.3793) (sigma 543.12))
(make polarizability
(omega 0.40408) (gamma 0.44764) (sigma 45.545))
(make polarizability
(omega 1.3381) (gamma 3.7271) (sigma 21.901))
(make polarizability
(omega 4.6095) (gamma 2.61) (sigma 1.3104))
)))
;Additional Information
;Normalization length=1e-06 in meter
;Material_used_is_Pdfrom Rakic et al.,Applied Optics (1998)
;Plasma Angular Frequency (and plasma wave vector,kp) in normalized units=4.5036

Platinum


\begin{picture}(9600,7680)(0,0)%
{%
\catcode\lq \%=14\relax\special {''
%!PS-Adob...
...0,0)[r]{\strut{}3}}%
\put(1128,951){\makebox(0,0)[r]{\strut{}2}}%
\end{picture}
(define myPt (make dielectric (epsilon 1)
(polarizations
 (make polarizability
(omega 1e-20) (gamma 0.064524) (sigma 1.9923e+41))
(make polarizability
(omega 0.62911) (gamma 0.41699) (sigma 28.872))
(make polarizability
(omega 1.0598) (gamma 1.4824) (sigma 35.102))
(make polarizability
(omega 2.5334) (gamma 2.9584) (sigma 5.099))
(make polarizability
(omega 7.4598) (gamma 6.8694) (sigma 3.8445))
)))
;Additional Information
;Normalization length=1e-06 in meter
;Material_used_is_Ptfrom Rakic et al.,Applied Optics (1998)
;Plasma Angular Frequency (and plasma wave vector,kp) in normalized units=4.4635

Titanium


\begin{picture}(9600,7680)(0,0)%
{%
\catcode\lq \%=14\relax\special {''
%!PS-Adob...
...0)[r]{\strut{}2}}%
\put(1128,976){\makebox(0,0)[r]{\strut{}1.5}}%
\end{picture}
(define myTi (make dielectric (epsilon 1)
(polarizations
 (make polarizability
(omega 1e-20) (gamma 0.066137) (sigma 5.1166e+40))
(make polarizability
(omega 0.62669) (gamma 1.8357) (sigma 79.136))
(make polarizability
(omega 1.2461) (gamma 2.0309) (sigma 8.7496))
(make polarizability
(omega 2.0236) (gamma 1.3413) (sigma 1.5787))
(make polarizability
(omega 1.5671) (gamma 1.4211) (sigma 0.014077))
)))
;Additional Information
;Normalization length=1e-06 in meter
;Material_used_is_Tifrom Rakic et al.,Applied Optics (1998)
;Plasma Angular Frequency (and plasma wave vector,kp) in normalized units=2.262

Tungsten


\begin{picture}(9600,7680)(0,0)%
{%
\catcode\lq \%=14\relax\special {''
%!PS-Adob...
...,0)[r]{\strut{}4}}%
\put(1128,1012){\makebox(0,0)[r]{\strut{}3}}%
\end{picture}
(define myW (make dielectric (epsilon 1)
(polarizations
 (make polarizability
(omega 1e-20) (gamma 0.05162) (sigma 2.3421e+41))
(make polarizability
(omega 0.80978) (gamma 0.42747) (sigma 9.3624))
(make polarizability
(omega 1.5462) (gamma 1.0332) (sigma 7.8945))
(make polarizability
(omega 2.8875) (gamma 2.6874) (sigma 9.6272))
(make polarizability
(omega 6.0475) (gamma 4.7071) (sigma 8.0514))
)))
;Additional Information
;Normalization length=1e-06 in meter
;Material_used_is_Wfrom Rakic et al.,Applied Optics (1998)
;Plasma Angular Frequency (and plasma wave vector,kp) in normalized units=4.8395

Useful Files

Changes to this Document

  1. 10/10/2011 Fixed incorrect plot units (Georg Wachter).

Bibliography

1
Aleksandar D. Rakic, Aleksandra B. Djurišic, Jovan M. Elazar, and Marian L. Majewski.
Optical properties of metallic films for vertical-cavity optoelectronic devices.
Appl. Opt., 37(22):5271-5283, Aug 1998.



Footnotes

... resonance.1
http://ab-initio.mit.edu/wiki/index.php/Dielectric_materials_in_Meep
... transformations2
Derivation and script by Bala Krishna Juluri http://juluribk.com/
... Montreal)3
http://mpl.mpg.de/mpf/php/bfp/aw/LD.m

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Laboratory of Biophotonics and Biosensing,   Max Planck Institute for the Science of Light