9 January 2015 Errata: Fitting the optical constants of gold, silver, chromium, titanium and aluminum in the visible bandwidth
Abstract
This paper [J. Nanophoton. 8(1), 083097 (2014)] was published on 6 January 2014. Thanks to a question by Yoann Brûlé from the Fresnel institute (Marseille, France), we found that the values of γL and γD were swapped in tables in Ref. 1. The problem comes from a bug in the automatic extraction of data from optimization method. Fortunately the curves in Ref. 1 are correct. This erratum gives a more readily available formulation of fitting for all considered metals and the corresponding criteria.
Barchiesi and Grosges: Errata: Fitting the optical constants of gold, silver, chromium, titanium and aluminum in the visible bandwidth

## The Combination of Drude and Lorentz Models

The function of fit ${ϵ}_{DL}$ of the relative permittivity of metal is written as the sum of the Drude and the Lorentz models:

## (1)

${ϵ}_{DL}\left(\omega \right)={ϵ}_{\infty }-\frac{{\omega }_{D}^{2}}{\omega \left(\omega +i{\gamma }_{D}\right)}-\frac{\mathrm{\Delta }ϵ{\omega }_{L}^{2}}{{\omega }^{2}-{\omega }_{L}^{2}+i{\gamma }_{L}\omega }.$

In the following the angular frequency $\omega \text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\mathrm{rad}/\mathrm{s}\right)$ that is used in formula falls within the visible domain $\left[2.354e15;4.709e15\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{rad}/\mathrm{s}$, corresponding to wavelengths in [400; 800] nm and photon energy in [1.55, 3.10] eV. Outside this domain, the quality of fitting can be impaired. This erratum gives us the opportunity to give better solutions to this hard problem of fitting, by investigating a wider space of search. The values of ${\sigma }_{R}$ and ${\sigma }_{I}$ are calculated according formula (8-9) in1, including the number of data used to compute the fitting equation.

## (2)

${ϵ}_{DL}^{{\mathrm{Au}}_{JC}}\left(\omega \right)=6.1599-\frac{1.8160\mathrm{E}32}{{\omega }^{2}+i7.2096\mathrm{E}13\omega }-\frac{4.5011\mathrm{E}31}{{\omega }^{2}-2.1732\mathrm{E}31+i1.6694\mathrm{E}15\omega },$
$C=0.99995,\phantom{\rule[-0.0ex]{1em}{0.0ex}}F=0.55,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\sigma }_{R}=0.40,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\sigma }_{I}=0.38.$

## (3)

${ϵ}_{DL}^{{\mathrm{Au}}_{P}}\left(\omega \right)=0.6888-\frac{1.5817\mathrm{E}33}{{\omega }^{2}+i7.3731\mathrm{E}15\omega }+\frac{9.3582\mathrm{E}32}{{\omega }^{2}-5.5354\mathrm{E}30+i4.9327\mathrm{E}15\omega },$
$C=0.24646,\phantom{\rule[-0.0ex]{1em}{0.0ex}}F=1.08,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\sigma }_{R}=0.95,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\sigma }_{I}=0.51.$

## (4)

${ϵ}_{DL}^{{\mathrm{Ag}}_{P}}\left(\omega \right)=0.0067526-\frac{1.7584\mathrm{E}32}{{\omega }^{2}+i1.0444\mathrm{E}14\omega }-\frac{9.9267\mathrm{E}32}{{\omega }^{2}-2.6509\mathrm{E}32+i7.3068\mathrm{E}15\omega },$
$C=0.80656,\phantom{\rule[-0.0ex]{1em}{0.0ex}}F=0.07154,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\sigma }_{R}=0.053,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\sigma }_{I}=0.048.$

## (5)

${ϵ}_{DL}^{{\mathrm{Al}}_{P}}\left(\omega \right)=0.13313-\frac{9.0588\mathrm{E}32}{{\omega }^{2}+i3.1083\mathrm{E}15\omega }+\frac{5.6526\mathrm{E}32}{{\omega }^{2}-1.2718\mathrm{E}31+i6.4539\mathrm{E}15\omega },$
$C=0.996,\phantom{\rule[-0.0ex]{1em}{0.0ex}}F=2.98,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\sigma }_{R}=2.49,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\sigma }_{I}=1.64.$

## (6)

${ϵ}_{DL}^{\mathrm{Cr}}\left(\omega \right)=2.7767-\frac{2.5306\mathrm{E}32}{{\omega }^{2}+i2.9966\mathrm{E}15\omega }-\frac{1.4736\mathrm{E}32}{{\omega }^{2}-1.1087\mathrm{E}31+i2.5764\mathrm{E}15\omega },$
$C=0.9998,\phantom{\rule[-0.0ex]{1em}{0.0ex}}F=0.947,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\sigma }_{R}=0.63,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\sigma }_{I}=0.71.$

## (7)

${ϵ}_{DL}\left(\omega \right)=-5.4742\mathrm{E}7-\frac{3.4555\mathrm{E}32}{{\omega }^{2}+i5.1502\mathrm{E}15\omega }-\frac{9.3068\mathrm{E}54}{{\omega }^{2}-1.7001\mathrm{E}47+i3.2120\mathrm{E}24\omega },$
$C=0.9665,\phantom{\rule[-0.0ex]{1em}{0.0ex}}F=0.57,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\sigma }_{R}=0.47,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\sigma }_{I}=0.33.$

## Conclusion

The proposed results of fitting of relative permittivities of metals are more accurate than those proposed in a previous paper4 and verify the criterion of compatibility with FDTD use. They can be used directly for any spectroscopic simulation5,6 and especially in FDTD codes, and for plasmonics7 and optimization where accurate positions of resonances should be found. The proposed method of fitting under constraint is a combination of PSO and Nelder-mead simplex methods appears to be efficient, even if the solution of the problem of fitting is not unique.

## References

1. D. BarchiesiT. Grosges, “Fitting the optical constants of gold, silver, chromium, titanium, and aluminum in the visible bandwidth,” J. Nanophoton. 8(1), 083097 (2014).JNOACQ1934-2608 http://dx.doi.org/10.1117/1.JNP.8.083097 Google Scholar

2. P. B. JohnsonR. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).PRBMDO0163-1829 http://dx.doi.org/10.1103/PhysRevB.6.4370 Google Scholar

3. E. D. Palik, Handbook of Optical Constants, Academic Press Inc., San Diego USA (1985). Google Scholar

4. A. VialA.-S. GrimaultD. MaciasD. BarchiesiM. Lamy de la Chapelle, “Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416–085423 (2005).PRBMDO0163-1829 http://dx.doi.org/10.1103/PhysRevB.71.085416 Google Scholar

5. T. GrosgesD. BarchiesiT. TouryG. Gréhan, “Design of nanostructures for imaging and biomedical applications by plasmonic optimization,” Opt. Lett. 33(23), 2812–2814 (2008).OPLEDP0146-9592 http://dx.doi.org/10.1364/OL.33.002812 Google Scholar

6. D. BarchiesiS. KessentiniN. GuillotM. Lamy de la ChapelleT. Grosges, “Localized surface plasmon resonance in arrays of nano-gold cylinders: inverse problem and propagation of uncertainties,” Opt. Express 21(2), 2245–2262 (2013).OPEXFF1094-4087 http://dx.doi.org/10.1364/OE.21.002245 Google Scholar

7. D. BarchiesiE. KremerV. P. MaiT. Grosges, “A Poincaré’s approach for plasmonics: The plasmon localization,” J. Microscopy 229(3), 525–532 (2008).0022-2720 http://dx.doi.org/10.1111/j.1365-2818.2008.01938.x Google Scholar

© 2014 Society of Photo-Optical Instrumentation Engineers (SPIE)
Dominique Barchiesi, Thomas Grosges, "Errata: Fitting the optical constants of gold, silver, chromium, titanium and aluminum in the visible bandwidth," Journal of Nanophotonics 8(1), 089996 (9 January 2015). https://doi.org/10.1117/1.JNP.8.089996 . Submission:
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