Errata: Fitting the optical constants of gold, silver, chromium, titanium and aluminum in the visible bandwidth

Dominique Barchiesi; Thomas Grosges

doi:10.1117/1.JNP.8.089996

9 January 2015 Errata: Fitting the optical constants of gold, silver, chromium, titanium and aluminum in the visible bandwidth

Dominique Barchiesi, Thomas Grosges

Author Affiliations +

Journal of Nanophotonics, Vol. 8, Issue 1, 089996 (January 2015). https://doi.org/10.1117/1.JNP.8.089996

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.

1. The Combination of Drude and Lorentz Models

The function of fit $ϵ_{D L}$ of the relative permittivity of metal is written as the sum of the Drude and the Lorentz models:

Eq. (1)

ϵ_{D L} (ω) = ϵ_{\infty} - \frac{ω_{D}^{2}}{ω (ω + i γ_{D})} - \frac{Δ ϵ ω_{L}^{2}}{ω^{2} - ω_{L}^{2} + i γ_{L} ω} .

In the following the angular frequency $ω (rad / s)$ that is used in formula falls within the visible domain $[2.354 e 15; 4.709 e 15] rad / s$ , 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 $σ_{R}$ and $σ_{I}$ are calculated according formula (8-9) in¹, including the number of data used to compute the fitting equation.

1.1.

Gold (Johnson & Christy²)

Eq. (2)

ϵ_{D L}^{{Au}_{J C}} (ω) = 6.1599 - \frac{1.8160 E 32}{ω^{2} + i 7.2096 E 13 ω} - \frac{4.5011 E 31}{ω^{2} - 2.1732 E 31 + i 1.6694 E 15 ω},

C = 0.99995, F = 0.55, σ_{R} = 0.40, σ_{I} = 0.38 .

1.2.

Gold (Palik³)

Eq. (3)

ϵ_{D L}^{{Au}_{P}} (ω) = 0.6888 - \frac{1.5817 E 33}{ω^{2} + i 7.3731 E 15 ω} + \frac{9.3582 E 32}{ω^{2} - 5.5354 E 30 + i 4.9327 E 15 ω},

C = 0.24646, F = 1.08, σ_{R} = 0.95, σ_{I} = 0.51 .

1.3.

Silver (Palik³)

Eq. (4)

ϵ_{D L}^{{Ag}_{P}} (ω) = 0.0067526 - \frac{1.7584 E 32}{ω^{2} + i 1.0444 E 14 ω} - \frac{9.9267 E 32}{ω^{2} - 2.6509 E 32 + i 7.3068 E 15 ω},

C = 0.80656, F = 0.07154, σ_{R} = 0.053, σ_{I} = 0.048 .

1.4.

Aluminum (Palik³)

Eq. (5)

ϵ_{D L}^{{Al}_{P}} (ω) = 0.13313 - \frac{9.0588 E 32}{ω^{2} + i 3.1083 E 15 ω} + \frac{5.6526 E 32}{ω^{2} - 1.2718 E 31 + i 6.4539 E 15 ω},

C = 0.996, F = 2.98, σ_{R} = 2.49, σ_{I} = 1.64 .

1.5.

Chromium (Palik³)

Eq. (6)

ϵ_{D L}^{Cr} (ω) = 2.7767 - \frac{2.5306 E 32}{ω^{2} + i 2.9966 E 15 ω} - \frac{1.4736 E 32}{ω^{2} - 1.1087 E 31 + i 2.5764 E 15 ω},

C = 0.9998, F = 0.947, σ_{R} = 0.63, σ_{I} = 0.71 .

1.6.

Titanium (Palik³)

Eq. (7)

ϵ_{D L} (ω) = - 5.4742 E 7 - \frac{3.4555 E 32}{ω^{2} + i 5.1502 E 15 ω} - \frac{9.3068 E 54}{ω^{2} - 1.7001 E 47 + i 3.2120 E 24 ω},

C = 0.9665, F = 0.57, σ_{R} = 0.47, σ_{I} = 0.33 .

2. 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). http://dx.doi.org/10.1117/1.JNP.8.083097 JNOACQ 1934-2608 Google Scholar

2.

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3.

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

4.

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5.

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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). http://dx.doi.org/10.1364/OE.21.002245 OPEXFF 1094-4087 Google Scholar

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Dominique Barchiesi and 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

Published: 9 January 2015

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KEYWORDS

Gold

Aluminum

Chromium

Silver

Titanium

Metals

Finite-difference time-domain method

1.

The Combination of Drude and Lorentz Models

Eq. (1)