The metals

Silver, gold, copper and aluminium: the mirror coatings and hollow-waveguide claddings of ultrafast optics. A metal’s optical constants are measured piecewise, each range by a different technique. So each recommended model is a table model: curated tables joined by range, one source per range, with the reason for each (TC-544, TC-619, TC-646, TC-647; Solids: conventions and models, Tables of optical constants).

from optmat.solid.models.registry import default_model, default_reason

silver = default_model("Ag")  # table/Ag
n = silver.refractive_index(800e-9)  # n + ik, k > 0
eps = silver.permittivity(800e-9)
default_reason("Al")  # what the aluminium table rests on

Read Solids: conventions and models for the complex-index convention and interpolation interface; the Quick start gives a first calculation. Source identifiers below can be looked up in the Source catalogue catalogue.

The recommended tables

Metal	Range (nm)	100 nm	400 nm	800 nm	1030 nm	1550 nm	10 um
Ag	4.1-12398	1.274 + 0.594i	0.053 + 2.140i	0.054 + 5.606i	0.073 + 7.387i	0.139 + 11.311i	4.155 + 71.291i
Au	3.5-12398	1.460 + 0.745i	1.569 + 1.838i	0.079 + 4.836i	0.102 + 6.609i	0.190 + 10.353i	5.863 + 68.134i
Cu	3.1-55556	1.112 + 0.731i	1.239 + 2.078i	0.061 + 5.135i	0.084 + 7.003i	0.186 + 10.949i	7.296 + 73.222i
Al	0.8-200000	0.034 + 0.718i	0.490 + 4.862i	2.795 + 8.449i	1.266 + 9.967i	1.444 + 15.957i	25.280 + 89.828i

Each join lies where the two neighbouring sources agree to a few per cent. The one exception is copper at 12.4 µm, where there is a step of 11 % in k (below).

Silver, gold and copper

Range	Silver	Gold	Copper
70.5 nm-12.4 µm	Babar & Weaver	Babar & Weaver	Babar & Weaver (from 41 nm)
25-70 nm	Werner et al., REELS	Werner et al., REELS
X-ray end to 25 nm	Hagemann et al., Table 4	Hagemann et al., Table 6	Hagemann et al., Table 3 (to 41 nm)
Beyond 12.4 µm			Ordal et al. (1985), to 55.6 µm

• Babar & Weaver (2015) analysed the absorptivity of polished single crystals, 0.1-6 eV, by Kramers-Kronig. Their paper prints 0.1-6 eV. Weaver’s website publishes the same analysis to 30 eV (the paper’s Table 1 is those values rounded; TC-634).

  • Above 6 eV it rests on other authors’ reflectance: Leveque, Olson & Lynch’s for silver, Olson & Lynch’s for gold, and Johnson & Christy’s then Hagemann et al.’s for copper.

  • Gold’s reflectance below 4 eV is Weaver’s unpublished absorptivity. The 1981 compilation lists it at 4.2 K, which the table itself does not state (TC-635).

• Werner, Glantschnig & Ambrosch-Draxl (2009) derived the permittivity from reflection electron energy-loss spectra (REELS). They give about 10 % accuracy above 5 eV.

• Hagemann, Gudat & Kunz’s DESY report SR-74/7 (1974) is the numerical tables behind their JOSA letter. Its core is their own transmission of evaporated films, 13-150 eV, analysed by Kramers-Kronig.

  • Above 65-91 eV it prints 1 - n instead of n (refractive_index_decrement). There its three decimals limit the tables: they end where the printed 1 - n is still 0.004 or more (300 eV for silver, 350 eV for gold, 400 eV for copper).

  • For gold the report gives two versions. Table 6, joined to Canfield et al.’s measured reflectance, agrees with REELS. Table 5, fitted to the transmission alone, is 30-40 % higher in k.

The vacuum ultraviolet is not well known. From 40 to 200 nm the curated sources for silver and gold differ by 20-50 % in k. Silver at 100 nm illustrates the spread:

Silver at 100 nm	n + ik	Method
Babar & Weaver (the table)	1.274 + 0.594i	Kramers-Kronig of reflectance
Werner et al.	1.288 + 0.919i	REELS
Hagemann et al.	1.627 + 0.608i	Kramers-Kronig of film transmission

At 128-206 nm Stahrenberg et al.’s ellipsometry of Ag(110) is 14-27 % above the table in k, and at 6 eV the silver sources give:

Silver at 6.0 eV (207 nm)	n + ik
Babar & Weaver	1.079 + 1.247i
Johnson & Christy	1.18 + 1.31i
Stahrenberg et al.	1.053 + 1.422i
Werner et al.	0.949 + 1.548i

Gold is similar. The joins were placed where independent methods agree:

• From 41 to 70 nm Babar & Weaver’s values and REELS agree to 3-16 % in n and k (29 % in silver’s k near 60 nm), and to 4 % at 70 nm, the join.

• The DESY films’ k is 25-100 % above both at 40-50 nm: two methods against one. Film transmission can include an oxide, as Shiles et al. suspected for aluminium.

• Babar & Weaver’s own k is likely least reliable at the top of their range, where their analysis extrapolates the reflectance beyond 30 eV. So the table leaves them at 70 nm.

Copper is simpler. Babar & Weaver’s copper above 6 eV is built on Hagemann et al.’s, and the two meet at 30 eV (41 nm) to 2 % in n and 5 % in k.

Beyond 12.4 µm only copper has a measurement: Ordal et al.’s (1985) infrared reflectance, 180-800 cm⁻¹. It is 9-10 % below Babar & Weaver in k where both tabulate, hence the step at the join.

• Silver and gold have only Drude curves there: Bennett & Bennett’s, which Ordal et al. (1983) reprint, and the Drude fits of Ordal et al. (1983, 1985).

• The tables hold their 12.4 µm values beyond it, with a warning. A caller that needs a refusal instead takes model.with_out_of_range("raise"): outside its knots the table then raises StateError (OEP 1, section 6.10). The status is kept, since the numbers it returns are the same.

Aluminium

Range	Source
0.79-31 nm	Rakic (1995), Table 2
31.8 nm-31 µm	Shiles, Sasaki, Inokuti & Smith, as tabulated by Weaver et al. (1981)
33-200 µm	Ordal et al. (1988)

Shiles et al.’s analysis is the core. It is a self-consistent Kramers-Kronig analysis of the measured reflectance, ellipsometry, transmission and energy-loss data of bulk aluminium, 0.04-1551 eV. It is tested against the sum rules, the dc conductivity and the stopping power.

• The paper itself (Phys. Rev. B 22, 1612 (1980)) says the table “will be given elsewhere”. It was printed in Weaver et al.’s Physics Data 18-2 (pp. 79-86).

• To make the sum rules hold, Shiles et al. reduced the measured absorption above the L edge by 14 %, suspecting the films’ oxide.

• Ordal et al.’s (1988) table rests on it above 320 cm⁻¹ and lies 1-3 % above it.

Rakic (1995) continues it into the EUV. Above the plasma edge Rakic’s k is Shiles et al.’s, from Palik’s handbook. Its n is Rakic’s own Kramers-Kronig result, within 0.3 % of Shiles’s. Its five figures resolve what the three-decimal tabulation cannot:

• above 40 eV the tabulation’s k is 0.00x, and it is 0.000 from 500 eV;

• Rakic samples the L edge (72.7 eV) every 0.05-0.1 eV.

Films against bulk

A mirror’s coating is an evaporated film, and films differ from bulk metal. The curated film data are kept as datasets, not as tables:

Source	Metals	Range	Against the tables
Johnson & Christy (1972), with stated errors	Cu, Ag, Au	188-1937 nm	k within -7/+4 % (Cu), -5/+23 % (Ag), -13/-1 % (Au); n 2-4 times higher in the infrared
McPeak et al. (2015), template-stripped	Ag, Au, Cu	300-1700 nm	Ag and Cu ε₁ within 3 %; Au |ε₁| 15-18 % larger; ε₂ higher (Au by 20-70 %)
McPeak et al. (2015), template-stripped	Al	200-1700 nm	|ε| 21-31 % smaller than Shiles et al.’s
Rakic (1995), intrinsic constants from films	Al	0.12 nm-200 µm	within 1-2 % in k at 5-10 eV; 2-4 % in k and 9 % in n below 2 eV

Babar & Weaver attribute the difference to the films’ morphology: polycrystalline, rough and “far from bulk-like”. For copper they find the films’ infrared ε₂ 2-4 times the crystal’s.

Source and interpolation differences

The choice of source and interpolant can change the optical constants appreciably:

• Silver, 70–207 nm: Werner et al.’s REELS data to 124 nm and Stahrenberg et al.’s ellipsometry to 207 nm differ from Babar & Weaver’s analysis, used by the recommended table. The latter is up to 18 % lower in \(k\).

• Aluminium, 150 nm–1.7 µm: McPeak et al.’s film is 12–17 % below the recommended table in \(k\).

• Aluminium, 1.8–31 µm: Ordal et al.’s data are 1–2 % above the table.

• Interpolation: Babar & Weaver tabulate in 0.05 eV steps. Near 10 µm the wavelengths are 4 µm apart; a cubic spline and the model’s monotone interpolant differ by 17 % in \(n\) and 0.4 % in \(k\). The latter dominates reflectance. The data do not establish which interpolant is closer to the material’s response; this spread indicates sensitivity to interpolation.

Limits

• Uncertainty. Few sources state it per point:

  • Johnson & Christy give Δn and Δk;

  • Weaver et al.’s compilation claims ±10 % “in most cases”;

  • Werner et al. claim about 10 % above 5 eV.

  In the vacuum ultraviolet the spread between sources (20-50 %) is the better measure.

• Stahrenberg et al.’s numbers reached the database privately. Whether they are the roughness-corrected dielectric function is not established (TC-637).

• Henke et al.’s atomic scattering factors are kept for the four metals (data/external/henke/), but no table uses them.

  • For aluminium with Rakic’s atom density, they follow Rakic’s k to about 2 % at 1 keV.

  • Near the plasma edge and the L edge they are 20-50 % off, as an independent-atom model must be.

  • The DESY report and Weaver et al. print no densities for the noble metals.

  A table can take a Henke segment (henke = "<element>" with an atom_density dataset) where that is wanted.

• Group-velocity dispersion of a metal from a table is the interpolant’s (see Solids: conventions and models). For a reflection the phase on reflection matters more, and it follows from n + ik.