First-principle model for the directional spectral absorptivity of gold-black in the near infrared

Nazia B. Munir; J. R. Mahan; Kory J. Priestley

doi:10.1364/JOSAA.36.001675

Journal of the Optical Society of America A
Vol. 36,
Issue 10,
pp. 1675-1689
(2019)
•https://doi.org/10.1364/JOSAA.36.001675

First-principle model for the directional spectral absorptivity of gold-black in the near infrared

Nazia B. Munir, J. R. Mahan, and Kory J. Priestley

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Nazia B. Munir, J. R. Mahan, and Kory J. Priestley, "First-principle model for the directional spectral absorptivity of gold-black in the near infrared," J. Opt. Soc. Am. A 36, 1675-1689 (2019)

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- Instrumentation, Measurement, and Metrology
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About this Article
History
- Original Manuscript: June 24, 2019
- Revised Manuscript: August 19, 2019
- Manuscript Accepted: August 21, 2019
- Published: September 13, 2019

Abstract

Gold-black coatings formed by evaporation and subsequent sublimation of pure gold in an inert low-pressure atmosphere are used to enhance absorption of thermal radiation. Previous attempts to predict the spectral absorptivity of gold-black coatings have typically assumed a granular continuum with effective bulk optical properties. In the current effort the principles of diffusion-limited aggregation (DLA) are applied to mimic the observed microstructure of actual gold-black layers consisting of a random fractal distribution of dendritic gold filaments that are postulated to behave as dipole antennas. Absorption of incident electromagnetic radiation by individual filaments is predicted using a unique time-dependent lossy antenna model drawing on the Drude–Sommerfeld free-electron theory. Single-filament spectral absorptivities are combined based on the DLA microstructure model to predict the spectral and, for the first time, the directional absorptivity of the gold-black layer. Results for normal spectral absorptivity are shown to be in good agreement with measurements reported in the literature.

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Wavelength $λ$ (µm)	Real Part of Refractive Index $n_{1}$ (Drude Theory)	Negative Real Part of Relative Permittivity $ε_{r, 1}$ (Drude Theory)	Real part of Electrical Conductivity $σ_{1}$ (Drude Theory)	$M = \frac{σ_{1}}{ε_{1}} \frac{Δ x n_{1}}{c_{0}}$	Calculation Time Step $n_{1} Δ x / c_{0}$ ( $10^{- 18} s$ )	Relaxation Time, $τ_{R}$ (ns)	Single-Filament Spectral Absorptivity, $α_{L} (λ)$
$Filament Length = 0.70 µ m$ ( $Δ x = 0.70 \times 10^{-8} m$ )
0.7	0.065095	22.0146	14555.1	0.00011342	1.518	0.006784	0.000688
1.4	0.255721	90.8645	58097.7	0.00043089	5.966	0.028066	0.000333
2.8	1.013369	363.385	230447.9	0.00169359	23.64	0.111756	0.000167
$Filament Length = 0.80 µ m$ ( $Δ x = 0.80 \times 10^{-8} m$ )
0.8	0.084562	29.0534	19006.6	0.00016661	2.255	0.006858	0.000778
1.6	0.333395	118.883	75817.5	0.00064038	8.890	0.028137	0.000379
3.2	1.320346	473.313	299969.6	0.00252025	35.20	0.111906	0.000191
$Filament Length = 0.90 µ m$ ( $Δ x = 0.90 \times 10^{-8} m$ )
0.9	0.106623	37.0271	24049.4	0.00023465	3.198	0.006901	0.000870
1.8	0.421326	150.579	95863.1	0.00090883	12.63	0.028184	0.000426
3.6	1.666682	596.998	378191.2	0.00357744	50.00	0.112258	0.000214
$Filament Length = 1.00 µ m$ ( $Δ x = 1.00 \times 10^{-8} m$ )
1.0	0.131275	45.9342	29682.5	0.00031936	4.375	0.006939	0.000961
2.0	0.519477	185.931	118221.0	0.00124350	17.31	0.028200	0.000473
4.0	2.051806	734.114	464907.6	0.00489190	68.39	0.112170	0.000238
$Filament Length = 1.10 µ m$ ( $Δ x = 1.10 \times 10^{-8} m$ )
1.1	0.158515	55.7733	35905.0	0.00042260	5.812	0.006973	0.001052
2.2	0.627804	224.916	142875.8	0.0016515	23.019	0.028232	0.000520
4.4	2.475090	884.307	559893.9	0.00648971	90.753	0.112271	0.000261
$Filament Length = 1.20 µ m$ ( $Δ x = 1.20 \times 10^{-8} m$ )
1.2	0.188338	66.5426	42715.9	0.00054619	7.53353	0.006988	0.001146
2.4	0.746263	267.506	169810.9	0.0021401	29.8505	0.028264	0.000566
4.8	2.935851	1047.192	662907.1	0.00839615	117.434	0.112271	0.000285
$Filament Length = 1.30 µ m$ ( $Δ x = 1.30 \times 10^{-8} m$ )
1.3	0.220741	78.2404	50113.8	0.00069198	9.56547	0.007007	0.001238
2.6	0.874802	313.673	199008.2	0.0027163	37.9081	0.028280	0.000613
5.2	3.433356	1222.359	773687.7	0.01063575	148.779	0.112233	0.000309
$Filament Length = 1.40 µ m$ ( $Δ x = 1.40 \times 10^{-8} m$ )
1.4	0.255721	90.8645	58097.7	0.00086178	11.9337	0.007017	0.001331
2.8	1.013369	363.385	230447.9	0.00338719	47.2906	0.028304	0.000660
5.6	3.966826	1409.374	891961.1	0.01323213	185.119	0.112271	0.000333
$Filament Length = 1.50 µ m$ ( $Δ x = 1.50 \times 10^{-8} m$ )
1.5	0.293274	104.4128	66666.1	0.00105744	14.6637	0.007027	0.001424
3.0	1.161904	416.611	264109.2	0.00415961	58.0952	0.028312	0.000707
6.0	4.535437	1607.781	1017439.4	0.01620806	226.772	0.112233	0.000357
$Filament Length = 1.60 µ m$ ( $Δ x = 1.60 \times 10^{-8} m$ )
1.6	0.333395	118.8831	75817.5	0.00128076	17.7811	0.007032	0.001518
3.2	1.320346	473.313	299969.6	0.00504051	70.4185	0.028312	0.000754
6.4	5.138326	1817.106	1149822.6	0.01958534	274.044	0.112233	0.000380
$Filament Length = 1.70 µ m$ ( $Δ x = 1.70 \times 10^{-8} m$ )
1.7	0.376081	134.2729	85550.4	0.00153357	21.3113	0.007042	0.001610
3.4	1.488628	533.456	338005.2	0.00603669	84.3556	0.028320	0.000801
6.8	5.774594	2036.860	1288801.1	0.02338483	327.227	0.112258	0.000404

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Equations (43)

Journal of the Optical Society of America A