Broadband Complex Refractive Indices of Ice and Water

Peter S. Ray

doi:10.1364/AO.11.001836

Applied Optics
Vol. 11,
Issue 8,
pp. 1836-1844
(1972)
•https://doi.org/10.1364/AO.11.001836

Broadband Complex Refractive Indices of Ice and Water

Peter S. Ray

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Peter S. Ray, "Broadband Complex Refractive Indices of Ice and Water," Appl. Opt. 11, 1836-1844 (1972)

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Abstract

A critical survey of the literature is presented. An empirical model of the complex refractive indices for ice and liquid water is constructed from this review. The model is applicable from −20°C to 0°C for ice and from −20°C to 50°C for water. The spectral interval for which the model applies extends from 2 μ to several thousand kilometers in wavelength for ice and from 2 μ to several hundred meters in wavelength for water.

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Investigator	Spectral interval (μ)	Temperature (°C)	Percent relative error ∑ × 100% real part/imaginary part
Irvine and Pollack	1–200	26	7.56/
Stanovich and Yaroslavskii	41.5–2067.0	room temp.	/18.33
Draegert	58–200	32	/4.03
Zolotarev et al.	1–5000/10–5000	25	4.87/6.38
Pontier and Dechambenay	1–40/2–40	35	3.08/67.90
Chamberlain	118,337/228–1340	25	0.57/4.86
Querry et al.	2–20	24	2.94/82.38
Rusk et al.	1.0–30.3/2.52–30.3/	24	1.84/32.36
Robertson and Williams	$\frac{2.286 - 36.364}{(cm)}$	24	/49.08
Collie et al.	1.27–10.0	0–60	1.71/2.73
Cook et al.	1.258–10.0	0–60	0.89/4.61
Grant et al.	1.26–52.0	0–60	0.33/7.84
Grant and Shack	8.6	0–40	2.33/3.10
Hasted and Sabeh	1.267–9.220	0–60	0.87/4.19
Lane and Saxton	0.62–3.21	−8–50	1.41/4.30
Rampollia et al.	3.1	20	2.52/7.23
Sandus and Lubitz	3.2	20–40	2.43/1.92
Saxton	1.58	0–30	4.06/6.99

Spectral interval (μ)	λ₀_i (μ)	β_i	Δ_i	γ_i
λ ≤ 2.97	2.97	0.27	0.025	2.0
	4.95	0.01	0.06	1.0
2.95 ≤ λ ≤ 4.95	2.97	0.27	0.04	2.0
	4.95	0.01	0.06	1.0
4.95 ≤ λ ≤ 6.1	4.95	0.01	0.05	1.0
	6.1	0.12	0.08	2.0
6.1 ≤ λ ≤ 17.0	6.1	0.12	0.042	0.6
	17.0	0.39	0.165	2.4
	62.0	0.41	0.22	1.8
17.0 ≤ λ ≤ 62.0	17.0	0.39	0.45	1.3
	62.0	0.41	0.22	1.8
	300.0	0.25	0.40	2.0
62.0 ≤ λ ≤ 300.0	17.0	0.39	0.45	1.3
	62.0	0.41	0.35	1.7
	300.0	0.25	0.40	2.0
300.0 ≤ λ ≤ 3,000.0	17.0	0.39	0.45	1.3
	62.0	0.41	0.35	1.7
	300.0	0.25	0.47	3.0

Spectral interval (μ)	α_i	ω₀_i	β_i	γ_i
(a) λ ≤ 6.0	1.79907	3352.27	99.914 × 10⁴	15.1963 × 10⁴
		1639.0	50.4835 × 10³	9246.27
		588.24	84.4697 × 10⁴	10.7615 × 10⁵
(b) 340.0λ ≥ 7.0	1.83899	1639.0	52,340.4	10,399.2
		688.24	345,005.0	259,913.0
		161.29	43,319.7	27,661.2
6.0 ≤ λ ≤ 7.0	n_r = n_r^a(7.0 − λ) + n_r^b(λ − 6λ)
340.0 ≤ λ ≤ 1000.0	$n_{r} = {n_{r}}^{D} (\frac{λ - 340.0}{660.0}) + {n_{r}}^{b} (\frac{1000.0 - λ}{660.0})$
1000.0 ≤ λ	n_r = n_r^D

Spectral interval (μ)	λ₀_i (μ)	β_i	Δ_i	γ_i
λ ≤ 1.5	1.5	0.00063	0.027	1.4
1.5 ≤ λ ≤ 2.0	1.5	0.00063	0.043	1.5
	2.0	0.00158	0.017	1.6
	4.5	0.0325	0.052	1.0
2.0 ≤ λ ≤ 3.075	2.0	0.00158	0.03	1.35
	3.075	0.3428	0.022	1.2
	4.5	0.0325	0.052	1.0
3.075 ≤ λ ≤ 4.5	3.075	0.3428	0.019	1.14
	4.5	0.0325	0.052	1.0
4.5 ≤ λ ≤ 6.05	4.5	0.0325	0.043	1.05
	6.05	0.064	0.032	1.32
	6.65	0.011	0.009	0.7
	9.0	0.008	0.06	1.3
	11.0	0.0855	0.038	0.91
6.05 ≤ λ ≤ 6.65	4.5	0.0325	0.043	1.05
	6.05	0.064	0.13	1.0
	6.65	0.011	0.009	0.7
	9.0	0.008	0.06	1.3
	11.0	0.0855	0.038	0.91
6.65 ≤ λ ≤ 9.0	4.5	0.0325	0.043	1.05
	6.05	0.064	0.13	1.0
	6.65	0.011	0.075	0.85
	9.0	0.008	0.06	1.3
	11.0	0.0855	0.038	0.91
9.0 ≤ λ ≤ 11.0	6.05	0.064	0.13	1.0
	6.65	0.011	0.075	0.85
	9.0	0.008	0.01	2.0
	11.0	0.0855	0.038	0.91
	44.8	0.581	0.06	0.90
11.0 < λ < 13.5	6.05	0.064	0.13	1.0
	6.65	0.011	0.075	0.85
	11.0	0.0855	0.0008	0.3
	13.5	0.087	0.0025	0.6
	44.8	0.58	0.06	0.9
13.5 < λ < 44.8	6.05	0.064	0.13	1.0
	6.65	0.011	0.075	0.85
	11.0	0.0855	0.0008	0.3
	13.5	0.087	0.14	1.1
	44.8	0.581	0.06	0.9
	62.0	0.242	0.04	1.0
44.8 < λ < 62.0	13.5	0.087	0.14	1.1
	44.8	0.581	0.055	1.0
	62.0	0.242	0.04	1.0
62.0 < λ	44.8	0.581	0.055	1.0
	62.0	0.242	0.23	1.6

Spectral interval (μ)	α_s	ω₀_i	β_i	γ_i
(a) λ ≤ 3.5	1.75	3252.27	10,18140.0	76,183.4
		1652.9	588,684.0	73,801.1
		909.09	46 × 10⁻¹¹	46 × 10⁻¹¹
(b) 4.0 ≤ λ ≤ 7.5	2.15622	3252.27	29.8023 × 10⁻⁹	10.7118 × 10¹⁸
		1652.9	744,332.0	1,100,350.0
		909.09	1,645,160.0	434,408.0
(c) 9.5 ≤ λ ≤ 200.0	1.2225	1652.9	1,120.820.0	46 × 10⁻¹¹
		909.09	416,441.0	118,852.0
		223.2	47,031.8	126,834.0
3.5 ≤ λ ≤ 4.0	$n_{r} = {n_{r}}^{b} (\frac{λ - 3.5}{0.5}) + {n_{r}}^{a} (\frac{4.0 - λ}{0.5})$
7.5 ≤ λ ≤ 9.5	$n_{r} = {n_{r}}^{c} (\frac{λ - 7.5}{2.0}) + {n_{r}}^{b} (\frac{9.5 - λ}{2.0})$
200.0 ≤ λ ≤ 800.0	$n_{r} = {n_{r}}^{D} (\frac{λ - 200.0}{600.0}) + {n_{r}}^{c} (\frac{800.0 - λ}{600.0})$
800.0 ≤ λ	n_r = n_r^D

	Spectral interval (9)	Temperature (°C)	Percent relative error ∑ × 100% real part/imaginary part
Irvine and Pollack	1–152/2–152	−50	1.64/22.95
Cumming	3.2 × 10⁴	−18–0	/1.70
Lamb	1.25 × 10⁴	−50–0	/22.19
Lamb and Turney	3.0 × 10⁴	−180–0	/84.81

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Equations (19)

Applied Optics