I. Introduction

The index of refraction n(λ) and the extinction coefficient k(λ) of water are, respectively, the real and imaginary parts of its spectral complex refractive index ñ(λ) = n(λ) − ik(λ), where λ is the wavelength of an electromagnetic wave in vacuum. These physical parameters together with the complex Fresnel equations and the generalized Fresnel equations provide the basis for computing the optical properties of water. In recent years knowledge of the optical properties of water has been of greater interest because of its application in (1) computing radiation transport through atmospheres containing water droplets and other aerosols or through oceans containing hydrosols, (2) development of optical remote sensing instruments for measuring the chemical and thermal quality or turbidity of environmental waters and for measuring the water content of soils, (3) computing the optical properties of plant leaves, and (4) investigations of the optical properties and optical constants of aqueous solutions.

In 1968 Irvine and Pollack[1] published results from a critical review of the existing literature on the optical properties of water for the 0.2–200-μm wavelength region. They tabulated measured values of both k(λ) and the Lambert absorption coefficient α(λ) = 4πk(λ)/λ from about thirty different papers appearing in the scientific literature. Next, using the only four reports on measurements of the reflectance of water that were available at that time, they tabulated values for the reflectance R(λ) measured at near normal incidence for a free water surface. The tabulated values for k(λ) and the generalized Fresnel reflectance equation for R(λ), i.e., the Cauchy equation, then provided values for n(λ).

Zolotarev et al.[2] in 1969 reported values for the optical constants of water throughout the spectral region 1–10⁶μm. They determined both k(λ) and n(λ) from their measurements of α(λ) and R(λ) for water at 25°C in the 2–50-μm region and measurements of internal reflectance spectra in the 2–10-μm region. Their measurements and reliable data from thirteen other papers in the scientific literature provided values of k(λ) throughout the spectral region 1–10⁶μm. Values of n(λ) were then obtained from a Kramers-Kronig (K-K) analysis of the k(λ) spectrum:

For making the integration indicated by Eq. (1) a model absorption band with central position at 100 nm was constructed for k(λ) in the uv region. The band was reported to have no significant affect on calculated values of n(λ) for λ ≥ 1 μm. Zolotarev et al. noted discrepancies in the 20–50-μm region between their values for n(λ) and those reported by Irvine and Pollack. They noted that in the 6.5–9-μm region the values of k(λ) measured by Pointier and Dechambenoy[3] were 30% smaller than their measured values for that quantity, and also that values for both n(λ) and k(λ) at the ir band centers were in disagreement with those measured by Pointier and Dechambenoy.

Since Zolotarev et al. completed their investigation of the optical constants of water in 1968 at least nine other papers[4]–[13] appearing in the literature have reported measurements of the optical properties of water in the vacuum uv, visible, ir, or microwave spectral regions. Of particular interest are the careful measurements reported by Robertson and Williams[6] for α(λ) in the 2.5–38.4-μm region, and the measurements of reflectance and subsequent computations of n(λ) and k(λ) reported by Painter et al.[9],[10] and Kerr et al.[8] which, when combined, extend through the 80–300-nm region of the vacuum uv.

We felt a need at this time for an updated review of the literature on the optical constants of water (1) because a current knowledge of the optical constants of water is essential to our investigations of aqueous solutions,[14]–[15] (2) because of the discrepancies between values reported for both n(λ) and k(λ), and (3) because of the additional measurements of these quantities that have been reported since 1968 for the ir and uv spectral regions. We therefore compiled values of k(λ) from the scientific literature and then point by point plotted graphs of these measured values for k(λ) vs λ through the microwave, far ir, ir, visible, x-ray, and part of the uv regions of the electromagnetic spectrum. A smooth continuous curve for k(λ) then was drawn manually through the plotted points. Values for k(λ) were postulated in the vacuum uv and soft x-ray regions where data were not available from the literature. A subtractive Kramers-Kronig (K-K) analysis of the k(λ) spectrum then provided values of n(λ) for the spectral region 200 nm ≤ λ ≤ 200 μm.

II. Acquisition of Data

A search was made for literature reporting measurements of the electromagnetic absorption characteristics of liquid water in any spectral region. Fifty-eight articles and books[1]–[13],[16]–[59] were selected from the literature of the past 81 years. The selected references were examined individually for specific values or for graphical or tabulated values of the extinction coefficient k(λ); Lambert absorption coefficient α(λ) = 4πk(λ)/λ; molecular absorption coefficient ξ(λ)_m = α(λ)/(2.3026C), where C is the concentration of the substance in units of moles/liter; absorption index K(λ) = k(λ)/n(λ); real ξ_r(λ) = 2n(λ)k(λ) and imaginary ξ_i(λ) = n(λ)² − k(λ)² parts of the complex dielectric constant; or mass absorption coefficient (μ/ρ)_λ = 4πk(λ)/λρ, where ρ = 1 g/cm³. A listing of all accumulated values for k(λ) ordered with respect to increasing λ was provided by an IBM 360/50 computer. In many cases significant discrepancies were noted between values of k(λ) at the same λ but obtained from different references. All points in the listing then were plotted manually on graphs of k(λ) vs λ. The best visual fit to the plotted data for k(λ) was obtained by manually drawing a smooth curve through the points while weighting the curve in favor of data for water at 25°C and in favor of data reported by authors who in our judgment used careful experimental procedures. Two postulated absorption bands of Gaussian shape were constructed for k(λ) in the soft x-ray and vacuum uv regions where data were not available from the literature, The final values for k(λ) are shown graphically by the solid-line curves in Figs. 1–5 and are tabulated in Table I at selected positions between 200 nm and 200 μm.

III. Discussion of Graphs for k(λ) vs λ

A. Figure 1, Graph 0–0.5 Å

The smooth curve shown in this graph was based on values for absorption coefficients of water for x rays and γ rays as reported by Allen.[27]

B. Figure 1, Graph 0–55 Å

The smooth solid-line curve shown in this graph was based on values for the absorption coefficients of x rays and γ rays as reported by Allen[27] for the 0.25–0.7-Å region and values for the mass absorption coefficients taken from Ref. [52] for the 0.71–2.5-Å region and from Engstrom[51] for the 5–22-Å region. The smooth dashed-line curve was the short wavelength side of an absorption band postulated for k(λ) in the soft x-ray region.

C. Figure 1, Graph 0–2000 Å

1. Spectral Region 22–849 Å

Data for this region were not available from the literature. The dashed-line curve for k(λ) in this region was postulated in the following subjective manner: The curve was made Gaussian shaped between 850 Å and about 300 Å. The band with peak value at about 80 Å was added in order to join continuously the solid-line curve ending at 22 Å with the Gaussian shaped curve at about 300 Å.

2. Spectral Region 849–1250 Å

The only data available for this region were those of Kerr et al.[8] (KHW). Their values for k(λ) were calculated by K-K analysis of reflectance data for water at 1°C. For λ ≥ 1250 Å their values of k(λ) are greater than those reported by other investigators. Therefore, we chose for this region the similar shaped but smaller-in-magnitude dashed-line curve that was thought to represent k(λ) for water at 25°C.

3. Spectral Region 1250–2000 Å

The heavy solid-line curve was based on data from Painter et al.[9],[10] and data from Refs. [42], [44], [46], [47], and [49]. Because the curve is rapidly rising in the 1700–1850-Å region, individual values of k(λ) read from this curve are subject to a significant amount of error.

D. Figure 2, Graph 180–1000 nm

The smooth solid-line curve was based mostly on values of k(λ) from Refs. [11], [16]–[19], [31], [34], and [40]. Notable exceptions to the curve selected for k(λ) are data of Lenoble and Saint-Guilly[31] (LSG) and of Tyler et al.[11] (TSW). The paper by TSW was a recent one predicting the optical properties of clear natural water. Data reported by LSG, when compared to our curve, are lower for λ < 370 nm and are greater for λ > 370 nm. At 400 nm, values of k(λ) from LSG were significantly greater than values of k(λ) from Clarke and James[40] (CJ) and from several of the other references. Data from CJ suggested a more highly structured curve for k(λ) than the one we constructed in the 380–580-nm region, but the scatter of data from other references allowed only the selected curve. Sullivan[18] provided a consistent set of data for k(λ) throughout the region 580–790 nm. The structure of our curve for k(λ), however, departed slightly from that suggested by Sullivan’s data in the region of the maximum for k(λ) at about 760 nm. In the 800–1000-nm region the curve for k(λ) was based primarily on data from Curcio and Petty[34] and Kondratyev.[19] Values for k(λ) from data reported by Bayly et al.[26] seemed to be consistently too large.

E. Figure 3, Graph 0.95–2.6 μm

The smooth solid-line curve for k(λ) was based on data from Refs. [1]–[7], [16], [17], [19], [20], [23], [25], [28], [34], [39], [53], and [54]. In the 0.95–2.0-μm region the primary references were Kondratyev,[19] Curcio and Petty,[34] and Zolotarev et al.[2] In the 2.0–2.6-μm region Centeno’s data were in poor agreement with data from several of the other references. The small shoulder band at 2.5 μm originally reported by Collins[39] and recently commented on by Robertson and Williams[6] was included in our curve for k(λ).

F. Figure 4, Graph 2.5–18.5 μm

The smooth solid-line curve for k(λ) was based on data from Refs. [1]–[7], [17], [19], [23], [25], [26], [28], [29], [38], and [54]. The better data seemed to be those of Robertson and Williams[6] and Zolotarev et al.[2] The maximum value of k(λ), k(17.2 μm) = 0.430, for the liberation band was estimated in both position and magnitude from data of Hale et al.[7] Values of k(λ) from Irvine and Pollack[1] agree fairly well with. the curve selected through this spectral region.

G. Figure 5, Graph 10–100 μm

Data through this spectral region were selected from Refs. [1]–[7], [17], [19], [22], [24], [28], [29], [50], and [54]. Many discrepancies existed between values for k(λ) reported by different investigators. Data from Rusk et al.[5] (RWQ) seemed to assign values for k(λ) that were too large; this was attributed to an inefficient polarizer for λ > 20 μm. Data from Robertson and Williams[6] (ROW) seemed to assign values for k(λ) that were too small; as noted in the original paper this was perhaps due in part to scattered radiant flux that remained undetected. Temperature influences the shape and position of the libration band as indicated by Hale et al.[7] and by Pointier and Dechambenoy[3] (PD, 1966). Temperature of the water was not indicated by ROW. Our smooth curve for k(λ) was based on a median estimate between ROW and PD; the estimate coincides very well with data from Zolotarev et al.[2] In the 58–84-μm region the curve for k(λ) follows closely data from Draegert et al.,[22] Zolotarev et al.,[2] and Irvine and Pollack.[1] Above 84 μm the curve was primarily based on data from Zolotarev et al.

H. Figure 5, Graph 50–100 μm

Data from Chamberlain et al.[21] and their references and from Zolotarev et al.[2] were used for k(λ) in this spectral region.

I. Figure 5, Graph 0–10 cm

Data from Rabinovich and Melentyev[13] (RBM) and from Zolotarev et al.[2] and from Refs. [12], [19], [30], [32], [35], and [41] were used for k(λ) in this spectral region.

J. Figure 5, Graph 0–1.0 m (curve begins at 10 cm)

The solid-line curve was a smooth fit to scattered data points selected from Refs. [2] and [37]. There were probably some regions of absorption that were not shown in the final curve.

IV. Index of Refraction

Values of the index of refraction n(λ) for the spectral region 200 nm to 200 μm were computed by applying a subtractive Kramers-Kronig analysis[58],[59] (SKK) to the continuous spectrum for k(λ) shown in Figs. 1–5. Accordingly, the index of refraction n(λ₀) at wavelength λ₀ is

where n(λ₁) is a known value for the index of refraction at wavelength λ₁, and Prin. denotes the Cauchy principal value of the integral. The integral was evaluated throughout the 10⁻⁴-nm to 1-m region by use of a combination of Simpson’s rule and trapezoidal numerical approximations that were programmed in Fortran IV for an IBM 360/50 computer. Contributions to the integral for λ ≤ 10⁻⁴ nm were assumed to be negligible, and those for λ ≥ 1m were obtained analytically by assuming k(λ) = k(1m) throughout the region λ ≥ 1m. It also was advantageous to choose λ₁ so that it did not coincide with λ at any point during the sequence of numerical approximations. We chose[52] n(λ₁) = n(589.3 nm) = 1.3325, whereas the data contained values of k(λ) at 585 nm and 590 nm. Values for n(λ) resulting from these integration procedures are given in Table I and are shown graphically by the solid-line curves in Fig. 6.

The influence of the shape and height of the postulated uv and soft x-ray bands for k(λ) on calculated values for n(λ) was investigated in four different ways. First, the SKK analysis was made with a straight line for k(λ) between the ends of the solid-line curves at 22 Å and 1250 Å. At 400 nm this gave n = 1.3558 as compared with 1.343 from Irvine and Pollack[1] and at 2 μm, 1.2947 as compared with 1.304 and 1.302 from Irvine and Pollack[1] and Zolotarev et al.,[2] respectively. Second, the SKK analysis was made for k = 1.55 × 10⁻³ throughout the 50–1200-Å region and then a straight line joining k = 1.55 × 10⁻³ at 1200 Å to the end of the solid-line curve at 1250 Å. This gave n = 1.336 at 400 nm, and n = 1.309 at 2 μm. Third, the SKK analysis was made with the curve marked KHW joined smoothly to the dashed-line curve. This gave n = 1.3449 at 350 nm as compared with 1.349 from Irvine and Pollack.[1] Fourth, the effect of the amplitude of the curve for k(λ) at about 880 Å was determined by raising only the peak value of the dashed-line curve to that for the KHW curve while other parts of the dashed-line curve remained fixed. This changed n at 200 nm from 1.3954 to 1.3957. Although these four tests indicate other postulated curves for k(λ) could yield equivalent results, the final values for n(λ) shown in Table I and Fig. 6 were obtained by applying the SKK analysis to the continuous curve described in Sec. III of this paper.

Values of n(λ) from Refs. [1], [2], [5], and [8] are compared graphically in Fig. 6 with n(λ) obtained during the present investigation. In the 0.2–0.6-μm region n(λ) from the present work are less than those from Irvine and Pollack[1] (IP) and from Kerr et al.[8] (KHW). In part this is attributed to the influence of the curve postulated for k(λ) in the uv on results of the SKK analysis in the 0.2–0.6-μm region. Additional investigations of the height, width, shape, and number of vacuum-uv absorption bands are needed in the future in order to resolve these discrepancies. In the 1.0–6.0-μm region n(λ) from IP and from Rusk et al.[5] (RWQ) are slightly lower than n(λ) determined during the present investigation. The maximum and minimum values of n(N) in the region of the water band centered at 2.95 μm are, respectively, less than and greater than values for similar quanitites from IP and from Zolotarev et al.[2] (ZMA). Values of n(λ) from ZMA and RWQ are in best agreement with n(λ) from the present investigations in the regions 3.1–6.0 μm and 7.5–9.0 μm, respectively. Values of n(λ) from Ref. [2] (ZMA) for the 10–200-μm region are in good general agreement while those from Ref. [1] (IP) do not agree with n(λ) determined during the present investigations. We feel the values from Ref. [1] are in error in the long wavelength region.

We repeat the encouragement[1] that authors in the future present their data in tabular form because it was very difficult to accurately read many of the graphs.

We thank Sue Riley for assisting with the plotting and reading of the graphs, Marie Light for typing the manuscript, and the staff of Linda Hall Library of Science and Technology, Kansas City, Missouri for their cooperation during the survey of the scientific literature.

This work was supported in part by the National Aeronautics and Space Administration grant NGR 25-001-012; by the U.S. Department of Interior, Geological Survey contract 14-08-0001-12636 and Office of Water Resources Research grant A-058-MO with the University of Missouri. Paper presented at the Annual Meeting of the Optical Society of America, San Francisco, October 1972 [J. Opt. Soc. Am. 62, 1381A (1972)].

Figures and Table

 

Fig. 1 Extinction coefficients of water for the 0–2000-Å spectral region. Notation and curves are described in the text.

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Fig. 2 Extinction coefficients of water for the 200–1000-nm spectral region. Well known data that were in notable disagreement with the curve finally selected for k(λ) are plotted as points on the graph. Notation and curves are described in the text.

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Fig. 3 Extinction coefficients of water for the 0.95–2.6-μm spectral region. Curves are described in the text.

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Fig. 4 Extinction coefficients of water for the 2.5–18.5-μm spectral region. Curves are described in the text.

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Fig. 5 Extinction coefficients of water for the 10–10⁶μm spectral region. Notation and curves are described in the text.

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Fig. 6 Index of refraction of water for the spectral regions 0.2–200 μm. Descriptions of the curves and the symbols are presented in the text.

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Table I. Optical Constants of Water

View Table

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