1. Introduction

The traditional transmission method and associated spectral devices operating in the UV-Vis spectral range provide a convenient way to get light absorption data from liquid (and solid) samples. Such apparatuses use well defined liquid cuvettes whose sample thickness typically varies between 1 and 100 mm. Obviously, data of the sample thickness is required when one calculates the extinction coefficient of the liquid. Thicknesses below 1 mm are often impractical in routine laboratory experiments. In the case of highly absorbing liquids, such as offset inks of this study, suitable path length is expressed in micrometers in order to obtain adequate transmission. Preparation of such a thin liquid (and solid) layer is subject to errors and thus knowledge of the actual sample thickness is often imprecise.

If the sample thickness is known accurately enough along with the transmittance or the reflectance, data handling based on the Kramers-Kronig (KK) analysis [1] or Maximum Entropy Model (MEM) [1] would give within a finite spectral range an estimate for the wavelength-dependent complex refractive index of a relative thin and strongly absorbing liquid layer. Then typically some additional information in anchor points, where the refractive index is determined at one or more wavelengths, is required [1]. However, the direct use of these methods is usually not possible due to the inadequate data obtained from a narrow wavelength range or lack of data in suitable anchor points.

Azzam formulated an equation for reflectance of s- and p-polarized light yielding information about the complex refractive index of media [2], and such an equation we have utilized in a previous study [3]. However, based on our experience the usefulness of the equation depends especially on the accurate measurement of the reflectance, and even small interference on reflectance can be a source of an error.

A refractometer is a device for determining the (real) refractive index of a liquid. Among several operation principles, commercial apparatuses, such as Abbe refractometer, typically utilize the determination of critical angle of total internal reflection. Such an angle can be clearly observed in reflection measurements as a border of bright and dark fields when dealing with non-absorbing and non-turbid liquids. However, the light absorption and scattering makes the use of a refractometer quite problematic because the critical angle is difficult to detect. Typically a refractometer operates at a certain fixed wavelengths, for example at 589.3 nm (Sodium D-line), hence continuous spectral data cannot be obtained with such devices.

There are liquids that strongly absorb light. In this study, we report on detection of the complex refractive index of offset inks. Inks present rather big business as concerns printed matters. When printing of characters and images on paper, card board and plastics the quality of the print is an issue and it is usually inspected on regular basis. One quality factor is the gloss and also the gloss mottling i.e. variation of the gloss. Gloss of an object such as the print depends on the light source, illumination and viewing angle, surface roughness and also on the spectral properties of the print in the visible spectral range. Knowledge of the complex refractive index of the ink, either in liquid or solid phase, is crucial if one wants to know the role of the spectral properties of the ink on the gloss and also color of the print.

Preston et al. [4,5] have studied the measurement of complex refractive index of magenta ink but for a fixed wavelength of red light. A multi-function optical device was utilized in the study of Elton and Day [6] who studied refractive index of low-extinction surfaces using a fixed wavelength. Niskanen et al. [7] investigated wavelength-dependent complex refractive index in high-extinction spectral range of magenta, cyan and yellow offset inks. Here we concentrate on the assessment of the wavelength-dependent complex refractive index of same inks as in [7] but applying different method of data analysis.

In this paper we propose an alternative method for those presented in [3–7] to obtain the complex refractive index of strongly absorbing liquids. The method requires data that includes the transmittance of liquid of unknown thickness and the reflectance determined for s-, p-, or for some well-defined intermediate polarization state. A fundamental property of the mutual dependence of absorption and dispersion is exploited in the optimization procedure of the method; a change that occurs in an extinction spectrum due to an absorption band equals to a change in the real refractive index (and vice versa). The use of this statement combined with the Fresnel’s light reflection theory gives us the complex refractive index as a function of wavelength, as well as, an estimate of the sample thickness of the light absorbing liquid in the transmittance measurement.

2. Theory

Let us assume that the extinction coefficient k(λ) of a sample, as a function of wavelength, depends only on the light absorption and thus light scattering is neglected. The extinction coefficient can be determined from the transmission spectra T = T(λ) and the sample thickness d using the Lambert-Beer equation, and the relation between the absorption coefficient α and the extinction coefficient k is as follows:

In our case of unknown sample thickness, only the product k(λ)d is readily calculable. The Fresnel’s equations that describe light reflection from the interface of two homogenous media yield the complex reflection coefficients r_s and r_p for s- and p-polarized light, respectively, as follows:

We start with a simple qualitative model for dielectrics namely the Lorentzian permittivity. This is only for the sake of clarity to demonstrate how the method proposed in this article is working. The real and imaginary parts of a single resonance Lorentzian relative permittivity ε_r of an insulator are expressed as follows [8]:

In the frame of the single resonance model of Lorentz we can set Re{ε(ω)}-1 = Im{ε(ω)} and using Eq. (3) get a quadratic equation which has roots ω_1,2 = - ½{Γ ± (Γ² + 4ω₀²)^½}. Physically reasonable roots are those that are positive numbers. In the case of a narrow band resonance we can approximate that the real and imaginary parts are equal at ω₁ = ω₀ –Γ/2. In a similar manner we can set also Re{ε(ω)}-1 = - Im{ε(ω)} (real part of the permittivity can take negative values at certain spectral ranges), which in turn after application of Eq. (3) leads to relevant roots that are given by the expression ω_1,2 = - ½{Γ ± (Γ_j²-4ω₀²)^½}. In the case of a narrow band single resonance the physically reasonable positive root is approximately ω₁ = ω₀ + Γ/2. Now it is quite evident that for the changes of the real and imaginary parts of the permittivity it holds that ΔReε = Reε(ω₀ –Γ/2) – (Reε(ω₀ + Γ/2)) = Im(ω₀ –Γ/2) + Imε(ω₀ + Γ/2) = ΔImε.

We can relax the assumption of a single resonance and deal with multiple resonances and even in the case of overlapping resonance bands. This means that the Lorentzian permittivity in Eq. (3) is replaced by a sum of contributions to the permittivity which is due to groups of electrons that experience resonance (absorption) at different wavelengths. In such a case too it is possible solve roots so that ΔReε = ΔImε and ΔReε = -ΔImε for an insulating medium although the analytic formulas for the roots is rather complicated already in the case of two resonances.

Naturally the model of Lorentzian permittivity is an approximation for real materials, but similar property for the dispersion and absorption, as described above for the permittivity, is suggested to hold for the complex refractive index as will be demonstrated in this article. More generally the KK-theory connects the (real) refractive index n and the extinction coefficient k together as follows [1]:

As an example, a simulation of an insulator possessing two absorption bands is shown in Fig. 1 . The simulation is based on the Lorentzian permittivity of Eq. (3) but allowing double resonance. In the simulation absorption bands are located in the visible part (600 nm) and in the UV-part (180 nm) of the spectrum as is often the case with real substances. For the clarity, the latter absorption band, being much stronger than the other one is not shown in the figure. However, its presence can be observed as an offset for the absorption band near 600 nm. Figure 1 also shows how the Δn and Δk are determined with the proposed procedure. The effect of the UV-absorption band is eliminated by a baseline-correction. In addition, the property Δn = Δk is also valid in this arbitrary case as can be observed from the figure.

 

Fig. 1 Lorentzian simulation of a substance which has two absoprtion bands located at 600 and 180 nm. The latter absorption band, being much stronger than the other one and located outside of the visible spectral range, is not shown in the figure. Vertical dashed line represents the baseline correction. λ_lo and λ_hi isolates the absorption band and defines the operational wavelength range for the procedure.

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Absorption bands (if any) can be readily observed from the data of k(λ)d. The spectral finger print of an insulator can be isolated thus defining the spectral range for the procedure by selecting two wavelengths (λ_lo and λ_hi as in Fig. 1) that include the whole band. Next a rather simple task is to optimize the value for d so that Δn = Δk in the selected operation range (here Δ is the difference between maximum and minimum values within the operation spectral range). A simplified flow chart of the procedure is shown in Fig. 2 .

 

Fig. 2 Flowchart of the procedure.

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3. Experiments

Data for the analysis were obtained using a home-built spectroscopic reflectometer (REM) designed for liquid analysis. Schematic illustration of the REM apparatus is shown in Fig. 3 . However, this is the first time when the transmission measurement mode of the apparatus is used. The transmission mode is available by placing a sample on the probe beam and turning the prism into an angle in which the total internal reflection occurs (i.e. the prism is acting like a mirror). The light source is a 75 W Xenon lamp and main adjustable parameters of the device are: the wavelength in a UV-Vis spectral range, the incidence angle of probe beam and the polarization state. HeNe laser, aperture and Det. 3 are for the calibration of the rotator. The adjustable aperture controls the detection of scattered light – for example, a small aperture allows only the observation of the specular reflection (used in this study). The probe beam is intensity modulated by a chopper (modulator in Fig. 3) to reduce the noise. In addition, the spectrum of the probe beam can be further modified by optical filters if needed. Det. 1 is a reference detector and Det. 2 monitors the reflected light from the prism–liquid interface. Detailed operation is explained in [9].

 

Fig. 3 Schematic illustration of REM reflectometer. Besides the reflection detecting mode, a liquid cuvette or holder for solid materials can be installed into the system for the transmission measurements.

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Thin layer of offset inks (magenta, yellow, cyan; products of Sun Chemical, Finland) were applied on glass plates. For this operation a special device was constructed. A sharp blade attached to the device creates a relatively even and thin layer when the blade is moved over the ink drop on a glass plate. By adjusting the distance between the glass plate and blade, suitable layer can be found in terms of adequate transmittance. Transmittances of these samples were measured by placing glass plates into the reflectometer. The wavelength was scanned from 340 to 800 nm with 5 nm step. The selected wavelength range corresponds approximately to the detection range of the human eye. Next we applied ink to one face of the prism to perform the reflectance measurement. In reflection measurement, the layer thickness is not an issue. One can get easily, for example, 1 mm thick layer to ensure that in practice refracted light is absorbed within the layer. In this study the reflectance was detected using even portions of s- and p-polarized light. The wavelength scanning parameters were the same as in the transmittance measurement and the incidence angle was set to 66 degrees which is below the critical angle in the system that uses prism made of F2 material. All measurements were carried out at room temperature c. 22 °C.

The transmittance and reflectance data were implemented to the procedure (explained above) and the wavelength dependent complex refractive indices of inks were calculated. In Figs. 4a -4c the complex refractive index of three offset inks is shown. The spectral features of the extinction curves in Fig. 4 are quite similar as those in [7]. One common feature that all ink samples share is overlapping absorption bands. In Fig. 4 a) there are two overlapping absorption bands in the range 500-600 nm. For yellow ink there are also at least two overlapping bands in the range between 400 and 500 nm. In the case of cyan in Fig. 4 c) it is probable that there are several overlapping bands in the spectral range 500-800nm.

 

Fig. 4 Refractive index (red curve) and extinction coefficient (blue curve) of a) magenta, b) yellow and c) cyan offset ink. Vertical lines show λ_lo and λ_hi for each ink.

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The real refractive index for magenta and yellow of this study are different from those presented in [7]. We suggest that the typical behavior of the refractive index in the region of “anomalous dispersion” in this study is better than in [7] for instance in the case of the magenta ink.

The suggested procedure gives us also information about the thickness of an ink layer in the transmission measurement. To verify whether these layer thickness values are reasonable or not, an independent interferometric measurement using WYKO NT 9300 optical profiler was carried out to inks on the glass plates. In these measurements the ink layers were more in a liquid than in a solid phase. A straight scratch by a surgeon knife was entered into the ink layer and the profile across the groove was detected from two different positions using the profiler’s step height measurement algorithm. Reasonable average d value was obtained by the two methods for magenta and cyan samples as shown in Table 1 , whereas in the case of yellow there is a disagreement. Due to the prism material (F2), the lowest usable wavelength of the system is approximately 340 nm. In the case of yellow ink, whose absorption band reaches below 340 nm, the baseline correction and definition of the operational wavelength range is a more difficult task than in the case of magenta or cyan and thus worsens the results.

Table 1. Layer thicknesses (μm) retrieved from the method and from the reference measurement. Uncertainties in the interferometric measurement refers to 95% level of confidence.

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4. Discussion

Sample dilution with water or other solvent is one commonly used procedure in the transmission measurement technique. The suitable dilution brings the transmittance to measureable range with practical liquid layer thicknesses. However, extensive dilution may interfere with the outcome and certainly alters the original sample. The proposed method does not need any dilution.

Because we are dealing with light absorbing, colored liquids, there is at least one notable absorption band in the visible spectral range. Naturally, this method prefers cases where the sample has no overlapping absorption bands. However, Figs. 4(a) and 4(c) indicate the method works also for a case in which absorption bands are closely packed. The isolation of the band i.e. selecting correct wavelength limits may affect the results. Thus one should carefully include into the operation spectral range the required minimum and maximum values of dispersion and extinction data. In addition, the baseline correction may serve as an error source. Yellow ink is a good example of these issues – part of its absorption band is omitted by insufficient measured wavelength range making especially the base line correction indefinable. These manual operations can be automated by realizing the fact that the location of the local minimum and maximum dispersion values due to the light absorption can be estimated in some cases with the aid of the Lorentz model, and by using commonly accepted background correction methods.

If the refractive index measurements of inks were carried out with a (commercial) Abbe refractometer that employs only one wavelength, namely 589 nm, an interesting observation can be made. From dispersion curves in Fig. 4 one can notice that the refractive index values (n_D) at 589 nm are located at different refractive index levels depending on the ink sample; in the case of magenta n_D reaches almost the maximum value, yellow the center value and cyan the minimum value in the detected wavelength range. This means that n_D does not necessarily give correct information on the average refractive index in close proximity of 589 nm.

The thin liquid layer may induce multiple reflection of light. However, multiple reflections can be reduced by adjusting the thickness of the strongly absorbing liquid layer thick enough so that the probability of double reflection is already low due to light absorption on a path that is two times the thickness of the liquid layer.

The complex refractive index of a substance in a liquid phase differs from the substance in a solid phase. This is also the case with inks [3]. In this study inks were in a liquid form during the measurements of the complex refractive index. The reference measurement concerning the layer thickness was performed at a different location than transmittance and reflectance measurements. During the shipment, inks on glass plates were partially dried affecting the actual thickness. Repeatability of the interferometric measurement is better than 10 nm while much higher measurement uncertainty comes entirely because of sample thickness variations. Rough analysis shows that in this case 0.1 μm change (error) in layer thickness changes the extinction coefficient and the refractive index by the order of 10⁻⁴ unit.

The refractive index range of the liquid sample is not restricted by the method, thus optically thin or thick liquids can be examined. However, the refractive index of the prism in reflectance measurement should be higher than that of the liquid to maintain the condition of the internal reflection.

5. Conclusions

We have proposed a practical method which is especially suitable for highly absorbing liquids in order to retrieve a wavelength dependent complex refractive index. As an example, offset inks were studied. Data for the method can be obtained using two separate spectrophotometers that operate in transmittance and reflection modes or using a single device in which both modes are integrated as in our case. Reference measurements support the validity of the method, however, providing that parameters for the optimization procedure are correctly chosen.

The method presented in this article has applications within scientific studies of strongly absorbing liquids. It can be also utilized in industry for the manufacturing processes and quality inspection of such liquids. Possible measurement applications, in addition to offset inks, include quality investigations of paints, red wine and inspection of optical properties of colloids consisting of low concentration of metallic nanoparticles in liquid matrix.