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We propose a simple method that uses a gradient shadow pattern (GSP) to measure the refractive index nL of liquids. A light source generates a “dark-bright-dark” GSP when it is projected through through the back of a transparent, rectangular block with a cylindrical chamber that is filled with a liquid sample. We found that there is a linear relationship between nL and the proportion of the bright region in a GSP, which provides the basic principle of the proposed method. A wide range 1.33≤nL≤1.46 of liquids was measured in the single measurement setup with error <0.01. The proposed method is simple but robust to illuminating conditions, and does not require for any expensive or precise optical components, so we expect that it will be useful in many portable measurement systems that use nL to estimate attributes of liquid samples.
Refractive index n is a fundamental physical properties of a substance. n determines the path of light when it crosses an interface between two media, so its measurement is essential when designing and fabricating optical instruments such as optical lenses1,2,3, prisms, mirrors, and optical fibers. In addition, because refractive index nL of a liquid strongly depends on the amounts or kinds of solutes in it4, measurement of nL provides important information in many fields (e.g., sugar content in foods, degeneration of industrial oils, and diagnoses of disease5,6,7,8,9).
To date, various existing methods measure n of a substance by evaluating refraction angle, critical angle, interference pattern, or dielectric constant10. Interferometers and ellipsometers provide comprehensive information of n, but are too expensive and complicated to use as a simple portable refractometer. More recently, microfluidic-11,12,13,14,15, diffractive-16, and holographic-based17 systems to measure n have been developed; they have several advantages such as low sample volume, but they entail complex configurations and require precise and expensive optical components. Therefore, these systems are also inappropriate for portable applications. The most widely-used portable refractometer applies triangular prisms. While passing through the interface between a solid triangular prism with known n and a liquid sample of which nL is to be determined, a beam of light is deviated, i.e., refracted or reflected. The amount of deviation is then converted to nL of the liquid sample by performing calculations that consider n of the prism and its dimensions. Despite its simple principle, a setup to estimate the deviation angle is complicated, because it is usually conducted by a rotating telescope, photodetector, or goniometer18. Also, due to the planar interface between the prism and liquid sample, incident angle θ1 at the interface does not change much, so the measurement range of nL is limited. To measure a wide range of nL, multiple prisms having different n should be used. Another solution to use a specialized illumination system that can widen the variation of θ1. Both of these solutions are expensive and increase the complexity of the measurement.
This paper presents an alternative simple method to measure nL. It does not require any expensive optical instruments or special illumination systems. The proposed method uses a conventional light source and a transparent, rectangular solid block with a cylindrical chamber that is filled with a liquid sample of interest. One novel feature is the circular interface between the solid block and the liquid sample that cause a continuous change of θ1 from 0° to ±90°. Projection of the light source through the block and the sample generates a “dark-bright-dark” spatial gradient shadow pattern (GSP). From a simple analysis of a GSP, a wide range 1.33≤nL≤1.46 was easily measured in a single measurement setup. The measurements agreed well with nL values obtained using a commercial refractometer, and the measurement error was <0.01. The effects of illumination conditions on the measurement accuracy were negligible. The proposed method has applications in portable measurement systems that require nL to estimate attributes of liquids.
The measurement system (Fig. 1a) consists of a light source, an observation camera, and a transparent rectangular poly(methyl methacrylate) (PMMA) block (length 12mm, width 10mm, height 14mm), into which a cylindrical chamber (radius r=3mm, depth 12mm) has been drilled. The block has refractive index nS=1.49 at wavelength of 589.3nm (sodium D line) at 20°C. The cylindrical chamber is filled with a liquid sample of interest. The cylindrical shape of the sample chamber causes a continuous change in θ1 at the interface between the block and a liquid sample.
A light source projected through the block generated a “dark-bright-dark” GSP (Fig. 1b), in which the width of bright region is represented by W. The left and right sides of the cylindrical chamber were observed to be dark, whereas the center region is bright. The proportion W/(2r) of bright region in the GSP depends only on nL and nS; the proposed method to measure nL exploits this relationship. One of the novel features of the proposed method is adopting the curved interface between the solid block and the liquid sample which is the main factor in generating a GSP. When a triangular prism chamber is used instead of the cylindrical chamber, no GSP appears (Supplementary Fig. S1). To simply analyze the generation of the GSP, we considered a circle on a plane and traced paths of rays entering and leaving the circular interface (Fig. 1c). We did not consider the planar interfaces between the solid block and air, because this interface did not contribute to formation of the GSP.
Different light rays incident on the circular interface refract differently from the cylindrical chamber, so their deviation angles θΔ, defined as the angle difference between the ray entering and leaving the circular interface (Fig. 1c), vary. As the light incidence position moves toward the sides of circular interface, θΔ increases sharply and thereby generates a nonlinear transformation of light intensity distribution. The light intensity is high (bright) at the center of the cylinder, but low (dim) at both sides. The paths of light rays can be classified into four regimes based on the position x of incident light on the circular interface (Fig. 1c).
When the light ray is illuminated at the central point (x=0) of circular interface, θ1=0° and the light ray travels straight. Thus, the central region of GSP is bright.
When 0<|x|<(nL/nS) the incident angle increases (Fig. 2a) as:
The incident ray is refracted twice (Fig. 1c) and the refraction angle
is obtained using Snell’s law. Deviation angle θΔ2(θ2−θ1) increases nonlinearly as path of the incident light ray approaches the sides of the chamber (Fig. 2b). Because of the spatial variation of θΔ, the incident rays diverge while traveling the chamber, so the brightness of transmitted light decreases in this region.
Beyond the critical points (nL/nS)≤|x|<r, θ1 exceeds the critical angle θcsin−1(nL/nS), so the incident light is totally reflected (Fig. 1c). The shadow that corresponds to this region is completely dark.
Light rays in the region of r≤|x| pass through and miss the cylindrical chamber, so θΔ=0. In this region, no shadow occurs (completely bright).
We analyzed the brightness profile in a GSP from the spatially-varying θΔ (Fig. 2c; details in Supplementary Information). In the central region (x=0) the brightness is high because θ1 and θΔ are close to 0°. In contrast, the brightness at the sides (x→±r) is low due to the deviated and spread light path. The trend of changes in θΔ and brightness profile in the GSP is strongly affected by the ratio nL/nS. When , θc becomes small; as a result, the portion of dark region in the GSP expands. In contrast, as nL/nS approaches 1, θc increases and θΔ decreases; this trend widens the bright region in the GSP widens (Fig. 2b,c). We do not consider nL/nS>1 in this study. The relationship between the proportion W/(2r) and nL/nS was almost perfectly linear (Supplementary Fig. S2c).
However, we ignored the refraction at the planar interfaces between the solid block and air, so a more realistic relationship between W/(2r) and nL/nS should be obtained. In this regards, instead of deriving a theoretical relationship, we deduced the relationship by analyzing the GSPs obtained using liquids of known nL. The width of the bright region was evaluated as the full width at half maximum (FWHM) brightness in the GSPs. Measured nL/nS ratio is found to have increased a linearly relationship with as W/(2r) increased proportion (R2=0.96, Fig. 3) as:
where α=0.21±0.03 and β=0.77±0.03.
To evaluate the versatility of the proposed nL measurement system, it was used to estimate nL of sucrose solutions with various concentration and the results compared to nL measured using the commercial refractometer. These sucrose solutions made different GSPs (Fig. 4a). Interestingly, the proportion of the bright region increased linearly with the sucrose concentration (R2=0.986, Fig. 4b). nL measured using the proposed method agreed quantitatively with nL obtained using the commercial refractometer (Fig. 4c) and reference nL of sucrose solutions19; measurement error was <0.01, which can be regarded as the resolution of the present measurement setup. This result confirmed the feasibility of the proposed method. The results also indicate that the proposed method to measure nL can be used quantify sugar content in foods, and perhaps to measure concentrations of other solutes in liquids.
Because the nL measurement is based on the empirically-derived linear relationship of Eq. (3), its applicable range is of practical importance. The experimental demonstration already confirmed the validity of the linearity between nL/nS and W/(2r) for 1.33≤nL≤1.46. This measurement range is appropriate for liquids, because 1.33 is the lowest nL value observed among common liquids19. We conducted the same measurement of air (n=1.0), but the result did not agree with the linear relationship of Eq. (3) (data not shown). Because we have assumed that nL<nS when calculating the divergence of the incident light, the upper limit of measurable nL is nS (here, 1.49). In this case, all of the GSP is bright.
We further investigated the reliability of estimates of nL obtained using the proposed method under different conditions of illuminance intensity, and direction of the light source. First, GSPs of DI water were collected at 16 illuminance intensities (Fig. 5a). There was no significant change in calculated nL under different illuminance intensities. The striking thing is that the calculated nL changed only 0.006, although W changed 0.12mm. Because the change of nL was lower than the resolution of the proposed measurement system, the effect of illuminance intensity can be neglected. Also, the linear relationship is still valid for various illuminance intensities.
Second, to examine the effect of illumination direction, we took GSPs of DI water obtained by rotating the solid block from −40° to 40° (Fig. 5b). As the angle of illumination direction increases, observed widths of the cylindrical hole and bright region decrease (Fig. S4). However, interestingly, the ratio of the two widths, or proportion of the bright region, was found to be almost constant regardless of the illumination angle (Fig. 5b). Maximum difference of calculated nL was only 0.008, which is lower than the resolution of the proposed method. Therefore, we can conclude that estimate of nL is not affected by the illumination angle from −40° to 40°. When the angle exceeds ±40°, a focused image of GSP cannot be obtained. Also, we found that the proposed method is little affected by the illumination color within the resolution range (±0.01) (Supplementary Fig. S3). However, in the absence of a color filter, estimate of nL was slightly overestimated (+0.02). These results demonstrate that the proposed nL measurement is not sensitive to illumination conditions, and imply that the proposed method is robust and can be used with natural light as a power-free portable device.
The resolution and measuring range of the proposed method are related to the quantity αnS/(2r) that relates W to nL (Eq. (3)). As this quantity decreases, the resolution would improve but the measuring range would narrow, and vice versa. The GSP-based nL measurement is simple but accurate, and is not sensitive to illumination conditions. It is also built using inexpensive components and does not requires an external power source or a specialized light source. Therefore, this device can be used in various power-free, portable, and even disposable devices. Also, it may be further modified for microfluidics applications such as detecting interfaces between two immiscible liquids or mixing interface between two miscible liquids that have different nL flowing in a microchannel.
In conclusion, we proposed and demonstrated a simple method to measure the refractive index nL of liquids. The method exploited a “dark-bright-dark” spatial GSP generated by a circular interface between solid and liquid. The circular interface caused a continuous change of incident angles from 0° to ±90°, and consequently generated a unique GSP. The proportion W/(2r) of bright region in the GSP was linearly related to nL/nS and this relationship is exploited to estimate nL. The proposed method was accurate over the range 1.33≤nL≤1.46, with error <0.01. Although the principle and structure of the proposed method is simple, it is accurate and robust to illumination conditions. Therefore, we expect that this device can be adapted to a variety of inexpensive (possibly disposable) portable instruments to measure nL.
Tests were conducted using liquid samples of various known nL (Dodecane (Sigma-Aldrich, anhydrous 99+%, nL=1.42), 40M glycerin (Daejung, 99+%) solution (nL=1.44), 6M CaCl2 (Kanto, anhydrous 95%) solution (nL=1.44), and a mixture of 40M glycerin and 6M CaCl2 (nL=1.46) to calibrate the measurement of nL. Mixtures of sucrose (Junsei, guaranteed reagent) and DI water with concentrations varying from 0 to 60wt%, of which nL vary linearly from 1.33 to 1.44 were used to evaluate the versatility of the proposed measurement system. A commercial hand-held refractometer (ATAGO®, R-5000) having a precision of 0.001 was used to check the accuracy of nL measurement by the proposed system. To avoid the effect of temperature on nL, all experiments were conducted at 25±1°C. A white light-emitting diode and a diffuser film were used to uniformly illuminate light. To avoid the dispersion effect and illuminate monochromatic light of wavelength 589nm, we used an orange-colored filter in front of the light source. Images of GSP were captured using a digital single-lens reflex camera (Canon, 650D) and macro lens (Canon, MP-E 65mm). The captured GSP images were processed using the Image Processing Toolbox™ of MATLAB® R2012a to analyze the brightness profile in the GSP.
How to cite this article: Kim, W. and Kim, D. S. Gradient shadow pattern reveals refractive index of liquid. Sci. Rep. 6, 28191; doi: 10.1038/srep28191 (2016).
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2014R1A2A1A01006527 and No. 2011-0030075).
Author Contributions W.K. conceived the original idea, designed and conducted the experiments. D.S.K. supervised the research. W.K. and D.S.K. wrote the manuscript. All authors contributed to discussing the results and editing of the final manuscript.