A method of liquid refractive index measurement based on image correlation coefficient

According to the difference of speckle image caused by the axial displacement due to the refraction of a transparent medium, the liquid refractive index is measured by using the image correlation coefficient. An optical measurement system based on laser speckle is built into the experiment. Firstly, the CCD camera is adjusted to move along the optical axis in air, and multiple speckle images at different positions are collected, which are correlated with the speckle image at zero position to obtain the image correlation coefficients, and then the exponential calibration relation between the correlation coefficient and the axial displacement can be obtained. Secondly, two speckle images are collected when the laser speckle spreads respectively through the liquid to be measured and water as the internal standard, and the correlation coefficient between them is calculated. Finally, according to the above calibration relation, the corresponding axial displacement between the liquid to be measured and water can be derived, and then the refractive index of the liquid to be measured is calculated using the axial displacement formula. The results show that this method can realize the measurement of liquid refractive index, and the optical structure is simple and easy to operate.


Introduction
The refractive index is a main parameter reflecting the characteristics of liquids, which has become more important in the fields of food hygiene and industrial production.Thus, simple and highprecision measurement methods of liquid refractive index are required for the liquid material analysis and also for its applications.At present, there are many optical methods for the measurement of liquid refractive index, including digital hologram method [1], Snell's law [2], grating diffraction method [3,4], Michelson interferometer method [5], and so on.
A laser speckle is a kind of irregularly distributed bright and dark image formed by the interference of scattered light when the laser irradiates the diffuser surface, which is widely used in surface roughness research, optical image processing, imaging quality evaluation, micro deformation measurement, and other fields [6][7][8].For the measurement of liquid refractive index, one of the common methods using laser speckle is the internal standard method [9], which has been developed [10] recently.In this paper, laser speckle is used to measure the refractive index of liquid through image correlation coefficient.According to the difference in speckle image caused by the axial displacement of the image due to the refraction of the medium, an optical system for measuring the liquid refractive index based on the correlation coefficient of the speckle image was designed.Firstly, the relationship of the image correlation coefficient and axial displacement was calibrated, and then through measuring the image correlation coefficient between the water and the liquid to be measured, the axial displacement between them was obtained from the calibration relationship.Finally, the liquid refractive index can be calculated theoretically.

Principle
As shown in Figure 1, when a laser speckle irradiates a vertically placed transparent medium, refraction will occur.From geometrical optics, it can be seen that the speckle images before and after refraction will have an axial displacement in the optical axis direction, and the value of axial displacement z0 relative to the air (refractive index of 1) can be expressed as where n is the refractive index of the transparent medium, and D is the thickness of the transparent medium.
In this paper, the transparent medium is a liquid, and water is used as the internal standard liquid.Then, the value of axial displacement z relative to the water can be deduced as where nw is the refractive index of water and the reference value is 1.3330.According to the axial displacement Formula (2), the difference in refractive index between the liquid to be measured and water will lead to different axial displacements, which cause the different speckle images collected by the CCD camera at the fixed position.Therefore, according to this image difference, the liquid refractive index can be measured by using the image correlation coefficient.

Experimental
Figure 2 shows the experimental setup for the refractive index measurement of transparent liquid.The light beam emitted from a He-Ne laser source is expanded by an extender lens (focal length of 4.5 cm) and collimated by a collimator lens (focal length of 100 mm) to the plane wave and transmits through a ground glass diffuser to form a speckle image.Then, a rectangular parallelepiped glass cell (crosssection area is 50mm×50mm, two types of cell width is D=5 mm and 10 mm) containing liquid is vertically placed at a fixed position between the ground glass diffuser and a CCD camera (MER-500-14U3M-L, number of pixels is 2592×1944 and pixel size is 2.2 μm×2.2 μm).Finally, the speckle images (e.g.liquids using water and solution) are collected by the CCD camera and the image correlation coefficient between them can be calculated by MATLAB.

Results and Discussion
In this paper, the refractive indices of liquids are measured by the image correlation coefficient method, and the results are compared with those derived by the Abbe refractometer.Sodium chloride solution and sucrose solution with different mass percent concentrations C are used as the liquids to be measured.

Calibration relationship
According to measurement principle, the correlation coefficient of speckle images is related to the axial displacement.To obtain this relationship, the CCD camera was adjusted to move along the optical axis every 30 μm in air, and multiple speckle images at different positions were collected, which were correlated with the speckle image at zero position to obtain the image correlation coefficients.Then the calibration curve of the correlation coefficient r changing with the value of axial displacement z is obtained in Figure 3, which can be fitted with an exponential function for simplicity.As long as the correlation coefficient value of speckle images collected from water and the liquid to be measured is calculated, the corresponding axial displacement between them can be obtained through Formula (3), and then the liquid refractive index can be calculated according to the axial displacement Formula (2).However, when the axial displacement exceeds 300 μm, the correlation coefficient will be less than 0.1, that is, the two speckle images are almost uncorrelated, which can increase the calculation error.In addition, the calibration curve can be influenced by the structure parameters of the optical system.If the structure parameters are changed, the calibration curve should be measured again.

Refractive index measurement
4.2.1.Sodium chloride solution.Placing the glass cell vertically between the ground glass diffuser and the CCD camera, two speckle images are collected when the laser speckle spreads through sodium chloride solution with different concentrations and water respectively, and the correlation coefficient between them can be measured.Then, the corresponding axial displacement is obtained according to Formula (3), and the refractive index of the sodium chloride solution is calculated through Formula ( 2), which is compared with the measured value of the Abbe refractometer.To facilitate discussion, the refractive index and concentration of sodium chloride solution are linearly fitted shown in Figure 4.
From Table 1 and Figure 4, it can be seen that the refractive indices of sodium chloride solution with different concentrations measured by this method are close to the measured values of the Abbe refractometer, and the relative errors are all within 1%.With the increase of solution concentration, the refractive index of the liquid increases, and the corresponding axial displacement also becomes large, leading to the increasing difference of speckle image collected by the CCD camera, as a result, reducing the correlation coefficient.In addition, according to the measurement principle, the axial displacement is related to the cell width.For the solution with the same concentration, the larger the cell width is, the greater the axial displacement is, and the smaller the correlation coefficient is.Selecting sucrose solution with different concentrations, the liquid refractive index is also measured using the same method.The measurement data are listed in Table 2, and the linear fitting curves of the refractive index and concentration of sucrose solution are shown in Figure 5.
Similar to sodium chloride solution, the measurement results of the refractive index of sucrose solution are also close to the measurement values of the Abbe refractometer, and the relative errors are almost within 1% at all concentrations except that the concentrations exceed 40% for the cell width of 10 mm.The reason is that the axial displacements for these concentrations are too large, and the correlation coefficient values of sucrose solution are less than 0.1 accordingly, leading to a decrease in measurement accuracy.Therefore, an appropriate cell width for measurement can further reduce the error.

Conclusion
In this paper, the refractive indices of sodium chloride and sucrose solution were measured by the image correlation coefficient method.First, the calibration relationship curve fitted with the exponential function between the correlation coefficient and the axial displacement is measured in air.
Then, the speckle image correlation coefficient value of water and the liquid to be measured (sodium chloride and sucrose solution) is calculated, and the corresponding axial displacement is obtained according to the above calibration relationship.Finally, the refractive index of the liquid can be derived from the axial displacement formula, which is close to the measured value of the Abbe refractometer.Thus it can be seen that the image correlation coefficient method can be used to measure the refractive index of liquid, and the optical system is simple and easy to operate.

Figure 1 .
Figure 1.Refractive characteristic of a transparent medium.

Figure 2 .
Figure 2. Schematic of the experimental setup of liquid refractive index measurement.

Figure 3 .
Figure 3. Calibration curve of image correlation coefficient and axial displacement.

Table 1 .
Refractive indices of sodium chloride solution with different concentrations.

Table 2 .
Refractive indices of sucrose solution with different concentrations.