1.

INTRODUCTION

The refractive index n of a material is a dimensionless number that is related to the speed of light when passing through a dielectric medium^1,2. This can be seen when light rays traveling from one medium to another bends at the interface. The speed of light in vacuum (c) is approximately 300,000 km/s, while in another transparent media it is lower than and its represented by υ, so that n of a material is expressed as

In a practical approach n can be calculated by using Snell’s law^3,4 (n₁sinθ₁ = n₂sinθ₂), where n₁ and n₂ are the refractive indexes of the two different media and θ₁, θ₂ are the incidence and refracted angles of a light ray traveling throw them. By following this principle and the use of diffraction gratings⁵, the refractive index of liquids can be calculated by using three diffraction gratings^6,7, a refractometer based on a sub-micron grating⁸. Some other methods are based on the interference generated by a Michelson interferometer^9,10.

In the present manuscript a low-cost and automated apparatus designed to estimate the refractive index of liquids is described, which is based on the sensing of the x-displacement of a refracted ray from a laser source when it passed through the testing liquid in accordance with the Snell’s law.

2.

METHODOLOGY AND IMPLEMENTATION

The developed optomechatronic device consists of a generic laser-diode beam that strikes (for implementation simplicity, at an incidence angle of θ₁ = 60°) the surface of a container with the liquid of the unknown refractive index. The beam of light hits the container and refracts both in the walls of the container and the liquid. Once refracted, a stepper motor moves the photodetector (LDR), which is mounted on a moving platform, until the output beam is detected. The total distance traveled on the x-axis is automatically computed in order to calculate the refractive index (in which the entire process as well as the user interface was carried out with LabVIEW and the electronic interface via Arduino UNO microcontroller).

By knowing the distances between the components of the system, as well as the thickness and material of the container, it is possible to calculate the refractive angle of the light in the liquid from the final opening distance. For this calculation, a diagram of the refractions of the light ray in the system is depicted in Figure 1, considering the three media (air, material of the container and given liquid) and all the involved variables to work with Snell’s law and trigonometric functions.

Figure 1.

Diagram of the incidence and refraction angles of the laser’s beam through different materials.

According to the ray diagram and the user defined assembly, the known constants are the angle of incidence θ₁ (starting from point P), the refractive indices of the air n₁ and the material of the container n₂ (which is considered as 1.49 since the material is acrylic); the vertical segments: the distance between the origin of the light beam and the container d₁, the thicknesses of the walls of the container d₂ and d₄, the opening of the container d₃ and the distance between the container and the receiver of the refracted light d₅; as well as the initial distance (L) traveled by the motor on the x-axis until it perpendicularly aligns to point P.

The cathetus (a,…,e) opposite to each angle (or the horizontal distances of each light segment) are defined now:

Snell’s law was also introduced for the relevant diffraction changes of the medium: from the air to the material of the container and from this to the contained liquid

Clearing θ₂ from Eq. (7)

and replacing in Eq. (3) and Eq. (5) to obtain

The total distance in the x-axis that the light beam travels is what the system will measure and is equal to the sum of the horizontal distances

Replacing Eq. (12) with Eqs. (2), (10), (4), (11) and (6) for a, b, c, d, e, we get

Clearing θ₃ from Eq. (13)

and clearing n₃ from Eq. (8)

Replacing Eq. (6) in Eq. (14)

From Eq. (16), the designed LabVIEW to obtain the refractive index after evaluating Eq. (14) to get the resulting angle. The schematic setup is shown in Figure 2, in which the most important components such as: a generic laser source (620nm), the container, the stepper motor and the light sensor mounted over a translation stage are indicated (the complete procedure can be observed in Media1).

Figure 2.

Schematic diagram of the apparatus, the ray tracing and formula to calculate θ₃.

In order to validate the performance of the optomechatronic device, several tests (35 per sample) were carried out to calculate the refractive index of distilled water, ethanol and glycerin. The obtained data was compared with the experimental value given by a refractometer (ATAGO Pal-1), except for glycerin, whose refractive index could not be measured with this apparatus. Nevertheless, for the latter, a theoretical value of n = 1,473 was considered. These results are presented in Figures 3-4.

Figure 3.

Measured refractive index for distilled water and ethanol.

Figure 4.

Measured refractive index for glycerin.

For the distilled water a refractive index of n = 1,3317 was measured with the commercial refractometer and the measurements made with the proposed apparatus showed an average of n = 1,3347, that is an error of 0,23%. On the other hand, ethanol presented a refractive index of n = 1,3609 and the average measurement was n = 1,3616, whose error of 0,05% is even lower than the one of the fatter. In addition, it is important to mention that the measurements of both tests showed a standard deviation of σ = ±0,012. Regarding to glycerin, the measurements made gave an average of n = 1,4675, with a standard deviation of σ = ±0,02. In this way, an error of 0.37% was obtained

For the distilled water a refractive index of n = 1,3317 was measured with the commercial refractometer and the measurements made with the proposed apparatus showed an average of n = 1,3347, that is an error of 0,23%. On the other hand, ethanol presented a refractive index of n = 1,3609 and the average measurement was n = 1,3616, whose error of 0,05% is even lower than the one of the fatter. In addition, it is important to mention that the measurements of both tests showed a standard deviation of σ = ±0,012. Regarding to glycerin, the measurements made gave an average of n = 1,4675, with a standard deviation of σ = ±0,02. In this way, an error of 0.37% was obtained.

3.

CONCLUSIONS

In this manuscript it has been presented a simple low-cost educational optomechatronic apparatus that computes the refractive index of liquids inside of an acrylic container, which is based on the application of the Snell’s law by measuring the x-distance in which the output beam is refracted. The sensing of the beam was achieved by a photoresistor (LDR) attached to a platform and moved by a stepper motor, controlled by Arduino UNO using a LabVIEW interface.

The refraction indexes of three liquids have been measured: ethanol, water and glycerin. The validation of the procedure was carried out by comparing the results with the ones measured by a commercial refractometer and depicted in Figs. (3-4).

It is worth mentioning that the value of this device is lower than $54.99 USD. In spite of this, the average measurement of the refractive index of any given substance is not further than 0,22% of the real value. Moreover, the accuracy of the proposed setup depends on the number of steps given by the stepper motor, as well as on the distance between threads in the translation screw which moves the LDR sensor.