A Wavefront Division Polarimeter for the Measurements of Solute Concentrations in Solutions

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Article A Wavefront Division Polarimeter for the Measurements of Solute Concentrations in Solutions

Sergio Calixto 1,*, Geminiano Martinez-Ponce 1 ID , Guillermo Garnica 1 and Susana Figueroa-Gerstenmaier 2

1 Centro de Investigaciones en Optica, Loma del Bosque 115, Leon 37150, Mexico; [email protected] (G.M.-P.); [email protected] (G.G.) 2 Departamento de Ingenierias Quimica, Electrónica y Biomedica, Division de Ciencias e Ingenierias, Universidad de Guanajuato Campus Leon, Loma del bosque 103, Leon 37150, Mexico; sﬁ[email protected] * Correspondence: [email protected]; Tel.: +52-477-441-4200

Received: 27 October 2017; Accepted: 30 November 2017; Published: 8 December 2017

Abstract: Polarimeters are useful instruments that measure concentrations of optically active substances in a given solution. The conventional polarimetric principle consists of measuring the rotation angle of linearly polarized light. Here, we present a novel polarimeter based on the study of interference patterns. A Mach–Zehnder interferometer with linearly polarized light at the input is used. One beam passes through the liquid sample and the other is a reference beam. As the linearly polarized sample beam propagates through the optically active solution the vibration plane of the electric ﬁeld will rotate. As a result, the visibility of the interference pattern at the interferometer output will decrease. Fringe contrast will be maximum when both beams present a polarization perpendicular to the plane of incidence. However, minimum visibility is obtained when, after propagation through the sample the polarization of the sample beam is oriented parallel to the plane of incidence. By using different solute concentrations, a calibration plot is obtained showing the behavior of visibility.

Keywords: polarimeter; interferometer; solutions concentrations; refractive index; speciﬁc rotation.

1. Introduction A substance is said to be optically active [1] if it rotates the vibration plane of linearly polarized light. The rotation angle depends, among other parameters, on the substance concentration and the optical path length. This optical property has been industrially used for a long time as a reference to quality control. Sugars [2] and essential oils [3] are among the substances that are measured using optical properties. The speciﬁc rotatory power of a substance is modeled through Biot’s law [4]:

T (α)λ = α/lc (1) where α is the rotation angle, l is the optical path length and c is the substance concentration. Solution temperature T and used wavelength λ are also related with the observed rotatory power. Instruments that measure the rotation of the plane of vibration produced by an optically active solution are known as polarimeters. In a basic configuration a polarimeter consists of: (a) a light source giving linearly polarized light with known orientation, or a light source with a fixed linear polarizer, (b) sample tubes of regularized length, where the liquid tested is introduced, and (c) an analyzer fixed in a graduated rotatory mount to measure the rotation angle following the null method principle. Polarizers could be of the sheet-type dichroic or polarizing (Nicol) prisms to mention but two. One type of polarimeters is based on the half-shadow angle phenomenon [5,6]. Polarimeters such as the biquartz, Laurent half wave plate, the Cornu-Jellet prism, Lippich polarizer and others are based

Sensors 2017, 17, 2844; doi:10.3390/s17122844 www.mdpi.com/journal/sensors Sensors 2017, 17, 2844 2 of 9 on the half-shadow angle. Some of them can be used with white light but others with monochromatic light. Lippich instruments could show an accuracy of 0.005◦. The polarimeters that we have mentioned in the last paragraph depend on a time sequential activity. That is, they take measurements one at a time. To avoid this time dependence polarimeters based on the wavefront division have been suggested [5]. They can measure the Stokes parameters. Different parts of the wavefront are analyzed with separate polarization elements and detectors. Among these wavefront division polarimeters are the four-channel polarimeter and the Azzam’s four detector photomultiplier. Other wavefront division polarimeters are based on gratings [5,7] and on the wavefront division by parallel plane glass plates [5]. In this article we suggest a new method, to our knowledge, to measure the concentration of an optically active substance based on a two beam interference pattern. When the linear polarization of two interference beams is parallel to each other, and perpendicular to the plane of incidence, the visibility of the fringes is maximum. As the plane of polarization of one beam rotates, due to the pass of light through an active medium, the visibility of the fringes decreases. The amount of rotation is a function of the solution concentration and the length of the optical path. The interference patterns are recorded with a CCD camera linked to a computer. Then it is possible to do the calculations and infer visibility. In Section2 we describe brieﬂy optical activity and some details of the materials used in the experiments. Also, an optical characterization regarding material’s refractive index is presented. In Section3 is described the interferometric optical conﬁguration. Section4 shows the results including the calibration plots, one for each material tested. Finally, in Section5 we conclude.

2. Optical Activity Usually when linearly polarized light passes through liquids its plane of polarization is unaffected [1]. However, there are some liquids that show optical activity, i.e. they rotate the plane of polarization. In such liquids each molecule can be thought as a small crystal with an optical axis. If plane polarized light is sent to the molecule its polarization plane will be rotated. Optical activity depends on the nature of the substance, the optical path length of the liquid column (l) in dm, the concentration (c) of the solution (g/100 mL), nature of the solvent, temperature (T) of the solution (Celsius degrees) and the wavelength (λ) of the light used, usually λ = 589 nm. The rotation is proportional to the inverse square of the wavelength. These parameters are considered in the Specific Rotation quantity given by Equation (1). To test our hypothesis, we have used fructose [8,9] 20 3 and glucose [8,9]. Former substance shows a large Specific Rotation, (α)D = −92 deg cm /(g dm) [8]. Glucose (C6H12O6)[10] is a monosaccharide, which is the simplest unit of a carbohydrate. It is the basic unit of polysaccharides: starch, cellulose, and glycogen, and it is the most abundant and most important sugar molecule in nature. Glucose is an aldohexose because its formula contains six-carbon atoms with a functional group aldehyde, and it is able to form a five or six-atom ring (including the oxygen) producing a furanose or a pyranose. Its 3D configuration leads to two different stereoisomers, I and II, where I deviates the plane of polarized light towards the right (+) and the II to the left (−). Moreover, the position of the chiral carbon of the aldehyde group and its configuration, gives the glucose D and L, depending if the OH group position is in the right or in the left of the main structure. Finally, the position of the OH group in the C1 produces an additional classification (that leads to a different chemical behavior) known as alpha or beta monosaccharide. In this work, the used glucose was α-D(+)-glucopyranose, which is a colorless powder, with molecular weight of ◦ 180.16 g/mol, solubility in water of 500 g/L and melting point of 146 C. Fructose (C6H12O6) is also a monosaccharide and a ketohexose, colorless powder and together with glucose makes the sucrose or common sugar. In this work, the β-D(−)-fructofuranose was employed. Solubility of fructose is 3.99 g/L at 20 ◦C and its melting point is lower than the one of glucose. The optical characterization study of fructose that we performed, besides its optical activity, included the behavior of refractive index as a function of solutions of fructose and distilled water. Refractive index measurements were done with an Abbe refractometer. Results showing the behavior of SensorsSensors 2017, 201717, 2844, 17, 2844 3 of 9 3 of 9 plots. We notice a steady increment of the refractive index as fructose increases in the solution of fructose–water.the refractive These index refractive measurements index aremeasurements shown in in have Section to4 be along done with once the for calibration each substance plots. tested We notice in ordera steady to build increment a standard of the curve. refractive Regarding index asthe fructose glucose increaseswe prepared in the solutions solution with of fructose–water. the same concentrationsThese refractive that indexfor the measurements fructose solutions have to and be done found once that for they each substancepresented testedalmost in the order same to build refractivea standard indices. curve. Regarding the glucose we prepared solutions with the same concentrations that for theTo investigate fructose solutions if linearly and polarized found that light they that presented passed almostthrough the the same substances refractive was indices. not affected in its linear Tostate investigate of polarization, if linearly except polarized for the light rota thattion passedof the plane through of polarization, the substances the was following not affected set in up wasits linearused. stateA sodium of polarization, lamp (λ = except589 nm) for was the used rotation as light of the source plane with of polarization, a polarizer in the front following of it. set Thenup light was passed used. Athrough sodium a lampglass cell (λ = 7 589 cm nm)long was that used contained as light the source solution. with At a polarizerthe end a in second front of it. polarizerThen light(analyzer) passed measured through athe glass light cell state. 7 cm The long liquids that contained study was the solution.made byAt examining the end a the second extinctionpolarizer light (analyzer) angle by measuredsight. All thelights light in state.the labo Theratory liquids were study turned was off made to have by examining a good adaptation the extinction of thelight eye angle to darkness. by sight. For All the lights liquid in thesamples laboratory tested were just turneda rotation off of to havethe plane a good of polarization adaptation of was the eye present.to darkness. No elliptical For polarized the liquid light samples was noticed. tested just a rotation of the plane of polarization was present. No elliptical polarized light was noticed. 3. Interferometric Optical Configuration and Optical Principle 3. Interferometric Optical Configuration and Optical Principle The optical configuration that we suggest for an interference polarimeter is based on a Mach-ZehnderThe optical interferometer configuration [11], Figure that we1. The suggest beamsplitters for an interferenceare of the absorption polarimeter type. isReference based on a and Mach-Zehndersample beams interferometerhad normalized [11 intensities], Figure1 .of The 0.714 beamsplitters and 0.286, arerespectively. of the absorption Linearly type. polarized Reference lightand comes sample from beams a He-Ne had normalizedlaser (polarizat intensitiesion ratio of 500:1) 0.714and with 0.286, a wavelength respectively. of Linearlyλ = 633 nm. polarized Beam light crosscomes intensity from profile a He-Ne has laser a Gaussian (polarization shape. ratio We 500:1)can call with trajectory a wavelength 1 the of“reference”λ = 633 nm. wave Beam and cross trajectoryintensity 2 the profile “sample” has awave. Gaussian Both shape. plane waves We can interfere call trajectory and because 1 the “reference” polarization wave of both and beams trajectory is linear,2 the parallel “sample” to each wave. other Both and plane perpendicular waves interfere to the and plane because of incidence, polarization the interference of both beams pattern is linear, will parallelconsist toof eachsinusoidal other andmodulated perpendicular fringes. to These the plane patterns of incidence, are captured the interference by a CCD pattern camera. will The consist interferenceof sinusoidal pattern modulated is modulated fringes. by Thesethe beam patterns Gaussian are captured intensity by profile. a CCD Next camera. we will The describe interference theoreticallypattern is the modulated modifications by the of beam the Gaussianinterference intensity pattern profile. when Next the plane we will of describe polarization theoretically of the the samplemodifications beam rotates. of the interference pattern when the plane of polarization of the sample beam rotates.

Figure 1. Mach–Zehnder interferometer. Figure 1. Mach–Zehnder interferometer. At the interferometer output, two mutually coherent light beams, linearly polarized, with plane wavefrontsAt are the interferometersuperimposed output,on an observation two mutually screen coherent parallel light beams,to the xy linearly plane polarized, located at with z = 0, plane

Figurewavefronts 2. The reference are superimposed beam with its on electric an observation field (, screen ) is perpendicularS parallel to theto thexy incidenceplane located plane at Σ,z = 0, → → (, ) whichFigure is parallel2. The to reference the xz plane. beam withOn the its other electric hand ﬁeld theE 1sample( r , t) is beam perpendicular has its electric to the field incidence plane → subtending an angle ψ with the reference beam. Incidence angles are symmetric (= =) with→ Σ, which is parallel to the xz plane. On the other hand the sample beam has its electric ﬁeld E2( r , t) respect to the normal of . subtending an angle ψ with the reference beam. Incidence angles are symmetric (θ = θ1 = θ2) with respect to the normal nˆ of S.

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Figure 2. Superposition of two light beams with linear polarization and different angular positions.

Thus, assuming equal GaussianGaussian amplitude distribution, (,) = exp ∙ −+ (2) → → → → → E1 r , t = E01 exp j k 1· r − ωt + ϕ1 (2) (,) = exp ∙ −+ (3) where: → → → → → E r , t = E exp j k · r − ωt + ϕ (3) 2 02 2 2 =exp − ̂ (4) where: ! → r2 E = E exp − 0 ˆ (4) = − 01 cos 0 sin ̂ + cos2 ̂ + sin sin exp w0 (5) ! → r2 h i is the radial distance from= the center− 0 of the Gaussian+ spot over+ the planeˆ wavefront, is the E02 E0 exp 2 cos θ sin ψıˆ cos ψˆ sin θ sin ψk (5) radius at which the electric field amplitudew0 falls to 1/e, and r0 is the radial distance from the center of∙ the = Gaussian( sin + spot cos over ) the plane wavefront, w0 is the radius(6) at which the electric ﬁeld amplitude falls to 1/e, and ∙ =(− sin + cos ) (7) → → where =2/ is the wavenumberk 1· rand= k (x sin is θthe+ z coswavelength.θ) In an amplitude division(6) interferometer, the output beams are emitted by the same source (1 = 2 = 0). Then, the resulting → → interference pattern is described by k 2· r = k(−x sin θ + z cos θ) (7) where k = 2π/λ is the wavenumber and λ is the wavelength. In an amplitude2 division interferometer, (, ) = + +2 cos cos(2 sin )exp − (8) the output beams are emitted by the same source (ϕ1 = ϕ2 = 0). Then, the resulting interference patternwhere is described= by , and the visibility of interference fringes is found to be , , 2 ! h p i 2r I(x, y) = I=+ I + 2 I =I cos ψ cos(, 20kx ≤sin θ ≤) 90°exp − 0 (9)(8) 1 21 2 2 w0 A theoretical plot showing the interference pattern intensity profile, for an angle ψ, is shown in 1 2 whereFigure I3a.1,2 =By2 εmeasuring0cE01,02, and the the maximum visibility intensity of interference and the fringes adjacent is found minimum to be intensity of different patterns, like the one shown in Figure 3a, we can√ get the visibility for each angle of rotation ψ. Plot in I − I 2 I I cos ψ Figure 3b shows this behavior.V = max min = 1 2 , 0 ≤ ψ ≤ 90◦ (9) Imax + Imin I1 + I2

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A theoretical plot showing the interference pattern intensity proﬁle, for an angle ψ, is shown in Figure3a. By measuring the maximum intensity and the adjacent minimum intensity of different patterns, like the one shown in Figure3a, we can get the visibility for each angle of rotation ψ. Plot in FigureSensors 20173b shows, 17, 2844 this behavior. 5 of 9

(a) (b)

Figure 3.3. (a(a)) Theoretical Theoretical cross cross section section Intensity Intensity pattern pattern as a functionas a function of length of length of an interference of an interference pattern; (pattern;b) Theoretical (b) Theoretical behavior ofbehavior interference of interference pattern visibility pattern as visibility a function as ofa function the rotation of the angle rotation (degrees) angle of the(degrees) sample of plane the sample polarized plane light. polarized light.

4. Results of Experimental Studies To ﬁndfind thethe calibrationcalibration plotsplots ofof VisibilityVisibility asas aa functionfunction ofof solutionssolutions concentrationsconcentrations a glass cell, 7 cm long, was inserted in the “sample” light trajecto trajectoryry of of the the Mach-Zehnder Mach-Zehnder interferometer, interferometer, Figure Figure 11.. Solutions with different mixing ratios of fructose-waterfructose-water or glucose-water were considered. Table 11 shows the concentration ofof eacheach sample.sample. Each amount of fructose or glucose was mixed in 4 mL of distilled water. Solutions were poured in the cell, oneTable at 1. aAmount time, and of fructose/ photographs glucose of in the solutions interference containing patterns 4 mL ofwere distilled taken. water. Some of these interference patterns are shown in Figure 4. Experiments were performed at room temperature. Solution Number Fructose or Glucose (g) Table 1. Amount of fructose/1 glucose in solutions containing 0.6 4 mL of distilled water. Solution Number2 Fructose or Glucose 1.2 (g) 3 1.8 1 40.6 2.4 2 51.2 3 3 61.8 3.6 4 72.4 4.2 8 4.8 5 93 5.4 6 103.6 6 7 4.2 8 4.8 Each amount of fructose or glucose was mixed in 4 mL of distilled water. Solutions were poured 9 5.4 in the cell, one at a time, and photographs of the interference patterns were taken. Some of these 10 6 interference patterns are shown in Figure4. Experiments were performed at room temperature.

(a)(b) (c)

Figure 4. Photographs of interference patterns. The parameter was the concentration of fructose-water solutions. Note that the fringes visibility decreases because as concentrations

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(a) (b)

Figure 3. (a) Theoretical cross section Intensity pattern as a function of length of an interference pattern; (b) Theoretical behavior of interference pattern visibility as a function of the rotation angle (degrees) of the sample plane polarized light.

4. Results of Experimental Studies To find the calibration plots of Visibility as a function of solutions concentrations a glass cell, 7 cm long, was inserted in the “sample” light trajectory of the Mach-Zehnder interferometer, Figure 1. Solutions with different mixing ratios of fructose-water or glucose-water were considered. Table 1 shows the concentration of each sample. Each amount of fructose or glucose was mixed in 4 mL of distilled water. Solutions were poured in the cell, one at a time, and photographs of the interference patterns were taken. Some of these interference patterns are shown in Figure 4. Experiments were performed at room temperature.

Table 1. Amount of fructose/ glucose in solutions containing 4 mL of distilled water.

Solution Number Fructose or Glucose (g) 1 0.6 2 1.2 3 1.8 4 2.4 5 3 6 3.6 7 4.2 8 4.8

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(a)(b) (c) Sensors 2017, 17, 2844 6 of 9 Figure 4.4.Photographs Photographs of interferenceof interference patterns. patterns. The parameter The parameter was the concentration was the concentration of fructose-water of fructose-watersolutions.increases the Note rotation solutions. that the angle fringes Note of linearly visibilitythat the polarized decreasesfringes lighvisibility becauset increases. asdecreases concentrations Amount because of fructose increases as concentrations in thethe rotationsolution anglewas: ( ofa) linearlyno fructose; polarized (b) 4.2 lightg; and increases. (c) 7.8 g. Amount of fructose in the solution was: (a) no fructose;

(b) 4.2 g; and (c) 7.8 g. It should be mentioned that when solutions have a low concentration of fructose they were pouredIt should in the be cell mentioned and after that a few when minutes solutions the haveinterference a low concentration pattern was ofsteady. fructose However, they were for poured heavy insolutions the cell andwe had after to a wait few some minutes minutes the interference so that the patternmixture was stabilizes. steady. It However, seems some for “currents” heavy solutions in the webulk had of tothe wait liquid some were minutes present. so We that had the mixtureto stir lightly stabilizes. the solution It seems to somemix it “currents” well. Then in photographs the bulk of theof the liquid interference were present. patterns We were had totaken stir and lightly they the were solution investigated to mix itwith well. the Then following photographs method. of the interferenceIn order patterns to calculate were taken the andfringes they Visibility were investigated of the interference with the following patterns method. photographs were recordedIn order and to analyzed calculate with the fringes a computer. Visibility The of images the interference had a resolution patterns in photographs pixels of 3456 were × recorded2304 and andcaptured analyzed in color. with However, a computer. the The intensity images values had a we resolutionre converted in pixels to gray of 3456 levels× ranging2304 and from captured 0 to 255. in color.Some However,samples the(sub-images) intensity values were weretaken converted to perform to graythe levelsanalysis. ranging In each from photograph 0 to 255. Some a region samples of (sub-images)interest was considered. were taken The to performregion consisted the analysis. of a horizontal In each photographstrip that passed a region through of interestthe center was of considered.the image. TheThen region to avoid consisted noise ofa alow-pass horizontal filter strip was that applied passed to through each column. the center This of thegave image. us a Thenone-dimensional to avoid noise vector a low-pass of size equal ﬁlter wasto the applied width toof eachthe image column. with This average gave values. us a one-dimensional This vector can vectorbe considered of size equal as a totypical the width cross-section of the image that withpasse averages through values. the center This vector of the can image. be considered Some plots as of a typicalthis vector cross-section are shown that in passes Figure through 5. The the abscissa center axis of the represents image. Some pixels, plots and of thisthe vectorordinate are axis shown the inintensity Figure5 values.. The abscissa axis represents pixels, and the ordinate axis the intensity values.

(a) (b)(c)

FigureFigure 5.5.Intensity Intensity proﬁles profiles as as a functiona function of of pixels pixels (length) (length) obtained obtained through through a computer a computer analysis analysis of the of interferencethe interference patterns patterns shown shown in Figure in Figure4;( a) Intensity4; (a) Intensity vs pixel vs number; pixel number; (b) Intensity (b) Intensity vs pixel numbervs pixel explanation;number explanation; and (c) Intensity and (c) Intensity vs pixel number vs pixel explanation. number explanation.

From the plots shown in Figure 5 we can calculate the visibility for each graph by taking the From the plots shown in Figure5 we can calculate the visibility for each graph by taking the maximum intensity value and the adjacent minimum. These visibility values were plotted as a maximum intensity value and the adjacent minimum. These visibility values were plotted as a function of the amount of fructose or glucose concentration in the solution and the refractive index function of the amount of fructose or glucose concentration in the solution and the refractive index of the solutions described in Section 2, and they are presented in Figures 6 and 7. These are the of the solutions described in Section2, and they are presented in Figures6 and7. These are the calibration plots. Thus, if one needs to know the concentration and/or the refractive index of a calibration plots. Thus, if one needs to know the concentration and/or the refractive index of a fructose fructose or glucose-water solution, a portion of it could be poured in the cell and through the or glucose-water solution, a portion of it could be poured in the cell and through the Visibility plot of Visibility plot of the interference patterns (the calibration plot) the concentration and refractive index the interference patterns (the calibration plot) the concentration and refractive index could be known. could be known. The length of error bars in the visibility axis of Figures 6 and 7 was calculated The length of error bars in the visibility axis of Figures6 and7 was calculated considering that the considering that the minimum difference of intensity measurements in the plots of Figure 6 was 0.008. Due to the use of a good balance to weight the amount of fructose/glucose, and the good instrument to measure the amount of water, the uncertainty in the concentration axis was assumed to be negligible. Regarding the sensitivity, dynamic range and resolution [12] of the polarimeter we have done the following calculations. Plots in Figures 6 and 7 have been considered. Figure 6 shows the calibration plot when fructose has been used. To calculate the sensitivity two points in the plot were chosen in the region where the plot shows a linear behavior. Saturation region has not been considered. Points have the following coordinates (1.8, 0.58) and (5.4, 0.25). The slope is found to be – 0.0916 (−5.23°). For the plot in Figure 7 that considers glucose the points have the coordinates (1.2, 0.7) and (3, 0.45). The slope is found to be −0.166 (−9.45°). The angles are taken from the abscissa axis. This sensitivity values represents the change in visibility as a function of the change in liquids concentration.

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From the same plots in Figures 6 and 7 we can get the dynamic ranges. That is the difference between the smallest and largest concentrations that can be measured. For the fructose we can get a dynamic range of 3.6 g/4 mL (9.22 g/10 mL). For the glucose the dynamic range is 2.59 g/4 mL (6.25 g/10 mL). To find out the concentration resolution we use the following process. In the plot of Figure 6 (fructose) we choose that the minimum change in visibility that we can distinguish is 0.01. Then we can distinguish a change in concentration of 150 mg/4 mL (37.5 mg/mL). This is the resolution in concentration. Regarding the calibration plot for the glucose, Figure 7, we can distinguish a minimum change of visibility of 0.013. This will give us a change in concentration of 90 mg/4 mL (22.5 mg/mL). Having calculated the dynamic range and the resolution for glucose we can compare them with other published work. This is shown in Table 2. Reference [13] mentions a method based on the effect of Surface-Plasmon Resonance (SPR) with a common-path heterodyne interferometry. Reference [14] shows an interferometric system with a measuring length host. Although these systems show better resolution their dynamic range is very short and besides this, the optical and electronic complexity, and cost, is very high.

Table 2. Comparison of Dynamic Ranges and Resolutions with other published works when glucose Sensorswas2017 considered., 17, 2844 7 of 9 Reference Dynamic Range Resolution minimum differenceRef. [13] of intensity measurements 45 mg/10 in the mL plots of Figure6 was 0.008. 0.141 mg/10 Due to mL the use of a good balanceRef. to[14] weight the amount of fructose/glucose, 200 mg/10 mL and the good instrument 48 mg/10 to mL measure the amount ofThis water, paper the uncertainty in the concentration 6250 mg/10 axismL was assumed to be 225 negligible. mg/10 mL

Sensors 2017, 17, 2844Figure 6. Calibration plot when solutions of fructose-water were considered. 8 of 9

Figure 7. CalibrationCalibration plot when solutions of glucose-water were considered.

InRegarding order to theknow sensitivity, the concentration, dynamic range a polynomial and resolution plot should [12] of be the fitted polarimeter to the experimental we have done data. the Forfollowing example, calculations. for fructose, Plots Figure in Figures 6, we 6have and 7fitted have a beenfourth considered. order polynomial Figure6 ofshows the type the calibration plot when fructose has been used. To calculate the sensitivity two points in the plot were chosen in the () = + + . (10) region where the plot shows a linear behavior. Saturation region has not been considered. Points have Withthe following the following coordinates result: (1.8, 0.58) and (5.4, 0.25). The slope is found to be – 0.0916 (−5.23◦). For the plot in Figure7 that considers glucose() the = points 0.643 have − 0.31 the coordinates+ 0.048 (1.2, 0.7) and (3, 0.45). The slope(11) is found to be −0.166 (−9.45◦). The angles are taken from the abscissa axis. This sensitivity values Equation (11) has been plotted along with the experimental data and is shown in Figure 8. represents the change in visibility as a function of the change in liquids concentration. The unknown solute concentration can be retrieved using the equation From the same plots in Figures6 and7 we can get the dynamic ranges. That is the difference between the smallest and largest concentrations that can be measured. For the fructose we can get − − −4( −) a dynamic range of 3.6 g/4 mL (9.22( ) = g/10 mL). For the glucose the dynamic range is 2.59 g/4(12) mL 2 (6.25 g/10 mL). where Vm is the measured visibility. The polynomial function in Equation (10), which is a fourth order approximation of the cosine function when a Taylor series is used, is proposed to predict visibility measurements as a function of solute concentration. The correlation coefficient between experimental and predicted visibility was R = 0.994.

Figure 8. Fourth order polynomial fitted to the experimental points (Fructose).

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To ﬁnd out the concentration resolution we use the following process. In the plot of Figure6 (fructose) we choose that the minimum change in visibility that we can distinguish is 0.01. Then we can distinguish a change in concentration of 150 mg/4 mL (37.5 mg/mL). This is the resolution in concentration. Regarding the calibration plot for the glucose, Figure7, we can distinguish a minimum change of visibility of 0.013. This will give us a change in concentration of 90 mg/4 mL (22.5 mg/mL). Having calculated the dynamic range and the resolution for glucose we can compare them with other published work. This is shown in Table2. Reference [ 13] mentions a method based on the effect of Surface-Plasmon Resonance (SPR) with a common-path heterodyne interferometry. Reference [14] shows an interferometric system with a measuring length host. Although these systems show better resolution their dynamic range is very short and besides this, the optical and electronic complexity, and cost, is very high. Figure 7. Calibration plot when solutions of glucose-water were considered. Table 2. Comparison of Dynamic Ranges and Resolutions with other published works when glucose wasIn order considered. to know the concentration, a polynomial plot should be fitted to the experimental data. For example, for fructose, Figure 6, we have fitted a fourth order polynomial of the type Reference Dynamic Range Resolution () = + + . (10) Ref. [13] 45 mg/10 mL 0.141 mg/10 mL With the following result:Ref. [ 14] 200 mg/10 mL 48 mg/10 mL This paper 6250 mg/10 mL 225 mg/10 mL () = 0.643 − 0.31 + 0.048 (11)

InEquation order to (11) know has thebeen concentration, plotted along a with polynomial the experimental plot should data be and ﬁtted is toshown the experimental in Figure 8. data. ForThe example,unknown for solute fructose, concentration Figure6, wecan have be retrieved ﬁtted a fourth using orderthe equation polynomial of the type

− − 2−4 (4 − ) V(c) = a0 + a2c + a3c . (10) () = (12) 2 With the following result: where Vm is the measured visibility. The polynomial function in Equation (10), which is a fourth order approximation of the cosineV (functionc) = 0.643 when− 0.31 a cTaylor2 + 0.048 seriesc4 is used, is proposed to predict(11) visibility measurements as a function of solute concentration. The correlation coefficient between experimentalEquation and (11) predicted has been plottedvisibility along was withR = 0.994. the experimental data and is shown in Figure8.

Figure 8. Fourth order polynomial fitted ﬁtted to the experimental points (Fructose).

5. Conclusions The unknown solute concentration can be retrieved using the equation An amplitude division interferometer has been used to assess the concentration of solutions by v q measuring the visibility of interferenceu fringes. This2 novel wavefront division polarimeter, of the u −a2 − a2 − 4a3(a0 − Vm) Mach–Zehnder type, comprisesc( Vcommon) = t optical elements in an optical laboratory and an image(12) m 2a 3 Sensors 2017, 17, 2844 9 of 9

where Vm is the measured visibility. The polynomial function in Equation (10), which is a fourth order approximation of the cosine function when a Taylor series is used, is proposed to predict visibility measurements as a function of solute concentration. The correlation coefﬁcient between experimental and predicted visibility was R = 0.994.

5. Conclusions An amplitude division interferometer has been used to assess the concentration of solutions by measuring the visibility of interference fringes. This novel wavefront division polarimeter, of the Mach–Zehnder type, comprises common optical elements in an optical laboratory and an image acquisition device. Calibration plot and digital image processing are key features in the proposed system. The selection of an optimally biased visibility value should allow the measurement of unknown (left) levo- as well as dextro-rotatory substance concentrations. In order to increase the sensitivity, spatial frequency in the interference pattern could be decreased keeping at least one full cycle into the camera view ﬁeld with the aim of achieving less uncertainty in the visibility measurement. As examples of studied solutes we have used fructose and glucose. These substances dissolved in water only rotate the plane of polarization. They do not produce elliptical polarization. We conclude that the described method is versatile because it can be adapted to the measurement of substances with different optical activity. For example, if a substance shows low optical activity, the length of the cell can be made longer to have better sensitivity. The opposite can be made when solutions with high optical activity are present.

Acknowledgments: We thank Raymundo Mendoza for the drawings. Author Contributions: S. Calixto conceived and designed the experiments and performed the experiments, G. Martinez-Ponce developed theory, G. Garnica analyzed the data, S. Figueroa-Gerstenmaier contributed with chemistry background, S, Calixto, G. Martinez-Ponce, G. Garnica and S.Figueroa-Gerstenmaier wrote the paper. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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