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  Vol.  129  (2016)  ACTA PHYSICA POLONICA A  No. 1 An Experimental Method for Determination of the RefractiveIndex of Liquid Samples Using Michelson Interferometer B. Abbas ∗ and M. Alshikh Khalil Atomic Energy Commission, P.O. Box 6091, Damascus, Syria (Received January 6, 2014; revised version May 26, 2015; in final form September 28, 2015) A new experimental and theoretical method has been presented for measuring the refractive index of liquidsand transparent solids. The experimental setup is based on a modified version of the Michelson interferometeremploying a rotation stage. This technique allows us the determination of the refractive index of a given liquid(or a transparent sample) accurately to high precision, with an accuracy limited only by the precision in thedetermination of the rotation angle.DOI: 10.12693/APhysPolA.129.59PACS: 78.20.Ci, 78.40.–q, 07.60.Ly 1. Introduction The refractive index is an important optical parame-ter in many fields such as in pharmaceuticals, food pro-duction, and other chemical-based industries to monitorpurity of the end product. The refractive index datafor some liquids cannot be found in reference books inmany cases, and must be measured if needed. Numerousmethods for measuring the refractive index of a liquidsolution are well documented in textbooks. However, acost effective method, proposed in this paper, providinghigh precision would be very attractive in a wide range of applications and industries. Also, it is often very impor-tant to be able to detect small differences of the refractiveindex between solutions [1]. There is a spectrum of tech-niques for determining the refractive index of liquids. Forexample, Grange et al. [2] and Singh et al. [3] accuratelymeasured the refractive index of liquid solutions via mea-suring the minimum angle of deviation of a light beampassing through a triangular cell filled up with a liquid.This method has been found to be relatively simple forobtaining the refractive index of liquid solutions whenneither high precision nor accuracy is required. How-ever, a telescope must often be used to locate the po-sition changes in the refracted beam. The probabilityof an error following determination of the angle is high.Leung and Vandiver [4] developed an automated refrac-tometer based on laser beam displacement passing theliquid. However, two measurements must be made: withthe cell full of liquid and with empty one. Other methodsrequire measurement of the critical angle at the bound-ary between the liquid and the cell containing it [4–6],and then an accurate value of the refractive index of thecell walls at the laser wavelength is needed.Recently, interferometric measurement techniqueshave been widely applied in determining the refractive ∗ corresponding author; e-mail:  pscientific@aec.org.sy index of fluids. Suhadolnik [7] developed an optical fiberinterferometeric refractometer, which is a combinationof Mach–Zehnder and Michelson interferometers. Kachi-raju and Gregory have designed two clever techniquesbased on a modified Michelson interferometer for mea-suring the refractive indices of liquids [8]. The first tech-nique consists in analysis of a single fringe, making itvery fast, although not very accurate. The second tech-nique, requiring more time, relies on the fringe-countingmethod as the optical path length through the liquid ischanged by a substantial amount. However, their tech-niques are time consuming and complicated because theyrequire the linearly moving mirror inside the liquid cell,as well as image processing software to record, analyze,and accurately count the interference fringes and deter-mine their FWHM. Ince et al. have presented a numericalmethod for determining refractive index of a glass sam-ple from an implicit function of its optical path withinthe sample arm of a Michelson interferometer [9]. Rota-tion of the sample from normal incidence causes the lightbeam to suffer increasing refraction since the optical pathin the air decreases, and increases in the glass sample.In the present work, we used a modified Michelsontechnique, in which the liquid-containing cell is placedon motorized rotating stage, with all mirrors fixed. Thenumber of the fringes varying during rotation of the sam-ple within a pre-determined range of angles is counted. 2. Experimental 2.1. Michelson interferometer  The experimental setup is a modified Michelson inter-ferometer as sketched in Fig. 1. A high power TEM 00 plane polarized Gaussian beam He–Ne laser (model 31-2140-000, Coherent) nominally 632.8 nm wavelength and5 mW of intensity was used as the light source. Thecollimated beam was split by a beam splitter into twobeams that travel perpendicular to each other. The ref-erence beam travels to a fixed mirror (M1), where it isreflected back to the beam splitter and to the detector.The other beam passing the liquid is reflected by the fixed(59)  60  B. Abbas, M. Alshikh Khalil  Fig. 1. Experimental setup: S is the liquid sample,B/S is a 50:50 beam splitter, EX’s are beam expanders,M’s are the interferometer’s mirrors, ND is a neutraldensity filter, and PD is a photodiode. mirror (M2) back through the beam splitter to the detec-tor. The two beams interfere at the detector and circularfringes are observed when the mirrors are aligned. Acentral bright fringe was observed when the optical pathdifference between the beams was (assumed to be) zero.Alternating central bright and dark fringes were observedby rotating the cell containing the liquid. The intensityvariation of the central fringe passing through the irisand reaching the photodiode was measured and readoutby a computer through a low noise current preamplifier(SR570, Stanford Research Systems), a lock-in ampli-fier (SR850, Stanford Research Systems), and an IEEEinterfacing card (National Instruments). The rotationstage model was 8MR150-1 (Standa, Inc.), supplied witha ZSS43.200.1.2 step motor, which provides a 0.6 arcminresolution, was connected to the PC via a 8SMC1-USBhcontroller (Standa, Inc.), which provides the possibilityof driving the rotational stage by 1/8 of step, which isequivalent to  0 . 00125 ◦ . An A/D card was used to controland record the experimental data along with a dedicatedsoftware program written in Borland C++. Finally, aplot of the light intensity versus the rotation angle wasgenerated. 2.2. The liquid cell  The liquid cell described in Fig. 1 was 10 mm rectangu-lar shaped with an optically flat quartz window on bothsides. The liquid cell was filled with the liquid for whichthe refractive index was being measured, and mountedon the rotation stage. The cell was aligned on the rota-tion stage in such a way that the clear faces of the liquidcell and the mirror were normal to the incident beam.Additionally, the rotational stage and the interfero-meter mirrors were insulated by a Plexiglas box withonly two apertures to allow the laser beam to enter andexit the interferometer section as illustrated in Fig. 1.The sample temperature was monitored by a thermocou-ple fixed to the sample, and connected to a heat con-troller (model 370, Lake Shore Cryotronics, Inc.). It wasfound that the fluctuation of the temperature inside thePlexiglas box in a closed laboratory was about 0.05 ◦ Cduring the measurement which typically took a few min-utes. Moreover, the experimental setup was placed on avibration-isolated table. 2.3. Calibration procedure  The calibration procedure started by manually settingthe sample on the rotation stage so that the laser lightwas incident on the sample face at normal angle. Then,the rotation stage was rotated to the desired startingangle automatically via the driving programme that wewrote. After that, the experimental procedure took placefor recording the variation of the light intensity as a func-tion of the rotating angle. As such, it was found that theautomatic moving of the sample to the starting anglemay be varied from one trial to another by an angle of  ± 0 . 0005 ◦ or less. This small error was seen when drawingthe experimental result using Excel or Origin packages,whereby a small shift in the central fringe was observed.This shift manifested the error in setting the starting an-gle of our experimental procedure, and was taken intoaccount as the experimental error of   φ . 3. Modeling The mathematicl derivation and desciption of the light-cell geometry is shown in Fig. 2. If the cuvette is absent,the light will trace the  PV    path. Placing the cuvettein the path of the light will force the light to change itscourse following the refraction law. The optical path dif- Fig. 2. The geometry of the experiment: symbols areexplained in the text. ference ( ∆ ) between these two paths is given by ∆ = 2 n q PQ + n liq QR − n air PV    (1)where QR  =  d liq cos θ liq , PQ  =  d q cos θ q ,Q  R  =  d liq cos θ liq cos( ϕ − θ liq ) ,PQ  =  R  V    =  d q cos( ϕ − θ q )cos θ q .  An Experimental Method for Determination of the Refractive Index...  61Substituting these relations in Eq. (1) and rearrangingthe resulting terms, the optical path difference will beexpressed as ∆ ϕ  = 2 n q d q cos θ q +  n liq d liq cos θ liq − 2 d q n air cos( ϕ − θ q )cos θ q − d liq cos θ liq n air  cos( ϕ − θ liq )  (2)or ∆ ϕ  = 4 d q   n 2q − n 2air  sin 2 ϕ − n air  cos ϕ  +2 d liq   n 2liq − n 2air  sin 2 ϕ − n air  cos ϕ  ,  (3)where  n q  is the refractive index of the quartz wall for thewavelength of the source light,  d q  is the thickness of thequartz wall,  d liq  is the thickness of the liquid,  n air  is therefractive index of air, and  ϕ  is the rotation angle. If thelight beam is incident at normal angle on the quartz face: ϕ  =  θ q  =  θ liq  = 0 , so that the optical path difference isreduced to ∆ normal  = 2 ×  2 n q d q  + n liq d liq − 2 d q n air − d liq n air  .  (4)Upon turning the cell from the initial normal positionthrough an angle  ϕ , the change in the path differencebetween the two beams is  ( ∆ ϕ − ∆ normal ) , which corre-sponds to  m  fringes appearing or disappearing at thecenter of the fringe pattern. In this case, 2 k 0  ( ∆ ϕ − ∆ normal ) = 2 πm ⇒ m  = 2 λ  ( ∆ ϕ − ∆ normal ) .  (5)This equation can be rearranged, so that a general equa-tion for the relationship between the number of fringes, m , appearing or disappearing at the center of the fringepattern, and the rotation angle will be m  = 2 λ  2 d q   n 2q − n 2air  sin 2 ϕ − n air  cos ϕ  + d liq   n 2liq − n 2air  sin 2 ϕ − n air  cos ϕ  − (2 n q d q  + n liq d liq − 2 d q n air − d liq n air )  .  (6)This equation is valid in both cases where the refractiveindex of the liquid is greater than or smaller than therefractive index of the cuvette.In the case of an empty cuvette, the refractive in-dex of air can be measured using Eq. (6) and replacing n liq  =  n air , which becomes m  = 2 λ  2 d q   n 2q − n 2air  sin 2 ϕ − n air  cos ϕ  − (2 n q d q − 2 d q n air )  .  (7)On the other hand, substituting Eq. (7) into Eq. (6) gives m  =  m  + 2 d liq λ   n 2liq − n 2air  sin 2 ϕ − n air  cos ϕ − n liq  + n air  ,  (8)where  m  is the number of fringes counted when the cu-vette is empty, and  m  is the number of fringes countedwhen the cuvette is filled with the liquid, in the samerange of rotating angles. This equation can be used tomeasure the refractive index of liquids without the needof knowledge of the refractive index of the cuvette.This method can also be used for measuring the refrac-tive index of a slab of crystals and glass plates, reducingEq. (6) to m  = 2 λ  d q   n 2q − n 2air  sin 2 ϕ − n air  cos ϕ  − ( n q d q − d q n air )  ,  (9)where  n q  is the refractive index of the glass (or crystal), d q  is its thickness.Additionally, a nonlinear least square fit procedure wascarried out, depending on the resultant light intensity of the interference, I   =  I  1  + I  2  + 2   I  1 I  2  cos  k (∆ ϕ − ∆ normal )  ,  (10)where  I  1  and  I  2  are the intensities of the beams followingthe two paths. The fitting procedure was executed byvarying  I  1 ,  I  2 , and  n liq  parameters. This procedure wasproven very useful in order to obtain the most accuratevalue of the refractive index with a very small deviation. 4. Uncertainty analysis Uncertainties in the refractive index measurement arisefrom systematic errors in angle measurements, and ran-dom errors caused by vibration, air disturbances, andlaser fluctuations. Random errors are accounted for us-ing statistical averaging techniques during the data ac-quisition. Systematic errors are determined from the fol-lowing error analysis. Referring to Fig. 2 and Eq. (6),uncertainties in  φ ,  n q ,  n air ,  d q , and  d liq  contribute to un-certainty in  n liq . Errors arising from  d q ,  n q ,  d liq , and n air  are considered negligible. The error of   φ  is taken ashalf of the measurement setup resolution,  ± 0 . 0005 ◦ . Er-ror in  n liq  increases with higher values of the refractiveindex of the liquids under investigation (varied between ± 0.0001 and  ± 0.003). These errors can be minimizedwith smaller steps (intervals) in the recorded data pointsas the rotation stage is rotated to the targeted angle.This procedure helps improving the accuracy of   φ . How-ever, the accuracy of  φ was still a main factor determiningprecision of the refractive index determination. 5. Results 5.1. Refractive index of water  Distilled water was chosen to validate the measurementtechnique discussed above. Mirrors M1 and M2 were ad- justed precisely to obtain good superposition of trans-mitted and reflected waves. Circular interference fringes  62  B. Abbas, M. Alshikh Khalil  of high visibility were then observed on the field of view.The interference intensities were simultaneously recordedcontinuously. The total number of central order fringecycles in the chosen rotation angle range was counted byrecording the experimental data in a spread sheet andcounting the number of peaks. The fringe count datawas then used in Eq. (8) to calculate the refractive indexof the water sample. The measured refractive index of the water,  n water , was 1.3316. Figure 3 represents themeasured fringes (circle points) and the correspondingnonlinear least square best fit (solid line) according toEq. (10). Fig. 3. The measured fringes (circle points) and thecorresponding nonlinear least square fit (solid line) forwater.TABLE IThe refractive indexes of different liquid and solid sam-ples at 632.8 nm laser light.Material refractive indexthis work othersair 1.00028 1.000267 a [22 ◦ C]  ± 0.00005  ± 0.000007 [21 ◦ C]liquid samplesdistilled 1.3316 1.33171 b water [22 ◦ C]  ± 0.0001  ± 0.00002 [21 ◦ C]benzene 1.4951 1.497108 b [22 ◦ C] 0.0002  ± 0.00002 [21 ◦ C]ethanol 1.3562 1.356039 c [20 ◦ C]  ± 0.0002  ± 0.000005 [20 ◦ C]dichloromethane 1.4195 –(DCM) [20 ◦ C]  ± 0.0002 –acetone 1.3585 1.35782 c [20 ◦ C]  ± 0.0002  ± 0.000005 [20 ◦ C]tetrahydrofurane 1.4034 –(THF) [20 ◦ C]  ± 0.0002 –solid slabsquartz 1.4571 1.457012 d [22 ◦ C]  ± 0.0001  ± 0.000001 [20 ◦ C]sapphire 1.7653 1.765894 e [22 ◦ C]  ± 0.0003  ± 0.000001 [20 ◦ C] a Ref. [10],  b Ref. [11],  c Ref. [5],  d Ref. [12],  e Ref. [13] 5.2. The refractive index of other materials  The refractive indices of different liquid samples andsolid transparent substrates were also measured. Theresults are shown in Table I, along with a comparisonwith those published data in the literature for some sub-stance samples we have used. It is noteworthy to saythat the noticed differences in the refractive index of anymaterial used by the researchers may be attributed tothe pollution, concentration, or impurity of this material,as well as the experimental conditions under which themeasurements were carried out. Therefore, one cannotexpect to get identical results or values for the refrac-tive index, which was what we have noticed many times.As an example: our material (benzene) was purchasedfrom Fluka, with a purity of   >  99 % and a density of 0.878 g/mL at 20 ◦ C, but there are different concentra-tions in the market such as that sold by Sigma-Aldrichwith purity of 99.8% and a density of 0.874 g/mL at25 ◦ C. Another factor which may be taken into accountfor the differences in the numbers found in literature isthe refractive index of the material at the sodium line.For example, our benzene has  n 20D  = 1 . 500  whereas thatsold by Sigma-Aldrich has the value of   n 20D  = 1 . 501 . How-ever, in the light of the agreement of our results withthose found in literature, we argue that the measuredvalue of the refractive index of the used benzene is ac-ceptable. 6. Conclusion In the present paper, we have derived the relationshipbetween the refractive index of liquid solutions and thenumber of central fringes appearing or disappearing asthe liquid samples are rotated within a certain range of angles. We have successfully demonstrated and deter-mined the refractive index of a wide range of solutions,as well as transparent solid slabs of crystals and glasses.On a different note, it is needless to say that thismethod is quite capable of measuring the thickness of the sample under test if its refractive index is known. Acknowledgments The authors would like to express their thanks to Prof.I. Othman for his continuous encouragement, guidanceand support. References [1] H.B. Thomas, K. Matsumoto, T. Eiju, K. Matsuda,N. Ooyama,  Appl. Opt.  30 , 745 (1991).[2] B.W. Grange, W.H. Stevenson, R. Viskanta,  Appl.Opt.  15 , 858 (1976).[3] V.K. Singh, B.B.S. Jaswal, V. Kumar, R. Prakash,P. Raib,  J. Integr. Sci. Technol.  1 , 13 (2013).[4] A.F. Leung, J.J. Vandiver,  Opt. Eng.  42 , 1128(2003).[5] E. Moreels, C. deGreef, R. Finsy,  Appl. Opt.  23 ,3010 (1984).[6] M.V.R.K. Murty, R.P. Shukla,  Opt. Eng.  18 , 177(1979).

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