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A single-element interferometer for measuring refractive index of transparent liquids
Optics Communications 332 (2014) 14–17
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
A single-element interferometer for measuring refractive index of transparent liquids Tao Zhang a, Guoying Feng a,n, Zheyi Song a, Shouhuan Zhou a,b a b
Institute of Laser & Micro/Nano Engineering, College of Electronics & Information Engineering, Sichuan University, Chengdu, Sichuan 610064, China North China Research Institute of Electro-Optics, Beijing 100015, China
art ic l e i nf o
a b s t r a c t
Article history: Received 10 March 2014 Received in revised form 16 June 2014 Accepted 16 June 2014 Available online 27 June 2014
A simple and stable method based on a single-element interferometer for accurately measuring refractive index of transparent liquids was demonstrated. The refractive index is measured by rotating a rectangular optical glass cell which contains sample liquid and air simultaneously, and by calculating interference fringe shift number which is detected from an interferogram. This method was successfully used to measure the refractive indices of various transparent liquids including distilled water, ethanol and NaCl–water and ethanol–water solutions at various concentrations. The temperature- dependent refractive index of distilled water was also measured. Furthermore, our method is simple to implement, vibration insensitive, and of high accuracy up to 10 4. Published by Elsevier B.V.
Keywords: Interferometer Liquid Refractive index measurement
1. Introduction Accurately measuring the refractive index of liquids is critical in various applications, such as biosensor [1], waveguide design [2], etc. Accurate knowledge of refractive index and its variation with concentration are used for material identification and characterization. Nowadays, different techniques have been developed for the refractive index measurement. The minimum angle of deviation method is a relatively simple method when high precision or accuracy is not required but the problem is that an error occurring in determining the angle is probably high [3]. Critical angle method measured the critical angle at the boundary between the prism and the liquid [4], which needs an accurate value of the refractive index of the prism. A transparent capillary filled with liquid is used as a cylindrical lens method based on an imageforming principle [5], while it needs an accurate value of the refractive index of the capillary or another standard liquid. The core-cladding mode interferometry method is simple and low cost but precision is only 0.001 [6]. Optical fiber sensors have also been used to measure the refractive index of liquid solutions for many advantages such as high sensitivity, small size and remote sensing capability. Most of them are based on the evanescent field interaction with the liquid, such as long period grating [7,8], etched fiber Bragg gratings [9], and photonic crystal waveguides
n
Corresponding author at: Tel.: þ 86 28 8546 3880; fax: +86 28 8546 3880 E-mail addresses: [email protected] (G. Feng), [email protected] (S. Zhou). http://dx.doi.org/10.1016/j.optcom.2014.06.028 0030-4018/Published by Elsevier B.V.
[10]. In these approaches, evanescent-field based sensors suffer from large temperature cross sensitivity and nonlinear refractive index response. Furthermore most of the sensors need to be fabricated by complex and bulky techniques. In addition, interferometric measurement techniques have been widely applied for measuring the refractive index of liquids. With interferometric methods, the accuracy is not affected because the real path length difference is measured. These methods with a higher resolution have also been proposed, such as Michelson interferometer [11], and Mach–Zehnder interferometer [12]. The disadvantages of these methods lie in the strict conditions on interferometer stability and in a critical dependence on the distance between the optical elements of the system and their position. The accuracy of the measurement is easily perturbed by the environment. A fiber sensor based on Fabry–Perot (FP) interferometer is another method [13]. Since the tip of the fiber sensor is immersed in the liquid sample, it may have an adverse effect on fiber when measuring some special solutions and the sample may be contaminated. In this work we proposed a new, simple, stable method for measuring the refractive index of transparent liquid sample which is based on the principle of a single-element interferometer [14]. The refractive index is determined by rotating a rectangular optical glass cell which contains sample liquid and the air simultaneously, and by calculating interference fringe shift number. The fringe shift number is easily detected from the interferogram formed by the object beam which passes through the liquid and the reference beam which passes through the air from a beam-splitter cube. We have mainly studied the refractive index of NaCl–water, ethanol– water solutions at various concentrations. Also the temperature-
T. Zhang et al. / Optics Communications 332 (2014) 14–17
dependent refractive index of distilled water was measured. This method is vibration insensitive, with high accuracy up to 10 4, and experimental realization is simple for measuring transparent liquid samples.
2. Experimental setup and principle of measurement The setup and principle of operation of the interferometric system are depicted in Fig. 1. The experimental setup is considerably simple. A laser beam (λ¼ 636.94 nm) which was expanded and collimated to a diameter of 20 mm is vertically incident on the sample along the z-axis. The sample is placed between the laser and beam-splitter cube with splitting ratio of 50:50. A rectangular optical glass cell with wall thickness of 1 mm and internal size of 20 mm 10 mm is fixed on a rotation stage which had a computer-controlled stepping motor with steps of 0.000671. The cell is divided into two parts by a partition, so that it can simultaneously contain sample liquid and the standard liquid (or air). Air is taken as a reference in this experiment. The surfaces of the cell have a flatness tolerance of less than 1 μm and their deviation from parallelism is of the order of 10 μm. Half of laser beam passes through the empty portion of the cell as a reference beam (Path 1). The other part of the laser beam passes through the filled liquid portion of the cell as an object beam (Path 2). And then the two beams pass through the beam-splitter cube and form two controllable parallel equidistant straight interference fringe patterns. The interference fringe patterns are divided into two by a beam-splitter, one passes through the lens and then arrives at a counter which was made by us to count the integer part of the interference fringe shift number, and the other arrives at the observation plane of a CCD camera (Microview, Model MVC-II1 MM, pixel size: 5.2 μm 5.2 μm) which captures the initial image and the final image at a certain rotation angle θ for counting the fractional part of the interference fringe shift number. On the other hand, the fringe spacing can be enlarged to an appropriate size by rotating the cube around an axis orthogonal to a plane of illustration in Fig. 1 in order to improve the accuracy of the measurement. The proposed single-element interferometer is depicted in illustration of Fig. 1. It shows the ray trajectories in the beamsplitter cube (BS). A light wave is incident along the z-axis on the beam-splitter cube with its central semi-reflecting layer placed along the propagation direction. The top half of the light beam
15
(Path 1) acts as the reference light and the other bottom half beam (Path 2) as the object light, or vice versa. Let the electric fields of two halves of the input beam at the plane ∑in be E1(x0 , y0 ) and E2(x0 , y0 ), the interference of the transmitted and reflected lights at the plane ∑out is obtained, and then intensity patterns of the Output 1 and the Output 2 are represented as follows: I Output 1 ¼ jE1 ðd x; yÞexp½iπ=2 þ E2 ðx d; yÞj2 ¼ jE1 ðd x; yÞj2 þ jE2 ðx d; yÞj2 þ 2jE1 ðd x; yÞjjE2 ðx d; yÞj cos ðξ1 ðd x; yÞ ξ2 ðx d; yÞ þ π=2Þ
ð1Þ
I Output 2 ¼ jE1 ðd þ x; yÞ þ E2 ð x d; yÞexp½iπ=2j2 ¼ jE1 ðd þ x; yÞj2 þ jE2 ð x d; yÞj2 þ 2jE2 ðd þ x; yÞjjE2 ð x d; yÞj cos ðξ1 ðd þ x; yÞ ξ2 ð x d; yÞ π=2Þ
ð2Þ
where ξ1, ξ2 are the phases of the fields E1, E2, respectively. (We neglect the splitter-ratio of the beam-splitter cube in above expressions.) When the incident light is plane wave of constant amplitude (E0), we have E1 ðx0 ; y0 Þ ¼ E0 ;
E2 ðx0 ; y0 Þ ¼ E0 exp½iξðx0 ; y0 Þ
ð3Þ
where ξðx'; y'Þ is phase distribution of the sample placed in Path 2. If the beam-splitter ratio is 50:50, Eqs. (1) and (2) can be rewritten as I Output 1 ¼ I 0 f1 þ cos ½ξðx d; yÞ π=2g ¼ I 0 f1 þ sin ξðx d; yÞg
ð4Þ
I Output 2 ¼ I 0 f1 þ cos ½ξð x d; yÞ þ π=2g ¼ I 0 f1 sin ξð x d; yÞg ð5Þ E20 .
where I 0 ¼ It is easy to see that the optical path difference between the two beams is changed when rotating the sample. Let the refractive index of a transparent liquid sample with thickness d be n, and rotation angle of the sample be θ, the interference fringe shift number N is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 N ¼ ð n2 n20 sin θ n0 cos θ n þ n0 Þ d=λ ð6Þ where n0 ¼ 1 is the refractive index of air, λ is the optical wavelength. We can obtain the expression of the refractive index
Fig. 1. Experimental setup. The illustration shows ray trajectories in the beam-splitter cube. d is the size of the working region in the x-direction.
16
T. Zhang et al. / Optics Communications 332 (2014) 14–17
of the liquid as follows: n¼
ðNλ=dÞ2 þ 2ð1 cos θÞðn20 n0 Nλ=dÞ 2½n0 ð1 cos θÞ Nλ=d
ð7Þ
3. Results and discussion During the measurement the environment temperature is kept stable at 25 1C. By recording the rotation angle θ and the interference fringe shift number N, we can calculate the refractive index of the liquids with Eq. (7). Table 1 lists the measured refractive indices of two different liquids. Comparing the measured refractive indices ns with the reference values nref yields a deviation of 0.0001 for ethanol and a deviation of 0.0002 for distilled water. To demonstrate the refractive index measurement capabilities of this technique, results of measured refractive index of NaCl– water and ethanol–water at different concentrations are shown in Figs. 2 and 3, respectively. Both of the black points are the measurements from the literature [17], but at a wavelength of 589 nm, where the refractive index is slightly lager. A very good agreement was found between all measured values and those reported in the literature. Fig. 2 shows a linear dependence on the concentration, on the contrary as shown in Fig. 3, which exhibits a strong nonlinear dependence on the concentration [17]. The uncertainty of the refractive index measurement made by the proposed method primarily depends on the errors in the thickness of the cell d, the rotation angle θ, and the interference fringe shift number N. The effects of these error sources are separately discussed below. Partially differentiating Eq. (7) with respect to θ, d and N, we have ∂n Nλ ðNλ=d 2Þ sin θ ¼ ∂θ d 2ð1 cos θ Nλ=dÞ2
ð8Þ
∂n Nλ Nλ sin 2 θ ¼ 2 ∂d 2d 2½dð1 cos θÞ Nλ2
ð9Þ
Table 1 Measured refractive indices of two liquids. Liquid
ns
nref [15, 16]
Distilled water Ethanol
1.33157 0.0002 1.35867 0.0001
1.3316 1.3587
Fig. 2. Refractive index of NaCl–water solution vs. concentration.
Fig. 3. Refractive index of ethanol–water solution vs. concentration.
Table 2 Sources of error in the measurement of refractive index of two liquids. Liquid
σθ
σd
σN
σn
Distilled water Ethanol
3.1 10 5 2.9 10 5
2.4 10 4 2.6 10 4
4.5 10 5 4.7 10 5
2.4 10 4 2.6 10 4
∂n λ 2ð1 cos θÞðNλ=d þ cos θÞ ðNλ=dÞ2 ¼ ∂N d 2ð1 cos θ Nλ=dÞ2
ð10Þ
It is obvious that the accuracy of the measured index is determined by the relative error of d, θ, and N. In distilled water, for example, substituting θ¼201, d ¼10 mm and N ¼ 245.3463 into Eqs. (8)–(10) yields |∂n/∂θ| ¼2.6564, |∂n/∂d| ¼24.4804, |∂n/∂N| ¼ 0.0018, respectively. The resolution of rotary table is 0.000671, so the angular deviation σθ ¼0.000671¼ 1.1694 10 5 leads to the deviation of the refractive index measurement with Δn ¼(|∂n/∂θ|) σθ E3.1 10 5. Employing a rotary table with high accuracy can greatly reduce the angular deviation. The thickness of the rectangular cell deviation σd ¼ 0.01 mm, it leads to the deviation of the refractive index measurement with Δn ¼(|∂n/∂d|)σd E 2.4 10 4. As measured fringe spacing is approximate 40 pixel of CCD, the minimum resolution of the interference fringe shift number is σN ¼1/40 ¼0.025; therefore it effects on the refractive index with Δn ¼(|∂n/∂N|)σN E5 10 5. Accordingly, the combined relative deviation of the refractive index measurement is σn ¼ [(3.1 10 5)2 þ(2.4 10 4)2 þ(5 10 5)2]1/2 E2.4 10 4, and the refractive index can be accurately determined to the fourth decimal place. The deviations of ethanol are shown in Table 2. Table 2 lists the sources of error in the measurement of refractive index of two liquids. We think the measurement error owing to the liquids wetting the glass wall of the cell will affect the measurment accuracy, but the impact is very small so we did not consider it in this paper. In addition, we also studied the device's capability to measure the temperature-dependent refractive index of liquid. Samples are kept in the original position with rotation angle θ¼0. A relative measurement method [18] was adopted, we assumed that n1 is the refractive index of liquid at temperature T, so n0 ¼n1 þΔn is the refractive index of liquid at temperature T þΔT. Since the refractive index change Δn caused by the temperature change, which leads to the optical path difference between the two beams, corresponds to the interference fringe shift number N, we neglect the influence
T. Zhang et al. / Optics Communications 332 (2014) 14–17
17
change can be observed, which is in good agreement with the measurement data reported previously [15]. To assess the stability of the proposed experimental setup against environmental disturbance such as mechanical vibrations, interferograms were continuously captured in the laboratory situation in a period of 40 min at intervals of 1 min. The fluctuation of the offset distance of interference fringe pattern over 40 min is shown in Fig. 5. The maximal deviation of the fluctuation during 40 min is 2.0014 pixels, which corresponds to the combined relative error of the refractive index measurement with 9 10 5. Obviously, the high stability and repeatability mainly can be attributed to the common-path property of the interferometer. 4. Conclussion
Fig. 4. The refractive index of distilled water vs. temperature. The solid lines are cubic polynomial fits to the measured data.
In conclusion, we demonstrated the feasibility of a stable laser refractometer based on a single-element interferometer constituted by a beam-splitter cube for highly sensitive refractive index measurement. The refractive index is measured by counting the interference fringe shift number when rotating the sample. The device was evaluated for refractive index measurement of various transparent liquids and the results matched well with the literature values. Moreover, once the relationship between concentration and refractive index is measured, the same technique can be used to measure the concentration of solutions. The proposed method requires no sensors to be immersed in liquid. Furthermore, the proposed method is very easy to implement in a laboratory with simple apparatus, with low cost, noncontact, and is of high precision, high, accuracy and quite insensitive to the atmospheric turbulence and vibration.
Acknowledgments This work was supported by Major Program of National Natural Science Foundation of China (60890200) (10976017). References Fig. 5. Stability of the experimental setup.
of the sample thermal expansion. ðn0 nÞd ¼ Δnd ¼ Nλ
ð11Þ
Δn ¼ Nλ=d
ð12Þ
The change was then added to the refractive index value at T to calculate the refractive index of liquid at that temperature. We measured the refractive index of distilled water at 25 1C, then made distilled water with a specific temperature dropped smoothly to 25 1C, and computed the refractive index change using Eq. (12) by measuring the interference fringe shift number during temperature changes. The refractive index of distilled water at that temperature was calculated by adding the change to the refractive index of distilled water at 25 1C. Thus the refractive index of distilled water at temperature from 25 1C to 65 1C at intervals of 2.5 1C was measured. Fig. 4 depicts the results indicating the dependence of the refractive index of distilled water on the increasing temperature. The nonlinear decrease of the amount and shape of the measured refractive index of distilled water
[1] C.H. Yeh, C.W. Chow, J.Y. Sung, P.C. Wu, W.T. Whang, F.-G, Sensors (Besel) 12 (2012) 481–488. [2] X. Li, J. Wu, A. Liu, Z. Li, Y. Soew, H. Huang, K. Xu, J. Lin, Appl. Phys. Lett. 93 (2008) 193901. [3] W. Yunus, A.b.A. Rahman, Appl. Opt. 27 (1988) 3341–3343. ̀ [4] A. Garcıa-Valenzuela, M. Peña-Gomar, C. Garcıa-Segundo, V. Flandes-Aburto, Sensors Actuators B 52 (1998) 236–242. [5] D. Káčik, P. Tvarožek, I. Martinček, K. Schuster, Optik 123 (2012) 1746–1749. [6] Q. Li, X. Pu, Appl. Opt. 52 (2013) 5318–5326. [7] J.H. Chong, P. Shum, H. Haryono, A. Yohana, M. Rao, C. Lu, Y. Zhu, Opt. Commun. 229 (2004) 65–69. [8] M. Han, F. Guo, Y. Lu, Opt. Lett. 35 (2010) 399–401. [9] W. Liang, Y. Huang, Y. Xu, R.K. Lee, A. Yariv, Appl. Phys. Lett. 86 (2005) 151122–151123. [10] N. Skivesen, A. Têtu, M. Kristensen, J. Kjems, L.H. Frandsen, P.I. Borel, Opt. Express 15 (2007) 3169–3176. [11] S.R. Kachiraju, D.A. Gregory, Opt. Laser Technol. 44 (2012) 2361–2365. [12] H. El-Kashef, G. Hassan, I. El-Ghazaly, Appl. Opt. 33 (1994) 3540–3544. [13] J. Tian, Y. Lu, Q. Zhang, M. Han, Opt. Express 21 (2013) 6633–6639. [14] J.A. Ferrari, E.M. Frins, Opt. Commun. 279 (2007) 235–239. [15] P. Schiebener, J. Straub, J.L. Sengers, J. Gallagher, Refractive index of water and steam as function of wavelength, temperature and density, American Chemical Society, 1990. [16] E. Moreels, C. De Greef, R. Finsy, Appl. Opt. 23 (1984) 3010–3013. [17] W.M. Haynes, D.R. Lide, T.J. Bruno, CRC Handbook of Chemistry and Physics 2012–2013, 93rd ed., CRC Press, 2012. [18] T. Wei, Y. Han, Y. Li, H.L. Tsai, H. Xiao, Opt. Express 16 (2008) 5764–5769.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
A single-element interferometer for measuring refractive index of transparent liquids Tao Zhang a, Guoying Feng a,n, Zheyi Song a, Shouhuan Zhou a,b a b
Institute of Laser & Micro/Nano Engineering, College of Electronics & Information Engineering, Sichuan University, Chengdu, Sichuan 610064, China North China Research Institute of Electro-Optics, Beijing 100015, China
art ic l e i nf o
a b s t r a c t
Article history: Received 10 March 2014 Received in revised form 16 June 2014 Accepted 16 June 2014 Available online 27 June 2014
A simple and stable method based on a single-element interferometer for accurately measuring refractive index of transparent liquids was demonstrated. The refractive index is measured by rotating a rectangular optical glass cell which contains sample liquid and air simultaneously, and by calculating interference fringe shift number which is detected from an interferogram. This method was successfully used to measure the refractive indices of various transparent liquids including distilled water, ethanol and NaCl–water and ethanol–water solutions at various concentrations. The temperature- dependent refractive index of distilled water was also measured. Furthermore, our method is simple to implement, vibration insensitive, and of high accuracy up to 10 4. Published by Elsevier B.V.
Keywords: Interferometer Liquid Refractive index measurement
1. Introduction Accurately measuring the refractive index of liquids is critical in various applications, such as biosensor [1], waveguide design [2], etc. Accurate knowledge of refractive index and its variation with concentration are used for material identification and characterization. Nowadays, different techniques have been developed for the refractive index measurement. The minimum angle of deviation method is a relatively simple method when high precision or accuracy is not required but the problem is that an error occurring in determining the angle is probably high [3]. Critical angle method measured the critical angle at the boundary between the prism and the liquid [4], which needs an accurate value of the refractive index of the prism. A transparent capillary filled with liquid is used as a cylindrical lens method based on an imageforming principle [5], while it needs an accurate value of the refractive index of the capillary or another standard liquid. The core-cladding mode interferometry method is simple and low cost but precision is only 0.001 [6]. Optical fiber sensors have also been used to measure the refractive index of liquid solutions for many advantages such as high sensitivity, small size and remote sensing capability. Most of them are based on the evanescent field interaction with the liquid, such as long period grating [7,8], etched fiber Bragg gratings [9], and photonic crystal waveguides
n
Corresponding author at: Tel.: þ 86 28 8546 3880; fax: +86 28 8546 3880 E-mail addresses: [email protected] (G. Feng), [email protected] (S. Zhou). http://dx.doi.org/10.1016/j.optcom.2014.06.028 0030-4018/Published by Elsevier B.V.
[10]. In these approaches, evanescent-field based sensors suffer from large temperature cross sensitivity and nonlinear refractive index response. Furthermore most of the sensors need to be fabricated by complex and bulky techniques. In addition, interferometric measurement techniques have been widely applied for measuring the refractive index of liquids. With interferometric methods, the accuracy is not affected because the real path length difference is measured. These methods with a higher resolution have also been proposed, such as Michelson interferometer [11], and Mach–Zehnder interferometer [12]. The disadvantages of these methods lie in the strict conditions on interferometer stability and in a critical dependence on the distance between the optical elements of the system and their position. The accuracy of the measurement is easily perturbed by the environment. A fiber sensor based on Fabry–Perot (FP) interferometer is another method [13]. Since the tip of the fiber sensor is immersed in the liquid sample, it may have an adverse effect on fiber when measuring some special solutions and the sample may be contaminated. In this work we proposed a new, simple, stable method for measuring the refractive index of transparent liquid sample which is based on the principle of a single-element interferometer [14]. The refractive index is determined by rotating a rectangular optical glass cell which contains sample liquid and the air simultaneously, and by calculating interference fringe shift number. The fringe shift number is easily detected from the interferogram formed by the object beam which passes through the liquid and the reference beam which passes through the air from a beam-splitter cube. We have mainly studied the refractive index of NaCl–water, ethanol– water solutions at various concentrations. Also the temperature-
T. Zhang et al. / Optics Communications 332 (2014) 14–17
dependent refractive index of distilled water was measured. This method is vibration insensitive, with high accuracy up to 10 4, and experimental realization is simple for measuring transparent liquid samples.
2. Experimental setup and principle of measurement The setup and principle of operation of the interferometric system are depicted in Fig. 1. The experimental setup is considerably simple. A laser beam (λ¼ 636.94 nm) which was expanded and collimated to a diameter of 20 mm is vertically incident on the sample along the z-axis. The sample is placed between the laser and beam-splitter cube with splitting ratio of 50:50. A rectangular optical glass cell with wall thickness of 1 mm and internal size of 20 mm 10 mm is fixed on a rotation stage which had a computer-controlled stepping motor with steps of 0.000671. The cell is divided into two parts by a partition, so that it can simultaneously contain sample liquid and the standard liquid (or air). Air is taken as a reference in this experiment. The surfaces of the cell have a flatness tolerance of less than 1 μm and their deviation from parallelism is of the order of 10 μm. Half of laser beam passes through the empty portion of the cell as a reference beam (Path 1). The other part of the laser beam passes through the filled liquid portion of the cell as an object beam (Path 2). And then the two beams pass through the beam-splitter cube and form two controllable parallel equidistant straight interference fringe patterns. The interference fringe patterns are divided into two by a beam-splitter, one passes through the lens and then arrives at a counter which was made by us to count the integer part of the interference fringe shift number, and the other arrives at the observation plane of a CCD camera (Microview, Model MVC-II1 MM, pixel size: 5.2 μm 5.2 μm) which captures the initial image and the final image at a certain rotation angle θ for counting the fractional part of the interference fringe shift number. On the other hand, the fringe spacing can be enlarged to an appropriate size by rotating the cube around an axis orthogonal to a plane of illustration in Fig. 1 in order to improve the accuracy of the measurement. The proposed single-element interferometer is depicted in illustration of Fig. 1. It shows the ray trajectories in the beamsplitter cube (BS). A light wave is incident along the z-axis on the beam-splitter cube with its central semi-reflecting layer placed along the propagation direction. The top half of the light beam
15
(Path 1) acts as the reference light and the other bottom half beam (Path 2) as the object light, or vice versa. Let the electric fields of two halves of the input beam at the plane ∑in be E1(x0 , y0 ) and E2(x0 , y0 ), the interference of the transmitted and reflected lights at the plane ∑out is obtained, and then intensity patterns of the Output 1 and the Output 2 are represented as follows: I Output 1 ¼ jE1 ðd x; yÞexp½iπ=2 þ E2 ðx d; yÞj2 ¼ jE1 ðd x; yÞj2 þ jE2 ðx d; yÞj2 þ 2jE1 ðd x; yÞjjE2 ðx d; yÞj cos ðξ1 ðd x; yÞ ξ2 ðx d; yÞ þ π=2Þ
ð1Þ
I Output 2 ¼ jE1 ðd þ x; yÞ þ E2 ð x d; yÞexp½iπ=2j2 ¼ jE1 ðd þ x; yÞj2 þ jE2 ð x d; yÞj2 þ 2jE2 ðd þ x; yÞjjE2 ð x d; yÞj cos ðξ1 ðd þ x; yÞ ξ2 ð x d; yÞ π=2Þ
ð2Þ
where ξ1, ξ2 are the phases of the fields E1, E2, respectively. (We neglect the splitter-ratio of the beam-splitter cube in above expressions.) When the incident light is plane wave of constant amplitude (E0), we have E1 ðx0 ; y0 Þ ¼ E0 ;
E2 ðx0 ; y0 Þ ¼ E0 exp½iξðx0 ; y0 Þ
ð3Þ
where ξðx'; y'Þ is phase distribution of the sample placed in Path 2. If the beam-splitter ratio is 50:50, Eqs. (1) and (2) can be rewritten as I Output 1 ¼ I 0 f1 þ cos ½ξðx d; yÞ π=2g ¼ I 0 f1 þ sin ξðx d; yÞg
ð4Þ
I Output 2 ¼ I 0 f1 þ cos ½ξð x d; yÞ þ π=2g ¼ I 0 f1 sin ξð x d; yÞg ð5Þ E20 .
where I 0 ¼ It is easy to see that the optical path difference between the two beams is changed when rotating the sample. Let the refractive index of a transparent liquid sample with thickness d be n, and rotation angle of the sample be θ, the interference fringe shift number N is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 N ¼ ð n2 n20 sin θ n0 cos θ n þ n0 Þ d=λ ð6Þ where n0 ¼ 1 is the refractive index of air, λ is the optical wavelength. We can obtain the expression of the refractive index
Fig. 1. Experimental setup. The illustration shows ray trajectories in the beam-splitter cube. d is the size of the working region in the x-direction.
16
T. Zhang et al. / Optics Communications 332 (2014) 14–17
of the liquid as follows: n¼
ðNλ=dÞ2 þ 2ð1 cos θÞðn20 n0 Nλ=dÞ 2½n0 ð1 cos θÞ Nλ=d
ð7Þ
3. Results and discussion During the measurement the environment temperature is kept stable at 25 1C. By recording the rotation angle θ and the interference fringe shift number N, we can calculate the refractive index of the liquids with Eq. (7). Table 1 lists the measured refractive indices of two different liquids. Comparing the measured refractive indices ns with the reference values nref yields a deviation of 0.0001 for ethanol and a deviation of 0.0002 for distilled water. To demonstrate the refractive index measurement capabilities of this technique, results of measured refractive index of NaCl– water and ethanol–water at different concentrations are shown in Figs. 2 and 3, respectively. Both of the black points are the measurements from the literature [17], but at a wavelength of 589 nm, where the refractive index is slightly lager. A very good agreement was found between all measured values and those reported in the literature. Fig. 2 shows a linear dependence on the concentration, on the contrary as shown in Fig. 3, which exhibits a strong nonlinear dependence on the concentration [17]. The uncertainty of the refractive index measurement made by the proposed method primarily depends on the errors in the thickness of the cell d, the rotation angle θ, and the interference fringe shift number N. The effects of these error sources are separately discussed below. Partially differentiating Eq. (7) with respect to θ, d and N, we have ∂n Nλ ðNλ=d 2Þ sin θ ¼ ∂θ d 2ð1 cos θ Nλ=dÞ2
ð8Þ
∂n Nλ Nλ sin 2 θ ¼ 2 ∂d 2d 2½dð1 cos θÞ Nλ2
ð9Þ
Table 1 Measured refractive indices of two liquids. Liquid
ns
nref [15, 16]
Distilled water Ethanol
1.33157 0.0002 1.35867 0.0001
1.3316 1.3587
Fig. 2. Refractive index of NaCl–water solution vs. concentration.
Fig. 3. Refractive index of ethanol–water solution vs. concentration.
Table 2 Sources of error in the measurement of refractive index of two liquids. Liquid
σθ
σd
σN
σn
Distilled water Ethanol
3.1 10 5 2.9 10 5
2.4 10 4 2.6 10 4
4.5 10 5 4.7 10 5
2.4 10 4 2.6 10 4
∂n λ 2ð1 cos θÞðNλ=d þ cos θÞ ðNλ=dÞ2 ¼ ∂N d 2ð1 cos θ Nλ=dÞ2
ð10Þ
It is obvious that the accuracy of the measured index is determined by the relative error of d, θ, and N. In distilled water, for example, substituting θ¼201, d ¼10 mm and N ¼ 245.3463 into Eqs. (8)–(10) yields |∂n/∂θ| ¼2.6564, |∂n/∂d| ¼24.4804, |∂n/∂N| ¼ 0.0018, respectively. The resolution of rotary table is 0.000671, so the angular deviation σθ ¼0.000671¼ 1.1694 10 5 leads to the deviation of the refractive index measurement with Δn ¼(|∂n/∂θ|) σθ E3.1 10 5. Employing a rotary table with high accuracy can greatly reduce the angular deviation. The thickness of the rectangular cell deviation σd ¼ 0.01 mm, it leads to the deviation of the refractive index measurement with Δn ¼(|∂n/∂d|)σd E 2.4 10 4. As measured fringe spacing is approximate 40 pixel of CCD, the minimum resolution of the interference fringe shift number is σN ¼1/40 ¼0.025; therefore it effects on the refractive index with Δn ¼(|∂n/∂N|)σN E5 10 5. Accordingly, the combined relative deviation of the refractive index measurement is σn ¼ [(3.1 10 5)2 þ(2.4 10 4)2 þ(5 10 5)2]1/2 E2.4 10 4, and the refractive index can be accurately determined to the fourth decimal place. The deviations of ethanol are shown in Table 2. Table 2 lists the sources of error in the measurement of refractive index of two liquids. We think the measurement error owing to the liquids wetting the glass wall of the cell will affect the measurment accuracy, but the impact is very small so we did not consider it in this paper. In addition, we also studied the device's capability to measure the temperature-dependent refractive index of liquid. Samples are kept in the original position with rotation angle θ¼0. A relative measurement method [18] was adopted, we assumed that n1 is the refractive index of liquid at temperature T, so n0 ¼n1 þΔn is the refractive index of liquid at temperature T þΔT. Since the refractive index change Δn caused by the temperature change, which leads to the optical path difference between the two beams, corresponds to the interference fringe shift number N, we neglect the influence
T. Zhang et al. / Optics Communications 332 (2014) 14–17
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change can be observed, which is in good agreement with the measurement data reported previously [15]. To assess the stability of the proposed experimental setup against environmental disturbance such as mechanical vibrations, interferograms were continuously captured in the laboratory situation in a period of 40 min at intervals of 1 min. The fluctuation of the offset distance of interference fringe pattern over 40 min is shown in Fig. 5. The maximal deviation of the fluctuation during 40 min is 2.0014 pixels, which corresponds to the combined relative error of the refractive index measurement with 9 10 5. Obviously, the high stability and repeatability mainly can be attributed to the common-path property of the interferometer. 4. Conclussion
Fig. 4. The refractive index of distilled water vs. temperature. The solid lines are cubic polynomial fits to the measured data.
In conclusion, we demonstrated the feasibility of a stable laser refractometer based on a single-element interferometer constituted by a beam-splitter cube for highly sensitive refractive index measurement. The refractive index is measured by counting the interference fringe shift number when rotating the sample. The device was evaluated for refractive index measurement of various transparent liquids and the results matched well with the literature values. Moreover, once the relationship between concentration and refractive index is measured, the same technique can be used to measure the concentration of solutions. The proposed method requires no sensors to be immersed in liquid. Furthermore, the proposed method is very easy to implement in a laboratory with simple apparatus, with low cost, noncontact, and is of high precision, high, accuracy and quite insensitive to the atmospheric turbulence and vibration.
Acknowledgments This work was supported by Major Program of National Natural Science Foundation of China (60890200) (10976017). References Fig. 5. Stability of the experimental setup.
of the sample thermal expansion. ðn0 nÞd ¼ Δnd ¼ Nλ
ð11Þ
Δn ¼ Nλ=d
ð12Þ
The change was then added to the refractive index value at T to calculate the refractive index of liquid at that temperature. We measured the refractive index of distilled water at 25 1C, then made distilled water with a specific temperature dropped smoothly to 25 1C, and computed the refractive index change using Eq. (12) by measuring the interference fringe shift number during temperature changes. The refractive index of distilled water at that temperature was calculated by adding the change to the refractive index of distilled water at 25 1C. Thus the refractive index of distilled water at temperature from 25 1C to 65 1C at intervals of 2.5 1C was measured. Fig. 4 depicts the results indicating the dependence of the refractive index of distilled water on the increasing temperature. The nonlinear decrease of the amount and shape of the measured refractive index of distilled water
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