Portable device based on beam
deflection for refractive index
mapping and diffusion coefficient
measurement
Vismay Trivedi
Mugdha Joglekar
Swapnil Mahajan
Vani Chhaniwal
Bahram Javidi
Arun Anand
Vismay Trivedi, Mugdha Joglekar, Swapnil Mahajan, Vani Chhaniwal, Bahram Javidi, Arun Anand, “Portable device based on beam deflection for refractive index mapping and diffusion coefficient
Portable device based on beam deflection for refractive
index mapping and diffusion coefficient measurement
Vismay Trivedi,aMugdha Joglekar,aSwapnil Mahajan,aVani Chhaniwal,aBahram Javidi,band Arun Ananda,*
aMaharaja Sayajirao University of Baroda, Faculty of Technology and Engineering, Applied Physics Department, Optics Laboratory,
Vadodara, India
bUniversity of Connecticut, Department of Electrical and Computer Engineering, Storrs, Connecticut, United States
Abstract. Nonuniform refractive index distributions in transparent mediums are of interest as it gives rise to a modification of the probe light beam passing through such mediums. Various properties of the probe beam can be used to quantify the modification happening to the probe beam. One of these properties is the deflection of the beam. This could be used to map and quantify the spatiotemporal evolution of refractive index distribution in such mediums. The deflections could be measured by imaging the deflection of structured line pattern projected through such a system. We describe the development of a compact, portable device for mapping of refractive index distributions as well measurement of the diffusion coefficient of liquid solutions. The method and device are demonstrated by the real-time display of the refractive changes as well as meas-urement of diffusion coefficients in diffusing binary liquid solutions.© 2019 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI:10.1117/1.OE.58.1.014101]
Keywords: diffusion; projection; Fourier image analysis; beam deflection.
Paper 181345 received Sep. 18, 2018; accepted for publication Dec. 6, 2018; published online Jan. 5, 2019.
1 Introduction
Nonuniform refractive index distributions appear in many processes including diffusion, especially in liquid solutions. Measurement and mapping of such refractive index will lead to better understanding of the phenomena involved as well as characterization and quantification of the system under investigation. Diffusion, which is the transfer of mass from one part of a system to another part of the same system due to random molecular motion, occurs in many physical, chemi-cal, and biological processes.1–11It plays an important role in many fields of physics and chemistry, as well as in fields such as mechanical engineering, crystal growth, pollution control, biological systems, separation of isotopes, etc.2–11 Diffusion process in binary liquids is of special interest as the quantification of this process leads to measurement of the properties of the mixture and its characterization. Quantitative measurements of the rate at which a diffusion process occurs are usually expressed in terms of a diffusion coefficient.1 The knowledge of diffusion and the
measure-ment of diffusivity of liquid solutions are important and can lead to detection of contamination of the solution, better design of chemical processing instruments, measurement of absorption process, screening of bioactive components, fine tuning of distillation process, adsorption measurement, qual-ity control of instruments, better design of artificial organs in medicine, design of membranes, etc.1–11The methods used to
measure the diffusion coefficient may be experimental meth-ods or empirical correlations, with experimental methmeth-ods providing accurate results than empirical methods. Among all experimental methods used to map refractive index distribution in diffusion process and the measurement of diffusion coefficient, the optical methods that measure the refractive index distribution or its derivatives are most
successful because they are easy to implement, noncontact, noninvasive, and are immune to harsh and corrosive environments.12–37 Added advantages of optical techniques include fast response time, larger field of view, and possibil-ity of remote measurements. Available optical methods for mapping of nonuniform refractive index include interfero-metric techniques,12–23 holography,24–30 electronic speckle pattern technique,31–34deflectometry,35–38and noninterfero-metric techniques.8,36–39 Interference methods including
holography are the most advanced methods to study mixing of liquid media. Holographic interferometry compares the wavefront interacting with the diffusing medium at two time instances for spatial mapping of the refractive index distribution.24–30 Digital version of holography directly provide the diffusion coefficient from the reconstructed phase profiles of the diffusing solutions at two time instances.27,29–30But the limitations of interferometric meth-ods including holography is that they require very strict con-ditions such as highly temporally coherent (lasers), vibration isolation to filter the external noise, and accurate adjustments of beam ratios for high contrast fringes. In addition, interfer-ence techniques involve many optical elements for splitting and combining of laser beams making them bulky. This makes them difficult to be implemented as portable device especially in harsh environments. Moreover, the need for specialized optical elements makes the setup expensive.
A light beam passing through a region with nonuniform refractive index will be deflected toward regions of higher refractive index.38 A measurement of this deflection can be used to quantify the refractive index distributions in such a region. Here we present a compact, low-cost, portable, noninterferometric optical device for mapping and visualiza-tion of refractive index distribuvisualiza-tions especially in mixing transparent binary liquid solutions and characterization of *Address all correspondence to Arun Anand: E-mail: aanand-apphy@
the resulting solution by the measurement the diffusion coef-ficient of the mixture. The developed device makes use of the principle of light beam deflection from nonuniform refrac-tive index distributions, quantified by the use of a structured line pattern printed on a paper.36,37The device is based on our earlier work,37in which we presented the usefulness of the projection use of printed line patterns for imaging of non-uniform refractive index distributions. The device is further enhanced by Fourier analysis of the imaged pattern through the mixing solutions to provide, fast, real-time visualization of the refractive index distribution as well as provide the diffusion coefficient of the solution. The camera of the device was also coupled to a smartphone and the data be sent to an off-site server that processes the data, computes the diffusion coefficients, and sends the measured values back to the user.
2 Device Based on Beam Deflection for Diffusivity Measurement
When light enters a region with nonuniform refractive index variation, it deflects toward regions of higher refractive index.39The refractive index distribution can be visualized and quantified if the beam deflection can be measured. Spatiotemporally varying refractive index distributions results, when binary liquid solutions diffuse into each other. The proposed refractive index profiling device images and measures diffusion coefficients by quantifying the beam deflection indirectly. Figure 1(a) shows the schematic of the experimental setup employed for the imaging of diffusion of binary liquid solutions.
The device comprises an experimental cell
(2.5 cm × 2.5 cm × 5 cm with longer dimension along the diffusion direction) to hold the solutions under investigation. Solution with lower concentration (C1) is introduced into the cell first and the solution with higher concentration (C2) of the same volume was then introduced below this using a funnel attached to a holding well. The rate at which the heavier concentration introduced is controlled by a stop-clock. The rate is adjusted to avoid a turbulent mixture of the two solutions.
As the solutions diffuse into each other, it will lead to set-ting up of a spatially varying refractive index profile inside the cell at an instant of time. This refractive index distribu-tion will lead to deflecdistribu-tion of light rays interacting with it. To measure the deflection an LED light source (central wave-length is 627 nm and linewidth is 30 nm) in conjunction with a line pattern printed on a normal paper was used. The pattern pasted on the entrance face of the experimental cell comprised a series of dark lines at 45 deg with the diffusion direction [which is assumed to be x as shown in Fig. 1(b)]. The light beam entering the experimental cell contains information about the line pattern. After passing through the medium, the light rays are deflected and this will reflect as a shift in the line pattern from its original posi-tion. The angle of line pattern with respect to the diffusion direction is important since it is impossible to measure the shift in a vertical (angle of 0 deg with respect to diffusion direction) or horizontal line pattern (angle of 90 deg with respect to diffusion direction) due to beam deflection. The shift in the pattern is best visible when the line pattern is at 45 deg with respect to the diffusion direction. Images of the line pattern were recorded using a VGA webcam every 20 s. The webcam was then connected to a smartphone through OTG cable. Smartphone acted as the recording and transmitting device. The recorded patterns were sent to an off-site server (through mobile internet), where the Fourier analysis code (n Python) was deployed.
After analysis, the processed images as well as the com-puted parameters were sent back to the user. The device and its working were tested using aqueous solutions of ammo-nium dihydrogen phosphate, which is a component of many artificial fertilizers.20
3 Diffusion Process
Fick’s second law governs the free diffusion process in a binary liquid system.1,21,22 Considering the diffusion to be one-dimensional along x axis, the solution of Fick’s sec-ond law provides the equation for spatiotemporally evolving concentration:
Fig. 1 (a) Schematic of the device for measurement of diffusion coefficient. (b) Line pattern pasted on the entrance face of the diffusion cell.
EQ-TARGET;temp:intralink-;e001;63;498 Cðx; tÞ ¼C1þ C2 2 þ C1− C2 2 erf x 2pffiffiffiffiffiffiDt ; (1)
where C1 and C2 are concentrations of lighter and heavier solutions, respectively (they are separated at x ¼ 0) and “erf” represents the error function and D is the diffusion coefficient of the solution. The refractive index can be con-sidered as a linear function of concentration:20
EQ-TARGET;temp:intralink-;e002;63;407
nðx; tÞ ¼ mCðx; tÞ þ n0: (2)
Here, n0is the constant, m is the mean value of the derivative of the concentration change.
Figure2shows the spatiotemporal variation in refractive index inside the experimental cell for ammonium dihydrogen
phosphate solution of average concentration of0.9959 mol l−1. It can be seen that as diffusion progresses the refractive index and its gradient changes. When light rays encounter the non-uniform refractive index variation given by Eq. (2), it will deflect toward regions of higher refractive index (Fig. 3). When a line pattern is projected through the cell containing this nonuniform refractive index, this deflection will lead to a shift of the line pattern seen by the sensor.
It should be noted that the line pattern has to be at an angle with the diffusion direction so that this shift can be seen. The amount of beam deflection and hence the amount of shift of the line pattern depends upon two parameters, (i) the refrac-tive index gradient∂n∕∂x (diffusion along x-direction) and (ii) the depth L of the experimental cell. The deflection angle (see Fig.3) is given as39
Fig. 2 Spatial distribution of refractive index distribution inside the experimental cell at several time instances. Ammonium dihydrogen phosphate solution of average concentration 0.9959 mol l−1 was considered for computation of refractive index distribution using Eqs. (1) and (2).
Fig. 3 Schematic of the change in ray deflection with position inside the cell.
Fig. 4 Beam deflection angle at each spatial point inside the cell at two time instancest1¼ 240 s and t2¼ 900 s. Dashed curve (- - - -) in the figure shows the change in deflection angle at each spatial point inside the cell between these two time instances (ammonium dihydro-gen phosphate solution of average concentration0.9959 mol l−1was considered for computation of the deflection angle).
EQ-TARGET;temp:intralink-;e003;63;752 ϕðx; tÞ ¼dx dz¼ L n ∂nðx; tÞ ∂x : (3)
In the above equation, n is the refractive index at each spatial point at time t. Since the gradient of refractive index is
maximum at the interface, the beam deflection will also be a maximum there. Figure4shows the change spatially varying deflection angle as a function of time computed using Eq. (3). Equation (3) can also be used to compute the change in deflection angle (Δϕ) with time.36Figure4also shows the
Fig. 6 Flowchart of the image analysis process.
Fig. 7 (a) Fourier spectrum of line pattern shown in Fig.5and (b) filtered Fourier spectrum containing information about shift of the line pattern.
change in deflection angle between two time instances as given by Eq. (3). The curve for the change in deflection angle between two instances (t1 and t2) has three extreme points. These extreme points are related to the diffusion coefficient through36 EQ-TARGET;temp:intralink-;e004;63;697 D ¼d 2½ð1∕t 2Þ − ð1∕t1Þ 16 lnðt3∕21 ∕t3∕22 Þ ; (4)
where d is the separation between the extreme points and t1 and t2 are the time instances at which the patterns were
recorded. Equation (4) can be used to determine the diffusion coefficient from the change in deflection angle. Analysis of the images recorded provides the deflection angle in terms of line pattern shift. Subtracting the obtained line pattern shifts at two time instances should lead to an image pattern con-taining the three extreme points shown in Fig.4.
4 Image Analysis
The deflection of the light ray will manifest as a spatially varying line pattern. Fourier analysis of images provides information about the spatial frequency component, which can then be filtered (or modified) to yield useful information.
From the Fourier analysis of the line pattern, shift in the line pattern from its equilibrium condition can be determined. This shift is then proportional to the bending angle.
From Fig. 4 and Eq. (3), it can be seen that the beam deflection depends upon the refractive index gradient. Since the beam deflects toward region of higher refractive index, it will reflect as a time evolving shift in the line pattern (Fig.5), compared to the reference line pattern (last image in Fig. 5), where no refractive index gradient exists.
Flowchart shown in Fig. 6 shows the procedure for Fourier analysis of the images (line patterns). Fourier trans-form of one of the images captured at time t1 after the dif-fusion process started generates the Fourier spectrum, which is similar to as shown in Fig.7(a). Obtained Fourier spectra contain information about frequency of the line pattern pro-jected through the chamber. The frequency of the sidebands of the spectrum for the line pattern recorded at time t1(when there was finite refractive index gradient) will contain infor-mation about the shift of the line pattern from the reference/ equilibrium condition. The shift of the line pattern from the reference condition is determined by first inverse Fourier transforming the filtered spectrum and then extracting the phase. This phase of the line pattern at time t1 is subtracted from the phase of the reference line pattern to get the shift of the line pattern and this shift is proportional to the deflection angleϕ1ðx; t1Þ at position x and time t1. Similarly deflection angleϕ2ðx; t2Þ is found by subtracting the phase of the line pattern recorded at time t2, obtained by Fourier filtering, from the reference phase. Difference between ϕ1 and ϕ2, i.e.,Δϕ will provide the information about the shift of the line pattern from reference position between time instances t1 and t2 and will have the three characteristic extremes (Fig. 4).
5 Results and Discussions
In the experiments, aqueous solutions of ammonium dihy-drogen phosphate were used. Initially, a lighter solution was introduced in the experimental cell. A heavier solution of same volume was introduced beneath the lighter solution by the use of the pipette. The time varying line patterns were recorded using the webcam every 20 s.
Figure5shows the variation of the line pattern for average concentration of 0.9959 mol l−1. These recorded images were Fourier analyzed and the phase is extracted. This phase is then subtracted from the phase obtained using the reference line pattern (line pattern for equilibrium) to find the spatial variation (shift) in the line pattern compared to reference pattern (Fig. 8). The beam deflection at each point is proportional to this variation.
Figure9shows the spatiotemporal evolution of diffusion process. From the figure it can be seen that as time pro-gresses the liquids diffuse more into each other trying to reach equilibrium and that with time the shift in the line pat-tern and hence beam deflection decreases. Figure10shows the change in maximum shift in the line pattern as a function of time.
From Figs.9 and 10, it can be seen that when the time involved is large, the concentration everywhere inside the experimental cell becomes average of the concentrations of the two diffusing solutions and hence the spatial variation of refractive index disappears and it becomes uniform throughout the cell. A light beam encountering uniform
refractive index will not be deflected. This results in a line pattern without the spatially varying shift from the refer-ence pattern (last image in Fig. 5). Since in the present method one already has the spatially varying shift in the line pattern (Fig.8), which is proportional to the spatially varying bending angle, a subtraction of the obtained line pat-tern shift at two time instances is expected to contain two extreme points. This is shown in Fig. 11 for various time instant pairs.
In Fig.11for all the plots the initial time instant t1 was 1200 s. Figure11also shows the diffusion coefficients com-puted from the spatial separation between the extreme points. As can be seen from Fig.11, the extreme points in the profile shift outward with time.
Figure 12 shows the separation between the extreme points as a function of time difference (Δt ¼ t2− t1). To have a better visualization of the change in bending angle, Fig.13shows the spatial evolution of the change in pattern shift as a function of different time differences. Figure13 along with Fig. 12 shows that as the diffusion progresses, the separation between the extreme points increases, indicat-ing that the concentration profile movindicat-ing toward the equilib-rium profile.
The measured diffusion coefficient for ammonium dihy-drogen phosphate of average concentration0.9959 mol l−1 was 7.47 × 10−6 0.11 × 10−6 cm2∕s (averaged over sev-eral runs). This is within 2% of the values measured using Gouy interferometer13, which is considered to be one of the most precise methods to measure diffusion coefficient. Since
Fig. 10 Time evolution of maximum pattern shift. Fig. 9 Three-dimensional rendering of the spatiotemporal evolution of diffusion process, which shows that as diffusion progresses, the shift in the line pattern decreases.
the bending angle is proportional to the gradient of refractive index, a numerical integration of the obtained line pattern shift profile will provide the shape of the refractive index distribution (Fig. 14), which shows the time evolution of the refractive index profile computed from the line shift
data. Figure 14 clearly shows the nonuniform nature of the refractive index distribution.
Longer distance of propagation of the light through the experimental cell will lead to larger beam deflection and hence larger shift of the fringe pattern, which could be observed/imaged for longer duration of time. In fact for
Fig. 13 Spatial variation of change in beam deflection as a function of time difference. Initial timet1for all the cases was 1200 s. Fig. 11 Spatial variation of the shift of the line pattern at different time instant pairs. In all the images,
t1was 1200 s.
the experimental cell used in the present study, beam deflec-tions could be observed for almost 9000 s after the introduc-tion of the heavier soluintroduc-tion.
6 Conclusions and Future Scopes
A compact, easy to use, low cost, and field portable device, which can be attached to a smartphone, to monitor spatio-temporal evolution of nonuniform refractive index distribu-tions was developed. It was tested on binary liquid mixtures to image and measure diffusion process. The method images a set of lines projected through the test object. For a binary liquid mixtures and finite time duration, a nonuniform refrac-tive index distribution develops inside the experimental cell, leading to a deflection of the light ray passing through the cell. This will lead to a spatiotemporally varying shift of the line pattern from a reference pattern that was recorded with solution of uniform refractive index inside the cell. This shift in the line pattern was quantified by Fourier image analysis of the recorded line patterns. The patterns were recorded using a webcam and a smartphone. The recorded patterns were sent to an off-site PC, which proc-esses the images in real-time and send the information about the refractive index distribution and measured diffu-sion coefficient back to the user. It was found that as time progresses the shift in the line pattern also changes, indicat-ing change in the deflection angle. Since the imagindicat-ing involves a single Fourier transformation of the recorded images, the beam deflection can be measured and displayed in real-time using this technique. It should be noted that the measurement of diffusion coefficient depends upon the iden-tification of the extreme points of the deflection map, which do not depend upon the period of the line pattern. But the use of line pattern with lower period will be useful for mapping rapidly varying (in space) refractive index variations.
The technique described required only a few low cost optical elements, its implementation is easy, and also it is immune to the mechanical vibrations that usually affect interferometric methods. The proposed technique was also found to provide accurate diffusion coefficient values, comparable to methods based on interference. Due to its compactness and ruggedness, the technique could be imple-mented in industrial environments, where mechanical vibra-tions are expected. From Eq. (3), it can be seen that beam deflection occurs whenever one has a gradient in optical
path length (n × L). So the technique might be useful in the optical path length mapping of transparent technical and biological specimen, which may introduce a change in path length of the probe beam. This spatially varying path length change will lead to a spatially varying probe beam deflection. Presently, work is progressing in this direction.
Acknowledgments
Authors Vani Chhaniwal and Arun Anand would like to acknowledge Abdus Salam International Center for Theoretical Physics (ICTP), Trieste, Italy for regular associate fellowship. DAE-BRNS (2013/34/11/BRNS/504), DST-FIST (SR/FST/PSI-106/2007).
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