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We present a method for dynamically measuring the refractive index distribution in a large range based on the combination of digital holographic interferometry and total internal reflection. A series of holograms, carrying the index information of mixed liquids adhered on a total reflection prism surface, are recorded with CCD during the diffusion process. Phase shift differences of the reflected light are reconstructed exploiting the principle of double-exposure holographic interferometry. According to the relationship between the reflection phase shift difference and the liquid index, two dimensional index distributions can be directly figured out, assuming that the index of air near the prism surface is constant. The proposed method can also be applied to measure the index of solid media and monitor the index variation during some chemical reaction processes.
© 2015 Optical Society of America
1. Introduction
A method for refractive index measurement with simple setup, large measuring range and high precision is of great significance in fields of material analysis, biologic sensing, and parameter design of scientific instruments. Abbe and Pulfrich refractometers are recognized as typically used apparatus [1]. Recently, various index sensors based on in-fiber devices with high sensitivity have been demonstrated [2–5 ]. But these devices usually need calibration previously and are difficult to achieve the full-field index distribution measurement of an inhomogeneous sample. In addition, the fiber based index sensors have a limited measuring range and the in-fiber devices mostly depend on complicated fabrication techniques.
D. Axelrod et al [6,7 ] demonstrated the total internal reflection (TIR) fluorescence microscopy to be an effective technique in studying cell-substrate contact. Since then, great interest has been expressed in TIR for its further applications in kinds of optical information acquisition [8–11 ]. M. Chiu et al proposed a method for index measurement in a large range, which is based on measuring the phase shift difference between the s- and p-polarization components of light beam in TIR using heterodyne interferometry [12]. But this method suffered from the disability of two dimensional (2D) index distribution measurements. To overcome this drawback, an alternative approach was presented based on TIR and phase-shifting interferometry (PSI) [13]. However, it requires successive phase shift steps and cannot achieve the dynamical measurement of index distribution. Then, a parallel PSI technique was introduced to solve this problem [14]. But the complicated setup and processing algorithm make the method difficult in practical applications.
In this paper, we present an effective method to dynamically measure 2D index distribution in a large range, which employs digital holographic interferometry (DHI) combined with TIR. In recent years, DHI has been widely used in index distribution measurement, flow field visualization, temperature measurement, and physics process monitoring etc [15–19 ], owing to its advantages of non-destructive, high-precision, real-time, and full-field measurement.
In the experiment, the information of the 2D index distribution of mixed liquids on the TIR prism surface is recorded continuously in form of digital holograms using CCD. By numerically simulating the diffraction of the holograms, the information of the reflection phase shift differences can be reconstructed. Since the index value of the air near the prism surface is known as 1, the index distribution of mixed liquids can be calculated directly, according to the relationship between the reflection phase shift difference and the index difference of the liquids and air. The experiment results indicate that this method possesses distinct capabilities and potential advantages.
2. Principles
In digital holography, the hologram recorded by CCD is an interferogram of the object and reference waves, from which the object wave O(x,y) can be numerically reconstructed by means of angular spectrum method [16, 19 ]. According to the principle of DHI, phase difference of the reconstructed object waves between time t (with liquid on the TIR prism surface) and the initial time t = 0s (without liquid on the prism surface) can be expressed as
where, Ot(x,y) and O 0(x,y) depict the complex amplitudes of the reconstructed object waves at time t and the initial time, respectively, and ϕ o t(x,y) and ϕ o0(x,y) are the corresponding phase distributions.According to Fresnel formula, TIR occurs when a light beam is reflected from a high index medium n 1 to a low index medium n 2 and the incident angle θ 1 is larger than the critical angle θ c = sin−1(n 2/n 1). In this case, the amplitude reflection coefficients become complex and can be expressed as
Where, r s and r p are the amplitude reflection coefficients of s- and p-components, and φ s and φ p represent the corresponding reflection phase shifts of s- and p-components, respectively.Equations (2) and (3) illustrate that the reflection phase shifts change with the index of the second medium for certain incident angle and prism, as shown in Fig. 1(a) . As a sample, we measure the 2D index distributions of mixed liquids adhered on the TIR prism surface. The liquid thickness is much larger than the penetration depth of the evanescent field. Therefore, the liquids on the prism surface only modulate the phase profile of the reflected light rather than its amplitude distribution. For a given incident angle θ 1, the index variation of the mixed liquids introduces an additional reflection phase shift difference, which eventually results in phase difference distribution of the reconstructed object waves Δϕ o t(x,y). For s-polarization, considering that the phase shift value φ s0 created by prism-air interface (n 2 = 1, without liquid on the prism surface) is constant, the additional reflection phase shift difference induced by the mixed liquids with index n 2(x,y) can be obtained as
Consequently, the 2D index distribution of the mixed liquids can be calculated byWhere, n 1 = 1.5195 and θ 1 = 72.7332° are known in advance.
Fig. 1 (a) TIR phase shift φ s (black) and φ p (red) versus index of the second medium n 2; (b) liquid index n 2 versus TIR phase shift difference Δφ s between two states; (c) measurement deviation of n 2 versus n 2; n 1 = 1.5195, θ 1 = 72.7332°.
Figure 1(b) illustrates that the liquid index n 2 changes with the reflection phase shift difference between two states (with and without liquid on the prism surface). The curve shows that the index n 2 is a monotonically increasing function of the reflection phase shift difference. It proves the validity of the proposed method for measuring the index distribution of mixed liquids.
To analyze the measurement accuracy of this method, we can calculate deviation of the measured index n 2(x,y) from Eq. (5) as
Where, δ[Δϕ o t(x,y)] denotes the measurement deviation of the phase difference of the reconstructed object waves, which is considered as ~0.25° by DHI [20]. Figure 1(c) depicts the relationship between the index deviation δn 2 and the index n 2.3. Experimental setup
Figure 2 shows the experimental setup for recording digital holograms, which consists of a TIR (rectangular) prism and a Mach-Zehnder interferometer. A vertically polarized thin laser beam with wavelength of 532 nm is expanded, filtered, and collimated via microscope objective MO, pinhole PH, and lens L, respectively. Then it is adjusted as horizontal polarization in free space by a half-wave plate HP, so that it is s-polarization when reflected in the TIR prism. After that it is split into two parts by beam splitter BS1. One of the beams acts as reference beam and the other one as object beam after undergoing TIR through the rectangular prism. The hypotenuse of the prism is upward and in the horizontal plane. The prism is accurately adjusted to a proper height to keep the output and incident beams coaxial. According to geometrical optics, the incident angle θ 1 is 72.7332° for the prism with index of 1.5195 (K9 glass), which ensures that TIR is effective for both prism-air and prism-liquid interfaces. After combined by beam splitter BS2, the reference and object beams interfere with each other in a small angle at the CCD target. The CCD used is a black and white type with 1280H × 960V pixels, and the pixel size is 4.65μm × 4.65μm.
Fig. 2 Experimental setup for recording digital holograms. MO: microscope objective; PH: pinhole; L: lens; HP: half-wave plate; BS1, 2: beam splitters; M1, 2: mirrors.
4. Experiment results and analysis
4.1 Measurement of homogeneous liquids
At first, we measure the indices of homogeneous glycerol-water mixtures with concentration of 40% (n 2 = 1.38413), 60% (n 2 = 1.41299), and 75% (n 2 = 1.43534), respectively. Utilizing the aforementioned experimental setup, a hologram is recorded firstly as the reference one before putting the mixture on the prism surface. Then the second hologram is captured immediately after adhering the mixture on the center of the prism surface. Using angular spectrum method and phase subtraction algorithm [19], the phase difference distributions of the reconstructed object waves are obtained, as shown in Fig. 3 . Clear boundaries between the regions of the mixture and air can be observed, which are attributed to their different reflection phase shifts. Due to the incident angle of 72.7332° in the experiment, the captured images are compressed 0.2968 times in one direction and the elliptic liquid drops are regarded as a distortion of the circle by unequal scale factors in two directions.
Fig. 3 Numerically reconstructed 2D phase difference distributions of homogeneous glycerol-water mixtures with different concentration. (a) 40%; (b) 60%; (c) 75%.
The indices of the mixture and air are identified under certain temperature, so the phase difference in each region of Fig. 3 should be uniform. Also, the index of the air region is unchanged (n 2 = 1) for the cases with and without mixture on the prism surface, thus the phase difference in this region should be zero by means of the phase subtraction operation. However, the actually calculated phase difference value in the air region from the numerically reconstructed object waves may be not zero, due to the unstable experimental environment. We can consider the calculated phase difference in the air region as background noise. So, two rectangle areas in the regions of the mixture and air are chosen to calculate the average phase differences, respectively. Then we subtract the average value in the air region from that in the mixture region to obtain the phase difference Δϕ o t(x, y) eventually.
Table 1 gives the measurement results of the homogeneous glycerol-water mixtures. It is shown that the measurement deviations of the reflection phase shift difference and the index are quite low, which fully prove the feasibility of the proposed method. The results also illustrate that larger index corresponds to higher measurement accuracy. It can be seen from Fig. 1 (c) that the index measurement accuracy increases with n 2 in case that the measurement accuracy of phase difference in DHI is steady. Note here that the incident angle also plays a crucial role in the measurement accuracy of the refractive index. In the experiment, we adjust the prism precisely to ensure that the output and incident beams are coaxial. So we can conclude that the incident angle is certain and the deviation of reflection phase shift difference is the only source of the measurement error.
Table 1. Measurement results of homogeneous glycerol-water mixtures
4.2 Measurement of 2D index distribution
Using the same experimental setup we further measure the 2D index distributions of inhomogeneous mixed liquids. Firstly, we record a hologram at the initial time before putting liquids on the prism surface. Then 112 holograms are recorded successively at the frequency of 7.5 frames per second immediately after adhering 75% glycerol-water mixture and water on the center of the prism surface. Accordingly, we acquire the 2D phase difference distributions of the reconstructed object waves during the liquids diffusion process, as shown in Fig. 4(a) (Visualization 1).
Fig. 4 Measurement results of the 2D index distributions of mixed liquids. (a) Numerically reconstructed 2D phase difference distributions without subtraction of the background noise (Visualization 1); (b) corresponding 2D phase difference distributions with subtraction of the background noise (Visualization 2); (c) calculated 2D index distributions (Visualization 3). The numbers 1-4 correspond the results at the time of 0s, 3.5s, 9.2s, and 15s, respectively.
From Fig. 4(a), we can obviously see that the phase difference in the air region varies as the liquids diffuse. Since the firstly recorded hologram is regarded as the reference one and the recording process of all holograms lasts long time, the disturbance of the experimental condition caused by several factors such as the unstable laser cavity may result in unexpected background noise. Thus, a rectangle area in the air region is chosen to calculate the average phase difference value. Then we subtract this average value from every value in the 2D phase difference distribution to eliminate the background noise, as displayed in Fig. 4(b) (Visualization 2). At last, the 2D index distributions of mixed liquids at different time are calculated according to Eq. (5), as shown in Fig. 4(c) (Visualization 3). The numbers 1-4 in Fig. 4 correspond the results at the time of 0s, 3.5s, 9.2s, and 15s, respectively.
Figure 4 shows that the index distribution of the mixed liquids changes during the diffusion process and approximately close to a certain value when the diffusion finishes. In order to observe how the liquids index varies with time in different regions, we choose two points A and B at top and bottom in the liquids regions in Fig. 4(c), respectively, and the results are plotted in Fig. 5(a) . We can see that the index in top region decreases gradually when water dissolves in the glycerol-water mixture, while the index in bottom region increases when water is adhered on the prism surface and during the liquids diffusion process. Moreover, we choose the central vertical lines in Figs. 4(c2)-(c4) to calculate the corresponding 1D index distributions at certain moments, as shown in Fig. 5(b). It is noticed that the index variations appear at the boundaries between the regions with different concentrations. As can be observed from the curve at 15s, the liquids index decreases slightly from top region to bottom, which is attributed to the corresponding concentration distribution.
Fig. 5 (a) Liquids index variation of two certain points during the diffusion process; (b) calculated 1D index distributions along with the central vertical lines in Figs. 4(c2)-(c4).
5. Conclusions
We have demonstrated an effective method to dynamically measure the 2D index distributions of inhomogeneous mixed liquids in a large range (1-1.4510 in our experiment, depend on the index of the prism) by use of DHI combined with TIR. According to the relationship between the reflection phase shift differences and the index differences of the liquids and air during TIR, the 2D index distributions of mixed liquids can be retrieved by DHI directly, when the incident angle and the indices of the prism and air are given. This method can also be applied to measure the index distributions of solid media as long as the samples can be pasted on the prism surface seamlessly, and it also shows potential in monitoring the 2D index variation during the processes of some chemical reactions and so on.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant No.61127011, 61405164), the Fundamental Research Funds for the Central Universities (Grant No.3102014KYJD025, 3102014JCQ01102) and the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant No.CX201507).
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