Polarized low-coherence interferometer based on a matrix CCD and birefringence crystal with a two-dimensional angle

Junfeng Jiang; Mengnan Xiao; Shuang Wang; Kun Liu; Xue Wang; Tiegen Liu

doi:10.1364/OE.25.015977

1. Introduction

Low-coherence interferometry, which was first proposed in the field of fiber optic sensing by Al-Chalabi et al. in 1983 [1], has become an important technique for absolute optical path measurement. This technique is widely used in optical coherence tomography [2], three-dimensional profiling [3] and optical fiber sensing applications. The physical parameters, such as pressure [4], temperature [5] and refractive index [6], etc., can be measured by interrogating the changed optical path with a high accuracy.

In order to improve the measured range, precision and stability, a variety of low-coherence interferometer (LCI) configurations have been explored. A typical LCI system includes a sensing interferometer and a demodulation interferometer. The latter is used to introduce an appropriate range of the optical path difference (OPD) related to the former. When the OPD in both of the interferometers matches, the low-coherence interferometric fringes appear [7]. Mechanical scanning LCI and electronic scanning LCI are two major kinds of approaches to obtain the correlogram of low-coherence interference. The mechanical scanning LCI commonly need a mechanical movement component in the demodulation interferometer, e.g., the reference mirror in a Michelson interferometer [8], to scan in time and achieve a proper OPD range. While the electronic scanning LCI, such as Fizeau interferometer [9], adapted Michelson interferometer [10], PLCI [11] and so on, scans the interferogram in space and project the interference pattern onto a linear charge-coupled-device (CCD) array. Without moving parts, the electronic scanning LCI usually has high mechanical stability and relatively low cost. The PLCI, which utilizes the birefringent wedge and polarize technology to produce the OPD, reduces the impact of the reflector surface quality and avoids the change in the relative angle between reflectors. Thus, the PLCI is especially suitable for the engineering applications. However, the measured range and resolution are restricted by the pixel count of a linear CCD.

The pixel count of a matrix CCD is hundreds of times as many as that of a linear CCD. Hence, we proposed an electronically scanned PLCI based on a matrix CCD and birefringence crystal with two-dimensional angle to obtain two-dimensional low-coherence interference fringes. Based on a designed two-dimensional angle of birefringence crystal, the compression and enlargement of interference fringes can be acquired at the same time, which are realized in different dimensions of the fringes pattern. As the fringes pattern is comprehensive demodulated by different algorithms, we can obtain a wide measurement range and guarantee low measurement deviation and high resolution. The presented system is then successfully applied to displacement measurement by using a high precision nanopositioning stage to generate varying OPD.

2. Theory and experimental setup

Figure 1 shows the experimental setup of the PLCI based on a matrix CCD and birefringence crystal with two-dimensional angle. The sensing interferometer is composed of the fiber end face and the glass surface. Light from a broadband source is guided into the sensing interferometer through a 2 × 1 fiber coupler. The output optical signal from the sensing interferometer launches into the demodulating interferometer, which consists of a collimation lens, a polarizer, birefringence crystal with two-dimensional angle, and an analyzer. The polarization axis of the polarizer and analyzer have a 45° angle with the optical axis of the birefringence crystal. When the OPD caused by the birefringence crystal is equal to that caused by the sensing interferometer, i.e., at the position of the matching-OPD, the low-coherence interference fringes (LCIF) appears. A matrix CCD is then used to catch the LCIF.

Fig. 1 Experimental setup of the proposed interferometer based on a matrix CCD and birefringence crystal with two-dimensional angle.

Download Full Size | PDF

When the light source spectrum conforms to the Gaussian shape, it can be expressed as

S (k) = \frac{2 \sqrt{\ln 2}}{\sqrt{π Δ k}} \exp [- \frac{4 \ln 2 {(k - k_{0})}^{2}}{Δ k^{2}}],

where

k

is the wave number,

k_{0}

is the central wave number, and

Δ k = 2 π Δ λ / λ_{0}^{2}

, where

λ_{0}

is the central wavelength,

Δ λ

is the full width at half maximum (FWHM) of the power spectrum in the wavelength domain. The fringe intensity of the low coherence interference can be described as

\begin{array}{l} I (d, h) = \int_{0}^{+ \infty} S (k) \cos [k (Δ n d - 2 h)] d k \\ \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \begin{matrix}  \end{matrix} = \exp [\frac{- Δ k^{2} {(Δ n d - 2 h)}^{2}}{16 \ln 2}] \cos [k_{0} (Δ n d - 2 h)] . \end{array}

where

d

is the thickness of the birefringence crystal,

Δ n

is the refractive index difference between ordinary light (o light) and extraordinary light (e light), h is the distance between the fiber end face and the glass surface.

The key of the proposed system is the design of two-dimensional angle of the birefringence crystal. Figure 2 shows the schematic diagram of the birefringence crystal with two-dimensional angle. Since the birefringence effect of the crystal, light in the crystal is divided into o light and e light, and the OPD $l (x, y)$ produced by the two components can be expressed as

l (x, y) = d (x, y) Δ n,

where x and y are the horizontal distance and vertical distance from the incident point to the vertex of the birefringence crystal, respectively. The wedge thickness

d (x, y)

at the incident point is given by

d (x, y) = x \tan (α) - y \tan (β) + d_{0},

where

α

and

β

are the wedge angles in horizontal and vertical dimension, respectively.

d_{0}

is the thickness at the vertex of the crystal.

Fig. 2 Schematic diagram of the birefringence crystal with two-dimensional angle.

Download Full Size | PDF

At horizontal dimension, the interference fringes are significantly compressed by designing a large angle $α$ . Thus the wide measurement range is realized. At vertical dimension, through the design of a small angle $β$ , part of the interference fringes is broadened, which guarantees the low measurement deviation and high resolution. However, using a single demodulation algorithm is difficult to achieve good demodulation of the two-dimensional signal at the same time. Therefore, an algorithm which consists of two steps is proposed. Firstly, we need to extract a line of signal in horizontal dimension. In order to ensure low demodulation error, the line should be selected in areas with high fringe contrast, such as the central area of interference pattern. The thickness of the birefringence crystal along the extracted line can be expressed as

d (x, y_{0}) = x \tan (α) - y_{0} \tan (β) + d_{0}

Due to sub-sampling at horizontal dimension, spatial-frequency domain analysis (SFDA) algorithm [12] is used to demodulate the signal after preprocesses of de-noise. An estimated position of the matching-OPD on the extracted line can be obtained by the first step of the algorithm, which corresponds to one pixel on the extracted line. Then go to the second step of the algorithm. The column which contains the pixel obtained by the first step is extracted for further analysis. The thickness along the column can be expressed as

d (x_{e p}, y) = x_{e p} \tan (α) - y \tan (β) + d_{0},

where

x_{e p}

is the estimated position of the matching-OPD gotten by the first step. Since the angle

β

is very small, the thickness of the birefringence crystal varies very slowly along the vertical dimension, which leads to fine-sampling at the column and part of the interference fringes is missing. By using the phase-shifting algorithm to process the extracted column, we can obtain a more accurate position of the matching-OPD

y_{a p}

along the analyzed column, which also corresponds to one pixel of the analyzed column. Theoretically, the more accurate position

y_{a p}

is very close to the position

y_{0}

proposed in the first step of the algorithm. Finally, the accurate position

x = x_{e p}, y = y_{a p}

of the matching-OPD can be obtained. The thickness of the birefringence crystal at the special point can be expressed as

d (x_{e p}, y_{a p}) = x_{e p} \tan (α) - y_{a p} \tan (β) + d_{0}

and the distance between the fiber end face and the glass surface is directly related to the OPD produced by the birefringence crystal, which can be expressed as

h = \frac{1}{2} d (x_{e p}, y_{a p}) Δ n .

The resolution of each pixel on horizontal and vertical dimensions of the matrix CCD can be respectively expressed as

Δ l_{x} = Δ n p_{x} \tan (α),

Δ l_{y} = Δ n p_{y} \tan (β),

where

p_{x}

and

p_{y}

represent the distance between the adjacent pixel centers in horizontal and vertical dimensions, respectively.

When $p_{x} = p_{y}$ , it is clear that $Δ l_{x} / Δ l_{y} = \tan (α) / \tan (β) = m$ , i.e., $Δ l_{x} = m Δ l_{y}$ . Obviously, in horizontal dimension of the matrix CCD, the interference fringes will produce a pixel shift every m lines. Correspondingly, the interference fringes between two adjacent columns in vertical dimension produce a shift of m pixels. The feature that the interference fringes moves at regular interval can be utilized to average denoising in a single frame signal.

In order to demonstrate the proposed concept, the PLCI based on a matrix CCD and birefringence crystal with two-dimensional angle is set up. The light source is a SLED (Hoyatek) with a central wavelength $λ_{0}$ of 755 nm and the FWHM is 19 nm. The birefringence crystal made of calcite (CaCO₃) is used, and the refractive index difference between o light and e light at 755 nm $Δ n_{755} \approx 0.1678$ [13]. The angles in horizontal and vertical dimension of the birefringence crystal are 11.98° and 0.64°, respectively. A piece of glass mounted on a high precision nanopositioning stage (PI, P-752.11C) forms a reflective surface. The nanopositioning stage has 0.03% linearity in close loop. It is about 4.5 nm over a closed-loop travel of 15 μm. The demodulation interferometer and sensing interferometer are connected via a 3 dB coupler, and the transmission fiber type is a single mode optical fiber with a core/cladding diameter of 5/125 μm. The interferogram is detected by a matrix CCD (Microvision, MV-EM510C/M) with 2456*2058 pixels, and the size of each pixel is 3.45 μm*3.45 μm.

Figure 3(c) and 3(d) show the simulation result, in which the parameters are set the same as the actual experimental setup, and d₀ is set to 300 μm, h is set to 90 μm. Figure 3(a) and 3(b) show the simulation result when vertical angle is 0°. From the simulation results, we can see that the interference fringes create an angle with horizontal dimension, due to the vertical angle of the birefringence crystal. Fringes appear in vertical dimension, and the fringes become denser as the vertical angle increases. Figure 3(e) shows the fringes of line 100 and line 1999 in Fig. 3(d). Since $\tan (11.98 °) / \tan (0.64 °) \approx 18.99$ , the interference fringes produce a pixel shift every 18.99 lines. With the central fringe maximum (CFM) as a mark, we can see that the interference fringes of line 100 and line 1999 have a translation of 100 pixels. Correspondingly, the interference fringes of two adjacent columns in vertical dimension have a translation of 18.99 pixels. Column 1060 and column 1100 in Fig. 3(d) are shown in Fig. 3(f). Similarly, the maximum value of the interference fringes is marked. Since there is no noise, the maximum value of column 1060 is obviously located at the 100th pixel in the column, which has a translation about 760 pixels from column 1100.

Fig. 3 Simulation results: (a)(b)When vertical angle of the birefringence crystal is 0°. (c)(d)When vertical angle of the birefringence crystal is 0.64°. (e) The interference fringes of line 100 and line 1999 in (d). (f) The interference fringes of column 1060 and column 1100 in (d).

Download Full Size | PDF

3. Experiments results and discussion

The experiment was carried out using the above-mentioned system to verify the feasibility of the proposed method. The position of the nanopositioning stage ranged from 0.5 μm to 14.5 μm, and the step was 0.2 μm. As the limitation of the light source power, it is necessary to increase the integration time of the matrix CCD. Of course, the shorter the integration time is, the less the effect caused by the environmental drift. In the experiment, when the integration time of the matrix CCD is in the range from 10 ms to 30 ms, there is no significant effect on the demodulation result. Below or beyond this range, the fringe contrast will be reduced or the intensity will be saturated.

Figure 4 shows the interferogram detected by the matrix CCD when the nanopositioning stage at 0.5 μm. In the experimental results, the number of fringes in vertical dimension is less than the simulation results. There are two main reasons. First, the machining error of the birefringence crystal is inevitable. Besides, the bottom surface of the birefringence crystal is not absolutely parallel to that of the CCD, which will also lead to the deviation. By demodulation processing, the deviation has no significant effect on the demodulation results.

Fig. 4 Interferogram detected by the matrix CCD.

Download Full Size | PDF

Taking the two-dimensional interferogram captured when the nanopositioning stage at 0.5 μm as an example, the two-step demodulation algorithm will be explained in detail. Here we assume that the pixel at bottom left corner of the matrix CCD is the coordinate system origin, and the corresponding birefringence crystal thickness is considered to be zero. First, line 1020 of the interferogram is extracted for analysis, as shown in Fig. 5(a). As the horizontal angle of the birefringence crystal is large, the interference fringes are greatly compressed, which leads to the number of sampling points on every fringe decrease. The envelope detection method is not applicable due to the distortion of the envelope. After preprocesses of filter, the SFDA-based algorithm is used to determine the estimated OPD of the air cavity. Details of this method are discussed elsewhere [12] and thus are not elaborated here. Birefringence dispersion compensation is also considered in the demodulation process [14]. Then an estimated position $N_{x} \approx 1454$ of the matching-OPD on line 1020 can be obtained, as shown in Fig. 5(c). After that, column 1454 is extracted for further analysis. The original data and filtered data of column 1454 are shown in Figs. 5(b) and 5(d), respectively. The small angle of the birefringence crystal in vertical dimension enlarges the local fringes, resulting in an increase in the number of sampling points on every fringe. As line 1020 was extracted for analysis in the first step of the algorithm, we focus on analyzing the fringe which contains the 1020th pixel in this column. Five-step phase-shifting algorithm [15], phase unwrapping, and linear fit are then used to determine the accurate position of matching-OPD, as shown in Fig. 5(d). Then a more accurate peak position $N_{y} \approx 1010 .35$ of the analyzed fringe can be obtained. Therefore, the position of matching-OPD when the nanopositioning stage at 0.5 μm corresponds to pixel $(N_{x}, N_{y})$ on the CCD plane. According to Eqs. (7) and (8), when the position of the nanopositioning stage changes,the displacement measurement can be obtained.

Fig. 5 (a) Original data of line 1020. (b) Original data of column 1454. (c) Filtered data of line 1020 and the estimated position of matching-OPD. (d) Filtered data of column 1454 and the accurate position of matching-OPD.

Download Full Size | PDF

As the nanopositioning stage moved toward the fiber end face, and the position changed from 0.5 μm to 14.5 μm at an interval of 0.2 μm, finally we got 71 frames of interferogram. The 71 frames of interferogram are all processed by the proposed demodulation algorithm. When only the horizontal demodulation, i.e., the first step of the proposed demodulation algorithm, is performed, the demodulation result is equivalent to that of using a traditional PLCI based on a linear CCD. When line 1020 of the 71 frames was extracted for analysis in the first step, Fig. 6(a) shows the two-dimensional demodulation result using the proposed algorithm and horizontal demodulation result using the SFDA-based algorithm. As the two-dimensional demodulation results have some correction based on the horizontal demodulation results, the results of the two demodulation methods are close. The linear fit is also shown in Fig. 6(a). The slopes of the two fitting lines are −0.9996 (two-step demodulation) and −1.003 (horizontal demodulation), respectively. It is clear that the two-step demodulation method is more accurate for the displacement measurement. The measurement deviations of the two demodulation results are shown in Fig. 6(b). The standard deviation (SD) of horizontal demodulation is ± 6.19 nm, while the SD of the two-step demodulation is ± 3.67 nm. From the comparison of measurement deviations, it is obvious that the measurement deviation of two-dimensional demodulation is nearly half of the horizontal demodulation. To the position of the nanopositioning stage, the measurement deviation of the two-step demodulation method is within ± 7.00 nm (the nanopositioning stage has about 4.5 nm linearity over a closed-loop travel of 15 μm). It proves that the proposed system with the two-step demodulation algorithm can guarantee the low measurement deviation and greatly expand the measurement range.

Fig. 6 When line 1020 is selected: (a) Two-dimensional demodulation result using the proposed algorithm and horizontal demodulation result using the SFDA-based algorithm. (b) Measurement deviations of the two demodulation results. When line 1010 is selected: (c) (d).

Download Full Size | PDF

Since the optical intensity on the CCD image plane is Gaussian distribution, in order to get a high fringe contrast, a line is selected in the center area of the matrix CCD. But no matter which line with high fringe contrast is selected, there is no obvious effect on the demodulation results. To demonstrate this point, for each of the 71 frames of interferogram, we select line 1010 in the first step to demodulate. The demodulation results are shown in Figs. 6(c) and 6(d). Compared with the demodulation results in Figs. 6(a) and 6(b), the displacement and measurement deviation are similar.

In order to reflect the measurement resolution of the proposed system, a standard block made by MgF₂ with a thickness of 7.5 mm was used instead of the sensing interferometer as a means for generating a fixed OPD. Place the standard block and the proposed system in a constant temperature environment. The data obtained within 40 seconds is shown in Fig. 7. The demodulated equivalent OPD is varied within 3.5 nm, and the standard deviation is 0.84 nm. The measurement resolution estimated as triple standard deviation is about 2.52 nm [16,17].

Fig. 7 Demodulation result of the standard block.

Download Full Size | PDF

In the proposed system, the number of pixels in horizontal dimension of the matrix CCD is 2456, and p_x = 3.45 μm. The horizontal angle of the birefringence crystal $α \approx {11.98}^{°}$ . Therefore, the measurement range of the proposed system can reach ~301 μm in theory. Since the measurement resolution is 2.52 nm, the dynamic range of the proposed system is approximately 101 dB. As the fringes in horizontal dimension are too dense to achieve low measurement deviation, the two-dimensional demodulation solves the problem. Thus, compared to the traditional PLCI based on a linear CCD, the proposed system can significantly expand the measurement range on the premise of guaranteeing low measurement deviation and high resolution.

4. Conclusion

In this paper, a PLCI based on a matrix CCD and birefringence crystal with two-dimensional angle is proposed. The proposed system can obtain two-dimensional interferogram for displacement measurement. By designing the two-dimensional angle of the birefringence crystal, the proposed system can greatly improve the measurement range and guarantee low measurement deviation and high resolution. The measurement range can reach ~301 μm with the measurement deviation kept within ± 7 nm, and a measurement resolution of 2.52 nm can be achieved.

Funding

National Natural Science Foundation of China (No. 61505139, 61227011, 61378043, 61675152 and 61475114); National Instrumentation Program of China (No. 2013YQ030915); Tianjin Natural Science Foundation (No.13JCYBJC16200, 16JCQNJC02000); Shenzhen Science and Technology Research Project (No. JCYJ20120831153904083); National Basic Research Program of China (No. 2010CB327802).

References and links

1. S. A. Al-Chalabi, B. Culshaw, and D. E. N. Davies, “Partially coherent sources in interferometric sensors,” in First International Conference on Optical Fibre Sensors (1983).

2. A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1), 67–74 (2002). [CrossRef]

3. T. Dresel, G. Häusler, and H. Venzke, “Three-dimensional sensing of rough surfaces by coherence radar,” Appl. Opt. 31(7), 919–925 (1992). [CrossRef] [PubMed]

4. B. Xu, Y. M. Liu, D. N. Wang, and J. Q. Li, “Fiber Fabry–Pérot interferometer for measurement of gas pressure and temperature,” J. Lightwave Technol. 34(21), 4920–4925 (2016). [CrossRef]

5. K. Li, M. Jiang, Z. Zhao, and Z. Wang, “Low coherence technique to interrogate optical sensors based on selectively filled double-core photonic crystal fiber for temperature measurement,” Opt. Commun. 389(15), 234–238 (2017). [CrossRef]

6. C. Gouveia, M. Zibaiic, H. Latific, M. J. B. Marques, J. M. Baptista, and P. A. S. Jorge, “High resolution temperature independent refractive index measurement using differential white light interferometry,” Sensor. Actuat. Biol. Chem. 188, 1212–1217 (2013).

7. Y. Rao and D. A. Jackson, “Recent progress in fibre optic low-coherence interferometry,” Meas. Sci. Technol. 7(7), 981–999 (1996). [CrossRef]

8. D. F. Murphy and D. A. Flavin, “Dispersion-insensitive measurement of thickness and group refractive index by low-coherence interferometry,” Appl. Opt. 39(25), 4607–4615 (2000). [CrossRef] [PubMed]

9. M. Tomić, Z. Đinović, and S. Petričević, “Fiber-optic pressure sensor based on Fizeau receiving interferometer,” in Proceedings of the 33rd International Convention MIPRO (IEEE, 2010), pp.100–104.

10. F. Depiereux, P. Lehmann, T. Pfeifer, and R. Schmitt, “Fiber-optical sensor with miniaturized probe head and nanometer accuracy based on spatially modulated low-coherence interferogram analysis,” Appl. Opt. 46(17), 3425–3431 (2007). [CrossRef] [PubMed]

11. S. Wang, T. Liu, J. Jiang, K. Liu, J. Yin, Z. Qin, and S. Zou, “Zero-fringe demodulation method based on location-dependent birefringence dispersion in polarized low-coherence interferometry,” Opt. Lett. 39(7), 1827–1830 (2014). [CrossRef] [PubMed]

12. J. Jiang, S. Wang, T. Liu, K. Liu, J. Yin, X. Meng, Y. Zhang, S. Wang, Z. Qin, F. Wu, and D. Li, “A polarized low-coherence interferometry demodulation algorithm by recovering the absolute phase of a selected monochromatic frequency,” Opt. Express 20(16), 18117–18126 (2012). [CrossRef] [PubMed]

13. G. Ghosh, “Dispersion-equation coefficients for the refractive index and birefringence of calcite and quartz crystals,” Opt. Commun. 163(1–3), 95–102 (1999). [CrossRef]

14. S. Wang, J. Jiang, T. Liu, K. Liu, J. Yin, J. Shi, S. Zou, and M. Zhang, “Birefringence-dispersion-induced frequency domain nonlinearity compensation for polarized low-coherence interferometry demodulation,” J. Lightwave Technol. 33(23), 4842–4848 (2015). [CrossRef]

15. K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13(4), 832–843 (1996). [CrossRef]

16. T. Liu, J. Yin, J. Jiang, K. Liu, S. Wang, and S. Zou, “Differential-pressure-based fiber-optic temperature sensor using Fabry-Perot interferometry,” Opt. Lett. 40(6), 1049–1052 (2015). [CrossRef] [PubMed]

17. L. Cao, H. Lin, and V. M. Mirsky, “Surface plasmon resonance biosensor for enrofloxacin based on deoxyribonucleic acid,” Anal. Chim. Acta 589(1), 1–5 (2007). [CrossRef] [PubMed]

Polarized low-coherence interferometer based on a matrix CCD and birefringence crystal with a two-dimensional angle

Abstract

1. Introduction

2. Theory and experimental setup

3. Experiments results and discussion

4. Conclusion

Funding

References and links

Cited By

Figures (7)

Equations (10)

Optics Express