Position-deviation-compensation demodulation method for multi-channel polarized low-coherence interferometry

1. Introduction

Low-coherence interferometry (LCI) is widely used in optical coherence tomography [1], three-dimensional profiling [2] and has been introduced into optical fiber sensing to measure parameters, such as pressure [3], temperature [4] and refractive index [5]. With the development of optical fiber sensing, multi-channel-based array sensing has attracted attention owing to its potential significant advantages, including low cost per sensing point and identical measurement accuracy for all sensor heads [6].

In recent years, several works have been performed for polarized LCI by reason of high mechanical stability and relatively low cost [7,8]. Meanwhile the multiplexing schemes based on polarized LCI also have been extensively devoted to establish compact and cost-effective multi-sensor networks, such as wavelength division multiplexing (WDM) [9] and time division multiplexing (TDM). The WDM uses the light sources with different center wavelength for each Fabry-Perot (F-P) sensor. However, this method is complex to implement and the sensor number is limited by the available LED spectrum. The principle of the TDM is that the light signal is routed into a multiple channels LCI and each sensor signal is separated in time from the others. Compared with the WDM technology, the TDM technology is a simple and practical method to realize the multi-sensor sensing because it can multiplex a large number of sensors in principle and can reduce the cost of whole sensing system greatly. However, the practical application of multi-channel TDM technology confronts the challenge of channel inadaptability of F-P sensor because the light beam emitting positions of sensing channels are different. To solve this problem, a common approach is calibrating a F-P sensor for a specific sensing channel [9], thus there is a one-to-one correspondence between a sensor and a sensing channel. In this case, the flexibility of system and the reusability of sensor are very poor. In addition, in harsh application circumstances such as aviation [10] and oil well [11], the F-P sensors are vulnerable to damage and difficult to repair or replace [12]. Therefore, the F-P sensors should be mounted in advance and it’s necessary to install redundant sensors, which serves as a backup in case the mounted sensor fails. Since one F-P sensor can only be demodulated accurately in the pairing channel, we have to install redundant sensors for each sensing channel, which not only wastes a certain amount of sensor, but also increases the installation difficulty. More serious is that if a sensing channel breaks down, the corresponding sensor is not available anymore, even if the sensor is undamaged. Supposing that the F-P sensor can be demodulated in any channels, we only need to install a handful of redundant sensors for all sensing channels. More importantly, the sensor can keep working even when the pairing channel in failure.

In this paper, we report a new method to improve the channel adaptability of F-P sensor for multi-channel LCI and establish a position-compensation model to illustrate the cause of position deviation between different channels. The method consists of an envelope peak position (EPP) retrieval process and a position compensation process. The EPP retrieval process uses the primitive EPP with low precision to identify the interference order and combines the phase information to recover the precise EPP. Position compensation process compensates the position deviation between sensing channel and reference channel through the position-compensation model. By combining these two processes, the method simplifies interference order identification processing greatly and enables F-P sensors to be demodulated in any channels without repetitive calibrating process while keeping the same high accuracy. Air pressure measurements are performed for demonstrating the effectiveness of the proposed method. The measured maximum measurement pressure error is no more than 0.13 kPa in a 4-channel polarized LCI, which indicated that the method is reliable for improving the channel adaptability of F-P sensors.

2. Theory analysis

The key of the demodulation system is the multi-channel LCI. Figure 1 is a schematic diagram of beam propagation paths in polarized LCI demodulator, which is similar to the structural principle in earlier reports [13]. The analysis of the reference paper based on single channel LCI structure are discussed deeply. However, the single channel model cannot describe the relationship of optical path difference (OPD) between sensing channels and cannot offer the principle basis of the compensation algorithm which will be discussed. Therefore, we extend its conclusion to multi-channel LCI structure.

 

Fig. 1 Schematic of the beam propagation paths in polarized LCI demodulator.

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In Fig. 1, the direction of X-axis is parallel to the charge-coupled device (CCD) surface, the direction of Y-axis is perpendicular to the CCD surface and coincides with the direction of channel 1 ($C_1$) in optical fiber array. In addition, positive angles are produced by the light beam clockwise rotates to normal and the angle marks in the graph are positive values. The polarization directions of polarizer and analyzer are vertical to each other and have a 45° angle with the optical axis of the birefringence crystal. The light beam’s optical path of an arbitrary sensing channel m ($C_m$) in polarized LCI demodulator can be written as

Since the refractive index $n_x$ of O light and E light are different, the geometric optical paths of ordinary (O) light and extraordinary (E) light are different when they project onto the same point on the CCD. In Fig. 1, the E light and O light rays are designated by $L_1$ and $L_2$, respectively. Therefore, there is an OPD caused by polarized LCI demodulator between O light and E light, which designated by ${\Delta }_L$.

Obviously, for each value of x, the corresponding value of ${\Delta }_L$ is determined. However, the relationship between x and ${\Delta }_L$ is difficult to defined by using an explicit function. In fact, the relationship is a set of logical equations consisting of implicit function, which expressed as

Figure 2(a) and Fig. 2(b) show the simulation results. For convenient, we use CCD pixel serial number $K$ to represent the X-value $x$ and the relationship between $K$ and $x$ could be expressed as $x=(K-K_o)\cdot a\text{μm}$, where $K_o$ is the CCD pixel serial number corresponding to the coordinate origin and $K_o=1638$ here, $a$ is the width of each CCD pixel and $a=7\text{μm}$ here. From Fig. 2(b) we can see that the $K$ of different channels are different and there is a position deviation $\Delta K$ between different channels under the same ${\Delta }_L$. According to the simulation result of Fig. 2(a), the relationship between $\Delta K$and ${\Delta }_L$ can be easily derived. Figure 2(c) shows the position deviation between reference channel $s$ and other channels with ${\Delta }_L$ as variable, $s\text{=}4$ here.

 

Fig. 2 (a) Characteristic curves of the $K$ with ${\Delta }_L$ as variable. (b) The partial enlarged image of Fig. 2(a). (c) characteristic curves of the $\Delta K$ between channel 4 and other channels with ${\Delta }_L$ as variable.

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As is known, the position of low-coherence interference fringes on the CCD reflects the OPD information and could be directly used for demodulation [14]. The output intensity of the polarized LCI can be seen as the weighted superposition of all monochromatic interference patterns over the source spectrum range [15]. In order to simplify the analysis, the low-coherence interference fringes on the CCD can be expressed as the following expression [16]

Considering there is a nor negligible position deviation $\Delta K$ between different channels under the same ${\Delta }_L$, in other words, under the same cavity length of F-P sensor, the corresponding EPP of different channels are different. Therefore, the F-P sensor is unable to be demodulated accurately in all channels. For this reason, the position compensation procedure can be performed by following steps:

 

Fig. 3 A simplified sequence flow diagram of Step (2) and Step (3). The red arrows represent solving simultaneous equations.

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The EPP obtained by Fourier transform method for extracting envelope [19] is designated by the primitive EPP $K_p$, which suffers from low precision due to the interference pattern is vulnerable to random noise and light source variation. In order to achieve higher precision, we combine the primitive EPP with the corresponding phase information to eliminate the EPP distortion. The phase distribution of Eq. (5) can be written as $\Phi =-w{z}_0$, where $w$ is the angular frequency in continuous form. Besides, $z_0=z_s+\text{d}z\cdot K_r$, here $z_s$ denotes the ${\Delta }_L$ which is corresponding to the first CCD pixel and $\text{d}z$ is the sampling interval of low-coherent interference fringes in spatial-domain. Owing to Discrete Fourier Transform (DFT) processing, $w$ should be replaced with a specific discrete monochromatic frequency $w_q=2\pi q/\left(N\cdot \text{d}z\right)$, where q is the DFT serial number, N is the total number of DFT serial number and it is equal to the total number of CCD pixels. Therefore, another expression for the phase-frequency characteristic $\Phi$ can be written as

From Eq. (7), we know that $\Phi$ is only related to EPP $K_r$ when we selected a monochromatic frequency q. As is known, the phase obtained by DFT is wrapped into the interval (-π, π) and 2π ambiguity is introduced. The key of the phase unwrapping procedure is to get the interference order. We utilize the low precision primitive EPP $K_p$ to estimate the phase for a given arbitrary interference fringe by ${\phi }_{est}=-2\pi q{K}_p/N$. Then the estimated interference order $l$ is identified by

The interference order is identified by $n=\text{round}(l)$, the $\text{round}()$ function returns an integer which is nearest to the parameter in the bracket. The interference order jump error occurs only if the distortion of the $K_p$ is bigger than $N/(2q)$. Then, the absolute phase $\phi$ could be retrieved by combining the relative phase and the identified interference order n.

Based on this, the EPP after position compensation can be worked out by Eq. (6) on condition that $K_r$ and $\Delta K$ are known.

3. Experimental setup and experiments results

Figure 4 shows the experimental setup of multi-channel optical fiber F-P sensing demodulation system. We put the F-P sensor in an air pressure chamber, in which the pressure can be precisely tuned by a controller with a control precision of 0.02 kPa. The F-P pressure sensor has a silicon diaphragm, which turns the change of external air pressure into the change of F-P cavity length. The light from a LED is launched into the sensor via a 3 dB coupler, and then the reflected signal from the sensor is routed into polarized LCI demodulator through a chosen sensing channel. The polarized multi-channel LCI demodulator, as shown in Fig. 1, has 4 sensing channels via a 4-channel fiber array, which is provided with 4 optical fibers that are arranged at equal interval in a cuboid glass column. The signal light is divided into E light and O light when it goes into the birefringent wedge. The interference fringes appear only if the ${\Delta }_L$ matches with twice the cavity length of the F-P sensor. The interference fringes then are received by a linear CCD array and digitized by a data acquisition card for further processing in a computer.

 

Fig. 4 Schematic of the multi-channel optical fiber F-P sensing demodulation system.

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We chose a channel to establish the relation between the EPP array and the calibration pressure array by linear fitting firstly. Here we chose channel 4 as the reference channel to establish the relation. The pressures were applied from 100 kPa to 200 kPa at interval of 5 kPa. The monochromatic spatial frequency was selected $w_{61}$ ($q=61$) and the total number of linear CDD array (N) is 3000.

Figure 5(a) shows the filtered low-coherence interference fringe and its envelope curve when air pressure is 155 kPa. To obtain the filtered interferogram from the original interferogram received by a linear CCD array, the Fourier transform filtering is adopted [19]. Considering the pattern distortion, the primitive EPP $K_p$ of interference fringes on the CCD has a low accuracy but it could be obtained easily. Figure 5(b) shows the change of the primitive EPP and the recovered EPP when air pressure is increasing from 100 kPa to 200 kPa at interval of 5 kPa, respectively. The primitive EPPs are deviate only slightly from the recovered EPPs, which keep a good linear relationship with air pressure. Obvious “ladder” feature can be seen in Fig. 5(c) and each “ladder” corresponds to an interference order.

 

Fig. 5 (a) The low-coherence interference fringes and its envelope when the air pressure is 155 kPa. (b) Characteristic curves of the primitive EPP and recovered EPP with increasing air pressure, respectively. (c) Characteristic curves of the estimated interference order and real interference order with increasing air pressure, respectively.

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Figure 6 is an illustrative graph interpretation for absolute phase recovery process. In Fig. 6, the estimated absolute phase ${\phi }_{est}$ which is estimated by the primitive EPP is expected to deviate only slightly from the real absolute phase. The phase difference between the estimated absolute phase and the relative phase is about $2\pi l$, where $l$ is the estimated interference order. Then it is easy to obtain the interference order n by rounding $l$ to the nearest integer. According to Eq. (7), the interference order jump error occurs only if the distortion of the primitive EPP is bigger than 24.59 when $q=61$ and $N=3000$. Finally, the real absolute phase could be retrieved by Eq. (9). The high demodulation precision is in fact determined by the relative phase, and the low precise primitive EPP is only used for interference order identification.

 

Fig. 6 Conceptual illustration of the phase recovery process by using the primitive EPP $K_p$ and relative phase $\varphi$ in theory.

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The demodulation experiments were carried out to verify the effectiveness of the proposed method. For each channel, we used the same sensor to collect experimental data independently. Pressures which differed from the calibration experiment were applied from 100 kPa to 200 kPa at interval of 0.5 kPa. Figure 7(a) shows the primitive EPPs of the four channels with increasing air pressure. The primitive EPPs have approximately linear relation to the increasing air pressure and the distortion of the primitive EPPs are obvious. Thus, it is necessary to carry out the EPP retrieval process to get high demodulation precision.

 

Fig. 7 (a) The primitive EPPs of different channels with increasing air pressure. (b) The recovered EPPs of different channels with increasing air pressure. (c) The compensative value $\Delta K$ of different channels. (d) The EPPs of different channels after position compensation.

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Figure 7(b) shows the recovered EPPs of different channels with increasing air pressure and the distortion of the primitive EPPs have been eliminated. There is a position deviation between different channels under the same pressure, which agreed well with the theoretical analysis. According to the principle of position compensation process, the compensation part $\Delta K$ was obtained by the recovered EPPs. Of course, the compensation is unnecessary for reference channel, i.e., channel 4. Figure 7(c) and Fig. 7(d) show the different channels’ EPP compensation values and the EPPs after position compensation respectively. After position compensation, the maximum EPP difference between reference channel and other channels is about 0.826.

We obtained the measured pressures by the relation which established in calibration process. The pressure measurement errors of different channels are shown as Fig. 8(a) and Table 1 summarizes the measurement results of the four channels. From Table 1, the proposed method could maintain a tolerance of no more than 0.13 kPa for all sensing channels. In order to analyze the random error of the measurement, experiments have been carried out for 20 times at the same pressure and the standard deviations of demodulation errors are shown as Fig. 8(b). The maximum standard deviations of demodulation errors of the four channels are 0.053 kPa, 0.046 kPa, 0.049 kPa and 0.076 kPa respectively.

 

Fig. 8 (a) The pressure errors of channel 1-4 within air pressure measuring range. (b) The standard deviations of channel 1-4.

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Table 1. Measurement Results of the Proposed Method

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In general, the proposed method is not only applicative for improving the channel adaptability of optical F-P sensors, but also theoretically suitable for multi-channel polarized LCI whose channels are more than 4.

4. Conclusion

In conclusion, a position-compensation method is proposed to enable F-P sensor to be demodulated in all channels without repetitive calibrating process. The method consists of two parts: an EPP retrieval process and a position compensation process. We have carried out pressure experiments to verify the effectiveness of the proposed method. The results showed the maximum measurement pressure error and standard deviation of demodulation error are 0.13 kPa and 0.076 kPa respectively in a 4-channel polarized LCI, which indicate that our approach can effectively improve the channel adaptability of F-P sensors and simplify the application of F-P sensor in the multi-channel sensing system.