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A demodulation algorithm based on absolute phase recovery of a selected monochromatic frequency is proposed for optical fiber Fabry-Perot pressure sensing system. The algorithm uses Fourier transform to get the relative phase and intercept of the unwrapped phase-frequency linear fit curve to identify its interference-order, which are then used to recover the absolute phase. A simplified mathematical model of the polarized low-coherence interference fringes was established to illustrate the principle of the proposed algorithm. Phase unwrapping and the selection of monochromatic frequency were discussed in detail. Pressure measurement experiment was carried out to verify the effectiveness of the proposed algorithm. Results showed that the demodulation precision by our algorithm could reach up to 0.15kPa, which has been improved by 13 times comparing with phase slope based algorithm.
©2012 Optical Society of America
1. Introduction
Low-coherence interferometry (LCI) circumvents the problem of 2π phase ambiguity and has an unlimited measurement range in principle [1]. The ability of absolute measurement with high precision makes it a good choice for three-dimensional surface profiling [2–4] and optical coherence tomography (OCT) [5] in early stage. It has also been introduced to optical fiber sensing field for measurement of many static physical quantity such as temperature, displacement and refractive index [6–8].
Peak position of the low-coherence interference envelope [9–11] and central fringe peak position [3, 12] are commonly used to acquire the shift information of the interferogram for demodulation. A problem involved in those fringe-pattern-based algorithms is the severe interferogram distortion due to factors of birefringence dispersion [13], non-vertical incidence of light, and so on in polarized LCI. Therefore, envelope peak could no longer accurately reflect the shift information, and it also greatly increases the difficulty of central fringe identification. Phase information can also be used for demodulation, for example, in 1995 de Groot et al. proposed spatial-frequency domain analysis (SFDA) algorithm [14] that utilizes phase slope between wavenumber and phase to achieve demodulation. In 1996 K.G. Larkin introduced phase-shifting technique to LCI [1] and many error-compensated phase-shifting algorithms [15, 16] have been proposed to improve the phase retrieval accuracy. In principle, phase-based-algorithms are more sensitive to measurand than fringe-pattern-based algorithms [17]. However, they suffer the same problem of fringe-order ambiguity as conventional phase-shifting interferometry (PSI) and correctly identified fringe-order is required to achieve the exact demodulation. In order to identify the fringe-order, P. Sandoz et al. constructed a ‘ladder-like’ function from the coherence term and computed relative phase to designate different fringe-order under the assumption that the coherence envelope is locally linear and the noise is low enough [3]. In 2002 de Groot et al. improved SFDA algorithm by using the phase gap between phase and coherence information to obtain fringe-order [18] and could dynamically compensate for optical aberrations, distortions, etc. In 2006 S. K. Debnath et al. proposed a spectrally resolved phase algorithm [19] which could narrow the scanning range to 2π through a temporal phase-shifting technique based on different scanning step to adapt for the limited focus depth of Mirau-type LCI. However, the two algorithms are complicated in the analysis of phase gap since they were both developed for three-dimensional surface profiling. In this paper we proposed a simplified demodulation algorithm for optical fiber Fabry-Perot pressure sensing system. Phase unwrapping and the selection of monochromatic frequency were discussed in detail, which is explored little in other literature, to our best knowledge. The algorithm is then successfully applied to the measurement of air pressure.
2. Experimental setup
Figure 1 shows the experimental setup of spatial polarized low-coherence interferometry which is similar to the system demonstrated by Dändliker [20] in 1992, and it needs no mechanical scanning device. Broadband light is guided into a 2 × 2 coupler and part of light reaches an optical Fabry-Pérot (F-P) sensor. The F-P sensor is placed into an air pressure chamber to sense air pressure and transfers it into cavity length. The light signal modulated by F-P sensor, which can be approximated as two-beam interference, reflects back to the coupler. And then it successively passes through a polarizer, a birefringent optical wedge, an analyzer, and eventually projects onto a linear charge-coupled device (CCD). The interference fringes are localized in the limited spatial region where the optical path difference (OPD) caused by the F-P sensor is compensated or equalized with the OPD caused by the thickness of the birefringent wedge. The CCD is used to record the interference fringes. Through a data acquisition card, the signal is digitalized for further processing in computer.
Fig. 1 Experimental setup of spatial polarized low-coherence interferometry used to measure air pressure with a Fabry-Pérot sensor.
Figure 2 shows schematic diagram of the F-P sensor. The silicon diaphragm and the multi-mode fiber end construct a F-P cavity. When air pressure is applied to the F-P sensor, the silicon diaphragm will has elastic deformation, which changes the cavity length. The relationship between the cavity length change and pressure change is [21]:
where Δd is the cavity length change, ΔP is the pressure change, r is the effective radius of the silicon diaphragm, h is its thickness, E and υ are the Young's modulus and Poisson's ratio of silicon, respectively. Equation (1) shows that the cavity length change is proportional to the air pressure.Broadband light source used in demodulation system is light emitting diode (LED), whose power spectrum is shown in Fig. 3(a) . It can be seen as the superposition of two Gaussian distributed spectrums that center on 458nm and 577nm, respectively. In experiment, air pressure increases from 30kPa to 170kPa at interval 1kPa, which is applied by an air pressure chamber with accuracy of 0.02kPa. For layout of the optical wedge and CCD in our experiment, the interferogram shifts from the right to the left on the CCD with the increase of air pressure. Figure 3(b) is a typical low-coherence interferogram acquired with F-P sensor under 100kPa in experiment, which consists of 3000 discrete data points.
Fig. 3 Spectrum of LED used in experiment and a typical interference signal acquired through a digital acquisition card. (a) LED spectrum. (b) Interference signal.
3. Theoretical analysis
When spectral profile of light source is Gaussian distribution, the output intensity of the spatial polarized low-coherence interferometry can be expressed as:
where α, β and γ are constants which are related to the optical system, xₒ is the coordinate on the wedge corresponding to zero-OPD, which shifts linearly with cavity length change Δd and thus is proportional to air pressure. For convenient, we use xₒ to represent air pressure. The interference fringes can be seen as a cosine signal modulated by a Gaussian function which is determined by the coherence length and center wavelength of the broadband light source. The model is also suitable for the spectrum of light source that can be decomposed into several Gaussian functions. In our system, the spectrum of LED can be approximated by two Gaussian functions.In order to simplify the analysis, we define:
Its Fourier transform is F(Ω), which is also a Gaussian function, and Ω is the frequency. According to the time-shift and frequency-shift feature of Fourier transform, Eq. (2)’s Fourier transform can be written as:In spectrum analysis, we only need to analyze the positive frequency components. Hence, the amplitude-frequency characteristic M(Ω) and phase-frequency characteristic φ(Ω) can be respectively written as:
From Eq. (5) and Eq. (6), we can see that M(Ω, xₒ) is also Gaussian function and φ(Ω, xₒ) is a direct proportion function. Especially, the phase slope of φ(Ω, xₒ) is the measurand of air pressure, i.e., xₒ, and we can simply make use of phase slope to achieve demodulation. In practice, the phase obtained by Discrete Fourier Transform (DFT) is wrapped into the interval (-π, π). Theoretical linear relationship between the phase and frequency is hidden in the aliasing phase. Therefore, phase unwrapping is required to reveal the subsistent linearity, which could be easily achieved through recursive comparison based on the theoretical monotonicity. Phase slope is then obtained through a linear fitting between the unwrapped phase and frequency. Phase slope is free of fringe-order ambiguity and easy to obtain. But it is hard to achieve high-precision demodulation because the strictly linear relationship in theoretical analysis could not be guaranteed in practice.
Instead of using the phase slope, we can also select a monochromatic frequency from the light source spectrum and recover its absolute phase for demodulation. It could achieve the same high precision as PSI. The key problem is to identify the interference-order or fringe-order of the monochromatic frequency. If Ωsf is the selected frequency of the interference fringes, then the absolute phase φ(Ω, xₒ) at the small frequency region centered on Ωsf can be written as - xₒΩ. If expressed with interference-order m, it can also be written as:
where Φ(Ω, xₒ) is the relative phase of Ω, the value of m depends on the measurand x0 and the specific Ω.Combing the two ways of absolute phase expression, we can get:
Using DFT processing, the selected frequency Ωsf in continuous form should be replaced with a specific discrete monochromatic frequency Ωk which has relative high amplitude, where subscript k is the DFT serial number. We unwrap the relative phase of the small frequency region centered on Ωk with the relative phase Φ(Ωk, xₒ) as the reference phase, and unwrapped phase is denoted as Φ’(Ωl, xₒ). Let n be the interference-order of monochromatic frequency Ωk, Eq. (8) is then changed to the following expression with discrete frequency being taken into account:
where k-p and k + q are the DFT serial number of starting frequency and ending frequency respectively. The unwrapped phase Φ’(Ωl, xₒ) could be seen as pseudo-absolute-phase. There is an overall 2nπ upwards translation from the theoretical absolute phase to the unwrapped phase. The intercept is then also moved from zero to 2nπ in theory. Therefore, we can easily calculate the interference-order n from the following formula:where T is the intercept obtained from the unwrapped phase linear fit curve with Ωl as variable. The function floor() returns the largest integer which is smaller or equal to the parameter in the bracket. Once the interference-order n is identified, we could get the absolute phase φ(Ωk, xₒ) at the selected frequency Ωk:Remembering the value of φ(Ωk, xₒ) is -xₒΩk and Ωk is fixed, the linear relationship between the measurand and the absolute phase at the selected frequency Ωk is therefore established and demodulation is realized.
Figure 4 is an illustrative frequency-domain graph interpretation for phase analysis of our algorithm. The amplitude-frequency characteristic and phase-frequency characteristic are shown in Fig. 4(a) and Fig. 4(b), respectively. Direct proportion curves ①-⑥ in Fig. 4(b) represent different φ(Ω, xₒ) curves with F-P sensor under different pressure, i.e. the F-P sensor has different xₒ, e.g. x01, x02…x06. Straight curves ①’-⑥’ represent corresponding unwrapped Φ’(Ωl, xₒ) curves after phase unwrapping process based on reference phase Φ(Ωk, xₒ). Curves ①’-⑥’ are parallel to curves ①-⑥, respectively, and the vertical translation is 2nπ. The unwrapped phase curves are divided into many clusters after phase unwrapping. Each cluster represents a specific interference-order n depending on xₒ and Ωk. Phase curves belonging to the same cluster have a common intercept. Only two clusters are symbolically shown in Fig. 4(b). In other words, the change of interference-order n is reflected on the 2π equi-spaced step change of phase intercept. Points A-F in Fig. 4(b) represent different absolute phase at Ωk with the change of air pressure, i.e. xₒ, e.g. x01, x02…x06, while points A’-F’ represent corresponding wrapped relative phase. Figure 4(c) shows the relative phase at Ωk obtained by DFT with xₒ as variable. The relative phase of a specific frequency changes periodically with the monotonous change of xₒ. Combining the identified interference-order n from Fig. 4(b) and the relative phase Φ(Ωk, xₒ) obtained by DFT in Fig. 4(c), we could recover the absolute phase φ(Ωk, xₒ), as shown in Fig. 4(d). The absolute phase curve is what we will use for demodulation.
Fig. 4 Conceptual illustration of amplitude-frequency and phase-frequency characteristics and the transforming relationship between the relative phase and absolute phase with pressure (xₒ) in theory.
The phase unwrapping is an important step for calculation of interference-order n through Eq. (9) and Eq. (10). We introduce a relative phase at a specific frequency Ωk as the reference phase to carry out phase unwrapping, which is different from traditional process. The small frequency range centered on Ωk is divide into two parts: downside frequency range (Ωk-p, Ωk) and upside frequency range (Ωk, Ωk + q). For the downside frequency range (Ωk-p, Ωk), relative phase unwrapping recursive formula is,
For the upside frequency range (Ωk, Ωk + q), relative phase unwrapping recursive formula is,The whole unwrapped phase-frequency curve is then obtained by simply connecting the two unwrapped parts. Figure 5(a) gives amplitude-frequency characteristic of the interference signal obtained by DFT. Figure 5(b) and Fig. 5(c) shows the relative phase before and after phase unwrapping process according to Eq. (12) and Eq. (13) when optical F-P sensor is under 100kPa pressure. The selected monochromatic frequency Ωk is Ω1558, which has been marked out in Fig. 5(c) and its relative phase is chosen as the reference phase for phase unwrapping. The unwrapped phase presents a good monotonic and linear relationship with frequency.
Fig. 5 Part amplitude-frequency characteristic curve obtained by DFT and phase-contrast curves before and after phase unwrapping. (a) Amplitude-frequency characteristic curve. (b) Wrapped phase-frequency curve. (c) Unwrapped phase-frequency curve.
In theory, we could select any monochromatic frequency with enough high amplitude after DFT and recover its absolute phase for demodulation. However, we found that: though the absolute phase of different monochromatic frequency could all be correctly recovered, their linearity with the measurand i.e., air pressure, varies greatly in practice. The frequency corresponding to the center wavelength of the light source does not necessarily have the best linearity, which may result from the dispersion, non-strict-parallel light illumination, and so on. Therefore, the selection of optimal monochromatic frequency is another important procedure in the proposed algorithm.
The absolute-phase-measurand curve is the connection of all the relative-phase-measurand curves with compensation of interference-order. Therefore, the overall linearity of absolute-phase-measurand curve depends on the local linearity of each relative-phase-measurand curve. Through experiments, we found that the difference between linearity of the relative-phase-measurand curves at the same monochromatic frequency but with different interference-order is little. Thus, it is feasible to make a good estimation of overall linearity through only a relative-phase-meaurand curve at any interference-order. So we can select an optimal frequency among the frequencies with enough high amplitude by comparing their relative-phase-measurand curve linearity at any interference-order. Figure 6 demonstrates the linearity difference between different frequencies with the same interference-order. It is impossible to choose a common pressure range that corresponds to a same interference-order for all the frequencies among the small frequency region. We use different air pressure range for different frequency to obtain the same interference-order. In practice, we first find the frequency with the highest amplitude after DFT and it is referred as center frequency here. Then a small frequency region around the center frequency is located by an amplitude threshold. For example, based on the amplitude-frequency characteristic as in Fig. 5(a), frequency region (Ω1547, Ω1572) is preliminarily determined. Next, we make a linearity quantitative comparison of the determined frequencies. The frequency with best linearity is the optimal frequency that we need for demodulation. Table 1 shows the comparative results. According to Table 1, we can see that the frequency Ω1558 has best linearity. Therefore, it is selected as the optimal frequency in our system.
Fig. 6 Linearity comparison charts. (a) DFT serial number is 1548. (b) DFT serial number is 1555. (c) DFT serial number is 1558. (d) DFT serial number is 1567.
Table 1. Quantitative linearity comparison table
4. Pressure measure experiment results and discussion
Pressure measurement over a range of 30kPa–170kPa has been performed at room temperature using the configuration in Fig. 1. The performance of the proposed algorithm is compared with the phase-slope-based algorithm. Figure 7(a) and Fig. 7(b) show the change of phase slope and intercept of the unwrapped phase-frequency linear fit curve with the increase of air pressure from 30kPa to 170kPa at interval 1kPa, respectively. Phase unwrapping is carried out for the small frequency range (Ω1547, Ω1572) using Eq. (12) and Eq. (13). The relative phase of Ω1558 which is the selected optimal frequency is chosen as the reference phase for phase unwrapping. From Fig. 7(a) and Fig. 7(b), it can be seen that the phase slope varies continuously while the intercept varies by step accompanying with the change of air pressure, as we expected from the theoretical analysis. Phase slope keeps a linear relationship with the air pressure on the whole, but linear response is locally distorted, which eventually leads to low-precision demodulation results. The intercept-pressure curve has obvious ladder feature and guarantees the validity of interference-order identification with a strong anti-noise capability.
Fig. 7 The slope and intercept curves obtained by least square fit of unwrapped-phase-frequency. (a) Characteristic curve between slope and pressure. (b) Characteristic curve between intercept and pressure.
Figure 8(a) and 8(b) show the relative phase of Ω1558 which is directly obtained by DFT and the identified interference-order by use of Eq. (10). The identified interference-order is exactly related to the relative phase of Ω1558. The absolute phase is then recovered by combining the relative phase and identified interference-order with Eq. (11), as shown in Fig. 8(c). It keeps a good linear relationship with air pressure whether from the overall or locally zoomed view. Compared with direct phase-slope-based algorithm, linearity of our algorithm has been improved evidently. Their linearity is 0.9998138 and 0.9999599, respectively.
Fig. 8 Relative phase of the selected frequency Ω1558 obtained by DFT, identified interference-order and the recovered absolute phase using the proposed algorithm. (a) Relative phase. (b) Identified interference order. (c) Recovered absolute phase.
Figure 9 shows the demodulation errors between the set pressure and demodulated pressure by use of the two algorithms. The proposed algorithm could maintain a tolerance of less than 0.15kPa or precision 0.11% full scale, while the tolerance of phase-slope-based algorithm is about 2kPa or precision 1.43% full scale. The demodulation precision has been improved 13 times.
Fig. 9 Demodulation error curves between the set pressure and demodulated pressure. (a) Phase-slope-based algorithm. (b) The proposed algorithm.
5. Conclusion
We proposed an algorithm to demodulate the low-coherence interferometry which is used for demodulation of optical fiber F-P pressure sensing. The algorithm is based on the absolute phase recovery of a selected monochromatic frequency Ωk. A theoretical model is established and we choose relative phase at a selected frequency Ωk as the reference phase to carry out phase unwrapping. The intercept of the unwrapped phase-frequency linear fit curve is used to identify the interference-order. The optimal monochromatic frequency is selected based on the linearity of relative-phase-measurand curve. Combining the relative phase obtained by DFT and the identified interference-order, the absolute phase is recovered. The algorithm is completely realized in frequency domain and needs no filtering process. Experimental results verified the effectiveness of our algorithm, and precision of pressure demodulation by our algorithm is improved by 13 times comparing with the direct phase-slope-based algorithm.
Acknowledgments
This work is supported by National Basic Research Program of China (Grant 2010CB327802), National Natural Science Foundation of China (Grant 11004150, 61108070), Tianjin Science and Technology Support Key Project (Grant 11ZCKFGX01900, 09ZCKFGX01400), New Faculty Fund for the Doctoral Program of Higher Education (Grant 200800561020), Scientific Research Foundation for the Returned Overseas Chinese Scholars, SEM, Shenzhen Key Laboratory of Sensor Technology Open Project (SST201013).
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