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Zero-fringe demodulation method based on location-dependent birefringence dispersion in polarized low-coherence interferometry Shuang Wang, Tiegen Liu, Junfeng Jiang,* Kun Liu, Jinde Yin, Zunqi Qin, and Shengliang Zou College of Precision Instrument & Optoelectronics Engineering, Tianjin University, Key Laboratory of Optoelectronics Information Technology, Tianjin 300072, China *Corresponding author:
[email protected] Received December 2, 2013; revised January 26, 2014; accepted February 18, 2014; posted February 20, 2014 (Doc. ID 202280); published March 20, 2014 We present a high precision and fast speed demodulation method for a polarized low-coherence interferometer with location-dependent birefringence dispersion. Based on the characteristics of location-dependent birefringence dispersion and five-step phase-shifting technology, the method accurately retrieves the peak position of zero-fringe at the central wavelength, which avoids the fringe order ambiguity. The method processes data only in the spatial domain and reduces the computational load greatly. We successfully demonstrated the effectiveness of the proposed method in an optical fiber Fabry–Perot barometric pressure sensing experiment system. Measurement precision of 0.091 kPa was realized in the pressure range of 160 kPa, and computation time was improved by 10 times compared to the traditional phase-based method that requires Fourier transform operation. © 2014 Optical Society of America OCIS codes: (120.3180) Interferometry; (120.2230) Fabry-Perot; (120.5475) Pressure measurement. http://dx.doi.org/10.1364/OL.39.001827
Phase-shifting interferometry (PSI) and low-coherence interferometry (LCI) are two effective techniques for displacement or optical path difference (OPD) measurement that have been widely used in profile measurement, optical coherence tomography, and optical fiber sensing applications such as for pressure and the refractive index [1–4]. The use of PSI with a monochromatic light source has the advantages of high-resolution and fast measurement speed [5]. However, the measurement range is restricted by 2π phase ambiguity and is generally only applied to measure smooth profiles. In contrast, LCI circumvents the phase ambiguity problem and has an unlimited measurement range in principle [6], which has made it a very important technique for the absolute measurement of displacement in the last few decades. However, the interference pattern is vulnerable to random noise and light source variation, which severely influence the accuracy of LCI if intensity-based peak detection algorithms are used. An extension of LCI is to utilize information about both the envelope peak and the phase at the peak’s position so that the same order of accuracy as PSI can be obtained while keeping the advantages of LCI [5,7]. The elimination of phase ambiguity benefits from the correct identification of the fringe order with envelope information. Unfortunately, the envelope peak and phase information are not always consistent. When the inconsistency is greater than 2π, misidentification of fringe order occurs and causes jump errors. The inconsistency is due to an imperfect system that includes dispersion and optical aberrations, for example, an interferometer with a location-dependent dispersion element [8]. In order to avoid misidentification of the fringe order, de Groot et al. [9] used the phase gap between the envelope peak and phase information to obtain the fringe order and compensated for the phase errors of the spatial-frequency domain analysis algorithm in surface profile measurements. But this surface profiling method needs a priori knowledge 0146-9592/14/071827-04$15.00/0
about the demodulation results of adjacent pixels, which are difficult to obtain for single-point measurements. In this Letter, we propose a new method, which is based on the characteristics of birefringence dispersion and a five-step phase-shifting algorithm, to identify the fringe order and retrieve the peak position of zero-fringe (ZF) at the central wavelength for polarized spatial LCI. We experimentally verified the effectiveness of the proposed method in an optical fiber Fabry–Perot (F-P) barometric pressure sensing system. The method obtains a high precision by using the sensitive phase information. The demodulation speed was approximately 10 times faster than that of traditional Fourier methods since we processed data only in the spatial domain and needed no time-consuming Fourier transform operation. The fast speed makes the proposed method more suitable for field applications using embedded hardware. To the best of our knowledge, this is the first report discussing the performance of such a phase-shifting technique for lowcoherence interference fringes (LCIF) with locationdependent birefringence dispersion. The polarized low-coherence interference configuration for the experiment is shown schematically in Fig. 1.
Fig. 1. Schematic layout of the optical fiber F-P barometric pressure sensing system based on polarized low-coherence interference with a birefringence wedge. © 2014 Optical Society of America
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Light from a broadband source, which consisted of a light-emitting diode (LED) in our experimental setup, was launched into the F-P sensor through a fiber coupler, and the fiber type was commercial multimode fiber with a core/cladding diameter of 62.5∕125 μm. The two reflecting surfaces of the F-P cavity form the sensing interferometer. This interferometric sensing arrangement senses the cavity length, which has a linear relationship with air pressure. The output optical signal from the F-P sensor projects into the demodulating interferometer, which is a modified version of the instrument demonstrated by Dändliker et al. [10] in 1992. The demodulating interferometer consists of a polarizer, birefringence wedge, and an analyzer. The polarization axis of the polarizer and analyzer makes a 45° angle with the optical axis of the birefringent wedge. Due to the birefringent effect of the wedge, it produces an ordinary (O) and extraordinary (E) optical ray whose optical path lengths along the wedge during light propagation are different; hence, the OPD is presented. When the OPD caused by the thickness of the birefringent wedge matches that of twice the cavity length of the F-P sensor, the LCIF is visible. The interferogram signals were acquired by a linear charge-coupled device (CCD), which consisted of 3000 discrete data points. In the experiment, the central wavelength λ0 of the LED light source was 613 nm and the full width at halfmaximum of the optical spectrum Δλ was 125 nm. The spectrum of LED approximates a Gaussian shape, which can be expressed as p p 2 ln 2 2 ln 2k − k0 2 ; (1) Ck p exp − Δk πΔk where k is the wave number, k0 2π∕λ0 is the central wave number, and Δk 2πΔλ∕λ20 . A MgF2 birefringence wedge was used in the experimental demodulation system. The dependence of the refractive index difference (RID) between E and O rays on the wave number can be expressed by a linear function nk n0 αk − k0 ;
(2)
where α is the birefringence dispersion slope and n0 is the RID at λ0 . For MgF2 material, α ≈ 0.047 nm∕rad and n0 ≈ 0.0118, which were calculated from the refractive index data in the spectral range of white LED [11]. The LCIF can be considered as the superposition of numerous interference signals with different k. With no birefringence dispersion being considered, the interference intensity corresponding to a cavity length h is Z Is
∞ 0
the location. The cosine oscillation in Eq. (3) represents the monochrome interference fringes (MIF) at λ0 . The peak position of ZF in MIF at λ0 is also the envelope peak of LCIF without birefringence dispersion, as shown in Fig. 2, and its value is twice the cavity length h. When considering birefringence dispersion, the intensity expression changes to Z∞ I 0 s Ck cosfks − 2h βk − k0 sgdk 0
2 −1∕4
1 η
Δk2 1 βk0 s − 2h2 exp − 4r1 η2
× cosΦ0 s;
(4)
where β α∕n0 , η βΔk2 s∕r, and Φ0 s Φs
arctan η ηΔk2 1 βk0 s − 2h2 − : 2 4r1 η2
(5)
The exponential term in Eq. (4) represents the envelope whose peak position s00 is equal to 2h∕1 βk0 . Comparing Eq. (3) with Eq. (4), we can see that the locationdependent birefringence dispersion introduces several effects on LCIF, such as envelope broadening, intensity decrease, envelope shift, and location-dependent phase shift. The last one is the most serious influence, which leads to the period distortion of fringes and makes the degree of inconsistency between envelope peak and phase information dependent on the location. In order to alleviate the dispersion effects and retrieve the peak position of ZF in MIF at λ0 , we performed five steps for demodulation, as shown in Fig. 2. ①: Utilize a centroid approach [12] to obtain the measured envelope peak position s0m with low precision, which includes a deviation s0ξ and s0m s00 s0ξ . Here, the measured cavity length hm equals 1 βk0 s0m ∕2. ②: Calculate the estimated peak position of the center fringe (CF) in LCIF s0cm and the fringe order m of the CF in MIF at λ0 based on the measured values. Here, CF is defined as the nearest fringe away from the envelop peak. ③: Obtain the accurate peak position of CF in LCIF s0c by use of the phase-shifting method. ④: Calculate the accurate CF peak in MIF at the central wavelength sc . ⑤: Retrieve
Ck cosks − 2hdk
Δk2 s − 2h2 cosΦs; exp − 4r
(3)
where r 4 ln 2, s n0 d, and Φs k0 s − 2h. d is the local geometrical path length of the ray passing through the birefringence wedge, and s represents the OPD between E and O rays at λ0 , which is linearly related to the serial number of CCD pixels and used to denote
Fig. 2. Simulated low-coherence interference showing the computation process based on the characteristics of locationdependent dispersion and the low precise envelop peak position.①–⑤ represent the demodulation process order.
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the accurate peak position of ZF in MIF at the central wavelength s0 based on sc and m. The estimated CF peak position in LCIF can be expressed as
With the accurate peak position s0c of CF in LCIF, we can get the peak position sc of CF in MIF at λ0 , and then retrieve the peak position s0 of ZF in MIF at λ0 by the following expression
s0cm 1 βk0 s0m − m ΔΦs0m ∕2πλ0 ;
s0 sc mλ0 s0c ΔΦs0c ∕k0 mλ0 ;
(6)
where m ints0m βk0 ∕λ0 . The function int() rounds the value in the brackets to the nearest integer. The phase difference (PD) between the LCIF and the MIF at λ0 changes little in a small region around the envelope peak position. Then, we can assume it is constant and, when evaluated at s0m , can be denoted as ΔΦs0cm
0
Φ
s0m
−
Φs0m
arctan βΔk2 s0m ∕r∕2:
(7)
In order to ensure s0cm belongs to the CF, the deviation s0ξ must be less than the maximum permissible error s0ξ max , which is equal to 1 − βk0 − φξ ψ ξ ∕2πλ0 ∕2, where φξ is the difference between the PD at s0m and s0c that is less than λ20 ln 2∕4l2c . The ψ ξ is the phase error that arises from the distortion period of LCIF, which is less than λ20 ln 21 βk0 2 ∕2l2c , where lc r∕Δk represents the coherence length. We can obtain the phase value ϕs in range −π; π for each point in CF with s0cm as the initial point and using the traditional five-step phase-shifting equation [13], tanϕs
2I 0 s − λ0 ∕4 − I 0 s λ0 ∕4 : 2I 0 s − I 0 s λ0 ∕2 − I 0 s − λ0 ∕2
(8)
The ϕs is calculated with a four-quadrant inverse tangent operation. After linearly fitting the phase values in CF, we acquired the accurate position s0c of the CF peak, which corresponds to the zero point of phase. The traditional PSI was used for MIF, whose envelope intensity and fringe period were both constant. But in LCIF with location-dependent dispersion, the envelope intensity varies and the fringe period is distorted, which cause the phase error. Figure 3(a) shows the phase error of the low-coherence phase-shifting method over the region s00 − 2λ0 ; s00 2λ0 in simulations with the experimental parameters. The phase error was in the range of −0.011–0.016 rad. In practice, the phase error on the fringe peaks in this region is much smaller, as shown in Fig. 3(b), which we were more concerned about. The simulation results show that the maximum error in the phase measurement was only 0.006 rad for the fringe peaks that were close to the envelop peak.
Fig. 3. Phase error of the low-coherence phase-shifting method (a) over the region s00 − 2λ0 ; s00 2λ0 and (b) on the fringe peaks around the envelop peak for the LCIF with location-dependent dispersion.
(9)
where ΔΦs0c
arctanβΔk2 s0c ∕r ηs Δk1βk0 s0c −s0m 2 : (10) − 2 4r1η2s
Here, ηs βΔk2 s0c ∕r. The cavity length is equal to s0 ∕2. We carried out a simulation to investigate the cavity length measurement error by the proposed method. The error was evaluated with the difference between this computed result and the exact cavity length, and the simulation results are shown in Fig. 4. It can be seen that the proposed method obtained a high measurement accuracy, which was less than 2.118 nm. In order to experimentally verify the effectiveness of the proposed method, we placed the F-P sensor on an air pressure chamber with a pressure control accuracy of 0.02 kPa. The 3000 pixels in the linear CCD correspond to the s range from 30.296 to 65.362 μm. In the experiment, we increased the air pressure from 5 to 165 kPa at intervals of 2 kPa and acquired 50 consecutive lowcoherence interference signals under each pressure for demodulation. The raw output of the CCD array, which consists of the interferogram and background, is shown in Fig. 5(a). By way of pixel-by-pixel division, the interferogram part was restored. We then carried out the processes of low precise envelop peak detection and CF peak identification in the LCIF, as shown in Fig. 5(b). The ZF peak in MIF at λ0 was retrieved by Eq. (9). The demodulation results by the proposed method are shown in Fig. 6. From this figure we can see that the linear response based on the centroid-based envelope detection method was locally distorted, and the fringe order identified by the traditional method of combining the envelope with PSI was incorrect, as the locationdependent birefringence dispersion caused the phase shift. The centroid-based envelope detection method had low precision demodulation results and the traditional combined method gave rise to jump errors. Both of these issues can be avoided by our proposed method.
Fig. 4. Error between the actual cavity length and the demodulation result under various cavity lengths with s0ξ in the region −s0ξ max ; s0ξ max by the proposed method.
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Fig. 7. Measure error (marks with error bars) between the demodulation results by the proposed method and the actual cavity length.
Fig. 5. Computation of s0c under 23 kPa pressure. (a) Raw interference signals and the background signals collected by CCD. (2) Computational process of identifying the CF peak position based on the interferogram without background.
Fig. 6. Relationship between the pressure and the detected cavity length. Top, proposed method (an offset of 400 nm was added to make the profiles distinguishable). Middle, combined method of envelope and PSI. Bottom, traditional centroid method (an offset of −400 nm was added).
Figure 7 shows the measurement error of cavity length by the proposed method, which was less than 6.155 nm and corresponded to 0.091 kPa with a sensitivity of 67.148 nm∕kPa. The error bars correspond to the standard deviation of the measurement error under each pressure, which was in the range of 0.097–0.302 nm and corresponded to 0.0014–0.0045 kPa. While for the traditional Fourier method [3], the demodulated error was less than 0.117 kPa and the standard deviation range was 0.0019–0.0085 kPa. The maximum standard deviation of the proposed method was reduced by almost two times. The error in the experimental results was larger than that of the simulation results shown in Fig. 4, which may have been caused by noise and imperfect optical elements since those effects were not taken into account in the analysis model. Furthermore, peak detection and phase calculation in the proposed method were both carried out only in the spatial domain. Thus, the proposed method can significantly accelerate the computation compared to the traditional phase-based method
that requires a Fourier transform in the spatial-frequency domain. The demodulation time for our method with the data size used in the experiment was 0.497 ms, while it was 5.224 ms for the traditional phase-based method (using VC 6.0 on a laptop with a 2.1 GHz CPU and 2 GB RAM). The speed was improved by more than 10 times. In conclusion, we proposed a new method to retrieve the accurate peak position of ZF in MIF at λ0 for demodulation of a low-coherence interference system with location-dependent birefringence dispersion. The method solved the ambiguity problem and reduced the computation load by using five-step phase-shifting technology. The method was demonstrated successfully in an optical fiber F-P barometric pressure sensing experiment system. The measurement error was less than 0.091 kPa, and the demodulation speed was improved by 10 times. This work was supported by the National Basic Research Program of China (Grant 2010CB327802), National Natural Science Foundation of China (Grants 61227011, 61378043, 11004150, 61108070), Tianjin Science and Technology Support Key Project (Grant 11ZCKFGX01900), Tianjin Natural Science Foundation (Grant 13JCYBJC162000), Science and Technology Key Project of Chinese Ministry of Education (Grant 313038), and Scientific Research Foundation for the Returned Overseas Chinese Scholars, SEM. References 1. Y. Ishii and R. Onodera, Opt. Lett. 16, 1523 (1991). 2. D. Sacchet, M. Brzezinski, J. Moreau, P. Georges, and A. Dubois, Appl. Opt. 49, 1480 (2010). 3. J. Jiang, S. Wang, T. Liu, K. Liu, J. Yin, X. Meng, Y. Zhang, S. Wang, Z. Qin, F. Wu, and D. Li, Opt. Express 20, 18117 (2012). 4. S. Kim, S. Lee, J. Lim, and K. Kim, Appl. Opt. 49, 910 (2010). 5. A. Harasaki, J. Schmit, and J. Wyant, Appl. Opt. 39, 2107 (2000). 6. K. Larkin, J. Opt. Soc. Am. A 13, 832 (1996). 7. P. Sandoz, R. Devillers, and A. Plata, J. Mod. Opt. 44, 519 (1997). 8. S. Wang, T. Liu, J. Jiang, K. Liu, J. Yin, and F. Wu, Opt. Lett. 38, 3169 (2013). 9. P. de Groot, X. de Lega, J. Kramer, and M. Turzhitsky, Appl. Opt. 41, 4571 (2002). 10. R. Dändliker, E. Zimmermann, and G. Frosio, Opt. Lett. 17, 679 (1992). 11. A. Duncanson and R. Stevenson, Proc. Phys. Soc. London 72, 1001 (1958). 12. S. Chen, A. Palmer, K. Grattan, and B. Meggitt, Appl. Opt. 31, 6003 (1992). 13. P. Hariharan, B. Oreb, and T. Eiju, Appl. Opt. 26, 2504 (1987).
