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Birefringence dispersion compensation demodulation algorithm for polarized low-coherence interferometry Shuang Wang, Tiegen Liu, Junfeng Jiang,* Kun Liu, Jinde Yin, and Fan Wu College of Precision Instrument & Optoelectronics Engineering, Tianjin University, Key Laboratory of Optoelectronics Information Technology, Tianjin 300072, China *Corresponding author:
[email protected] Received May 29, 2013; revised July 22, 2013; accepted July 22, 2013; posted July 23, 2013 (Doc. ID 191425); published August 15, 2013 A demodulation algorithm based on the birefringence dispersion characteristics for a polarized low-coherence interferometer is proposed. With the birefringence dispersion parameter taken into account, the mathematical model of the polarized low-coherence interference fringes is established and used to extract phase shift information between the measured coherence envelope center and the zero-order fringe, which eliminates the interferometric 2π ambiguity of locating the zero-order fringe. A pressure measurement experiment using an optical fiber Fabry–Perot pressure sensor was carried out to verify the effectiveness of the proposed algorithm. The experiment result showed that the demodulation precision was 0.077 kPa in the range of 210 kPa, which was improved by 23 times compared to the traditional envelope detection method. © 2013 Optical Society of America OCIS codes: (120.3180) Interferometry; (050.2230) Fabry-Perot; (120.5475) Pressure measurement. http://dx.doi.org/10.1364/OL.38.003169
As an effective technique to measure absolute displacement, low-coherence interferometry is widely used in surface profile measurement, optical coherence tomography (OCT), and optical fiber sensing applications such as pressure, refractive index, and thickness of transparent material [1–4]. The process algorithm of the low-coherence interference fringe is critical for obtaining a good measurement. Intensity-based peak detection algorithms [3,5] are simple and can acquire the shift information of the interference fringe pattern directly for demodulation, while phase-based detection algorithms, such as the phaseshifting algorithm [6] and the spatial-frequency domain analysis algorithm [7], can retrieve the shift information with higher precision when the fringe order is accurately identified. Combining the envelope information with the corresponding phase information is an effective way to overcome the fringe order ambiguity problem [8,9]. However, the envelope peak and phase information are not always consistent, which leads to the misidentification of the fringe order and causes jump errors, for example, the ghost step errors when measuring smooth surfaces. Unbalanced dispersion is the main reason for the inconsistencies in the process of low-coherence interference fringes, since a broadband light source is utilized and the two optical paths cannot be balanced for all wavelengths [1,10,11]. The sources of dispersion include asymmetrical beam splitting prisms, not perfectly identical microscope objectives, a transparent measuring object [2], and the evident unbalanced dispersion elements such as an additional glass plate in one arm of a Michelson/Linnik interferometer and the birefringent wedges in an interferometer/spectrometer [12,13]. In 2004, Pavliček and Soubusta investigated the influence of dispersion on the Michelson interferometer by inserting a different thickness plate of BK7 glass in one arm and using a different light source [10]. Because the thickness of one specific glass plate is constant, the dispersion influence is thickness independent. In 2010, Lehmann discussed the influence of dispersion on the Mirau interferometer and explained the resulting 0146-9592/13/163169-04$15.00/0
systematic measuring errors [11]. It is a natural thought to alleviate the dispersion effects in low-coherence interferometers with a special hardware design or with a compensation algorithm based on a priori/posteriori knowledge. Depth-dependent dispersion compensation in OCT has been carried out by optimizing frequencydomain optical delay line hardware with static tilted grating [14] and a numerical postprocess based on a depth-variant kernel [2]. However, the compensation mainly focuses on the envelope full width at halfmaximum (FWHM), not the envelope peak position. de Groot et al. used the phase gap between the phase and coherence information as a priori knowledge to obtain fringe order and compensate for the errors that arose from the dispersion in the surface profile measurement [15]. But the a priori/posteriori knowledge is frequently difficult to obtain. Furthermore, birefringence dispersion, which cannot be neglected in a high precision measurement, is less discussed. In this Letter, we propose an algorithm based on the thickness-dependent dispersion characteristics of a birefringence wedge for polarized low-coherence interferometry, which is used for Fabry–Perot (FP) cavity length demodulation. The phase shift information between the measured coherence envelope center and the zero-order fringe introduced by birefringence dispersion is used, and the 2π ambiguity of locating the zero-order fringe is avoided. So we can use very sensitive phase information to measure absolute distances larger than λ∕2. The algorithm is then successfully applied to the measurement of air pressure. We experimentally verify the accuracy of our algorithm, which is enhanced 23 times more than the result of the traditional envelope detection method during pressure demodulation, and the proposed algorithm does not introduce an additional computation burden. Figure 1 is the experiment setup of an optical fiber FP pressure sensing system based on polarized lowcoherence interference with a birefringence wedge, which is a modified version of the instrument demonstrated by Dändliker et al. in 1992 [12]. The FP sensor © 2013 Optical Society of America
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where k0 2π∕λ0 is the central wave number, and λ0 is the central wavelength, Δk 2πΔλ∕λ20 , where Δλ is the FWHM of the power spectrum in the wavelength domain. The birefringence dispersion is small, and the dependence of the RID on the wave number can be expressed as a linear function nk nk0 αk − k0 ;
(3)
where α is the birefringence dispersion slope. Substituting Eqs. (2) and (3) into Eq. (1), Id; h can be rewritten as
Fig. 1. Schematic layout of an optical fiber FP pressure sensing system based on the polarized low-coherence interference setup: CCD, charge coupled device; LED, light-emitting diode; h, actual FP cavity length.
placed on an air pressure chamber is used to sense and transfer air pressure into the cavity length. The cavity length change Δh and pressure change ΔP is linearly related by Δh 31 − υ2 r 4 ΔP∕16Et3 , where r and t represent the effective radius and thickness of the silicon diaphragm, respectively. E is the Young modulus of silicon and υ is the Poisson ratio of silicon [7]. The interference fringes are visible in the limited spatial region only if the optical path difference (OPD) caused by the thickness of the birefringent wedge matches with twice the cavity length of the FP sensor. The ordinary (O) and extraordinary (E) rays that project onto the same CCD pixel have an equal geometrical path length in the birefringent wedge. However, the birefringence dispersion, i.e., birefringence varies with the wavelength, leads to a shift of the envelope peak and an additional phase shift between the maximum of the envelope and zero-order fringe peak. That will give rise to erroneous measurement results or even a misidentification of fringe order when this relative phase shift is greater than 2π. The low-coherence interferogram in the experiment can be regarded as an approximate double-beam interference. The interference can be described as Z Id; h
∞ 0
Sk cosfknkd − 2hgdk;
where γ 4 ln 2, η αΔk2 d∕γ, z Nk0 d − 2h, and Nk0 nk0 αk0 . Nk0 is termed the group RID and z is the group-velocity OPD. The phase Φd; h is given by Φd; h zk0 − αk20 d 1∕2 arctan η −
ηΔk2 z2 : (5) 4γ1 η2
It can be seen from Eq. (4) that the low-coherence interferogram is a cosine oscillation with a Gaussianshaped envelope. The position shift of the envelope peak caused by the birefringence dispersion is αk0 d0 ∕nk0 , and the relative phase shift of the zero-order fringe peak is αk20 d0 − 1∕2 arctanαΔk2 d0 ∕γ, where d0 is the actual envelope peak position of the low-coherence interferogram with dispersion, as shown in Fig. 2(a). The relative phase shift depends on d0 , i.e., when the zero groupvelocity OPD position in the spatially distributed interferogram changes, the zero-order fringe position relative to the envelope peak changes, as shown in Fig. 2.
(1)
where d is the local geometrical path length of the ray through the birefringence wedge for the vertical incidence, which is also the local thickness of the birefringence wedge; h is the cavity length of the FP sensor, and k is the wave number. Sk is the light source spectrum, nk is the refractive index difference (RID) between the E ray and O ray. To simplify the calculation of the dispersion characteristic of the interference pattern, the light source is modeled with a Gaussian spectrum p p 2 ln 2 2 ln 2k − k0 2 Sk p exp − ; π Δk Δk
zΔk2 cosΦd; h; Id; h 1 η2 −1∕4 exp − 4γ1 η2 (4)
(2)
Fig. 2. Simulated low-coherence interference with different cavity length showing that the location of the zero-order fringe moves relative to the envelope peak. (a) Cavity length is 12.5 μm, (b) cavity length is 1.9 μm, and (c) cavity length is 25 μm.
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It is obvious that when z 0, the envelope function has the maximum value and the measured FP cavity length hpeak can be calculated by the following expression: hpeak Nk0 dpeak ∕2;
φξ Φdpeak ; hpeak − Φdpeak ; h 2k0 hξ
ηpeak Δk2 h2ξ
γ1 η2peak
;
(7)
where ηpeak αΔk2 dpeak ∕γ. Φdpeak ; h and Φdpeak ; hpeak represent the absolute phase at the measured envelope peak position dpeak of the interference fringe under h and hpeak , respectively. With Eq. (7), we can obtain hξ if φξ is known. When 400 nm < λ0 < 1000 nm and Δλ < 150 nm, the coefficient of the quadratic item of Eq. (7) is less than 0.1% of the linear term coefficient; hence, the quadratic item can be neglected, i.e., hξ φξ ∕2k0 . The φξ can also be expressed by the relative phase φ when φξ is in the region −π; π [see Fig. 3(c)], φξ φdpeak ; hpeak − φdpeak ; h − 2mπ;
8 π < φdpeak ; hpeak − φdpeak ; h < 2π < 1; π ≤ φdpeak ; hpeak − φdpeak ; h ≤ π m 0; : : −1; −2π < φdpeak ; hpeak − φdpeak ; h < −π (9)
(6)
where dpeak is the measured envelope peak position by the envelope detection method. Because the envelope is sensitive to random noise and vulnerable to variation of the light source, dpeak usually has a deviation from the actual value d0 , which leads to a measurement error hξ between hpeak and actual cavity length h, i.e., h hpeak hξ . This cavity length offset ultimately give rise to the phase deviation φξ at the measured envelope peak position dpeak , as shown in Fig. 3. φξ can be calculated from the absolute phase Φ with Eq. (5),
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Here, φdpeak ; h and φdpeak ; hpeak , which are both in the interval 0; 2π, represent the relative phase at the measured envelope peak position dpeak of the interference fringe under h and hpeak , respectively. φdpeak ; h can be obtained from a raw interference signal, as shown in Fig. 4. Dividing the filtered interference signal by the envelope signal to obtain the cosine signal cosM d; h, the relative phase value of cosM d; h at dpeak is φdpeak ; h. φdpeak ; hpeak can be calculated by φdpeak ; hpeak Φdpeak ; hpeak − 2π × floorΦdpeak ; hpeak ∕2π:
(10)
The function floor returns the nearest integer which is smaller or equal to the value in the brackets. Thus,φξ is obtained from Eq. (8) to Eq. (10), and then hξ can be calculated by hξ φξ ∕2k0 . Due to the method based on the condition that φξ is in the region −π; π, the maximum permissible error between h and hpeak is approximately λ0 ∕4. In our polarized low-coherence interferometer experiment setup, the central wavelength λ0 of the LED light source is 613 nm and Δλ is 125 nm. The cavity length demodulated by the proposed algorithm will be reliable if hξ is less than λ0 ∕4 ≈ 153 nm, which corresponds to the demodulate pressure error 2.8 kPa and the envelope detection method accuracy can assure this [3,7]. An MgF2 birefringence wedge is used and the birefringence dispersion slope α ≈ 0.0468 rad∕nm, which is calculated with refractive index data in the spectral
(8)
where
Fig. 3. Interference fringe under (a) actual cavity length h and (b) measured cavity length hpeak . (c) Phase information corresponding to (a) and (b).
Fig. 4. Computation of the relative phase at the measured envelope peak position under 138 kPa pressure. (a) Raw interferogram collected by CCD, envelope, and filtered interferogram. (b) The cosine signal and (c) the relative phase of the interference fringe under h (solid curve) and hpeak (dashed curve) near the measured envelope peak position.
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Fig. 5. Relationship between the pressure and the detected cavity length. Top, proposed method (an offset of 300 nm is added to make the two profiles distinguishable). Bottom, envelope detection method.
region from 551 to 675 nm [16]. A linear CCD with 3000 pixels samples 3000 thicknesses ranging from 1.706 to 4.628 cm, corresponding to the relative phase shift of the zero-order fringe from 8.325 to 22.590 rad. The change of the relative phase shift is more than 2π, which will give rise to the erroneous evaluation of the zeroorder fringe peak and misidentification of the fringe order when using the algorithm that is dependent on the position or phase of the fringe peak. This error can be avoided by the proposed algorithm. In the experiment, the air pressure increases from 50 to 260 kPa at interval 2 kPa, which is applied by an air pressure chamber with accuracy of 0.02 kPa. The performance of the proposed algorithm is compared with the envelope detection algorithm. An overall linear relationship exists between pressure and demodulated cavity length by the use of two algorithms, but the linear response with the envelope detection algorithm is locally distorted. After compensating the measurement error of the envelope detection method by the proposed algorithm, the linearity has been greatly improved, as shown in Fig. 5. We acquired 50 consecutive interference signals under each pressure for demodulation. Figure 6 shows the error of the demodulated pressure by the envelope detection algorithm and by the proposed algorithm, respectively. The demodulated error is less than 0.077 kPa, corresponding to 0.037% accuracy for the full scale by the proposed algorithm, while the demodulated error is about 1.789 kPa, corresponding to 0.85% accuracy by the envelope detection algorithm. The error bars in Fig. 6 correspond to the standard deviation (SD) of the demodulated pressure error under each pressure. For the proposed method, the SD is in the range of 0.0042– 0.0118 kPa, while for the envelope detection method, the range is 0.0703–0.1406 kPa. It clearly demonstrates that in the same experimental conditions, the demodulated results of the proposed method are much more precise and stable. To summarize, we have presented an algorithm which can correct the measurement error of the traditional envelope detection method by the use of the dispersive characteristics of the birefringence wedge in polarized low-coherence interferometry. It can effectively improve the accuracy and stability without an additional computation burden, which has been verified by the experimental results. The measurement precision is improved by 23 times compared with the envelope detection algorithm.
Fig. 6. Error (marks with error bars) between the set pressure and demodulated pressure for 50 consecutive interference data under each pressure by (a) the envelope detection algorithm and (b) the proposed algorithm.
This work was supported by National Basic Research Program of China (Grant No. 2010CB327802), National Natural Science Foundation of China (Grant Nos. 61227011, 11004150, and 61108070), Tianjin Science and Technology Support Key Project (Grant No. 11ZCKFGX01900), Tianjin Natural Science Foundation (Grant No. 13JCYBJC162000), Science and Technology Key Project of Chinese Ministry of Education (Grant No. 313038), and Scientific Research Foundation for the Returned Overseas Chinese Scholars, SEM. References 1. A. Pförtner and J. Schwider, Appl. Opt. 40, 6223 (2004). 2. A. Fercher, C. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, Opt. Commun. 204, 67 (2002). 3. S. Wang, J. Jiang, T. Liu, K. Liu, J. Yin, X. Meng, and L. Li, IEEE Photon. Technol. Lett. 24, 1390 (2012). 4. M. Haruna, M. Ohmi, T. Mitsuyama, H. Tajiri, H. Maruyama, and M. Hashimoto, Opt. Lett. 23, 966 (1998). 5. P. Sandoz, Opt. Lett. 22, 1065 (1997). 6. K. Hibino, B. Oreb, D. Farrant, and K. Larkin, J. Opt. Soc. Am. A 14, 918 (1997). 7. 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). 8. P. Sandoz, R. Devillers, and A. Plata, J. Mod. Opt. 44, 519 (1997). 9. Y. Ghim and A. Davies, Appl. Opt. 51, 1922 (2012). 10. P. Pavliček and J. Soubusta, Appl. Opt. 43, 766 (2004). 11. P. Lehmann, Opt. Lett. 35, 1768 (2010). 12. R. Dändliker, E. Zimmermann, and G. Frosio, Opt. Lett. 17, 679 (1992). 13. J. Li, J. Zhu, and H. Wu, Opt. Lett. 35, 3784 (2010). 14. E. Smith, A. Zvyagin, and D. Sampson, Opt. Lett. 27, 1998 (2002). 15. P. de Groot, X. Lega, J. Kramer, and M. Turzhitsky, Appl. Opt. 41, 4571 (2002). 16. A. Duncanson and R. Stevenson, Proc. Phys. Soc. London 72, 1001 (1958).
