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Brillouin power spectrum analysis for partially uniformly strained optical fiber
ARTICLE IN PRESS Optics and Lasers in Engineering 47 (2009) 976–981
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Brillouin power spectrum analysis for partially uniformly strained optical fiber Dan Zhang a,, Hongzhong Xu b, Bin Shi a, Haibo Sui a, Guangqing Wei a a b
Center for Engineering Monitoring with Opto-Electronic Sensing (CEMOES), Nanjing University, 210093 Nanjing, China College of Civil Engineering, Nanjing University of Technology, 210009 Nanjing, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 2 August 2007 Received in revised form 25 February 2009 Accepted 16 April 2009 Available online 17 May 2009
Due to the restriction of the spatial resolution, about 1 m for current commercially available system, strain distribution measured by Brillouin optical time domain reflectometer (BOTDR) is slightly different from the actual one. In this paper, the equation of the Brillouin power spectrum for partially uniformly strained fiber within the spatial resolution is theoretically derived. Based on the derived results, investigation has been made on the shape characteristics of the superposed Brillouin power spectrum, as well as the dependence of the calculated strain of BOTDR on the actual strain of the fiber. It was found that the difference between the calculated strain and the actual strain depends mainly on the strain value of the fiber and the strained length within the spatial resolution for the given distributed sensing system. & 2009 Elsevier Ltd. All rights reserved.
Keywords: BOTDR Spatial resolution Power spectrum Strain measurement Superposition
1. Introduction Distributed fiber optic sensor based on Brillouin scattering is currently attracting considerable research interest in various fields, such as composite material, civil structure, pipeline and geotechnical engineering, owing to its unrivalled capability to provide a continuous strain distribution along the sensing fiber [1–3]. Brillouin optical time domain reflectometer (BOTDR) and Brillouin optical time domain analysis (BOTDA) are two common and commercially available distributed sensing systems. Though the set-ups of the two systems are different, strain and temperature of the fiber are similarly calculated by fitting the peak of Brillouin backscattered light power spectrum (BS) on the basis of the strain or temperature dependence on the Brillouin frequency shift. Theoretical investigations show that the shape of the BS changes not only depending on the pulse width but also on the shape of the launched light [4,5]. And an in-depth study on the signal processing has been conducted widely as power spectrum shape analysis is an important subject for BOTDR/BOTDA system to determine the peak frequency correctly [6,7]. Actually, due to the restriction of spatial resolution, the spectrum shape also changes with the strain distribution along the fiber. In such aspect, Naruse et al. did researches on the deformed BS caused by linear and parabolic strain distribution and analyzed their influences on the strain measurement [8,9].
Corresponding author. Tel.: +86 25 83597888.
E-mail address: [email protected] (D. Zhang). 0143-8166/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2009.04.008
In practice, it is found that, when the fiber is uniformly strained and the strained length is larger than the spatial resolution of BOTDR/BOTDA, the measured stain is equal to the actual value of the fiber. Whereas, if the fiber is non-uniformly strained or the strained length is less than the spatial resolution, Brillouin power spectrum is distorted, and even several peaks can be found in the BS. This leads to inaccurate measurement of the peak frequency and the measured strain distribution is slightly different from the actual one. Meanwhile, strained length obtained by BOTDR/BOTDA is also usually larger than the actual one when the fiber is slightly strained. These indicate that it is impossible to detect inhomogeneous strain distribution accurately or to determine the position and length of the strained section along the fiber exactly with current sensing system due to the limitation of the spatial resolution. In order to enhance the performance of BOTDR/BOTDA, several methods have been proposed to improve the spatial resolution without further narrowing the pulse width of the incident light [10–14]. Therein, spectral decomposition is a method based on the further analysis of the measured Brillouin spectrum, which is considered as the superposition of all the Brillouin gain spectra within the spatial resolution. Moreover, some new systems based on Brillouin fiber sensing are proposed for distributed measurement with very high spatial resolution, such as Brillouin echo distributed sensing (BEDS) with a spatial resolution of 10 cm and Brillouin optical correlation-domain reflectometry (BOCDR) with a spatial resolution of 13 mm [15,16]. Potentially, these systems are possibly available for engineering applications in the near future.
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In this paper, we give the theoretical basis for the spectral decomposition method used to improve the spatial resolution of the current BOTDR system as well as the explanation of the difference between the actual strain distribution and the calculated strain distribution. Here, the equation of the superposed BS for partially uniformly strained optical fiber is first proposed. Then, shape analyses on BS are performed for different strain and strained length, and discrepancy between the actual strain and the calculated strain by fitting the superposed spectrum is also illustrated.
2. Brillouin backscattered light For the BOTDR system [17], pulsed light is launched into one end of a single-mode optical fiber, and the Brillouin backscattered light power dPB (z, v) that is produced in a small section dz of the fiber and detected at the same end is given by Eq. (1) without the consideration of the pump depletion. dP B ðz; vÞ ¼ gðv; vB Þ
gðv; vB Þ ¼
c pðzÞ dz expð2az zÞ, 2n
hðw=2Þ2 ðv vB Þ2 þ ðw=2Þ2
,
(1)
(2)
z ¼ ct=ð2nÞ,
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sampling position can be determined by Z zi þDz=2 Gi ðnÞ ¼ gfn; nB ½ðzÞg dz,
(5)
zi Dz=2
where zi is the position of the i-th sampling point along the fiber. 3. Superposition of Brillouin power spectrum A simplified case, as shown in Fig. 1, is investigated here to reveal the characteristics of the superposed BS: the length of strained fiber is less than the spatial resolution of BOTDR, that is to say, the fiber within the spatial resolution is partially strained and the fiber’s strain is uniform. The measured Brillouin spectrum by BOTDR is actually the superposition of Brillouin gain spectrum within the spatial resolution, including Brillouin gain spectrum of the strained fiber and that of the unstrained fiber. It can be found that the unstrained length within the spatial resolution and the strain value of the strained section exert an influence on the shape of the superposed spectrum, which results in the difference between the actual strain and the calculated stain by fitting the highest peak of the superposed spectrum with Lorentzian curve. The analysis on the shape of the superposed spectrum is based on the followings assumptions:
(3)
where z is the distance along the fiber from the light input end, p(z) is the launched light power at z, v is the optical frequency of the Brillouin backscattered light, c is the velocity of light in vacuum, n is the refractive index of the fiber, az is the attenuation coefficient of the fiber, g(v, vB) is the Brillouin gain spectrum given by a Lorentzian function, vB is the frequency at which g(v, vB) has a peak value h, w is the FWHM of g(v, vB) and t is the time interval between the pulse light’s being launched and the scattered light’s being detected. According to the former researches, Eqs. (1) and (2) are valid in the presented form as long as the pulse width is larger than the phonon lifetime (E10 ns) [5,7]. Just like other OTDR technology, the spatial resolution of BOTDR is determined by the pulse width of the incident light, which can be expressed as
DZ ¼ ct=ð2nÞ,
(4)
where t is the pulse width of the incident light. The strain or temperature at every sampling point is calculated by fitting the BS with Lorentzian function. The BS of the i-th
(1) The electric field of the Brillouin backscattered light produced at every scattering point in the fiber does not exhibit any phase correlation, so the principle of superposition holds with respect to the backscattered light power. (2) The dependence of the Brillouin line-width and peak power on strain and temperature is ignored. (3) The Brillouin gain coefficient is constant over the length of a fiber section of constant strain. If the strained length is less than the spatial resolution, namely, if the fiber is subjected to two distinct but uniform strains (the strains are e and 0, respectively, as shown in Fig. 1) over the spatial resolution, according to Eq. (5), the BS is given by Z zi Dz=2þl Z zi þDz=2 Gi ðnÞ ¼ gfn; nB ½ðzÞg dz þ g½n; nB ð0Þ dz, (6) zi Dz=2
zi Dz=2þl
where l, in the range of 0 and Dz, is the length of the strained fiber. As the fiber’s stain within the spatial resolution is uniform, the BS measured by BOTDR system can be presented theoretically by
Strain
Brillouin Scattered Power
Strained Section 1:BS of Unstrained Section 2:BS of Strained Section 3:BS measured by BOTDR
ε=0
Distance Optical Fiber
Spatial Resolution
Frequency Fig. 1. Demonstration of the Brillouin power spectra when the strained length is less than the spatial resolution.
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the following equation, which is changed from Eq. (6): Gi ðnÞ ¼
g 0 ðDnB =2Þ2 ðn nB ð0Þ DnðÞÞ2 þ ðDnB =2Þ2 þ
the actual strain of the fiber and the real roots of Eq. (12), by which the measured strain is obtained is illustrated as follows.
l 4. Analysis of the superposed Brillouin power spectrum
g 0 ðDnB =2Þ2 2
2
ðn nB ð0ÞÞ þ ðDnB =2Þ
ðDz lÞ,
(7)
where Dn(e) is the Brillouin frequency shift of the strained fiber. Being divided by Dz on both sides, Eq. (7) is shown as follows: Gi ðnÞ g 0 ðDnB =2Þ2 ¼ ð1 rÞ Dz ðn nB ð0ÞÞ2 þ ðDnB =2Þ2 g 0 ðDnB =2Þ2
r ðn nB ð0Þ DnðÞÞ2 þ ðDnB =2Þ2 " 1r ¼ g 0 ðDnB =2Þ2 ðn nB ð0ÞÞ2 þ ðDnB =2Þ2 # r , þ ðn nB ð0Þ DnðÞÞ2 þ ðDnB =2Þ2 þ
4.1. Brillouin spectrum when r is equal to 0.5
(8)
where parameter r is defined as the ratio of the strained length l to the spatial resolution Dz. By differentiating Gi(v)/Dz given by Eq. (8) with respect to v, the optical frequency of the Brillouin backscattered light, we get ! dGi ðnÞ 2 ð1 rÞðn nB ð0ÞÞ ¼ dn g 0 Dn2B Dz ½ðn nB ð0ÞÞ2 þ ðDnB =2Þ2 2 þ
rðn nB ð0Þ DnðÞÞ ½ðn nB ð0Þ DnðÞÞ2 þ ðDnB =2Þ2 2
.
(9)
The peak frequency of the measured BS is obtained from the zeros of derivative of Gi(v)/Dz with respect to v as shown in Eq. (10) ! dGi ðnÞ 2 ¼ 0. dn g 0 Dn2B Dz
(10)
Then, Eq. (9) is given as ð1 rÞx ½x2 þ a2 2
þ
rðx bÞ ½ðx bÞ2 þ a2 2
¼ 0,
(11)
where x ¼ (nnB(0)), a2 ¼ (DnB/2)2 and b ¼ Dn(e). And Eq. (11) can be rewritten as x5 þ Ax4 þ Bx3 þ Cx2 þ Dx þ E ¼ 0.
Based on the above analysis, the shape of the BS depends not only on the strained length within the spatial resolution but also on the strain value of the fiber. In order to investigate the characteristics of the superposed BS of partially and uniformly strained fiber, analyses on the shape of the BS and difference between the calculated strain and actual strain are conducted when r is 0.4, 0.5 and 0.6, respectively.
Fig. 2 shows the shape of the BS when r, the ratio of the strained length to the spatial resolution, is equal to 0.5. The power and the sweep frequency of the spectrum are normalized by g0Dn2B Dz and nB(0), respectively. The strain of the fiber is normalized by e0, the minimal actual stain which makes the measured BS contains two peaks. We use eNA and eNC to denote the normalized actual strain and normalized calculated strain by BOTDR, respectively. The shape of the normalized BS deforms that of the Lorentzian function of Eq. (2) as it is a superposition of spectrum of the strained section and unstrained section. It is also seen from Fig. 2 that the superposed spectrum is symmetric and shows two peaks when the stain is large enough because of the equal contribution of the strained section and the unstrained section. By fitting the right peak of the spectrum with Lorentzian function, the fiber’s strain can be calculated approximately with the obtained peak frequency. By investigating the curves in Fig. 2 and the relationship in Fig. 3, it can be found that the superposed spectrum contains only one peak when the stain is small and shows a linear relationship between the normalized calculated strain eNC and the normalized actual strain eNA. The actual strain can be estimated with the calculated strain of BOTDR by the linear relationship. Nevertheless, when the fiber’s strain becomes larger, the superposed spectrum contains two peaks. By fitting the peak of higher frequency in Fig. 2, the calculated strain is close to the actual strain with the increase of the strain, as the A branch shown in Fig. 3. And if we fit the peak of lower frequency, the calculated strain is close to zero, as the B branch shown.
(12)
Lorentzian curve
1.0
εNA=0.35
ε=0
0.70
2
2
B ¼ 6rb þ 6b þ 2a2 3
3
C ¼ rð4b þ 2ba2 Þ 4b 4ba2 2
4
2
4
D ¼ rð2a2 b þ b Þ þ a4 þ 2a2 b þ b 4
E ¼ ra b.
1.05 1.40 1.75
0.8
2
A ¼ 3rb 4b
Normalized power G/[g0ΔνΒ Δz]
The coefficients of Eq. (12) are given by
(13)
Eq. (12) is the theoretical BS when the fiber is partially and uniformly strained within the spatial resolution. It may have one real root or three real roots, which depends on the parameters A to E of the equation. Generally, if the strain within the spatial resolution is small, the measured BS, namely, the superposed Brillouin spectrum contains one peak and Eq. (12) has one real root accordingly. However, in case the strain within the spatial resolution is large enough, the measured BS usually contains two peaks and Eq. (12) has three real roots. The relationship between
2.10
0.6
0.4
0.2
0.0 -3
-2
-1
0
1
2
3
4
Normalized sweep frequency [ν−νΒ(0)]/0.1 Fig. 2. Shape of the BS caused by different strain when r is 0.5.
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4.2. Brillouin power spectrum when r is less than 0.5 Fig. 4 shows the superposed BS of the strained section and unstrained section within the spatial resolution when r is equal to 0.4. The normalized actual strain eNA in Fig. 3 is 0.35, 0.7, 1.05, 1.4, 1.75 and 2.1, respectively. It is found that the shape of the superposed spectrum deforms and is not symmetric with respect to the peak frequency, which is different with that of Lorentzian function. The Brillouin line-width is also broadened. With the increase of fiber’s strain, two peaks appears in the superposed spectrum, of which, approximately, the peak of lower frequency (the left peak with higher power) coincides with the spectrum of the unstrained section and peak of higher frequency (the right peak with lower power) coincides with the spectrum of the strained section within the spatial resolution. Therefore, the fiber’s real strain can be estimated with the peak frequency of the right peak. However, in order to automate the data processing, the strain of the fiber is calculated by picking the highest peak of the BS in most commercially available BOTDR system. According to this method, the calculated strain with Fig. 4 will deviate from the actual stain of the fiber undoubtedly and the
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error becomes bigger and bigger with the increase of the fiber’s actual strain. The difference between the measured strain calculated with the superposed spectrum and the actual strain is shown in Fig. 5 when r is 0.1, 0.2, 0.3 and 0.4, respectively. With the increase of the fiber’s actual strain, the influence of the Brillouin power of the strained fiber decreases on the superposed spectrum and the calculated strain with the peak frequency of the superposed spectrum is diminished to zero. It is also seen from Fig. 5 that the calculated strain is close to the actual strain according to the increase of r. However, the difference between them is very large and cannot be ignored whenever the fiber’s actual strain is large or small. Similar to Fig. 3, the relationship between the calculated strain and the actual strain is linear approximately when the fiber is slightly strained (normalized actual strain eNA is less than about 0.3).
4.3. Brillouin power spectrum when r is larger than 0.5 Fig. 6 shows the superposed BS of the strained section and unstrained section within the spatial resolution when r is equal to
Actual strain (r = 1.00)
0.5 Calculated strain with superposed spectrum εNC
Calculated strain with superposed spectrum εNC
2.0 Actual strain (r = 1.00) A 1.5
1.0 r = 0.5 0.5
0.4 0.40
0.3
0.30 0.2
0.20 r = 0.10
0.1
B 0.0 0.0
0.5 1.0 1.5 Actual strain within the spatial resolution εNA
2.0
Fig. 3. Relationship between the calculated strain with the superposed BS and the actual strain of the fiber within the spatial resolution when r is 0.5.
Lorentzian curve ε=0
1.0
1.5
2.0
εΝΑ=0.35
Lorentzian curve ε=0
0.70 1.05 1.40 1.75
0.8
2
1.05 1.40 1.75
0.6
1.0
Fig. 5. Relationship between the calculated strain with the superposed BS and the actual strain of the fiber within the spatial resolution when r is less than 0.5.
εΝΑ = 0.35 0.70
0.8
0.5
Actual strain whitin the spatial resolution εNA
Normalized power G/[g0ΔνΒ Δz]
Normorlized power G/[g0ΔνΒ2Δz]
1.0
0.0 0.0
2.10 0.4
0.2
0.0
2.10 0.6
0.4
0.2
0.0
-3
-2 -1 0 1 2 3 4 Normorlized sweeep frequency [ν−νΒ (0)]/0.1
Fig. 4. Shape of the BS caused by different strain when r is 0.4.
5
-3
-2
-1
0
1
2
3
4
Normalized sweep frequency [ν−νΒ(0)]/0.1 Fig. 6. Shape of the BS caused by different strain when r is 0.6.
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0.6. Similar to Fig. 4, the line-width of the BS is also broadened and the profile of it is asymmetric. In contrast, the peak power with higher frequency, which coincides with that of Brillouin spectrum of strained section, is larger. Therefore, it is possible to roughly estimate the actual strain of the fiber with the peak frequency of the superposed spectrum for most commercially available BOTDR system. Fig. 7 shows the deviation of the calculated stain with the superposed spectrum from the actual stain when r is 0.6, 0.7, 0.8 and 0.9, respectively. It is seen from Fig. 7 that the calculated strain is almost close to the actual strain whenever the strain is small or large. And the error between them is relatively large when the normalized strain eNA is approximately within the range 0.3–1.4. Moreover, with the increase of r, the calculated strain with the superposed spectrum approaches the fiber’s actual strain.
100
εNA = 0.9 εNA = 0.7
80
RE %
980
60 εNA = 0.5 εNA = 0.3
40
εNA = 0.1 20 0 0.0
0.2
0.4
0.6
0.8
1.0
r
We conducted an analysis of the errors on determining the fiber’s actual strain while fitting the peak of the superposed BS by introducing the relative error relation RE ¼ |(eNAeNC)/eNA|, where eNA is the normalized actual strain and eNC is the normalized calculated strain of BOTDR as above. Results of the error analysis are showed in Figs. 8 and 9 and can be summarized as follows: (1) As the ratio r increases, the relative error RE decreases without reference to the value of eNA. (2) When the normalized actual strain eNA is less than 0.3, the RE decreases approximately linearly with the increase of r, as shown in Fig. 8. In other words, we can establish a linear relation between normalized eNA and eNC with the known ratio r if the strain of fiber is small. Similar results can also be found in the researches of Ohsaki. It can be concluded that for a small strain of less than about 103 there held a simple relation of xp ¼ a r between the apparent strain xp obtained by BOTDR and the accurate strain a, where r denotes the ratio of the strained length within the spatial resolution [11]. Therefore, by using the equation proposed by Ohsaki, the actual strain can be approximately estimated with the measured strain of BOTDR. However, when the normalized actual strain eNA is larger than 0.3, the linear relationship is
Calculated strain with superposed spectra εNC
Actual strain (r = 1.00) 1.5 0.90 0.80 0.70 r = 0.60 0.5
0.0 0.0
εΝΑ = 2.0
100 80 60
εΝΑ = 1.8 εΝΑ = 1.6 εΝΑ = 1.4 εΝΑ = 1.2 εΝΑ = 1.0 εΝΑ = 1.0
40
εΝΑ = 1.2 εΝΑ = 1.4 εΝΑ = 1.6 εΝΑ = 1.8
20 εΝΑ = 2.0
0 0.0
0.2
0.4
0.6
0.8
1.0
r Fig. 9. Dependence of the relative error on r when the normalized actual strain eNA is larger than 1.0.
not tenable any more and the nonlinearity become larger according to the increase of eNA. (3) As the actual strain of the fiber is large (the normalized strain eNA is larger than 1.0), the relationship between the RE and r is shown in Fig. 9. If r is larger than 0.5, the relative error ER is close to zero according to the increase of actual strain and r. The calculated strain eNC is almost equal to the actual strain eNA. In contrast, when the r is less than 0.5, the relative error ER is approaching 100%. In another word, the normalized calculated strain eNC is close to zero as the BOTDR system automatically fits the peak of higher power which coincides with the Brillouin spectrum of the unstrained section, although the superposed BS contains two peak.
2.0
1.0
Fig. 8. Dependence of relative error on r when the normalized actual strain eNA is less than 1.0.
RE %
4.4. Discussion
5. Conclusions 0.5 1.0 1.5 Actual strain within the spatial resolution εNA
2.0
Fig. 7. Relationship between the calculated strain with the superposed BS and the actual strain of the fiber within the spatial resolution when r is larger than 0.5.
According to the derived equation of the BS for partially uniformly strained fiber within the spatial resolution, investigation has been made on the shape characteristics of the superposed BS with the changes of the fiber’s strain and the ratio r, as well as
ARTICLE IN PRESS D. Zhang et al. / Optics and Lasers in Engineering 47 (2009) 976–981
the dependence of the measured strain on the actual strain of the fiber. This investigation yields the following results: (1) If the length of the strained fiber is less than the spatial resolution, the measured BS by BOTDR is a superposed spectrum of the strained section and the unstrained section within the spatial resolution. The superposed spectrum has an asymmetric shape except r, ratio of the strained length to the spatial resolution, is equal to 0.5. (2) The shape of the superposed spectrum deforms that of the Lorentzian function and is determined mainly by the strain value of the fiber and the length of the strained section within the spatial resolution, that is, the ratio r. (3) When the strain is large enough and r is more than 0.5, the superposed spectrum contains two peaks and the calculated strain with the peak frequency of the superposed spectrum is almost equal to the actual strain. (4) When the strain is small, the superposed spectrum contains only one peak. The linear equation proposed by Ohsaki is tenable. The actual stain can be approximately obtained by modifying the calculated strain of BOTDR with a ratio parameter. The above theoretical analysis is of significance for improving the spatial resolution with the method of spectral decomposition proposed by the author [14], which has been successfully verified by the experimental results conducted by the author. Similar experimental results can also be found in the researches of Ohsaki and Murayama [11,18].
Acknowledgments It is gratefully noted that the project is supported by the National Natural Science Foundation of China (40702045), National Science Fund for Distinguished Young Scholars of China (40225006) and Nanjing University Talent Development Foundation. We would like to express our thanks to Prof. Xiaoyi Bao, Department of Physics of the University of Ottawa, Canada and her group for instructive discussing with them.
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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Brillouin power spectrum analysis for partially uniformly strained optical fiber Dan Zhang a,, Hongzhong Xu b, Bin Shi a, Haibo Sui a, Guangqing Wei a a b
Center for Engineering Monitoring with Opto-Electronic Sensing (CEMOES), Nanjing University, 210093 Nanjing, China College of Civil Engineering, Nanjing University of Technology, 210009 Nanjing, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 2 August 2007 Received in revised form 25 February 2009 Accepted 16 April 2009 Available online 17 May 2009
Due to the restriction of the spatial resolution, about 1 m for current commercially available system, strain distribution measured by Brillouin optical time domain reflectometer (BOTDR) is slightly different from the actual one. In this paper, the equation of the Brillouin power spectrum for partially uniformly strained fiber within the spatial resolution is theoretically derived. Based on the derived results, investigation has been made on the shape characteristics of the superposed Brillouin power spectrum, as well as the dependence of the calculated strain of BOTDR on the actual strain of the fiber. It was found that the difference between the calculated strain and the actual strain depends mainly on the strain value of the fiber and the strained length within the spatial resolution for the given distributed sensing system. & 2009 Elsevier Ltd. All rights reserved.
Keywords: BOTDR Spatial resolution Power spectrum Strain measurement Superposition
1. Introduction Distributed fiber optic sensor based on Brillouin scattering is currently attracting considerable research interest in various fields, such as composite material, civil structure, pipeline and geotechnical engineering, owing to its unrivalled capability to provide a continuous strain distribution along the sensing fiber [1–3]. Brillouin optical time domain reflectometer (BOTDR) and Brillouin optical time domain analysis (BOTDA) are two common and commercially available distributed sensing systems. Though the set-ups of the two systems are different, strain and temperature of the fiber are similarly calculated by fitting the peak of Brillouin backscattered light power spectrum (BS) on the basis of the strain or temperature dependence on the Brillouin frequency shift. Theoretical investigations show that the shape of the BS changes not only depending on the pulse width but also on the shape of the launched light [4,5]. And an in-depth study on the signal processing has been conducted widely as power spectrum shape analysis is an important subject for BOTDR/BOTDA system to determine the peak frequency correctly [6,7]. Actually, due to the restriction of spatial resolution, the spectrum shape also changes with the strain distribution along the fiber. In such aspect, Naruse et al. did researches on the deformed BS caused by linear and parabolic strain distribution and analyzed their influences on the strain measurement [8,9].
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E-mail address: [email protected] (D. Zhang). 0143-8166/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2009.04.008
In practice, it is found that, when the fiber is uniformly strained and the strained length is larger than the spatial resolution of BOTDR/BOTDA, the measured stain is equal to the actual value of the fiber. Whereas, if the fiber is non-uniformly strained or the strained length is less than the spatial resolution, Brillouin power spectrum is distorted, and even several peaks can be found in the BS. This leads to inaccurate measurement of the peak frequency and the measured strain distribution is slightly different from the actual one. Meanwhile, strained length obtained by BOTDR/BOTDA is also usually larger than the actual one when the fiber is slightly strained. These indicate that it is impossible to detect inhomogeneous strain distribution accurately or to determine the position and length of the strained section along the fiber exactly with current sensing system due to the limitation of the spatial resolution. In order to enhance the performance of BOTDR/BOTDA, several methods have been proposed to improve the spatial resolution without further narrowing the pulse width of the incident light [10–14]. Therein, spectral decomposition is a method based on the further analysis of the measured Brillouin spectrum, which is considered as the superposition of all the Brillouin gain spectra within the spatial resolution. Moreover, some new systems based on Brillouin fiber sensing are proposed for distributed measurement with very high spatial resolution, such as Brillouin echo distributed sensing (BEDS) with a spatial resolution of 10 cm and Brillouin optical correlation-domain reflectometry (BOCDR) with a spatial resolution of 13 mm [15,16]. Potentially, these systems are possibly available for engineering applications in the near future.
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In this paper, we give the theoretical basis for the spectral decomposition method used to improve the spatial resolution of the current BOTDR system as well as the explanation of the difference between the actual strain distribution and the calculated strain distribution. Here, the equation of the superposed BS for partially uniformly strained optical fiber is first proposed. Then, shape analyses on BS are performed for different strain and strained length, and discrepancy between the actual strain and the calculated strain by fitting the superposed spectrum is also illustrated.
2. Brillouin backscattered light For the BOTDR system [17], pulsed light is launched into one end of a single-mode optical fiber, and the Brillouin backscattered light power dPB (z, v) that is produced in a small section dz of the fiber and detected at the same end is given by Eq. (1) without the consideration of the pump depletion. dP B ðz; vÞ ¼ gðv; vB Þ
gðv; vB Þ ¼
c pðzÞ dz expð2az zÞ, 2n
hðw=2Þ2 ðv vB Þ2 þ ðw=2Þ2
,
(1)
(2)
z ¼ ct=ð2nÞ,
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sampling position can be determined by Z zi þDz=2 Gi ðnÞ ¼ gfn; nB ½ðzÞg dz,
(5)
zi Dz=2
where zi is the position of the i-th sampling point along the fiber. 3. Superposition of Brillouin power spectrum A simplified case, as shown in Fig. 1, is investigated here to reveal the characteristics of the superposed BS: the length of strained fiber is less than the spatial resolution of BOTDR, that is to say, the fiber within the spatial resolution is partially strained and the fiber’s strain is uniform. The measured Brillouin spectrum by BOTDR is actually the superposition of Brillouin gain spectrum within the spatial resolution, including Brillouin gain spectrum of the strained fiber and that of the unstrained fiber. It can be found that the unstrained length within the spatial resolution and the strain value of the strained section exert an influence on the shape of the superposed spectrum, which results in the difference between the actual strain and the calculated stain by fitting the highest peak of the superposed spectrum with Lorentzian curve. The analysis on the shape of the superposed spectrum is based on the followings assumptions:
(3)
where z is the distance along the fiber from the light input end, p(z) is the launched light power at z, v is the optical frequency of the Brillouin backscattered light, c is the velocity of light in vacuum, n is the refractive index of the fiber, az is the attenuation coefficient of the fiber, g(v, vB) is the Brillouin gain spectrum given by a Lorentzian function, vB is the frequency at which g(v, vB) has a peak value h, w is the FWHM of g(v, vB) and t is the time interval between the pulse light’s being launched and the scattered light’s being detected. According to the former researches, Eqs. (1) and (2) are valid in the presented form as long as the pulse width is larger than the phonon lifetime (E10 ns) [5,7]. Just like other OTDR technology, the spatial resolution of BOTDR is determined by the pulse width of the incident light, which can be expressed as
DZ ¼ ct=ð2nÞ,
(4)
where t is the pulse width of the incident light. The strain or temperature at every sampling point is calculated by fitting the BS with Lorentzian function. The BS of the i-th
(1) The electric field of the Brillouin backscattered light produced at every scattering point in the fiber does not exhibit any phase correlation, so the principle of superposition holds with respect to the backscattered light power. (2) The dependence of the Brillouin line-width and peak power on strain and temperature is ignored. (3) The Brillouin gain coefficient is constant over the length of a fiber section of constant strain. If the strained length is less than the spatial resolution, namely, if the fiber is subjected to two distinct but uniform strains (the strains are e and 0, respectively, as shown in Fig. 1) over the spatial resolution, according to Eq. (5), the BS is given by Z zi Dz=2þl Z zi þDz=2 Gi ðnÞ ¼ gfn; nB ½ðzÞg dz þ g½n; nB ð0Þ dz, (6) zi Dz=2
zi Dz=2þl
where l, in the range of 0 and Dz, is the length of the strained fiber. As the fiber’s stain within the spatial resolution is uniform, the BS measured by BOTDR system can be presented theoretically by
Strain
Brillouin Scattered Power
Strained Section 1:BS of Unstrained Section 2:BS of Strained Section 3:BS measured by BOTDR
ε=0
Distance Optical Fiber
Spatial Resolution
Frequency Fig. 1. Demonstration of the Brillouin power spectra when the strained length is less than the spatial resolution.
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the following equation, which is changed from Eq. (6): Gi ðnÞ ¼
g 0 ðDnB =2Þ2 ðn nB ð0Þ DnðÞÞ2 þ ðDnB =2Þ2 þ
the actual strain of the fiber and the real roots of Eq. (12), by which the measured strain is obtained is illustrated as follows.
l 4. Analysis of the superposed Brillouin power spectrum
g 0 ðDnB =2Þ2 2
2
ðn nB ð0ÞÞ þ ðDnB =2Þ
ðDz lÞ,
(7)
where Dn(e) is the Brillouin frequency shift of the strained fiber. Being divided by Dz on both sides, Eq. (7) is shown as follows: Gi ðnÞ g 0 ðDnB =2Þ2 ¼ ð1 rÞ Dz ðn nB ð0ÞÞ2 þ ðDnB =2Þ2 g 0 ðDnB =2Þ2
r ðn nB ð0Þ DnðÞÞ2 þ ðDnB =2Þ2 " 1r ¼ g 0 ðDnB =2Þ2 ðn nB ð0ÞÞ2 þ ðDnB =2Þ2 # r , þ ðn nB ð0Þ DnðÞÞ2 þ ðDnB =2Þ2 þ
4.1. Brillouin spectrum when r is equal to 0.5
(8)
where parameter r is defined as the ratio of the strained length l to the spatial resolution Dz. By differentiating Gi(v)/Dz given by Eq. (8) with respect to v, the optical frequency of the Brillouin backscattered light, we get ! dGi ðnÞ 2 ð1 rÞðn nB ð0ÞÞ ¼ dn g 0 Dn2B Dz ½ðn nB ð0ÞÞ2 þ ðDnB =2Þ2 2 þ
rðn nB ð0Þ DnðÞÞ ½ðn nB ð0Þ DnðÞÞ2 þ ðDnB =2Þ2 2
.
(9)
The peak frequency of the measured BS is obtained from the zeros of derivative of Gi(v)/Dz with respect to v as shown in Eq. (10) ! dGi ðnÞ 2 ¼ 0. dn g 0 Dn2B Dz
(10)
Then, Eq. (9) is given as ð1 rÞx ½x2 þ a2 2
þ
rðx bÞ ½ðx bÞ2 þ a2 2
¼ 0,
(11)
where x ¼ (nnB(0)), a2 ¼ (DnB/2)2 and b ¼ Dn(e). And Eq. (11) can be rewritten as x5 þ Ax4 þ Bx3 þ Cx2 þ Dx þ E ¼ 0.
Based on the above analysis, the shape of the BS depends not only on the strained length within the spatial resolution but also on the strain value of the fiber. In order to investigate the characteristics of the superposed BS of partially and uniformly strained fiber, analyses on the shape of the BS and difference between the calculated strain and actual strain are conducted when r is 0.4, 0.5 and 0.6, respectively.
Fig. 2 shows the shape of the BS when r, the ratio of the strained length to the spatial resolution, is equal to 0.5. The power and the sweep frequency of the spectrum are normalized by g0Dn2B Dz and nB(0), respectively. The strain of the fiber is normalized by e0, the minimal actual stain which makes the measured BS contains two peaks. We use eNA and eNC to denote the normalized actual strain and normalized calculated strain by BOTDR, respectively. The shape of the normalized BS deforms that of the Lorentzian function of Eq. (2) as it is a superposition of spectrum of the strained section and unstrained section. It is also seen from Fig. 2 that the superposed spectrum is symmetric and shows two peaks when the stain is large enough because of the equal contribution of the strained section and the unstrained section. By fitting the right peak of the spectrum with Lorentzian function, the fiber’s strain can be calculated approximately with the obtained peak frequency. By investigating the curves in Fig. 2 and the relationship in Fig. 3, it can be found that the superposed spectrum contains only one peak when the stain is small and shows a linear relationship between the normalized calculated strain eNC and the normalized actual strain eNA. The actual strain can be estimated with the calculated strain of BOTDR by the linear relationship. Nevertheless, when the fiber’s strain becomes larger, the superposed spectrum contains two peaks. By fitting the peak of higher frequency in Fig. 2, the calculated strain is close to the actual strain with the increase of the strain, as the A branch shown in Fig. 3. And if we fit the peak of lower frequency, the calculated strain is close to zero, as the B branch shown.
(12)
Lorentzian curve
1.0
εNA=0.35
ε=0
0.70
2
2
B ¼ 6rb þ 6b þ 2a2 3
3
C ¼ rð4b þ 2ba2 Þ 4b 4ba2 2
4
2
4
D ¼ rð2a2 b þ b Þ þ a4 þ 2a2 b þ b 4
E ¼ ra b.
1.05 1.40 1.75
0.8
2
A ¼ 3rb 4b
Normalized power G/[g0ΔνΒ Δz]
The coefficients of Eq. (12) are given by
(13)
Eq. (12) is the theoretical BS when the fiber is partially and uniformly strained within the spatial resolution. It may have one real root or three real roots, which depends on the parameters A to E of the equation. Generally, if the strain within the spatial resolution is small, the measured BS, namely, the superposed Brillouin spectrum contains one peak and Eq. (12) has one real root accordingly. However, in case the strain within the spatial resolution is large enough, the measured BS usually contains two peaks and Eq. (12) has three real roots. The relationship between
2.10
0.6
0.4
0.2
0.0 -3
-2
-1
0
1
2
3
4
Normalized sweep frequency [ν−νΒ(0)]/0.1 Fig. 2. Shape of the BS caused by different strain when r is 0.5.
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4.2. Brillouin power spectrum when r is less than 0.5 Fig. 4 shows the superposed BS of the strained section and unstrained section within the spatial resolution when r is equal to 0.4. The normalized actual strain eNA in Fig. 3 is 0.35, 0.7, 1.05, 1.4, 1.75 and 2.1, respectively. It is found that the shape of the superposed spectrum deforms and is not symmetric with respect to the peak frequency, which is different with that of Lorentzian function. The Brillouin line-width is also broadened. With the increase of fiber’s strain, two peaks appears in the superposed spectrum, of which, approximately, the peak of lower frequency (the left peak with higher power) coincides with the spectrum of the unstrained section and peak of higher frequency (the right peak with lower power) coincides with the spectrum of the strained section within the spatial resolution. Therefore, the fiber’s real strain can be estimated with the peak frequency of the right peak. However, in order to automate the data processing, the strain of the fiber is calculated by picking the highest peak of the BS in most commercially available BOTDR system. According to this method, the calculated strain with Fig. 4 will deviate from the actual stain of the fiber undoubtedly and the
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error becomes bigger and bigger with the increase of the fiber’s actual strain. The difference between the measured strain calculated with the superposed spectrum and the actual strain is shown in Fig. 5 when r is 0.1, 0.2, 0.3 and 0.4, respectively. With the increase of the fiber’s actual strain, the influence of the Brillouin power of the strained fiber decreases on the superposed spectrum and the calculated strain with the peak frequency of the superposed spectrum is diminished to zero. It is also seen from Fig. 5 that the calculated strain is close to the actual strain according to the increase of r. However, the difference between them is very large and cannot be ignored whenever the fiber’s actual strain is large or small. Similar to Fig. 3, the relationship between the calculated strain and the actual strain is linear approximately when the fiber is slightly strained (normalized actual strain eNA is less than about 0.3).
4.3. Brillouin power spectrum when r is larger than 0.5 Fig. 6 shows the superposed BS of the strained section and unstrained section within the spatial resolution when r is equal to
Actual strain (r = 1.00)
0.5 Calculated strain with superposed spectrum εNC
Calculated strain with superposed spectrum εNC
2.0 Actual strain (r = 1.00) A 1.5
1.0 r = 0.5 0.5
0.4 0.40
0.3
0.30 0.2
0.20 r = 0.10
0.1
B 0.0 0.0
0.5 1.0 1.5 Actual strain within the spatial resolution εNA
2.0
Fig. 3. Relationship between the calculated strain with the superposed BS and the actual strain of the fiber within the spatial resolution when r is 0.5.
Lorentzian curve ε=0
1.0
1.5
2.0
εΝΑ=0.35
Lorentzian curve ε=0
0.70 1.05 1.40 1.75
0.8
2
1.05 1.40 1.75
0.6
1.0
Fig. 5. Relationship between the calculated strain with the superposed BS and the actual strain of the fiber within the spatial resolution when r is less than 0.5.
εΝΑ = 0.35 0.70
0.8
0.5
Actual strain whitin the spatial resolution εNA
Normalized power G/[g0ΔνΒ Δz]
Normorlized power G/[g0ΔνΒ2Δz]
1.0
0.0 0.0
2.10 0.4
0.2
0.0
2.10 0.6
0.4
0.2
0.0
-3
-2 -1 0 1 2 3 4 Normorlized sweeep frequency [ν−νΒ (0)]/0.1
Fig. 4. Shape of the BS caused by different strain when r is 0.4.
5
-3
-2
-1
0
1
2
3
4
Normalized sweep frequency [ν−νΒ(0)]/0.1 Fig. 6. Shape of the BS caused by different strain when r is 0.6.
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0.6. Similar to Fig. 4, the line-width of the BS is also broadened and the profile of it is asymmetric. In contrast, the peak power with higher frequency, which coincides with that of Brillouin spectrum of strained section, is larger. Therefore, it is possible to roughly estimate the actual strain of the fiber with the peak frequency of the superposed spectrum for most commercially available BOTDR system. Fig. 7 shows the deviation of the calculated stain with the superposed spectrum from the actual stain when r is 0.6, 0.7, 0.8 and 0.9, respectively. It is seen from Fig. 7 that the calculated strain is almost close to the actual strain whenever the strain is small or large. And the error between them is relatively large when the normalized strain eNA is approximately within the range 0.3–1.4. Moreover, with the increase of r, the calculated strain with the superposed spectrum approaches the fiber’s actual strain.
100
εNA = 0.9 εNA = 0.7
80
RE %
980
60 εNA = 0.5 εNA = 0.3
40
εNA = 0.1 20 0 0.0
0.2
0.4
0.6
0.8
1.0
r
We conducted an analysis of the errors on determining the fiber’s actual strain while fitting the peak of the superposed BS by introducing the relative error relation RE ¼ |(eNAeNC)/eNA|, where eNA is the normalized actual strain and eNC is the normalized calculated strain of BOTDR as above. Results of the error analysis are showed in Figs. 8 and 9 and can be summarized as follows: (1) As the ratio r increases, the relative error RE decreases without reference to the value of eNA. (2) When the normalized actual strain eNA is less than 0.3, the RE decreases approximately linearly with the increase of r, as shown in Fig. 8. In other words, we can establish a linear relation between normalized eNA and eNC with the known ratio r if the strain of fiber is small. Similar results can also be found in the researches of Ohsaki. It can be concluded that for a small strain of less than about 103 there held a simple relation of xp ¼ a r between the apparent strain xp obtained by BOTDR and the accurate strain a, where r denotes the ratio of the strained length within the spatial resolution [11]. Therefore, by using the equation proposed by Ohsaki, the actual strain can be approximately estimated with the measured strain of BOTDR. However, when the normalized actual strain eNA is larger than 0.3, the linear relationship is
Calculated strain with superposed spectra εNC
Actual strain (r = 1.00) 1.5 0.90 0.80 0.70 r = 0.60 0.5
0.0 0.0
εΝΑ = 2.0
100 80 60
εΝΑ = 1.8 εΝΑ = 1.6 εΝΑ = 1.4 εΝΑ = 1.2 εΝΑ = 1.0 εΝΑ = 1.0
40
εΝΑ = 1.2 εΝΑ = 1.4 εΝΑ = 1.6 εΝΑ = 1.8
20 εΝΑ = 2.0
0 0.0
0.2
0.4
0.6
0.8
1.0
r Fig. 9. Dependence of the relative error on r when the normalized actual strain eNA is larger than 1.0.
not tenable any more and the nonlinearity become larger according to the increase of eNA. (3) As the actual strain of the fiber is large (the normalized strain eNA is larger than 1.0), the relationship between the RE and r is shown in Fig. 9. If r is larger than 0.5, the relative error ER is close to zero according to the increase of actual strain and r. The calculated strain eNC is almost equal to the actual strain eNA. In contrast, when the r is less than 0.5, the relative error ER is approaching 100%. In another word, the normalized calculated strain eNC is close to zero as the BOTDR system automatically fits the peak of higher power which coincides with the Brillouin spectrum of the unstrained section, although the superposed BS contains two peak.
2.0
1.0
Fig. 8. Dependence of relative error on r when the normalized actual strain eNA is less than 1.0.
RE %
4.4. Discussion
5. Conclusions 0.5 1.0 1.5 Actual strain within the spatial resolution εNA
2.0
Fig. 7. Relationship between the calculated strain with the superposed BS and the actual strain of the fiber within the spatial resolution when r is larger than 0.5.
According to the derived equation of the BS for partially uniformly strained fiber within the spatial resolution, investigation has been made on the shape characteristics of the superposed BS with the changes of the fiber’s strain and the ratio r, as well as
ARTICLE IN PRESS D. Zhang et al. / Optics and Lasers in Engineering 47 (2009) 976–981
the dependence of the measured strain on the actual strain of the fiber. This investigation yields the following results: (1) If the length of the strained fiber is less than the spatial resolution, the measured BS by BOTDR is a superposed spectrum of the strained section and the unstrained section within the spatial resolution. The superposed spectrum has an asymmetric shape except r, ratio of the strained length to the spatial resolution, is equal to 0.5. (2) The shape of the superposed spectrum deforms that of the Lorentzian function and is determined mainly by the strain value of the fiber and the length of the strained section within the spatial resolution, that is, the ratio r. (3) When the strain is large enough and r is more than 0.5, the superposed spectrum contains two peaks and the calculated strain with the peak frequency of the superposed spectrum is almost equal to the actual strain. (4) When the strain is small, the superposed spectrum contains only one peak. The linear equation proposed by Ohsaki is tenable. The actual stain can be approximately obtained by modifying the calculated strain of BOTDR with a ratio parameter. The above theoretical analysis is of significance for improving the spatial resolution with the method of spectral decomposition proposed by the author [14], which has been successfully verified by the experimental results conducted by the author. Similar experimental results can also be found in the researches of Ohsaki and Murayama [11,18].
Acknowledgments It is gratefully noted that the project is supported by the National Natural Science Foundation of China (40702045), National Science Fund for Distinguished Young Scholars of China (40225006) and Nanjing University Talent Development Foundation. We would like to express our thanks to Prof. Xiaoyi Bao, Department of Physics of the University of Ottawa, Canada and her group for instructive discussing with them.
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