Compact micro-displacement sensor with high sensitivity based on a long-period fiber grating with an air-cavity

doi:10.1364/OE.21.003193

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Compact micro-displacement sensor with high sensitivity based on a long-period fiber grating with an air-cavity

Liang Qi, Chun-Liu Zhao, Yunpeng Wang, Juan Kang, Zaixuan Zhang, and Shangzhong Jin »View Author Affiliations

Optics Express, Vol. 21, Issue 3, pp. 3193-3200 (2013)
http://dx.doi.org/10.1364/OE.21.003193

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▸ Article Outline
– Abstract
1. Introduction
2. Experimental setup
3. Sensing principle
4. Experimental results
5. Conclusion
– References and links

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Abstract

We proposed and experimentally demonstrated a compact micro-displacement sensor with high sensitivity based on a long-period fiber grating (LPG) with an air-cavity. The sensor head is obtained by composing an air-cavity with the ends of a LPG and a single mode fiber (SMF). The wavelength shift of the LPG has a linear relationship with the length of the air gap which agrees well with the theoretical analysis. The experimental results show that the sensitivity is ~0.22 nm/µm within the micro-displacement range of 0 to 140 µm.

1. Introduction

Displacement measurement is one of the key factors in various fields of nanotechnology systems, such as micro-manufacture, precisely positioning, scanning tunneling microscopy (STM), atomic force microscopy (AFM) and so on. Several types of displacement sensors, such as capacitive sensors and inductive sensors, have been proposed [1

H. G. Xu, T. Ono, and M. Esashi, “Precise motion control of a nanopositioning PZT microstage using integrated capacitive displacement sensors,” J. Micromech. Microeng. 16(12), 2747–2754 (2006). [CrossRef]

, 2

S. Fericean and R. Droxler, “New noncontacting inductive analog proximity and inductive linear displacement sensors for industrial automation,” IEEE Sens. J. 7(11), 1538–1545 (2007). [CrossRef]

]. Compared with displacement sensors based on electronic components, fiber optical displacement sensors have important advantages of electrically-passive operation and immunity to electromagnetic interference. Recently, fiber grating based displacement sensors have been widely studied, which mainly use fiber Bragg grating (FBG) [3

J. H. Ng, X. Zhou, X. Yang, and J. Hao, “A simple temperature-insensitive fiber Bragg grating displacement sensor,” Opt. Commun. 273(2), 398–401 (2007). [CrossRef]

–8

T. Guo, C. Chen, and J. Albert, “Non-uniform-tilt-modulated fiber Bragg grating for temperature-immune micro-displacement measurement,” Meas. Sci. Technol. 20(3), 034007 (2009). [CrossRef]

], tilted fiber Bragg gratings (TFBGs) [9

Q. Jiang and D. Hu, “Microdisplacement sensor based on tilted fiber Bragg grating transversal load effect,” IEEE Sens. J. 11(9), 1776–1779 (2011). [CrossRef]

] and long-period gratings (LPGs) [10

J. M. Baptista, S. F. Santos, G. Rego, O. Frazão, and J. L. Santos, “Micro-displacement or bending measurement using a long-period fibre grating in a self-referenced fibre optic intensity sensor,” Opt. Commun. 260(1), 8–11 (2006). [CrossRef]

]. Most of the proposed FBG displacement sensors are structures that glue a FBG on a cantilevered beam [4

X. Y. Dong, X. Yang, C.-L. Zhao, L. Ding, P. Shum, and N. Q. Ngo, “A novel temperature-insensitive fiber Bragg grating sensor for displacement measurement,” Smart Mater. Struct. 14(2), N7–N10 (2005). [CrossRef]

–8

T. Guo, C. Chen, and J. Albert, “Non-uniform-tilt-modulated fiber Bragg grating for temperature-immune micro-displacement measurement,” Meas. Sci. Technol. 20(3), 034007 (2009). [CrossRef]

]. Those sensors need employ relatively complicated force transfer components, leading to a large error introduced to the sensor, so that they are not practical for micro-displacement measurement.

Due to a large period of perturbations along longitudinal direction of optical fiber, LPG couples light from the fundamental core mode to some cladding modes inducing a series of attenuation bands in the transmission spectrum. LPGs have been employed to measure the displacement variation utilizing the property of LPGs offering high bending sensitivity [10

]. However, the large volume and complicated fabrication of the LPG displacement sensor is unsuitable for applications in confined spaces such as precision instruments, piezoelectric device etc.

In this paper, a compact micro-displacement sensor with high sensitivity based on a long-period fiber grating with an air-cavity is proposed and experimentally demonstrated. The cavity is composed by the ends of an LPG and a single mode fiber. The transmission spectrum of the LPG is modulated by the air-cavity leading to the dip wavelength of the LPG gradually shift with the length of the air-cavity increasing. By monitoring the dip wavelength shift of the LPG, we can obtain the variation of the length of the air-cavity. Experimental results show that the proposed LPG-based displacement sensor works well, and the sensitivity of the sensor within the measurement range of 0 to 140 μm is about 0.22 nm/μm. Compared with FBG displacement sensors, the proposed displacement sensor based on LPG with an air cavity has a high sensitivity.

2. Experimental setup

Figure 1 shows the proposed micro-displacement sensor that consists of a LPG and a SMF. In the experiment, the LPG and the SMF are packaged inside a fiber capillary with inner diameter, outer diameter and length of 127 μm, 300 μm and 3cm, respectively, which is used to align the two fibers and to protect the sensor from environmental damage. The SMF is glued on the fiber capillary, while the LPG can be moved axially along the capillary. The LPG and the fiber capillary are clamped in two holders which are fixed on two translation stages. The gap between the LPG’s end and the SMF’s end is used as the sensing part, and is controlled within 200 μm by adjusting one translation stage. The transmission spectra are measured by using a superluminescent light-emitting diode light source (SLED) whose wavelength range is from 1200 nm to 1700 nm, and an optical spectrum analyzer (OSA, YOKOGAWA735301) with a spectral resolution of 0.02 nm. The LPG used in the experiment is fabricated by using a high-frequency pulsed CO₂-laser (CO2-H10, Han's Laser) with a maximum average output power of 10 W. The detail manufacturing process refers to our team's relavent works [11

Y. Wang, C.-L. Zhao, L. Hu, X. Dong, Y. Jin, C. Shen, and S. Jin, “A tilt sensor with a compact dimension based on a long-period fiber grating,” Rev. Sci. Instrum. 82(9), 093106 (2011). [CrossRef] [PubMed]

]. The grating pitch of the LPG is about 530 μm and the grating length is 20.12 mm (40 grating periods).

Fig. 1 Experimental setup of the proposed sensor.

Figure 2 shows the transmission spectra of the used LPG and the proposed sensor. For the original LPG, a resonant dip can be observed at about 1559 nm wavelength and the depth of the dip is nearly 35 dB which is due to the fundamental core mode coupling to one cladding mode. For the proposed structure, the light from the end of the LPG directly launches into the air cavity and interferes in the air cavity formed by the ends of the LPG and the SMF. It is shown that the output intensity of the proposed sensor in the whole wavelength range is lower than that of the original LPG due to the air cavity induced loss. The resonant wavelength shifts towards a shorter wavelength direction, and the depth becomes smaller. Meanwhile, the interference fringes appear in the range of 1400nm to 1500nm, as shown in the dashed rectangle in Fig. 2.

Fig. 2 Transmission spectra of the original LPG (in red), and the LPG with air cavity (in black).

3. Sensing principle

In the proposed structure, the air cavity is cascaded closely to the LPG’s grating area. Thus, both lights distributed in the core and the cladding at the end of the LPG, which are just called the “core” mode and the “cladding” modes and are formed by mode coupling with phase matching condition, enter into the air cavity and interfere simultaneously. The transmission lights from the interference cavity launch into the SMF (right), then propagate along the SMF. Since large propagation losses of cladding modes, only core mode exists in the SMF after propagating several centimeters. The diagrammatic sketch of the above process is shown in the lower right of Fig. 1.

According to the mode coupling theory, when the light propagates through the LPG by the length of L, the amplitudes of the core mode and the v-th co-propagating cladding mode can be expressed as following [12

Y. G. Han, B. H. Lee, W. T. Han, U. C. Peak, and Y. Chung, “Controllable transmission characteristics of multi-channel long period fiber gratings,” IEICE Trans. Electron. E 84-C(5), 610–614 (2001).

,13

X. Shu, L. Zhang, and I. Bennion, “Sensitivity characteristics of long period fiber gratings,” J. Lightwave Technol. 20(2), 255–266 (2002). [CrossRef]

{\begin{cases} a_{c o} (L)= \exp (i (β_{co} - \frac{1}{2} Δ β^{ν})L) [a_{c o} (0) (\cos s L +i \frac{Δ β^{ν}}{2s} \sin s L) + a_{c l}^{ν} (0) \frac{i κ}{s} \sin s L] \\ a_{c l}^{ν} (L)= \exp (i (β_{c l}^{ν} - \frac{1}{2} Δ β^{ν})L) [a_{c o} (0) \frac{i κ^{*}}{s} \sin s L + a_{c l}^{ν} (0) (\cos s L - i \frac{Δ β^{ν}}{2s} \sin s L)] \end{cases}

(1)

where

s

and

Δ β^{ν}

are defined as

Δ β^{ν} = β_{c o} - β_{c l}^{ν} - \frac{2 π}{Λ}

(2)

s = {(κ^{*} κ + \frac{Δ β^{ν}^{2}}{4})}^{\frac{1}{2}}

(3)

and

κ

Λ

β_{c o}

and

β_{c l}^{ν}

are, respectively, the coupling coefficient, grating period, the propagation constant of the core mode and the

ν

number co-propagating cladding mode, and the initial conditions are

a_{c o} (0) = 1

a_{c l}^{ν} (0) = 0

. Because light intensity equals to the product of amplitude and its conjugate, Eq. (1) can describe the modified power distribution between the core mode and the co-propagating higher order modes at the output end of a LPG. For one given LPG, the amplitudes of the core mode and the cladding mode at one certain wavelength are determinate. Then those modes launch into the air cavity simultaneously, and the interference happens here. Here, the matrix method is used for analysis the function of the air cavity since the air cavity can be considered as a phase shift which applies on all the light inside.

For an air cavity, its matrix can be expressed as [14

W. Zhou, X. Dong, L.-Y. Shao, C. C. Chan, C.-L. Zhao, and P. Shum, “Compact refractometer based on extrinsic-phase-shift fiber Bragg grating,” Sens. Actuators A Phys. 168(1), 46–50 (2011). [CrossRef]

]

M= [\begin{array}{l} A \\ E \end{array} \begin{array}{l} B \\ F \end{array}] / t_{g a p}^{2}

(4)

A = e^{i \cdot f} / α - α \cdot r_{g a p}^{2} \cdot e^{-i \cdot f}

(5)

B = - r_{g a p} \cdot e^{i \cdot f} / α + α \cdot r_{g a p} \cdot e^{-i \cdot f}

(6)

E = r_{g a p} \cdot e^{i \cdot f} / α - α \cdot r_{g a p} \cdot e^{-i \cdot f}

(7)

F = - r_{g a p}^{2} \cdot e^{i \cdot f} / α + α \cdot e^{-i \cdot f}

(8)

where the

r_{g a p}

t_{g a p}

and

f

are defined as

r_{g a p} = \frac{n_{eff} {-n}_{gap}}{n_{eff} {+n}_{gap}}

(9)

t_{g a p} = \sqrt{1- r_{g a p}^{2}}

(10)

f = \frac{2 π n_{gap} L_{g a p}}{λ}

(11)

and the

α

n_{neff}

n_{gap}

L_{g a p}

and

λ

are, respectively, the gap-induced loss, the effective index of the core mode, the refractive index and the length of the air gap and the operating wavelength. Therefore, the amplitudes of the light propagating through the LPG and the air cavity, can be calculated by

[\begin{array}{l} a_{c o} \\ a_{c l}^{ν} \end{array}] = M \cdot [\begin{array}{l} a_{c o} (L) \\ a_{c l}^{ν} (L) \end{array}]

(12)

According to Eq. (9), (10), and (11), the parameters of

a

r_{g a p}

t_{g a p}

in Eq. (4) are constants. For one given wavelength, the transmission matrix of

M

is only dependent on the parameter of

L_{g a p}

. In long-distance transmission, because of the attenuation of cladding modes, only core mode can be detected at the receiving terminal. So the above equation can be simplified as

[a_{c o}] = [A B] / t_{g a p}^{2} \cdot [\begin{array}{l} a_{c o} (L) \\ a_{c l}^{ν} (L) \end{array}]

(13)

Equation (13) contains a periodic factor of

e^{\pm i \cdot f}

which makes the

a_{c o}

a periodic equation of

f

, so

a_{c o}

has its minimum value where

f

equals to a certain constant

C

. In addition, the

L_{g a p}

consists of an initial value

L_{g a p 0}

and the displacement variation of

Δ L_{g a p}

. Then Eq. (11) can be expressed as

λ = \frac{2 π n_{g a p} Δ L_{g a p}}{C} + \frac{2 π n_{g a p} L_{g a p 0}}{C}

(14)

Then the resonant wavelength

λ

has a linear relationship with the displacement variation. Due to the periodicity of

e^{\pm i \cdot f}

, the

C

can be chosen with a small constant so that the slope coefficient in Eq. (14) will become bigger and then the sensitivity is enhanced.

4. Experimental results

Figure 3 shows the transmission spectra of the proposed sensor when displacements variation step of 10 μm in a range of 0-140 μm is used in the experiment. Increasing the length of the air cavity causes the resonant wavelength to shift towards a longer wavelength direction. The resonant wavelength of the interference fringe, shown in the dashed rectangle, also shift towards a longer wavelength direction, but the shift of the wavelength is not strong.

Fig. 3 Transmission spectra of the LPG with air cavity at different displacement variations.

With the displacement variation

Δ L_{g a p}

increasing from 0 to 140 μm, about 30 nm wavelength shift is achieved. According to the experiment, we get constant

C

equaling to 29.16. The resonant wavelength responding to displacement variation is plotted in Fig. 4 and shows a good linearity which agrees well with Eq. (14). The sensitivity is just the slope of the curve in blue and as high as ~0.216 nm/μm. Furthermore, the intercept of the ordinate axis in Fig. 4 can be controlled by changing the initial cavity length

L_{g a p 0}

. Because of the phase modulation factor of

e^{\pm i \cdot f}

, the transmission intensity of the resonant wavelength has certain periodicity. It can also be seen from Fig. 3 that the transmission intensity at resonant wavelength changes with the increasing length of the air cavity, and reaches the minimum value when the displacement variation is about 50 μm. This trend can also be used as intensity demodulation.

Fig. 4 The relationships of the dip wavelength with the displacement variations of the air cavity (in red) and cavity with LPG (in blue).

For comparison, we also investigated the performance of the air cavity formed only by two SMF ends, as shown in Fig. 5 . The interference extinction ratios are low and about 0.25 dB. As the variation of displacement increases, the interference spacing becomes smaller and the transmission intensity reduces. The interference peak responding to displacement variation is also plotted in Fig. 4 and the sensitivity is ~0.127 nm/μm. Obviously, the proposed LPG with the air cavity has the high extinction ratio and is easy to distinguish the resonant wavelength change. Table 1 illustrates that the proposed sensor has a high sensitivity and is more suitable for micron-dimension-displacement measurement.

Fig. 5 Transmission spectra of the air cavity at different displacement variation. The measured dip wavelength is marked by black arrows.

Table 1 Performance Comparison of Optical Fiber Grating Displacement Sensors

Type	Sensitivity	Displacement range
Bending FBG [3 J. H. Ng, X. Zhou, X. Yang, and J. Hao, “A simple temperature-insensitive fiber Bragg grating displacement sensor,” Opt. Commun. 273(2), 398–401 (2007). [CrossRef] ]	0.6234 nm/mm	0-6 mm
FBG on cantilever [4 X. Y. Dong, X. Yang, C.-L. Zhao, L. Ding, P. Shum, and N. Q. Ngo, “A novel temperature-insensitive fiber Bragg grating sensor for displacement measurement,” Smart Mater. Struct. 14(2), N7–N10 (2005). [CrossRef] ]	0.36 nm/mm	0-22 mm
Compressing TFBG [9 Q. Jiang and D. Hu, “Microdisplacement sensor based on tilted fiber Bragg grating transversal load effect,” IEEE Sens. J. 11(9), 1776–1779 (2011). [CrossRef] ]	0.121 nm/mm	0-1.1 mm
Bending LPG [10 J. M. Baptista, S. F. Santos, G. Rego, O. Frazão, and J. L. Santos, “Micro-displacement or bending measurement using a long-period fibre grating in a self-referenced fibre optic intensity sensor,” Opt. Commun. 260(1), 8–11 (2006). [CrossRef] ]	Average 0.023nm/µm	0-240 µm
Fiber air cavity	0.127 nm/µm	0-155 µm
Proposed sensor	0.216 nm/µm	0-140 µm

In practical applications, the conventional LPG can be replaced by temperature-insensitive one in order to eliminate temperature-induced wavelength shift. Then both the temperature-insensitive LPG and the SMF can be packaged in ZrO2 Ceramic Ferrule Bore which does not exceed 4 cm. After that, utilizing the transmission intensity of the LPG increasing with displacement variation at a fixed wavelength (for example, at 1536 nm in experiment), a low cost micro-displacement sensor can be obtained basing on an intensity measurement technique [15

C.-L. Zhao, L. Xiao, J. Ju, M. S. Demokan, and W. Jin, “Strain and temperature characteristics of a long-period grating written in a photonic crystal fiber and its application as a temperature-insensitive strain sensor,” J. Lightwave Technol. 26(2), 220–227 (2008). [CrossRef]

], in which a single wavelength light source (such as a distributed-feedback-Bragg laser, DFB) and an optical power meter is used. Comparing with FBG-based displacement sensors, the LPG-based micro-displacement sensor has advantages of simple, small and low cost.

5. Conclusion

In conclusion, we have proposed and demonstrated a compact micro-displacement sensor with high sensitivity based on a long-period fiber grating with an air-cavity. The sensing system can detect the displacement varies with a high sensitivity of 0.2155 nm/μm in the measurement range of 0 to 140 μm. Due to compact dimension of the sensor, the proposed displacement sensor has great applications in confined space.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant No. 61108058, the National Basic Research Program of China (973 Program) under Grant No. 2010CB327804, the National Key Technology R&D Program 2011BAF06B02 and the Science and Technology Commission of Shanghai Municipality of China under Grant No. 10595812300.

References and links

1.	H. G. Xu, T. Ono, and M. Esashi, “Precise motion control of a nanopositioning PZT microstage using integrated capacitive displacement sensors,” J. Micromech. Microeng. 16(12), 2747–2754 (2006). [CrossRef]
2.	S. Fericean and R. Droxler, “New noncontacting inductive analog proximity and inductive linear displacement sensors for industrial automation,” IEEE Sens. J. 7(11), 1538–1545 (2007). [CrossRef]
3.	J. H. Ng, X. Zhou, X. Yang, and J. Hao, “A simple temperature-insensitive fiber Bragg grating displacement sensor,” Opt. Commun. 273(2), 398–401 (2007). [CrossRef]
4.	X. Y. Dong, X. Yang, C.-L. Zhao, L. Ding, P. Shum, and N. Q. Ngo, “A novel temperature-insensitive fiber Bragg grating sensor for displacement measurement,” Smart Mater. Struct. 14(2), N7–N10 (2005). [CrossRef]
5.	Y. Zhu, P. Shum, C. Lu, M. Lacquet, P. Swart, A. Chtcherbakov, and S. Spammer, “Temperature insensitive measurements of static displacements using a fiber Bragg grating,” Opt. Express 11(16), 1918–1924 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-16-1918. [CrossRef] [PubMed]
6.	Y. Zhao, H. Huang, and Q. Wang, “Interrogation technique using a novel spectra bandwidth measurement method with a blazed FBG and a fiber-optic array for an FBG displacement sensor,” Sens. Actuators A Phys. 165(2), 185–188 (2011). [CrossRef]
7.	C. Shen and C. Zhong, “Novel temperature-insensitive fiber Bragg grating sensor for displacement Measurement,” Sens. Actuators A Phys. 170(1–2), 51–54 (2011). [CrossRef]
8.	T. Guo, C. Chen, and J. Albert, “Non-uniform-tilt-modulated fiber Bragg grating for temperature-immune micro-displacement measurement,” Meas. Sci. Technol. 20(3), 034007 (2009). [CrossRef]
9.	Q. Jiang and D. Hu, “Microdisplacement sensor based on tilted fiber Bragg grating transversal load effect,” IEEE Sens. J. 11(9), 1776–1779 (2011). [CrossRef]
10.	J. M. Baptista, S. F. Santos, G. Rego, O. Frazão, and J. L. Santos, “Micro-displacement or bending measurement using a long-period fibre grating in a self-referenced fibre optic intensity sensor,” Opt. Commun. 260(1), 8–11 (2006). [CrossRef]
11.	Y. Wang, C.-L. Zhao, L. Hu, X. Dong, Y. Jin, C. Shen, and S. Jin, “A tilt sensor with a compact dimension based on a long-period fiber grating,” Rev. Sci. Instrum. 82(9), 093106 (2011). [CrossRef] [PubMed]
12.	Y. G. Han, B. H. Lee, W. T. Han, U. C. Peak, and Y. Chung, “Controllable transmission characteristics of multi-channel long period fiber gratings,” IEICE Trans. Electron. E 84-C(5), 610–614 (2001).
13.	X. Shu, L. Zhang, and I. Bennion, “Sensitivity characteristics of long period fiber gratings,” J. Lightwave Technol. 20(2), 255–266 (2002). [CrossRef]
14.	W. Zhou, X. Dong, L.-Y. Shao, C. C. Chan, C.-L. Zhao, and P. Shum, “Compact refractometer based on extrinsic-phase-shift fiber Bragg grating,” Sens. Actuators A Phys. 168(1), 46–50 (2011). [CrossRef]
15.	C.-L. Zhao, L. Xiao, J. Ju, M. S. Demokan, and W. Jin, “Strain and temperature characteristics of a long-period grating written in a photonic crystal fiber and its application as a temperature-insensitive strain sensor,” J. Lightwave Technol. 26(2), 220–227 (2008). [CrossRef]

OCIS Codes
(060.2340) Fiber optics and optical communications : Fiber optics components
(060.2370) Fiber optics and optical communications : Fiber optics sensors
(280.4788) Remote sensing and sensors : Optical sensing and sensors

ToC Category:
Sensors

History
Original Manuscript: November 19, 2012
Revised Manuscript: January 23, 2013
Manuscript Accepted: January 24, 2013
Published: February 1, 2013

Citation
Liang Qi, Chun-Liu Zhao, Yunpeng Wang, Juan Kang, Zaixuan Zhang, and Shangzhong Jin, "Compact micro-displacement sensor with high sensitivity based on a long-period fiber grating with an air-cavity," Opt. Express 21, 3193-3200 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-3193

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References

H. G. Xu, T. Ono, and M. Esashi, “Precise motion control of a nanopositioning PZT microstage using integrated capacitive displacement sensors,” J. Micromech. Microeng.16(12), 2747–2754 (2006). [CrossRef]
S. Fericean and R. Droxler, “New noncontacting inductive analog proximity and inductive linear displacement sensors for industrial automation,” IEEE Sens. J.7(11), 1538–1545 (2007). [CrossRef]
J. H. Ng, X. Zhou, X. Yang, and J. Hao, “A simple temperature-insensitive fiber Bragg grating displacement sensor,” Opt. Commun.273(2), 398–401 (2007). [CrossRef]
X. Y. Dong, X. Yang, C.-L. Zhao, L. Ding, P. Shum, and N. Q. Ngo, “A novel temperature-insensitive fiber Bragg grating sensor for displacement measurement,” Smart Mater. Struct.14(2), N7–N10 (2005). [CrossRef]
Y. Zhu, P. Shum, C. Lu, M. Lacquet, P. Swart, A. Chtcherbakov, and S. Spammer, “Temperature insensitive measurements of static displacements using a fiber Bragg grating,” Opt. Express11(16), 1918–1924 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-16-1918 . [CrossRef] [PubMed]
Y. Zhao, H. Huang, and Q. Wang, “Interrogation technique using a novel spectra bandwidth measurement method with a blazed FBG and a fiber-optic array for an FBG displacement sensor,” Sens. Actuators A Phys.165(2), 185–188 (2011). [CrossRef]
C. Shen and C. Zhong, “Novel temperature-insensitive fiber Bragg grating sensor for displacement Measurement,” Sens. Actuators A Phys.170(1–2), 51–54 (2011). [CrossRef]
T. Guo, C. Chen, and J. Albert, “Non-uniform-tilt-modulated fiber Bragg grating for temperature-immune micro-displacement measurement,” Meas. Sci. Technol.20(3), 034007 (2009). [CrossRef]
Q. Jiang and D. Hu, “Microdisplacement sensor based on tilted fiber Bragg grating transversal load effect,” IEEE Sens. J.11(9), 1776–1779 (2011). [CrossRef]
J. M. Baptista, S. F. Santos, G. Rego, O. Frazão, and J. L. Santos, “Micro-displacement or bending measurement using a long-period fibre grating in a self-referenced fibre optic intensity sensor,” Opt. Commun.260(1), 8–11 (2006). [CrossRef]
Y. Wang, C.-L. Zhao, L. Hu, X. Dong, Y. Jin, C. Shen, and S. Jin, “A tilt sensor with a compact dimension based on a long-period fiber grating,” Rev. Sci. Instrum.82(9), 093106 (2011). [CrossRef] [PubMed]
Y. G. Han, B. H. Lee, W. T. Han, U. C. Peak, and Y. Chung, “Controllable transmission characteristics of multi-channel long period fiber gratings,” IEICE Trans. Electron. E84-C(5), 610–614 (2001).
X. Shu, L. Zhang, and I. Bennion, “Sensitivity characteristics of long period fiber gratings,” J. Lightwave Technol.20(2), 255–266 (2002). [CrossRef]
W. Zhou, X. Dong, L.-Y. Shao, C. C. Chan, C.-L. Zhao, and P. Shum, “Compact refractometer based on extrinsic-phase-shift fiber Bragg grating,” Sens. Actuators A Phys.168(1), 46–50 (2011). [CrossRef]
C.-L. Zhao, L. Xiao, J. Ju, M. S. Demokan, and W. Jin, “Strain and temperature characteristics of a long-period grating written in a photonic crystal fiber and its application as a temperature-insensitive strain sensor,” J. Lightwave Technol.26(2), 220–227 (2008). [CrossRef]

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