Abstract

We demonstrate a fiber-optic 3D vector displacement sensor based on the monitoring of Bragg reflection from an eccentric grating inscribed in a depressed-cladding fiber using the femtosecond laser side-illumination and phase-mask technique. The compact sensing probe consists of a short section of depressed cladding fiber (DCF) containing eccentrically positioned fiber Bragg gratings. The eccentric grating breaks the cylindrical symmetry of the fiber cross-section and further has bending orientation-dependence. The generated fundamental resonance is strongly sensitive to bending of the fiber, and the direction of the bending plane can be determined from its responses. When integrated with axis strain monitoring, the sensor achieves a 3D vector displacement measurement via simple geometric analysis.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Vector-displacement information is extremely useful for the fields of mechanical engineering, crack growth monitoring, and robotic arms. Over the years, various compact fiber-bending-based devices have been developed to realize displacement measurements; such devices were found to exhibit excellent performances [1–3]. More importantly, some of these devices can simultaneously respond to both the amplitude and direction of the applied displacement; such capability is urgently required in real applications. Such direction-dependent property usually benefits from a specific asymmetric geometric structure or mode excitation. Utilizing the specially positioned fiber core, the eccentric core fiber or the multicore fiber can be inscribed with a fiber Bragg grating (FBG) to achieve the vector curvature measurement [4–6]. The asymmetric cladding geometry has also been proposed and demonstrated, e.g., a fiber long period grating (LPG) written in D-shaped fiber [7]. Meanwhile, fiber interferometers exploiting special configurations can obtain great directional response because of their well-designed core-cladding coupling mechanism [8,9]. All of the works mentioned above are subject to the strict mode coupling conditions or process parameters. Moreover, the modified fiber of some complicated configurations is somewhat harder to assemble and less robust than normal fiber. The titled fiber Bragg grating (TFBG) sensor, to a certain extent, eliminates these problems and provides a more advanced sensing method with monitoring bending-sensitive cladding modes for displacement measurement [10,11]. However, the TFBG requires long-running exposure processing and a nontrivial inscription technique. The TFGB also does not meet the demands for real application.

In this paper, we propose an eccentric FBG inscribed over part of the core of a depressed-cladding fiber (DCF) for three-dimension (3D) vector displacement measurement. The fiber used has an unusual profile, consisting of one core with a core dip and one depressed cladding [12,13], and the grating structure is easily formed in part of the fiber core by using precise femtosecond laser side-illumination. The formed eccentric FBG breaks the cylindrical symmetry of the fiber. Such a novel profile and FBG position enable realization of high bending responsiveness and sinusoidal bending-direction response in a two-dimension (2D) plane. Moreover, the direction-finding capacity of the proposed device is much improved to 3D space via introduction of the additional one-dimension (1D) fiber-longitudinal-stretch measurement. The sensing mechanism is primarily based on the fiber-bending induced lateral mode-field shift and fiber-tension induced longitudinal strain variation. As a result, the orientation and amplitude of the displacement (bending) can be simultaneously monitored by analyzing the intensity fluctuation and wavelength shift of reflection spectra of FBG.

2. Fabrication and principle

As shown in Fig. 1(a), the grating inscription region is eccentrically positioned in the center dip core. With this configuration, a strong fundamental mode is achieved. The original circular symmetrically fundamental mode from the core of the input SMF is partly coupled into the dip core and then asymmetrically reflected by the eccentric FBG, finally being recoupled back in the input SMF. In contrast to earlier reports of the normal Bragg fundamental mode, this fundamental mode is easy to “leak” out and is determined by fiber-bending induced mode deviation because of its special refractive index profile and eccentric grating location. The detailed fabrication of sensor and sensing mechanisms are presented in the following.

 figure: Fig. 1

Fig. 1 (a) Schematic of the eccentric grating. (b) Photomicrograph of the DCF cross section and the position illustration for normal inscription (white dash). (b) Diffraction patterns of the laser after passing through the fiber core for normal inscription (upper) and off-axis inscription (below).

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2.1 Fabrication of sensor

Figure 1(b) shows the optical micrographs of the DCF cross section; from the inside to the outside, it has a ring core diameter of 8 μm with an inside dip core of 1.6 μm diameter and a depressed inner cladding diameter of 26 μm, with an outer cladding diameter of 101 μm. A short section of DCF will be self-aligned and spliced with the leading-in SMF using a commercial compact fusion splicer for fabrication. Writing the Bragg grating in the entire core can generate a fundamental core resonant mode and several high-order modes in reflection, of which the fundamental mode is easily disturbed by fiber-bending via mode coupling fluctuation and the weak waveguide condition between the core dip and the ring core, as discussed in our previous report [14]. However, the normal FBG written in this DCF cannot respond to the fiber-bending direction because of the symmetrical grating distribution relatively close to the fiber core axis. That behavior has certain superiorities in some applications that diminish its practicability and flexibility to a greater extent. Introducing an asymmetric configuration will directly and efficiently improve the situation. In particular, positioning the FBG away from the core axis can break the cylindrical symmetry of the fiber and sequentially enable it to sense the off-axis strain induced by fiber bending [15,16].

Generally, the eccentric grating structure is fabricated by a tightly focused femtosecond pulse laser and the use of a high-precision translation stage [17,18]. However, in this work, the more efficient and time-saving phase-mask technology is used in combination with femtosecond laser side-illumination for fabrication. The FBG fabrication system is the same as the setup described in [19]. A Ti:sapphire laser system emitting 100 fs pulses of linearly polarized light at the central wavelength of approximately 800 nm (TEM00 spatial mode, 1 kHz repetition rate) is used as the light source. The laser beam was focused to a 3.5 μm-wide line beam by a cylindrical lens with a 25.5 mm focal distance, which is the key process because the micro beam is the premise for precise fabrication. Before the FBG fabrication, the DCF was treated using a 5-day hydrogen-loading process (at 60° and 10 MPa) for improving the fiber photosensitivity. The femtosecond laser is known for fabricating the type-II FBG [20], resulting in dramatic grating extension and refractive index (RI) modification due to the use of high-intensity and ultrashort pulses. Therefore, we strictly control the laser pulse energy (fixed at 0.4 mJ) and exposure time (last 90 s) during fabrication for writing neat type-I FBG; we slightly deviate the laser beam away from the fiber center via adjustment of the height position of fiber, as shown in Fig. 1(b), so that the FBG can be smoothly and accurately formed in part of the fiber core, as shown in the schematic diagram of Fig. 1(a). We easily implement control of the exposure position by observing the laser diffraction pattern after it passes through the fiber (as shown in Fig. 1(c)). An eccentric FBG is easily formed using this fabrication setup. After the FBG fabrication, the refractive-index modification degree is stronger on the fiber upper center than that on the other region because of the adjusted laser exposure. A normal and an eccentric FBG were successively fabricated under identical conditions, as shown in Fig. 2, the normal fiber achieved a sole core mode, whereas the eccentric generated a fundamental core mode and few high-order modes at the shorter wavelength side due to depressed index profile. The grating in the ring core causes those high-order modes in reflection, in a manner similar to the inscription over fiber core and cladding [19]. However, here, the reflection intensity of high-order modes is quite low because of the relatively weak refractive-index modification induced by the mild process conditions.

 figure: Fig. 2

Fig. 2 Reflection spectra of the normal FBG (blue) and eccentric FBG (red).

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2.2 Sensing mechanism (radial bending)

For such a single mode fiber structure, fiber-bending can induce an additional perturbation of the RI profile over the fiber cross section and further cause the lateral mode fields to shift along a certain direction of the bending plane [21,22]. As shown in Fig. 1(a), the fundamental mode is well reflected by the downstream FBG in the dip core and is recoupled back to the upstream SMF. However, the fiber bending has a negative impact on the power of the fundamental mode because of the weak waveguiding condition between the dip core and the ring core. Accordingly, the E-field of fundamental mode of bent fiber generates a departure from fiber center and further induces an obvious reflected power loss. In contrast, the E-field distribution in the ring core is confined via the step-index configuration between ring core and depressed-cladding. Thus, the fundamental mode presents a great response to fiber bending. In the following, for depicting how the fundamental mode intensity of fiber responds to the direction for bending more vividly, the E-field distributions of the fundamental mode with and without applied fiber bending were simulated by the finite element analysis methods [23,24].

Based on the method of conformal mapping [25,26], the circularly bent fiber can be transformed to an equivalent straight fiber with modified refractive index distribution by coordinate transformation, as shown in Fig. 3(a). Assuming the fiber-bending radius R is much greater than the fiber radius, the modified RI of the bent fiber cross-section can be described by [27]

nm=(n+Δn)(1+xR)
Here xR is assumed with respect to relatively moderate bends, and n and Δn are the RI distribution of the straight fiber, and the RI change of fiber, respectively. Through the photoelastic effect, the RI change can be given by [28]
Δn=(n32)(p12νp12νp11)xR
pij is the component of elasto optical tensor (i and j = 1, 2) and ν is the Poisson radio. Figure 3(b) shows the variation of RI profile with fiber-bending in the x-y plane. Combining Eqs. (1) and (2), the equivalent bend radius can be defined as

 figure: Fig. 3

Fig. 3 (a) Schematic of a circularly bent fiber. (b) Refractive index profile distribution of an undisturbed (black line) and a bent fiber (red).

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Reff=R1(n22)(p12νp12νp11)xR

Finally, the refractive index distribution of equivalent straight waveguide is expressed as

nm=n(1+xReff)
This is the refractive index distribution for the mode simulations. As shown in Fig. 4(b), the E-field is mainly concentrated on the core region (ring core and dip core) initially. After applying fiber bending, the mode field intensity will shift toward the opposite bending direction from the fiber center (seen in the enlarged simulated diagram of field distributions of core region in Figs. 4(a) and 4(c)). The FBG is position-fixed and off-center, therefore, the total change of backward fundamental mode intensity (reflected by the FBG) is directly determined by the field deviation. As a result, not only fiber-bending amplitude but also fiber-bending direction dominates the reflection intensity; i.e., the refection intensity decrement induced by fiber-bending coming from FBG side is less than that from FBG offside. Figure 4(d) illustrates the results of fiber bending toward the FBG side and FBG offside; the results are perfectly matched with our discussion and simulation.

 figure: Fig. 4

Fig. 4 The E-field distribution of straight fiber versus different bending condition: (a) bending toward FBG side; (b) unbent; (c) bending backward FBG side. (d) The reflection spectra corresponding to different bending condition: blue (a); red (b); yellow (c).

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2.3 Sensing mechanism (longitudinal tension)

The axial tension regarding the FBG is a common and simple solution to the displacement measurement. Assuming the fiber center axis is parallel to the z-axis, a longitudinal tension produces an inertial displacement along the z-axis, and the deformation proportion of axial effective-length is identical to that of grating-period, i.e., ΔLL=ΔΛΛ, within the linear elastic range (see Fig. 5). According to materials theory [29], the axial strain ε of the FBG is defined by ε=ΔLL, and the RI change can be given by

Δneff=(neff32)(p12νp12νp11)ΔLL=(neff32)(p12νp12νp11)ΔΛΛ
For an FBG device, the fundamental resonance mode wavelength meets the Bragg condition: λB=2neffΛ. Therefore, the Bragg wavelength shift ΔλB of an FBG has the following relationship with the axial displacement increment
ΔλB=λBL[1(neff22)(p12νp12νp11)]ΔL
Thus, the Bragg wavelength of the FBG can be well used to monitor the axis displacement.

 figure: Fig. 5

Fig. 5 Schematic of an axially stretched fiber.

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3. Measurements and discussion

The schematic configuration for the 3D vector displacement measurement is shown in Fig. 6(a). Both sides of the sensor probe (total 10 cm in length, with the grating region located in the middle) are fixed in rotators (change fiber-bending direction relative to the FBG plane), of which one is fixed on a vertical lifting stage for horizontal adjustment and the other is on a translation stage for displacement control. The arbitrary vector displacement in cylindrical coordinate can be recovered by two measurements, as shown in Fig. 6(b): the projection in the horizontal plane and the projection in vertical axis against horizon. Thus, to clearly analyze the direction responses of sensor in 3D space, we decompose the 3D measurement into one radial 2D component and one longitudinal 1D component (cylindrical coordinate). The vector information of the displacement can be easily recovered by superposing the components.

 figure: Fig. 6

Fig. 6 (a) Schematic diagram of the displacement measurement. (b) Vector decomposition in circular cylindrical coordinates.

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3.1 Measurement in radial 2D plane

As Fig. 6(a) and 6(b) show, the fiber axis is defined as z-axis, and the x-y plane is defined as the 2D plane. The grating fiber is placed between two splints for ensuring the relative bending direction parallel to x-axis direction and can be bent in the x-axis direction when the translation stage moves along the z-axis because both sides of the fiber are fixed, as shown in Fig. 6(a). Here, except for the deflection of mode field intensity as discussed in section 2.1, the bending deformation also has a negative impact on the forward fundamental core mode coupling at SMF-to-DCF splicing junction. Moreover, the RI variation induced by the deformation stress of fiber cross section affects the backward fundamental core mode field recoupling at the SMF-DCF splicing junction.

For studying the performance of the eccentric FBG in the radial 2D plane, we conducted a series of measurements. The sensor was applied with increasing displacement (step of 0.01 mm) toward FBG side. The spectra versus different displacements are shown in Fig. 7(a), the linear sensitivity of −124.17 dB/mm is lower than that of normal FBG [14] due to the weak RI modification and smallish grating size. However, it can be effectively improved by further sensibilization packaging and optimized fabrication.

 figure: Fig. 7

Fig. 7 (a) Reflection spectra versus different 2D-plane displacements. (b) Reflection spectra versus a certain 2D-plane displacement in different directions.

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The change of displacement direction was achieved by the rotators (with a rotation step of 20°). The spectrum of the sensor presents significant intensity fluctuations with fiber-bending variations. Applying a certain displacement, the intensity responses of fundamental mode to fiber-bending for angle θ from 0° to 360° presents a regular sinusoidal trajectory, as shown in Fig. 7(b). This response characteristic offers us an available mechanism of direction identification for 2D in-plane displacement measurement. In detail, we can obtain displacement direction by the following approach: the sensor reveals a corresponding initial intensity variation when the displacement has changed; we rotate the sensor direction and record the intensity in real time to recover a sine curve of intensity and then locate the angle θ corresponding to initial intensity from the sine.

The direction responses of a sensor applied with different displacement amplitudes were also obtained for comparison. Using the results of the comparison experiment, we plot the intensity difference as a function of fiber-bending angle in a polar coordinate in Fig. 8. The experimental data of each test are perfectly matched with the above discussion. The radial 2D component information is achieved via the above measurements.

 figure: Fig. 8

Fig. 8 The fundamental mode intensity changes versus different 2D-plane displacements (0-360°) with amplitude of 0.05 mm, 0.08 mm, and 0.11 mm.

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3.2 Measurement in longitudinal 1D axis

The change of displacement in the 1D axis is provided by moving the translation stage away from the lifting stage, resulting in the corresponding strain inside the fiber. The spectrum of the sensor exhibits a wavelength redshift trend with increasing stretch (axis displacement), as we expected (see Fig. 9). The intensity response of the fundamental mode remains stable for the axis displacement measurement. The measurement indicates that the RI variation induced by longitudinal strain will not have a significant effect on the mode field distribution of fiber cross section. Thus, separate component monitoring is available for displacement measurement. The experimental results show that the sensor demonstrates great linear wavelength response versus longitudinal displacement. Finally, the longitudinal 1D component information is measured with monitoring the wavelength responses to fiber axis strain.

 figure: Fig. 9

Fig. 9 The fundamental mode wavelength shift and reflection spectra (insert) versus different axis displacements.

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3.3 Recover of overall 3D vector displacement measurement

Based on above 2D and 1D displacement measurements, we can recover the overall 3D vector displacement. The procedure of 3D vector displacement recovery is similar to the reverse process of the initial vector decomposition. Throughout the entire measurement processes, the 2D component is the vector projection in the x-y plane and the 1D component is the vector projection in z-axis. Therefore, by integrating the results of the 2D component information obtained from section 3.1 with the 1D component information obtained from section 3.2, then the amplitude and the direction of overall 3D vector displacement can be obtained from Pythagorean theorem calculation and vector synthesis, as shown in Fig. 6(b).

4. Conclusion

In summary, a novel 3D vector displacement sensor was proposed and demonstrated. The proposed sensor consists of an eccentric fiber Bragg grating (FBG) fabricated in part of the core of a special depressed cladding fiber (DCF). The femtosecond laser side-illumination technique and phase-mask technology were utilized to implement precisely grating inscription. Taking advantage of the extremely regular sinusoidal response for radial 2D plane displacement to fiber bending and the linear response to longitudinal displacement induced by axial strain, the sensor can recover the 3D information of displacement according to their vectorial resultant, step by step. Because of the convenient fabrication and tailoring of the properties of this grating, it shows promise for application in the optical fiber sensing field for 3D sensing.

Funding

National Natural Science Foundation of China (NSFC) (Nos. 60727004, 61077060, 61605159); Natural Science Foundation of Shaanxi Province (No. S2016YFJQ0899); Natural Science Foundation of Northwest University (No. 338020009); Graduate Innovation Project of Northwest University (No. YZZ17090).

References and links

1. A. Mehta, W. Mohammed, and E. G. Johnson, “Multimode interference-based fiber-optic displacement sensor,” IEEE Photonics Technol. Lett. 15(8), 1129–1131 (2003). [CrossRef]  

2. X. Y. Dong, Y. Liu, Z. G. Liu, and X. Y. Dong, “Simultaneous displacement and temperature measurement with cantilever-based fiber Bragg grating sensor,” Opt. Commun. 192(3-6), 213–217 (2001). [CrossRef]  

3. Q. Z. Rong, X. G. Qiao, J. Zhang, R. H. Wang, M. L. Hu, and Z. Y. Feng, “Simultaneous Measurement for Displacement and Temperature Using Fiber Bragg Grating Cladding Mode Based on Core Diameter Mismatch,” J. Lightwave Technol. 30(11), 1645–1650 (2012). [CrossRef]  

4. K. M. Yang, J. He, C. Liao, Y. Wang, S. Liu, K. Guo, J. Zhou, Z. Li, Z. Tan, and Y. Wang, “Femtosecond Laser Inscription of Fiber Bragg Grating in Twin-core Few-mode Fiber for Directional Bend Sensing,” J. Lightwave Technol. 35(21), 4670–4676 (2017). [CrossRef]  

5. G. M. H. Flockhart, W. N. MacPherson, J. S. Barton, J. D. C. Jones, L. Zhang, and I. Bennion, “Two-axis bend measurement with Bragg gratings in multicore optical fiber,” Opt. Lett. 28(6), 387–389 (2003). [CrossRef]   [PubMed]  

6. X. Chen, C. Zhang, D. J. Webb, R. Suo, G. D. Peng, and K. Kalli, “Optical bend sensor for vector curvature measurement based on Bragg grating in eccentric core polymer optical fibre,” Proc. SPIE 7503, 750327 (2009). [CrossRef]  

7. D. Zhao, X. Chen, K. Zhou, L. Zhang, I. Bennion, W. N. MacPherson, J. S. Barton, and J. D. C. Jones, “Bend sensors with direction recognition based on long-period gratings written in D-shaped fiber,” Appl. Opt. 43(29), 5425–5428 (2004). [CrossRef]   [PubMed]  

8. J. Kong, X. W. Ouyang, A. Zhou, and L. B. Yuan, “Highly Sensitive Directional Bending Sensor Based on Eccentric Core Fiber Mach–Zehnder Modal Interferometer,” IEEE Sens. J. 16(18), 6899–6902 (2016). [CrossRef]  

9. S. Zhang, W. Zhang, S. Gao, P. Geng, and X. Xue, “Fiber-optic bending vector sensor based on Mach-Zehnder interferometer exploiting lateral-offset and up-taper,” Opt. Lett. 37(21), 4480–4482 (2012). [CrossRef]   [PubMed]  

10. L. Y. Shao, L. Y. Xiong, C. K. Chen, A. Laronche, and J. Albert, “Directional Bend Sensor Based on Re-Grown Tilted Fiber Bragg Grating,” J. Lightwave Technol. 28(18), 2681–2687 (2010). [CrossRef]  

11. D. Feng, W. Zhou, X. Qiao, and J. Albert, “Compact Optical Fiber 3D Shape Sensor Based on a Pair of Orthogonal Tilted Fiber Bragg Gratings,” Sci. Rep. 5(1), 17415 (2015). [CrossRef]   [PubMed]  

12. L. Dong, L. Reekie, J. L. Cruz, J. E. Caplen, J. P. de Sandro, and D. N. Payne, “Optical fibers with depressed claddings for suppression of coupling into cladding modes in fiber Bragg gratings,” IEEE Photonics Technol. Lett. 9(1), 64–66 (1997). [CrossRef]  

13. C. W. Haggans, H. Singh, W. F. Varner, and J. S. Wang, “Narrow-depressed cladding fiber design for minimization of cladding mode losses in azimuthally eccentric fiber Bragg gratings,” J. Lightwave Technol. 16(5), 902–909 (1998). [CrossRef]  

14. Q. Rong, T. Guo, W. Bao, Z. Shao, G. D. Peng, and X. Qiao, “Highly sensitive fiber-optic accelerometer by grating inscription in specific core dip fiber,” Sci. Rep. 7(1), 11856 (2017). [CrossRef]   [PubMed]  

15. C. Waltermann, A. Doering, M. Köhring, M. Angelmahr, and W. Schade, “Cladding waveguide gratings in standard single-mode fiber for 3D shape sensing,” Opt. Lett. 40(13), 3109–3112 (2015). [CrossRef]   [PubMed]  

16. D. Feng, X. Qiao, and J. Albert, “Off-axis ultraviolet-written fiber Bragg gratings for directional bending measurements,” Opt. Lett. 41(6), 1201–1204 (2016). [CrossRef]   [PubMed]  

17. K. Chah, V. Voisin, D. Kinet, and C. Caucheteur, “Surface plasmon resonance in eccentric femtosecond-laser-induced fiber Bragg gratings,” Opt. Lett. 39(24), 6887–6890 (2014). [CrossRef]   [PubMed]  

18. K. Chah, D. Kinet, and C. Caucheteur, “Negative axial strain sensitivity in gold-coated eccentric fiber Bragg gratings,” Sci. Rep. 6(1), 38042 (2016). [CrossRef]   [PubMed]  

19. W. J. Bao, X. G. Qiao, Q. Z. Rong, N. F. Hu, H. Z. Yang, Z. Y. Feng, and M. L. Hu, “Sensing Characteristics for a Fiber Bragg Grating Inscribed Over a Fiber Core and Cladding,” IEEE Photonics Technol. Lett. 27(7), 709–712 (2015). [CrossRef]  

20. C. Smelser, S. Mihailov, and D. Grobnic, “Formation of Type I-IR and Type II-IR gratings with an ultrafast IR laser and a phase mask,” Opt. Express 13(14), 5377–5386 (2005). [CrossRef]   [PubMed]  

21. L. Faustini and G. Martini, “Bend Loss in Single-Mode Fibers,” J. Lightwave Technol. 15(4), 671–679 (1997). [CrossRef]  

22. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66(3), 216–220 (1976). [CrossRef]  

23. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976). [CrossRef]  

24. A. W. Snyder and J. D. Love, “Optical Waveguide Theory,” (Chapman & Hall, Upper Saddle River, 1983). B.

25. R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007). [CrossRef]  

26. M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11(5), 313 (1975).

27. A. M. Smith, “Birefringence induced by bends and twists in single-mode optical fiber,” Appl. Opt. 19(15), 2606–2611 (1980). [CrossRef]   [PubMed]  

28. U. L. Block, M. J. F. Digonnet, M. M. Fejer, and V. Dangui, “Bending-Induced Birefringence of Optical Fiber Cladding Modes,” J. Lightwave Technol. 24(6), 2336–2339 (2006). [CrossRef]  

29. E. J. Hearn, Mechanics of Materials 2: The mechanics of elastic and plastic deformation of solids and structural materials (Butterworth-Heinemann, 1997).

References

  • View by:

  1. A. Mehta, W. Mohammed, and E. G. Johnson, “Multimode interference-based fiber-optic displacement sensor,” IEEE Photonics Technol. Lett. 15(8), 1129–1131 (2003).
    [Crossref]
  2. X. Y. Dong, Y. Liu, Z. G. Liu, and X. Y. Dong, “Simultaneous displacement and temperature measurement with cantilever-based fiber Bragg grating sensor,” Opt. Commun. 192(3-6), 213–217 (2001).
    [Crossref]
  3. Q. Z. Rong, X. G. Qiao, J. Zhang, R. H. Wang, M. L. Hu, and Z. Y. Feng, “Simultaneous Measurement for Displacement and Temperature Using Fiber Bragg Grating Cladding Mode Based on Core Diameter Mismatch,” J. Lightwave Technol. 30(11), 1645–1650 (2012).
    [Crossref]
  4. K. M. Yang, J. He, C. Liao, Y. Wang, S. Liu, K. Guo, J. Zhou, Z. Li, Z. Tan, and Y. Wang, “Femtosecond Laser Inscription of Fiber Bragg Grating in Twin-core Few-mode Fiber for Directional Bend Sensing,” J. Lightwave Technol. 35(21), 4670–4676 (2017).
    [Crossref]
  5. G. M. H. Flockhart, W. N. MacPherson, J. S. Barton, J. D. C. Jones, L. Zhang, and I. Bennion, “Two-axis bend measurement with Bragg gratings in multicore optical fiber,” Opt. Lett. 28(6), 387–389 (2003).
    [Crossref] [PubMed]
  6. X. Chen, C. Zhang, D. J. Webb, R. Suo, G. D. Peng, and K. Kalli, “Optical bend sensor for vector curvature measurement based on Bragg grating in eccentric core polymer optical fibre,” Proc. SPIE 7503, 750327 (2009).
    [Crossref]
  7. D. Zhao, X. Chen, K. Zhou, L. Zhang, I. Bennion, W. N. MacPherson, J. S. Barton, and J. D. C. Jones, “Bend sensors with direction recognition based on long-period gratings written in D-shaped fiber,” Appl. Opt. 43(29), 5425–5428 (2004).
    [Crossref] [PubMed]
  8. J. Kong, X. W. Ouyang, A. Zhou, and L. B. Yuan, “Highly Sensitive Directional Bending Sensor Based on Eccentric Core Fiber Mach–Zehnder Modal Interferometer,” IEEE Sens. J. 16(18), 6899–6902 (2016).
    [Crossref]
  9. S. Zhang, W. Zhang, S. Gao, P. Geng, and X. Xue, “Fiber-optic bending vector sensor based on Mach-Zehnder interferometer exploiting lateral-offset and up-taper,” Opt. Lett. 37(21), 4480–4482 (2012).
    [Crossref] [PubMed]
  10. L. Y. Shao, L. Y. Xiong, C. K. Chen, A. Laronche, and J. Albert, “Directional Bend Sensor Based on Re-Grown Tilted Fiber Bragg Grating,” J. Lightwave Technol. 28(18), 2681–2687 (2010).
    [Crossref]
  11. D. Feng, W. Zhou, X. Qiao, and J. Albert, “Compact Optical Fiber 3D Shape Sensor Based on a Pair of Orthogonal Tilted Fiber Bragg Gratings,” Sci. Rep. 5(1), 17415 (2015).
    [Crossref] [PubMed]
  12. L. Dong, L. Reekie, J. L. Cruz, J. E. Caplen, J. P. de Sandro, and D. N. Payne, “Optical fibers with depressed claddings for suppression of coupling into cladding modes in fiber Bragg gratings,” IEEE Photonics Technol. Lett. 9(1), 64–66 (1997).
    [Crossref]
  13. C. W. Haggans, H. Singh, W. F. Varner, and J. S. Wang, “Narrow-depressed cladding fiber design for minimization of cladding mode losses in azimuthally eccentric fiber Bragg gratings,” J. Lightwave Technol. 16(5), 902–909 (1998).
    [Crossref]
  14. Q. Rong, T. Guo, W. Bao, Z. Shao, G. D. Peng, and X. Qiao, “Highly sensitive fiber-optic accelerometer by grating inscription in specific core dip fiber,” Sci. Rep. 7(1), 11856 (2017).
    [Crossref] [PubMed]
  15. C. Waltermann, A. Doering, M. Köhring, M. Angelmahr, and W. Schade, “Cladding waveguide gratings in standard single-mode fiber for 3D shape sensing,” Opt. Lett. 40(13), 3109–3112 (2015).
    [Crossref] [PubMed]
  16. D. Feng, X. Qiao, and J. Albert, “Off-axis ultraviolet-written fiber Bragg gratings for directional bending measurements,” Opt. Lett. 41(6), 1201–1204 (2016).
    [Crossref] [PubMed]
  17. K. Chah, V. Voisin, D. Kinet, and C. Caucheteur, “Surface plasmon resonance in eccentric femtosecond-laser-induced fiber Bragg gratings,” Opt. Lett. 39(24), 6887–6890 (2014).
    [Crossref] [PubMed]
  18. K. Chah, D. Kinet, and C. Caucheteur, “Negative axial strain sensitivity in gold-coated eccentric fiber Bragg gratings,” Sci. Rep. 6(1), 38042 (2016).
    [Crossref] [PubMed]
  19. W. J. Bao, X. G. Qiao, Q. Z. Rong, N. F. Hu, H. Z. Yang, Z. Y. Feng, and M. L. Hu, “Sensing Characteristics for a Fiber Bragg Grating Inscribed Over a Fiber Core and Cladding,” IEEE Photonics Technol. Lett. 27(7), 709–712 (2015).
    [Crossref]
  20. C. Smelser, S. Mihailov, and D. Grobnic, “Formation of Type I-IR and Type II-IR gratings with an ultrafast IR laser and a phase mask,” Opt. Express 13(14), 5377–5386 (2005).
    [Crossref] [PubMed]
  21. L. Faustini and G. Martini, “Bend Loss in Single-Mode Fibers,” J. Lightwave Technol. 15(4), 671–679 (1997).
    [Crossref]
  22. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66(3), 216–220 (1976).
    [Crossref]
  23. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976).
    [Crossref]
  24. A. W. Snyder and J. D. Love, “Optical Waveguide Theory,” (Chapman & Hall, Upper Saddle River, 1983). B.
  25. R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007).
    [Crossref]
  26. M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11(5), 313 (1975).
  27. A. M. Smith, “Birefringence induced by bends and twists in single-mode optical fiber,” Appl. Opt. 19(15), 2606–2611 (1980).
    [Crossref] [PubMed]
  28. U. L. Block, M. J. F. Digonnet, M. M. Fejer, and V. Dangui, “Bending-Induced Birefringence of Optical Fiber Cladding Modes,” J. Lightwave Technol. 24(6), 2336–2339 (2006).
    [Crossref]
  29. E. J. Hearn, Mechanics of Materials 2: The mechanics of elastic and plastic deformation of solids and structural materials (Butterworth-Heinemann, 1997).

2017 (2)

K. M. Yang, J. He, C. Liao, Y. Wang, S. Liu, K. Guo, J. Zhou, Z. Li, Z. Tan, and Y. Wang, “Femtosecond Laser Inscription of Fiber Bragg Grating in Twin-core Few-mode Fiber for Directional Bend Sensing,” J. Lightwave Technol. 35(21), 4670–4676 (2017).
[Crossref]

Q. Rong, T. Guo, W. Bao, Z. Shao, G. D. Peng, and X. Qiao, “Highly sensitive fiber-optic accelerometer by grating inscription in specific core dip fiber,” Sci. Rep. 7(1), 11856 (2017).
[Crossref] [PubMed]

2016 (3)

D. Feng, X. Qiao, and J. Albert, “Off-axis ultraviolet-written fiber Bragg gratings for directional bending measurements,” Opt. Lett. 41(6), 1201–1204 (2016).
[Crossref] [PubMed]

J. Kong, X. W. Ouyang, A. Zhou, and L. B. Yuan, “Highly Sensitive Directional Bending Sensor Based on Eccentric Core Fiber Mach–Zehnder Modal Interferometer,” IEEE Sens. J. 16(18), 6899–6902 (2016).
[Crossref]

K. Chah, D. Kinet, and C. Caucheteur, “Negative axial strain sensitivity in gold-coated eccentric fiber Bragg gratings,” Sci. Rep. 6(1), 38042 (2016).
[Crossref] [PubMed]

2015 (3)

W. J. Bao, X. G. Qiao, Q. Z. Rong, N. F. Hu, H. Z. Yang, Z. Y. Feng, and M. L. Hu, “Sensing Characteristics for a Fiber Bragg Grating Inscribed Over a Fiber Core and Cladding,” IEEE Photonics Technol. Lett. 27(7), 709–712 (2015).
[Crossref]

C. Waltermann, A. Doering, M. Köhring, M. Angelmahr, and W. Schade, “Cladding waveguide gratings in standard single-mode fiber for 3D shape sensing,” Opt. Lett. 40(13), 3109–3112 (2015).
[Crossref] [PubMed]

D. Feng, W. Zhou, X. Qiao, and J. Albert, “Compact Optical Fiber 3D Shape Sensor Based on a Pair of Orthogonal Tilted Fiber Bragg Gratings,” Sci. Rep. 5(1), 17415 (2015).
[Crossref] [PubMed]

2014 (1)

2012 (2)

2010 (1)

2009 (1)

X. Chen, C. Zhang, D. J. Webb, R. Suo, G. D. Peng, and K. Kalli, “Optical bend sensor for vector curvature measurement based on Bragg grating in eccentric core polymer optical fibre,” Proc. SPIE 7503, 750327 (2009).
[Crossref]

2007 (1)

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007).
[Crossref]

2006 (1)

2005 (1)

2004 (1)

2003 (2)

G. M. H. Flockhart, W. N. MacPherson, J. S. Barton, J. D. C. Jones, L. Zhang, and I. Bennion, “Two-axis bend measurement with Bragg gratings in multicore optical fiber,” Opt. Lett. 28(6), 387–389 (2003).
[Crossref] [PubMed]

A. Mehta, W. Mohammed, and E. G. Johnson, “Multimode interference-based fiber-optic displacement sensor,” IEEE Photonics Technol. Lett. 15(8), 1129–1131 (2003).
[Crossref]

2001 (1)

X. Y. Dong, Y. Liu, Z. G. Liu, and X. Y. Dong, “Simultaneous displacement and temperature measurement with cantilever-based fiber Bragg grating sensor,” Opt. Commun. 192(3-6), 213–217 (2001).
[Crossref]

1998 (1)

1997 (2)

L. Dong, L. Reekie, J. L. Cruz, J. E. Caplen, J. P. de Sandro, and D. N. Payne, “Optical fibers with depressed claddings for suppression of coupling into cladding modes in fiber Bragg gratings,” IEEE Photonics Technol. Lett. 9(1), 64–66 (1997).
[Crossref]

L. Faustini and G. Martini, “Bend Loss in Single-Mode Fibers,” J. Lightwave Technol. 15(4), 671–679 (1997).
[Crossref]

1980 (1)

1976 (2)

1975 (1)

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11(5), 313 (1975).

Albert, J.

Angelmahr, M.

Bao, W.

Q. Rong, T. Guo, W. Bao, Z. Shao, G. D. Peng, and X. Qiao, “Highly sensitive fiber-optic accelerometer by grating inscription in specific core dip fiber,” Sci. Rep. 7(1), 11856 (2017).
[Crossref] [PubMed]

Bao, W. J.

W. J. Bao, X. G. Qiao, Q. Z. Rong, N. F. Hu, H. Z. Yang, Z. Y. Feng, and M. L. Hu, “Sensing Characteristics for a Fiber Bragg Grating Inscribed Over a Fiber Core and Cladding,” IEEE Photonics Technol. Lett. 27(7), 709–712 (2015).
[Crossref]

Barton, J. S.

Bennion, I.

Block, U. L.

Caplen, J. E.

L. Dong, L. Reekie, J. L. Cruz, J. E. Caplen, J. P. de Sandro, and D. N. Payne, “Optical fibers with depressed claddings for suppression of coupling into cladding modes in fiber Bragg gratings,” IEEE Photonics Technol. Lett. 9(1), 64–66 (1997).
[Crossref]

Caucheteur, C.

K. Chah, D. Kinet, and C. Caucheteur, “Negative axial strain sensitivity in gold-coated eccentric fiber Bragg gratings,” Sci. Rep. 6(1), 38042 (2016).
[Crossref] [PubMed]

K. Chah, V. Voisin, D. Kinet, and C. Caucheteur, “Surface plasmon resonance in eccentric femtosecond-laser-induced fiber Bragg gratings,” Opt. Lett. 39(24), 6887–6890 (2014).
[Crossref] [PubMed]

Chah, K.

K. Chah, D. Kinet, and C. Caucheteur, “Negative axial strain sensitivity in gold-coated eccentric fiber Bragg gratings,” Sci. Rep. 6(1), 38042 (2016).
[Crossref] [PubMed]

K. Chah, V. Voisin, D. Kinet, and C. Caucheteur, “Surface plasmon resonance in eccentric femtosecond-laser-induced fiber Bragg gratings,” Opt. Lett. 39(24), 6887–6890 (2014).
[Crossref] [PubMed]

Chen, C. K.

Chen, X.

X. Chen, C. Zhang, D. J. Webb, R. Suo, G. D. Peng, and K. Kalli, “Optical bend sensor for vector curvature measurement based on Bragg grating in eccentric core polymer optical fibre,” Proc. SPIE 7503, 750327 (2009).
[Crossref]

D. Zhao, X. Chen, K. Zhou, L. Zhang, I. Bennion, W. N. MacPherson, J. S. Barton, and J. D. C. Jones, “Bend sensors with direction recognition based on long-period gratings written in D-shaped fiber,” Appl. Opt. 43(29), 5425–5428 (2004).
[Crossref] [PubMed]

Cole, J. H.

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007).
[Crossref]

Cruz, J. L.

L. Dong, L. Reekie, J. L. Cruz, J. E. Caplen, J. P. de Sandro, and D. N. Payne, “Optical fibers with depressed claddings for suppression of coupling into cladding modes in fiber Bragg gratings,” IEEE Photonics Technol. Lett. 9(1), 64–66 (1997).
[Crossref]

Dangui, V.

de Sandro, J. P.

L. Dong, L. Reekie, J. L. Cruz, J. E. Caplen, J. P. de Sandro, and D. N. Payne, “Optical fibers with depressed claddings for suppression of coupling into cladding modes in fiber Bragg gratings,” IEEE Photonics Technol. Lett. 9(1), 64–66 (1997).
[Crossref]

Digonnet, M. J. F.

Doering, A.

Dong, L.

L. Dong, L. Reekie, J. L. Cruz, J. E. Caplen, J. P. de Sandro, and D. N. Payne, “Optical fibers with depressed claddings for suppression of coupling into cladding modes in fiber Bragg gratings,” IEEE Photonics Technol. Lett. 9(1), 64–66 (1997).
[Crossref]

Dong, X. Y.

X. Y. Dong, Y. Liu, Z. G. Liu, and X. Y. Dong, “Simultaneous displacement and temperature measurement with cantilever-based fiber Bragg grating sensor,” Opt. Commun. 192(3-6), 213–217 (2001).
[Crossref]

X. Y. Dong, Y. Liu, Z. G. Liu, and X. Y. Dong, “Simultaneous displacement and temperature measurement with cantilever-based fiber Bragg grating sensor,” Opt. Commun. 192(3-6), 213–217 (2001).
[Crossref]

Faustini, L.

L. Faustini and G. Martini, “Bend Loss in Single-Mode Fibers,” J. Lightwave Technol. 15(4), 671–679 (1997).
[Crossref]

Fejer, M. M.

Feng, D.

D. Feng, X. Qiao, and J. Albert, “Off-axis ultraviolet-written fiber Bragg gratings for directional bending measurements,” Opt. Lett. 41(6), 1201–1204 (2016).
[Crossref] [PubMed]

D. Feng, W. Zhou, X. Qiao, and J. Albert, “Compact Optical Fiber 3D Shape Sensor Based on a Pair of Orthogonal Tilted Fiber Bragg Gratings,” Sci. Rep. 5(1), 17415 (2015).
[Crossref] [PubMed]

Feng, Z. Y.

W. J. Bao, X. G. Qiao, Q. Z. Rong, N. F. Hu, H. Z. Yang, Z. Y. Feng, and M. L. Hu, “Sensing Characteristics for a Fiber Bragg Grating Inscribed Over a Fiber Core and Cladding,” IEEE Photonics Technol. Lett. 27(7), 709–712 (2015).
[Crossref]

Q. Z. Rong, X. G. Qiao, J. Zhang, R. H. Wang, M. L. Hu, and Z. Y. Feng, “Simultaneous Measurement for Displacement and Temperature Using Fiber Bragg Grating Cladding Mode Based on Core Diameter Mismatch,” J. Lightwave Technol. 30(11), 1645–1650 (2012).
[Crossref]

Flockhart, G. M. H.

Gao, S.

Geng, P.

Grobnic, D.

Guo, K.

Guo, T.

Q. Rong, T. Guo, W. Bao, Z. Shao, G. D. Peng, and X. Qiao, “Highly sensitive fiber-optic accelerometer by grating inscription in specific core dip fiber,” Sci. Rep. 7(1), 11856 (2017).
[Crossref] [PubMed]

Haggans, C. W.

Harris, J. H.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11(5), 313 (1975).

He, J.

Heiblum, M.

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11(5), 313 (1975).

Hu, M. L.

W. J. Bao, X. G. Qiao, Q. Z. Rong, N. F. Hu, H. Z. Yang, Z. Y. Feng, and M. L. Hu, “Sensing Characteristics for a Fiber Bragg Grating Inscribed Over a Fiber Core and Cladding,” IEEE Photonics Technol. Lett. 27(7), 709–712 (2015).
[Crossref]

Q. Z. Rong, X. G. Qiao, J. Zhang, R. H. Wang, M. L. Hu, and Z. Y. Feng, “Simultaneous Measurement for Displacement and Temperature Using Fiber Bragg Grating Cladding Mode Based on Core Diameter Mismatch,” J. Lightwave Technol. 30(11), 1645–1650 (2012).
[Crossref]

Hu, N. F.

W. J. Bao, X. G. Qiao, Q. Z. Rong, N. F. Hu, H. Z. Yang, Z. Y. Feng, and M. L. Hu, “Sensing Characteristics for a Fiber Bragg Grating Inscribed Over a Fiber Core and Cladding,” IEEE Photonics Technol. Lett. 27(7), 709–712 (2015).
[Crossref]

Johnson, E. G.

A. Mehta, W. Mohammed, and E. G. Johnson, “Multimode interference-based fiber-optic displacement sensor,” IEEE Photonics Technol. Lett. 15(8), 1129–1131 (2003).
[Crossref]

Jones, J. D. C.

Kalli, K.

X. Chen, C. Zhang, D. J. Webb, R. Suo, G. D. Peng, and K. Kalli, “Optical bend sensor for vector curvature measurement based on Bragg grating in eccentric core polymer optical fibre,” Proc. SPIE 7503, 750327 (2009).
[Crossref]

Kinet, D.

K. Chah, D. Kinet, and C. Caucheteur, “Negative axial strain sensitivity in gold-coated eccentric fiber Bragg gratings,” Sci. Rep. 6(1), 38042 (2016).
[Crossref] [PubMed]

K. Chah, V. Voisin, D. Kinet, and C. Caucheteur, “Surface plasmon resonance in eccentric femtosecond-laser-induced fiber Bragg gratings,” Opt. Lett. 39(24), 6887–6890 (2014).
[Crossref] [PubMed]

Köhring, M.

Kong, J.

J. Kong, X. W. Ouyang, A. Zhou, and L. B. Yuan, “Highly Sensitive Directional Bending Sensor Based on Eccentric Core Fiber Mach–Zehnder Modal Interferometer,” IEEE Sens. J. 16(18), 6899–6902 (2016).
[Crossref]

Laronche, A.

Li, Z.

Liao, C.

Liu, S.

Liu, Y.

X. Y. Dong, Y. Liu, Z. G. Liu, and X. Y. Dong, “Simultaneous displacement and temperature measurement with cantilever-based fiber Bragg grating sensor,” Opt. Commun. 192(3-6), 213–217 (2001).
[Crossref]

Liu, Z. G.

X. Y. Dong, Y. Liu, Z. G. Liu, and X. Y. Dong, “Simultaneous displacement and temperature measurement with cantilever-based fiber Bragg grating sensor,” Opt. Commun. 192(3-6), 213–217 (2001).
[Crossref]

MacPherson, W. N.

Marcuse, D.

Martini, G.

L. Faustini and G. Martini, “Bend Loss in Single-Mode Fibers,” J. Lightwave Technol. 15(4), 671–679 (1997).
[Crossref]

Mehta, A.

A. Mehta, W. Mohammed, and E. G. Johnson, “Multimode interference-based fiber-optic displacement sensor,” IEEE Photonics Technol. Lett. 15(8), 1129–1131 (2003).
[Crossref]

Mihailov, S.

Mohammed, W.

A. Mehta, W. Mohammed, and E. G. Johnson, “Multimode interference-based fiber-optic displacement sensor,” IEEE Photonics Technol. Lett. 15(8), 1129–1131 (2003).
[Crossref]

Ouyang, X. W.

J. Kong, X. W. Ouyang, A. Zhou, and L. B. Yuan, “Highly Sensitive Directional Bending Sensor Based on Eccentric Core Fiber Mach–Zehnder Modal Interferometer,” IEEE Sens. J. 16(18), 6899–6902 (2016).
[Crossref]

Payne, D. N.

L. Dong, L. Reekie, J. L. Cruz, J. E. Caplen, J. P. de Sandro, and D. N. Payne, “Optical fibers with depressed claddings for suppression of coupling into cladding modes in fiber Bragg gratings,” IEEE Photonics Technol. Lett. 9(1), 64–66 (1997).
[Crossref]

Peng, G. D.

Q. Rong, T. Guo, W. Bao, Z. Shao, G. D. Peng, and X. Qiao, “Highly sensitive fiber-optic accelerometer by grating inscription in specific core dip fiber,” Sci. Rep. 7(1), 11856 (2017).
[Crossref] [PubMed]

X. Chen, C. Zhang, D. J. Webb, R. Suo, G. D. Peng, and K. Kalli, “Optical bend sensor for vector curvature measurement based on Bragg grating in eccentric core polymer optical fibre,” Proc. SPIE 7503, 750327 (2009).
[Crossref]

Qiao, X.

Q. Rong, T. Guo, W. Bao, Z. Shao, G. D. Peng, and X. Qiao, “Highly sensitive fiber-optic accelerometer by grating inscription in specific core dip fiber,” Sci. Rep. 7(1), 11856 (2017).
[Crossref] [PubMed]

D. Feng, X. Qiao, and J. Albert, “Off-axis ultraviolet-written fiber Bragg gratings for directional bending measurements,” Opt. Lett. 41(6), 1201–1204 (2016).
[Crossref] [PubMed]

D. Feng, W. Zhou, X. Qiao, and J. Albert, “Compact Optical Fiber 3D Shape Sensor Based on a Pair of Orthogonal Tilted Fiber Bragg Gratings,” Sci. Rep. 5(1), 17415 (2015).
[Crossref] [PubMed]

Qiao, X. G.

W. J. Bao, X. G. Qiao, Q. Z. Rong, N. F. Hu, H. Z. Yang, Z. Y. Feng, and M. L. Hu, “Sensing Characteristics for a Fiber Bragg Grating Inscribed Over a Fiber Core and Cladding,” IEEE Photonics Technol. Lett. 27(7), 709–712 (2015).
[Crossref]

Q. Z. Rong, X. G. Qiao, J. Zhang, R. H. Wang, M. L. Hu, and Z. Y. Feng, “Simultaneous Measurement for Displacement and Temperature Using Fiber Bragg Grating Cladding Mode Based on Core Diameter Mismatch,” J. Lightwave Technol. 30(11), 1645–1650 (2012).
[Crossref]

Reekie, L.

L. Dong, L. Reekie, J. L. Cruz, J. E. Caplen, J. P. de Sandro, and D. N. Payne, “Optical fibers with depressed claddings for suppression of coupling into cladding modes in fiber Bragg gratings,” IEEE Photonics Technol. Lett. 9(1), 64–66 (1997).
[Crossref]

Rong, Q.

Q. Rong, T. Guo, W. Bao, Z. Shao, G. D. Peng, and X. Qiao, “Highly sensitive fiber-optic accelerometer by grating inscription in specific core dip fiber,” Sci. Rep. 7(1), 11856 (2017).
[Crossref] [PubMed]

Rong, Q. Z.

W. J. Bao, X. G. Qiao, Q. Z. Rong, N. F. Hu, H. Z. Yang, Z. Y. Feng, and M. L. Hu, “Sensing Characteristics for a Fiber Bragg Grating Inscribed Over a Fiber Core and Cladding,” IEEE Photonics Technol. Lett. 27(7), 709–712 (2015).
[Crossref]

Q. Z. Rong, X. G. Qiao, J. Zhang, R. H. Wang, M. L. Hu, and Z. Y. Feng, “Simultaneous Measurement for Displacement and Temperature Using Fiber Bragg Grating Cladding Mode Based on Core Diameter Mismatch,” J. Lightwave Technol. 30(11), 1645–1650 (2012).
[Crossref]

Schade, W.

Schermer, R. T.

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007).
[Crossref]

Shao, L. Y.

Shao, Z.

Q. Rong, T. Guo, W. Bao, Z. Shao, G. D. Peng, and X. Qiao, “Highly sensitive fiber-optic accelerometer by grating inscription in specific core dip fiber,” Sci. Rep. 7(1), 11856 (2017).
[Crossref] [PubMed]

Singh, H.

Smelser, C.

Smith, A. M.

Suo, R.

X. Chen, C. Zhang, D. J. Webb, R. Suo, G. D. Peng, and K. Kalli, “Optical bend sensor for vector curvature measurement based on Bragg grating in eccentric core polymer optical fibre,” Proc. SPIE 7503, 750327 (2009).
[Crossref]

Tan, Z.

Varner, W. F.

Voisin, V.

Waltermann, C.

Wang, J. S.

Wang, R. H.

Wang, Y.

Webb, D. J.

X. Chen, C. Zhang, D. J. Webb, R. Suo, G. D. Peng, and K. Kalli, “Optical bend sensor for vector curvature measurement based on Bragg grating in eccentric core polymer optical fibre,” Proc. SPIE 7503, 750327 (2009).
[Crossref]

Xiong, L. Y.

Xue, X.

Yang, H. Z.

W. J. Bao, X. G. Qiao, Q. Z. Rong, N. F. Hu, H. Z. Yang, Z. Y. Feng, and M. L. Hu, “Sensing Characteristics for a Fiber Bragg Grating Inscribed Over a Fiber Core and Cladding,” IEEE Photonics Technol. Lett. 27(7), 709–712 (2015).
[Crossref]

Yang, K. M.

Yuan, L. B.

J. Kong, X. W. Ouyang, A. Zhou, and L. B. Yuan, “Highly Sensitive Directional Bending Sensor Based on Eccentric Core Fiber Mach–Zehnder Modal Interferometer,” IEEE Sens. J. 16(18), 6899–6902 (2016).
[Crossref]

Zhang, C.

X. Chen, C. Zhang, D. J. Webb, R. Suo, G. D. Peng, and K. Kalli, “Optical bend sensor for vector curvature measurement based on Bragg grating in eccentric core polymer optical fibre,” Proc. SPIE 7503, 750327 (2009).
[Crossref]

Zhang, J.

Zhang, L.

Zhang, S.

Zhang, W.

Zhao, D.

Zhou, A.

J. Kong, X. W. Ouyang, A. Zhou, and L. B. Yuan, “Highly Sensitive Directional Bending Sensor Based on Eccentric Core Fiber Mach–Zehnder Modal Interferometer,” IEEE Sens. J. 16(18), 6899–6902 (2016).
[Crossref]

Zhou, J.

Zhou, K.

Zhou, W.

D. Feng, W. Zhou, X. Qiao, and J. Albert, “Compact Optical Fiber 3D Shape Sensor Based on a Pair of Orthogonal Tilted Fiber Bragg Gratings,” Sci. Rep. 5(1), 17415 (2015).
[Crossref] [PubMed]

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007).
[Crossref]

M. Heiblum and J. H. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quantum Electron. 11(5), 313 (1975).

IEEE Photonics Technol. Lett. (3)

W. J. Bao, X. G. Qiao, Q. Z. Rong, N. F. Hu, H. Z. Yang, Z. Y. Feng, and M. L. Hu, “Sensing Characteristics for a Fiber Bragg Grating Inscribed Over a Fiber Core and Cladding,” IEEE Photonics Technol. Lett. 27(7), 709–712 (2015).
[Crossref]

A. Mehta, W. Mohammed, and E. G. Johnson, “Multimode interference-based fiber-optic displacement sensor,” IEEE Photonics Technol. Lett. 15(8), 1129–1131 (2003).
[Crossref]

L. Dong, L. Reekie, J. L. Cruz, J. E. Caplen, J. P. de Sandro, and D. N. Payne, “Optical fibers with depressed claddings for suppression of coupling into cladding modes in fiber Bragg gratings,” IEEE Photonics Technol. Lett. 9(1), 64–66 (1997).
[Crossref]

IEEE Sens. J. (1)

J. Kong, X. W. Ouyang, A. Zhou, and L. B. Yuan, “Highly Sensitive Directional Bending Sensor Based on Eccentric Core Fiber Mach–Zehnder Modal Interferometer,” IEEE Sens. J. 16(18), 6899–6902 (2016).
[Crossref]

J. Lightwave Technol. (6)

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

X. Y. Dong, Y. Liu, Z. G. Liu, and X. Y. Dong, “Simultaneous displacement and temperature measurement with cantilever-based fiber Bragg grating sensor,” Opt. Commun. 192(3-6), 213–217 (2001).
[Crossref]

Opt. Express (1)

Opt. Lett. (5)

Proc. SPIE (1)

X. Chen, C. Zhang, D. J. Webb, R. Suo, G. D. Peng, and K. Kalli, “Optical bend sensor for vector curvature measurement based on Bragg grating in eccentric core polymer optical fibre,” Proc. SPIE 7503, 750327 (2009).
[Crossref]

Sci. Rep. (3)

K. Chah, D. Kinet, and C. Caucheteur, “Negative axial strain sensitivity in gold-coated eccentric fiber Bragg gratings,” Sci. Rep. 6(1), 38042 (2016).
[Crossref] [PubMed]

D. Feng, W. Zhou, X. Qiao, and J. Albert, “Compact Optical Fiber 3D Shape Sensor Based on a Pair of Orthogonal Tilted Fiber Bragg Gratings,” Sci. Rep. 5(1), 17415 (2015).
[Crossref] [PubMed]

Q. Rong, T. Guo, W. Bao, Z. Shao, G. D. Peng, and X. Qiao, “Highly sensitive fiber-optic accelerometer by grating inscription in specific core dip fiber,” Sci. Rep. 7(1), 11856 (2017).
[Crossref] [PubMed]

Other (2)

A. W. Snyder and J. D. Love, “Optical Waveguide Theory,” (Chapman & Hall, Upper Saddle River, 1983). B.

E. J. Hearn, Mechanics of Materials 2: The mechanics of elastic and plastic deformation of solids and structural materials (Butterworth-Heinemann, 1997).

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Figures (9)

Fig. 1
Fig. 1 (a) Schematic of the eccentric grating. (b) Photomicrograph of the DCF cross section and the position illustration for normal inscription (white dash). (b) Diffraction patterns of the laser after passing through the fiber core for normal inscription (upper) and off-axis inscription (below).
Fig. 2
Fig. 2 Reflection spectra of the normal FBG (blue) and eccentric FBG (red).
Fig. 3
Fig. 3 (a) Schematic of a circularly bent fiber. (b) Refractive index profile distribution of an undisturbed (black line) and a bent fiber (red).
Fig. 4
Fig. 4 The E-field distribution of straight fiber versus different bending condition: (a) bending toward FBG side; (b) unbent; (c) bending backward FBG side. (d) The reflection spectra corresponding to different bending condition: blue (a); red (b); yellow (c).
Fig. 5
Fig. 5 Schematic of an axially stretched fiber.
Fig. 6
Fig. 6 (a) Schematic diagram of the displacement measurement. (b) Vector decomposition in circular cylindrical coordinates.
Fig. 7
Fig. 7 (a) Reflection spectra versus different 2D-plane displacements. (b) Reflection spectra versus a certain 2D-plane displacement in different directions.
Fig. 8
Fig. 8 The fundamental mode intensity changes versus different 2D-plane displacements (0-360°) with amplitude of 0.05 mm, 0.08 mm, and 0.11 mm.
Fig. 9
Fig. 9 The fundamental mode wavelength shift and reflection spectra (insert) versus different axis displacements.

Equations (6)

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n m =(n+Δn)(1+ x R )
Δn=( n 3 2 )( p 12 ν p 12 ν p 11 ) x R
R eff = R 1( n 2 2 )( p 12 ν p 12 ν p 11 ) x R
n m =n(1+ x R eff )
Δ n eff =( n eff 3 2 )( p 12 ν p 12 ν p 11 ) ΔL L =( n eff 3 2 )( p 12 ν p 12 ν p 11 ) ΔΛ Λ
Δ λ B = λ B L [ 1( n eff 2 2 )( p 12 ν p 12 ν p 11 ) ]ΔL

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