Abstract

Recently there is a growing interest in developing few-mode fiber (FMF) based distributed sensors, which can attain higher spatial resolution and sensitivity compared with the conventional single-mode approaches. However, current techniques require two lightwaves injected into both ends of FMF, resulting in their complicated setup and high cost, which causes a big issue for geotechnical and petroleum applications. In this paper, we present a single-end FMF-based distributed sensing system that allows simultaneous temperature and strain measurement by Brillouin optical time-domain reflectometry (BOTDR) and heterodyne detection. Theoretical analysis and experimental assessment of multi-parameter discriminative measurement techniques applied to distributed FMF sensors are presented. Experimental results confirm that FM-BOTDR has similar performance with two-end methods such as FM-BOTDA, but with simpler setup and lower cost. The temperature-induced expansion strain (TIES) in response to different modes is discussed as well. Furthermore, we optimized the FMF design by exploiting modal profile and doping concentration, which indicates up to fivefold enhancement in measurement accuracy. This novel distributed FM-sensing system endows with good sensitivity characteristics and can prevent catastrophic failure in many applications.

© 2015 Optical Society of America

1. Introduction

The fiber distributed strain and temperature measurement based on the Brillouin scattering has been intensively studied [1], for one optical fiber can potentially replace a large number of closely spaced point sensors [2], which have attracted great interest for assessing the performance of civil and geotechnical structures [3]. One of the most imperative issues for such techniques is how to discriminate the change occurred in temperature and strain applied to the optical fiber simultaneously [4]. Since the Brillouin frequency shift (BFS) is proportional to both temperature and strain variations, it’s theoretically impossible to separate these two effects by only measuring one BFS. Many methods have been proposed in order to solve this issue in the past two decades [5]. Earlier approaches mostly used single-mode fibers (SMF). Nevertheless, these approaches either led to poor sensing accuracy by measuring both BFS and the Brillouin power level in SMFs [6], or added extra noise and complexity to the system, such as spatially resolving both spontaneous Raman and Brillouin signals [7]. Other groups proposed to use large effective-area fiber (LEAF) to achieve simultaneous temperature and strain sensing, which creates multiple BFSs within a multiple composition fiber core [8]. Nonetheless, this approach leads to poor spatial resolution, limited sensing accuracy, and short sensing distance, compared with the frequency and intensity analysis in the SMF, due to large interference between different wavelengths. To overcome this, a specially designed multicore optical fiber (MCF) has been proposed for distributed sensing recently [9], using different BFSs from the central core and outer cores. However, one drawback could be its complex drawing processes and thus high expenses. Besides, larger space between the cores must be provided in MCF with a 250 μm cladding diameter to minimize core-to-core crosstalk, thus it might behave just like parallel SMF channels, with relatively low achievable information capacity per unit area.

The spectacular advances of space-division multiplexing (SDM) originate from the utilization of few-mode fibers (FMF) to increase capacity for optical communication [10]. Recently there is a growing interest in developing FMF-based distributed fiber sensors, which can attain higher spatial resolution and sensitivity compared with conventional single-mode approaches [11,12]. In particular, Brillouin dynamic gratings (BDG) and Brillouin optical time-domain analysis (BOTDA) thru generating stimulated Brillouin scattering (SBS) in FMF have been proposed successively for sensing purposes [13,14]. Nevertheless, though the FMF-based BDG and BOTDA have good sensing performance and multi-parameter discrimination capability, two lightwaves must be injected into both ends of the fiber under test (FUT) when operating with these techniques, resulting in their complicated setup and high cost. Besides, this might cause a big issue for geotechnical and petroleum applications, such as the automated monitoring of oil or gas pipelines. If part of the FUT cracks, the measurement can no longer be performed, which could lead to pipeline leakage and explosion. To resolve these problems, we propose and experimentally demonstrate the FMF-based Brillouin optical time domain reflectometry (BOTDR) based on the combination of spontaneous Brillouin scattering (SPBS) and heterodyne detection, which can measure the distribution of strain and temperature simultaneously along an FUT by light injection only from one end of the fiber, based on their dependence on the BFS. The schematic diagram of BOTDR system is presented in Fig. 1(a), with υB and υo as the altered and reference BFS respectively. The backscatter spectrum after Lorentzian fitting is symmetrical around the incident lightwave frequency and the new peaks are equally spaced by BFS, as shown in Fig. 1(b).

 figure: Fig. 1

Fig. 1 (a) Operational Principle of BOTDR Sensing System; (b) Schematic of Brillouin frequency shift.

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Moreover, the measurement accuracy of few-mode (FM) Brillouin sensing systems depends on the BFS difference ∆υB. Though it’s theoretically impossible to separate strain and temperature effects by only measuring one BFS, their discrimination can be achieved using two BFSs from different spatial modes, under the condition that the BFS difference between two modes ∆υB is sufficiently large to identify. Though we experimentally verified that FM-BOTDR has similar performance with ∆υB~22 MHz, compared with FM-BDG or FM-BOTDA where ∆υB is around 19~27 MHz [12, 14], there is still room for improvement, for the potential of space-division multiplexing (SDM) hasn't been fully utilized for such distributed Brillouin sensing systems. Since the index profiles of conventional FMFs are designed to provide low differential group delay (DGD) and limited nonlinear effects for transmission purposes [15], they might not be optimized for differential frequency shift between modes, which is the major consideration for sensing application. In this paper, to optimize the design of our Brillouin sensing system using FMF, we further studied the signal-to-noise ratio (SNR) of FM-BOTDR, the temperature-induced expansion strain (TIES) in response to different modes, as well as the impact of refractive index (RI), modal profile and rare-earth doping concentrations on the overall sensing performance, which conveys valuable information for developing practical next-generation optical sensing systems. Finally, an optimal design of double-index highly-nonlinear few-mode fiber (HNL-FMF) is proposed for simultaneous temperature and strain measurement, which indicates up to fivefold increase of ΔυΒ by adjusting the doping level of GeO2 and F2.

In this paper, we present a single-end FMF-based distributed sensing system that allows simultaneous temperature and strain measurement by BOTDR and heterodyne detection. The performance is optimized by exploiting modal profile and doping concentration. An overview of basic theory as well as an experimental assessment of FM-BOTDR will be presented in following sections. Section 2 covers the detailed theoretical analysis and numerical modeling of Brillouin gain spectrum (BGS) characterization as well as multi-parameter discriminative measurement techniques. Theoretical predictions are then verified by experiments that each spatial mode in FMF may have slightly different Brillouin properties, as presented in Section 3-4. This scheme employs the differential BFS subtraction of LP01 and LP11 mode, allowing for an inversion in the linear equation system which calculates the temperature and strain variations, with their proportionality constants obtained. Furthermore, a Ge/F co-doped double-index HNL-FMF design is purposed for enhanced discriminative sensing performance in Section 5. Lastly, the conclusion is reached in Section 6.

2. Principle

Brillouin scattered light is caused by nonlinear interaction between the incident light and phonons that are thermally excited within the light propagation medium, which is shifted in frequency by BFS and propagates in the opposite direction [16]. The scalar wave equation of the optical field is written as:

d2fodr2+1rdfodr+ko2(no2(r)noeff2)fo=0.
where fo represents the optical field distribution as a function of the radial position r, while the subscript o stands for the optical field. Here ko is the optical wave number associated with the optical wavelength λ by 2π/λ; no (r) defines the optical refractive index for the fundamental optical mode, while no eff is the effective refractive index of the optical guided modes.

The thermally excited mechanical vibrations can propagate as guided acoustic modes in optical fibers, and a fraction of the light is backscattered due to the interaction between the acoustic modes on the optical fiber modes. The acoustic refractive index na (r) is defined as [17]:

na(r)=VCladVL(r).
where VL(r) indicates the longitudinal acoustic velocity as a function of radial position r, and VClad denotes the longitudinal acoustic velocity in fiber cladding. The acoustic wavelength is:
λa=λ2noeff.
Correspondingly, the acoustic wave number associated with the acoustic wavelength λa is:
ka=2πλa=4πnoeffλ.
And the effective refractive index of the acoustic guided modes is described as [18]:
naeff=VCladVeff.
where Veff is the effective longitudinal velocity. Accordingly, the scalar wave equation for acoustic modes with no azimuthal variations can be expressed in the same form as Eq. (1):
d2fadr2+1rdfadr+ka2(na2(r)naeff2)fa=0.
where the subscript a denotes the longitudinal acoustic field. Thus the optical refractive index delta profile Δo can be described as:
Δo=no2(r)noClad22no2(r)×100%.
where the optical refractive index for the fiber cladding no Clad = 1.445. Similarly, the acoustic refractive index delta profile Δa stands as:
Δa=na2(r)naClad22na2(r)×100%.
where the acoustic refractive index for the cladding naClad = 1.

When light propagates through a FMF, the Brillouin scattering yields either frequency down-shifted (Stokes) or up-shifted (anti-Stokes) photons. The corresponding BFS is given by:

νB=2noeffλVeff=VCladλ2noeffnaeff.
Here the longitudinal acoustic velocity in cladding VClad equals to 5944 in the unit of m/s [19].

The thermally excited mechanical vibrations can propagate as guided acoustic modes in optical fibers, and a fraction of the light is backscattered due to the interaction between the acoustic modes on the optical modes [1,2]. The normalized modal overlap integral between optical and acoustic fields Iu is defined as [20]:

Iu=(EoEo*ρu*rdrdθ)2(EoEo*)2rdrdθρρ*rdrdθ.
where the integral brackets denote the integration over the polar coordinates r and θ with the electric field distribution of optical modes Eo and acoustic density variation ρ for the acoustic mode of order u. This overlap integral can be controlled through fiber refractive index profile design and acoustic velocity profile design.

Figures 2(a) and 2(b) depict the intensity profiles of optical/acoustic modes in FMF, from which we can clearly see that for LP01 mode, the optical and acoustic profiles match well, but the overlap integral of optical/acoustic profiles for LP11 mode is much smaller. This may explain why each spatial mode in FMF has slightly different Brillouin property, for Brillouin scattering depends on the strong correlation between the longitudinal acoustic and optical modes. The optical refractive index of the FMF core as a function of temperature and strain variation, as well as the GeO2/F dopant concentration, is expressed by the following equation:

no(ΔT,Δε,ωGe,ωF)=noClad[1+(1×103+3×106ΔT+1.5×107Δε)ωGe+(3.3×103+3.6×106ΔT+7.5×107Δε)*ωF].
Here ωGe/ωF are the mole percent of germanium and fluorine dopant, and no Clad = 1.445.

 figure: Fig. 2

Fig. 2 The intensity profiles of optical/acoustic modes in FMF. (a) Normalized Intensity Profile for LP01; (b) Normalized Intensity Profile for LP11.

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Likewise, the acoustic refractive index of the FMF core can be described as [20]:

na(ΔT,Δε,ωGe,ωF)=naClad[1+(7.2×1034.7×105ΔT2.1×106Δε)ωGe+(2.7×1031.8×105ΔT3.8×106Δε)*ωF].
where naClad = 1. In accordance with Eqs. (11)(12), both the optical and acoustic refractive index in fiber cores rise with the GeO2 dopant. In contrast, F dopant decreases the optical index, while increasing the acoustic index.

The operation principle of BOTDR is based on the fact that BFS can be affected by both tensile strain and temperature. One of the major challenges in distributed fiber optic sensing is how to separate the temperature and strain effects, thus avoiding the measurement cross-sensitivity. The BFSs of optical spatial mode one and mode two, ΔνBMode 1 and ΔνBMode 2, are related to the temperature change ∆T and strain variation ∆ε by the following equations:

(ΔνBMode1ΔνBMode2)=(CνTMode1CνεMode1CνTMode2CνεMode2)(ΔTΔε).
If the optical fiber can support up to N spatial modes, then Eq. (13) can be expanded into:
(ΔνBMode1ΔνBModeN)=(CνTMode1CνεMode1CνTModeNCνεModeN)(ΔTΔε).
where CνTModei, CνεModei (i = 1,2,…N) are the temperature and strain coefficients for mode i in the FMF, calculated by CνTModei = (∂νBMode i/T)ε ; CνεModei = (∂νBMode i/ε)T. Two spatial modes are necessary for the discriminative measurement of temperature and strain. In principle, N spatial modes in FMF can be applied to separate N different environmental parameters. Besides, additional modes can be used to do the error correction upon the same parameter measurement, thus enabling higher spectral resolution and faster time response. To get started, we choose N = 2 here. Thus the changes in temperature and strain can be derived from Eq. (13), given by the following equations, respectively:
ΔT=CνεMode2ΔνBMode1CνεMode1ΔνBMode2CνεMode2CνTMode1CνεMode1CνTMode2.
Δε=CνTMode2ΔνBMode1CνTMode1ΔνBMode2CνTMode2CνεMode1CνTMode1CνεMode2.
By solving the simultaneous equations Eqs. (15)(16) for the distance z along the optical fiber, it is possible to discriminate between strain and temperature effects, and henceforward obtain the distribution data along the FMF. Figure 3 shows the sample spectrum of optical and acoustic-wave Brillouin scattering in a FMF that supports LP01 and LP11 modes.

 figure: Fig. 3

Fig. 3 Brillouin Gain Spectrum of BOTDR in FMF for LP01/11 modes.

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where ΔPB is the power level difference of Brillouin scattered light caused by strain or temperature applied to the FMF. Since each spatial mode possesses a different effective refractive index (ERI), chromatic dispersion (CD) and loss, the SPBS for various LP modes will create different BGSs and the corresponding BFSs.

Furthermore, if we ignore the uncertainty in the temperature and strain coefficients, and only account for the uncertainty induced by the BFSs from different modes, the maximum errors for the temperature and strain measurements can be determined as follows:

δT=|CνεMode2|δνBMode1+|CνεMode1|δνBMode2|CνεMode2CνTMode1CνεMode1CνTMode2|.
δε=|CνTMode2|δνBMode1+|CνTMode1|δνBMode2|CνTMode2CνεMode1CνTMode1CνεMode2|.
If we assume that the probability distributions for ∆T and ∆ε are Gaussian functions, the root mean square (RMS) values for ∆T and ∆ε can be further derived as:
RMS(ΔT)={[CνεMode2RMS(ΔνBMode1)CνεMode2CνTMode1CνεMode1CνTMode2]2+[CνεMode1RMS(ΔνBMode2)CνεMode2CνTMode1CνεMode1CνTMode2]2}1/2
RMS(Δε)={[CνTMode2RMS(ΔνBMode1)CνTMode2CνεMode1CνTMode1CνεMode2]2+[CνTMode1RMS(ΔνBMode2)CνTMode2CνεMode1CνTMode1CνεMode2]2}1/2
From Eqs. (19) and (20), it can be seen that the accuracy of the temperature and strain measurements depends on the RMS values of ΔνBMode 1 and ΔνBMode 2. In MCF, the BFS differences between the cores have to be maintained by choosing certain material compositions, and careful doping concentrations [9], which inevitably increase the RMS values of ΔνBMode 1 and ΔνBMode 2. On the other hand, the spatial modes in FMF are within the same guiding medium structure with slightly different propagation constants, so they will provide more accurate and stable results.

3. Experimental setup

The experimental setup of the FM-BOTDR system is shown in Fig. 4. A 1550 nm distributed feedback (DFB) laser diode (LD) is used as a light source, the output of which is divided into two arms by a 50:50 coupler. The pump wave is modulated by an electro-optical switch (OS) driven with 50 ns Gaussian pulse from the electrical pulse generator (EPG). One path is first amplified by an erbium doped fiber amplifier (EDFA), and the mode converter (MC) using phase plate makes sure the pump is launched into any desired higher order LP modes. The single-end setup leads to simpler configuration and thus lower cost.

 figure: Fig. 4

Fig. 4 Experimental Setup of Few-mode Brillouin Sensing System. DFB-LD, Distributed Feedback Laser Diode; OS, Optical Switch; EPG, Electrical Pulse Generator; PM-EDFA, Polarization Maintaining Erbium-doped Fiber Amplifier; FPC, Fiber Polarization Controller; MC, Mode Converter; BS, Beam Splitter; MMUX, Mode Multiplexer; FUT, Fiber Under Test; MDEMUX, Mode De-Multiplexer; RE: Reflective End; LO, Local Oscillator; OH, 90° Optical Hybrid; BR, Balanced Receiver; TDS, Time Domain Oscilloscope.

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After the mode multiplexer (MMUX) in the pump path, a 4 km circular-core step-index FMF is used as the fiber under test (FUT), including one stressed section of 0.5 m, one heated section of 20 m, separated by a long loose fiber, with a reflective end (RE) attached to the other side. More details about the FUT parameters can be found in Table 1.

Tables Icon

Table 1. Parameters for Custom-designed Step-index FMF [21]

The weak Stokes light backscattered from the FUT at frequency υo-υB is amplified again by another EDFA. The relative polarization state of the two light beams is optimized by adjusting polarization controllers manually. The overall sampling rate of the measurement for a single position is 50 Hz. The other path is passing through a polarization-maintaining EDFA (PM-EDFA) as the local oscillator (LO) for coherent detection. The optical beat signal of the reference light and the Stokes light is detected by balanced photo-diodes (PDs) and converted to an electrical signal. Eventually, the electrical signals comprising the in-phase (I) and quadrature (Q) components of different LP modes are sampled by a 40-GSa/s time-domain sampling scope (TDS), which is then synchronized and triggered by the EPG with the same frequency of the pump pulse, thus enabling the distributed measurement of BGS.

4. Results and discussion

The operation principle of few-mode Brillouin sensing systems is shown in Fig. 5(a), while the Brillouin backscatter spectrum is displayed in Fig. 5(b). The measurement data are then transferred to a personal computer (PC) for further analysis.

 figure: Fig. 5

Fig. 5 (a) The operation principle of few-mode Brillouin sensing systems; (b) Sample of experimental Brillouin spectrum.

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The measurement results of the BGS distribution for LP01 and LP11 modes are depicted in Fig. 6(a) respectively. With the inset of Fig. 6(a), which reports the acquired mode patterns, we can clearly see that each spatial mode in FMF may have different Brillouin properties, even though they are propagating through the same guiding medium structure with slightly different propagation constants. The BFS is then determined from the differential spectrum and used to determine the effects of strain or temperature along the sensing fiber. The SNR of BOTDR system can be expressed as the ratio of maximum and minimum of Lorentzian fitting curve for all the amplitude data at a fixed frequency. Since single measurement’s SNR of BOTDR is relatively low, heterodyne detection is applied to boost the system sensitivity; while averaging is performed to improve the SNR. The SNR distribution is illustrated in Fig. 6(b), where fluctuation results from the amplitude distribution variability, and it shows that the fundamental mode experiences higher gain compared to the high-order mode, due to their dissimilar optical/acoustic correlation profiles shown in Fig. 2.

 figure: Fig. 6

Fig. 6 (a) Output Spectra of SPBS in FMF for LP01 (left) and LP11 (right) modes, with the corresponding mode patterns shown in the inset; (b) SNR comparison between LP01 and LP11 modes along the sensing fiber for FM-BOTDR system after 20 times averaging.

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In order to separate the effects of thermally-induced strain and manually added strain, we first kept the temperature unchanged when measuring the strain coefficients, and vice versa. The BFS dependence of LP01 and LP11 modes on temperature and strain are presented in Figs. 7-8. The initial temperature To is set as 25 °C, and the initial strain value εo is 0 με. In the inset of Fig. 7 and Fig. 8, a least squares fitting of linear regression has been illustrated in order to calculate the proportionality coefficients described in Eq. (13).

 figure: Fig. 7

Fig. 7 Calibration of Temperature Coefficient for different modes in FMF.

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 figure: Fig. 8

Fig. 8 Calibration of Strain Coefficient for different modes in FMF.

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The proportionality coefficients differ slightly depending on the type of optical fiber and the coating material being used, as well as the instinct Brillouin properties for each spatial mode caused by different differential group delay (DGD), chromatic dispersion, loss and effective refractive index [1, 4].

Furthermore, the differential BFS between two modes was measured when we increased the temperature, which also shows a linear relationship with an average slope of 34.35 kHz/°C in the inset of Fig. 9. Correspondingly, the average slope between the differential BFS and strain is 0.89 kHz/με through linear regression, as depicted in the inset of Fig. 10. It can be seen in Fig. 9 and Fig. 10 that the differential BFS increases with the maximum measurable temperature and strain, which makes FM-BOTDR an excellent candidate for sensing requirements in harsh environment applications, such as temperature/pressure sensing for the petroleum industry.

 figure: Fig. 9

Fig. 9 Differential BFS Dependence on Temperature in FMF.

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 figure: Fig. 10

Fig. 10 Differential BFS Dependence on Strain in FMF.

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The 3D Brillouin gain spectra as well as the received signal strength for different modes are shown in Fig. 11(a) and 11(b), when the temperature and strain changes are performed at the same time, from which we can see that LP01 mode has slightly higher gain compared with LP11 mode, because of their diverse optical/acoustic overlap integrals.

 figure: Fig. 11

Fig. 11 Three-Dimensional BGS Diagrams (upper), Received signal strength vs. sensing distance (bottom); (a) for LP01 mode; (b) for LP11 mode.

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To understand the strain change under different temperature condition, the 3D plots of Brillouin frequency shift υB vs. simultaneous temperature T and total strain ∑ε variations for two modes are depicted in Fig. 12. The surface stands for numerical prediction, and the red dots represent the experimental data when we change temperature and strain simultaneously. The relation between BFS and strain is not completely linear, because a small fraction of total strain ∑ε was thermally-induced, instead of manually added. The temperature-induced expansion strain (TIES) is guided by εΔT = α ΔT, where α serves as the coefficient of thermal expansion of pure silica [22]. We can clearly see that the experimental points match well with the numerical results.

 figure: Fig. 12

Fig. 12 3D plots of Brillouin frequency shift υB vs. simultaneous temperature T and total strain ∑ε changes, where the surface stands for numerical prediction, and the red dots represent the experimental data when we change temperature and strain simultaneously. (a) υB vs.T vs.∑ε for LP01; (b) υB vs.T vs.∑ε for LP11. The nonlinear relation between υB and ∑ε is due to the temperature-induced expansion strain.

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In contrast to the proportionality coefficients in standard silica SMF at 1550 nm [1,17], which are averagely 1.08 MHz/°C and 43 kHz/με respectively, both the strain and temperature coefficients (f-ε and f-T) in FMF are slightly larger, as denoted in Table 2 below, which is caused by the difference of structural deformation in FMF. The LP01 mode has slightly larger coefficients, because its intensity profile has stronger correlation between optical and acoustic modes. Besides, since the thermal expansion coefficient α is extremely small for FMF, the coefficient according to the thermal expansion CεΔT is minor compared to the other ones.

Tables Icon

Table 2. Comparison of f-T and f-ε coefficients in FMF

Repeated experiments indicate that the measurement uncertainties for BFSs of LP01/11 modes are around 0.8 MHz, which could be used to estimate the temperature and strain accuracies by Eqs. (17) and (18) as roughly ± 1.7 °C in temperature and ± 39 με in strain. Furthermore, the measurement accuracy of FM-Brillouin sensing systems depends on the BFS difference ΔυB. Since conventional FMFs are designed for transmission purposes, the largest ΔυB that could be generated through experiments was only around 22 MHz, compared with FM-BDG or FM-BOTDA where ΔυB is around 19~27 MHz [11, 12, 14], there is still room for improvement, for the potential of space-division multiplexing (SDM) hasn't been fully utilized for such distributed Brillouin sensing systems. So we further purposed and numerically verified novel designs of rare-earth-doped double-index highly nonlinear few-mode fibers (HNL-FMF) to enhance the sensing performance by analyzing various modal profiles and doping concentrations. According to Eqs. (11) and (12), the GeO2 dopant increases both the optical and acoustic refractive indices in fiber cores, while the F2 dopant decreases the optical index, while increasing the acoustic index. Therefore various FMF designs with different refractive index profiles can be fully characterized by simultaneously adjusting the doping level of GeO2 and F2, as displayed in Fig. 13(a)-13(f), while the designed fiber retains a step optical refractive index profile.

 figure: Fig. 13

Fig. 13 Refractive Index Profiles of Various Designs; n vs. Radius r (μm).

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The specific data of HNL-FMF designs in Fig. 13 regarding the GeO2/F2 dopant concentration ranging from 0 up to 10 wt.%, profile radius (μm), propagation constants for optical/acoustic modes, BFSs for LP01/LP11 modes (GHz), ΔυB (MHz) between two modes are exhibited by Table 3 above. We tried more than two dozen designs in order to reach the optimum, yet only the most typical ones are listed in Fig. 13 and Table 3. Figure 14(a) schematically illustrates the cross-section and modal RI profile of type (f). The fiber core diameter is around 15 μm, while the cladding and coating diameters are 125 μm and 250 μm respectively. This novel Ge/F co-doped HNL-FMF has exceptionally higher germanium concentration to produce the desired Brillouin outputs within the 1550 nm window. In addition, from Fig. 14(b) we can see that, by carefully adjusting the doping level of GeO2 and F2, the correlation between the longitudinal acoustic and optical modes can be boosted, which indicates up to fivefold enhancement in measurement accuracy. This design conveys valuable information for developing practical next-generation few-mode optical sensing systems.

Tables Icon

Table 3. Comparison of various mode profiles and doping design in FMF

 figure: Fig. 14

Fig. 14 Specifications of HNL-FMF core. (a) Cross section and profile for HNL-FMF core; (b) Refractive index and modal profiles of HNL-FMF.

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5. Conclusion

A space-division multiplexed BOTDR sensing system requiring only one accessible end is investigated that realizes simultaneous temperature and strain sensing based on spontaneous Brillouin scattering and heterodyne detection. Detailed theoretical analysis is provided for the multi-parameter discriminative measurement technique of Brillouin FM-sensing systems. This scheme employs the differential BFS subtraction of LP01 and LP11 mode, allowing for an inversion in the linear equation system which calculates the temperature and strain variations, with their proportionality constants obtained. It is then verified by experiments that FM-BOTDR achieves comparable performance with two-end approaches such as FM-BOTDA or FM-BDG, but with simpler setup and lower cost. The measurement accuracy is further optimized by exploiting modal profile and doping concentration. Finally, a Ge/F co-doped double-index HNL-FMF design is purposed for enhanced discriminative sensing performance with up to fivefold increase of ∆υB. This novel single-end sensing system proves to have great potential for geotechnical and petroleum applications, such as the automated monitoring of oil or gas pipelines.

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16. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).

17. H. H. Kee, G. P. Lees, and T. P. Newson, “All-fiber system for simultaneous interrogation of distributed strain and temperature sensing by spontaneous Brillouin scattering,” Opt. Lett. 25(10), 695–697 (2000). [CrossRef]   [PubMed]  

18. A. B. Ruffin, M.-J. Li, X. Chen, A. Kobyakov, and F. Annunziata, “Brillouin gain analysis for fibers with different refractive indices,” Opt. Lett. 30(23), 3123–3125 (2005). [CrossRef]   [PubMed]  

19. M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007). [CrossRef]   [PubMed]  

20. K. Oh and U.-C. Paek, Silica Optical Fiber Technology for Devices and Components: Design, Fabrication, and International Standards (Wiley, 2012).

21. E. Ip, N. Bai, Y.-K. Huang, E. Mateo, F. Yaman, M.-J. Li, S. Bickham, S. Ten, J. Linares, C. Montero, V. Moreno, X. Prieto, Y. Luo, G.-D. Peng, G. Li, and T. Wang, “6×6 MIMO transmission over 50+25+10 km heterogeneous spans of few-mode fiber with inline erbium-doped fiber amplifier,” in Optical Fiber Communication Conference, 2012 OSA Technical Digest Series (Optical Society of America, 2012), paper OTu2C.4.

22. X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012). [CrossRef]  

References

  • View by:

  1. X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors (Basel) 11(12), 4152–4187 (2011).
    [Crossref] [PubMed]
  2. T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
    [Crossref]
  3. W. R. Habel and K. Krebber, “Fiber-optic sensor applications in civil and geotechnical engineering,” Photonic Sensors 1(3), 268–280 (2011).
    [Crossref]
  4. M. Alahbabi, Y. T. Cho, and T. P. Newson, “Comparison of the methods for discriminating temperature and strain in spontaneous Brillouin-based distributed sensors,” Opt. Lett. 29(1), 26–28 (2004).
    [Crossref] [PubMed]
  5. S. M. Maughan, H. H. Kee, and T. P. Newson, “Simultaneous distributed fiber temperature and strain sensor using microwave coherent detection of spontaneous Brillouin backscatter,” Meas. Sci. Technol. 12(7), 834–842 (2001).
    [Crossref]
  6. T. R. Parker, M. Farhadiroushan, V. A. Handerek, and A. J. Rogers, “Temperature and strain dependence of the power level and frequency of spontaneous Brillouin scattering in optical fibers,” Opt. Lett. 22(11), 787–789 (1997).
    [Crossref] [PubMed]
  7. M. N. Alahbabi, Y. T. Cho, and T. P. Newson, “Simultaneous temperature and strain measurement with combined spontaneous Raman and Brillouin scattering,” Opt. Lett. 30(11), 1276–1278 (2005).
    [Crossref] [PubMed]
  8. X. Liu and X. Bao, “Brillouin spectrum in LEAF and simultaneous temperature and strain measurement,” J. Lightwave Technol. 30(8), 1053–1059 (2012).
    [Crossref]
  9. X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).
  10. G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photon. 6(4), 413–487 (2014).
    [Crossref]
  11. A. Li, Y. Wang, Q. Hu, and W. Shieh, “Few-mode fiber based optical sensors,” Opt. Express 23(2), 1139–1150 (2015).
    [Crossref]
  12. K. Y. Song and Y. H. Kim, “Characterization of stimulated Brillouin scattering in a few-mode fiber,” Opt. Lett. 38(22), 4841–4844 (2013).
    [Crossref] [PubMed]
  13. S. Li, M.-J. Li, and R. S. Vodhanel, “All-optical Brillouin dynamic grating generation in few-mode optical fiber,” Opt. Lett. 37(22), 4660–4662 (2012).
    [Crossref] [PubMed]
  14. A. Li, Q. Hu, and W. Shieh, “Characterization of stimulated Brillouin scattering in a circular-core two-mode fiber using optical time-domain analysis,” Opt. Express 21(26), 31894–31906 (2013).
    [Crossref] [PubMed]
  15. D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
    [Crossref]
  16. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).
  17. H. H. Kee, G. P. Lees, and T. P. Newson, “All-fiber system for simultaneous interrogation of distributed strain and temperature sensing by spontaneous Brillouin scattering,” Opt. Lett. 25(10), 695–697 (2000).
    [Crossref] [PubMed]
  18. A. B. Ruffin, M.-J. Li, X. Chen, A. Kobyakov, and F. Annunziata, “Brillouin gain analysis for fibers with different refractive indices,” Opt. Lett. 30(23), 3123–3125 (2005).
    [Crossref] [PubMed]
  19. M.-J. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. A. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15(13), 8290–8299 (2007).
    [Crossref] [PubMed]
  20. K. Oh and U.-C. Paek, Silica Optical Fiber Technology for Devices and Components: Design, Fabrication, and International Standards (Wiley, 2012).
  21. E. Ip, N. Bai, Y.-K. Huang, E. Mateo, F. Yaman, M.-J. Li, S. Bickham, S. Ten, J. Linares, C. Montero, V. Moreno, X. Prieto, Y. Luo, G.-D. Peng, G. Li, and T. Wang, “6×6 MIMO transmission over 50+25+10 km heterogeneous spans of few-mode fiber with inline erbium-doped fiber amplifier,” in Optical Fiber Communication Conference, 2012 OSA Technical Digest Series (Optical Society of America, 2012), paper OTu2C.4.
  22. X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
    [Crossref]

2015 (1)

2014 (2)

X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).

G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photon. 6(4), 413–487 (2014).
[Crossref]

2013 (3)

2012 (3)

S. Li, M.-J. Li, and R. S. Vodhanel, “All-optical Brillouin dynamic grating generation in few-mode optical fiber,” Opt. Lett. 37(22), 4660–4662 (2012).
[Crossref] [PubMed]

X. Liu and X. Bao, “Brillouin spectrum in LEAF and simultaneous temperature and strain measurement,” J. Lightwave Technol. 30(8), 1053–1059 (2012).
[Crossref]

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

2011 (2)

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors (Basel) 11(12), 4152–4187 (2011).
[Crossref] [PubMed]

W. R. Habel and K. Krebber, “Fiber-optic sensor applications in civil and geotechnical engineering,” Photonic Sensors 1(3), 268–280 (2011).
[Crossref]

2007 (1)

2005 (2)

2004 (1)

2001 (1)

S. M. Maughan, H. H. Kee, and T. P. Newson, “Simultaneous distributed fiber temperature and strain sensor using microwave coherent detection of spontaneous Brillouin backscatter,” Meas. Sci. Technol. 12(7), 834–842 (2001).
[Crossref]

2000 (1)

1997 (1)

1995 (1)

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Alahbabi, M.

Alahbabi, M. N.

Annunziata, F.

Bai, N.

Bao, X.

X. Liu and X. Bao, “Brillouin spectrum in LEAF and simultaneous temperature and strain measurement,” J. Lightwave Technol. 30(8), 1053–1059 (2012).
[Crossref]

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors (Basel) 11(12), 4152–4187 (2011).
[Crossref] [PubMed]

Burgess, D. T.

X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).

Chen, L.

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors (Basel) 11(12), 4152–4187 (2011).
[Crossref] [PubMed]

Chen, X.

Cho, Y. T.

Crowley, A. M.

Demeritt, J. A.

Farhadiroushan, M.

Fini, J. M.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Gray, S.

Habel, W. R.

W. R. Habel and K. Krebber, “Fiber-optic sensor applications in civil and geotechnical engineering,” Photonic Sensors 1(3), 268–280 (2011).
[Crossref]

Handerek, V. A.

Hines, M.

X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).

Horiguchi, T.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Hu, Q.

Kee, H. H.

S. M. Maughan, H. H. Kee, and T. P. Newson, “Simultaneous distributed fiber temperature and strain sensor using microwave coherent detection of spontaneous Brillouin backscatter,” Meas. Sci. Technol. 12(7), 834–842 (2001).
[Crossref]

H. H. Kee, G. P. Lees, and T. P. Newson, “All-fiber system for simultaneous interrogation of distributed strain and temperature sensing by spontaneous Brillouin scattering,” Opt. Lett. 25(10), 695–697 (2000).
[Crossref] [PubMed]

Kim, Y. H.

Kobyakov, A.

Koyamada, Y.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Krebber, K.

W. R. Habel and K. Krebber, “Fiber-optic sensor applications in civil and geotechnical engineering,” Photonic Sensors 1(3), 268–280 (2011).
[Crossref]

Kurashima, T.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Lees, G. P.

Li, A.

Li, G.

Li, J.

X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).

Li, M.-J.

Li, S.

Li, X.

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

Liang, J.

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

Lin, S.

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

Liu, A.

Liu, X.

Maughan, S. M.

S. M. Maughan, H. H. Kee, and T. P. Newson, “Simultaneous distributed fiber temperature and strain sensor using microwave coherent detection of spontaneous Brillouin backscatter,” Meas. Sci. Technol. 12(7), 834–842 (2001).
[Crossref]

Nelson, L. E.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Newson, T. P.

Oigawa, H.

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

Parker, T. R.

Richardson, D. J.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Rogers, A. J.

Ruffin, A. B.

Shieh, W.

Shimizu, K.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Song, K. Y.

Sun, X.

X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).

Tateda, M.

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

Ueda, T.

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

Vodhanel, R. S.

Walton, D. T.

Wang, J.

Wang, Y.

Xia, C.

Zenteno, L. A.

Zhang, Y.

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

Zhao, N.

Zhu, B.

X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).

Adv. Opt. Photon. (1)

IEEE Photonics J. (1)

X. Li, S. Lin, J. Liang, Y. Zhang, H. Oigawa, and T. Ueda, “Fiber-Optic Temperature Sensor Based on Difference of Thermal Expansion Coefficient Between Fused Silica and Metallic Materials,” IEEE Photonics J. 4(1), 155–162 (2012).
[Crossref]

J. Lightwave Technol. (2)

T. Horiguchi, K. Shimizu, T. Kurashima, M. Tateda, and Y. Koyamada, “Development of a distributed sensing technique using Brillouin scattering,” J. Lightwave Technol. 13(7), 1296–1302 (1995).
[Crossref]

X. Liu and X. Bao, “Brillouin spectrum in LEAF and simultaneous temperature and strain measurement,” J. Lightwave Technol. 30(8), 1053–1059 (2012).
[Crossref]

Meas. Sci. Technol. (1)

S. M. Maughan, H. H. Kee, and T. P. Newson, “Simultaneous distributed fiber temperature and strain sensor using microwave coherent detection of spontaneous Brillouin backscatter,” Meas. Sci. Technol. 12(7), 834–842 (2001).
[Crossref]

Nat. Photonics (1)

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Opt. Express (3)

Opt. Lett. (7)

Photonic Sensors (1)

W. R. Habel and K. Krebber, “Fiber-optic sensor applications in civil and geotechnical engineering,” Photonic Sensors 1(3), 268–280 (2011).
[Crossref]

Proc. SPIE (1)

X. Sun, J. Li, D. T. Burgess, M. Hines, and B. Zhu, “A multicore optical fiber for distributed sensing,” Proc. SPIE 9098, 90980W (2014).

Sensors (Basel) (1)

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors (Basel) 11(12), 4152–4187 (2011).
[Crossref] [PubMed]

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).

K. Oh and U.-C. Paek, Silica Optical Fiber Technology for Devices and Components: Design, Fabrication, and International Standards (Wiley, 2012).

E. Ip, N. Bai, Y.-K. Huang, E. Mateo, F. Yaman, M.-J. Li, S. Bickham, S. Ten, J. Linares, C. Montero, V. Moreno, X. Prieto, Y. Luo, G.-D. Peng, G. Li, and T. Wang, “6×6 MIMO transmission over 50+25+10 km heterogeneous spans of few-mode fiber with inline erbium-doped fiber amplifier,” in Optical Fiber Communication Conference, 2012 OSA Technical Digest Series (Optical Society of America, 2012), paper OTu2C.4.

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Figures (14)

Fig. 1
Fig. 1 (a) Operational Principle of BOTDR Sensing System; (b) Schematic of Brillouin frequency shift.
Fig. 2
Fig. 2 The intensity profiles of optical/acoustic modes in FMF. (a) Normalized Intensity Profile for LP01; (b) Normalized Intensity Profile for LP11.
Fig. 3
Fig. 3 Brillouin Gain Spectrum of BOTDR in FMF for LP01/11 modes.
Fig. 4
Fig. 4 Experimental Setup of Few-mode Brillouin Sensing System. DFB-LD, Distributed Feedback Laser Diode; OS, Optical Switch; EPG, Electrical Pulse Generator; PM-EDFA, Polarization Maintaining Erbium-doped Fiber Amplifier; FPC, Fiber Polarization Controller; MC, Mode Converter; BS, Beam Splitter; MMUX, Mode Multiplexer; FUT, Fiber Under Test; MDEMUX, Mode De-Multiplexer; RE: Reflective End; LO, Local Oscillator; OH, 90° Optical Hybrid; BR, Balanced Receiver; TDS, Time Domain Oscilloscope.
Fig. 5
Fig. 5 (a) The operation principle of few-mode Brillouin sensing systems; (b) Sample of experimental Brillouin spectrum.
Fig. 6
Fig. 6 (a) Output Spectra of SPBS in FMF for LP01 (left) and LP11 (right) modes, with the corresponding mode patterns shown in the inset; (b) SNR comparison between LP01 and LP11 modes along the sensing fiber for FM-BOTDR system after 20 times averaging.
Fig. 7
Fig. 7 Calibration of Temperature Coefficient for different modes in FMF.
Fig. 8
Fig. 8 Calibration of Strain Coefficient for different modes in FMF.
Fig. 9
Fig. 9 Differential BFS Dependence on Temperature in FMF.
Fig. 10
Fig. 10 Differential BFS Dependence on Strain in FMF.
Fig. 11
Fig. 11 Three-Dimensional BGS Diagrams (upper), Received signal strength vs. sensing distance (bottom); (a) for LP01 mode; (b) for LP11 mode.
Fig. 12
Fig. 12 3D plots of Brillouin frequency shift υB vs. simultaneous temperature T and total strain ∑ε changes, where the surface stands for numerical prediction, and the red dots represent the experimental data when we change temperature and strain simultaneously. (a) υB vs.T vs.∑ε for LP01; (b) υB vs.T vs.∑ε for LP11. The nonlinear relation between υB and ∑ε is due to the temperature-induced expansion strain.
Fig. 13
Fig. 13 Refractive Index Profiles of Various Designs; n vs. Radius r (μm).
Fig. 14
Fig. 14 Specifications of HNL-FMF core. (a) Cross section and profile for HNL-FMF core; (b) Refractive index and modal profiles of HNL-FMF.

Tables (3)

Tables Icon

Table 1 Parameters for Custom-designed Step-index FMF [21]

Tables Icon

Table 2 Comparison of f-T and f-ε coefficients in FMF

Tables Icon

Table 3 Comparison of various mode profiles and doping design in FMF

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

d 2 f o d r 2 + 1 r d f o dr + k o 2 ( n o 2 ( r ) n oeff 2 ) f o =0.
n a ( r )= V Clad V L ( r ) .
λ a = λ 2 n oeff .
k a = 2π λ a = 4π n oeff λ .
n aeff = V Clad V eff .
d 2 f a d r 2 + 1 r d f a dr + k a 2 ( n a 2 ( r ) n aeff 2 ) f a =0.
Δ o = n o 2 ( r ) n oClad 2 2 n o 2 ( r ) ×100%.
Δ a = n a 2 ( r ) n aClad 2 2 n a 2 ( r ) ×100%.
ν B = 2 n oeff λ V eff = V Clad λ 2 n oeff n aeff .
I u = ( E o E o * ρ u * rdrdθ ) 2 ( E o E o * ) 2 rdrdθ ρ ρ * rdrdθ .
n o ( ΔT,Δε, ω Ge , ω F )= n oClad [1+( 1× 10 3 +3× 10 6 ΔT+1.5× 10 7 Δε ) ω Ge +( 3.3× 10 3 +3.6× 10 6 ΔT+7.5× 10 7 Δε )* ω F ].
n a ( ΔT,Δε, ω Ge , ω F )= n aClad [1+( 7.2× 10 3 4.7× 10 5 ΔT2.1× 10 6 Δε ) ω Ge +( 2.7× 10 3 1.8× 10 5 ΔT3.8× 10 6 Δε )* ω F ].
( Δ ν B Mode1 Δ ν B Mode2 )=( C νT Mode1 C νε Mode1 C νT Mode2 C νε Mode2 )( ΔT Δε ).
( Δ ν B Mode1 Δ ν B ModeN )=( C νT Mode1 C νε Mode1 C νT ModeN C νε ModeN )( ΔT Δε ).
ΔT= C νε Mode2 Δ ν B Mode1 C νε Mode1 Δ ν B Mode2 C νε Mode2 C νT Mode1 C νε Mode1 C νT Mode2 .
Δε= C νT Mode2 Δ ν B Mode1 C νT Mode1 Δ ν B Mode2 C νT Mode2 C νε Mode1 C νT Mode1 C νε Mode2 .
δT= | C νε Mode2 |δ ν B Mode1 +| C νε Mode1 |δ ν B Mode2 | C νε Mode2 C νT Mode1 C νε Mode1 C νT Mode2 | .
δε= | C νT Mode2 |δ ν B Mode1 +| C νT Mode1 |δ ν B Mode2 | C νT Mode2 C νε Mode1 C νT Mode1 C νε Mode2 | .
RMS( ΔT )= { [ C νε Mode2 RMS( Δ ν B Mode1 ) C νε Mode2 C νT Mode1 C νε Mode1 C νT Mode2 ] 2 + [ C νε Mode1 RMS( Δ ν B Mode2 ) C νε Mode2 C νT Mode1 C νε Mode1 C νT Mode2 ] 2 } 1/2
RMS( Δε )= { [ C νT Mode2 RMS( Δ ν B Mode1 ) C νT Mode2 C νε Mode1 C νT Mode1 C νε Mode2 ] 2 + [ C νT Mode1 RMS( Δ ν B Mode2 ) C νT Mode2 C νε Mode1 C νT Mode1 C νε Mode2 ] 2 } 1/2

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