Multi-functional fiber optic sensors based on mode division multiplexing

Yi Weng; Ting Wang; Zhongqi Pan

doi:10.1364/OME.7.001917

1. Introduction

Brillouin based optical fiber sensors have attracted substantial interest for more than two decades due to many eminent advantages such as their corrosion-free features and immunity to electromagnetic interference, enhanced measuring accuracy attributable to frequency revolved interrogation over kilometer range, manifold sensitivities of external perturbations (strain, temperature etc.), as well as the dead-zone-free characteristics in an integral or distributed sensing format compared with conventional point sensors like fiber Bragg gratings (FBG) and/or inline Fabry-Perot resonators [1, 2]. Nonetheless, one of the main concerns for such fiber-optic sensors is how to discriminate various measurands simultaneously, for the stimulated Brillouin scattering (SBS) of light, as the foundation of the classical sensing mechanism originated from the intrinsic nonlinearity in monomode optical fibers, relies on the precise detection of the Brillouin frequency shift (BFS), which is proportional to the variations of both temperature and strain, hence creating one equation with two variables and making it hypothetically impossible to separate these two effects by measuring only one BFS [3]. Though quite a couple of techniques have been proposed to resolve this problem in the past decades, previous methods utilizing the single-mode fibers (SMF) indicated either inferior sensing accuracy or extra noise and complexity to the system architecture [4, 5]. Alternatively, the few-mode fibers (FMF) were initially proposed to overcome the capacity crunch in the telecom field via the spatial division multiplexing (SDM) technologies, and interest has lately arisen in the design of FMF-based distributed fiber sensors [6, 7]. In our preliminary experimental demonstrations, we managed to apply one mode for temperature sensing and the other one for strain monitoring, so that one equation with two unknowns is now converted into a 2 × 2 matrix, and hence simultaneous multi-functional sensing can be achieved by means of mode division multiplexing (MDM) [8]. Similar proof has also been attained by other groups at the same time [9]. Hypothetically, the matrix dimensions increase with more spatial modes, and hence more parameters can be measured via an FMF and differentiated thru multiple-input multiple-output (MIMO) signal processing at the receiver side to realize multi-functionality within one optical fiber, whose potential is far beyond that of the conventional SMF-based optical sensors [10]. Henceforward, such MDM-based sensing technologies are involved in a wide variety of industrial applications to smart materials and smart structures with significant potential, as illustrated in Fig. 1, including the oil and gas pipeline monitoring, health monitoring for dam, bridge and building, as well as fire and earthquake disaster alerts [11].

Fig. 1 Schematic of MDM multi-functional sensors for various industrial applications.

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Nonetheless, the FMF-based optical sensors mainly faced the following concerns. To begin with, the responses to the equivalent external stress or perturbations for dissimilar spatial modes must be distinguishable; otherwise the unprecedented multi-functionality might suffer from severe spatiotemporal instability resulting from lack of sufficient resolution [12]. On the other hand, the key parameter for Brillouin distributed FMF sensors is the differential BFS, for it’s been verified by experiments that each spatial mode in an FMF has slightly altered Brillouin properties due to the interactions between guided optical modes and acoustic modes in the fiber [13]. In accordance with the experimental data so far, the measured differential BFS is merely around 30 MHz, making it exceedingly dependent on high-resolution receivers and optical heterodyne detection, which becomes the root cause of high costs and operating complexity [14, 15]. In the end of [8], we mentioned an effective solution with preliminary data, which is to increase the differential BFS via an optimized FMF design by exploiting modal profiles and doping concentrations. However, such approach hasn’t been discussed in detail due to space constraints. More importantly, the few-mode fiber-optic sensors are attributable to intermodal nonlinear backscattering, whose variability depends on the spatial temporal evolution due to mode coupling, birefringence and optical-acoustic interactions [16, 17]. There is therefore a strong need for further optimization of multi-material multimode fibers to realize better multi-functionality in structural health monitoring (SHM) and other industrial applications.

In this paper, the performance of few-mode Brillouin sensors is analytically evaluated via diverse refractive index profiles and dopant material compositions to achieve five to ten folder improvements on the measurement accuracy. The rest of this paper is outlined as follows. Section 2 covers the detailed theoretical analysis and numerical modeling of Brillouin gain spectrum (BGS) characterization owing to mode coupling, birefringence and optical-acoustic interactions. Then the system configuration and preliminary experimental demonstrations are presented in Section 3. The optimization of refractive index profiles and dopant material compositions in the FMF for advanced multi-functionality are explored in Section 4 to 5. In the end, a conclusion is reached in Section 6.

2. Theoretical modeling of few-mode Brillouin sensors

To begin with, the BFS can be expressed as a function of the acoustic speed in silicon v_A, wavelength λ and fiber modal birefringence B_f , as shown in the following [3]:

ν_{B} = (\frac{2 \cdot v_{A}}{λ}) \cdot B_{f} .

where the longitudinal acoustic velocity in silicon is equivalent to roughly 5945 in the unit of m/s. As aforementioned, the Brillouin backscattering is attributable to the interaction between the guided acoustic modes upon the optical modes owing to the thermally excited mechanical vibrations in the FMF, which can also be interpreted as a “dynamic Brillouin grating”; with the normalized intermodal overlap integral concerning the optical and acoustic fields I_u as:

I_{u} = \frac{{(\int E_{o} E_{o}^{*} ρ_{u}^{*} r d r d θ)}^{2}}{\int {(E_{o} E_{o}^{*})}^{2} r d r d θ \cdot \int ρ ρ^{*} r d r d θ} .

Here the integral brackets designate the integration over the polar coordinates r and θ with the electric field distribution of optical modes E_o and the acoustic density variation ρ for the acoustic mode of order u, which can be managed by means of fiber refractive index profile design. The modal birefringence of the FMF B_f can be determined by [18]:

B_{f} = {[P^{x} \cdot (n_{C o}^{2} - n_{C l}^{2}) + n_{C l}^{2}]}^{\frac{1}{2}} + {[P^{y} \cdot (n_{C o}^{2} - n_{C l}^{2}) + n_{C l}^{2}]}^{\frac{1}{2}} .

Here n_Co and n_Cl denote the refractive indices of the core and cladding, while P ^x and P ^y stand for the normalized propagation constants on x or y propagation direction respectively, which can be hypothetically categorized into the zero-order normalized propagation constants of rectangular waveguide P₀ ^x,y, the first-order perturbation of shape-correction P_I^x,y, as well as the second-order perturbation of stress-correction P_II^x,y that is associated with the stress or bending upon the fiber core, defined as:

P^{x} = P_{0}^{x} + P_{I}^{x} + P_{I I}^{x} .

P^{y} = P_{0}^{y} + P_{I}^{y} + P_{I I}^{y} .

The zero-order normalized propagation constant on the x propagation direction is given by:

P_{0}^{x} = \frac{β_{0}^{x} - k_{0}^{2} \cdot n_{C l}^{2}}{k_{0}^{2} \cdot (n_{C o}^{2} - n_{C l}^{2})} .

where β₀ ^x represents the propagation constant on the x propagation direction for LP_11a mode, and k₀ signifies the wave-number.

The first-order perturbation regarding the shape-correction on the x propagation direction can be described as [19]:

P_{I}^{x} = (\frac{1}{n_{C o}^{2} - n_{C l}^{2}}) \cdot \frac{\iint {| ψ_{x} (x, y) |}^{2} \cdot | n_{C o}^{2} - n_{C l}^{2} | d x d y}{\iint {| ψ_{x} (x, y) |}^{2} d x d y} .

Here ψ_x (x, y) denotes the two-dimensional optical field distribution along the x propagation direction.

As for the second-order perturbation of stress-correction, its form on the x propagation direction can be written as:

P_{I I}^{x} = (\frac{1}{n_{C o}^{2} - n_{C l}^{2}}) \cdot \frac{\iint {| ψ_{x} (x, y) |}^{2} \cdot [2 n_{C o} \cdot Δ n_{x} + Δ n_{x}^{2}] d x d y}{\iint {| ψ_{x} (x, y) |}^{2} d x d y} .

where Δn_x symbolizes the refractive-index change in the stressed FMF on the x propagation direction for LP_11a mode.

Due to micro-bending in the cable in addition to the random longitudinal index fluctuation induced by manufacturing process, mode coupling is random distributed along the FMF in location and strength. Such intermodal coupling is enormously important for the performance of MDM sensors with more dynamic spatial channels. When facing non-directional outside influence such as thermal perturbation that is equally sensitive along the fiber, the mode coupling distribution is illustrated in Fig. 2(a), whereas the red boxes at the crossing point of two LP modes designate the strong coupling mode pairs, and the purple ones stand for the weaker coupling combines.

Fig. 2 Mode coupling distribution in the FMF: (a) under thermal perturbation; (b) for x-z plane strain; (c) for y-z plane strain.

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Since all modes are propagating within the same core of FMF and its cross-section is symmetric, the perturbations experienced by the fiber core for different spatial modes would be principally the same. On the other hand, the optical waves with the x and y directions will experience different SBS frequencies, owing to their different refractive indices, for the light energy of the fundamental mode LP₀₁ is most intense at the center, while the power of the higher-order modes like LP_11a_/_b are located along the edges of the FMF core. Henceforward in principle, the LP_11a mode propagating along the x-z dimension would most likely experience more stress on the x propagation direction than the LP₀₁ at the center and the perpendicular LP_11b mode.

Consequently, the two principal axes of the fiber birefringence yield different BFS during the Brillouin scattering, while the mode pairs between which couplings arise for x-z plane and y-z plane perturbations are displayed in Fig. 2(b) and Fig. 2(c) respectively, with blue boxes at the junctions indicating the corresponding strong intermodal coupling primarily between the next adjacent higher and lower mode groups.

Similarly, the zero-order normalized propagation constant on the y propagation direction can be expressed as:

P_{0}^{y} = \frac{β_{0}^{y} - k_{0}^{2} \cdot n_{C l}^{2}}{k_{0}^{2} \cdot (n_{C o}^{2} - n_{C l}^{2})} .

Here β₀ ^y characterizes the corresponding propagation constant on the y propagation direction for LP_11b mode.

The first-order perturbation of shape-correction on the y propagation direction is defined as:

P_{I}^{y} = (\frac{1}{n_{C o}^{2} - n_{C l}^{2}}) \cdot \frac{\iint {| ψ_{y} (x, y) |}^{2} \cdot | n_{C o}^{2} - n_{C l}^{2} | d x d y}{\iint {| ψ_{y} (x, y) |}^{2} d x d y} .

where ψ_y (x, y) signifies the optical field distribution along the y propagation direction. And the second-order perturbation stress-correction on the y propagation direction for LP_11b mode can be determined as:

P_{I I}^{y} = (\frac{1}{n_{C o}^{2} - n_{C l}^{2}}) \cdot \frac{\iint {| ψ_{y} (x, y) |}^{2} \cdot [2 n_{C o} \cdot Δ n_{y} + Δ n_{y}^{2}] d x d y}{\iint {| ψ_{y} (x, y) |}^{2} d x d y} .

Here Δn_y represents the refractive-index variation in the perturbed FMF on the y propagation direction for LP_11b mode. Likewise, since LP_11b mode is propagating along the y-z dimension, it's customarily corresponding to the perturbations on the y propagation direction than the other modes, and henceforth it would have little influence on the fiber birefringence along the x direction. The refractive-index change in the stressed fiber along x/y propagation direction is dependent on the stress-optic coefficient and the stress along x/y propagation direction, while the stress is a function of the strain, Young’s modules, the bending distances as well as the bending angles along the x/y propagation direction.

The refractive-index variation within the perturbed FMF along the x propagation direction for LP_11a mode is shown as [20]:

Δ n_{x} = C_{x} \cdot σ_{x} + C_{y} \cdot σ_{y} .

where C_x and C_y denote the stress-optic coefficients along the x/y propagation directions for LP_11a/b modes, while σ_x and σ_y stand for the corresponding stress for LP_11a/b modes.

In the meantime, the refractive-index change in the FMF along the y propagation direction for LP_11b mode can be determined by:

Δ n_{y} = C_{x} \cdot σ_{y} + C_{y} \cdot σ_{x} .

The stress experienced by different modes can be expressed as a function of the Young’s module and strain along the fiber in the form of:

σ_{x} = E_{x} \cdot ε_{x} .

σ_{y} = E_{y} \cdot ε_{y} .

Here E_x and E_y designate the Young’s module along the x/y propagation direction for LP_11a/b modes, while ε_x and ε_y represent the corresponding strain for LP_11a/b modes.

The strain for LP_11a mode ε_x is defined as:

ε_{x} = - \frac{r \cdot \cos ϕ_{x}}{1 - d S_{x} \cdot \sin \frac{ϕ_{x}}{2}} .

where r symbolizes the fiber core radius, ϕ_x signifies the bending angle on the x propagation direction, while S_x stands for the consistent bending distance for LP_11a mode.

Likewise, the corresponding strain for LP_11b mode ε_y is given by:

ε_{y} = - \frac{r \cdot \cos ϕ_{y}}{1 - d S_{y} \cdot \sin \frac{ϕ_{y}}{2}} .

Here ϕ_y indicates the bending angle along the y propagation direction, while S_y denotes the consistent bending distance for LP_11b mode.

3. Preliminary experimental demonstrations

In our preliminary experimental demonstrations, a single-end FMF-based distributed sensing system was proposed by means of Brillouin optical time-domain reflectometry (BOTDR) and heterodyne detection that allows simultaneous temperature and strain measurement. The high-level experimental setup is presented in Fig. 3, in which a 50 ns Gaussian pulse was emitted through a phase-plate-based mode coupler into a multi-channel BOTDR along a 4 km step-index FMF (SI-FMF) supporting LP₀₁ and LP_11a_/_b modes, and the crosstalk in-between was approximately −22 dB [8]. The backscattered lights from both fundamental mode and higher-order modes of the FMF were then detected simultaneously by the multi-channel synchronous BOTDR via spatial light modulators (SLM) with polarization diversity.

Fig. 3 Experimental configuration for simultaneous multi-functional sensing via multi-channel BOTDR using an FMF; CL_1,2,3, collimating lens; M_1,2, turning mirror; SLM_1,2,3, spatial light modulator; PBS, polarizing beam splitter; BS, beam splitter.

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As aforementioned, BFS is linearly proportional to both temperature and strain variations as presented in the following:

ν_{B} (T, ε) = ν_{B 0} + C_{T}^{B F S} (T - T_{0}) + C_{ε}^{B F S} (ε - ε_{0}) .

where C_T^BFS and C_ε^BFS are the temperature and strain proportionality coefficients in the FMF.

Due to the divergent optical and acoustic correlation profiles for LP₀₁ and LP₁₁ modes, the power spectral density (PSD) of the BGS for each mode might experience different spatial properties, as depicted in Fig. 4, where LP₁₁ mode was considered a combination of LP_11a_/_b modes caused by modal degeneracy rotation, while the BFS can be measured from the differential BGS and used to determine the effects of temperature or strain along the sensing fiber.

Fig. 4 Measured BGS of LP₀₁ and LP₁₁ modes in the FMF; MUX, multiplexer.

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In accordance with our preliminary experimental demonstration, the relationship between the BFS versus temperature and strain variants for the fundamental mode can be described as:

ν_{B} {(T, ε)}_{L P 01} = 10.91 G H z + 1.29 M H z / {}^{o}C \cdot (T - T_{0}) + 57.6 K H z / μ ε \cdot (ε - ε_{0}) .

Similarly, the correlation between the BFS for LP₁₁ mode versus temperature and strain variations is expressed in the form of:

ν_{B} {(T, ε)}_{L P 11} = 10.90 G H z + 1.25 M H z / {}^{o}C \cdot (T - T_{0}) + 58.5 K H z / μ ε \cdot (ε - ε_{0}) .

Therefore, these two effects can be discriminated by solving the simultaneous equations above, as shown in Fig. 5(a) and 5(b) respectively, whereas the corresponding color legends represent the normalized magnitude of Brillouin scattering. The two distinct vertical strips in the figures at different distances were caused by different variations of temperature and strain upon the FMF correspondingly. In addition, the temperature and strain accuracy is limited by the minimum detectable BFS difference, which was averagely around 25 MHz in our preliminary experimental demonstrations. The temperature accuracy of ± 0.5 °C and strain accuracy of ± 50 $μ ε$ were achieved by coherent heterodyne detection along with the FMF-based multi-parameter discriminative measurement technique introduced in this manuscript.

Fig. 5 (a) Temperature distribution attained by converting two BFSs from LP₀₁ and LP₁₁ modes; (b) Strain distribution obtained by adapting two BFSs.

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4. Optimization of refractive index profiles for multi-functionality

Although the preliminary verification has been accomplished vis-à-vis the multi-parameter discriminative measurement capability thru mode multiplexing; most of the MDM-based sensing systems were just utilizing the commercial FMFs that were originally fabricated for long-haul transmission [9, 17]. Some FMF designs to reduce mode coupling and crosstalk between modes are beneficial for reducing the complexity of the receiving end as well as for enhancing overall sensing resolution [21]. Nevertheless, certain fiber materials and refractive index profiles might not be favorable to the distributed sensing functionality, or even disadvantageous in some cases. For instance, some large-mode-area (LMA) fibers were designed via a co-doping scheme to increase the SBS threshold for telecom and high-power laser applications [22]. As for the MDM sensors, on the other hand, the performance is directly associated with the intensity of backscattered Brillouin signals resulting from the interaction of optical wave with acoustic modes thru electrostriction, while the fiber core material is compressed in the presence of an electrical field. Therefore, it is necessary to enhance fiber nonlinearity and reduce SBS threshold instead. In order to fill such temporary gap in this emerging field, in the next two sections, we systematically investigated for multi-functional MDM sensing systems.

In conventional design and fabrication, FMFs can be categorized into three major types, including the SI-FMF, graded-index FMF (GI-FMF) and triangular-index FMF (TI-FMF). As the most customary, widely used, and easily fabricated class, the SI-FMF core has a uniform refractive index all over with an abrupt alteration of the refractive index right at the core-cladding boundary, as shown in Fig. 6(a). Such sudden refractive index change would retain substantial modal diversity and hence cause differential mode group delay (DMGD) [23].

Fig. 6 Sketch of various FMF refractive index profiles: (a) SI-FMF; (b) GI-FMF; (c) TI-FMF; (d) SI-FMF with trenches; (e) GI-FMF with trenches; (f) TI-FMF with trenches; (g) SI-FMF with rings; (h) GI-FMF with rings; (i) TI-FMF with rings.

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To reduce this intermodal dispersion in high-speed SDM transmission systems, GI-FMFs and TI-FMFs were fabricated to ensure all rays travelling inside have nearly equivalent delays, as illustrated in Fig. 6(b) and 6(c), for the refraction index varies across the diameter of the core smoothly but remains constant in the cladding. Similarly, an additional trench near the core, as designated in Fig. 6(d-f), results in even lower DMGD [24]. Thus an FMF index profile with a graded-index core and an outer trench, like Fig. 6(e), would be the optimum choice for MDM optical communication. However, such advantages might not be as efficient as for the SDM sensing, because the differential BFS is the vital parameter for Brillouin distributed FMF sensors, which requires a relatively larger modal diversity instead of a smaller one. In other words, graded-index core or outer trench might be unfavorable to the multi-parameter discriminative measurement capability, for such refractive index profiles would differential BFS smaller, and even incapacitate the MDM multi-functionality in harsh environment, i.e. under strong stress.

The corresponding multi-functionalities with different FMF designs are then validated thru detailed examples of numerical modeling based on optical mode solver along with correlated acoustic velocity profiles. We can clearly notice that, under the same peripheral temperature and strain variations, the differential BFS remains steady within SI-FMF as depicted in Fig. 7; while in Fig. 8 and Fig. 9, the BFS difference would dramatically shrink especially under strong strain. This is because, due to dedicated smaller modal diversity, bending deforms the mode field distribution and thus reduces the overlap integral between the optical and acoustic fields, thus weakening the MDM multi-functionality. In a sense, since superior BFS depends on greater differences in the relative refractive index between two modes, SI-FMF might create better sensitivity for Brillouin distributed fiber sensors than GI-FMF and TI-FMF, which incidentally explains why most state-of-the-art MDM sensors were using SI-FMFs.

Fig. 7 Numerical analysis in SI-FMF for LP₀₁ and LP₁₁ modes: (a) BFS vs. Temperature; (b) BFS vs. Strain.

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Fig. 8 Numerical analysis in GI-FMF for LP₀₁ and LP₁₁ modes: (a) BFS vs. Temperature; (b) BFS vs. Strain.

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Fig. 9 Numerical analysis in TI-FMF for LP₀₁ and LP₁₁ modes: (a) BFS vs. Temperature; (b) BFS vs. Strain.

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In addition, an additional ring near the core could enhance the interaction of optical wave with acoustic modes, as exemplified in Fig. 6(g-i), by adjusting the width of the ring d₁ as well as the distance between the ring and the core center d₂ for matching the modal phase profiles. Hence the optimum FMF profile for MDM optical sensing applications would be in the form of Fig. 6(g).

As denoted in Table 1, with the same FMF core diameter of 9.374 μm and the same GeO₂ dopant concentration 2.395%, different combinations of ring width d₁ and distance from ring to core center d₂ are provided along with their subsequent differential BFS values, whereas the normalized frequency V₀ is in the value of 3.8, and numerical aperture (NA) is 0.1 in the SI-FMF.

Table 1. Comparison of various mode profiles and doping design in SI-FMF

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In theory, the SI-FMF, GI-FMF and TI-FMF are all candidates for the multi-functional fiber optic sensors based on MDM. In some cases, the base values of differential BFS for a GI-FMF or a TI-FMF might appear even a bit larger than that of a SI-FMF based on different fiber designs, as depicted in Fig. 7 to Fig. 9. Nonetheless, this could be utterly resolved via the optimization of FMF profile design and material compositions according to Table 1 and Section 5 below. Secondly and more importantly, certain refractive index profiles might not be favorable to the distributed sensing functionality, or even disadvantageous in some cases, because the GI-FMF and TI-FMF have different slopes in terms of BFS versus temperature or strain, which could cause the differential BFS decreases as temperature or strain increases. Therefore, to ensure a stable base value of differential BFS for further enhancement via the optimization of FMF profile design and material compositions, it is reasonable to choose SI-FMF over GI-FMF and/or TI-FMF structures for MDM sensing systems.

5. Optimization of dopant material compositions for multi-functionality

In addition to small core size, long interaction length and optimized fiber profiles, the multi-functional performance of MDM sensors can be further enhanced by adjusting the relative doping level in the FMF core. As aforementioned, the Brillouin backscattered light is downshifted in frequency attributable to the Doppler shift related to the index grating moving at the acoustic velocity. Hereafter, thru the modification of the material density variations, the overlap integral between the optical and acoustic modes can be manipulated via the co-doping scheme, so that excessive enhancements on the measurement accuracy associated with the intermodal differential BFS can be achieved. Still it is perceptible that the influence of material compositions is exceedingly reliant on fiber refractive index profile structure. The evolutions of the effective optical and longitudinal acoustic refractive indices for various FMF profiles are presented in Fig. 10. In other words, the right FMF configuration would strengthen the SBS threshold enhancement, while the deprived one burdens the improvement of MDM multi-functionality by means of glass composition designs.

Fig. 10 Schematic of the effective optical and longitudinal acoustic refractive indices: (a) for SM-FMF; (b) for GI-FMF; (c) for TI-FMF.

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The equation for the optical field can be written as:

\frac{d^{2} f_{o}}{d r^{2}} + \frac{1}{r} \frac{d f_{o}}{d r} + k_{o}^{2} [n_{o}^{2} (r) - n_{o e f f}^{2}] f_{o} = 0.

Meanwhile, the corresponding equation for the longitudinal acoustic fields is in the form of [25]:

\frac{d^{2} f_{a}}{d r^{2}} + \frac{1}{r} \frac{d f_{a}}{d r} + \frac{k_{a}^{2}}{c} [n_{a}^{2} (r) - n_{a e f f}^{2}] f_{a} = 0.

where the effective optical refractive index

n_{o e f f}

can be defined as:

n_{o e f f} = \frac{β}{k_{o}} = \frac{λ β}{2 π} .

while the effective longitudinal acoustic index

n_{a e f f}

is given by:

n_{a e f f} = \frac{V_{C l a d}}{V_{e f f}} .

In line with the coupled mode formalism, one optical mode could be corresponding to multiple longitudinal acoustic modes. In the conventional step-index SMF with pure GeO₂ doping, all higher order acoustic modes with azimuthal variations are disregarded, concerning that the overlap integrals between the fundamental optical mode and these acoustic modes are virtually zero. Nevertheless, for FMF cases with varying material compositions, it is entirely conceivable for a higher order optical mode to be coupled with another higher order acoustic mode, after a few rounds of iterations.

For an FMF with a core diameter of 11.841 μm and the GeO₂ dopant concentration of 2.395%, Table 2 provides the overlap between the optical and longitudinal acoustic fields under different temperature and strain values, with the V number as 4.799 and an optical cutoff wavelength of 3093.710 nm. From this table we can notice that the optical LP₀₁ mode is coupled with acoustic LP₀₁ mode, while the optical LP₁₁ mode is coupled with acoustic LP₀₂ mode instead of acoustic LP₁₁ mode. Besides, such tendency grows solider with higher temperature or stronger strain on the FMF.

Table 2. Overlap between the optical and acoustic modes under different temperature and strain

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The common material dopants in the optical fiber comprise GeO₂, F₂, TiO₂ and B₂O₃, etc [23]. The effects of different dopants upon the optical and acoustic refractive indices might be quite different. For instance, GeO₂ and TiO₂ dopants increase both the optical and acoustic indices with different fractions, while B₂O₃ and F₂ dopants have opposite effects on the optical and acoustic indices. Therefore it is hypothetically promising that significant multi-functional performance enhancement can be accomplished by fine-tuning the relative contributions of refractive index profiles from diverse dopants while keeping a step-index profile for the FMF core.

Furthermore, the inner core region can be doped with GeO₂ and/or TiO₂, while the outer ring region is doped with B₂O₃ and/or F₂, as depicted in Fig. 6(g), to maximally exploit the improvements on the measurement accuracy associated with the intermodal differential BFS in MDM sensing systems, for the optical refractive index profile is retained to be a step-index one for thoroughgoing spatial diversity, while the acoustic index profile is in an inversed W shape.

The optical refractive index of the FMF core as a function of temperature and strain variants, as well as various dopant compositions, can be expressed by the following equation:

n_{o} = n_{o C l a d} \cdot [\begin{array}{l} 1 + (1 \times 10^{- 3} + 3 \times 10^{- 6} \cdot Δ T + 1.5 \times 10^{- 7} \cdot Δ ε) \cdot w_{G e O_{2}} \\ + (2.1 \times 10^{- 3} + 5.4 \times 10^{- 6} \cdot Δ T + 3.2 \times 10^{- 7} \cdot Δ ε) \cdot w_{T i O_{2}} \\ + (- 4.5 \times 10^{- 3} + 1.8 \times 10^{- 6} \cdot Δ T + 2.3 \times 10^{- 7} \cdot Δ ε) \cdot w_{B_{2} O_{3}} \\ + (- 3.3 \times 10^{- 3} + 3.6 \times 10^{- 6} \cdot Δ T + 7.5 \times 10^{- 7} \cdot Δ ε) \cdot w_{F_{2}} \end{array}] .

Here

w_{G e O_{2}}

,

w_{T i O_{2}}

,

w_{B_{2} O_{3}}

,

w_{F_{2}}

represent the mole percent of Germanium, Titanium, Boron and Fluorine dopant, and n_o _Clad = 1.445.

Similarly, the acoustic refractive index of the FMF core is described as:

n_{a} = n_{a C l a d} \cdot [\begin{array}{l} 1 + (7.2 \times 10^{- 3} + 4.7 \times 10^{- 5} \cdot Δ T + 2.1 \times 10^{- 6} \cdot Δ ε) \cdot w_{G e O_{2}} \\ + (5.6 \times 10^{- 3} + 8.2 \times 10^{- 5} \cdot Δ T + 3.5 \times 10^{- 6} \cdot Δ ε) \cdot w_{T i O_{2}} \\ + (4.5 \times 10^{- 3} - 1.3 \times 10^{- 5} \cdot Δ T - 7.6 \times 10^{- 6} \cdot Δ ε) \cdot w_{B_{2} O_{3}} \\ + (2.7 \times 10^{- 3} - 1.8 \times 10^{- 5} \cdot Δ T - 3.8 \times 10^{- 6} \cdot Δ ε) \cdot w_{F_{2}} \end{array}] .

where n_a_Clad = 1. The manipulation of the optical-acoustic overlap by adjusting the relative doping level in the core of the FMF is provided in Table 3, in which we may clearly observe five to ten folder improvements on the differential BFS, compared to our preliminary experimental demonstrations whereas the differential BFS was averagely around 25 MHz, which is corresponding to the second case in Table 1, thus enhancing the measurement accuracy of the multi-functional MDM sensing systems. Such combination of material dopants leads to an optimized FMF design with a much larger differential BFS value compared with preliminary data, and hence reducing cost and operating complexity while realizing multi-functionality in SHM and other industrial applications.

Table 3. Differential BFS vs. various doping compositions in the FMF

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6. Conclusion

In summary, the SDM systems utilizing FMFs provide unprecedented properties and multi-functionalities for structure health monitoring as well as oil and gas pipeline monitoring based on mode multiplexing. In this paper, the performance of few-mode Brillouin sensors has been analytically evaluated by means of diverse refractive index profiles and dopant material compositions, which are affected by intermodal coupling, birefringence and optical-acoustic interactions. We found the SI-FMF structure plus side rings might provide better sensitivity for Brillouin distributed fiber sensors than GI-FMF and TI-FMF. Besides, five to ten folder improvements can be achieved on the measurement accuracy compared to our preliminary experimental demonstrations, attributable to the differential BFS via a co-doping material composition of GeO₂, F₂, TiO₂ and B₂O₃.

References and links

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d₁ (μm)	d₂ (μm)	υ_B ₀₁ (GHz)	υ_B ₁₁ (GHz)	Δυ_B (MHz)
1	4	10.471	10.459	12
2	3	10.578	10.553	25
3	3	10.641	10.608	33
2	5	10.665	10.619	46

Overlap between Optical and Acoustic Modes	0 ͦ C	25 ͦ C	45 ͦ C	65 ͦ C	1000 με	2000 με	3000 με
Optical LP(0,1) with Acoustic LP(0,1):	0.795	0.812	0.829	0.862	0.825	0.876	0.981
Optical LP(0,1) with Acoustic LP(0,2):	0.160	0.150	0.139	0.117	0.142	0.106	0.019
Optical LP(0,1) with Acoustic LP(0,3):	0.036	0.031	0.026	0.018	0.027	0.015	0.000
Optical LP(0,1) with Acoustic LP(0,4):	0.008	0.006	0.005	0.003	0.005	0.002	0.000
Optical LP(0,1) with Acoustic LP(0,5):	0.002	0.001	0.001	0.001	0.001	0.000	0.000
Optical LP(0,1) with Acoustic LP(0,6):	0.000	0.000	0.000	0.000	0.000	0.000	0.000
Optical LP(0,1) with Acoustic LP(0,7):	0.000	0.000	0.000	0.000	0.000	0.000	0.000
Optical LP(1,1) with Acoustic LP(0,1):	0.076	0.081	0.087	0.101	0.085	0.105	0.179
Optical LP(1,1) with Acoustic LP(0,2):	0.270	0.281	0.292	0.314	0.288	0.316	0.364
Optical LP(1,1) with Acoustic LP(0,3):	0.183	0.178	0.173	0.159	0.174	0.153	0.116
Optical LP(1,1) with Acoustic LP(0,4):	0.086	0.079	0.073	0.060	0.075	0.059	0.000
Optical LP(1,1) with Acoustic LP(0,5):	0.035	0.031	0.028	0.027	0.029	0.000	0.000
Optical LP(1,1) with Acoustic LP(0,6):	0.014	0.012	0.014	0.000	0.015	0.000	0.000
Optical LP(1,1) with Acoustic LP(0,7):	0.007	0.000	0.000	0.000	0.000	0.000	0.000

$w_{G e O_{2}}$ (%)	$w_{T i O_{2}}$ (%)	$w_{B_{2} O_{3}}$ (%)	$w_{F_{2}}$ (%)	$Δ ν_{B 01}$ (MHz)	$Δ ν_{B 11}$ (MHz)	$Δ ν_{B}$ (MHz)
2.50	1.50	0.00	0.25	10.521	10.424	97
3.00	0.75	5.00	0.00	10.531	10.407	124
2.50	1.45	3.10	1.50	10.623	10.455	168
2.50	1.20	5.60	0.40	10.687	10.475	212

d₁ (μm)	d₂ (μm)	υ_B ₀₁ (GHz)	υ_B ₁₁ (GHz)	Δυ_B (MHz)
1	4	10.471	10.459	12
2	3	10.578	10.553	25
3	3	10.641	10.608	33
2	5	10.665	10.619	46

Overlap between Optical and Acoustic Modes	0 ͦ C	25 ͦ C	45 ͦ C	65 ͦ C	1000 με	2000 με	3000 με
Optical LP(0,1) with Acoustic LP(0,1):	0.795	0.812	0.829	0.862	0.825	0.876	0.981
Optical LP(0,1) with Acoustic LP(0,2):	0.160	0.150	0.139	0.117	0.142	0.106	0.019
Optical LP(0,1) with Acoustic LP(0,3):	0.036	0.031	0.026	0.018	0.027	0.015	0.000
Optical LP(0,1) with Acoustic LP(0,4):	0.008	0.006	0.005	0.003	0.005	0.002	0.000
Optical LP(0,1) with Acoustic LP(0,5):	0.002	0.001	0.001	0.001	0.001	0.000	0.000
Optical LP(0,1) with Acoustic LP(0,6):	0.000	0.000	0.000	0.000	0.000	0.000	0.000
Optical LP(0,1) with Acoustic LP(0,7):	0.000	0.000	0.000	0.000	0.000	0.000	0.000
Optical LP(1,1) with Acoustic LP(0,1):	0.076	0.081	0.087	0.101	0.085	0.105	0.179
Optical LP(1,1) with Acoustic LP(0,2):	0.270	0.281	0.292	0.314	0.288	0.316	0.364
Optical LP(1,1) with Acoustic LP(0,3):	0.183	0.178	0.173	0.159	0.174	0.153	0.116
Optical LP(1,1) with Acoustic LP(0,4):	0.086	0.079	0.073	0.060	0.075	0.059	0.000
Optical LP(1,1) with Acoustic LP(0,5):	0.035	0.031	0.028	0.027	0.029	0.000	0.000
Optical LP(1,1) with Acoustic LP(0,6):	0.014	0.012	0.014	0.000	0.015	0.000	0.000
Optical LP(1,1) with Acoustic LP(0,7):	0.007	0.000	0.000	0.000	0.000	0.000	0.000

Multi-functional fiber optic sensors based on mode division multiplexing

Abstract

1. Introduction

2. Theoretical modeling of few-mode Brillouin sensors

3. Preliminary experimental demonstrations

4. Optimization of refractive index profiles for multi-functionality

5. Optimization of dopant material compositions for multi-functionality

6. Conclusion

References and links

Cited By

Figures (10)

Tables (3)

Equations (26)

Optical Materials Express