1. Introduction

Among existing paradigms for quasi-distributed sensing, the fiber optic approach is particularly attractive, given its numerous advantages such as miniature sizes, immunity to electromagnetic interference, and capability of large scale multiplexing and harsh environment operation [1–3]. Typically, a quasi-distributed fiber optic sensor network utilizes time-division-multiplexing (TDM) and / or wavelength-division-multiplexing (WDM) to collect and demodulate responses produced by individual sensors [4,5]. Up to date, most sensor networks are based on single mode fibers (SMF) that support only a single linearly polarized (LP) mode [6–8]. Consequently, the operation characteristics of such a senor network cannot be easily tuned. For absorption-based sensors, this traditional approach carries a significant drawback — the inability to achieve large scale multiplexing. As an example, consider a simple sensor network that includes N identical sensors connected in series. In a traditional SMF-based sensor network, if each sensor can induce α dB in optical absorption, the maximum attenuation for the entire sensor network is αN. Obviously, under this scenario, the requirements for high sensitivity (i.e., large α) and large-scale multiplexing (i.e., α as small as possible) are mutually exclusive. Therefore, under the current paradigm of SMF-based sensors, in order to achieve large-scale multiplexing of absorption based sensors, one often needs to resort to very complicated network designs [9,10].

Due to the strong and often uncontrollable intermodal coupling, multimode fibers (MMFs) are traditionally regarded as unsuitable for optical sensing. This traditional paradigm has recently begun to change with research on few mode fiber (FMF) based sensors [11–13]. Compared with MMFs with hundreds of guided modes, by using FMFs that support only a few LP modes, one can counteract the effect of intermodal coupling through techniques such as multiple-input-multiple-output (MIMO) equalization, originally developed for optical communications [14]. On the other hand, with multiple guided LP modes, FMF-based sensors can accomplish functionalities that cannot be easily achieved using standard single-mode-fiber based sensors, such as a sensor network capable of simultaneously monitoring temperature and strain [15–17]. Fiber Bragg gratings (FBGs) written in FMFs can also be used to detect bending and displacement [18,19] which cannot be sensed by traditional FBGs written in SMFs. Recently, we experimentally developed an adaptive optics (AO) technique that can precisely control the mode composition of optical signals in few-mode fibers (FMFs) [20,21]. Given these recent advances, FMF sensors may, under certain circumstances, become superior to the traditional SMF sensors.

In this paper, we aim to theoretically analyze one such example: large scale multiplexing of absorption-based sensor network through MDM. We develop a matrix approach that allows us to analyze the performance of such a sensor network, which contains multiple fiber Bragg gratings (FBGs) and absorption-based sensing segments with significant mode-dependent loss (MDL). The performance of the proposed sensor network is analyzed under different operation parameters, including different levels of intermodal coupling. We find that under many scenarios, we can simultaneously achieve high sensitivity as well as large-scale multiplexing by using the low-loss LP mode for signal delivery, and the high-loss LP mode for absorption-based sensing. The capacity and potential limitations of the sensor network are also discussed. The work presented here can serve as a proof-of-concept demonstration showing that FMF-based sensor networks can be superior to SMF-based sensor networks, at least for applications that rely on optical absorption for sensing.

2. Design and analysis of the MDM quasi-distributed sensing network

This section is organized as follows. First, we discuss the operation principle of a single sensor in section 2.1. Then, in section 2.2, we describe how to multiplex such sensors together and form a quasi-distributed sensor network. Finally, we present a few possible implementations of such sensors and sensor networks in section 2.3.

2.1 Operation principle of a single sensor

The design of a single absorption-based sensor is shown in Fig. 1(a). It includes a sensing segment with significant MDL, and multiple fiber Bragg gratings (FBGs) for mode control, mode-conversion, and sensing signal generation. (The attenuation within the sensing segment can be due to gas absorption, plasmonic excitations, as well as other mechanisms.) For quantitative analysis, we focus on a fiber network that supports four LP modes: LP_01, LP₁₁ and LP₀₂, and LP₂₁. The two degenerate LP₁₁ (or LP₂₁) modes are labelled as LP_11a and LP_11b (or LP_21a and LP_21b), respectively. The choice of a four-mode fiber is not critical, and the same design principle can be applied to other FMFs. The key feature of our design is that by utilizing the low-loss LP₀₁ mode for signal transmission and the potentially high-attenuation LP₀₂ mode for sensing, we can simultaneously achieve both high sensitivity and large scale multiplexing. We begin our analysis by assuming that at the operation wavelength: 1) FBG 2 converts a small percentage of the LP₀₁ mode to the LP₀₂ mode and vice versa; 2) FBG 3 and FBG 4 reflect the LP₀₂ mode only. Justifications for these requirements are given later. The inclusion of an ultra-weak FBG 1 is optional and is intended for AO-based mode control only [20]. As a result, we will ignore FBG 1 in our quantitative analysis.

 

Fig. 1 (a) A schematic drawing of the single sensor design. For any interrogation pulse, FBG 1, 3, and 4 will produce three reflection pulses shown in (b). Pulse 1 is for AO-based mode control [20], pulse 2 and 3 are for sensing. (c) Simplify the sensor design in (a) for transfer matrix analysis. The presence of FBG 1 is ignored.

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Sensor interrogation is carried out using pulsed signals. For any input pulse, we assume that all FBGs are placed sufficiently far apart and produce distinct reflection pulses, as shown in Fig. 1(b). Hence we can ignore the effect of multiple reflections. By applying the AO algorithm we recently demonstrated [20], we can use the pulse produced by FBG 1 (i.e., pulse 1) to ensure that the incident interrogation signal is dominated by the LP₀₁ component. Then,FBG 2 converts a small percentage of the incident LP₀₁ signal into the LP₀₂ mode for sensing. The two sensing FBGs (FBG 3 & 4), which only reflect the LP₀₂ mode, produce pulse 2 and 3 in Fig. 1(b). As pulses 2 and 3 travel backwards, FBG 2 again ensures that the back-propagating sensing signals contain the low-loss LP₀₁ mode. Since pulse 3 passes through the sensing segment twice whereas pulse 2 does not, we can extract the attenuation of the LP₀₂ mode by comparing the peak power of the two pulses. Notice that in this design, we utilize the LP₀₁ mode for signal transmission and the attenuation of the LP₀₂ mode for sensing.

To simplify our analysis, we assume that the FMF is polarization maintaining, and all optical fields are polarized along the same direction and can be regarded as scalar fields. As a result, the optical field in the fiber E(x,y) can be expressed as a superposition of the LP modes:

To avoid confusion, we adopt the following conventions for our notations. First, we distinguish parameters associated with the forward-propagating (i.e., along the + z direction) or the backward-propagating (i.e., along the ‒z direction) signals using a superscript “+” or “‒”. For example, the vector $A_{in}^{+}$ contains the mode coefficients of the forward-propagating optical signals at the sensor input. For the reflected signal, we introduce a subscript specifying the source of reflection. Under this convention, $A_{g3}^{-}$ and $A_{g4}^{-}$denote mode coefficients of the signals reflected by FBG 3 and FBG 4, respectively. For the forward-propagating (or backward-propagating) waves, the effect of transmission through a single sensor is described by a 6 × 6 matrices ${\widehat{T}}^{+}$(or ${\widehat{T}}^{-}$). For reflection matrices, we add the subscript “g1”, “g3”, or “g4” to distinguish the source of reflection. For example, ${\widehat{R}}_{g3}^{+}$is the reflection matrix associated with FBG 3, and ${\widehat{R}}_{g4}^{+}{A}_{in}^{+}$ gives us the complex mode coefficient (at the sensor input) for the signal reflected by FBG 4, with input signal being $A_{in}^{+}$. We distinguish vectors and matrices by adding an overhat ^ for matrices.

All FBG parameters (e.g., reflection / transmission coefficients) are labeled using a subscript such as “g1”. For example, t_g3 represents FBG 3 power transmission coefficient. To reduce simulation parameters, we ignore the presence of FBG 1 in our numerical calculations. Also, for simplicity, we assume equal distance d_g between FBG 2, 3, and 4.

The transmission and reflection characteristics of a single sensor can be modeled using transfer matrices. Specifically, we divide a single sensor into five sections and give each section its own transfer matrices, as in Fig. 1(c). Section 1 contains only FBG 2, since the ultra-weak FBG 1 is ignored. Section 2 is the FMF (total length d_g) between FBG 2 and 3. Section 3 is FBG 3. Section 4 (total length d_g) is the FMF between FBG 3 and 4, and includes the sensing segment. Section 5 is FBG 4. Again, we use a letter T or R to denote transfer matrices associated transmission or reflection, a superscript “+” or “‒” to distinguish forward or backward incident signal, and a subscript “s1” to “s5” to represent section 1 to 5.

For section 2 and 4, it is easy to find the corresponding transfer matrices, ${\widehat{T}}_{s2}^{\pm }$ and ${\widehat{T}}_{s4}^{\pm }$, as:

The transmission matrix associated with section 1 (${\widehat{T}}_{s1}^{\pm }$) can be expressed as [22]:

Since FBG 3 and 4 only impact the LP₀₂ mode, we use $r_{g3}^{02}$ / $t_{g3}^{02}$ and $r_{g4}^{02}$ / $t_{g4}^{02}$ to denote their power reflection / transmission coefficients, respectively. Note that we explicitly use the superscript “02” to emphasize that FBG reflection / transmission are for the LP₀₂ mode only. Also, we assume all FBGs to be lossless, i.e., κ_g2 + t_g2 = $r_{g3}^{02}$ + $t_{g3}^{02}$ = $r_{g4}^{02}$ + $t_{g4}^{02}$ = 1. The transmission / reflection matrices associated with section 3 and 5 are [22]:

Equations (3) to (9) give the transmission / reflection matrices of all five sections of a single sensor. Using these notations, the transfer matrices that describe the behavior of a single sensor shown in Fig. 1 can be expressed as:

We can use Eq. (10) to calculate the optical powers of the reflected pulse 2 ($p_{g3}^{-}$, produced by FBG 3) and pulse 3 ($p_{g4}^{-}$, by FBG 4). Assuming the input signal is $A_{in}^{+}$, the reflected mode amplitudes for pulse 2 and 3 are ${\widehat{R}}_{g3}^{+}{A}_{in}^{+}$ and${\widehat{R}}_{g4}^{+}{A}_{in}^{+}$, respectively. The power ratio of the two pulses, i.e., $p_{g4}^{-}$/$p_{g3}^{-}$, is given by:

2.2 Sensing network architecture

The sensor described in section 2.1 can be multiplexed together and form a quasi-distributed MDM sensor network shown in Fig. 2. In our analysis, in addition to the convention described previously, we introduce an additional subscript “n” to denote quantities associated with the nth sensor, where $1\le n\le N$ and N is the total number of sensors used in the network. For instance, $A_n^{+}$ describes the mode coefficients of the forward-propagating signals at the input of the nth sensor. For reflected signals, we use two subscripts to specify the source of reflection as well as the physical location of the reflected signals. As a specific example, $A_{i,ng3}^{-}$ denotes the mode coefficients of the backward-propagating signal generated by FBG 3 of the nth sensor (thus “ng3” in the subscript) and measured at the input of the ith sensor. (Obviously, $1\le i\le n$.) As in section 2.1, the effects of forward (or backward) optical transmission through the nth sensor are represented by ${\widehat{T}}_n^{+}$ (or ${\widehat{T}}_n^{-}$). Similar to the previous section, ${\widehat{R}}_{n,g3}^{+}$ is the reflection matrix associated with FBG 3 of the nth sensor. We use ${\widehat{P}}_{n-1}^{+}$ and ${\widehat{P}}_{n-1}^{-}$ to describe the forward-direction and the backward-direction intermodal coupling induced by the FMF between the (n‒1)th and the nth sensor. Finally, we introduce another set of matrices ${\widehat{X}}_{n-1}^{+}$ and ${\widehat{X}}_{n-1}^{-}$ to describe additional perturbations not included in the intermodal coupling matrices ${\widehat{P}}_{n-1}^{+}$ and ${\widehat{P}}_{n-1}^{-}$.

 

Fig. 2 Matrix approach of the designed MDM quasi-distributed sensing network.

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Using the notations given above, we can readily calculate the mode coefficients of optical signals in the network. For example, using $A_1^{+}$ as the input signal, the corresponding forward-propagating signal at the input of the nth sensor is given by:

Similar to Section 2.1, we use$p_{1,ng3}^{-}$ and $p_{1,ng4}^{-}$ to denote optical powers associated with the reflection signals generated by FBG 3 and FBG 4 of the nth sensor, measured at the input of the entire sensor network (i.e., sensor 1), respectively. The ratio $p_{1,ng4}^{-}$/$p_{1,ng3}^{-}$ is given by:

3. MDM sensor network modeling

In order to quantitatively analyze the MDM sensor network, we need to specify the transmission / reflection characteristics of key components such as the sensing segment and the FMF that connects adjacent sensors. The modeling of such components is described here.

3.1 Sensing segment

A simple yet practical model for the sensing segment is a silica fiber taper in a sealed gas chamber and operated at the absorption peak of the gas, as illustrated in Fig. 3(a). Other implementation, such as a silica fiber covered with plasmonic nanoparticles (Fig. 3(b)), is also possible. To simplify our calculations, here we consider the sensing segment to be a fiber taper with uniform radius and placed in an absorptive gas.

 

Fig. 3 Schematic drawing for using silica fiber taper as a sensing-segment in (a) gas and (b) plasmon-based sensing networks. (c) The radial distributions of the LP₀₁ and the LP₀₂ field intensity when propagating inside a silica fiber taper of 1 μm radius at 1550 nm wavelength. (d) The ratio of α⁰² / α⁰¹ at different values of fiber taper radius.

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In our model, we consider a silica fiber taper with almost negligible LP₀₁ mode transmission loss (i.e., α⁰¹ ≈1) and a much higher gas-induced attenuation of the LP₀₂ mode. We can estimate the α^lm values of various LP_lm modes as:

According to Eq. (18), the mode dependent loss of the sensing segment can be calculated from the percentage of evanescent field in the cladding region. Figure 3(c) gives a representative example of the LP₀₁ and the LP₀₂ mode in a fiber taper with 1 µm radius and core refractive index n₁ = 1.46 at wavelength λ = 1.55μm. Based on Eq. (18), we can also find out the dependence of α⁰² / α⁰¹ on fiber taper radius, as shown in Fig. 3(d). As expected, α⁰² / α⁰¹ can become quite small (~0.3) near the cutoff radius of the LP₀₂ mode. Our analysis shall focus primarily in this regime, where optical absorption is highly mode dependent.

3.2 Intermodal coupling in the FMF

Perhaps the most critical element in our analysis is how to account for intermodal coupling within the MDM network. In our model, we follow [23] and assume fiber bending to be the primarily source of intermodal coupling. In this case, the coupled mode equation along the direction of light propagation (z) is [23,24]:

Let us now consider the FMF that connects the (n‒1)th and the nth sensor. In our model, we divide the FMF into M segments, where σ_x and σ_y remain constant within each segment. Equation (19) can be solved for each segment separately to give the following solution in its matrix form for the FMF:

3.3 Impact of modal noise produced by the reflecting FBGs

In the ideal case, we assume that FBG 3 and 4 only reflect the LP₀₂ mode. In reality, however, reflection of other LP modes is not zero. For such cases, we modify the transmission / reflection matrices of section 3 (FBG 3) and section 5 (FBG 4) as:

The presence of non-ideal FBG 3 and 4 has no impact on the transfer matrices of sections 1, 2, and 4. Therefore, their transfer matrices remain the same. Given these considerations above, for a single sensor with non-ideal FBG 3 and 4, we have:

3.4. Other effects

Many assumptions we introduced earlier can be relaxed by using an appropriate form for the “perturbation” matrix $\widehat{X}$ in Eqs. (12) to (16). For example, in Fig. 1(c), we assume the distance between FBG 2, 3, and 4 to be d_g. In reality, there will inevitably be some deviation. To account for fiber length variations, we can introduce random phase shifts to individual LP modes and describe this effect in the matrix $\widehat{X}$. Additionally, up to now, we have assumed the transmission through FBG 1 to be unity. If we want to relax this assumption and include the effect of small but non-zero FBG 1 reflection, we can introduce a small modal loss to the LP₀₁ mode. In general, we define the residual perturbation matrix $\widehat{X}$ as:

4. Numerical analysis for the designed MDM sensing network

In this section, we numerically analyze the performance of the MDM sensor network under different operation conditions. For simplicity, we assume all individual sensors within the network to be identical.

4.1 Ideal MDM sensing network

We start by analyzing an ideal sensor network, where we assume: 1) no intermodal coupling (e.g., σ_x = σ_y = 0 in Eq. (21)), 2) ideal reflection characteristics for FBG 3 and 4 (i.e., they only reflect the LP₀₂ mode), and 3) no perturbation effects (i.e.,$\widehat{X}$ is an identity matrix).

For the MDM sensing network in Fig. 2, we first consider the design criteria on FMF length. For simplicity, we assume an interrogation signal of a non-return-to-zero (NRZ) modulation format and 1 ns pulse duration. Within each sensor, we choose the distance d_g (refer to Fig. 1) to be 0.5 m and assume that FBG 1 and 2 are almost at the same location. In this case, the time delay between pulse 1, 2, and 3 is ~5 ns, which should be sufficient for time domain separation. If a 5-m-long FMF separates two adjacent sensors, the time delay between their responses should be ~50 ns. This brief analysis suggests that it should be possible to distinguish sensor responses in time domain.

We assume that the FMF is a step index fiber with 10 μm core radius, with core and cladding refractive indices being 1.462 and 1.457, respectively. (A few important parameters of the FMF are listed in Table 1.) Setting the operation wavelength at 1550 nm, this FMF supports LP₀₁, LP_11a, LP_11b, LP₀₂, LP_21a, and LP_21b modes. For the sensing segment, we assume it to be a fiber taper (1 μm radius) in air. Using Eq. (18) and assuming LP₀₁ mode transmission coefficient to be α⁰¹ ≈1, the transmission coefficients of the remaining modes can be estimated as α¹¹ ≈0.92 α⁰¹, α²¹ ≈0.75 α⁰¹, and α⁰² ≈0.56 α⁰¹. (This assumption means that we always consider the worst case scenario, where all sensors experience high absorption.) For FBG 2, we assume the mode conversion coefficient κ_g2 is 1%. Regarding signal detection, we use the parameters of a commercial InGaAs detector (Thorlabs, DET08C) with noise-equivalent power (NEP) of 2.0 $\times$ 10 ⁻¹⁵ W/$\sqrt{Hz}$ at 5 GHz bandwidth. This bandwidth is sufficient for separating pulse 1, 2 and 3. At this bandwidth, the minimum detectable power is 0.14 nW. In our analysis, we increase the detection threshold to 0.5 nW. Finally, we assume that the interrogation pulse is purely LP₀₁, and its input power is 10 mW or less.

Table 1. Parameters of the four mode fiber used in our analysis.

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We start by estimating the values of $r_{g3}^{02}$ and $r_{g4}^{02}$ (i.e., LP⁰² mode reflection coefficient of FBG 3 and 4) that maximizes the total sensor number. This is achieved by using Eqs. (15) and (16) to calculate $A_{1,N_{\max }g3}^{-}$ and $A_{1,N_{\max }g3}^{-}$, which then give the power of the reflected pulses at the network input (i.e., $p_{1,N_{\max }g3}^{-}$ and $p_{1,N_{\max }g4}^{-}$) for different values of $r_{g3}^{02}$ and $r_{g4}^{02}$. The maximum sensor number N_max is determined by equating the smallest reflected signal power (in this case, either $p_{1,N_{\max }g3}^{-}$ or $p_{1,N_{\max }g4}^{-}$) with the detection threshold. Figure 4(a) shows the change of N_max at different values of $r_{g3}^{02}$ and $r_{g4}^{02}$. As this figure demonstrates, up to 561 sensors can be used when $r_{g3}^{02}$ = 5.95% and $r_{g4}^{02}$ = 20.80%. Using these values for $r_{g3}^{02}$ and $r_{g4}^{02}$, we also consider the relationship between the maximum number of sensors (N_max) and the input signal power. The result is shown in Fig. 4(b), and exhibits interesting “nonlinear” behaviors. Fundamentally, this is likely due to the fact that the “effective” attenuation per sensor is not a constant. Using this effect, we may be able to use more sophisticated network topology to further increase the maximum number of sensors in the MDM network.

 

Fig. 4 (a) Maximum number of sensors allowed for the ideal quasi-distributed sensing network, calculated using different $r_{g3}^{02}$ and $r_{g4}^{02}$. κ_g2 is fixed at 1%. (b) Maximum sensor number versus input power.

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4.2 Impact of intermodal coupling

To investigate the impact of intermodal coupling on sensor network performance, we follow the general framework in [23]. Specifically, we divide the 5-m-long connecting fiber into M = 500 segments, each 1 cm long. We assume the bending curvature σ (σ_x or σ_y) of each segment to be an independent and identically distributed (i.i.d.) random variable such that its probability density function (pdf) f(σ,s_N) is the positive side of a normal pdf N(0,$s_N^2$), as f(σ,s_N) = [(2/π)^0.5/s_N]exp[‒(σ/s_N)²/2], σ > 0. The mean curvature $〈\sigma 〉$ and the variance s² of this pdf are $〈\sigma 〉$ = (2/π)^0.5s_N and s² = (1‒2/π)$s_N^2$, respectively. Obviously, we can adjust the strength of intermodal coupling by changing the value of s.

First, we quantify the impact of intermodal coupling on the transfer matrix ${\widehat{P}}_{n-1}^{+}$ defined in Eq. (25). Since it is not easy to investigate all 36 matrix elements, we consider perhaps the most relevant case, where the input field is purely LP₀₁. Here, we assign a randomly varying σ_x and σ_y to each of the 500 segments according to f(σ,s_N), calculate the transfer matrix of each segment according to Eq. (22), and multiply all transfer matrices together using Eq. (25) to obtain the transfer matrix of the 5-m-long fiber. Assuming the input mode is purely LP₀₁ with power p_01i and phase θ_01i, we can easily calculate the power and phase of different LP modes at the output of the 5-m-long fiber. Figures 5(a) and 5(b) shows the mean and the standard deviation of different LP mode powers at the fiber output (p_lmf), normalized by the input power (p_01i). Due to the similarities between the LP_11a (or LP_21a) and the LP_11b (or LP_21b) mode, only the results for the LP_11a (or LP_21a) mode are shown in Fig. 5(b). Similarly, Fig. 5(c) shows the phase shift (θ_lmf ‒ θ_01i) associated with the intermodal coupling, where θ_lmf is the phase of the LP_lm mode at the fiber output and θ_01i is the phase of the incident LP₀₁ mode. The general behavior in this figure indicates that increasing the s values results in more power coupling from the fundamental mode to the higher order modes. For the LP₀₁ mode, the bending-induced phase shift increases as s increases. Phase shifts associated with remaining LP modes, however, do not exhibit consistent s dependence.

 

Fig. 5 Change of the power of (a) LP₀₁, (b) LP_11a, LP₀₂, and LP_21a modes at the end of the 5-m-long connection fiber at different s values. (c) Phase shifts of different LP modes at the end of the 5-m-long fiber at different s values.

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After obtaining the intermodal coupling matrix, we can readily calculate the powers of the signals reflected by various FBGs. By imposing the condition that all pulses must remain above the detection threshold (p_min = 0.5 nW), we can determine the maximum number of sensors N_max at different s values. The results are shown in Fig. 6(a). The average and standard deviation of N_max are calculated by running 50 simulations at each s value. (All connecting fibers are modeled independently but share the same s value.) The results in Fig. 6(a) suggests three distinct regimes, weak coupling regime (s < 0.8 m^‒1), intermediate regime (0.8 m^‒1 ≤ s ≤ 1.0 m^‒1), and strong coupling regime (s > 1.0 m^‒1)). The behavior in the weak coupling regime can be easily explained. As shown in Figs. 5(a) and 5(b), in the weak coupling regime, a large s value leads to stronger coupling from the LP₀₁ mode to the higher order modes, which experiences higher absorption / reflection and reduces the power received by the detector. Consequently, N_max should decrease as s increases. The behaviors of the sensor networks is more complex for the intermediate- and strong-coupling regime, where the FBG-induced and bending-induced coupling may compete and interfere. We do notice, however, that in the strong coupling regime, the maximum number of sensors N_max saturates around 41.

 

Fig. 6 (a) The maximum number of sensors in the quasi-distributed network at different intermodal coupling strength. (b) and (c) give the power of the forward-propagating (b) LP₀₁, LP₀₂, (c) LP_11a, and LP_21a modes at the nth sensor. (d) The normalized power of the back-propagating optical signal at the network input, produced by different sensor number (n) and calculated using different intermodal coupling strength (s). (e) The power of optical signals reflected by the N_max sensor, at different sensor locations.

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To further understand the properties of this sensor network, we consider the distribution of optical power among the various LP modes in this network. Follow our previous notation, we use the superscript “+” or “–“ and “lm” to denote the propagation direction and the type of theLP mode, and the subscript “n” to indicate that the power is measured at the input of the nth sensor. Figures 6(b) and 6(c) show the average $p_n^{+,lm}$ normalized by the input power ($p_1^{+,01}$) at different s values. The behaviors of the forward-propagating LP₀₁ and LP₀₂ mode can explain many aspects of the sensor network performance. First, as shown in Fig. 6(b), relatively far from the network input, the per-sensor attenuation for the forward LP₀₁ and LP₀₂ mode are very similar. For example, for s = 0.2 m^‒1 and n > 10, the attenuation of the LP₀₁ and LP₀₂ mode are almost identical: ‒0.0159 dB per sensor. This is to be expected, since the LP₀₁ and the LP₀₂ mode are closely coupled together by FBG 2. This conclusion also holds for cases with stronger intermodal coupling. For example, for s = 0.7 m^‒1 and n > 10, the per sensor attenuations for the LP₀₁ and the LP₀₂ mode are ‒0.1623 and ‒0.1629, respectively. (At sensors near the network input, the LP₀₂ mode power does not follow the constant attenuation behavior, since its input value is zero and needs to build up.)

Equation (17) suggests that the ultimate capacity of the sensor network is limited by the power of pulse 3 produced by the last (i.e., Nth) sensor, i.e., $p_{1,Ng4}^{-}$. Figure 6(d) shows the power of back-propagating pulse 3 ($p_{1,ng4}^{-}$), reflected by the nth sensor, measured at the sensor network input, and normalized with respect to the detection threshold p_min. The calculations are carried out under different intermodal coupling strength by changing the value of s. To make figures easier to see, we only show average of 50 simulations for any given s value. (For each simulation, we randomly assign the values of σ_x and σ_y using the previously defined pdf.) The results in Fig. 6(d) suggest that generally, the reflected sensor powers exhibit two distinct regimes: a fast decay regime for sensors near the input, and a much slower decay for sensors further away.

To understand the origin of the multi-decay behaviors, we consider the behaviors of the back-propagating sensor signals for a specific case of s = 0.2 m^‒1, which are shown in Fig. 6(e). Specifically, as the signal produced by the N_max sensor travels towards the network input, the power and the mode composition of the reflected signal must continuously change, due to bending-induced intermodal mode coupling, FBG reflection / coupling at each sensor location, and optical absorption within each sensing segment. Initially, the reflected power is dominated by the LP₀₂ mode, which experiences large per-sensor attenuation of ~‒3.80 dB (‒1.01 dB due to FBG 4 transmission, ‒2.52 dB due to sensor absorption, ‒0.27 dB due to FBG 3 transmission, and negligible bending-induced coupling at this small s value). However, due to the large per-sensor attenuation of the LP₀₂ mode, within ~7 sensors, the reflected power becomes dominated by the LP₀₁ mode, which decays at a much slower rate of ‒0.0135 dB per sensor. This value is less than the 1% mode conversion of FBG 2, which translates into ‒0.04 dB attenuation. This is reasonable, since optical power coupled into other LP modes are not completely absorbed, and may couple back into the LP₀₁ mode.

For sensors relatively close to the input port, the reflected signals are dominated by the LP₀₂ mode. This behavior can be directly seen in Fig. 6(d), where the per-sensor attenuation is ‒3.69 dB within the fast decay rate regime at s = 0.2 m^‒1. This value is almost the same as the one (‒3.80 dB) we extracted from the high LP₀₂ mode attenuation regime in Fig. 6(e). Additionally, in Fig. 6(d), for the same s value but within the slow decay regime, the per-sensor attenuation is ‒0.0317 dB, which is almost double the value of single-pass attenuation (‒0.0135 dB) for the LP₀₁ mode in Fig. 6(b) and 6(e). Again, this result makes sense.

4.3 Impact of non-ideal FBGs

In practical implementations, FBG 3 and 4 will likely reflect more than just the LP₀₂ mode, due to the presence of side-lobes in their reflection spectra. The impact of non-ideal FBG 3 and 4 are investigated here, where we assume that for the LP₀₁, LP₁₁, and LP₂₁ modes, optical reflection by FBG 3 and FBG 4 are 40 dB weaker than that of the LP₀₂ mode, with $r_{g3}^{01}$ = $r_{g3}^{11}$ = $r_{g3}^{21}$ = 10^‒4 $r_{g3}^{02}$ and $r_{g4}^{01}$ = $r_{g4}^{11}$ = $r_{g4}^{21}$ = 10^‒4 $r_{g4}^{02}$. (We note that apodization can reduce the reflectivity of FBG side-lobes by 80 dB [26].)

For non-ideal FBGs, Eq. (11) no longer strictly holds. Consequently, we define a measurement error ε% as the percentage difference between the ideal result as given by Eq. (11), and the power ratio of pulses reflected by FBG 3 and 4:

Our previous analysis shows that the interrogation signals in the sensor network are dominated by the LP₀₁ component. Therefore, if FBG 2 possesses a small mode conversion coefficient κ_g2, the measurement error ε% can be quite large, since even a small percentage of LP₀₁ mode reflected by FBG 3 and 4 can be comparable with the LP₀₂ signals produced by the same pair of FBGs. We can reduce the relative significance of LP₀₁ reflection by increasing the value of κ_g2. This strategy, however, will reduce N_max, since a higher percentage of the LP₀₂ component leads to higher signal attenuations. In Fig. 7, we quantify the tradeoff between N_max and ε% by changing the values of κ_g2 while keeping the rest of sensor network parameters the same. Four different sets of intermodal coupling parameter s are considered. Again, for each s value, we randomly assign σ_x and σ_y using the previously defined pdf, and calculate ε% at each sensor location. After repeating this process 50 times, we show the average and the standard deviation of the simulated ε% in Fig. 7. In our simulations, all network parameters, except for κ_g2, are the same. We increase the value of κ_g2 from 1% to more than 10% to ensure that ε% is ~10% or less. Fluctuations in ε% are likely caused by the interference between different LP components of the optical signals. Note that even in these non-ideal cases, we should be able to multiplex ~50 sensors in a single network.

 

Fig. 7 The percentage error defined in Eq. (33), at different sensor location (n) and calculated for network with different s values: (a) s = 0, (b) s = 0.4 m^‒1, s = 0.8 m^‒1, and s = 1 m^‒1.

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5. Discussion

The main novelty of our work is theoretically demonstrating the possibility of using MDM to construct an absorption-based optical sensor network that combines high sensitivity with the capability for large scale multiplexing. Since we are mainly interested in describing the fundamental principle of the MDM network, our discussion is not limited to any specific sensor designs. In this context, higher sensitivity means that the sensor signal (i.e., $p_{1,ng4}^{-}/p_{1,ng3}^{-}$ in Eq. (17)) responds more sensitively to changes in the to-be-measured external parameters. For the FMF network considered in section 4, we can define the enhancement of sensitivity in terms of optical attenuation associated with the LP₀₁ and the LP₀₂ mode as (1 ‒ α⁰²) / (1 ‒ α⁰¹). Theoretically, the enhancement of sensor sensitivity can be extremely high. However, in reality, this enhancement factor is likely limited by practical constrains such as sensor insertion loss and detector noises.

In order to quantitatively evaluate the performance of the MDM sensor network, we inevitably need to make many simplifying assumptions. For example, we assume that fiber bending is the only source of intermodal coupling. In reality, other sources of perturbation, such as pressure and strain, may also lead to significant intermodal coupling. The main results of the proposed sensor network, however, should still hold qualitatively, since our analytical framework does not depend on the nature of intermodal coupling.

A practical MDM network may exhibit strong MDL or mode dependent gain (MDG) [27]. However, the existence of MDL or MDG should not significantly impact the performance of the proposed sensor network. This is because we extrapolate sensor response by comparing the power of two pulses that possess almost exactly the same mode composition, travel through the same optical path, and are only nanoseconds apart from each other.

Depending on the strength of intermodal coupling, our results in Fig. 7 suggest that a network of ~80 sensors at s = 0.4 m^‒1 should be feasible. Given our simulation parameters, the total length of the sensor network is less than 500 m. Assuming a commercial four-mode step-index fiber with ~2 ps/m modal group delay, the delay between the LP₀₁ and LP₀₂ in the backward direction is ~1 ns, which is comparable to the 1 ns pulse duration assumed in our calculations. In practical implementations, we may adopt many tools developed for MDM communication network to manage the challenge of modal dispersion. Alternatively, we may also use graded-index FMF to reduce the impact of modal dispersion. In fact, the main reason that we assume a step index fiber in simulations is to simplify calculation of the attenuation of different LP modes, i.e., α^lm.

For the proposed sensor network, perhaps the most critical limiting factor in the insertion loss for individual sensors: If even the lowest loss mode (e.g., LP₀₁) suffers from high attenuation, large scale multiplexing is clearly impossible. If this is the case, one can always introduce optical amplifiers to compensate high insertion loss. However, analyzing a FMF sensor network with optical amplification is beyond the scope of this work.

The performance of the proposed sensor network may also be impacted by many other practical considerations. For example, LP mode polarization may become an issue, since strain and pressure may induces polarization changes even if polarization maintaining fiber is used [28]. Another potential challenge is the possibility of significant temperature / strain variations at sensor locations, which can induce significant shift in the reflection spectra of the FBGs. If this becomes an issue, one may need to design a sensor package that can mitigate significant variations in temperature, pressure, or strain.

Finally, we point out that FBGs written in FMFs and MMFs have been investigated in [19,29–34]. For future development, it should be possible to establish a more realistic model for FMF- based FBGs to simulate the performance of the proposed sensor network.

6. Conclusion

We theoretically consider the possibility of using FMF to construct a quasi-distributed network of absorption based fiber optical sensors through MDM. In such a network, we can use the low-attenuation LP₀₁ mode for sensing signal delivery and collection, and the high-attenuation LP₀₂ mode for absorption-based optical sensing. We develop a transfer matrix approach that can simulate important processes such as intermodal coupling within the FMF, optical reflection / mode coupling produced by the FBGs, and mode-dependent optical attenuation within the sensing segment. We find that the MDM design can indeed significantly increase the maximum number of sensors in a point-to-point network.