Coherence-length-gated distributed optical fiber sensing based on microwave-photonic interferometry

Liwei Hua; Yang Song; Baokai Cheng; Wenge Zhu; Qi Zhang; Hai Xiao

doi:10.1364/OE.25.031362

1. Introduction

One of the unique advantages of optical fiber sensing is its ability to acquire spatially distributed information. The combination of ultra-low loss optical fibers and high-speed electronics now make it possible to continuously monitor various spatially varying parameters over tens of kilometers or even longer distance. The applications have also extended from the structural health monitoring (SHM) [1] to other areas such as the monitoring of geophysical properties [2], chemical/biological species [3], and physiological parameters [4].

In general, distributed optical fiber sensing can be categorized into two groups. The first is the so called quasi-distributed sensing which cascades many discrete devices (e.g., fiber Bragg gratings (FBGs) [5,6]) along the fiber. These cascaded sensor devices share the same signal processing instrument but sample the entire space at discrete points. It has the advantages of flexible deployment, multi-agent capability and high detection sensitivity. However, most of the existing systems can only multiplex a limited number of sensors.

Another category is the so-called fully distributed optical fiber sensing technology which is commonly based on the time domain reflectometry (TDR) of various kinds. In a traditional optical time domain reflectometry (OTDR) system, a short optical pulse launches into an optical fiber and the back scatterings of the light are recorded by the photodetector in the order of time of arrival. The scatterings can be the Rayleigh scattering of the fiber or the nonlinear signals such as Raman (ROTDR) and Brillouin (BOTDR) scatterings [7]. The characteristics (e.g., the intensity and wavelength shift) of these scatterings are sensitive to the parameters to be measured (e.g., temperature or strain). However, the TDR based technologies have relatively low signal to noise ratio (SNR). The system needs to perform hundreds of averages to get decent sensing performance. The strain sensitivity for TDR type distributed sensing technology is generally around 10 με [8].

Optical interference is very sensitive to the optical path difference (OPD) change. Phase-OTDR uses a coherent light source in a traditional OTDR system. The optical interference of the distributed Rayleigh scatterings within the duration of the light pulse is collected and processed. The strain sensitivity of phase-OTDR can be as high as 4 nε [8,9]. However, phase-OTDR has the difficulty to quantitatively link the obtained interference signal to the specific parameters of interest unless phase recovery methods are deployed, because the random nature of the Rayleigh scattering. In addition to the traditional OTDR technique, the optical frequency domain reflectometry (OFDR) has also been developed for distributed optical fiber sensing [10,11]. OFDR also uses a frequency-swept coherent light source. The time-of-arrival information is obtained by the Fourier transform of the optical signal of the frequency sweeping range. OFDR has much higher SNR and spatial resolution compared with the traditional OTDR. OFDR can resolve hundreds nε in strain [8,12]. However, its measurement range is short.

In the past few years, microwave-photonics technologies have been investigated for optical fiber sensing [13]. By introducing microwave modulation into the optical system, the optical detection is synchronized with the microwave modulation frequency. As a result, the system has a high SNR and thus an improved detection limit. In addition, the phase of the microwave-modulated light can be easily obtained and Fourier transformed to find the time-of-arrival information for distributed sensing. The microwave photonics technology has been demonstrated for both quasi-distributed [14] and fully-distributed sensing [15–17]. Recently, an incoherent optical carrier based microwave interferometry (OCMI) technique has been demonstrated for fully distributed sensing with high spatial resolution and large measurement range [15]. The system used a microwave modulated incoherent (broadband) light source to interrogate cascaded intrinsic Fabry-Perot interferometers formed by adjacent weak reflectors inside an optical fiber. When the distance between two adjacent reflectors was larger than the coherence length of the light source, the optical interference components in the received signal became zero and the microwave terms were processed to form a microwave interferogram, which was further analysed to calculate the optical path difference between any two reflectors along the fiber. Similar concept has also been successfully validated using large core multimode silica fibers [18] and single crystal sapphire fibers [19]. The method has a number of unique advantages including high signal quality, relieved requirement on fabrication, low dependence on the types of optical waveguides, insensitive to the variations of polarization, high spatial resolution, and fully distributed sensing capability. However, the reported OCMI system only read the interference in microwave domain. As such the sensing resolution was low (in tens of με), limited by the long wavelength of the microwave signal and the intermedia frequency of the microwave source.

In this paper, we propose a new coherent microwave-photonics interferometry (CMPI) system for distributed optical fiber sensing. The system uses a coherent light source to obtain the optical interference signal from the cascaded weak reflectors for much improved sensitivity. In addition, the coherence length of the light source is carefully chosen or controlled to gate the signal so that distributed sensing can be achieved.

2. Description of the method

The proposed CMPI system is schematically shown in Fig. 1. A continuous wave (CW) laser with full width at half maximum (FWHM) of $Δ ω$ is used as the light source with its electrical field given by

E_{0} (ω, t) = A (ω) \cos (ω t),

where ω is the optical frequency, t is the time variable, and

A (ω)

is the amplitude of the light. The light intensity is modulated by the microwave signal given by

V_{0} (Ω, t) = V_{0} (Ω) \cos (Ω t),

where

V_{0} (Ω)

is the amplitude of the microwave signal and Ω is the microwave frequency. The intensity modulated lightwave is launched into the single mode fiber (SMF) and the electric field of the lightwave becomes

E_{i n} (Ω, ω, t) = \sqrt{1 + M \cos [Ω t + Φ_{0} (Ω)]} \cdot A (ω) \cos [ω t + ϕ_{0} (ω)],

where

ϕ_{0} (ω)

and

Φ_{0} (Ω)

are the initial phases of the optical carrier and the microwave at the launch port respectively, and

M = m \cdot V_{0} (Ω)

where m is the modulation coefficient of the electro-optic modulator (EOM) and M is small.

Fig. 1 Schematic illustration of coherent microwave-photonics interferometry (CMPI). EOM: electro-optic modulator; PD: photodetector; S₂₁(Ω) = V(Ω)/V₀(Ω).

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If there are N weak reflectors fabricated along the optical fiber, the reflected lightwave from the ith reflector can be expressed as

E_{i} (Ω, ω, t) = \sqrt{1 + M \cos [Ω t + Φ_{i} (Ω)]} \cdot A_{z_{i}} (ω) \cos [ω t + ϕ_{i} (ω)],

where

A_{z_{i}} (ω) = A (ω) \cdot Γ_{i}

and

Γ_{i}

is the amplitude reflective coefficient of the ith reflector.

The optical phase and the microwave phase for the lightwave before the detector are $ϕ_{i} (ω) = ϕ_{0} (ω) - \frac{ω z_{i} n}{c}$ and $Φ_{i} (Ω) = Φ_{0} (Ω) - \frac{Ω z_{i} n}{c}$ respectively, where c is the speed of light in vacuum, n is the refractive index of the fiber, z_i is the distance that the light travels from the electro-optic modulator (EOM) to the ith reflector and then back to the photodetector. The total reflected signal power received by the photodetector is given by

I (Ω, t) = {| \int_{- \infty}^{\infty} \sum_{i = 1}^{N} E_{i} (Ω, ω, t) d ω |}^{2} .

The beat among different optical frequency components creates low level noise [20], which is neglected in this work, so the Eq. (5) can be expressed as

I (Ω, t) = \frac{1}{Δ ω} \int_{- \infty}^{\infty} {| \sum_{i = 1}^{N} E_{i} (Ω, ω, t) |}^{2} d ω = I_{s e l f} (Ω, t) + I_{c r o s s} (Ω, t),

where I_self(Ω, t) and I_cross(Ω, t) are the self and cross products terms, respectively. Because the optical frequency is much higher than that of the photodetector, the photodetector output is the time-averaged signal over the optical period, given by

〈 I_{s e l f} (Ω, t) 〉 = 〈 \int_{- \infty}^{\infty} \sum_{i = 1}^{N} E_{i}^{2} d ω 〉 = \frac{1}{2} \sum_{i = 1}^{N} {\int_{- \infty}^{\infty} A_{z_{i}} {(ω)}^{2} d ω [1 + M \cos (Ω t - \frac{Ω n z_{i}}{c})]},

\begin{array}{r} 〈 I_{c r o s s} (Ω, t) 〉 = 〈 \int_{- \infty}^{\infty} (\sum_{i = 1}^{N} \sum_{j \neq i}^{N} E_{i} E_{j}) d ω 〉 = \sum_{i = 1}^{N} \sum_{j \neq i}^{N} {\frac{1}{2} \int_{- \infty}^{\infty} A_{z_{i}} (ω) A_{z_{j}} (ω) \cos (ϕ_{i} - ϕ_{j}) d ω \\ \cdot \sqrt{[1 + M \cos (Ω t - \frac{Ω n z_{i}}{2 c})] [1 + M \cos (Ω t - \frac{Ω n z_{j}}{2 c})]}} . \end{array}

The microwave photonics system synchronizes the detection and only measures the amplitude and phase of the signal at the microwave frequency Ω. The other frequency components (e.g., the DC term and the 2Ω terms) are excluded in the vector microwave detection. Applying Taylor expansion to Eq. (8), we can find the fundamental frequency component of $〈 I_{c r o s s} (Ω, t) 〉$ , which is given by:

\begin{array}{l} {〈 I_{c r o s s} (Ω, t) 〉 |}_{at Ω} \\ = \frac{M}{2} \sum_{i = 1}^{N} \sum_{j \neq i}^{N} {\int_{- \infty}^{\infty} A_{z_{i}} (ω) A_{z_{j}} (ω) \cos (ϕ_{i} - ϕ_{j}) d ω \cdot \cos \frac{Ω n (z_{i} - z_{i})}{2 c} \cdot \cos [Ω t - \frac{Ω n (z_{i} + z_{j})}{2 c}]} . \end{array}

Thus, the complex frequency response S₂₁ of the system, i.e., complex reflectivity normalized with respect to the input signal, is

\begin{matrix} S_{21} (Ω) \\ = \frac{m}{2} \sum_{i = 1}^{N} \int_{- \infty}^{\infty} A_{z_{i}} {(ω)}^{2} d ω \cdot e^{- j \frac{Ω n z_{i}}{c}} + \frac{m}{2} \sum_{i = 1}^{N} \sum_{j = 1, j \neq i}^{N} \int_{- \infty}^{\infty} A_{z_{i}} (ω) A_{z_{j}} (ω) \cos (ϕ_{i} - ϕ_{j}) d ω \cdot \cos \frac{Ω n (z_{i} - z_{j})}{2 c} e^{- j \frac{Ω n}{2 c} (z_{i} + z_{j})} \\ = \underset{self product term}{\underset{︸}{\frac{m}{2} \sum_{i = 1}^{N} \int_{- \infty}^{\infty} A_{z_{i}} {(ω)}^{2} d ω \cdot e^{- j \frac{Ω n z_{i}}{c}}}} + \underset{cross product term}{\underset{︸}{\frac{m}{4} \sum_{i = 1}^{N} \sum_{j = 1, j \neq i}^{N} \int_{- \infty}^{\infty} A_{z_{i}} (ω) A_{z_{j}} (ω) \cos (ϕ_{i} - ϕ_{j}) d ω \cdot (e^{- j \frac{Ω n z_{i}}{c}} + e^{- j \frac{Ω n z_{j}}{c}})}} \end{matrix}

By applying complex Fourier transform to S₂₁(Ω), we obtain the time domain signal F(t_z)

F (t_{z}) = \sum_{i = 1}^{N} I_{N} (z_{i}) δ (t_{z} - \frac{n z_{i}}{c}),

where t_z denotes the time variable.

F (t_{z})

is the superposition of N delta functions with different time delays

n z_{i} / c

,

I_{N} (z_{i})

is expressed as

\begin{matrix} I_{N} (z_{i}) = \frac{m}{2} \int_{- \infty}^{\infty} A_{z_{i}} {(ω)}^{2} d ω + \frac{m}{2} \sum_{j = 1, j \neq i}^{N} \int_{- \infty}^{\infty} A_{z_{i}} (ω) A_{z_{j}} (ω) \cos (ϕ_{i} - ϕ_{j}) d ω \\ = \frac{m}{2} \int_{- \infty}^{\infty} Γ_{i}^{2} A_{0} {(ω)}^{2} d ω + \frac{m}{2} \sum_{j = 1, j \neq i}^{N} \int_{- \infty}^{\infty} Γ_{i} Γ_{j} A_{0} {(ω)}^{2} \cos \frac{ω n (z_{i} - z_{j})}{c} d ω, \end{matrix}

As shown in Eq. (12), the amplitude of each time domain pulse is the sum of two parts. The first part is a constant determined by the reflectivity of the reflector and the input light power. The second part is the superposition of the optical interference among the reflected lightwaves of the ith and any other reflectors in the system. The intensity of the interference of any two waves changes sinosoidally as function of phase difference between them. The amplitude of the sinusoidal function approaches zero as the phase difference increases. The amplitude decay rate is decided by the spectrum of the light carrier. Let’s assume the light carrier has Lorentzian shape spectrum and centered at with maximum amplitude of, so can be expressed as [21]

A_{0} {(ω)}^{2} = A_{0}^{2} \frac{Δ ω^{2}}{4 {(ω - \bar{ω})}^{2} + Δ ω^{2}} .

By plugging Eq. (13) into Eq. (12) and integrating over all frequencies [21,22], we can get the expression of $I_{N} (z_{i})$ as

I_{N} (z_{i}) = \frac{m Δ ω A_{0}^{2}}{16} [Γ_{i}^{2} + \sum_{j = 1, j \neq i}^{N} Γ_{i} Γ_{j} \cdot \exp \frac{- Δ ω n | z_{i} - z_{j} |}{2 c} \cdot \cos \frac{\bar{ω} n (z_{i} - z_{j})}{c}] .

Now let’s consider that the Lorentzian shape light source has a coherence length of L_c, which is inversely proportional to $Δ ω$ , given by [21]

L_{c} = \frac{2 c}{Δ ω} .

Equation (12) can be simplified as

I_{N} (z_{i}) = \frac{m Δ ω A_{0}^{2}}{16} [Γ_{i}^{2} + \sum_{j = 1, j \neq i}^{N} Γ_{i} Γ_{j} \cdot \exp \frac{- n | z_{i} - z_{j} |}{L_{c}} \cdot \cos \frac{\bar{ω} n (z_{i} - z_{j})}{c}] .

The coherence length L_c performs as a truncating function, which limits the number of reflectors contributing to the amplitude of the time domain pulse. If the OPD between the jth and ith reflectors is much larger than the coherence length of the light source L_c, the jth reflector’s contribution becomes very small and can be neglected in Eq. (16). On the other hand, if the OPD between the ith and jth reflector is smaller than L_c, slightly change of the OPD can change the value of the second part dramatically. Thus, the amplitude variation of the time domain pulse can provide a sensitive indication of the OPD changes.

We can engineer the system to make sure that at any fiber location there are only two reflectors within the coherence length of the light source. It becomes apparent that we can either control the linewidth (∆ω) of the optical source or the distance between two adjacent reflectors to include or exclude the reflectors to contribute to the second term of $I_{N} (z_{i})$ . If we further assume that the OPD between the two reflectors is much smaller than L_c, or $n (z_{i} - z_{j}) π < < L_{c}$ , then Eq. (16) can be simplified and approximated to be

\begin{matrix} I_{2} (z_{i}) \approx \frac{m Δ ω A_{0}^{2}}{16} [Γ_{i}^{2} + Γ_{i} Γ_{j} \cdot \exp \frac{- n | z_{i} - z_{j} |}{L_{c}} \cdot \cos \frac{\bar{ω} n (z_{i} - z_{j})}{c}] . \\ \approx \frac{m Δ ω A_{0}^{2}}{16} [Γ_{i}^{2} + Γ_{i} Γ_{j} \cdot \cos \frac{\bar{ω} n (z_{i} - z_{j})}{c}] \end{matrix}

According to Eq. (17), $I_{2} (z_{i})$ varies sinusoidally as a function of the OPD between the two reflectors. The period of the sinusoidal function equals to the average optical wavelength. A minor change in OPD between the two reflectors can produce a detectable change in the amplitude of the time domain pulses. The locations of the reflectors can be identified by their corresponding position in the time axis. Therefore, the coherence gated microwave-photonic system can be used for distributed sensing with very high sensitivity.

In our previous work [15], we used an incoherent light source whose bandwidth is much smaller than the optical path difference (OPD) between any two reflectors. As such, the cross-product term becomes zero because the integration $\int_{- \infty}^{\infty} A_{z_{i}} (ω) A_{z_{j}} (ω) \cos (ϕ_{i} - ϕ_{j}) d ω$ over the optical bandwidth in Eq. (9) is practically zero when the coherence length of the light source is short. The self-product term given in Eq. (10) was used to reconstruct the interferogram in the microwave domain and the microwave phase shift was used to derive the optical path difference. For this reason, we named it the optical carrier based microwave interferometry (OCMI).

When a coherent source is used, the optical interference between the ith and its neighbor reflectors makes the value of $I_{N} (z_{i})$ a very sensitive function of the OPD change between those reflectors. On the other hand, because the microwave frequency (Ω) is much smaller than the light frequency (ω), a small change in distance z_i would cause an insignificant change in the microwave phase $Φ_{i} (Ω)$ than that in the optical phase $ϕ_{i} (ω)$ . Thus, under the assumption of small change of z_i, the microwave phase terms in Eq. (10) become constant.

In reality, the sweeping microwave frequency has a limited bandwidth of Ω_b at the center frequency of Ω_c. To consider the limited bandwidth the time domain signal expressed in Eq. (11) should be modified to be

\begin{matrix} F^{'} (t_{z}) = [Ω_{b} \sin c (Ω_{b} t_{z}) e^{- j Ω_{c} t_{z}}] * F (t_{z}) \\ = \sum_{i = 1}^{N} Ω_{b} \sin c [Ω_{b} (t_{z} - \frac{n z_{i}}{c})] e^{- j Ω_{c} (t_{z} - \frac{n z_{i}}{c})} I_{N} (z_{i}), \end{matrix}

If the reflectors are far away from each other, the side lobes of the sinc functions can be ignored. The signal at the distance z_i can be approximated to be

F'_{i} (t_{z}) \approx Ω_{b} \sin c [Ω_{b} (t_{z} - \frac{n z_{i}}{c})] e^{- j Ω_{c} (t_{z} - \frac{n z_{i}}{c})} I_{N} (z_{i}) .

Besides, the received signal should be scaled by a factor which is the product of the optical gain from the EDFA G, responsivity of the photodetector $ℜ$ , and link loss $σ$ as

F''_{i} (t_{z}) \approx Ω_{b} \sin c [Ω_{b} (t_{z} - \frac{n z_{i}}{c})] e^{- j Ω_{c} (t_{z} - \frac{n z_{i}}{c})} I_{N} (z_{i}) \cdot G \cdot ℜ \cdot σ .

3. Experiments, results and discussions

We have analytically shown that the CMPI system can effectively convert the OPD change between reflectors into the intensity change variation of the time domain pulses corresponding to the specific reflectors. The intensity of the time pulse changes as a function of OPD according to the optical interferometry formula which can be used to measure very small OPD variations for distributed sensing. To validate the proposed concept, we designed two sets of experiments. The first set of experiments using a pair of in-fiber reflectors to study the effects of the coherence length and cavity length to the performance of the sensor. The second set of experiments use an array of in-fiber reflectors to demonstrate the CMPI’s capability for highly sensitive distributed sensing.

3.1 System configuration

The experiment setup to validate the CMPI concept for distributed sensing is shown in Fig. 2. Two types of light sources were used to study the coherence dependence of the system. The two sources had different linewidths, which allow us to study the coherence length effect on the system. The first source was a tunable laser that had a Lorentzian shape spectrum with FHWM about 100 kHz (81640A, Agilent) whose coherence length was about 0.954 km at the center wavelength of 1543nm. The second light source was a F-P laser (81554, HP). A single longitude mode of laser was filtered out by adding a 1 nm band pass filter (BPF) after the laser. The single longitude mode had a Lorentzian shape spectrum with FHWM about 1.5 GH at the center wavelength of 1543 nm, which had a coherence length of 6 cm.

Fig. 2 Schematic of the system configuration for concept demonstration. Two types of light sources were used to study the coherence length effect on the system. EOM: Electro-optic modulator, EDFA: Erbium-doped fiber amplifier, PD: photodetector, BPF: band pass filter, RF Amp: RF amplifier, BPF: band pass filter; VNA: vector network analyzer.

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The light from the source was intensity modulated by a microwave signal via an electro-optic modulator (EOM). An inline polarization controller (Thorlabs, US) was used to optimize the modulation depth of the EOM which was connected to the port 1 of a vector network analyzer (VNA Agilent E8364B). The microwave-modulated light output from the EOM was first amplified by an EDFA, and then launched into the port 1 of a fiber circulator. Port 2 of the fiber circulator was connected to the sensing fiber on which cascaded weak reflectors with reflectivity around −30 dB were fabricated using femtosecond laser micromachining [23,24]. Each two adjacent reflectors formed a weak reflection FPI. The purpose of making low reflectivity reflectors is to minimize the optical power loss at each reflector and avoid multi-path interference, so we could cascade hundreds of reflectors along the fiber for long distance distributed sensing. The reflected signals travelled back to the port 3 of the circulator and were further amplified by another EDFA. Another BPF was used to pass the signal and cut down the amplified spontaneous emission (ASE) of the EDFA. The filtered and amplified signal was detected by a high-speed photo-detector, which converted the optical signal into an electrical signal. The electrical signal was then connected to the port 2 of the VNA, which measured the amplitude and phase of the signal at the modulation frequency. After the VNA swept through the designated microwave bandwidth, the S₂₁ spectrum was obtained which was further processed according to the method outlined in Section 2 to realize distributed sensing. Because S₂₁ refers to the voltage ratio between the received and the sent-out microwave signal, the amplitude variation of the microwave source won’t affect the sensing signal. In all the experiments below, the VNA was set to have the number of sampling points of 3201 in the microwave band and the intermediate frequency bandwidth (IFBW) of 5 kHz.

3.2 System validation using a pair of reflectors

To validate the signal processing method given in Section 2, a single pair of reflectors separated by 10 cm were fabricated on a SMF by femtosecond laser micromachining. The femtosecond laser beam was focused into the fiber core to slightly modify the refractive index of the focusing area without striping the polymer coating of the fiber [23,24]. The two reflectors had very close reflectivity and they formed a weak reflection FPI with a cavity length of 10 cm. The two fiber ends of the FPI were glued onto two motorized translation stages (PM500, Newport) respectively. The two fixing points were separated by 1.55 m and the FPI was positioned in the middle of the two stages. Axial strains were applied to the FPI by moving one stage at 1μm (corresponding to about 0.6 μɛ) per step. The sweeping microwave bandwidth of the VNA was from 0.1 GHz to 4.1 GHz, and the tunable laser source (Option 1) was used in the experiment.

The amplitudes of the time domain signals under different applied strains are shown in Fig. 3(a). The insert shows the amplitudes of the two peaks (at the time points t _s1 and t _s2) as function of the applied strain. The microwave bandwidth was large enough to separate the two reflectors as two time-domain pulses. It is obvious that the amplitudes of the two reflector pulses changed sensitively as a function of the applied strain. The changes of the two peak amplitudes were in phase. Figure 3(b) plots the real part of the signal and the insert shows the peak of the real part at the time points t_s1 and t_s2 as a function of the applied strain. The amplitudes of the real part of the signal were also in phase. A close comparison of the Figs. 3(a) and 3(b) appears that the amplitude changed faster than the real part of the signal. The amplitude of the time domain signal shown in Fig. 3(a) is the absolute value of the cosine function given in Eq. (17) while the real (or imaginary) part of the signal is the sinusoidal function itself. It has been reported that a cross phase modulation might be resulted in the intensity modulated light using an EOM [25]. The cross phase modulation is neglected in our derivations but deserves a detailed study in the future. The period of the sinusoidal function is 7.2 µɛ, which is slightly larger than the theoretical period 6.5 µɛ got from Eq. (17), where we assume the effective strain-optic coefficient P_eff is about 0.204 [18]. Several possibly reasons lead to the mismatch. The major reason is that the fiber was glued on the stage without striping off the coating, so the applied strain was not 100 percent transferred to the sensor due to the shear deformation at the coating and adhesive [26].

Fig. 3 Strain response of an FPI consisting of two reflectors separated by a distance of 10 cm using a microwave bandwidth of 4 GHz. (a) Amplitude of the time-domain pulse under various applied strains. Inset: amplitudes of the two peaks as a function of the applied stain. (b) Real parts of the time-domain signals shown in (a). Inset: amplitudes of the two peaks as function of the applied stain.

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3.3 Dependence on the microwave bandwidth

When the microwave bandwidth was chosen to be 4 GHz, the two reflectors spaced 10 cm apart showed as two separate pulses in the time domain plot. It is worth to know what will happen if the microwave bandwidth becomes small and the individual reflectors cannot be resolved in the time domain. To experimentally investigate this, we reduced the microwave bandwidth to 0.8 GHz (from 1.1 GHz to 1.9 GHz). With the reduced bandwidth, the two pulses were inseparable. However, because the time domain signals of the two reflectors (and) have the same optical inference information and they are in phase, the combined time domain signal of these two reflectors should have the same response to the OPD change (in this case the applied strain) as the individual pulses. Figures 4(a) and 4(b) show the amplitude response and the real part of the signal as a function of the applied strain. The two pulses practically merged into one pulse in the time domain plots. The inserts of Figs. 4(a) and 4(b) plot the amplitudes of the time pulses as a function of the applied strain. It is obvious that the response of the merged pulse is very similar to that of the individual pulses, providing the same sensing information. This becomes very useful because smaller bandwidth means less data points to be processed and small cavities can also be read by the system. Because the two pulses merged into one when the reduced microwave bandwidth was used, the amplitude changing range of the merged peak in Fig. 4 is about two times of that of the single peak shown in Fig. 3. In addition, the microwave-photonic system has different responses at different microwave frequencies. As a result, the amplitudes in Figs. 3 and 4 are different.

Fig. 4 Strain response of an FPI consisting of two reflectors separated by a distance of 10 cm using a microwave bandwidth of 0.8 GHz. (a) Amplitude of the time-domain pulse under various applied strains. Inset: amplitudes of the two peaks as a function of the applied stain. (b) Real parts of the time-domain signals shown in (a). Inset: amplitudes of the two peaks as function of the applied stain.

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3.3 Gating by coherence length

The coherence length of the light source limits the maximum distance between two adjacent reflectors that participate in optical interference. We can thus use the coherence length to select the specific reflector pair that will contribute to optical interference. In a general design, the cavity length should be smaller than the coherence length to obtain a high interference contrast. The distance between two adjacent FPIs (i.e., reflector pairs) needs to be much larger than the coherence length of the light source to avoid cross talks. Under this design, the optical coherence length determines the spatial resolution when the system is used for distributed sensing.

To verify the coherence length effect, we used the filtered F-P laser source (Option 2 in Fig. 2) whose single longitudinal mode had a linewidth of about 1.5 GHz, corresponding to a coherence length of about 6 cm. Axial strains were applied to an FPI with a cavity length of 10 cm by using the same translation stages as described in section 3.2 (with the applied strain resolution of 1/1.55μɛ per step). The microwave bandwidth was set to be 0.8 GHz (from 1.1 GHz to 1.9 GHz). Figure 5(a) shows the comparison of using the two light sources, where the peak real values were normalized and plotted as a function of the applied strain. Using the tunable laser, the signal showed a clear interference pattern with a contrast close to 100%. When interrogated using the filtered F-P laser source, the signal was almost flat with a variation less than 1% because the OPD of the FPI was larger than the coherence length of the filtered F-P laser source. Figure 5(b) shows the normalized peak real values change as a function of the applied strain for an FPI with a cavity length of 1 cm, interrogated by the filtered F-P laser source with the same VNA setup. This time, the contrast was about 80% because the coherence length (6 cm) was larger than the OPD (~2.9 cm). A similar experiment was performed to use the filtered F-P laser source to interrogate an FPI of 1 mm cavity length, whose OPD is much smaller than the coherence length and resulted in increased contrast of close to 100%.

Fig. 5 Normalized real part of time pulses as function of strain (a) for the time domain pulse generated by the 10-cm cavity FPI by using two different linewidth light sources; (b) for the time domain pulse generated by the 1-cm cavity FPI by using filtered F-P laser.

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3.4 Distributed strain measurement by using an array of reflectors

The distributed sensing capability of the proposed CMPI technique was studied using an array of 29 cascaded reflectors as schematically shown in Fig. 6(a). Three different separation distances of 1mm, 1.5 cm and 30 cm between two adjacent reflectors were planned as shown in Fig. 6(a). In theory, any two reflectors can form an FPI. However, only those two with a separation distance smaller than the coherence length of the light source can produce interference fringes with good visibility. The light source used in the experiment was the filtered F-P laser with a linewidth of 1.5 GHz (Option 2 in Fig. 2). As a result, closely spaced two reflectors (i.e., separation distances of 1mm and 1.5 cm) will produce an optical interference signal, and the reflectors separated by a distance of 30 cm will not produce an optical interference. The microwave bandwidth was set to be 2 GHz, scanning from 2 to 4 GHz. Figure 6(b) shows the time domain signal of the distributed sensors where only 19 pulses can be seen because the small gapped reflectors (e.g., 1 mm and 1.5 cm) merged together when the microwave bandwidth was 2 GHz.

Fig. 6 (a) Schematic of SMF distributed sensors with 29 cascaded reflectors. (b) Amplitude of the time domain signal, where the pulses with separation distance 1 mm and 1.5 cm from each other merged together.

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Axial strains of 0.6 μɛ per step were applied to the range (1.55 m total length) marked as purple rectangular section shown in Figs. 6(a) and 6(b) by pulling the fiber at a small distance of 1 μm per step using a translation stage. In this section, there are 6 time domain pulses shown in the Fig. 7(a), where I₄, I₅ and I₆ are the pairs of reflectors and the rest are single reflectors. Pulses I₄ and I₆ were formed by a pair of reflectors with a separation distance of 1.5 cm, and Pulse I₅ was formed by a pair of reflectors with a separation distance of 1 mm. Because these three paired reflectors (I₄, I₅ and I₆) had separation distance smaller than the coherence length of the light source, their time domain pulse amplitudes varied as a function of the applied strain as a result of effective optical interference. On the other hand, the pulse amplitudes of the single reflectors (located at R₄, R₅ and R₆) had shown negligible variation when the applied strain changed, because their distances to the adjacent reflectors were much larger than the coherence length of the light source.

Fig. 7 (a) Amplitude of the time domain signal under different applied strain within the strained section regime. I₄, I₅, and I₆ are the three merged pulses formed by the FPIs with cavity length of 1.5 cm, 1mm, and 1.5 cm respectively. (b) Normalized amplitude (real part) changes of the 19 pulses as a function of the applied strain. (c) Normalized amplitude (real part) changes for pulse I₄, I₅, and I₆ as function of strain around the quadrature point on the strain spectrum of I₅, which is circled in (c). (d) The zoomed in circled regime in (c).

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Figure 7(b) plots the amplitudes (real part) of all time domain pulses as a function of strain and distance in 3D. The amplitudes of the I₄, I₅ and I₆ reflector pairs varied sinusoidally as the applied strain changed, while those of single reflectors remained almost unchanged as the applied strain varied. The experiment result proved that the coherence length of the light source could be used to optically isolate the reflectors based on their separations and optical interference contrasts. Using the microwave-photonic system, the locations of these reflector pairs can be clearly identified in the plot, indicating the distributed sensing capability of the system. The spatial resolution of the sensing system is determined by both the coherence length of the light source and microwave bandwidth, which are about 6 cm in this experiment. In real applications, the coherence length of the light source, the microwave bandwidth and the separation of the cascaded reflectors can be varied to fulfill the specific requirement in spatial resolution.

Figure 7(c) shows the zoomed-in plots of the amplitude (real value) variations of the I₄, I₅ and I₆ reflector pairs as a function of the applied strains (i.e., the circled region in Fig. 7(b)). The amplitudes of the I₄, I₅ and I₆ pulses changed apparently with the applied strains as predicted by Eq. (17). Figure 7(d) shows the further zoomed-in plot of the amplitude of pulse I₄ (i.e., the circled region in Fig. 7(c)) in response to the applied strains of 7.2 μɛ (or 12 steps) in total. The results clearly indicated that even a small strain step of 0.6 μɛ could be resolved without ambiguity using the FPI with a cavity length of 1.5 cm, proving the very high measurement sensitivity of the system. The slightly offset of some individual points may be caused by several reasons. The first is the system error and the second is the inaccurate applied strain. In our case, the strain was applied using a translation stage with a nominal resolution of 0.1 μm and a repeatability of 0.2 μm. Such inaccuracy accounted for the major contribution of the offset in the experiment. It is worth noting that temperature variations would also cause the OPD change. The strain-temperature cross talk of a typical fused silica SMF is about 9.57 με/°C [24]. The strain-temperature cross-sensitivity needs to be considered when applying the sensor and system in real applications. Nevertheless, the sub-micron strain resolution is high comparing with other types of distributed sensing techniques, especially when considering the gauge length of the sensor is only 1.5 cm.

3.5 Strain sensitivity and dynamic range

Using Eq. (17), we can calculate the strain sensitivity of the individual sensors. If we assume that the i_th sensor (i.e., reflector pair) has a cavity length of, the signal of the i_th sensor is

I_{2} (z_{i}) \approx \frac{m Δ ω A_{0}^{2}}{16} [Γ_{i}^{2} + Γ_{i} Γ_{j} \cdot \cos \frac{\bar{ω} n L_{i}}{c}]

By taking the partial derivative of nL_i in Eq. (21), we have

\frac{\partial I_{2} (z_{i})}{\partial (n L_{i})} = - \frac{\bar{ω} m Δ ω A_{0}^{2}}{16 c} Γ_{i} Γ_{j} \cdot \sin \frac{\bar{ω} n L_{i}}{c}

By plugging in the strain definition (

ε_{i} = Δ L_{i} / L_{i}

) and P_eff into Eq. (22), we obtain

Δ I_{2} (z_{i}) = - [\frac{\bar{ω} m Δ ω A_{0}^{2}}{16 c} n L_{i} Γ_{z_{i}} Γ_{z_{j}} \sin (\frac{\bar{ω} n L_{i}}{c}) (1 - P_{e f f})] ε_{i}

Equation (23) indicates that the change of signal amplitude is a sinusoidal function of the applied strain, meaning the sensitivity is nonlinear. In addition, the strain sensitivity is proportional to the initial cavity length (L_i) of the sensor, which is the distance between two adjacent reflectors in our case. To further increase the strain sensitivity, we can increase the gauge length of the sensor by increasing the cavity length. This can be clearly seen in Fig. 7(c), with the same amount of applied strain of about 65 μɛ, the amplitudes of long-cavity FPIs (I₄ and I₆, with a cavity length of 1.5 cm) changed more than an entire sinusoidal period while that of shorter cavity FPI (I₅ with a cavity length of 1 mm) varied much less than a sinusoidal period.

It took about 50 µɛ to produce one period of change in amplitude when using the FPI with a cavity length of 1.5 cm as shown in Fig. 7(c). This 1.5 cm cavity length FPI could clearly resolve 0.6 µɛ in strain measurement as shown in Fig. 7(d). For the FPI with a cavity length of 10 cm, the amplitude of the pulse changed an entire sinusoidal period with the amount of applied strain of about 7.2 µɛ as shown in Fig. 3(b). This indicates that the 10-cm cavity FPI had much higher strain sensitivity than the 1.5-cm cavity FPI. We expect that the strain measurement sensitivity will further increase when the cavity length of the FPI increases, potentially reaching the nɛ level when the cavity length reaches 1m. However, the sensitivity is not linear as indicated by Eq. (23). The sensitivity is maximum in the quadrature region of the sinusoidal response curve, but it can become zero at the peak or valley of the sinusoidal curve. When using the proposed CMPI system for distributed sensing, it needs to be proper calibrated. One way to calibrate the FPIs is to tune the center optical frequency of the laser source and record the amplitude of each time pules at each laser frequency after installing the fiber into the host environment. The amplitude as a function of the optical frequency could be a useful indicator to find the strain sensitivity of each FPI.

Another issue is that the high sensitivity comes with a sacrifice in the dynamic range due to the sinusoidal nature of the signal, in which the 2π ambiguity prevents the continuous tracking of the response curve in a range that is larger than the period. In general, there is a tradeoff between the sensitivity and the dynamic range. A long cavity length FPI has a high strain resolution by a small dynamic range. The situation becomes opposite when the cavity length of the FPI is small.

Knowing this allows one to optimize the distributed sensing system in terms of sensitivity and dynamic range by choosing the proper separations of the reflector pairs (i.e., the cavity length of the FPI) at specific locations. This provides the desired flexibility to design a distributed sensing system to satisfy the different sensitivity and dynamic range needs at different locations in a specific application. In addition, we can use a combination of FPIs with different cavity lengths at a close proximity to achieve both large dynamic range and high sensitivity, if these FPIs are under the same applied strain. In our experiments, we didn’t observe noticeable optical interference signal fluctuations or contrast variations. We believe that this is mainly because the cavity length of the FPIs are much smaller than the typical birefringence beat length of the SMF and the FPIs were kept straight during experiments. However, polarization fading effects might be significant once we increase the cavity length [27]. In addition, unstable fiber bending will also lead to significant fiber birefringence of the FPI and thus an unstable interference signal, which should be considered in real applications. Another interesting observation that can be made by comparing the two FPIs I₄ and I₆ which have about the same cavity length of 1.5 cm, in response to the applied strain as shown in Fig. 6(d). First to notice is that their magnitudes are different which is due to the different reflectivity of the reflectors. The second is that their changing periods are about the same because they have about the same cavity length. The third is that the two response curves have a phase difference, which is caused by the initial cavity length difference between the two FPIs. This phase difference, if they can be adjusted to π/2 (quadrature phase shift), might be potentially to expand the dynamic measurement range based on bi-directional counting of the interference fringes.

CMIP is based on the measurement of the time-domain pulse amplitude of each FPI. The power fluctuations of the light source and microwave signal could induce measurement errors. To minimize the source power variation induced errors, a single reflector R_i (where i = 1, 2, 3..) just before each FPI I_i, as shown in Fig. 6(a), can be added to the system for compensation. The ratio between the amplitude of the single reflector and the reflector pair is not depended on the input power level. To verify this, we varied the input optical power by changing the gain of EDFA 1. Figure 8(a) shows the time pulses before and after changing the power of the light source where the amplitudes changed significantly. However, the ratio between the amplitude of the reflector pair R_i, and that of the single reflector I_i just before it remained practically unchanged as shown in Fig. 8(b). Because each FPI has a reference reflector located just before it, the power compensation should work when there is an optical loss in the middle of the fiber such as that caused by fiber bending.

Fig. 8 Compensation for power fluctuation. (a) Time pulses at different power levels of the light source, showing as much as 2.7 times in power difference. (b) Power ratio between the FPI pair (I_i) and the single reflector(R_i) before it before and after input optical power change.

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4. Conclusion

This paper presents the studies on demonstration of a new concept based on coherent microwave-photonics interferometry (CMPI) for distributed sensing. The CMPI uses a microwave modulated coherent light source to interrogate cascaded interferometers for distributed measurement. By scanning the microwave frequencies, the complex microwave spectrum is obtained and converted to time domain pulses at known locations by complex Fourier Transform. Because a coherent light source is used, the amplitudes of these time domain pulses are a function of the optical path differences of the distributed interferometers and can be processed for sensing applications. The optical interference can measure very small OPD changes. As a result, CMPI offers the key advantage of the high sensing resolution.

To demonstrate the concept, cascaded fiber Fabry-Perot interferometers were fabricated in a SMF using femtosecond laser micromachining. Our modeling and analysis provided the relation between the time-domain pulse amplitude and the OPD of the cascaded interferometers. By carefully select the separation distances among the cascaded reflectors, the coherence length can gate the signal so that only two adjacent reflectors participate in the optical interference and ambiguity can be avoided to achieve distributed sensing. The experimental results indicated that the strain measurement resolution can be better than 0.6 µε using a FPI with a cavity length of 1.5 cm. Further improvement of the strain resolution to the nε level is achievable by increasing the cavity length of the FPI to over 1m. However, the increase of the cavity length may lead to other issues such as polarization fading and larger temperature cross sensitivity, which need to be considered in real applications. The theoretical analyses and experimental results showed that the signal was a sinusoidal function of the OPD. As a result, the sensitivity is nonlinear and there is a tradeoff between the sensitivity and dynamic range due to the 2π ambiguity. A longer cavity length will result in a higher resolution but smaller dynamic range. Knowing this tradeoff allows the strategy and flexibility to design a distributed sensing system to satisfy the different sensitivity and dynamic range needs at different locations in a specific application. To minimize the optical power instability (either from the light source or the fiber loss) induced errors, a single reflector can be added in front of an individual FPI as an optical power reference for the purpose of compensation. The theoretical analyses and experimental results all showed that CMPI could have a great potential for distributed sensing, especially for its high sensitivity, though many aspects of the concept still remain for further research.

Funding

National Science Foundation (NSF) (CMMI-1335163); Department of Energy (DOE) (DE-FOA-0001445)

References and links

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Coherence-length-gated distributed optical fiber sensing based on microwave-photonic interferometry

Abstract

1. Introduction

2. Description of the method

3. Experiments, results and discussions

3.1 System configuration

3.2 System validation using a pair of reflectors

3.3 Dependence on the microwave bandwidth

3.3 Gating by coherence length

3.4 Distributed strain measurement by using an array of reflectors

3.5 Strain sensitivity and dynamic range

4. Conclusion

Funding

References and links

Cited By

Figures (8)

Equations (23)

Optics Express