Three-wavelength passive demodulation technique for the interrogation of EFPI sensors with arbitrary cavity length

Jingshan Jia; Yi Jiang; Hongchun Gao; Liuchao Zhang; Yuan Jiang

doi:10.1364/OE.27.008890

1. Introduction

The fiber optic extrinsic Fabry-Perot interferometric (EFPI) sensor is one of the most sensitive and compact sensors [1–5] and is an efficient solution for the measurement of dynamic signals [5–9]. As a key part of the sensing system, signal demodulation largely determines the performance of the entire system [9]. The demodulation technique for the dynamic signal, which extracts the cavity length change of the EFPI sensor, is different from the white light interferometry (WLI) technique for the absolute cavity length measurement [10]. Dynamic signal demodulation can be performed either by an active demodulation method [11] or by a passive demodulation method. An active demodulation method, such as the phase generated carrier (PGC) demodulation technique [11], requires a carrier signal, and the carrier frequency determines the upper frequency of the method [12]. A passive demodulation method, such as the linear demodulation technique [6,7], the quadrature phase-shifted demodulation based passive technique [8], [13], and the three-wavelength-based passive quadrature digital phase-demodulation technique [14], can be used for high-frequency measurements. However, the linear demodulation technique has the disadvantage of a limited dynamic range ( $< λ / 4$ ) and requires the sensor to be operated at a quadrature point [15,16]. For the quadrature phase-shifted demodulation based passive technique and the three-wavelength-based passive quadrature digital phase-demodulation technique, the quadrature condition is required; therefore, the cavity length of the EFPI and wavelengths of the light sources must be matched, and these techniques can only interrogate an EFPI sensor with a fixed cavity length [8,14,17]. Moreover, the quadrature phase-shifted demodulation based passive technique can be performed only if the DC component of the interferometric fringe is eliminated [8,13]. DC elimination can be achieved by measuring the DC value directly and regarding it as a fixed parameter [8,13] or by using a high-pass filter (HPF) [18]. However, if the phase modulation is smaller than $2 π$ , the DC value cannot be measured directly. An HPF is capable of eliminating the DC component only when the phase modulation is larger than $2 π$ .

To overcome these problems, we have proposed a phase compensation demodulation scheme [19] and a dual-wavelength DC compensation demodulation technique [17]. The frequency range of the two passive demodulation methods is limited only by the signal processing electronics. However, the phase compensation demodulation scheme is only capable when the phase modulation is larger than $2 π$ . For the dual-wavelength DC compensation demodulation technique, the influence of the DC component is eliminated, and two orthogonal signals are generated to recover the applied dynamic signal. However, the DC calculation is sensitive to the initial cavity length for this technique, which will lead to a deviation in the demodulated signal if the measurement accuracy of the initial cavity length is not good enough. Furthermore, the DC calculation is performed in advance, whereas the DC component of the interferometric fringe may have fluctuations caused by the bending loss of the leading fiber, which may result in unstable output signals. Moreover, the sensor may be modulated continuously if it has been installed, which leads to difficulties in the DC calculation [17].

In this paper, we present a three-wavelength passive demodulation technique for the interrogation of EFPI sensors with arbitrary cavity length. This technique is significantly more suitable for engineering applications and more reliable compared with demodulation techniques have been proposed. In this demodulation technique, three distributed feedback laser diodes (DFB-LD) with different wavelengths are introduced to calculate the DC component online. Orthogonal signals are generated by the DC compensation and the signal calibration. Then the applied dynamic signal is extracted. For the technique, the cavity length of the EFPI is not required to match the wavelengths of the light sources; therefore, theoretically, EFPI sensors with arbitrary cavity length can be demodulated. The demodulation technique is robust with respect to the bending loss of the leading fiber. It has properties of high frequency, a large dynamic range and high sensitivity.

2. Principle

Figure 1 illustrates the schematic of the three-wavelength passive demodulation technique for the interrogation of EFPI sensors with arbitrary cavity length. Light beams emitted from three DFB-LDs with different wavelengths, $λ_{1}$ , $λ_{2}$ , and $λ_{3}$ , are combined by a wavelength division multiplexer (WDM), WDM1. The EFPI sensor is illuminated by the combined light beam through a coupler. The reflected interference signal is split into three beams by means of another WDM, WDM2. Three interferometric signals at each wavelength are obtained by three photodiodes (PD), PD1, PD2, and PD3. These voltage signals are then acquired by an analog-digital converter (ADC). Digital signal processing is then carried out with a personal computer. The EFPI sensor, as shown in the inset of Fig. 1, is formed by placing a cleaved fiber end face in front of a mirror coupled to a piezoelectric transducer (PZT).

Fig. 1 Schematic of the three-wavelength passive demodulation technique for the interrogation of EFPI sensors with arbitrary cavity length.

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Three interferometric signals received by the ADC can be described by

f_{i} = A + B \cos (\frac{4 n π}{λ_{i}} d_{t}), (i = 1, 2, 3 ​ ​ ​),

where A is the DC component of the interferometric fringe, B is the interferometric fringe visibility, n is the effective refractive index of the EFPI cavity, here

n = 1

, and d_t is the cavity length modulated by the vibration signal. The vibration signal that is used to drive the EFPI is written as

y = D \cos (ω t + ϕ),

where D is the amplitude of the vibration signal,

ω

is the frequency of the vibration signal, and

ϕ

is the initial phase of the vibration signal. The cavity length d_t can then be expressed as

d_{t} = d_{0} + Δ d_{t} = d_{0} + k * y = d_{0} + k D \cos (ω t + ϕ),

where d₀ is the initial length of the cavity, which is measured by the WLI technique [10], ∆d_t is the vibration of the cavity length, and k is the sensitivity of the sensor. Then Eq. (1) can be expressed as

f_{i} = A + B \cos (θ_{t} + δ_{i}), (i = 1, 2, 3),

where

θ_{t} = 4 n π d_{t} / λ_{1}

,

δ_{1} = 0

,

δ_{2} = 4 n π d_{t} / λ_{2} - 4 n π d_{t} / λ_{1}

δ,

δ_{3} = 4 n π d_{t} / λ_{3} - 4 n π d_{t} / λ_{1}

. Since the vibration amplitude of the cavity length is much less than the initial length of the EFPI (

k D ≪ d_{0}

) in Eq. (3), then

d_{t} \approx d_{0}

, and we get

δ_{2} \approx 4 n π \frac{λ_{1} - λ_{2}}{λ_{1} λ_{2}} d_{0},

δ_{3} \approx 4 n π \frac{λ_{1} - λ_{3}}{λ_{1} λ_{3}} d_{0} .

Let

I_{1} = f_{2} - f_{1} = B (\cos δ_{2} - 1) \cos θ_{t} - B \sin δ_{2} \sin θ_{t},

I_{2} = f_{3} - f_{1} = B (\cos δ_{3} - 1) \cos θ_{t} - B \sin δ_{3} \sin θ_{t},

then

L = \sin δ_{3} I_{1} - \sin δ_{2} I_{2} = B (\sin δ_{3} \cos δ_{2} - \cos δ_{3} \sin δ_{2} + \sin δ_{2} - \sin δ_{3}) \cos θ_{t} .

Let

P = sin δ_{3} cos δ_{2} - cos δ_{3} sin δ_{2} + sin δ_{2} - sin δ_{3}

, where

δ_{2}

,

δ_{3}

are constants and are given by Eq. (5) and Eq. (6). Then

L = P B \cos θ_{t},

H = P f_{1} - L = P A .

The DC value, A, is derived by Eq. (11),

A = H / P .

Thus, the DC component, A, is calculated in real time, and the calculated value of A changes with the bending loss of the leading fiber. In this work, the DC calculation is associated with the phase difference,

δ_{2}

and

δ_{3}

; therefore, the influence of the initial cavity length error on the DC calculation is significantly decreased compared with the previously reported technique, which is associated with the initial phase,

θ_{0} = 4 n π d_{0} / λ_{1, 2}

[17],

Δ_{δ} = \frac{4 n π (λ_{i} - λ_{1})}{λ_{1} λ_{i}} Δ_{c l}, (i = 2, 3),

Δ_{θ} = \frac{4 n π}{λ_{j}} Δ_{c l}, (j = 1, 2),

Δ_{δ} / Δ_{θ} = \frac{(λ_{1} - λ_{i}) λ_{j}}{λ_{1} λ_{i}},

where

Δ_{c l}

is the initial cavity length error,

Δ_{δ}

is the error of

δ_{2, 3}

, and

Δ_{θ}

is the error of

θ_{0}

. The wavelength of the DFB-LDs we used is around 1550 nm. In our experiments,

λ_{i} - λ_{1} \leq 5.6 nm

, as shown in section 3 and 4. Then,

Δ_{δ} / Δ_{θ} \leq 3.6 \times 10^{- 3} .

Therefore, the DC calculation for this technique is more robust.

Three signals without the DC component are obtained by Eq. (4) and Eq. (12)

F_{i} = f_{i} - A = B \cos (θ_{t} + δ_{i}), (i = 1, 2, 3),

where

δ_{i}

has been calculated. Then, F₂ and F₃ can be expressed as

F_{2} = B \cos θ_{t} \cos δ_{2} - B \sin θ_{t} \sin δ_{2},

F_{3} = B \cos θ_{t} \cos δ_{3} - B \sin θ_{t} \sin δ_{3} .

Then [17],

P_{1} = \cos δ_{2} F_{1} - F_{2} = \sin δ_{2} B \sin θ_{t},

P_{2} = \sin δ_{2} F_{1} = \sin δ_{2} B \cos θ_{t} .

P₁ and P₂ are orthogonal to each other. The signal

θ_{t}

can be extracted from P₁ and P₂ with the differential cross multiplier (DCM) algorithm [20],

θ_{t} = \int \frac{P_{1}^{'} P_{2} - P_{1} P_{2}^{'}}{P_{1}^{2} + P_{2}^{2}} .

Then,

Δ d {}_{t}= \frac{λ_{1}}{4 n π} θ {}_{t}.

The signal ∆d_t can also be extracted from F₃ and F₁ with Eq. (20) to Eq. (23).

3. Simulation

To demonstrate the calculation principle of the DC component, we first conducted a numerical simulation. The parameters in Eq. (1) to Eq. (3) were set as in Table 1. The frequency of the input signal, d_t, was 1 kHz, and it was sampled at 100 kHz. The DC component, A, was extracted using Eq. (12). The value of A for different initial cavity lengths was measured with a step of 1μm. The simulation result of the DC component is shown in Fig. 2. The calculated DC components for initial cavity lengths from 100 μm to 300 μm resulted in an excellent fit to the input value of A (1 V). The simulation demonstrates the feasibility of the calculation principle of the DC component.

Table 1. Parameters of the Simulation

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Fig. 2 Simulation result for the DC component.

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To investigate the accuracy of this demodulation technique, the initial cavity length, d₀, was set to 150 μm, and the vibration amplitude of the cavity length, kD, was set to change from 1 nm to 500 nm with a step of 1 nm in a simulation. The frequency of the input signal, d_t, was 1 kHz, and it was sampled at 100 kHz. The other parameters were also set as in Table 1 in the simulation. The simulation result is shown in Fig. 3. The amplitude of the output signal shows good agreement with that of the input signal at different magnitudes, as shown in Fig. 3(a). Figure 3(b) displays the amplitude deviation of the output signal from the input signal. The amplitude deviation is still less than 4 nm when the vibration amplitude reaches 500 nm ( $Vpp = 1 μm$ ). The simulation demonstrates that the demodulation technique can accurately extract the applied signal from an EFPI.

Fig. 3 (a) The output signal amplitude as a function of the input signal amplitude. (b) The amplitude deviation of the output signal from the input signal.

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4. Experiment

The calculation principle of the DC component was then experimentally demonstrated. A demodulator based on this technique was used in experiments as shown in Fig. 1. The wavelengths of the DFB-LDs are 1540.30 nm ( $λ_{1}$ ), 1550.12 nm ( $λ_{2}$ ), and 1552.90 nm ( $λ_{3}$ ). The power of each DFB-LD is 10 mW. The sampling frequency of the ADC is 160 kHz per channel. A series of EFPI sensors with different cavity lengths were tested. The DC component can be measured directly by PDs only if the phase modulation is larger than 2π, so the EFPI was driven by a sinusoidal signal with a large amplitude. The measured DC component is half of the peak-to-peak amplitude added with the minimum value of the interferometric signal, which was compared with the calculated DC component of Eq. (12). The result is shown in Table 2, and the measured DC components are consistent with the calculated DC components.

Table 2. Measurement and Calculation of the DC Components

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To evaluate the properties of the demodulation technique, EFPI sensors with different cavity lengths were tested at different driving signals. These sensors were interrogated by the same demodulator. An EFPI with a cavity length of 129.28 μm was driven with a 100 Hz sinewave. Three interferometric signals, f₁, f₂, and f₃, sampled with the ADC are shown in Fig. 4(a). The Lissajous figure for f₁ and f₂, the Lissajous figure for f₂ and f₃, and the Lissajous figure for f₁ and f₃ are shown in Fig. 4(b). Figure 4(b) shows that the phase modulation is larger than $2 π$ . Figure 4 confirms that the quadrature condition is not required for this demodulation technique, since Lissajous figures are not circles [20]. This demodulator was also used to interrogate a 700 Hz sinusoidal signal loaded on a 149.33 μm EFPI and a 2 kHz sinusoidal signal loaded on a 242.47 μm EFPI. The demodulated results are shown in Fig. 5. Figure 5(a) is the output signal d_t demodulated from the EFPI with a cavity length of 129.28 μm. Figure 5(b) is the output signal demodulated from the EFPI with a cavity length of 149.33 μm. Figure 5(c) is the output signal demodulated from the EFPI with a cavity length of 242.47 μm. Power spectrum plots of these demodulated signals are shown in Fig. 6. Frequencies of demodulated signals are 100 Hz, 700 Hz, and 2 kHz, respectively. The frequency of the demodulated result is consistent with that of the sinusoidal driving signal, which means vibration signals loaded on EFPIs with different cavity lengths are demodulated successfully by the demodulator. A signal-to-noise ratio (SNR) of approximately 50 dB is observed at 700 Hz. The Lissajous figure for f₁ and f₂, the Lissajous figure for f₂ and f₃, and the Lissajous figure for f₃ and f₁ were recorded when the 242.47 μm EFPI was driven with a 2 kHz sinusoidal signal, as shown in Fig. 7. From Fig. 7, the phase modulation of the signal loaded on the 242.47 μm EFPI is smaller than $2 π$ . However, the applied signal is extracted perfectly. Experimental results demonstrate that the demodulator can accurately extract an applied signal from EFPIs with different cavity lengths, regardless of whether the phase modulation is larger than $2 π$ .

Fig. 4 (a) Three interferometric signals of the 129.28 μm EFPI working at 100 Hz, (b) the Lissajous figure for f₁ and f₂, the Lissajous figure for f₂ and f₃, and the Lissajous figure for f₁ and f₃.

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Fig. 5 Demodulated results: (a) the 100 Hz output signal demodulated from the 129.28 μm EFPI, (b) the 700 Hz output signal demodulated from the 149.33 μm EFPI, (c) the 2 kHz output signal demodulated from the 242.47 μm EFPI.

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Fig. 6 Power spectrum plots of demodulated signals.

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Fig. 7 Lissajous figures of the 242.47 μm EFPI working at 2 kHz.

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The minimum cavity length of an EFPI detected by the demodulator was 22.96 μm, and the maximum cavity length of an EFPI detected by the demodulator was 1.0023 mm, as shown in Fig. 8. Figure 8(a) shows the three interferometric signals received by the ADC when the 22.96 μm EFPI was driven with a 200 Hz sinewave, and Fig. 8(b) shows the output signal detected by the demodulator. Figure 8(c) shows the three interferometric signals received by the ADC when the 1.0223 mm EFPI was driven with a 200 Hz sinewave, and Fig. 8(d) shows the output signal detected by the demodulator. The frequency of the demodulated signal is consistent with that of the driving signal. It is clear that the demodulator can successfully extract an applied signal from an EFPI in the cavity length range of 22.96-1002.3 μm.

Fig. 8 (a) Three interferometric signals of the 22.96 μm EFPI working at 200 Hz, (b) the output signal demodulated from the 22.96 μm EFPI, (c) three interferometric signals of the 1.0023 mm EFPI working at 200 Hz, (d) the output signal demodulated from the 1.0023 mm EFPI.

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The independence of the demodulation technique to the bending loss of the leading fiber was experimentally demonstrated. An EFPI with a cavity length of 135.68 μm was driven with a 200 Hz sinewave, and a part of the leading fiber was curved with different curvature angles. The experimental results are shown in Fig. 9. The interferometric fringe visibility B and the DC component A of $f_{1, 2, 3}$ decrease as the curvature angle is increased, as shown in Fig. 9(a). However, the output signals are unaffected by the bending loss of the leading fiber, as shown in Fig. 9(b). Experimental results demonstrate that the technique is robust with respect to the bending loss of the leading fiber.

Fig. 9 (a) DC components and interferometric fringe visibilities of signals received by the ADC, (b) output signals.

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5. Conclusion

In conclusion, a three-wavelength passive demodulation technique for the interrogation of EFPI sensors with arbitrary cavity length has been demonstrated. For the technique, the cavity length of the EFPI is not required to match the wavelengths of the light sources. The DC component is measured and compensated online, and orthogonal signals are generated using a signal calibration algorithm. The demodulation technique can extract dynamic signals, regardless of whether the phase modulation is larger than $2 π$ . Theoretically, EFPI sensors with arbitrary cavity length can be demodulated by the demodulation technique, and EFPI sensors with cavity lengths in the 22.96-1002.3 μm range are detected successfully by the same demodulator in experiments. Experiment results demonstrate the potential of this technique for the measurement of dynamic signals in different environments. The demodulation technique provides a robust and accurate method to measure dynamic signals for EFPI sensors.

Funding

National Key R&D Program of China (2018YFB1107200); 863 Program of China (2015AA043504); National Natural Science Foundation of China (NSFC) (61575021, 61775020).

References

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Cavity length (μm)	Measurement (V)	Calculation (V)
126.66	0.678	0.675
129.28	0.691	0.699
149.33	0.603	0.591
180.23	0.502	0.491
190.98	0.429	0.425
239.77	0.424	0.420
242.47	0.399	0.400

Cavity length (μm)	Measurement (V)	Calculation (V)
126.66	0.678	0.675
129.28	0.691	0.699
149.33	0.603	0.591
180.23	0.502	0.491
190.98	0.429	0.425
239.77	0.424	0.420
242.47	0.399	0.400

Three-wavelength passive demodulation technique for the interrogation of EFPI sensors with arbitrary cavity length

Abstract

1. Introduction

2. Principle

3. Simulation

4. Experiment

5. Conclusion

Funding

References

Cited By

Figures (9)

Tables (2)

Equations (23)

Optics Express

Symbol	Value	Description
A	1 V	the DC component
B	0.8 V	the interferometric fringe visibility
n	1	the effective refractive index
λ₁	1547.30 nm	wavelength
λ₂	1550.12 nm	wavelength
λ₃	1552.90 nm	wavelength
d₀	100 to 300 μm	the initial length of the cavity
kD	400 nm	the change of the cavity length
ϕ	0	the initial phase