Quadrature phase-shifted optical demodulator for low-coherence fiber-optic Fabry-Perot interferometric sensors

Bing Qi; Drew E. Winder; Yun Liu

doi:10.1364/OE.27.007319

1. Introduction

A fiber-optic extrinsic Fabry-Perot Interferometer (EFPI) is a highly sensitive, versatile sensing element which has been used to measure various physical quantities, such as vibration, sound, pressure, and strain [1–3]. A typical fiber-optic EFPI-based sensor can be constructed by bonding two pieces of optical fibers in a glass tube. The two flat ends of the optical fibers form an etalon with a gap sensitive to the measurand. The measurement of physical quantities is realized by tracking the change of the gap of the EFPI. The fiber-optic EFPI-based sensors have been widely used in a variety of applications owing to their robustness against harsh environment and immunity to electromagnetic field [4–6].

A variety of demodulation schemes have been developed to recover the signal from a fiber-optic EFPI-based sensor [7–15]. When a highly coherent source is employed (the coherence time of the light source is larger than the round-trip time of the sensor gap), reflections from the two end-faces of the sensor head will directly interfere with each other. Information encoded on the gap of the sensor will be transformed into the intensity of the reflected light and can be read out by using a photodetector. While this method can provide high speed and high resolution, its dynamic range is limited to a fraction of the wavelength of the source, due to the periodic nature of the interference signal. To achieve a large dynamic range and resolve the absolute value of the sensor gap, spectroscopic methods are commonly applied by using a broadband light source and a spectrometer, or a tunable laser source and a single detector [10–13]. The absolute value of the sensor gap can be determined from the spectrum, for example, by performing a Fourier transformation [11]. The limitations of the spectroscopic method include system complexity and slow measurement speed.

To achieve high-speed demodulation with a large dynamic range, quadrature phase-shifted demodulation scheme has been proposed and demonstrated [16–20]. This scheme is especially useful in applications where the primarily interested parameter (such as displacement, strain, pressure) is the change of the senor gap rather than its absolute value. Previously, we proposed and demonstrated a high-speed quadrature phase-shifted demodulation scheme by using a free-space Fizeau interferometer to interrogate the reflected signal from an EFPI-based strain sensor [21]. While large measurement range (15 µm) and fast response (100 kHz) have been demonstrated in vibration measurement, the achieved measurement resolution of the EFPI sensor gap (7 nm) is nevertheless limited by the stability and the low power efficiency of the free-space interferometer [21].

In this paper, we propose and demonstrate a high-precision, large dynamic range, fast quadrature phase-shifted demodulation scheme for fiber-optic EFPI-based sensors by using a fiber-optic Mach-Zehnder Interferometer (MZI) constructed with a 1 × 2 fiber coupler and a 3 × 3 fiber coupler. The fiber-optic MZI has symmetric optical paths and the path length difference is tunable to match the gap of the EFPI sensor. Thanks to the all-fiber design, the demodulator presents great long-term stability, as demonstrated by running the system continuously over 11 days. Actual measurements of vibrations are conducted. Both a large dynamic range of 45 dB and a high resolution of 0.67 nm are obtained. Note that while a 3 × 3 fiber coupler has been used to demodulate the signals from a fiber MZI previously [20], the present work is the first to apply this scheme in the demodulation of fiber-optic EFPI-based sensors.

2. Theory

Figure 1 shows a schematic diagram of the proposed phase-shifted demodulation scheme. The continuous wave output from a low-coherence light source is coupled to an EFPI sensor by a fiber circulator (or alternatively, a fiber coupler). Since the coherence length of the light source is much shorter than twice of the gap of the Fabry-Perot cavity in the sensor, no direct inference is expected between the two reflected light waves. The light reflected from the two end-faces of the EFPI sensor is detoured to the demodulator by the same fiber circulator. The main part of the demodulator is an MZI constructed with a 1 × 2 fiber coupler and a 3 × 3 fiber coupler. The path length difference of the MZI is adjusted to match with the zero point of the EFPI sensor, i.e., twice of the gap distance of the Fabry-Perot etalon when no external force is applied to the sensor. The output lights from the 3 × 3 fiber coupler are detected with three photodetectors.

Fig. 1 Schematic diagram of fiber-optic phase-shifted demodulator for a low-coherence fiber-optic EFPI-based sensor. SLD: superluminescent laser diode, FC: fiber coupler, VDL: variable delay line, PS: phase shifter, PC: polarization controller, PD: photodetector.

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We first examine the resulting light intensities from each photodetector. To start with, we denote the complex electrical field of the reflected light from the EFPI as:

E_{r} (t) = r_{1} E (t) + r_{2} E (t - T_{S}),

where

E (t)

is the electric field of the light source, and we denote

I (t) = 〈 {| E (t) |}^{2} 〉

the optical intensity of the incoming light to the sensor, T_S is the round-trip time of light in the gap of the EFPI sensor, r₁ and r₂ are (amplitude) reflectivities of the first and second end-faces of the EFPI, respectively. Without loss of generality, r₁ and r₂ are assumed to be real. Furthermore, both r₁ and r₂ are assumed to be much less than one, so we neglect multiple reflections.

As shown in Fig. 1, the reflected light from the sensor is equally split into two paths through a 1 × 2 fiber coupler. A relative path delay difference of T_d is introduced by using fiber-optic delay lines. The two beams are then recombined at a 3 × 3 fiber coupler. Here, we calculate the intensities of the lights received by the three detectors by using the transfer matrix of a symmetric 3 × 3 fiber coupler given in [22]:

M = [\begin{matrix} a & b & b \\ b & a & b \\ b & b & a \end{matrix}],

where

a = \frac{e^{- j ϕ} + 2 e^{j ϕ}}{3}

,

b = \frac{e^{- j ϕ} - e^{j ϕ}}{3}

and ϕ is the coupling coefficient of the fiber coupler and equals to π/3 for a lossless symmetric 3 × 3 coupler. The electric fields of the light output from the 3 × 3 fiber coupler can be calculated from

[\begin{matrix} E_{D 1} (t) \\ E_{D 2} (t) \\ E_{D 3} (t) \end{matrix}] = M [\begin{matrix} 0 \\ \sqrt{C_{1}} E_{r} (t) \\ - j \sqrt{C_{2}} E_{r} (t - T_{d}) \end{matrix}] = [\begin{matrix} b \sqrt{C_{1}} E_{r} (t) - j b \sqrt{C_{2}} E_{r} (t - T_{d}) \\ a \sqrt{C_{1}} E_{r} (t) - j b \sqrt{C_{2}} E_{r} (t - T_{d}) \\ b \sqrt{C_{1}} E_{r} (t) - j a \sqrt{C_{2}} E_{r} (t - T_{d}) \end{matrix}],

where C₁ and C₂ represent optical power ratios in the two beam paths of MZI. For simplicity, we assume a 50/50 beam splitting ratio, i.e., C₁ = C₂ = 0.5. The effect of C₁ and C₂ will be discussed at the end of this section. In this case, the light intensity received by the first detector is therefore given by

I_{D 1} (t) = 〈 {| E_{D 1} (t) |}^{2} 〉 = {| b |}^{2} I_{r} (t) + \frac{1}{2} {| b |}^{2} [〈 j E_{r} (t) E_{r}^{*} (t - T_{d}) 〉 - 〈 j E_{r}^{*} (t) E_{r} (t - T_{d}) 〉] .

Here

I_{r} (t) = 〈 {| E_{r} (t) |}^{2} 〉 = 〈 {| E_{r} (t - T_{d}) |}^{2} 〉

is the optical intensity of the reflected light from the EFPI sensor (the light source is assumed to be stationary), T_d is the delay time difference between the two paths of the MZI, and

〈 E_{r} (t) E_{r}^{*} (t - T_{d}) 〉 = (r_{1}^{2} + r_{2}^{2}) Γ (T_{d}) + r_{1} r_{2} Γ (Δ T) + r_{1} r_{2} Γ (T_{d} + T_{S}),

where

Δ T = T_{d} - T_{S}

and

Γ (x) = 〈 E (t + x) E^{*} (t) 〉

is the field autocorrelation function of the light source. For a light source with a Gaussian spectrum, we further assume

Γ (x) = \exp (- \frac{2 x^{2}}{τ_{c}^{2}} - j ω x) I (t)

where ω is the angular frequency and τ_c is the coherence time of the light source.

Since we only focus on the signal demodulation scheme based on the low-coherence interferometry, two assumptions are made in this paper: i) the coherence time of the light source is much shorter than the round-trip time of the sensing EFPI, i.e., τ_c << T_s; and ii) the optical path difference of the local MZI in the demodulator matches with that of the sensing EFPI, i.e.,

| Δ T | = | T_{d} - T_{S} | < τ_{c} .

Note, Eq. (6) places a constraint on the maximum measurement range of a demodulator with a fixed path delay difference.

Under the above conditions, the only term in Eq. (5) that does not disappear is $r_{1} r_{2} Γ (Δ T)$ . Therefore, Eq. (4) can be rewritten as

\begin{array}{l} I_{D 1} (t) = {| b |}^{2} I_{r} (t) + \frac{1}{2} {| b |}^{2} r_{1} r_{2} [j Γ (Δ T) - j Γ (- Δ T)] \\ = {| b |}^{2} (r_{1}^{2} + r_{2}^{2}) I (t) [1 + k \exp (\frac{- 2 Δ T^{2}}{τ_{c}^{2}}) \sin (ω Δ T)], \end{array}

where

k = \frac{r_{1} r_{2}}{r_{1}^{2} + r_{2}^{2}}

is an amplitude coefficient determined by the EFPI sensor. Note the relation

I_{r} (t) = (r_{1}^{2} + r_{2}^{2}) I (t)

is used in the derivation of Eq. (7). We remark that the maximum achievable interference visibility in Eq. (7) is limited to 0.5 (when

r_{1} = r_{2}

and

Δ T = 0

). This is due to the constant background contributed by non-interference lights (light reflected by the first end-face of the EFPI sensor and passing the shorter arm of the MZI, and light reflected by the second end-face of the sensor and passing the longer arm of the MZI).

The lights received by the second and the third detector can be calculated in a similar way

\begin{array}{l} I_{D 2} (t) = 〈 {| E_{D 2} (t) |}^{2} 〉 \\ = \frac{1}{2} ({| a |}^{2} + {| b |}^{2}) (r_{1}^{2} + r_{2}^{2}) I (t) [1 + m k \exp (\frac{- 2 Δ T^{2}}{τ_{c}^{2}}) \sin (ω Δ T - Δ φ)], \end{array}

\begin{array}{l} I_{D 3} (t) = 〈 {| E_{D 3} (t) |}^{2} 〉 \\ = \frac{1}{2} ({| a |}^{2} + {| b |}^{2}) (r_{1}^{2} + r_{2}^{2}) I (t) [1 + m k \exp (\frac{- 2 Δ T^{2}}{τ_{c}^{2}}) \sin (ω Δ T + Δ φ)], \end{array}

where

Δ φ = \arg (a b^{*})

is the phase shift between different output ports of the 3 × 3 fiber coupler, and

m = \frac{2 | a b |}{{| a |}^{2} + {| b |}^{2}}

is referred to the modulation depth of detector outputs 2 and 3 with respect to that of detector output 1. By continuously sampling the outputs of the three detectors, the phase change of the EFPI sensor can be determined unambiguously, as shown in our previous work [21]. For a fixed path length difference in the demodulator, the non-ambiguity range is limited only by the coherence length of the light source. This allows us to further determine the change of the gap of the EFPI and thus the physical quantity to be measured.

Note that Δφ and m are fully determined by the 3 × 3 fiber coupler. For an ideal symmetric 3 × 3 coupler, Δφ equals 2π/3 and m equals 1. Figure 2 shows the simulation results of phase shift Δφ and modulation depth m as functions of the fiber coupler coefficient ϕ. Clearly, even when ϕ is varied around ± 20% of its ideal value (π/3), we have the values of modulation depth m larger than 0.8 and the phase shift Δφ only slowly changes around 2π/3. This suggests that our demodulation scheme can tolerate relatively high fabrication errors of the 3 × 3 coupler.

Fig. 2 Simulation results of phase shift Δφ (in unit of π) and modulation depth m of demodulator as functions of the coupling coefficient of a 3 × 3 fiber coupler.

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We have studied the effects of power splitting ratios in MZI by including C₁ and C₂ in the modeling. Specifically, we investigated the variation of parameters Δφ, κ, and m in Eqs. (7)-(9) as a function of the ratio C₂/C₁. The detailed derivation is omitted but the main results are summarized in the following. (i) The phase difference Δφ is independent on C₂/C₁. (ii) The amplitude coefficient κ in Eq. (7) shows a very weak dependence on C₂/C₁. In fact, κ changes less than 1% when C₂/C₁ changes from 1 to 0.8. (iii) The values of the modulation depth m obtained from Eq. (8) and Eq. (9) will be different for C₂/C₁≠1. However, the difference is generally very small within a quite large range of C₂/C₁ and Δφ. For example, for C₂/C₁ ≥ 0.8 and 115° ≤ Δφ ≤125°, the difference is within 4%. The analysis is verified by our experimental results described in the next section.

3. Experimental results

We conduct experiments to evaluate the performance of the proposed demodulation scheme. A superluminescent laser diode (Thorlabs S5FC1018S) with a center wavelength of 1320 nm and a spectral bandwidth of 40 nm is employed as the low-coherence light source (Fig. 1). Its coherence length is calculated to be about 30 µm. The light from the source is coupled to a fiber-optic vibration sensor that is essentially an EFPI formed by a polished end-face of an incoming single-mode fiber and a vibrating reflector (with a fixed vibration frequency of 3.5KHz). The default gap is typically set at 0.1 mm which yields a round-trip time T_s of 0.66 ps. The light reflected from the EFPI sensor is coupled into the fiber MZI through the fiber circulator. Identical components are placed in both optical paths of the MZI to reduce its temperature dependence: each path consists of a compact tunable fiber delay line (General Photonics VDL-004, 130 ps tuning range) and a fiber phase shifter (General Photonics FPS-001). By adjusting the tunable delay lines, we match the path length difference of the MZI to the gap of the sensor. An in-line fiber optic polarization controller (Thorlabs CPC900) is directly applied to the ϕ900-µm tight-buffer fiber in one path of MZIs. All fiber components are commercially available single-mode products and no polarization maintaining fibers are used. The symmetric configuration of MZI greatly reduces path length/polarization instabilities due to temperature change or mechanical noises. The fiber phase shifter is used only for the system calibration purpose, as described below. The fiber-coupled photodiodes are followed by a 3-channel custom-designed transimpedance amplifiers with a gain about 10⁵ V/W and a bandwidth of 1MHz.

3.1. Long-term stability of the demodulator

We first estimate values of the key parameters (e.g. Δφ, κ, m) in Eqs. (7)-(9) from the outputs of three photodetectors. In this experiment, the gap in the vibration sensor remains fixed. One of the phase shifters in the demodulator is driven by a periodic electrical signal, which introduces a phase oscillation larger than 2π. Figure 3(a) shows the output waveforms from three detectors. By plotting the detector outputs against one another, we obtain Lissajous curves in Figs. 3(b) and (c). The curves take elliptic profiles and the phase shift between each pair can be calculated from the respective ellipse. Experimentally, the phase shift Δφ between channel 1 and 2, and the one between channel 2 and 3, have been determined to be 121.21° and 118.74° respectively, which are very close to the theoretical value of 120°.

Fig. 3 (a) Raw signals from 3 photodetectors, (b) Lissajous plot of I_D2 versus I_D1 output, (c) Lissajous plot of I_D3 versus I_D2. The detector outputs are normalized in (b) and (c).

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The value of parameter k in Eq. (7) is estimated from the visibility of I_D₁. The k value determined from I_D₁ is 0.379, which is lower than the ideal value of 0.5. We consider that the deviation is mainly due to the unequal reflectivities (i.e., r₁ ≠ r₂) from the two surfaces in the EFPI sensor. Nevertheless, since we perform system calibration by scanning the phase shift over 2π range to determine the maximum and minimum of the interfere signal, our demodulation scheme is not sensitive to the absolute value of k. The values of m in Eqs. (8) and (9) are estimated from I_D₂ and I_D₃ using the value of k. Experimentally, the values of m estimated from I_D₂ and I_D₃ are 0.985 and 0.961, respectively. Note that the difference between the two values is only about 2%, which verified an excellent symmetry of the MZI. We have evaluated the Allan variance of the phase shift over different measurement time periods. The calculated Allan deviation is proportional to 1/τ (τ: sampling time period), which indicates that the flicker noise of the photodetector is the main noise source.

We further investigate the stability of the optical demodulator by continuously running the above measurement over 11 days. The detector outputs are sampled at a 1-minute step. Figure 4(a) shows the two phase-shift values which are calculated between I_D₂ and I_D₁, and between I_D₃ and I_D₂, respectively. Figure 4(b) shows the variations of the parameters κ and m. All parameters show very small drifts over the test period of 11 days. Note that the demodulator is installed in an environment with no temperature stabilization or vibration isolation, our measurement results demonstrate that the demodulation scheme is very robust against the ambient temperature or mechanical instabilities. In addition, the value of modulation depth m is very close to unity and the measured phase shifts are close to 120°. The maximum phase shift error is 2°, which shows that the 3 × 3 fiber coupler used in the experiment is highly symmetric. On the other hand, since our phase recovery algorithm works on arbitrary phase shift, the individual phase shift values will not affect the measurement result.

Fig. 4 Measured parameters Δφ, κ, m from the fiber-optic demodulator over 11 days.

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3.2. Linearity of the demodulator

In our previous work [21], we proposed a digital demodulation algorithm and demonstrated phase recovery using a pair of photo detectors implemented in a free-space local interferometer. The present optical processor provides better performance than the previous one thanks to the compact all-fiber MZI design: the phase shift between the detectors is extremely stable and the overall optical transmission efficiency is significantly improved. Therefore, the same digital algorithm is applied to the present setup for phase demodulation. To this end, two phase-shifted photodetector outputs are connected to a data acquisition module (National Instrument NI 5751-B, 50 MS, 14 bit) and the digitized signals are processed in a computer to calculate the phase change in the sensor.

To determine the linearity of our demodulator, we mount the single mode fiber of the EFPI sensor on a compact Nano-positioner stage driven by a piezoelectric actuator (Thorlabs NF15AP25). This allows us to control the sensor gap precisely. The motion of the stage is controlled by the voltage applied to the piezoelectric actuator, which has a pre-calibrated slope of 185.6 nm/V over the driving voltage range of 7.5 – 60 V. Limited by the speed of the piezoelectric actuator, the modulation frequency in this experiment is set to 100 Hz. Figure 5 shows the measured vibration amplitude (determined by our demodulator) as a function of the amplitude of the driving signal on the piezo actuator. The standard deviation of each measured point is within 0.1% of the nominal value. A good linear fit has been achieved from the measurement results, and the measured slop of 183 nm/V matches well with the pre-calibrated value.

Fig. 5 Measured vibration amplitude versus driving voltage of the piezoelectric actuator. Dots are measurement data and line is the linear fit. The measured slop of 183 nm/V matches well with the pre-calibrated value of 185.6 nm/V.

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3.3. Sensitivity of the demodulator

In this experiment, the sensor gap is modulated by driving the vibrating reflector (piezo-driven buzzer) of the EFPI sensor. The vibration frequency is around 3.5 kHz, and the vibration amplitude is controlled by the amplitude of the driving signal. Figures 6(a) and 6(b) show an example of the measurement result of the vibrations generated from the buzzer at an applied voltage of 4 V. Figure 6(a) shows waveforms directly output from the photodetectors. A total phase change of about 38π can be found from the waveforms. Due to the problem of ambiguous phase discrimination, such a large displacement cannot be determined from the output of a single detection channel. Using the quadrature phase-shifted optical demodulator, we can successfully recover the vibration waveform shown in Fig. 6(b). The vibration amplitude (peak-to-peak) shown in Fig. 6(b) is 12.3 µm. We repeated the same measurement more than 100 times. The measured vibration amplitude shows a very high repeatability with a variation less than 0.1%.

Fig. 6 Measurement results of 3.5 kHz vibrations. (a) Output signals from PD1 (top) and PD2 (bottom). For better visualization, the PD2 output waveform is shifted vertically. (b) Demodulated vibration waveform. The vibration amplitude (peak-to-peak value) is 12.3 µm.

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The dynamic range of the demodulator is defined as the ratio of the maximum vibration amplitude that can be accurately demodulated to the minimum measurable amplitude for a given bandwidth. The maximum measurable amplitude is limited by the coherence length of the light source (which is about 30 µm in our setup). In the current experiment, we obtained a maximum measurable amplitude of 18 µm at a bandwidth of 10 Hz with a very small distortion in the waveform. The minimum measurable amplitude is limited by many factors, such as the bandwidth of the electronics, the noise of the detectors, and the stability of the optical processor. To eliminate the effect from instabilities of the EFPI sensor itself, we experimentally investigate the minimum measurable amplitude by fixing the sensor gap at 0.1mm and using the phase shifter in the demodulator setup to introduce small path length changes. Figures 7(a) and 7(b) show the measured waveform induced by the phase shifter modulated at 10 mV (peak to peak) at 1-Hz. The detector output signals are processed through a 10-Hz low-pass filter. Small but clear 1-Hz oscillation waveforms are obtained from both detectors as shown in Fig. 7(a). The recovered waveform from the optical demodulator is shown in Fig. 7(b). The amplitude of the oscillation is about 0.67 nm which agrees well with the specification value of the phase shifter (2π/10V or 0.65 nm/V at the wavelength of 1320 nm). Clearly, a vibration amplitude of 0.6 nm can be easily resolved with our demodulator at this bandwidth. Therefore, a conservative estimation of the dynamic range of the current demodulator is about 45 dB (0.6 nm over 18 µm range) at a bandwidth of 10 Hz.

Fig. 7 Measurement results of 1-Hz vibrations. Signal is processed at 10-Hz bandwidth. (a) Output signals from PD1 (top) and PD2 (bottom). (b) Demodulated vibration waveform with a peak-to-peak amplitude of 0.67 nm.

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The maximum measurement bandwidth is limited by the response of the photodetector and the speed of the data acquisition system. The detectors used in the current experiment have a 1 MHz bandwidth and a ~10⁵ V/W gain. Recently, we demonstrated the measurement of fast dynamic strain waveforms with a rising edge of 10 µs using the similar data acquisition module with a free-space optical demodulator [23]. It is noted that the gain-bandwidth product of the photodetectors used in the current fiber-optic demodulator has been increased by more than four times compared to the one used in [23]. Therefore, we expect the measurement bandwidth to be well beyond 100 kHz. Measurement results of high-frequency strain pulses on a Spallation Neutron Source mercury target will be reported elsewhere.

Finally, we briefly comment on a variation of the current demodulator design. The MZI in the demodulator can be replaced by a Michelson interferometer where the reflected lights from the EFPI sensor are directly connected to one input port of the 3 × 3 fiber coupler and split into three beams. Two output beams from the coupler are reflected by using fiber Faraday mirrors and the reflected light beams interfere with each other at the same 3 × 3 coupler. The phase-shifted interference signals can be obtained at the two unused input ports of the coupler. The fiber coupler and two Faraday mirrors formed a Michelson interferometer and a variable path difference can be introduced by inserting fiber delay lines and phase shifters in the two arms of the interferometer. The same phase demodulation algorithm presented in this paper can be applied directly in the Michelson setup. We have experimentally investigated the Michelson scheme. Similar quadrature phase-shifted detection has been confirmed. The advantage of the Michelson interferometer scheme is that no polarization controller is needed since the polarization rotations due to single mode fibers are automatically compensated by using Faraday mirrors. The drawbacks of the Michelson scheme are that the optical power efficiency is dropped by a factor of three and the path length is doubled, which makes it more sensitive to the ambient temperature/mechanical instabilities.

4. Conclusions

In conclusion, we have designed and implemented an all-fiber optical demodulator for low-coherence Fabry-Perot-based fiber-optic sensors. The demodulator provides a quadrature phase-shifted detection scheme that has high flexibility to sensors with a wide range of gaps. Over an experiment period of 11 days, the phase shift generated from the demodulator varies within 0.5°, and the detector output fluctuations are within 0.3%. The demodulator has been applied to vibration measurements and the results demonstrated successful recovery of vibrations with amplitudes up to 18 μm and a dynamic range of 45 dB. Such systems when combined with Fabry-Perot strain gauges allow for measurement of the dynamic response of structures in particle accelerators, where electrical resistance strain gauges can suffer from electro-magnetic interference. Other applications of Fabry-Perot cavities are as pressure and temperature sensors.

Funding

U.S. Department of Energy Office of Science (DE-AC05-00OR22725).

Acknowledgments

We acknowledge A. Webster, S. Murray III, and C. Long for their technical help, and M. Wendel and A. Aleksandrov for their support.

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Quadrature phase-shifted optical demodulator for low-coherence fiber-optic Fabry-Perot interferometric sensors

Abstract

1. Introduction

2. Theory

3. Experimental results

3.1. Long-term stability of the demodulator

3.2. Linearity of the demodulator

3.3. Sensitivity of the demodulator

4. Conclusions

Funding

Acknowledgments

References

Cited By

Figures (7)

Equations (9)

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