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Abstract
A symmetrical demodulation method is developed for the recovery of dynamic signals. Extrinsic Fabry–Perot interferometers (EFPIs) with different cavity lengths can be interrogated by a same demodulator. In the demodulation technique, three interferometric signals are introduced by selecting three specified laser wavelength, two of the three signals are symmetrical about the third signal. The dynamic signal is recovered by the proposed method from the three interferometric signals. EFPI sensors in a wide cavity length range (>1000 µm) can be demodulated without dead zone. The calculated amplitude error of the demodulated signal is less than 0.25% with the cavity length in the 20-1005 µm range. The proposed demodulation technique is adapted to the measurement of EFPIs with unsteady cavity lengths and unknown cavity lengths.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Fiber optic extrinsic Fabry–Perot interferometers (EFPIs) are desirable for the measurement of vibrations [1–4] or acoustics [5–8] since they are compact, temperature insensitive, high resolution, high sensitivity, and immune to electromagnetic interference. One method to recover a dynamic signal is the linear demodulation method [1–3,6–7], the change of a cavity length is converted into the optical power change of a interferometric signal by a linear region of the interference fringe [1,9]. The linear demodulation method is concise, straightforward, and low-cost [10]; however, the linear demodulation method can be performed only if the dynamic range of the sensor is smaller than $\lambda /4$, $\lambda $ is the laser wavelength [1,10]. Moreover, the orthogonal working point is always drift since the cavity length of the EFPI is affected by fluctuations of the temperature, thus, a static working point stabilizing technology is required to compensate for the orthogonal working point drift [1,7]. EFPI sensors are also demodulated by the orthogonal signals based passive homodyne demodulation technique [11–13]. For the orthogonal signals based demodulation technique, the cavity length of the sensor must be matched with wavelengths of the lasers to obtain two orthogonal signals [11–13]. Therefore, a demodulator based on the technique is only suitable for the demodulation of an EFPI with a specific cavity length [11–13]. Moreover, for the technique, the DC compensation must be performed, which is difficult if the phase shift of the sensor is smaller than $2\pi $ [11–13].
A dual-wavelength DC compensation demodulation technique [14] and a three-wavelength passive demodulation technique [15] have been proposed to interrogate EFPIs with different phase modulations and different cavity lengths. For the two demodulation techniques, phase differences between different interferometric signals with different wavelengths are calculated using wavelengths of the laser sources and the initial cavity length of the EFPI [14,15]. Then, these phase differences are used to perform the DC compensation and quadrature signals reconstruction [14,15]. The dynamic signal is extracted from the quadrature signals [14,15]. Thus, for the two demodulation techniques, the initial cavity length of an EFPI must be measured before demodulation. Therefore, the output will deviate from the measurand if the cavity length is changed caused by the fluctuation of the temperature.
In this paper, we develop a symmetrical demodulation method for the measurement of dynamic signals. For the symmetrical demodulator, three interferometric signals are introduced by selecting three specified laser wavelength, two of the three signals are symmetrical about the third signal. The measurand is recovered by the demodulator from the three interferometric signals. Unlike the dual-wavelength DC compensation demodulation technique [14] and the three-wavelength passive demodulation technique [15], EFPIs with different phase modulations and different cavity lengths can be interrogated by the demodulation technique without measuring the initial cavity lengths. The proposed demodulation technique is adapted to the measurement of EFPIs with unsteady cavity lengths and unknown cavity lengths. The demodulation technique is simple and suitable for practical engineering applications.
2. Principle
A schematic of the symmetrical demodulation system is shown in Fig. 1. Three laser diodes are used to produce three interferometric signals, wavelengths of which are ${\lambda _1}$, ${\lambda _2}$, and ${\lambda _3}$, respectively. The three light beams are injected into the sensor by a coupler and a circulator. Three interferometric signals at different wavelengths are extracted by a 3-CH wavelength division multiplexer (WDM). Three photodiodes (PDs) are used to convert the three interferometric signals into voltage signals. An analog-to-digital converter (ADC) is used to sample the voltage signals. Then, the phase demodulation is performed by a computer. An EFPI sensor is consisted of a single mode fiber (SMF) and a mirror, as shown in Fig. 1. The end of the SMF is polished. The SMF is fixed in a ceramic ferrule which is coupled to a translation stage. Therefore, the cavity length of the EFPI can be adjusted continuously. The mirror is fixed on a piezoelectric transducer (PZT) to modulate the EFPI by a driving signal. The driving signal is generated by a function generator.
A low-finesse EFPI is normally a two-beam interferometer. The interferometric signal of the EFPI at different wavelengths can be described as
where ${I_1}$, ${I_2}$ are two reflected light intensities of the EFPI, n is the effective refractive index, ${d_t}$ is the cavity length modulated by the driving signal, and $i = 1, 2, 3$. For the proposed phase demodulator, the phase different between ${f_1}$ and ${f_2}$ should be consistent with that between ${f_3}$ and ${f_2}$, which means that ${f_1}$ and ${f_3}$ are symmetrical about ${f_2}$. Let $A = {I_1} + {I_2}$, $B = 2\sqrt {{I_1}{I_2}} $, ${\varphi _t} = 4n\pi {d_t}/{\lambda _2}$. Then where3. Simulation
The proposed demodulation technique was firstly demonstrated by numerical simulations. The wavelengths of lasers were selected according to Eq. (6), where ${\lambda _1} = 1546.93\; \textrm{nm}$, ${\lambda _2} = 1549.79\; \textrm{nm}$, and ${\lambda _3} = 1552.66\; \textrm{nm}$. The EFPI was modulated by a 1 kHz sinewave. The amplitude of the cavity length vibration was 1000 nm. The cavity length of the EFPI was set to change from $5 \;\mu \textrm{m}$ to $1005\; \mu \textrm{m}$ with a step of 50 nm. The variation of the cavity length was demodulated by the symmetrical demodulation method. Figure 2(a) shows amplitude errors of output signals with different cavity lengths. The error was calculated by $({{A_o} - {A_i}} )/{A_i}$, where ${A_o}$ is the output amplitude and ${A_i}$ is the input amplitude. Amplitude errors are less than $1{\%}$ in Fig. 2(a) except for dead zones where $\delta = j\pi , j = 0, \pm 1, \pm 2 \cdots $. For dead zones, three interferometric signals are degenerate into two signals or one signal, thus, the demodulation technique will collapse. From Eq. (5), the interval between two dead zones is increases as the wavelength interval is decreased. Thus, EFPI sensors in wide cavity length range can be interrogated without dead zone. Wavelengths of the three lasers were changed as 1550.52 nm (${\lambda _1}$), 1550.92 nm (${\lambda _2}$), and 1551.32 nm (${\lambda _3}$), respectively. Then, amplitude errors of output signals were calculated and plotted in Fig. 2(b). From Fig. 2(b), EFPIs with cavity lengths in the 5-1005 µm range can be demodulated without dead zone. The amplitude error of the demodulated signal is less than $0.25{\%}$ when the cavity length is larger than $20\; \mu \textrm{m}$.
Fig. 2. (a) Amplitude errors of output signals with different cavity lengths, (b) Amplitude errors of output signals with decreased laser wavelength intervals.
The amplitude measuring error of the demodulated signal was investigated. The EFPI was modulated by a 1 kHz sinewave. The initial cavity length of the EFPI in the simulation was 290 µm. The amplitude of the input signal was set to change from 10 nm to 1000 nm with a step of 1 nm. Wavelengths of the three lasers were 1546.93 nm, 1549.79 nm, and 1552.66 nm, respectively. Figure 3 shows the simulation result of the amplitude error. From Fig. 3, most of the measuring errors with the input amplitude in the range of 10-1000 nm are less than ${\pm} 0.05{\%}$.
The measuring error at different input frequency was then investigated by a simulation. The initial cavity length of the EFPI was set to 290 µm. The EFPI was modulated by a sinewave. The input frequency changed from 10 Hz to 20000 Hz with a step of 10 Hz. The amplitude of the cavity length vibration was set as 200 nm. For the simulation, ${\lambda _1} = 1546.93\; \textrm{nm}$, ${\lambda _2} = 1549.79\; \textrm{nm}$, and ${\lambda _3} = 1552.66\; \textrm{nm}$. Figure 4 shows the simulation result of the amplitude error. From Fig. 4, the output amplitudes with frequencies in the range of 10-20000 Hz is consistent with the input amplitudes. The amplitude error of the demodulated signal is less than 0.01%. Simulations demonstrate that the proposed symmetrical phase demodulator can recover dynamic signals from EFPIs with different phase modulations and different cavity lengths accurately.
4. Experiment
A demodulator based the proposed demodulation technique was fabricated, in which ${\lambda _1} = 1546.93\; \textrm{nm}$, ${\lambda _2} = 1549.79\; \textrm{nm}$, and ${\lambda _3} = 1552.66\; \textrm{nm}$. The power of each laser was 10 mW. The interferometric signals were sampled at a sampling frequency of 160 kHz/channel. An EFPI shown in Fig. 1 was used in the experiment. The initial cavity length of the EFPI was measured by the white light interferometry (WLI) technique [16], which was 301.252 µm. The initial cavity length will be used to calculate the change in cavity length after the experiment. The EFPI was driven by a 300 Hz sinewave which was generated by the signal generator. The amplitude and the frequency of the sinusoidal signal kept stable during the experiment. The translation stage of the sensor was adjusted to change the cavity length during the experiment. The sensor was demodulated by the demodulator. The demodulated signal is shown in Fig. 5(a). The modulation of the PZT and the adjustment of the translation stage were recovered by the proposed demodulation technique, as shown in Fig. 5(a). The enlarged images of areas indicated by solid boxes in Fig. 5(a) show in Fig. 5(b), Fig. 5(c), and Fig. 5(d), respectively. The adjusted cavity length was measured by the WLI after the experiment, which was 264.427µm. The peak-to-peak amplitude of Fig. 5(b), 837.3 nm, is consistent with that of Fig. 5(d), 836.6 nm. The difference in the DC value between Fig. 5(b) and Fig. 5(d) is 36.6 µm which is consistent with the cavity length difference measured by the WLI, 36.8 µm. The amplitude and the frequency of the output signal kept stable during the experiment. From Fig. 5, EFPI based sensors with different cavity lengths can be interrogated by the demodulation technique without measuring the cavity lengths.
Fig. 5. (a) Output signal when the cavity length is changed from 301.252 µm to 264.427 µm, (b)-(d) Enlarged images of areas indicated by the solid boxes in Fig. 5(a).
The translation stage was adjusted to decrease the cavity length. A 200 Hz sinewave was generated by the signal generator, which was used to drive the EFPI. The EFPI was demodulated by the same demodulator. The three interferometric signals of the sensor are plotted in Fig. 6(a). Figure 6(a) shows that two signals, ${f_1}$ and ${f_3}$, are symmetrical about ${f_2}$. Figure 6(b) plots the output signal of the demodulator. The fast Fourier transform (FFT) was used to convert the demodulated signal into the power spectrum plot which is plotted in Fig. 6(c). From Fig. 6(b) and Fig. 6(c), the frequency of the output sinusoidal signal is 200 Hz, which is consistent with the input signal. Figure 6(a) shows that the phase shift of the sensor is less than $2\pi $. The cavity length of the sensor was measured by the WLI after the experiment, which is 9.12 µm. Figure 6(c) shows that the signal to noise ratio (SNR) of the output signal is 40 dB. When the cavity length was adjusted to be less than 9.12 µm, the output of the demodulator is significantly distorted. The cavity length of 9.12 µm is close to the lower limit of the demodulator.
Fig. 6. Experimental data for the 9.12 µm EFPI operating at 200 Hz: (a) three interferometric signals, (b) the demodulated signal, (c) the power spectrum plot.
The passive demodulator can be used to interrogate high frequency signals. The translation stage was adjusted. A 25 kHz sinewave was generated by the signal generator, which was used to drive the EFPI. The EFPI was demodulated by the same demodulator. The three interferometric signals of the sensor are plotted in Fig. 7(a). Figure 7(b) plots the output signal of the demodulator. The FFT was used to convert the demodulated signal into the power spectrum plot which is plotted in Fig. 7(c). From Fig. 7(b) and Fig. 7(c), the frequency of the output sinusoidal signal is 25 kHz, which is consistent with the input signal. Figure 7(c) shows that the SNR of the output signal is 70 dB. The phase shift of the sensor is less than $2\pi $. The cavity length was measured by the WLI after the experiment, which is 339.95 µm.
Fig. 7. Experimental data for the 339.95 µm EFPI operating at 25 kHz: (a) three interferometric signals, (b) the demodulated signal, (c) the power spectrum plot.
The translation stage was adjusted to increase the cavity length. A 200 Hz sinewave was generated by the signal generator, which was used to drive the EFPI. The EFPI was demodulated by the same demodulator. The three interferometric signals of the sensor are plotted in Fig. 8(a). Figure 8(b) plots the output signal of the demodulator. The FFT was used to convert the demodulated signal into the power spectrum plot which is plotted in Fig. 8(c). From Fig. 8(b) and Fig. 8(c), the frequency of the output sinusoidal signal is 200 Hz, which is consistent with the input signal. Figure 8(a) and Fig. 8(b) show that the phase shift of the sensor is larger than $2\pi $. The cavity length was measured by the WLI after the experiment, which is 1160.28 µm. When the cavity length was adjusted to be larger than 1160.28 µm, the output of the demodulator is significantly distorted. The cavity length of 1160.28 µm is close to the upper limit of the demodulator. Limitations on the cavity length range of the demodulator will be set by laser wavelengths, laser optical powers, and the reflector reflectivity of the EFPI.
Fig. 8. Experimental data for the 1160.28 µm EFPI operating at 200 Hz: (a) three interferometric signals, (b) the demodulated signal, (c) the power spectrum plot.
5. Conclusion
In conclusion, a symmetrical demodulation method for the phase recovery of EFPIs is proposed and experimentally demonstrated. For the demodulation technique, two interferometric signals are symmetrical about the third interferometric signal. The measurand is extracted by the demodulator from the three interferometric signals. EFPIs with different cavity lengths can be interrogated by one phase demodulator without measuring the initial cavity lengths. An EFPI with the cavity length ranging from 9.12 µm to 1160.28 µm is interrogated successfully by a demodulator. The symmetrical demodulation method for the dynamic signal recovery is adapted to the measurement of EFPIs with unsteady cavity lengths and unknown cavity lengths.
Funding
National Key R&D Program of China (2018YFB1107200); National Natural Science Foundation of China (61775020, 61575021).
Disclosures
The authors declare no conflicts of interest.
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