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Abstract
A four-wavelength passive demodulation algorithm is proposed and experimentally demonstrated for the interrogation of the one cavity in a dual-cavity extrinsic Fabry-Perot interferometric (EFPI) sensor. The lengths of two cavities are adjusted to generate four quadrature signals for each individual cavity. Both simulation and experimental results are presented to validate the performance of this technique. The experimental results demonstrate that dynamic signals at frequencies of 100 Hz, 200 Hz, and 300 Hz with varying amplitude are successfully extracted from a dual-cavity EFPI sensor with initial lengths of 93.4803 µm and 94.0091 µm. The technique shows the potential application to measure dynamic signals in dual-cavity fiber-optic EFPI sensors.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Fiber-optic vibration, acceleration, and acoustic sensors [1–4] find extensive applications in industries such as rail transportation, aerospace, and oil and gas extraction due to their unique advantages of anti-electromagnetic interference, high sensitivity, and remote sensing capabilities. Among them, the single-cavity fiber-optic extrinsic Fabry-Perot interferometer (EFPI) sensors [5–7], comprising a fiber tip and a reflective diaphragm, are considered an optimal solution due to their compact structure, high response frequency, and elimination of the need for fiber stretching. However, in engineering applications, the measurement accuracy of single-cavity fiber EFPI sensors can be influenced by environmental factors such as temperature and pressure. To address this issue, dual-cavity fiber-optic EFPI sensors have been developed for multi-parameter measurement [8–11]. Besides, the adoption of a dual-cavity scheme is frequently unavoidable due to material and sensor manufacturing constraints. For instance, in the case of a sapphire pressure sensor fabricated via the high-temperature bonding technique, an air cavity and sapphire wafer substrate combine to form a dual-cavity structure [12,13]. The white light Interferometry (WLI) technique is commonly employed for the demodulation of dual-cavity EFPI sensors, including peak tracking [8–10], Fourier transform [11,14–16], and cross-correlation algorithm [17,18]. However, a common characteristic of these methods is their reliance on acquiring light intensity across a broad wavelength range, as a result, achieving high-speed demodulation becomes challenging. High-speed spectrometers are considered a promising solution; however, their current cost remains prohibitively high, their current speed fails to meet the demands of many high-speed measurement occasions, and their measurement resolution is low.
The laser interferometric demodulation technique is commonly used in the high-speed demodulation of single-cavity fiber-optic EFPI sensors, which can be mainly divided into two categories: the active demodulation methods and the passive demodulation methods. An active demodulation technique, the phase-generated carrier (PGC) method [19–21], has inherent limitations in handling phase modulation loads. To maintain signal distortion within acceptable limits, it becomes necessary to increase the carrier frequency. However, the carrier frequency cannot be infinitely increased due to practical limitations arising from device and circuit constraints. Furthermore, this method’s measurement accuracy is influenced by power fluctuations of the light source as well as amplitude drift and fluctuations of the mixing signal. The passive demodulation technology encompasses the intensity-based quadrature point (Q-point) technology [22–24], the dual-wavelength quadrature phase demodulation technique [25–27] and the three-wavelength passive demodulation technology [28–31], etc. The Q-point of the interferometer must be fixed to achieve optimal sensitivity and linearity, which is often affected by random phase drift resulting from thermal EFPI cavity length fluctuations. The dual and three-wavelength demodulation algorithms compensate for the limitations of the Q-point method, thereby addressing the challenges associated with high-speed demodulation of fiber-optic EFPI sensors with arbitrary or unknown cavity lengths. The passive dual and three-wavelength demodulation technologies are considered the most practical methods for high-speed demodulation owing to their wide frequency response, heightened sensitivity, and expansive dynamic range. However, these techniques cannot be applied to the demodulation of dual-cavity fiber-optic EFPI sensors due to the complex three-beam interference model associated with such sensors. In 2021, O.R. et al. proposed an indirect approach to address the high-speed demodulation of multi-cavity fiber-optic EFPI sensors by manipulating the coherence length of the light source. The interference is exclusively confined to the shortest cavity, and the three-beam interference model is simplified into the two-beam interference model [32].
In this work, a novel four-wavelength passive quadrature demodulation algorithm is proposed for the high-speed demodulation of one cavity in the dual-cavity fiber-optic EFPI sensors. Four distributed feedback lasers (DFB-LD) with approximately equal frequency intervals are multiplexed as coherent light sources and propagated through a dual-cavity fiber-optic EFPI sensor, in which one cavity of the interferometer is modulated by a piezoelectric ceramic transducer (PZT) to generate four interference signals with different wavelengths, while the other cavity is fixed. By analyzing the three-beam interference model and the relationship of four interferometric signals, the direct current (DC) component and phase associated with the non-modulating cavity are eliminated, then two quadrature signals related to the modulated cavity are obtained. Finally, the input signal is demodulated by the arctangent method (ATAN). Both the simulation and experiment are accomplished to evaluate the demodulation performance of this algorithm. This four-wavelength passive quadrature demodulation algorithm with the advantages of simple, high resolution, has potential applications in the high-speed demodulation of dual-cavity fiber-optic EFPI sensors.
2. Principle
The schematic diagram of the optical path is illustrated in Fig. 1. The light emitted by four distributed feedback lasers (DFB-LD) with wavelengths $\mathrm {\lambda }_1$, $\mathrm {\lambda }_2$, $\mathrm {\lambda }_3$, and $\mathrm {\lambda }_4$ is multiplexed through wavelength division multiplexer 1 (WDM1). The multiplexed light passes through a coupler and reaches a dual-cavity fiber-optic EFPI sensor consisting of three reflecting surfaces: a reflective surface positioned at a specific location within the single-mode fiber $\text {S}_1$, the fiber tip $\text {S}_2$, and a mirror attached to PZT $\text {S}_3$. The cavity formed by $\text {S}_1$ and $\text {S}_2$ is a non-modulated cavity with a length of $d_1$, while the cavity comprising $\text {S}_2$ and $\text {S}_3$ is a modulated cavity with an initial length of $d_2$. The reflected interferometric signal returns to the coupler and is demultiplexed by WDM2. Four interferometric signals with different wavelengths are converted into electrical signals by photodetectors (PDs), which are collected via high-speed analog-to-digital conversion (ADC). Data acquisition and real-time demodulation processes are performed using computer software. According to the principle of three-beam interference, four interference signals can be represented as:
The amplitude of the sinusoidal signal is denoted by $C$, the frequency by $\omega$, and the initial phase of the sinusoidal signal by $\phi$. To ensure the orthogonality of each individual phase among four interference signals, the relationship between the wavelengths and the $d_1$ is designed as:
Considering the sinusoidal signal’s amplitude is significantly smaller than the initial cavity length ($C \ll d_2$), it can be approximated that the length of the modulated cavity is equal to that of the initial length ($d_t \approx d_2$). To ensure orthogonality, a relationship between $d_2$ and wavelengths is satisfied:
Assume that:
According to formula (4), (5), and (6), formula (1) can be expressed as:
Based on formula (6), the $\theta _1$ is a known quantity, so it can be obtained from formula (8):
The amplitude of $D_1$ is $2kB_1B_2$, and the amplitude of signal $D_2$ is $kB_1$, where $0<k<2$. The $B_1$ can be get:
Then the $\theta _t$ can be obtained by the ATAN:
Then:
3. Simulation and analysis
To validate the impact of cavity length offset on the demodulation performance of the four-wavelength orthogonal algorithm, several simulation tests are conducted based on the fundamental principles of the algorithm. The input signal frequency is set at 200 Hz with a sampling number of 5000 and a sampling rate of 40 kHz. The specific parameters employed in this simulation are presented in Table 1. Figure 2 demonstrates the original signal loaded during simulation, the four interference signals, and the demodulation signal obtained through the four-wavelength passive quadrature demodulation algorithm.
Fig. 2. (a) The input modulated signal; (b) The four simulated interferometric signals driven by a cosine signal; (c) The simulated demodulated signal.
Table 1. Simulation parameter
By utilizing formula (4), (5) and considering the wavelengths of the four DFB-LDs, the initial optical length for modulated and non-modulated cavities can be calculated:
Firstly, the simulation is conducted to evaluate the demodulation error caused by optical length deviation of the non-modulated cavity. The initial optical length of the modulated cavity is fixed at 93.1643 µm, while the optical length of the non-modulated cavity varies within a range of ${\pm }5\;\mathrm{\mu}\textrm {m}$ (from 88.1643 µm to 98.1643 µm) with an accuracy of 1 nm. After each demodulation process, the ratio of signal amplitude deviation (represented as $(A_i-A_0)/A_0$) is calculated, where $A_i$ denotes the demodulation amplitude and $A_0$ represents the amplitude of the input signal. Figure 3(a) illustrates these simulation results, indicating that as non-modulated cavity length offset increases, demodulation error exhibits oscillatory behavior with a maximum error reaching approximately 5%. To evaluate the impact of modulation-cavity length deviation on demodulation performance, a simulation of modulation-cavity demodulation error is conducted. The optical length of the non-modulated cavity is set at 93.1643 µm, while the optical length of the modulated cavity increases from 88.1643 µm to 98.1643 µm (${\pm }5\;\mathrm{\mu}\textrm {m}$) with an accuracy of 1 nm. The simulation result is shown in Fig. 3, which demonstrates that the maximum demodulation error remains below 2%.
Fig. 3. (a) Demodulation error of deviations in the non-modulated cavity; (b) Demodulation error of deviations in the modulated cavity.
To examine the effect of simultaneous deviations in both modulating and non-modulating cavities on demodulation performance, a dual-cavity deviation error simulation is performed. Both modulated and non-modulated cavity lengths varied within a range of ${\pm }5\;\mathrm{\mu}\textrm {m}$ around 88.1643 µm to 98.1643 µm with an accuracy of increments by 1nm. The modulation errors are calculated as absolute values representing percentage deviations, as shown in Fig. 4, simulation results indicate that the maximum error does not exceed 6%. The three simulation results indicate that, given the same degree of error, deviations in the initial cavity length have a more obvious impact on the algorithm for the non-modulated cavity compared to that for the modulated cavity.
4. Experiments and results
To evaluate the demodulation performance of the four-wavelength orthogonal algorithm, a dual-cavity fiber-optic EFPI sensor is tested. Before the experiment, as illustrated in Fig. 5, white light interference technology with a resolution of $\pm$1 nm was employed to measure the length of the non-modulated cavity in the dual-cavity EFPI sensor, yielding 93.4803 µm. Additionally, the compound cavity length was measured at 187.4894 µm and subsequently enabled obtaining an initial length of the modulated cavity at 94.0091 µm. The experimental setup is illustrated in Fig. 1, with four DFB-LDs at wavelengths of 1543.710 nm, 1546.914 nm, 1550.131 nm, and 1553.362 nm respectively being utilized for this experiment while maintaining a sampling rate set at 40 kHz. The system was evaluated without a carrier signal, and the demodulation results illustrated a resolution of less than $\pm$1.5 nm.
To evaluate the demodulation performance of the algorithm for phase modulation amplitude below 2$\mathrm {\pi }$, a cosine signal with an amplitude of 422.8 nm (measured by the Three-wavelength passive demodulation technique [29]) and a frequency of 200 Hz is generated using a PZT. The four interferometric signals sampled by the ADC are shown in Fig. 6(a). The demodulated signal is illustrated in Fig. 6(b), and its corresponding power spectrum is presented in Fig. 6(c).
Fig. 6. (a) The four interferometric signals driven by a low-amplitude signal with a frequency of 200 Hz; (b) Waveform of demodulation results; (c) Power spectrum plots of the demodulated signal.
Conversely, to assess the demodulation performance of the algorithm for phase modulation values above 2$\mathrm {\pi }$, a cosine signal with an amplitude of 1556.6 nm (measured by the Three-wavelength passive demodulation technique [29]) and the same frequency of 200 Hz is driven by PZT. The four interferometric signals acquired by ADC are shown in Fig. 7(a), while its corresponding demodulated signal can be observed in Fig. 7(b), along with its power spectrum displayed in Fig. 7(c). The optical spectrum of the single-cavity EFPI is similar with the cosine model, as shown in Fig. 5(a), while the optical spectrum of a two-cavity EFPI is irregular, as illustrated in Fig. 5(b). Due to the distinct spectral characteristics between the dual-cavity EFPI and single-cavity EFPI, even when the modulation amplitude exceeds 2$\mathrm {\pi }$, there are differences in amplitude ranges among the four interference signals. The results of these two experiments confirm the efficacy of the four-wavelength passive quadrature demodulation algorithm in achieving demodulation, regardless of whether the phase modulation amplitude exceeds or remains below 2$\mathrm {\pi }$.
Fig. 7. (a) The four interferometric signals driven by a high-amplitude signal with a frequency of 200 Hz; (b) Waveform of demodulation results; (c) Power spectrum plots of the demodulated signal.
To explore the response characteristics of the algorithm to input signals with varying frequencies, sinusoidal signals with the same amplitude at frequencies of 100 Hz, 200 Hz, and 300 Hz are sequentially generated by a PZT. The demodulation results are presented in Fig. 8(a), demonstrating successful signal demodulation. The power spectrum is depicted in Fig. 8(b). These experimental findings indicate that the proposed algorithm exhibits excellent frequency response.
Fig. 8. Demodulation results of different frequencies; (b) Power spectrum plots of the demodulated signal.
5. Conclusion
In conclusion, a four-wavelength passive quadrature algorithm is proposed for the demodulation of dual-cavity fiber-optic EFPI sensors. A theoretical framework for laser interferometry is established to address the three-beam interference model through mathematical derivation. With this technology, high-speed demodulation of a single cavity change in dual-cavity EFPI sensors can be achieved without any additional control on the light source. Moreover, the presented technique can extract dynamic signals regardless of whether the phase modulation is larger than 2$\mathrm {\pi }$. In the experiment, sinusoidal signals ranging from 100 Hz to 300 Hz with varying amplitudes are successfully recovered, which demonstrates the potential of this technique for measuring dynamic signals in dual-cavity EFPI sensors.
Funding
National Natural Science Foundation of China (U20B2057, 62305265).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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