Fiber-optic interferometric sensors have been intensively developed for various field applications owing to their unique advantages such as remote sensing, small size, light weight, high sensitivity, passivity, immunity to electromagnetic interference, environmental ruggedness, etc [1, 2]. However, one of the major problems impeding the performance of interferometric sensors is signal fading, where the signal amplitude varies unpredictably due to random changes in the phase bias state of interfering optical waves [3, 4].

For the practical use of interferometric sensors, the signal fading problem must be solved so that the sensing output is stable without temporal fluctuation. Regarding this, previous approaches used a piezoelectric transducer with a phase tracking loop to fix the phase bias state [5], or an optical amplitude modulator for heterodyne detection to extract optical phase information instead of measuring intensity [6, 7]. Although these techniques are effective with interferometric signal fading, the bulky piezoelectric transducer and the costly electro-optic modulators are undesirable in many applications. Homodyne schemes using a passive 3 × 3 coupler are known to suppress fading, but complex signal demodulation is required [8, 9]. A phase feedback control scheme in a Michelson interferometer and a balanced Sagnac interferometer [10, 11] were recently proposed, but they require the additional use of optical components and nontrivial control units, thereby making the interferometric sensors less practical.

In this paper, we describe and analyze an all-optical homodyne interferometric vibration sensor where the output signal is completely independent of the state of phase bias. A 90° optical hybrid and two balanced photo-detectors are used to obtain in-phase and quadrature (I/Q) signals with a π/2 phase bias difference. No additional phase-shifting devices, electronics, feedback circuits, or optical active components for signal modulation and demodulation are needed in this interferometer. A fade-free output can be obtained by simply applying the Hilbert transform to one in-phase signal to shift its phase by π/2, and then adding this to the quadrature signal. The final compound output was found to be stable without signal fading for any arbitrary phase bias state. This π/2 phase-shifting technique with the quadrature signals enables direct measurement of the interferometer response corresponding to external disturbances without complex signal modulation and demodulation steps, and signal processing tasks are thus significantly reduced.

2.1. Fade-free Interferometry

Figure 1 shows a schematic of the proposed fade-free optical fiber vibration sensor based on a Mach–Zehnder interferometer. A 1,550 nm optical input from the LD (CoSF-D; Connect Laser Technology Co., Shanghai, China) is split into two interferometer arms (sensor arm and reference arm) by a fiber coupler (5:5), and then recombined by a 90° optical hybrid (COH24-X; Kylia, Paris, France). The 90° optical hybrid is a passive device composed of beam splitters and half-wave and quarter-wave plates to generate four quadrature states of interference (S + R, S − R, S + i_R, S − i_R) between the signal (S) and reference (R) light. The optical hybrid then sends the four light signals to two pairs of balanced detectors, resulting in two equal amplitude interference signals (i₁, i₂) that are both quadrant. The Hilbert transform is then applied to one of the interference signals i₁ to shift its phase by π/2 (= H{i₁}). The final fade-free output is constructed by the simple summation of i₂ and π/2 phase shifted i₁ (= i₂ + H{i₁}).

Figure 1. Configuration of phase bias independent fade-free optical fiber interferometric vibration sensor.

The light intensity of the output of the 90° optical hybrid based Mach–Zehnder interferometer can be expressed as follows:

where ∆φ is the optical phase shift, φ_e is the phase bias, and dφ is the signal induced phase delay, respectively. In the case where a small dynamic disturbance such as sinusoidal vibration/acoustic energy is applied to the interferometer’s sensor arm, dφ under a small-signal approximation can be expressed as

where A is the scale factor for the vibration intensity and ω is the angular velocity corresponding to the vibration frequency, respectively. Therefore, the AC terms of the two balanced photo-detector outputs can be expressed as follows:

The following step is to shift the phase of the in-phase output signal i₁ by π/2. The Hilbert transform, which is a linear operator that produces a π/2 phase shift in a signal, was used as a phase shifter in the interferometric system. Under the small-signal approximation when dφ is small enough to assume a linear response of the sinusoidal functions, the Hilbert transform of i₁ will be a function similar to the one as follows:

It should be noted that Eq. (6) is a simplified form under the small-signal assumption, which has been used to estimate the state of the optical signals for understanding the working principle of the proposed fade-free interferometry. Figure 2 shows three-dimensional plots of the temporal amplitude of i₂ and H{i₁} with respect to the phase bias state. The scale factor A was set to 0.01 in the small signal approximation analysis. From the graphs, one can see that the amplitudes of the output signals change for different phase biases; this is called phase bias-induced signal fading. For example, the amplitude of i₂ is lowest at a phase bias (φ_e) of π/2 rad and highest at a φ_e of π rad, while the amplitude of H{i₁} is highest at a φ_e of π/2 rad and lowest at a φ_e of π rad.

Figure 2. Three-dimensional plots of the temporal amplitude of (a) i₂ and (b) Hilbert transform of i₁ (H{i₁}) with respect to phase bias state.

As mentioned above, a fade-free output can be obtained by summation of i₂ and H{i₁}, and is expressed as

Figure 3 shows a three-dimensional plot of the amplitude of the fade-free signal (i₂ + H{i₁}) with respect to time and the phase bias state. Contrary to the individual graphs for i₂ and H{i₁}, the compound output signal (i₂ + H{i₁}) exhibited no variation in signal amplitude with respect to the phase bias state; only a shift in the initial phase of the output signal was observed. To clarify the results, the signals i₁, i₂, i₁ + i₂, i₁ − i₂, and the output signal i₂ + H{i₁} for different phase bias states (φ_e = 0, π/8, π/4, 3π/8, π/2, 5π/8, 3π/4, 7π/8, π) were plotted, respectively, as shown in Fig. 4. The amplitudes of the in-phase signal i₁ and the quadrature signal i₂ decrease as the phase bias states approach nπ (n = 0, 1, 2, …) and (n + 1)π/2(n = 0, 1, 2, …), respectively. Summation of the in-phase and quadrature signals (i₁ + i₂) and subtraction of the signals (i₁ − i₂) also result in fading regions with phase bias states at π/4 and 3π/4, respectively. In contrast, the amplitude of the final output signal (i₂ + H{i₁}) was found to remain unaffected by the phase bias state. In other words, the output signal is completely independent of the phase bias state of interfering light waves.

Figure 3. Three-dimensional plot of the temporal amplitude of summation of i₂ and H{i₁} (= i₂ + H{i₁}) with respect to phase bias state.

Figure 4. Plots of the amplitudes of the signals (a) i₁, (b) i₂, (c) i₁ + i₂, (d) i₁ − i₂, and the output signal (e) i₂ + H{i₁} for different phase bias states (φ_e = 0, π/8, π/4, 3π/8, π/2, 5π/8, 3π/4, 7π/8, π).

To demonstrate the proposed fade-free interferometry, vibration measurement was performed using the interferometer, as shown in Fig. 5. A sensor coil was made with a 5 m length of non-polarization-maintaining single mode optical fiber cable. A piezoelectric transducer (PA25LEW; Thorlabs, NJ, USA) was used as a vibration exciter and a 10 kHz CW-vibration (Vpp = 10 mA) was applied to the sensor coil. High-pass filtering in software was used after the balanced detection to suppress the undesirable DC components in the optical signals during measurement.

Figure 5. Experimental setup for measuring vibration using the fade-free optical fiber interferometer.

Figure 6 shows the long-term (600 ms) measured output signal i₂ + H{i₁} as well as the signal i₁ for comparison. Since the phase bias state randomly changes in the interferometer, amplitudes of i₁ and i₂ fluctuated with time during vibration measurement. However, the amplitude of the compound signal i₂ + H{i₁} was consistently maintained during the measurement although each of the signal components i₁, i₂, and H{i₁} fluctuated by temporal change of the phase bias state. Since the amplitude of the i₂ varied from maximum to minimum, or from minimum to maximum in the CW-vibration measurement, the i₂ and the quadrature i₁ experienced all the phase bias state changes that may occur during long-term measurement.

Figure 6. Temporal amplitude variation of the signals (a) i₂ and (b) i₂ + H{i₁} during vibration measurement (600 ms).

Figure 7 shows short-term (700 μs) sampling data graphs for i₁, i₂, and H{i₁}, and i₂ + H{i₁} captured by a data acquisition device during the 10 kHz CW-vibration measurement. The phase bias changed randomly during the measurement, and so the data graphs represent instantaneous signals at different phase bias states. From the phase and amplitude differences between i₁ and i₂ in the graphs, one may speculate the phase bias state of interfering waves at a given moment. For example, in the case of Fig. 7(a), the amplitude of i₁ was very low relative to that of i₂, which indicates that the instantaneous phase bias state was near the π radian. In the case of Fig. 7(b), the amplitudes for i₁ and i₂ were similar to opposite phases, which means that the phase bias state would be at π/4 radian. In the case of Fig. 7(c), the amplitude of i₁ was high, whereas i₂ was fully faded, which clearly indicates that the phase bias was very close to π/2 radian. In the case of Fig. 7(d), both the amplitude and phase for i₁ and i₂ were identical, which means that the phase bias was 3π/4.

Figure 7. Short-term (700 μs) measured data for the interferometer’s signals [(a) i₁, (b) i₂, (c) H{i₁}, and (d) i₂ + H{i₁}] during vibration measurement.

Although i₁, i₂, and H{i₁} fluctuated or faded during the measurement due to the unstable phase bias state, the compound outputs i₂ + H{i₁} were consistent by maintaining the amplitude and frequency responses at different phase bias states, as shown in Figs. 7(a)–7(d). The measurement results clearly show that i₂ and H{i₁} were completely complementary to each other in terms of phase bias induced fading, which corresponds with the simulation results shown in Figs. 3 and 4.

The performance of fade-free interferometry can be affected by the degree of the phase delay corresponding to the intensity of the induced vibration. Signal analysis was carried out under small signal approximation where the scale factor A of the signals (i₁, i₂) was 0.01. To investigate degradation related to the intensity of the phase delay, we plotted fade-free outputs (i₂ + H{i₁}) with larger A values (A = 0.1, 1), as shown in Fig. 8.

Figure 8. Plots for the amplitudes of the output signal i₂ + H{i₁} for phase bias state (0, π/4, π/2, 3π/4, π) with different A values (a) 0.1 and (b) 1.

In the case where A is 0.1, no significant differences in the output amplitude and frequency response for different phase biases (0, π/4, π/2, 3π/4, π) appeared. On the other hand, the output was distorted in the case where the A value is 1. This is because the phase delay term is too large to approximate the linear behavior of the signals i₁ and i₂ in the interferometry. Therefore, the interferometer’s sensor arm should be designed in consideration of sensing limitations such as sensitivity and dynamic range to avoid signal distortions due to excessively large phase delay, and so guarantee reliable fade-free output of the interferometer.

This technique is expected to be very useful from a practical perspective, as it can effectively suppress signal fading in interferometric vibration measurements caused by dynamic variations of phase bias due to internal and/or external environmental changes. Potential applications of this sensor technology include early detection of internal defects in power plant equipment, partial discharge monitoring and loose-part monitoring. Furthermore, the proposed signal processing technique can be used to enhance the resolution in interferometric imaging fields or to address signal fading issues in distributed sensor technologies.

We have experimentally demonstrated phase bias independent vibration measurement using homodyne optical fiber interferometry without a phase modulator. In-phase (i₁) and quadrature (i₂) interferometric signals with a π/2 phase bias difference were obtained using 90° optical hybrid, and the Hilbert transform was applied to one of the signals to shift its phase by π/2 (H{i₁}). The fade-free output was obtained by summation of the two signals (i₂ + H{i₁}) and its amplitude and frequency response for 10 kHz CW-vibration measurement were consistent for any arbitrary phase bias state during the measurement.

The proposed interferometric scheme is simple and does not require active devices such as an optical modulator or complex signal processing, but nonetheless effectively solves signal fading problems. Although it is more likely to be applied to micro-vibration measurement since the output is degraded with a larger phase delay, it is also possible to achieve reliable fade-free output for strong vibration measurement by using a properly designed interferometer sensor arm with adequate sensitivity.

This work was supported by the National Research Foundation of Korea (NRF) funded by the Korean government (Ministry of Science and ICT) (Grant No. RS-2022-00144110 & RS-2023-00258052).

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The data that support the findings of this study are available from the corresponding author upon reasonable request.