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Abstract
In the reference interferometer demodulation scheme, it's difficult to guarantee in practice that both interferometers have the same optical path length difference (OPD), which makes the phase modulation depth different in different interferometers with the same laser modulation. The random shift of phase modulation depth also affects the demodulation results. An improved phase-generated carrier (PGC) technique is proposed based on an auxiliary reference interferometer and the ellipse fitting algorithm (EFA). The technique ensures the correct fitting of the EFA for small amplitude signals by introducing a sinusoidal signal as an additional phase modulation. The combination of the reference interferometer and EFA can eliminate the effect of different phase modulation depths of the two interferometers caused by different OPDs, the non-linear distortion caused by phase modulation depth shifts, and improve the accuracy of the demodulation results. The experiment results are consistent with the theoretical analysis, and the method extends the application of the EFA in the reference interferometer phase demodulation technique.
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1. Introduction
Due to their high sensitivity, diversified structure, anti-electromagnetic interference, wide dynamic range, etc [1–3], fiber-optic interferometric sensors (FOIS) are gaining more and more attention. It's widely used in national defense and civil fields [4,5]. The demodulation method applied to the FOISs will directly influence the performance of the demodulation system. The heterodyne demodulation scheme, the 3 × 3 fiber coupler method and the phase-generated carrier (PGC) technology are commonly used phase demodulation methods [6–8]. The PGC technology is widely used for its advantages of high linearity, high accuracy of phase measurement, and high sensitivity [9]. The typical PGC techniques mainly include the differential and cross multiplication method (PGC-DCM) and the arctangent method (PGC-Arctan). But the PGC-DCM method is affected by light intensity disturbance (LID), carrier phase delay, and phase modulation depth shift, which directly result in low reproduction of demodulation results [7,10]. The PGC-Arctan method can eliminate the effect of LID and suppress intensity noise but produce severe harmonic distortion if phase modulation depth deviation from the ideal value [11–13]. The nonlinear distortion can be effectively reduced using some modified PGC methods. For example, a differential self-multiplication method was applied to PGC technology (PGC-DSM) to suppress the influence of the C-value shift [14]. This method achieves lower harmonic distortion when the C-value is in the range of 1.5 rad to 3.5 rad. Previous studies have used the ellipse fitting algorithm to compensate for the nonlinear distortion caused by LID, the carrier phase delay, and the shift of phase modulation depth [15–19]. The amplitude of the measured signal should be larger than π/4 rad, otherwise, the EFA can’t operate properly [18]. Another part of the researcher used triangular wave signals as additional modulation signals by an EOM to apply the EFA for measuring tiny displacements. This method demodulation correctly while the amplitude of the measured signal is less than π/4 rad [20].
The noise problem in FOIS is also an important factor limiting the performance of demodulation systems [21]. In an unbalanced interferometer, the phase noise caused by the frequency instability of the light source makes the largest contribution to the noise level [22]. A highly stable light source helps to reduce phase noise [23], but such sources can be quite expensive. The known effective solution for reducing phase noise is to use the auxiliary reference interferometer [24–25]. Previous researchers have proposed that the phase noise level is reduced by approximately 40 dB when using two interferometers with an optical path length difference (OPD) of 80 m [24]. It is necessary that both used interferometers had the same OPD. Otherwise, the OPD causes different phase modulation depths in different interferometers under the same light source and increases system instability. In our previous work, we proposed an ameliorated PGC algorithm based on a reference interferometer that can eliminate the effect of phase modulation depth drift [26]. The method doesn’t consider the effect of the non-strictly equal OPD producing different phase modulation depths on the demodulation results. It is difficult to guarantee that two interferometers have the same OPD in practice [27]. Therefore, it is necessary to propose a scheme for stable demodulation results under non-strictly equal OPD of two interferometers.
In this paper, an improved PGC technique based on a reference interferometer and the EFA is proposed. The technique ensures the correct fitting of the EFA for small amplitude signals by introducing an additional sinusoidal signal as phase modulation. The combination of the reference interferometer and EFA can eliminate the effect of different phase modulation depths of the two interferometers caused by different OPDs, the non-linear distortion caused by phase modulation depth shifts, and improve the accuracy of the demodulation results. Theoretical analysis and simulation are combined to design a system for experimental, to verify the effectiveness of the new method.
2. Theory and principles of operation
2.1 Theory of operation
Figure 1. shows a schematic of the PGC demodulation technique based on a reference interferometer and the EFA. It is called the Ref-PGC-EFA method. One interference signal Is is generated by the signal interferometer and the other interference signal Ir is generated by the reference interferometer. The two interference signals Is, Ir can be expressed as
Fig. 1. The schematic of Ref-PGC-EFA (LPF: Low-pass-filter; HPF: High-pass-filter; EF: ellipse fitting).
The phase modulation depth is generated by the modulating laser.
Where c is the speed of light in a vacuum, n is the refractive index of the fiber core, v is the maximum value of the laser frequency shift, ΔL is the physical path length difference between interferometer arms. For different interferometers, even a small difference in OPD, the same laser will produce different C-values. Cs and Cr are the phase modulation depth in the signal interferometer and the reference interferometer respectively.The interference signal Is multiplied with a fundamental carrier Gcos(ω0t) and a second-harmonic carrier Hcos(2ω0t), respectively. Then, the high frequencies are filtered out by two low-pass filters, and a pair of non-strict quadrature components SS1 and SS2 are obtained.
The signal Ir is processed in the same way, a pair of non-strict quadrature components SR1 and SR2 can also be obtained.
Where J1 (Cs), J1 (Cr), and J2 (Cs), J2 (Cr) are the first-order and second-order Bessel functions respectively. In the ideal case, including the carrier amplitudes, G and H are fixed to the same value, optimal phase modulation depth, and no phase delay. Many demodulation methods can be used to obtain the correct demodulation results. However, in actual applications, factors such as different C-values for the two interferometers, phase delay, and non-ideal performance of the low-pass filter can lead to DC offset and fluctuation of AC amplitude of quadrature components, resulting in inaccurate demodulation results. The EFA has been introduced into the phase demodulation technique to deal with the deviation and imbalance of the direct current component, and the phase difference of the interference signals between the channels. In general, the output of signal interferometer SS1, SS2 can be rewritten by Where a and c are the DC offsets, b and d are the AC amplitudes, and φz = φs+φe+Δφs is the sum of the measured phase signal. θ is the phase difference between SS1 and SS2.Apparently, the dynamic drift of a, b, c, d, and θ causes a shift in the quadrature component and the resulting periodic non-linear distortion. The EFA consists of two parts, ellipse fitting and ellipse correction [7,16]. When the variation of φz is large enough, such as 2π, SS1 and SS2 form a Lissajous figure that is a full ellipse. Therefore, we can obtain the parameters by ellipse fitting. Then, the ellipse correction part corrects the ellipse to a standard circle and obtains a pair of strict quadrature signals.
Take the signal interferometer as an example. The signal coefficients are normalized after the EFA and a pair of strictly quadrature signals are obtained. The effect of phase modulation depth errors due to the different OPDs of the two interferometers and phase modulation depth fluctuations is eliminated. The corrected signal can be expressed as follows
In the same way, the output of the reference interferometer can be rewritten by
For two interferometers with the same parameters, the phase noise levels induced by the same laser are approximately equal. Therefore, Δφs ≈ Δφr [23]. After the trigonometric operation, the additional signal φe is eliminated and no longer affects the subsequent demodulation process.
Equations (14) and (15) are divided, then the arctangent function operation is performed. The demodulation result of the Ref-PGC-EFA method is obtained.
Finally, the phase changes caused by low-frequency environmental noise φr in the interferometer are filtered out by the HPF to recover the sensed signal φs. At the same time, the residual values of the additional phase modulation signal are filtered out by the HPF if it exists to ensure a stable demodulation result. It can be concluded from the previous theory that the result of the Ref-PGC-EFA method is no longer related to the phase modulation depth.
For the traditional reference interferometer demodulation scheme based on the PGC-Arctan method without using EFA, the phase modulation depth is not strictly equal which can seriously affect the demodulation results. The equation for the demodulation results of the traditional method with guaranteed G = H, and Bs = Br is shown below:
It is evident from the equation that the demodulation results are very dependent on the phase modulation depth, and the non-strict matching of the phase modulation depths of the two interferometers will eventually lead to errors in the demodulation results.
2.2 EFA describe
EFA is divided into two parts: ellipse fitting and ellipse correction, which are used to suppress nonlinear distortion. The general expression of the nonlinear distortion can be expressed as Eqs. (8) and (9) [16]. Due to sin2φz + cos2φz = 1, Eqs. (8) and (9) can be rewritten by
The general form of the elliptic equation can be expressed as
Comparing Eqs. (18) and (19) can obtain
Therefore, using two data sources S1 and S2, the ellipse fitting parameters B, C, D, E, and F are calculated using the least-squares fitting method. And the ellipse correction parameters a, b, c, d, and θ can be obtained. After correction, a pair of strictly quadrature signals containing the required phase information is obtained as follows
Finally, the DC offset is eliminated, the AC amplitude is normalized, and a pair of non-strictly quadrature signals are corrected to a pair of strictly quadrature signals while increasing a portion of the intensity noise carried in the non-strictly quadrature signals. The EFA doesn’t work properly under extreme conditions. For example, the phase modulation depth deviates too much from the ideal value causing the Lissajous figure not to be elliptical and the relevant parameters cannot be calculated.
3. Simulation analysis
In this chapter, we will compare the Ref-PGC-EFA method with the traditional method without EFA from the simulation aspect and evaluate the performance of the EFA. Many aspects will be verified, including the correction for phase modulation depth caused by non-strict equal OPD, the insensitivity of the C-value, and high stability. A sinusoidal signal φe(t) is set as an additional modulation signal, where φe(t) = Kcos(2πfet). The additional signal amplitude K is set to π rad and the frequency fe is set to 1 Hz. This additional signal will act on the reference arm of both the signal interferometer and the reference interferometer at the same time. Another cosine signal φs(t) is set as a test signal, where φs(t) = Scos(2πfst). The test signal amplitude S is set to 1 rad, the frequency fs is set to 1 kHz and the carrier frequency is 20 kHz.
3.1 OPD influence on phase modulation depth
In practice, it is difficult to provide two interferometers with the same OPD, which leads to some demodulation methods that are meaningful only under some restrictive conditions. When the signal interferometer is the ideal phase modulation depth Cs = 2.63rad, changing the OPD of the reference interferometer, $\mathrm{\Delta}C=C_s-C_r$ varies with the $\mathrm{\Delta}OPD={OPD}_s-{OPD}_r$ difference as shown in Fig. 2. OPDs is the OPD of the signal interferometer, OPDr is the OPD of the reference interferometer. In this case, the ΔC is linearly related to the ΔOPD, the larger ΔOPD, the greater the difference in C-value. As shown in Eqs. (10), (11), and Eqs. (12), (13), the EFA normalizes the AC amplitude, and the effect of the different OPD of two interferometers on the C-value can be eliminated.
3.2 Performance of EFA with an additional modulation signal
The traditional EFA can only measure large phase variation signals. Previous studies have shown that the range of measured phase variation is at least π/4 rad to satisfy the normal fitting of the EFA. In our design, we ensure the normal operation of the EFA in measuring small signals by adding compensated sinusoidal signal. This sinusoidal signal introduces a periodic phase shift signal through the PZT applied to the reference arms of the two interferometers. The phase variation range of this signal is π rad, which satisfies the requirement of ellipse fitting. When the phase modulation depth is 2.63 rad from both interferometers of the same OPD, the Lissajous figure of the two quadrature signals obtained after the low-pass filter without adding the test signal is all shown in Fig. 3(a). After adding the 1 rad test signal, the DC and AC components are corrected by EFA, after which the trigonometric operation eliminates the additional sinusoidal modulated signal and the Lissajous figure of the measured result is shown in Fig. 3(b).
Fig. 3. (a). The Lissajous figure of SS1 and SS2 before trigonometric operations, (b) The Lissajous figure of S1 and S2 after trigonometric operations in simulation.
We set different phase modulation depths to show the performance of the EFA. Take the signal interferometer as an example, when the C-values are 1.5 rad, 2.63 rad, and 3.5 rad, the result of the Lissajous figure after the low-pass filter is shown in Fig. 4(a). Due to the existence of nonlinear distortion caused by the unsatisfactory modulation depth, the Lissajous figure by components S1 and S2 is an ellipse before using the EFA. And after EFA, the components S1 and S2 are corrected, and the Lissajous figure is a standard unit circle, as shown in Fig. 4(b).
Fig. 4. (a). The Lissajous figure of SS1 and SS2 before EFA, (b) The Lissajous figure of S1 and S2 after EFA in simulation.
The variation of elliptic parameters with C-value shift is shown in Table 1. a, b, c, d, θ are elliptic fitting parameters before EFA, a’, b’, c’, d’, θ’ are elliptic fitting parameters after EFA, and e and e’ are elliptic eccentricities of the Lissajous figure before EFA and after EFA, respectively. The smaller the ellipse eccentricity is, the closer the ellipse is to a circle. And when e = 0, the Lissajous figure is a standard circle. It can be seen from Table 1 that the elliptical eccentricity turns from a non-zero value to zero after EFA. Meanwhile, the DC offset is effectively suppressed to 0 V, and the AC amplitude is corrected to 1 V, and the effect of phase modulation depth is eliminated. The phase difference θ is corrected from non-strict quadrature state to a strict quadrature state. The corrected Lissajous figure is a standard unit circle so that a pair of strictly quadrature signals with amplitude 1 V is obtained.
Table 1. Comparison of EFA parameters in simulation
3.3 Simulation results
Figure 5 shows the frequency spectrum of the demodulation results of different methods when the phase modulation depth is different. By comparison, the Ref-PGC-EFA method has almost no harmonic distortion compared to the traditional PGC method without using EFA when the C-value is 1.5 rad or 3.5 rad. Meanwhile, the fluctuation of the C-value can significantly reduce noise reduction capacity. The Ref-PGC-EFA method has a better suppression effect on noise.
Fig. 5. The resulting spectrogram of different PGC methods at different C-values in simulation (Blue: Ref-PGC-EFA, Red: Traditional PGC method).
The total harmonic distortion (THD) results of the different PGC demodulation methods are shown in Fig. 6. As the C-value increases from 1 rad to 3.5 rad, the traditional PGC method without EFA has a wide range of THD fluctuations than the Ref-PGC-EFA method. By contrast, the Ref-PGC-EFA methods have the same low level of THD when C-values are different. The above methods will reach the lowest at C = 2.63 rad. The signal to noise and distortion ratio (SINAD) is another method to calculate the results to evaluate the harmonic distortion performance. The SINAD results of different PGC demodulation methods are shown in Fig. 7. When the C-value is between 1 rad and 3.5 rad, the Ref-PGC-EFA method effectively suppresses harmonic distortion and shows higher stability compared to the traditional PGC methods. The traditional method causes severe harmonic distortion when the C-value deviates from 2.63 rad. And the range of SINAD variation is larger for the traditional method. When C = 2.63 rad, the Ref-PGC-EFA’s SINAD reaches 50.27 dB, while the traditional method SINAD reaches 50.36 dB, which is slightly higher than the Ref-PGC-EFA method because the ellipse correction operation could lose some data accuracy. The above data shows that the Ref-PGC-EFA method has superior performance in both THD and SINAD than the traditional method.
4. Experimental setup and results
4.1 Experimental setup
In this work, an experimental setup based on a dual Michelson fiber optic interference system is established. The schematic is shown in Fig. 8. The light emitted from a semiconductor laser (RIO, Orion laser module) with a central wavelength of 1550 nm is used as the light source, where a cosine signal with a frequency of 20 kHz is used to modulate the light source. The light from the laser is divided into two beams of equal intensity by the coupler and enters port 1 of the two optical circulators respectively. Then, light is emitted from port 2 of the two optical circulators into two Michelson interferometers with equal arm length difference (ΔL = 5 m). The sensing arm of the signal interferometer is wrapped around a piezoelectric ceramic (PZT1) and uses a cosine signal driven with an amplitude of 1 rad and a frequency of 1 kHz as the sensing signal. The reference arms of the signal interferometer and reference interferometer are twin wound on the same piezoelectric ceramic (PZT2), and the PZT2 is driven by a sinusoidal signal with an amplitude of π rad and a frequency of 1 Hz as an additional phase modulation signal. Faraday rotating mirror (FRM, 90◦ rotation angle) connected at the end of each optical fiber. Finally, light is reflected from the FRM, and emits from port 3 of the optical circulator. Two interference signals are detected by a photodetector (THORLABS, PDB450C), and two voltage signals are captured by the data acquisition card (NI, Pxle-5170R) and sent to the computer for calculation.
4.2 Impact of different OPDs on demodulation results
In a reference interferometer-based demodulation scheme, the different OPDs of the two interferometers cause different modulation depths and reduce the effect of phase noise reduction. To verify the effect of phase modulation depth on demodulation results generated by different OPDs of two interferometers, several interferometers with different OPDs were designed for experiments. When the OPD of the signal interferometer is maintained at 5 m and the phase modulation depth is at an ideal value, the OPD of the reference interferometer is changed. The results of the SINAD of the two demodulation methods with different OPD are shown in Fig. 9. ΔOPD is the difference between the OPD of the signal interferometer and the reference interferometer. The greater the value of ΔOPD, the greater the SINAD fluctuation of the traditional method, and the fluctuation of the EFA method is much smaller. When ΔOPD = 0, the SINAD of the traditional method is slightly higher than the EFA method because the ellipse correction operation will lose part of the data accuracy. The EFA suppresses harmonic distortion and the effect of C-values caused by different OPDs but does not affect phase noise caused by different OPDs of the two interferometers.
4.3 Experimental results
Take the signal interferometer part as an example. We design two interferometers of the same OPD under different phase modulation depths to evaluate the real-time correction performance of the EFA on the quadrature components. The Lissajous figure before the correction operation is shown in Fig. 10(a). The Lissajous figure after correction operation is shown in Fig. 10(b). The real-time correction operation corrects the ellipse to a unit circle for different C-values. The ellipse fitting parameters, ellipse correction parameters, and ellipse eccentricity are shown in Table 2. After EFA, the DC component is controlled at a very low level. The AC amplitude is corrected to 1 V or approximately equal to 1 V and the elliptical eccentricity e is 0.00219 at maximum. Within the error tolerance, the Lissajous figure is regarded as a standard unit circle. A pair of non-strictly quadrature components are corrected to a pair of strictly quadrature components.
Fig. 10. (a). The Lissajous figure of SS1 and SS2 at different C-values before EFA. (b) The Lissajous figure of S1 and S2 after EFA in the experiment.
Table 2. Comparison of EFA parameters in the experiment
In the reference interferometer part, there is the same correction operation process. The different phase modulation depths of the two interferometers due to the phase modulation depth shift and non-strict equal OPD are corrected to 1 V by the coefficient normalization operation, and the influence of the carrier phase delay is further reduced at the same time.
Figure 11. shows the frequency spectrum of demodulation results for the different PGC methods at different phase modulation depths. When the C-value deviates from 2.63 rad, the demodulation result of the traditional PGC method will have severe harmonic distortion and a significantly higher noise level. The Ref-PGC-EFA method can effectively suppress harmonic distortion under different C-values, and significantly suppress the noise level due to the deviation of the C-value. While maintaining the same maximum dB value for both methods, the SNR obtained by the Ref-PGC-EFA method is higher than the traditional method, and the variation with the fluctuation of the C-value is minimal.
Fig. 11. The resulting spectrogram of different PGC methods at different C-values in the experiment (Blue: Ref-PGC-EFA, Red: Traditional PGC method).
The THD of the different PGC demodulation methods with different results for different C-values is shown in Fig. 12. The EFA is effective in suppressing harmonic distortion compared to the traditional PGC method when the phase modulation depth deviates from the ideal value. When the C-value is in the range of 1 rad to 3.5 rad, the traditional PGC method only achieves an average THD of 5.20% in the demodulation result. And the Ref-PGC-EFA method achieves an average THD of 0.22%.
Figure 13 shows the SINAD of the demodulation results for different demodulation methods with the variation of the phase modulation depths. The Ref-PGC-EFA method can keep the SINAD change at the minimum when the C-value changes compared to the traditional method without EFA. The Ref-PGC-EFA method achieves a maximum SINAD of 25.71 dB in demodulation results, which is slightly lower than the 26.23 dB of the traditional PGC method, in line with the simulation results. When the C-value is in the range of 1 rad to 3.5 rad, the Ref-PGC-EFA method achieves an average SINAD of 23.86 dB, which is 9.95 dB higher than the traditional PGC method.
5. Conclusion
In the reference interferometer demodulation scheme, for the different phase modulation depths generated by the different OPDs of the two interferometers and the effect of random fluctuations in the phase modulation depth, we proposed an improved PGC technique based on a reference interferometer and the EFA and verified by simulation and experiment. The introduction of additional sinusoidal phase modulation ensures the proper use of the EFA in the reference interferometer and ensures that the Ref-PGC-EFA method can demodulate small amplitude signals. By using the EFA to correct the non-strict quadrature signal into the strict quadrature signal, the coefficients are normalized to remove the effect of different phase modulation depths caused by non-strict equal OPDs of the two interferometers and phase modulation depths shift on the demodulation result, which suppresses the nonlinear harmonic distortion of the demodulation result. The Ref-PGC-EFA method has significant advantages over the traditional PGC method based on a reference interferometer in reducing non-linear distortion. When the phase modulation depth ranges from 1 rad to 3.5 rad, experimental results show that the Ref-PGC-EFA method’s average THD is always around 0.22%. By comparison, the average THD only is 5.2% of the traditional PGC method. The Ref-PGC-EFA method achieves an average SINAD of 23.86 dB, which is 9.95 dB higher than the traditional PGC method.
Funding
the University Synergy Innovation Program of Anhui Province (GXXT-2020-050, GXXT-2020-052).
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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