## Abstract

In order to reduce the nonlinearity caused by an error of phase modulation depth, carrier phase delay and non-ideal performance of the low pass filters in the sinusoidal phase modulating interferometer (SPMI), a modified EOM-based SPMI is proposed in this paper to realize real-time normalization of the quadrature components for the arctangent approach of phase generated carrier (PGC-Arctan) demodulation. To verify the effectiveness of the real-time normalization technique, a fixed-phase-difference detection method is presented to evaluate the periodic nonlinearity in real time. The modified EOM-based SPMI is consisted of a monitor interferometer and a probe interferometer. The two interferometers share a reference corner cube, which is mounted on a slowly moving stage, thus periodic interference signals are generated for real-time normalization of the quadrature components in PGC demodulation. Subtracting the demodulated phase of the monitor interferometer from the phase of the probe interferometer, the phase to be measured can be obtained. The fixed-phase-difference detection method is realized by detecting an interference signal with two photodetectors, which are placed at an interval of quarter fringe, and the variation of the fixed-phase-difference can reflect the nonlinear error in PGC demodulation. Experiments of real-time normalization, nonlinear error evaluation of PGC demodulation, and displacement measurement were implemented to demonstrate the effectiveness of the proposed method. Experimental results show that the nonlinear error of phase demodulation reduced to less than ± 1° with real-time normalization, and nanometer displacement measurement is realized.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The phase modulation of an electro-optic modulator (EOM) is produced by a voltage-induced change in the refractive index of the electro-optic material (e.g. lithium niobate [1], silica [2] and polymers [3]). As having the properties of high modulating frequency and high intrinsic modulation bandwidth, the EOM is widely used in the phase modulating interferometers. L. Yan *et al*. applied a period sawtooth voltage to an EOM to construct a linear phase modulated dual-homodyne interferometer without polarizing optics, and two sinusoidal interference signals were obtained for zero-crossing phase detecting [4]. This interferometer can eliminate the periodic nonlinear error and has the ability of anti-interference, while the sawtooth signal is not appropriate to high frequency modulation, thus the measurement speed is limited. An alternative to linear phase modulation is sinusoidal phase modulation, which facilitates high-speed measuring system. For instance, U. Minoni *et al*. proposed a high-frequency sinusoidal phase modulating interferometer (SPMI) with an EOM operating at frequencies up to 10 MHz, and the target can speed up to 1 m/s [5].

For conventional SPMI, the sinusoidal phase modulation can be realized by adding a piezoelectric (PZT) modulator on one arm of the interferometer [6–8], or applying a current modulation to a laser diode [9]. Compared with phase modulation with EOM, the PZT modulation cannot realize high speed measurement because of the limited vibrating frequency of the PZT, and the laser diodes current modulation is accompanied with optical intensity modulation which will cause instability and distortion of phase demodulation output. Coupled with the merits of stable performance and large dynamic range, sinusoidal phase modulation with EOM has been used in optical feedback interferometer [10] and self-mixing interferometer [11] for high speed displacement measurement with large range.

With sinusoidal phase modulation, a phase carrier is generated which up-converts the desired phase signal onto the sidebands of the carrier frequency and the corresponding homodyne detection scheme is phase generated carrier (PGC) demodulation. Due to the distinct advantages of simple construction, high sensitivity and the capability of real-time demodulation, the PGC demodulation has been used to fiber optical interferometric sensors [7, 12] and the measurements of displacement, vibration and surface topography [13–15]. Generally, the fundamental and second harmonic of the interference signals are chosen to acquire a pair of quadrature components containing the desired phase signal. The PGC algorithms employing the differential-and-cross-multiplying approach (PGC-DCM) [16] and the arctangent approach (PGC-Arctan) [17] have been well developed to recover the phase from the quadrature components. The PGC-DCM approach has relatively low harmonic distortion, while its result is influenced by the light intensity disturbance (LID) and the visibility of the interference fringes. The scheme was improved with the PGC-Arctan approach, which is immune to the changes of the LID and the visibility of the interference fringes [18]. However, a phase delay between the carrier component of the detected interference signal and the carrier signal, a deviation of the phase modulation depth used for signal processing from the actual value and the non-ideal performance of the low pass filters will introduce periodic nonlinearity in phase demodulation.

Several methods have been proposed to eliminate the influences of the phase delay and the phase modulation depth. K. Wang *et al*. employed the orthogonal detection method to obtain the phase delay introduced by the fiber-optic interferometric sensor (FIOS), while the signals with non-perfect orthogonality will introduce an error for phase detection [19]. A phase compensator based on PGC-DCM approach was designed by S. C. Huang *et al*. to compensate the phase delay caused by the transmission of the laser light in the fibers for FOISs, while the phase delay correction was carried out offline and the compensate angle cannot be changed automatically when the phase delay varied [20]. In our previous work, a real-time phase delay compensator was proposed to track the change of the phase delay and the proposed compensator is appropriate to both PGC-DCM approach and PGC-Arctan approach [21]. Nevertheless, to ensure the PGC demodulating work properly, the methods mentioned above required the phase modulation depth at certain value (2.37 rad for PGC-DCM approach and 2.63 rad for PGC-Actan approach [22]) or the modulation depth used for signal processing equal to the actual value strictly. J. He *et al*. presented an ameliorated approach based on arctangent function and differential-self-multiplying (DSM) to remove the effect of the phase modulation depth [23], while the results of differential operation will be impacted seriously by the noise of the interference signal in ultra-low speed or static measurement. A. V. Volkov *et al*. proposed a phase modulation depth evaluation and correction technique based on an integral control feedback to provide the phase modulation depth stabilization to a certain predefined optimal value [24], while except for the phase modulation depth, the carrier phase delay and some other factors may occur simultaneously in the experiments. Normalizing the quadrature components obtained by low pass filters can eliminate the influences induced by the factors mentioned above at a time. C. Ni *et al*. presented an ellipse fitting method which can estimate the parameters to correct the measurement error by moving the target to produce several periods of interference signals [25]. However, when the target is in stationary or the movement of the target is much less than half of a laser wavelength, the ellipse fitting will not be implemented correctly or the result of the ellipse fitting is inaccuracy. Considering that the phase delay varies with the optical path, and the phase modulation depth is also not a constant during the measurement (e.g. the phase modulation depth of the EOM will drift with the half-wave voltage which is related to the environmental temperature), a universal method which can depress the influences mentioned above simultaneously and timely is on the urgent need.

In this work, a real-time normalization method of PGC-Arctan demodulation is proposed based on a modified SPMI with an EOM to suppress the periodic nonlinearity. A fixed-phase-difference detection method for evaluating the nonlinearity of the phase demodulation in real time is also presented. The theory of real-time normalization and evaluation of the periodic nonlinearity in PGC-Arctan demodulation is described in section 2. In section 3, the verifying experiments and results are given.

## 2. Theory

#### 2.1 PGC-Arctan demodulation with real-time normalization

With sinusoidal phase modulation, the optical path difference of the interferometer is modulated and a high-frequency sinusoidal phase carrier is introduced to the interferometer. Supposing the carrier signal is

where*A*and

*ω*are the amplitude and frequency of the signal. The interference signal is given by

_{c}*S*

_{0}and

*S*

_{1}are the amplitudes of the DC component and the AC component, respectively;

*z*is the phase modulation depth,

*θ*is the phase delay of the carrier component and

*φ*(

*t*) is the phase to be demodulated. Expending Eq. (1), we can obtain

*J*

_{(2}

_{m}_{-1)}(

*z*) and

*J*

_{2}

*(*

_{m}*z*) denote the odd- and even-order Bessel functions, respectively. The schematic of PGC-Arctan demodulation is shown in Fig. 1. Firstly, multiplying the interference signal

*S*(

*t*) with the carrier

*Vω*(

_{c}*t*) and the second-harmonic carrier

*V*2

*ω*(

_{c}*t*), respectively. Then, filtering out the high-frequency signals including the fundamental carrier and all harmonic carrier frequencies by low-pass filters (LPFs) and dividing the theoretical value of the Bessel functions, a pair of quadrature components can be obtained as

*K*(

_{i}*i*= 1,2) is the total gain of the multiplier and the low-pass filter,

*z*′ is the theoretical value of the phase demodulation depth used for signal processing. Obviously, the Lissajous figure of

*P*

_{1}(

*t*) and

*P*

_{2}(

*t*) should be a circle if

*K*

_{1}=

*K*

_{2},

*z*=

*z*′ and

*θ*=

*kπ*(

*k*= 0,1,2,…). However, in experiment,

*K*

_{1}is not exactly equal to

*K*

_{2}due to the non-ideal performance of the multipliers and the LPFs. Moreover, a deviation of the phase demodulation depth

*z*′ used for signal processing from the actual value

*z*and a phase delay between the carrier and the detected interference signal often occurs, and they may vary with the environmental temperature and the measuring displacement. All of these factors make the Lissajous figure of

*P*

_{1}(

*t*) and

*P*

_{2}(

*t*) to be an ellipse, and a periodic nonlinearity will be induced to the demodulated phase. Supposing that the major axis and minor axis of the ellipse are

*α*

_{1}= −

*J*

_{1}(

*z*)

*K*

_{1}

*S*

_{1}

*A*cos

*θ*/

*J*

_{1}(

*z*′) and

*α*

_{2}= −

*J*

_{2}(

*z*)

*K*

_{2}

*S*

_{1}

*A*cos2

*θ*/

*J*

_{2}(

*z*′), respectively, they are defined as the normalized parameters of the quadrature components. If

*α*

_{1}and

*α*

_{2}are estimated by ellipse fitting of the Lissajous figure or calculating the amplitudes of

*P*

_{1}(

*t*) and

*P*

_{2}(

*t*), Eqs. (4) and (5) can be normalized as Therefore, with real-time normalization, all the factors affecting the phase demodulation are eliminated and the Lissajous figure of the quadrature components transforms to a unit circle. After the operations of division and arctangent, the demodulated phase is described asIt can be deduced that, the method of real-time normalization can also eliminate the influences induced by light intensity disturbance (LID) and the visibility varying of interference fringes in the DCM approach of PGC demodulation.

In order to realize real-time normalization in PGC demodulation, a modified EOM-based SPMI consisting of a monitor interferometer and a probe interferometer is constructed. The two interferometers share a reference corner cube which is mounted on a stage moving back and forth with an ultra-low speed. Then periodic interference signals are generated for both the monitor and probe interferometers, and the normalized parameters can be always estimated to realize real-time normalization in PGC demodulation whether the target is in dynamic or static state.

The schematic of the modified SPMI is shown in Fig. 2. The laser beam emitted from a He-Ne laser is linearly polarized by a polarizer (P) along the vertical direction which is the electro-optic crystal's extraordinary axis direction, and split into two parts by a beam splitter (BS_{1}). The reflected part is introduced to the common reference arm consisting of an EOM and a reference corner cube (CC_{1}) mounted on a slowly moving stage, then propagates to another beam splitter (BS_{2}). The transmitted part passes through the third beam splitter (BS_{3}) and propagates to the measurement corner cube (CC_{2}). The reflected beams from BS_{2} and BS_{3} are combined at the forth beam splitter (BS_{4}) and interfere with each other, then the interference signal of the monitor interferometer is generated and detected by a photodetector (PD_{1}). The transmitted beams from BS_{2} and CC_{2} are combined at BS_{1}, and the interference signal of the probe interferometer is generated and detected by two photodetectors (PD_{2}, PD_{3}) which are placed at an interval of a quarter of interference fringe.

If the modulating signal applied to the EOM is

where*β*is the amplification factor of a high voltage amplifier (HVA) used to drive the EOM. Then the phase shift caused by the EOM is given bywhere

*V*

_{π}is the half-wave voltage (the voltage required to shift the output phase by

*π*radians) of the EOM. Assuming that the initial optical path differences of the monitor interferometer and the probe interferometer are

*l*

_{1}and

*l*

_{2}, respectively, the detected interference signals of the monitor and probe interferometers can be expressed as

*S*

_{0}

*and*

_{j}*S*

_{1}

*(*

_{j}*j*= 1,2) are the amplitudes of the DC component and AC component, respectively,

*θ*is the carrier phase delay,

_{j}*z = πβA*/

*V*is the sinusoidal phase modulation depth,

_{π}*l*(

*t*) is the displacement of the slowly moving stage in the common reference arm, and

*d*(

*t*) is the displacement to be measured.

*φ*

_{1}(

*t*) = 2

*π*[

*l*

_{1}+ 2

*l*(

*t*)]/

*λ*and

*φ*

_{2}(t) = 2

*π*[

*l*

_{2}+ 2

*l*(

*t*) + 2

*d*(

*t*)]/

*λ*are phases to be demodulated corresponding to the monitor interferometer and probe interferometer, respectively. As the demodulated phase varies with the

*l*(

*t*), the quadrature components

*P*

_{1}(

*t*) and

*P*

_{2}(

*t*) in Eqs. (4) and (5) will vary periodically, with which the normalized parameters

*α*

_{1}and

*α*

_{2}can be always estimated by ellipse fitting of the Lissajous figure or calculating the amplitudes of the quadrature components. Then real-time normalization of the PGC demodulation is realized and the periodic nonlinearity can be reduced significantly. Subtracting the demodulated phases of the two interferometers, we can obtain

*ϕ*

_{0}= 2

*π*[

*l*

_{2}−

*l*

_{1}]/

*λ*is the initial phase-difference between the monitor and probe interferometers. It can be seen that, the phases induced by

*l*(

*t*) are counteracted and the phase

*ϕ*(

*t*) is only related to the displacement

*d*(

*t*).

#### 2.2 Real-time evaluation of the periodic nonlinearity using fixed-phase-difference detection method

According to Eqs. (4) and (5), without normalization of the quadrature components in PGC demodulation, the demodulated phase *Φ*(*t*) can be expressed as

*J*

_{1}(

*z*)

*J*

_{2}(

*z′*)

*K*

_{1}cos

*θ*]/[

*J*

_{1}(

*z′*)

*J*

_{2}(

*z*)

*K*

_{2}cos2

*θ*] is the proportional coefficient determined by the deviation of the theoretical phase modulation depth from the actual value, the performance of the multipliers and LPFs, and the carrier phase delay. Equation (14) can be expended and simplified into a linear component and a nonlinear component [23]:It can be seen that, a nonlinear component with the period of π will be introduced if

*ν*≠1, and the amplitude of the nonlinear component reaches its maximum of (

*v*−1)/2 rad when

*φ*(

*t*) =

*kπ*/2 +

*π*/4, (k = 0,1,2…). Generally, the total harmonic distortion (THD) is used to evaluate the nonlinearity of PGC phase demodulation. With a single-frequency sinusoidal signal

*φ*(

*t*), the expansion of Eq. (15) is combined of the fundamental and harmonics of the phase signal, and the THD is defined as the ratio of equivalent root-mean-square (RMS) amplitude of all harmonics to the amplitude of the fundamental frequency. It is an effective method to evaluate the nonlinearity of phase demodulation. However, the phase signal is limited to single-frequency sinusoidal signal, which is inconvenient to implement in practical applications and is not appropriate to real-time evaluation. In addition, the value of THD is related to the initial phase and the dynamic range of the phase signal, and the error of phase demodulation cannot be reflected directly.

In order to realize real-time evaluation of the nonlinear phase error caused by demodulation, we use two photodetectors (PDs) placed at different probe points to detect the same interference signal. Supposing that the phase-difference induced by the interference fringe interval is *δ*, according to Eq. (15), the phase demodulated by another PD is

*∆*Φ comprises a fixed phase

*δ*and a nonlinear component caused by

*v*≠1, and the amplitude of the nonlinear component reaches its maximum of (

*v*−1) rad when |sin

*δ*| = 1 (

*δ*= 90° or 270°). Therefore, for the modified EOM-based SPMI in Fig. 2, PD

_{2}and PD

_{3}are placed at an interval of quarter fringe to generate a fixed-phase-difference of 90° or 270° and a variation of the fixed-phase-difference can be detected if the nonlinear error is exist. As the common reference corner cube is mounted on a slowly moving stage, the phase-difference can be observed all the time during the measurement, thus the value of (

*v*−1) can be obtained. The phase error caused by nonlinear demodulation can be deduced from the phase-difference

*∆*Φ no matter the target is in dynamic or static state.

## 3. Experiments and results

#### 3.1 Experimental setup of the modified EOM-based SPMI

The experimental setup of the modified SPMI with an EOM was constructed as shown in Fig. 3. The laser source was a single-frequency He-Ne laser (XL80, Renishaw) with the wavelength of 632.990577 nm. The sinusoidal phase modulation was introduced by an EOM (EO-PM-NR-C1, Thorlabs) with lithium niobate crystal, and the displacement *d*(*t*) was provided by a nano-positioning stage (P-753.1CD, Physik Instrument) with a range of 15 μm and a resolution of 0.05 nm. The movement of the common reference corner cube *l*(*t*) is provided by another P-753.1CD stage, which moves back and forth with a speed of 100 nm/s in the range of 15 μm. The mechanical vibration induced by the moving stage with such a low speed can be neglected. The carrier signal and the detected interference signals were converted into digital signals by a 20 MHz simultaneous 4-chanel data acquisition card (PCI-9812, Adlink), and the real-time normalization and the periodic nonlinearity evaluation methods of PGC demodulation were processed in a computer based on Labview workbench.

#### 3.2 Experimental verification for real-time normalization in PGC demodulation

During the displacement measurement, the quadrature components of the two interferometers varied continuously with the moving of CC_{1}, and the normalization of the quadrature components is implemented automatically when either amplitude of the quadrature components has a deviation of 0.01 from 1 (i.e. the corresponding phase error is larger than 0.3°). An experiment was performed to verify the effectiveness of real-time normalization in PGC demodulation. In order to observe the effect of real-time normalization clearly, a discrepancy of 0.1 rad was set between the value of the phase demodulation depth used for signal processing and the actual value. CC_{1} was moving with a speed of 100 nm/s, and CC_{2} was in stationary. As shown in Fig. 4(a) and 4(c), *φ*_{1}(*t*) and *φ*_{2}(*t*) are demodulated phases of the monitor interferometer and the probe interferometer, respectively, and *φ*_{2}(*t*)-*φ*_{1}(*t*) is the phase-difference between them which is set as 200°. Before real-time normalization, the Lissajous figure of the quadrature components is an ellipse and the phase-difference varies periodically in 200 ± 10°. Figure 4(b) and 4(c) show that, after real-time normalization from the moment *t*_{1}, the Lissajous figure transforms into a unit circle and the phase-difference varies in 200 ± 1°. Figure 4(d) shows the corresponding demodulated displacements with CC_{1} moving from 0 to 1.4 μm. *d*_{1}(*t*) and *d*_{2}(*t*) are demodulated displacements of the monitor and the probe interferometers, respectively, and *d*_{2}(*t*)−*d*_{1}(*t*) is the displacement difference between them. We can see that a periodic nonlinear error is added on the demodulated displacements of the two interferometers, and the displacement difference fluctuates in ± 10 nm before real-time normalization. After real-time normalization, the displacement difference is maintained to ± 1 nm.

#### 3.3 Experimental verification for real-time evaluation of the periodic nonlinearity

According to section 2.2, the periodic nonlinearity of PGC demodulation can be evaluated in real time by the fixed-phase-difference detection method and the optimal value of the phase-difference is 90° or 270°. In order to verify the effectiveness of the fixed-phase-difference detection method, the THD was also analyzed for the same demodulated displacement. Technically, the fixed-phase-difference detection method has no special requirement for the movement forms of the target, while in order to compare with the THD analysis, a sinusoidal signal is applied to the P-753.CD stage to make the probe corner cube CC2 has a sinusoidal movement with a frequency of 50 Hz and dynamic range of 10 rad. The two photodetectors of PD_{2} and PD_{3} in Fig. 3 were used to evaluate the periodic nonlinearity of PGC demodulation, and the phase-difference should be a fixed value around 90° in ideal demodulation. The phase-difference can be observed in real time. At the same time, the demodulated phase of PD_{2} was analyzed to obtain the THD.

In the first experiment, so as to observe the phenomenon of nonlinear demodulation clearly, a discrepancy of 0.4 rad was set between the value of the phase demodulation depth used for signal processing and the actual value. The experimental results are shown in Fig. 5, in which *φ*_{2}(*t*) and *φ*_{3}(*t*) are demodulated phases of PD_{2} and PD_{3}, respectively, and *φ*_{2}(*t*)−*φ*_{3}(*t*) is the phase-difference between them. Experimental results show that without real-time normalization, the phase-difference varies periodically in 90 ± 35° [Fig. 5(a)], and a periodic nonlinear error is added on the wrapped phase of *φ*_{2}(*t*) [Fig. 5(b)]. Figure 5(c) shows that the THD of the wrapped phase of *φ*_{2}(*t*) is 2.260%, and the signal-to-noise-and-distortion (SINAD) is 29.12 dB. In the second experiment, the quadrature components were normalized in real time and the experimental results are shown in Fig. 6. It can be seen that the phase-difference reduces to 90 ± 1° [Fig. 6(a)], and the wrapped phase of *φ*_{2}(*t*) becomes a relative smooth sinusoidal signal [Fig. 6(b)]. The corresponding THD is reduced to 0.933%, and the SINAD is improved to 40.40 dB as shown in Fig. 6(c).

#### 3.4 Nanometer displacement measurement

Nanometer displacement measurement was performed to verify the method of real-time normalization in PGC demodulation. CC_{1} was moving with a speed of 100 nm/s during the measurement and the quadrature components were normalized in real time. The nanometer displacement measurement was conducted by moving CC_{2} with a step of 20 nm for 256 times (5.12 μm), and the measurement results are shown in Fig. 7. The displacement of the stage, the demodulated displacement and the deviation of the displacements between them are shown in Fig. 7(a). According to the measurement results, with real-time normalization, the average of the deviation is almost zero. The maximum error is 1.25 nm and the standard deviation is 0.56 nm. Figure 7(b) shows the FFT analysis of the displacement deviation shown in Fig. 7(a). According to Eq. (15), if a nonlinear error with the period of π occurs in the phase demodulation, there will be a peak at the second harmonic components of fringe. As shown in Fig. 7(b), no apparent peak emerges at the second harmonic components of fringe, and displacement measurement with nanometer accuracy can be realized without detectable nonlinearity beyond 0.1 nm in the modified EOM-based SPMI.

## 4. Conclusion

In this paper, real-time normalization in PGC demodulation has been realized in a modified EOM-based SPMI consisting of a monitor interferometer and a probe interferometer with a common slowly moving reference corner cube. In addition, a fixed-phase-difference detection method is proposed to evaluate periodic nonlinearity in PGC-Arctan demodulation. Several experiments were implemented for the feasibility verification of the proposed methods. The experimental results show that: (1) The phase error induced by the nonlinear demodulation can be significantly reduced by normalizing the quadrature components in real time. (2) With real-time normalization, the phase error caused by nonlinear demodulation is reduced to less than ± 1° and the corresponding THD is 0.933% with SINAD of 40.40 dB for the test sinusoidal movement with the frequency of 50 Hz and dynamical range of 10 rad. (3) Nanometer displacement measurement is realized with real-time normalization in PGC demodulation. Limited by the sampling rate of the data acquisition card and the data processing capability of the host computer, the measurement speed of the proposed SPMI in this paper is limited to several millimeters per second. If using high-speed data acquisition card with data processed by high performance FPGA, the proposed SPMI has a great potential in industrial applications such as machine-tool control and displacement sensor calibration.

## Funding

National Natural Science Foundation of China (NSFC) (51527807, 51375461 and 51475435); Natural Science Foundation of Zhejiang Province (LZ18E050003); Program for Changjiang Scholars and Innovative Research Team in University (IRT_17R98).

## Acknowledgments

Authors acknowledge the financial support from the 521 Talent Project and Science Foundation of Zhejiang Sci-Tech University.

## References and links

**1. **A. Guarino, G. Poberaj, D. Rezzonico, R. Degl’Innocenti, and P. Günter, “Electro-optically tunable microring resonators in lithium niobate,” Nat. Photonics **1**(7), 407–410 (2007). [CrossRef]

**2. **Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature **435**(7040), 325–327 (2005). [CrossRef] [PubMed]

**3. **D. H. Park, V. R. Pagán, T. E. Murphy, J. Luo, A. K. Jen, and W. N. Herman, “Improved carrier-to-sideband ratio for free space millimeter wave-coupled electro-optic polymer high speed phase modulators,” in Proceedings of CLEO: Science and Innovations (Optical Society of America, 2015), STh3F. 7.

**4. **L. Yan, B. Chen, Z. Chen, J. Xie, E. Zhang, and S. Zhang, “Phase-modulated dual-homodyne interferometer without periodic nonlinearity,” Meas. Sci. Technol. **28**(11), 115006 (2017). [CrossRef]

**5. **U. Minoni, E. Sardini, E. Gelmini, F. Docchio, and D. Marioli, “A high-frequency sinusoidal phase-modulation interferometer using an electro-optic modulator: Development and evaluation,” Rev. Sci. Instrum. **62**(11), 2579–2583 (1991). [CrossRef]

**6. **O. Sasaki and K. Takahashi, “Sinusoidal phase modulating interferometer using optical fibers for displacement measurement,” Appl. Opt. **27**(19), 4139–4142 (1988). [CrossRef] [PubMed]

**7. **Q. Lin, L. Chen, S. Li, and X. Wu, “A high-resolution fiber optic accelerometer based on intracavity phase-generated carrier (PGC) modulation,” Meas. Sci. Technol. **22**(1), 015303 (2011). [CrossRef]

**8. **Q. Li, H. Huang, F. Lin, and X. Wu, “Optical micro-particle size detection by phase-generated carrier demodulation,” Opt. Express **24**(11), 11458–11465 (2016). [CrossRef] [PubMed]

**9. **H. Y. F. S. C. Huang and F. H. Hwang, “An improved sensitivity normalization technique of PGC demodulation with low minimum phase detection sensitivity using laser modulation to generate carrier signal,” Sensor Actuat. A-Phys. **191**, 1–10 (2013).

**10. **N. Servagent, T. Bosch, and M. Lescure, “Design of a phase-shifting optical feedback interferometer using an electrooptic modulator,” IEEE J. Sel. Top. Quantum Electron. **6**(5), 798–802 (2000).

**11. **W. Xia, Q. Liu, H. Hao, D. Guo, M. Wang, and X. Chen, “Sinusoidal phase-modulating self-mixing interferometer with nanometer resolution and improved measurement velocity range,” Appl. Opt. **54**(26), 7820–7827 (2015). [CrossRef] [PubMed]

**12. **C. K. Kirkendall and A. Dandridge, “Overview of high performance fibre-optic sensing,” J. Phys. D Appl. Phys. **37**(18), R197–R216 (2004). [CrossRef]

**13. **T. Suzuki, K. Kobayashi, and O. Sasaki, “Real-time displacement measurement with a two-wavelength sinusoidal phase-modulating laser diode interferometer,” Appl. Opt. **39**(16), 2646–2652 (2000). [CrossRef] [PubMed]

**14. **T. Suzuki, “Real-time vibration measurement using a feedback type of laser diode interferometer with an optical fiber,” Opt. Eng. **36**(9), 2496 (1997). [CrossRef]

**15. **O. Sasaki, J. Xin, S. Choi, and T. Suzuki, “Profile measurement of thin films by backpropagation of multiple-wavelength optical fields with two sinusoidal phase-modulating interferometers,” Opt. Commun. **356**, 578–581 (2015). [CrossRef]

**16. **A. Dandridge, A. Tveten, and T. Giallorenzi, “Homodyne demodulation scheme for fiber optic sensors using phase generated carrier,” IEEE J. Quantum Electron. **18**(10), 1647–1653 (1982). [CrossRef]

**17. **T. R. Christian, P. A. Frank, and B. H. Houston, “Real-time analog and digital demodulator for interferometric fiber optic sensors,” North American Conference on Smart Structures and Materials (International Society for Optics and Photonics), 324–336 (1994). [CrossRef]

**18. **L. Wang, M. Zhang, X. Mao, and Y. Liao, “The arctangent approach of digital PGC demodulation for optic interferometric sensors,” Proc. SPIE **6292**, 62921E (2006). [CrossRef]

**19. **K. Wang, M. Zhang, F. Duan, S. Xie, and Y. Liao, “Measurement of the phase shift between intensity and frequency modulations within DFB-LD and its influences on PGC demodulation in a fiber-optic sensor system,” Appl. Opt. **52**(29), 7194–7199 (2013). [CrossRef] [PubMed]

**20. **S.-C. Huang and H. Lin, “Modified phase-generated carrier demodulation compensated for the propagation delay of the fiber,” Appl. Opt. **46**(31), 7594–7603 (2007). [CrossRef] [PubMed]

**21. **S. Zhang, L. Yan, B. Chen, Z. Xu, and J. Xie, “Real-time phase delay compensation of PGC demodulation in sinusoidal phase-modulation interferometer for nanometer displacement measurement,” Opt. Express **25**(1), 472–485 (2017). [CrossRef] [PubMed]

**22. **C. McGarrity and D. A. Jackson, “Improvement on phase generated carrier technique for passive demodulation of miniature interferometric sensors,” Opt. Commun. **109**(3–4), 246–248 (1994). [CrossRef]

**23. **J. He, L. Wang, F. Li, and Y. Liu, “An ameliorated phase generated carrier demodulation algorithm with low harmonic distortion and high stability,” J. Lightwave Technol. **28**(22), 3258–3265 (2010).

**24. **A. V. Volkov, M. Y. Plotnikov, M. V. Mekhrengin, G. P. Miroshnichenko, and A. S. Aleynik, “Phase Modulation Depth Evaluation and Correction Technique for the PGC Demodulation Scheme in Fiber-Optic Interferometric Sensors,” IEEE Sens. J. **17**(13), 4143–4150 (2017). [CrossRef]

**25. **C. Ni, M. Zhang, Y. Zhu, C. Hu, S. Ding, and Z. Jia, “Sinusoidal phase-modulating interferometer with ellipse fitting and a correction method,” Appl. Opt. **56**(13), 3895–3899 (2017). [CrossRef] [PubMed]