## Abstract

As the phase delay between the carrier component of the detected interference signal and the carrier has adverse effect for phase generated carrier (PGC) demodulation, it is essential to compensate the phase delay to improve the accuracy of precision displacement measurement in sinusoidal phase-modulation interferometer (SPMI). In this paper, a real-time phase delay compensation method is proposed by regulating a compensating phase introduced to the carrier to maximize the output of the low pass filter so as to make the carrier synchronize with the interference signal. The influence of phase delay for PGC demodulation is analyzed and the method for real-time phase delay compensation is described in detail. The simulation of the method was performed to verify the validity of the phase delay compensation algorithm. A SPMI using an EOM was constructed and several comparative experiments were carried out to demonstrate the feasibility of the proposed method. The experimental results show that the phase delay can be compensated accurately in real time, and nanometer accuracy is achieved for precision displacement measurement.

© 2017 Optical Society of America

## 1. Introduction

The phase generated carrier (PGC) homodyne detection scheme is widely used in optical interferometers for its high sensitivity, wide dynamic range and good linearity [1–4]. The phase demodulation of sinusoidal phase modulating interferometer (SPMI) is a typical application of PGC technique [5], and SPMI has seen tremendous growth in the measurement of displacement, vibration, surface topography and so on because of its simple configuration, high accuracy and immunity to the optical and electronic noises [6–9]. PGC homodyne detection needs high frequency phase carrier signals with fixed amplitude, which can be generated by laser diodes with current modulation or external phase modulators. Laser diodes current modulation will unavoidably generate an accompanied optical intensity modulation, which causes distortion of PGC demodulation output [10–12]. External phase modulation can be realized by a piezoelectric (PZT) phase modulator or an electro-optic modulator (EOM). For PZT modulation, the mechanical vibration will result in measurement error, and the measuring speed is limited by the vibrating frequency of PZT. For phase modulation using EOM, the modulation frequency can be further increased up to tens of megahertz, so the measuring speed can be improved greatly. With the advantages of stable performance and the absence of moving parts, SPMI with EOM has been used in optical interferometers for precision displacement measurement [13–15].

In the PGC demodulation scheme, the high-frequency carrier phase signal applied to SPMI up-converts the desired phase-shift signal corresponding to the measured displacement onto the sidebands of the carrier frequency. A pair of quadrature components containing the phase-shift signal can be acquired by extracting the fundamental and the second harmonic of PGC interference signal. In order to recover the phase-shift from the quadrature components, the PGC algorithm employing the differential-and-cross-multiplying approach (PGC-DCM) [1] or the arctangent approach (PGC-Arctan) [16] has been well developed. However, due to the time delay induced by optical path transmission, circuit transmission and photoelectric conversion, a phase delay between the carrier component of the detected interference signal and the carrier often occurs, which will result in adverse effect for PGC demodulation and the demodulated result could be completely wrong in some special cases. What’s more, for the same time delay, the phase delay will increase significantly with increasing modulating frequency.

Several methods have been proposed to eliminate the influence of the phase delay. U. Minoni *et al*. employed a phase shifter using a composite amplifier structure in order to synchronize the photo detector signal to the carrier signals [5,17], while the full analog circuit is rather complex and susceptible to electromagnetic interference and temperature. A carrier phase advance technique was developed by T. Lan *et al*., in which a phase-advanced carrier was generated according to the time delay estimated beforehand [18]. However, the time delay introduced by every part of the demodulation system (e.g. the amplifier, the phase modulator, the detector and so on) is difficult to be measured accurately which limits the accuracy of the phase delay compensation. K. Wang *et al*. adopted the orthogonal detection method to obtain the phase delay in the measurement of the phase shift between intensity and frequency modulations with distributed feedback lasers to eliminate the errors induced by accompanied optical-intensity modulation in fiber-optic interferometric sensor (FOIS) [19]. An orthogonal signal of the carrier was needed for the detection method, and the system calibration was required to overcome the problem of the π ambiguity. S. C. Huang *et al*. designed a phase compensator based on PGC-DCM approach to compensate the phase delay caused by the transmission of the laser light in the fibers for FOISs [20]. The phase delay was determined by using another PZT phase modulator, and the phase delay correction was carried out before the actual detection of FOIS. So the compensate angle cannot be changed automatically when the phase delay varies with optical path, modulating frequency and so on, and the method can be only applied to the PGC-DCM approach.

In this paper, a new method of phase delay compensation for the PGC demodulation is proposed to compensate the phase delay in real time and it is suitable to both the PGC-DCM and PGC-Arctan approaches. The influence and compensation of phase delay in PGC demodulation are analyzed in section 2. Simulations and analysis of the phase delay compensation are present in section 3. The experimental setup and the verified experiments are given in section 4. In section 5, the error sources and the uncertainty of the interferometer are discussed.

## 2. Theory

#### 2.1 The influence of phase delay in PGC demodulation

In a SPMI, the sinusoidal phase modulation can be introduced by laser diodes with current modulation or external phase modulators. The phase modulating signal of SPMI can be taken as the carrier in PGC demodulation. Assuming that the carrier is expressed as

where*A*and

*ω*are the amplitude and frequency of the carrier, respectively. When the sinusoidal phase modulation is introduced into the SPMI, the optical path difference of the interferometer is modulated, and the interference signal which carries the information of the measured displacement is given bywhere

_{c}*S*

_{0}and

*S*

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

*z*cos

*ω*is the carrier component caused by phase modulation;

_{c}t*z*denotes the sinusoidal phase modulation depth;

*φ*(

*t*) is the phase shift to be demodulated which is composed of the initial phase and the phase shift caused by the measured displacement.

Expanding Eq. (2), we obtain

*J*

_{(}

_{2m-1}_{)}(

*z*) and

*J*(

_{2m}*z*) denote the odd- and even-order Bessel functions, respectively.

In order to demodulate the phase shift *φ*(*t*), the PGC-DCM approach and the PGC-Arctan approach are generally adopted. The schematic of the two approaches are shown in Fig. 1. The second-harmonic carrier

*V*(

_{ωc}*t*) by the doubled frequency transformer (2FT). The interference signal

*S*(

*t*) is multiplied with

*V*(

_{ωc}*t*) and

*V*

_{2}

*(*

_{ωc}*t*), respectively. The high-frequency signals including the fundamental carrier and all harmonic carrier frequencies are filtered out by low-pass filters (LPFs), and then a pair of quadrature components can be obtained by:

*K*

_{1}and

*K*

_{2}denote the gains of the multiplier and the low-pass filter.

For the PGC-DCM approach, after the operations of differential-and-cross-multiplying, subtracting, integrating and high-pass filtering, the demodulated signal is described as

For the PGC-Arctan approach, after the operations of division and arctangent, the demodulated signal is described as

Considering the time delay caused by the optical path, circuit transmission and the photodetection, Eq. (2) can be rewritten as

*θ = ω*

_{c}Δ

*t*is the phase delay introduced by the time delay Δ

*t*. It’s obvious that the phase delay increases significantly with increasing carrier frequency

*ω*, especially in high speed measurement. The quadrature components become:

_{c}The demodulated signal of the PGC-DCM approach becomes

*θ*cos2

*θ*≤1 is induced by the phase delay

*θ*, which will result in a decreased signal-to-noise ratio (SNR) of the demodulated signal, and the amplitude of the demodulated signal will be zero in the most serious case.

The demodulated signal of the PGC-Arctan approach becomes

*θ*, and the value of

*P*

_{1}or

*P*

_{2}will be zero when cos

*θ =*0 or cos2

*θ =*0 in which cases the phase cannot be demodulated rightly.

According to the above analysis, the phase delay has adverse effects on the demodulation of phase shift corresponding to the measured displacement in SPMI. Thus this problem should be solved for precision displacement measurement.

#### 2.2 Method of real-time phase delay compensation

In order to eliminate the influences caused by the phase delay *θ*, a compensating phase is introduced to the carrier to make the interference signal synchronize with the carrier. Assuming the compensating phase is *α*, the carrier becomes

*B*

_{1}=

*-K*

_{1}

*S*

_{1}

*AJ*

_{1}(

*z*)sin

*φ*(

*t*) and

*B*

_{2}=

*-K*

_{2}

*S*

_{1}

*AJ*

_{2}(

*z*)cos

*φ*(

*t*) are constant coefficients at a given time

*t*=

*t*

_{0}, and the values of

**|**

*P*

_{1}

**|**and

**|**

*P*

_{2}

**|**are considered as functions of

*α*. The values of

**|**

*P*

_{1}

**|**and

**|**

*P*

_{2}

**|**will approach to maximum on the condition of

**|**cos(

*θ*-

*α*)

**|**= 1, then the influence of the phase delay can be eliminated. Since maximizing

**|**

*P*

_{1}

**|**is the sufficient condition of maximizing

**|**

*P*

_{2}

**|**, a compensating phase measurement (CPM) algorithm is designed to determine the accurate value of

*α*corresponding to the maximum of

**|**

*P*

_{1}

**|**by using look-up tables (LUT).

Although the phase delay *θ* ranges from 0 to 2π, the maximum of **|***P*_{1}**|** can be always found when *α* changes from 0 to π. If *θ*∈[0, π), the value of **|***P*_{1}**|** reaches to maximum when *α = θ*, and if *θ*∈[π, 2π), **|***P*_{1}**|** reaches to maximum when *α = θ*-*π*. Therefore, the change range of the compensating phase *α* is only from 0 to π. The compensating phase measurement algorithm is performed as follows:

- Step 1: Coarse measurement, regulate the compensating phase from 0 to 180̊ with step of 1̊, and record the value of the compensating phase corresponding to the maximum of
**|**P_{1}**|**which is denoted as*α*^{1}; - Step 2: Fine measurement, regulate the compensating phase from
*α*^{1}-1̊ to*α*^{1}+ 1̊ with step of 0.1̊, and record the value of the compensating phase corresponding to the maximum of**|**P_{1}**|**which is denoted as*α*^{2}; - Step 3: Accurate measurement, regulate the compensating phase from
*α*^{2}-0.1̊ to*α*^{2}+ 0.1̊ with step of 0.01̊, and record the value of the compensating phase corresponding to the maximum of**|**P_{1}**|**which is denoted as*α*^{3}.*α*^{3}is the measured compensating phase for phase delay compensation of PGC demodulation.

Because the change of the phase delay is generally small during actual displacement measurement, the coarse measurement of compensating phase is executed before each displacement measurement starts. The fine measurement and accurate measurement of compensating phase are executed all the time so as to keep in pace with the changes of the phase delay during displacement measurement.

As shown in Fig. 2, after the compensating phase is obtained by the CPM module, it is induced to the carrier by a phase shifter. The phase delay compensator (PDC) is composed of the compensating phase measurement module and the phase shifter. The compensation process is implemented by regulating the compensating phase introduced to the carrier to maximize the output of the low pass filter so as to make the carrier synchronize with the interference signal.

As mentioned above, the resolution of compensating phase is 0.01̊, and the progress of PDC is executed repeatedly during the displacement measurement. Thus the proposed compensation method is sensitive to every change of the phase delay caused by the varying optical path and electronics condition, so the real-time compensation of the phase delay is realized. What’s more, the PDC is performed before the executing of PGC-DCM and PGC-Arctan approaches, so it’s suitable to both of them for PGC demodulation.

## 3. Simulation and analysis

In SPMI, assuming that the phase delay *θ* = 30° and the carrier frequency is 10 kHz. The measured target moves 950 nm (about 3 interference fringes) at a constant velocity of 100 μm/s. Thus, the time of the motion is 9.5 ms and the Doppler frequency caused by the motion is about 316 Hz. The cutoff frequency of the LPFs is 500Hz, which is lower than one tenth of the carrier frequency to eliminate the effect of the carrier components sufficiently. The sampling rate is 100 kHz and the modulation depth is set to *z* = 2.63. Comparative simulation experiments were carried out to verify the effectiveness of the proposed method, and the PGC-Arctan approach is adopted here for PGC demodulation.

Considering the compensating phase *α*, Eqs. (10) and (11) can be transformed into

The peak-to-peak values of *P*_{1}/*J*_{1}(*z*) and *P*_{2}/*J*_{2}(*z*) are denoted as *V _{pp}*

_{1}and

*V*

_{pp}_{2}, and we set

*K*

_{1}=

*K*

_{2}in the experiments. After phase delay compensation,

**|**cos(

*θ-α*)

**|**=

**|**cos2(

*θ-α*)

**|**= 1 is satisfied, so

*V*

_{pp}_{1}and

*V*

_{pp}_{2}should be equal. Thus the equality of

*V*

_{pp}_{1}and

*V*

_{pp}_{2}can be taken as the evaluation criterion for the effectiveness of the compensation. As shown in Fig. 3(a),

*V*

_{pp}_{1}= 1.600,

*V*

_{pp}_{2}= 0.924, then

*V*

_{pp}_{1}≈1.73

*V*

_{pp}_{2}before compensation (because cos

*θ*≈1.73cos2

*θ*in the case of

*θ*= 30°). After compensation as shown in Fig. 3(b),

*V*

_{pp}_{1}= 1.847,

*V*

_{pp}_{2}= 1.849, then

*V*

_{pp}_{1}≈

*V*

_{pp}_{2}, which indicates that the phase delay compensation method is effective.

The simulation results of displacement measurement are shown in Figs. 3(c) and 3(d). Before compensation, the demodulated displacement increases non-linearly with the time and the displacement error fluctuates periodically in ± 15 nm, while the theoretical and demodulated displacement are synchronous, and the error between them is less than ± 0.02 nm after compensation.

## 4. Experiments and results

#### 4.1 The SPMI with an EOM

In order to verify the feasibility of the proposed method, a SPMI with an EOM based on Michelson interferometer is designed for precision displacement measurement as shown in Fig. 4. The laser beam emitted from a He-Ne laser becomes linearly polarized light after a polarizer (P). It is split into two parts by a beam splitter (BS). The reflected part passes through an EOM and propagates to a reference corner cube (M_{1}), and the transmitted one is directed to a measurement corner cube (M_{2}). The beams reflected by M_{1} and M_{2} are recombined in BS where they interfere with each other. The interference signal is detected by a photodetector (PD).

In Fig. 4, *l _{r}* is the optical path between BS and M

_{1}, and

*l*

_{0}is the optical path between BS and M

_{2}. M

_{2}is mounted upon a measured object and displaced by a distance

*d*(

*t*) along the optical axis. The phase of the laser beam is modulated linearly by applying a modulating signal to the EOM, and the voltage required to shift the output phase by

*π*radians is called the half-wave voltage (

*V*

_{π}). The modulating signal applied to the EOM is expressed as

*Vω*(

_{c}*t*) is the output of a generator which is taken as the carrier, and

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

The interference signal detected by PD is expressed as

Equation (21) has the same form as Eq. (2), and *z = πβA*/*V _{π}* is the sinusoidal phase modulation depth. The phase shift to be demodulated can be expressed as

*d*(

*t*) and the initial phase of the interferometer

*φ*(

_{0}= 4π*l*-

_{0}*l*)

_{r}*/λ*.

The experimental setup of the SPMI with an EOM was constructed as shown in Fig. 5. 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), and the measured displacement *d*(*t*) was provided by a nano-positioning linear stage (P-753.1CD, Physik Instrument) with a range of 15μm and a resolution of 0.05 nm. The carrier *Vω _{c}*(

*t*) were converted into digital signals by a 20 MHz simultaneous 4-chanel data acquisition card (PCI-9812, Adlink), and the real-time phase delay compensation algorithm was processed in a computer.

#### 4.2 Experimental verification of real-time phase delay compensation

Three displacement measurement experiments before and after phase delay compensation with different phase delays were performed in this section. In the experiments, the measured displacement of M_{2} was provided by moving the P-753.1CD stage with 50 nm/s for a distance of 950 nm. The modulation frequency of the EOM is 1 kHz, the sampling rate of the acquisition card is 100 kHz, the cutoff frequency of the LPFs is 10Hz, and set *K*_{1} = *K*_{2}. Since the input voltage range of the acquisition card is −5 V~ + 5 V and the half-wave voltage of the EOM is 135 V, the corresponding modulation depth is *z* = 2.33 according to Eq. (21).

The first experiment was carried out to determine the inherent phase delay of the measurement system. The phase shift was demodulated in real time when the P-753.1CD stage was moving. The measured displacement before phase delay compensation was obtained by accumulating the demodulated phase without running the phase delay compensator. The displacement after phase delay compensation was given according to the phase delay measurement and compensation steps described in section 2.2. The experimental results are shown in Fig. 6. The real-time measured compensating phases are shown in Fig. 6(b), which have an average of *α*_{0} = 0.940°. In fact, the phase delay of the system is *θ*_{0}≈*α*_{0} + 180° = 180.940° and the phase of 180° is introduced by the inverting amplifier of the HVA. In Fig. 0.6(a) we can see that *V _{pp}*

_{1}is close to

*V*

_{pp}_{2}before phase delay compensation, since the inherent phase delay of the measurement system is close to

*kπ*(

*k*= 0,1,2…) and

**|**cos

*θ*

_{0}

**|**≈

**|**cos2

*θ*

_{0}

**|**≈1, the compensation effect is not obvious. The displacement measurement results and the error (

*ε*) between the measured displacement and the displacement of the linear stage are shown in Fig. 6(c) and (d). The max error and the standard deviation of the error are −2.01 nm and 0.63 nm before phase delay compensation. After compensation, they become 1.19 nm and 0.56 nm, respectively.

The second and third experiments were performed to verify the feasibility of the proposed method at different phase delays. In practical applications, especially in long distance transmission of the laser light in the fibers, the phase delay could be any value from 0 to 2*π*. In order to illuminate the effectiveness of the proposed method at any phase delay and observe the compensation effect more clearly, an additional phase delay is introduced by a dual channel function generator (Tektronix, AFG 3102). One channel (CH1) provides the modulating voltage for the EOM, and the signal generated by another channel (CH2) is sent to the acquisition card directly and taken as the fundamental carrier for PGC demodulation. The signals generated by CH1 and CH2 have been set as the same amplitude and frequency, and the given phase difference between them can be considered as an additional phase delay.

In the second experiment, the additional phase delay was *δ*_{1} = 30°. Similar to the steps of the first experiment, the experimental results before and after phase delay compensation are shown in Fig. 7. The phase delay of the experimental system became *θ*_{1}≈*θ*_{0}-*δ*_{1} = 150.940°. From Fig. 7(b), the average of the measured compensating phases is α_{1} = 150.939°, thus α_{1}≈*θ*_{1} is obtained. The changes of *V _{pp}*

_{1}and

*V*

_{pp}_{2}before and after phase delay compensation are shown in Fig. 7(a) and 7(b). Before compensation,

*V*

_{pp}_{1}= 2.15,

*V*

_{pp}_{2}= 1.30, and

*V*

_{pp}_{1}≈1.65

*V*

_{pp}_{2}which is in accordance with the relation of

**|**cos

*θ*

_{1}

**|**≈1.65

**|**cos2

*θ*

_{1}

**|**. After compensation,

*V*

_{pp}_{1}= 2.45,

*V*

_{pp}_{2}= 2.44. The displacement measurement results are shown in Figs. 7(c) and 7(d). Before compensation, the max error is 12.89 nm and the corresponding standard deviation is 8.85 nm, while they decrease to −2.05 nm and 0.48 nm after compensation. The experimental results demonstrate the effectiveness of the phase delay compensation method.

In the third experiment, the additional phase delay of *δ*_{2} = 90° was introduced to the system. Figure 8 shows the experimental results and the phase delay of the system became *θ*_{2}≈*θ*_{0}-*δ*_{2} = 90.940°. From Fig. 8(b), the average of the measured compensating phase delay is *α*_{2} = 90.943°, which is in accordance with the phase delay *θ*_{2}. Because cos*θ*_{2}≈-0.017, *V _{pp}*

_{1}is close to zero, as shown in Fig. 8(a), and the measured displacement cannot be demodulated rightly before compensation [Fig. 8(c)]. After compensation, the difference of

*V*

_{pp}_{1}and

*V*

_{pp}_{2}is less than 0.01 [Fig. 8(b)], meanwhile the max displacement error and standard deviation reduce to −1.26 nm and 0.47 nm, respectively [Fig. 8(d)]. This shows that the phase delay compensation is essential for PGC demodulation, especially in some serious cases (cos

*θ*or cos2

*θ*close to zero).

The above experimental results are summarized in Table 1, from which we can see that the standard deviation is less than 1 nm with different phase delays after compensation for the displacement measurement of 950 nm. The experimental results prove that the proposed method has good compensation effect for any phase delay that could be introduced to the system.

#### 4.3 Stepping displacement measurement

Experiments of stepping displacement measurement were carried out to further verify the phase delay compensation method by introducing an additional phase delay of 30°. One experiment is for nanometer step displacement measurement which was conducted by moving the linear stage with a step of 20 nm for 50 times. The measured displacement and the measurement error compared with the stepping displacements are shown in Fig. 9(a). With the phase delay compensation, the maximum error is 1.27 nm and the average error is −0.07 nm with a standard deviation of 0.59 nm. Another experiment is for micrometer displacement measurement which was performed by moving the linear stage of 30 times with a step increment of 0.5 μm. As shown in Fig. 9(b), with the phase delay compensation, the maximum error is 3.95 nm and the average error is 0.93 nm with a standard deviation of 2.00 nm. The two experiments demonstrate that the displacement measurement with nanometer accuracy can be realized with the phase delay compensation in the SPMI with an EOM.

## 5. Discussion

From Eq. (22), the measured displacement *D* can be expressed as

*t*

_{1}and

*t*

_{2}are the points of time corresponding to the start and stop position, respectively,

*N*is the integer fringe number and ∆φ =

*φ*(

*t*

_{2})-

*φ*(

*t*

_{1}) is the phase difference between the start and stop position. Assuming that the integer fringe number

*N*can be counted rightly, the main error sources which limit the displacement measurement accuracy are the laser source stability and the error caused by the phase demodulation. According to the rules of the International Organization for Standardization’s guide to the expression of uncertainty in measurement [21], the standard uncertainty of measured displacement is

*u*(

*λ*)/

*λ*is the relative uncertainty of the wavelength and

*u*(Δ

*φ*) is the uncertainty in determination of the phase difference which can be expressed as where

*u*[

*φ*(

*t*)] is the uncertainty of the demodulated phase.

As the experiments of displacement were carried out in air, while the wavelength used in the experiments is the wavelength in vacuum without air refractive index correction. So the error caused by the refractive index of air should be considered. In term of the measurement of air refractive index in our previous work [22] and using the updated Edlén formulae [23], the wavelength relative uncertainty of the wavelength is estimated as *u*(*λ*)/*λ =* 2.65 × 10^{−4} with the influence the air refractive index considered, thus the error induced by the first term of Eq. (24) is about 2.65 × 10^{−4}*D*.

Next, we analyze the uncertainty of the demodulated phase *u*[*φ*(*t*)] to estimate the error induced by the second term of Eq. (24). As the algorithm we adopted in the experiment is PGC-Arctan, from Eqs. (10), (11) and (13), the main error sources of the phase demodulation are the phase delay *θ*, the nonideal performance of LPFs and the error in determining the sinusoidal phase modulation depth *z*.

According to the simulation and experimental results, the error caused by the phase delay is dominated when the phase delay deviates from *kπ* which makes cos*θ*/cos2*θ* ≠1. For example, the demodulated phase could be completely wrong when the phase delay is close to 90°. From the experiment result shown in Fig. 6, the effect of the phase delay is not obvious when the compensating phase is less than 1°. As described in section 2.2, the measurement of the compensating phase is independent on other parameters except the phase delay *θ*, and the resolution of the compensating phase is 0.01°, so the error caused by the phase delay can be neglected after compensation. For the LPFs used in the experiments, the cutoff frequency (10 Hz) was set to be lower than one tenth of the modulating frequency (1 kHz) to eliminate the effect of the carrier components sufficiently, and the Doppler frequency caused by the moving target (0.15 Hz corresponding to 50 nm/s) is lower than one tenth of the cutoff frequency of the LPF, so the error caused by the nonideal performance of the LPFs can be ignored. From Eq. (13), the values of Bessel function *J*_{1}(*z*) and *J*_{2}(*z*) which determined by the sinusoidal phase modulation depth (*z* = π*βA*/*V _{π}*) have a direct impact on the result of demodulated phase. The amplification factor of the HVA (

*β*) and the amplitude of the carrier (

*A*) can be precisely known. The half-wave voltage (

*V*) of the EOM is measured with uncertainty of 10

_{π}^{−3}which leads to a consequence of

*u*[

*φ*(

*t*)] = 0.49°, corresponding to 0.61 nm for the second term of Eq. (24).

Combining of all the error sources analyzed above, the standard uncertainty of measured displacement is

In our experiment, the stepping displacements of *D* = 0.5 μm and *D* = 20 nm were measured, and the standard uncertainty estimated by Eq. (27) are 0.62 nm and 0.61 nm, respectively, which are mainly contributed by the phase measurement. The results of actual experiments are on the same level but a little larger than the estimated value, which could be caused by the noise and airflow in the experimental environment.

## 6. Conclusion

The phase delay caused by the optical path transmission, circuit transmission and photoelectric conversion has always affected PGC demodulation. The influence of phase delay on PGC demodulation with PGC-DCM approach and PGC-Arctan approach are analyzed, and a real-time phase delay compensation method is proposed by regulating a compensating phase introduced to the carrier to maximize the output of the low pass filter so as to make the carrier synchronize with the interference signal. The phase delay measurement algorithm is described in detail. Simulation with phase delay of 30° was performed and the effectiveness of the proposed method is verified. The experimental setup of the SPMI based on Michelson interferometer with an EOM was constructed. The experimental results of contrast experiments before and after compensation at different phase delays show that the proposed method has good compensation effect for any phase delay. Stepping measurement results with nanometer and micrometer displacements show that the displacement measurement with nanometer accuracy is realized using the proposed phase delay compensation method.

## Funding

National Natural Science Foundation of China (51527807, 51375461 and 51475435); Program for Changjiang Scholars and Innovative Research Team in University (PCSIRT) (IRT13097).

## Acknowledgments

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

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