## Abstract

A precision PGC demodulation for homodyne interferometer modulated with a combined sinusoidal and triangular signal is proposed. Using a triangular signal as additional modulation, a continuous phase-shifted interference signal for ellipse fitting is generated whether the measured object is in static or moving state. The real-time ellipse fitting and correction of the AC amplitudes and DC offsets of the quadrature components in PGC demodulation can be realized. The merit of this modulation is that it can eliminate thoroughly the periodic nonlinearity resulting from the influences of light intensity disturbance, the drift of modulation depth, the carrier phase delay, and non-ideal performance of the low pass filters in the conversional PGC demodulation. The principle and realization of the signal processing with the combined modulation signal are described in detail. The experiments of accuracy and rate evaluations of ellipse fitting, nanometer, and millimeter displacement measurements were performed to verify the feasibility of the proposed demodulation. The experimental results show that the elliptical parameters of the quadrature components can be achieved precisely in real time and nanometer accuracy was realized in displacement measurements.

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

## 1. Introduction

Recently, some key fabrication technologies, such as the semiconductor industry, precision optics fabrication, micro-technology and nanotechnology, have experienced a continuously rapid progress and brought about ambitious challenges for displacement measurement technology used in these high-tech fields [1–3]. As a common configuration for optical interferometry, Michelson interferometer has always played a significant role in the precision displacement measurement because of its high accuracy and direct traceability to the length standard. In order to improve the performance of Michelson interferometer, many kinds of displacement detection schemes have been presented [4–6]. Among them, the phase modulating homodyne interferometer (PMHI) has been widely applied in precise measurements of displacement, vibration, surface profile and so on since it was proposed by Basile *et al*. in 1991 [7–12]. Usually, the phase modulation is implemented by using a piezoelectric actuator (PZT) to vibrate a reference mirror or by using an electro-optic modulator (EOM) to modulate the phase of reference wave. Due to the limited vibrating frequency of the PZT, the PZT modulation is not appropriate for high-speed measurement. Moreover, in PZT modulation, an inevitable vibrating noise introduced to the interferometer will deteriorate the measurement accuracy. Compared with PZT, the EOM is more suitable for high frequency modulation without mechanical vibration. Theoretically, both linear phase modulation and sinusoidal phase modulation can be applied to EOM. Linear modulation can be realized by applying a triangular/sawtooth signal to EOM [13]. As the triangular/sawtooth signal is not appropriate for high frequency modulation, linear phase modulation is only suitable for low-speed measurement. On the other hand, the sinusoidal phase modulation is more appropriate for high frequency modulation. For example, M. Aketagawa *et al*. proposed a PMHI with an EOM operating at frequencies up to 4 MHz to measure the fluctuation of air refractive index [14].

For sinusoidal phase modulation, the phase generated carrier (PGC) is the most widely used demodulation algorithm because it has many advantages such as a wide dynamic range and high sensitivity. In the PGC demodulation scheme, a high-frequency sinusoidal phase modulation is applied to EOM to generate a phase carrier, which up-converts the desired phase signal onto the sidebands of the carrier frequency. Generally, a pair of quadrature components containing the expected phase information are acquired by detecting the odd and even harmonics of the interference signals. The differential-and-cross-multiplying approach (PGC-DCM) [15] and the arctangent approach (PGC-Arctan) [16] are typical algorithms used to obtain the measurement results. Usually, the demodulation result of the PGC-DCM approach is influenced by the light intensity disturbance (LID) and the visibility of the interference fringes. Moreover, a phase delay between the carrier component of the detected interference signal and the carrier signal and a deviation of the phase modulation depth used for signal processing from the true value will introduce periodic nonlinearity in PGC-Arctan demodulation.

The nonlinearity of PGC demodulation can be reduced a lot by using the improved PGC-DCM and PGC- Arctan approaches. For example, by introducing a reference signal to eliminate the influence of LID, the SNR of demodulation result obtained from an reference compensation method (PGC-RCM) achieves a gain of 11.78 dB over PGC-DCM algorithm [17]. And a differential-self-multiplying approach (PGC-DSM-Arctan) has achieved low harmonic distortion and high stability in a wide range of modulation depths (1.5 to 3.5 rad) and larger LID depth (26.5dB) [18]. For the influence of the phase delay, methods such as the carrier phase advance technique [19], the carrier frequency tracking [20] and real-time phase delay compensation [21] have been presented to synchronize the interference signals to the carrier signals, thereby to decrease the nonlinearity of PGC-Arctan demodulation algorithm. In addition, M. Zhang utilized a look-up table to estimate the dynamic value of the effective phase modulation depth to eliminate the error caused by the amplitude modulation and deviation of the phase modulation depth [9]. A. V. Volkov proposed a phase modulation depth evaluation and correction technique for PGC demodulation scheme based on the integral control feedback [22]. All of these schemes have shown superior abilities to eliminate the influences of partial parameters such as LID, the phase delay and the phase modulation depth. However, in most applications, the changes of these parameters and some other factors may occur simultaneously. Considering that the phase delay, the LID and the phase modulation depth are dynamically changing in real time, a universal method which can depress the influences mentioned above simultaneously and timely is on the urgent need.

Ellipse fitting and correction of the quadrature components is an effective method to resolve these problems. For example, the Heydemann model is widely used in homodyne interferometers to correct the offsets, amplitudes and the phase lag of two quadrature interference signals [23–25]. C. Ni *et al*. proposed an off-line ellipse fitting and correction method realized in a personal computer (PC) based on Matlab software and achieved a 3σ error of 16.73 nm in 4 ms and a 3σ error of 199.18 nm in 2 s, respectively [26]. In our previous work, we proposed a modified EOM-based sinusoidal phase modulating interferometer (SPMI), which introduced an additional PZT to drive the reference corner cube to generate periodic interference signals for real-time normalization of the quadrature components [27]. However, to counteract the phase shift induced by PZT, another monitor interferometer is required. In addition, as the interference processing and ellipse fitting are processed in PC based on Labview workbench, it is not appropriate for dynamic measurement.

In this paper, a precision PGC demodulation with a combined sinusoidal and triangular phase modulating scheme is proposed for homodyne interferometer. The low-frequency linear modulation is used to generate a continuous phase-shifted interference signal for real-time ellipse fitting and correction of the quadrature components of PGC demodulation. The optical configuration is given in section 2. In section 3, the all-digital PGC-Arctan demodulation algorithm based on the combination of field programmable gate array and advanced RISC machines (FPGA + ARM) is described in detail. In the last section, the experiments and results are given.

## 2. System configuration

The system configuration for the homodyne interferometer modulated with a combined sinusoidal and triangular signal is shown in Fig. 1. An elliptically polarized beam emitted from a single frequency He-Ne laser is adjusted to a linearly polarized beam after passing through a polarizer (P). This linearly polarized beam is divided into two beams by a beam splitter (BS), one directed to a fixed corner cube (M_{1}) and the other to a moving corner cube (M_{2}). An EOM is placed between BS and M_{1} to modulate the phase of the laser beam reflected from BS to M_{1}. The reflected laser beams from M_{1} and M_{2} are recombined at BS and interfere with each other. Then the modulated interference signal is detected by a photodetector (PD). This interference signal is sampled by an analog to digital converter (ADC) controlled by a FPGA for further phase demodulation. A combined sinusoidal and triangular signal is generated by FPGA and then converted to analog signal by a digital to analog converter (DAC). The output signal of DAC is amplified by a high voltage amplifier and then applied to the EOM.

In an ideal case, the modulated interference signal is expressed as

*I*

_{0}and

*I*

_{1}are the DC offset and the AC amplitude, respectively;

*λ*is the laser wavelength;

*d*(t) is the displacement to be measured;

*V*is the modulation voltage applied to EOM; V

_{eom}_{π}is the voltage required to produce a phase shift of π radians and is also called half-wave voltage. The electric field is applied along the crystal axis of EOM transverse to the direction of optical propagation. The polarization direction of laser beam reflected from BS to M

_{1}is aligned with the crystal axis.

## 3. Signal processing

#### 3.1 Phase demodulation with combined sinusoidal and triangular modulation signal

Figure 2 is the principle of phase demodulation with combined sinusoidal and triangular modulation signal. The fundamental carrier $\mathrm{cos}({\omega}_{c}t)$ and second-harmonic carrier $\mathrm{cos}(2{\omega}_{c}t)$are generated by the DDS module and the triangle waveform $Tri({\omega}_{t}t)$is generated by the TWG module. When a combined modulation signal ${V}_{eom}={A}_{c}\mathrm{cos}({\omega}_{c}t)+{B}_{t}Tri({\omega}_{t}t)$ is applied to EOM, the modulated interference signal can be rewritten by

*A*and

_{c}, B_{t}*ω*are the amplitudes and angular frequencies of the carrier signal and the triangular signal, respectively; $C={A}_{c}\pi /{V}_{\pi}$ denotes the phase modulation depth; $\varphi (t)=\phi (t)+\epsilon (t)$ is the phase shift to be demodulated. The phase shift $\phi (t)=4\pi d(t)/\lambda $ is related to the displacement to be measured and the phase shift $\epsilon (t)={B}_{t}Tri({\omega}_{t}t)\pi /{V}_{\pi}$ is introduced by the modulation of triangular signal.

_{c}, ω_{t}Expending Eq. (2), we can obtain

*J*

_{2n}(C) and

*J*

_{(2n-1)}(C) denote the odd- and even-order Bessel functions, respectively.

As shown in Fig. 2, in order to demodulate the phase shift $\varphi (t)$, the modulated interference signal *S*(t) is down-conversion mixed with $\mathrm{cos}({\omega}_{c}t)$ and $\mathrm{cos}(2{\omega}_{c}t)$, respectively. After filtering the high-frequency signals such as the fundamental carrier and all harmonic carrier frequencies with two LPFs, a pair of quadrature components can be obtained by:

*K*

_{1}and

*K*

_{2}denote the total gains of the multiplier and the low-pass filter. Under ideal circumstance, with the assumption of constant gains

*K*

_{1}and

*K*

_{2}, AC amplitude

*I*

_{1}, accurate modulation depth

*C*and synchronous demodulation process, $\varphi (t)$ can be obtained by using conventional PGC-Arctan and PGC-DCM demodulation approaches. However, in real application, the factors including the phase delay, intensity modulation, the fluctuation of

*V*

_{π}, non-ideal performance of the low pass filters and other electronic noise of interference signal will result in the dynamical change of DC offsets and AC amplitudes of the quadrature components. Therefore,

*I*(

_{x}*t*) and

*I*(

_{y}*t*) can be rewritten by where

*a*(

*t*),

*b*(

*t*) and

*x*(

_{0}*t*),

*y*(

_{0}*t*) are the AC amplitudes and DC offsets of

*I*(

_{x}*t*) and

*I*(

_{y}*t*), respectively.

Obviously, the dynamical change of *a*(*t*), *b*(*t*), *x _{0}*(

*t*) and

*y*(

_{0}*t*) will introduce a periodic nonlinearity to the demodulated phase $\varphi (t)$. When the variation of $\varphi (t)$ is large enough, such as 2π, the Lissajous figure of

*I*(

_{x}*t*) and

*I*(

_{y}*t*) is an ellipse. Therefore, we can adopt ellipse fitting to determine the parameters

*a*(

*t*),

*b*(

*t*),

*x*(

_{0}*t*) and

*y*(

_{0}*t*) to eliminate the nonlinearity in real time. According to the previous research [25], data corresponding to at least 1

*/*4 ellipse arc is required for an effective ellipse fitting algorithm. Moreover, the determination of the ellipse parameters, using 3

*/*4 of the ellipse, is sufficiently accurate compared to the parameters obtained with a full ellipse. This means that the changing range of $\varphi (t)$should be at least larger than π/2 and preferable up to 3π/2. For traditional SPMI, the $\varphi (t)$is only related to the measured displacement

*d*(

*t*). When the target is in stationary or the movement of the target is much less than half of a laser wavelength, the ellipse fitting cannot be implemented correctly or the result of the ellipse fitting is inaccurate.

In our design, in order to generate continuous phase-shifted interference signal for real-time ellipse fitting, a periodic phase shift *ε*(*t*) is introduced to $\varphi (t)$by applying a triangular modulation signal to EOM. The amplitude of triangular signal is chosen about 3*V _{π}*/4, and the phase shift range of

*ε*(

*t*) ($\varphi (t)$) is almost up to 3π/2, which meets the requirement for a perfect ellipse fitting. For real implementation, the sampled data sets of

*I*(

_{x}*t*) and

*I*(

_{y}*t*) in a period of triangular signal are adopted for each ellipse fitting. After determining the parameters of

*a*(

*t*),

*b*(

*t*),

*x*(

_{0}*t*) and

*y*(

_{0}*t*), we can obtain a pair of perfect quadrature components

After the operations of division, arctangent and phase unwrapping, the demodulated phase is expressed as

*M*is the count for the phase unwrapping in arctangent calculation.

It should be noted that the demodulated phase ${\varphi}^{\prime}(t)=\phi (t)+\epsilon (t)$ includes the phase shift $\phi (t)=4\pi d(t)/\lambda $ related to the displacement to be measured and the phase shift $\epsilon (t)$ introduced by the modulation of triangular signal. In order to obtain $\phi (t)$, we should subtract $\epsilon (t)$ from ${\varphi}^{\prime}(t)$. As it is well- known, for a bipolar periodic triangular signal, the sum of the sampled data of a triangular signal during one cycle is zero. Here, in order to remove the influence of $\epsilon (t)$, we average the values of ${\varphi}^{\prime}(t)$ in one cycle of a bipolar triangular signal in real time by using a first-in first-out buffer on FPGA. Therefore, we can obtain the $\phi (t)$ in every clock period.

#### 3.2 Realization of the least-squares ellipse fitting

From above description, it can be seen that accurate ellipse fitting is a key technique for demodulating the phase shift $\varphi (t)$. Generally, parameters *a*(*t*), *b*(*t*), *x _{0}*(

*t*) and

*y*(

_{0}*t*) change slowly, thus, they can be considered unchanged in each ellipse fitting process when the rate of ellipse fitting is fast enough. Denoting

*a*(

*t*)

*= a*,

_{n}*b*(

*t*)

*= b*,

_{n}*x*(

_{0}*t*)

*= x*,

_{0n}*y*(

_{0}*t*)

*= y*during

_{0n}*t*∈(

*t*Δ

_{n}, t_{n}+*t*) (where

*t*is the starting moment and

_{n}*t*Δ

_{n}+*t*is the ending moment of the

*n*-th ellipse fitting, respectively; Δ

*t*is the period of triangular signal), then

*I*(

_{x}*t*) and

*I*(

_{y}*t*) can be expressed as

Thus, during *t*∈(*t _{n}, t_{n} +* Δ

*t*),

*I*(

_{x}*t*) and

*I*(

_{y}*t*) satisfy the following implicit equation, which can be used to describe the ellipse to be fitted

*A*

_{1},

*A*

_{2},

*A*

_{3}and

*A*

_{4}are defined as following

Assuming that the measurement deviations obey the normal distribution, it is known that the maximum likelihood estimation of the parameters (*A*_{1}, *A*_{2}, *A*_{3} and *A*_{4}) are given by the orthogonal resp. *total least squares fit* (TLS) [28], which is based on minimization of the function

*I*(

_{x}*i*),

*I*(

_{y}*i*)) (

*i*= 1, 2, 3…

*N*) denote

*N*measurement points sampled in one triangular signal period, ${\sigma}_{i}$ denotes the geometric (orthogonal) distance from the measurement point (

*I*(

_{x}*i*),

*I*(

_{y}*i*)) to the fitted curve. In mathematics, $\Omega $ satisfies the following partial derivative equations

Then we obtain four linear equations, which can be expressed into the following matrix form

For convenience, Eq. (17) is rewritten as ** PA = F**, where

**is the coefficients matrix,**

*P***is the constants vector and**

*F***= [**

*A**A*

_{1}

*A*

_{2}

*A*

_{3}

*A*

_{4}]

^{T}is the unknown parameters vector. It can be seen that matrix

**is symmetric and there are eleven elements in matrix**

*P***and vector**

*P***. After calculating**

*F***and**

*P***with the**

*F**N*sampled points (

*I*(

_{x}*i*),

*I*(

_{y}*i*)), the vector

**can be determined by solving the linear equations.**

*A*The real-time ellipse fitting and correction based on FPGA + ARM is shown in Fig. 3. Firstly, *N* sampled points (*I _{x}*(

*i*),

*I*(

_{y}*i*)) are self- or cross- multiplied and synchronized in the module of SCS, then the synchronized eleven terms are accumulated to obtain eleven elements in matrix

**and vector**

*P***in FPGA. It should be noted that as each operation in FPGA is activated by the global clock signal, the signal obtained after each multiplication operation is delayed by one clock cycle. In order to ensure the correctness of the calculation, it is important to synchronize the signals involved in the same operation. In our design, we use the digital time delayer (DTD) to realize the delay of the signals. For example, as shown in Fig. 4, because the output signal of MUL**

*F*_{1}is delayed by one clock (denoting as D

_{1}[

*I*(

_{x}*i*) ×

*I*(

_{y}*i*)]), the signal

*I*(

_{y}*i*) needs to be delayed by one clock (denoting as D

_{1}[

*I*(

_{y}*i*)]) with a DTD. Then, the two signals D

_{1}[

*I*(

_{y}*i*)] and D

_{1}[

*I*(

_{x}*i*) ×

*I*(

_{y}*i*)] are synchronized and can be operated in MUL

_{2}. Secondly, matrix

**and vector**

*P***are transmitted to ARM. In ARM, four linear equations are reconstructed and the Gaussian-Jordan elimination algorithm is adopted to solve vector**

*F***. The parameters (**

*A**a*,

_{n}*b*,

_{n}*x*,

_{0n}*y*) can be calculated according to Eq. (14). Considering that the division operation is difficult to be realized in FPGA, the reciprocals of

_{0n}*a*and

_{n}*b*are also calculated in ARM. At last, the parameters (1/

_{n}*a*, 1/

_{n}*b*,

_{n}*x*,

_{0n}*y*) are transmitted back to FPGA, and

_{0n}*I*(

_{x}*i*) and

*I*(

_{y}*i*) are corrected to

*I′*(

_{x}*i*) and

*I′*(

_{y}*i*) after subtraction and multiplication operations. It should be noted that, as the parameters drift slowly, we use the parameters (

*a*

_{n}_{-1},

*b*

_{n}_{-1},

*x*

_{0n}_{-1},

*y*

_{0n}_{-1}) calculated in the last triangular signal period(

*t*∈(

*t*Δ

_{n}-*t, t*)) to correct the sampled points (

_{n}*I*(

_{x}*i*),

*I*(

_{y}*i*)). Therefore, real-time demodulated results can be obtained.

## 4. Experiments

To validate the effectiveness of the proposed phase demodulation, an experimental setup of a homodyne interferometer modulated with combined sinusoidal and triangular signal was constructed as shown in Fig. 5.

The laser source is a single-frequency He-Ne laser (XL80 laser kit, Renishaw) with the wavelength of 632.990577 nm. The combined phase modulation was introduced by an EOM (EO-PM-NR-C1, Thorlabs) with half-wave voltage *V*_{π} about 135 V. The signal processing board is an ADC & DAC card (STEMlab 125-14, Red Pitaya), which is based on a SOC (Zynq7010, Xilinx), integrating a FPGA and a double-core ARM Cortex-A9 MPCore, i.e. FPGA + ARM. FPGA includes abundant programmable logics, such as 80 DSP Slices and 17600 LUTs. These DSP Slices are capable of fulfilling multiplication in parallel in one clock, which can greatly enhance the efficiencies of LPFs and ellipse fitting. The development platform of ARM is C language, which facilitates the equation solving of ellipse fitting. In addition, this ARM is a hard processor with frequency up to 667 MHz, whose execution efficiency overwhelms the soft processor core, such as MicroBlaze, embedded in FPGA. The combined modulation signal from STEMlab board was amplified from ± 1 V to ± 12 V and then to ± 200 V by a self-made amplifier in tandem with a high voltage amplifier (HVA200, Thorlabs) with a voltage gain of *β* = 20. Both the amplitudes of sinusoidal and triangular signals applied to EOM were ± 93.5V, and the frequencies were 244.14 kHz and 956.67 Hz, respectively. Thus, the amplitude of the combined signal was ± 187 V. The measured displacement *d*(*t*) was provided by a nano-positioning stage (P-753.1CD, Physik Instrument) whose movement range is 15 μm and bidirectional repeatability is ± 1 nm and a precision linear stage (UPS-150, Physik Instrument) whose movement range is 300 mm and unidirectional repeatability is 15 nm, respectively. A commercial homodyne interferometer (XL80, Renishaw) was used to measure the displacement of the linear stage for comparison.

#### 4.1 Experimental verification for ellipse fitting algorithm

### A. Accuracy evaluation of ellipse fitting algorithm

Firstly, to demonstrate that the phase shifts induced by the triangular modulation and the measured displacement *d*(*t*) have the same influences on *I _{x}*(

*t*) and

*I*(

_{y}*t*), measurement points (

*I*(

_{x}*i*),

*I*(

_{y}*i*)) were recorded and their Lissajous figures were plotted in Fig. 6 when M

_{2}was in stationary and moving states, respectively. It can be seen that the two ellipse trajectories are consistent except that few points deviate from the trajectory in Fig. 6(b). We speculate that this was caused by the mechanical vibration noise arising from the moving of M

_{2}.

Later, to evaluate the accuracy of ellipse fitting algorithm designed by ourselves, the results of ellipse fitting fulfilled by FPGA + ARM and Labview workbench are compared and listed in Table 1. From Table 1, it can be seen that, for measurement points in Fig. 6(a) and Fig. 6(b), the maximal difference between the fitted parameters obtained by FPGA + ARM and that by Labview workbench is only −0.0006. This demonstrates that the performance of the ellipse fitting implemented by FPGA + ARM is considerably well. In addition, comparing the fitted parameters obtained from Fig. 6(a) and Fig. 6(b) by FPGA + ARM, although the trajectories are not same due to the influence of random noise and the sampled data in Fig. 6(a) only makes up 3/4 ellipse, yet the differences are still very small and the maximal discrepancy is 0.0027. These results indicate that the triangular phase modulation can be used to estimate the elliptical parameters whether M_{2} is in stationary or moving state. Thus, *I _{x}*(t) and

*I*(t) can be corrected dynamically by using the parameters (

_{y}*a*,

_{n}*b*,

_{n}*x*,

_{0n}*y*) obtained with the ellipse fitting.

_{0n}### B. Rate evaluation of ellipse fitting algorithm

For the efficiency test of the ellipse fitting implemented by FPGA + ARM, two counters are used to monitor the total time used for ellipse fitting and record the number of ellipse fitting times, respectively. Here, each whole ellipse fitting takes into account of reading coefficient matrix ** P** and vector

**from FPGA, solving the linear equations in ARM and writing parameters (1/**

*F**a*, 1/

_{n}*b*,

_{n}*x*,

_{0n}*y*) to FPGA. In the signal processing system, the ARM is a

_{0}*Cortex-A9*based processor with the CPU frequency of 333.3 MHz and the clock frequency of FPGA is 125 MHz. The experimental results are listed in Table 2. The period of each ellipse fitting is about 12.05 μs, which means that the real-time correction of

*I*(t) and

_{x}*I*(t) is feasible.

_{y}### C. Necessity verification of ellipse fitting

To demonstrate the necessity of the real-time correction, an experiment was carried out to observe the drifts of AC amplitudes *a*(*t*), *b*(*t*) and DC offsets *x _{0}*(

*t*),

*y*(

_{0}*t*) without correcting these parameters. In this experiment, M

_{2}was mounted on the linear stage which moved a range of 300 mm with a speed of 1 mm/s. During the moving of stage, the parameters

*a*(

*t*),

*b*(

*t*),

*x*(

_{0}*t*) and

*y*(

_{0}*t*) are calculated using the ellipse fitting in real time but only corrected at the beginning. The drifts of these parameters are plotted in Fig. 7. It can be seen that the difference between

*a*(

*t*) and

*b*(

*t*) becomes larger with the increasing of displacement even though their drifting trends are roughly consistent. And the

*x*(

_{0}*t*) and

*y*(

_{0}*t*) are also fluctuating. All these factors will lead to a larger nonlinearity in the phase demodulation and deteriorate the accuracy of displacement measurement. Therefore, real-time correction of these parameters in the application of precision displacement measurement is necessary.

#### 4.2 Experimental result of real-time ellipse fitting and correction effect

In order to verify the effectiveness of real-time correction for displacement measurement, a comparison experiment was carried out. To observe the effect of real-time correction clearly, *I _{x}*(t) and

*I*(t) were set about 1.2 and 0.8 artificially by tuning the phase demodulation depth of EOM. The actual position of the P-753.CD stage was used as a reference for comparison. At the beginning of experiment, the P-753.CD stage was located at the position of 2 µm, the readout of the demodulated displacement was clear to zero. Then, the stage was moved with a speed of 100 nm/s in a range of 2.5 µm. The stage′s position and the demodulated displacement were recorded simultaneously every 50 ms. As it is shown in Fig. 8(a), the Lissajous figure of the quadrature components

_{y}*I*(t) and

_{x}*I*(t) is apparently an ellipse before real-time correction (dotted red line), while it changes into a unit circle (dotted blue line) after correcting from the moment

_{y}*t*. Correspondingly, as shown in Fig. 8(b), before correction, the difference between the demodulated displacement obtained with proposed method and that provided by the stage ranges in ± 7.5 nm with a period of λ/4 ≈158 nm, and after the real-time correction from the moment

*t*, the displacement difference is optimized to ± 1.5 nm.

#### 4.3 Experiment of nanometer displacement measurement

In this experiment, M_{2} was mounted on the P-753.CD stage which moved with a speed of 1 μm/s and a displacement steps of 10 nm for 300 times (3 μm) and 20 nm for 256 times (5.12 μm), respectively. The period of triangular signal is about 1 ms, which corresponds to 1 nm displacement of the measured stage. The measured range cover both the periods of PGC demodulation and triangular signal. For each measurement, the stage′s position and the demodulated displacement were recorded simultaneously until the stage stopped completely. Experimental results are plotted in Fig. 9 and Fig. 10, respectively. According to the experimental results shown in Fig. 9(a) and Fig. 10(a), the standard deviations of the displacement deviation are 0.47 nm and 0.45 nm, respectively. Furthermore, fast Fourier transforms (FFT) of the displacement deviation were performed to evaluate the nonlinearity. Theoretically, if nonlinear errors with the period of λ/4 or λ/2 occurs in the phase demodulation, there will be peaks around the sequence number of [300 × 10 nm/(633 nm/4)] ≈19 or [300 × 10 nm/(633 nm/2)] ≈9 in Fig. 9(b). And the nonlinear peaks should be around the sequence number of [256 × 20 nm/(633 nm/4)] ≈32 or [256 × 20 nm/(633 nm/2)] ≈16 in Fig. 10(b). As shown in Fig. 9(b) and Fig. 10(b), no apparent nonlinearity peaks can be observed around these sequence numbers. Therefore, displacement measurement with nanometer accuracy can be realized without detectable nonlinearity beyond 0.1 nm in the proposed homodyne interferometer modulated with combined sinusoidal and triangular signal.

#### 4.4 Experiment of dynamic measurement in millimeter range

In this experiment, M_{2} was mounted on UPS-150 stage which moved for a distance of 300 mm divided into 150 steps with a speed of 5 mm/s. These displacements were measured simultaneously with the proposed interferometer and Renishaw XL80 interferometer for comparison. The experimental results are plotted in Fig. 11. In Fig. 11(a), the deviations are the differences between the actual displacement of UPS-150 stage and the measurement results with the proposed interferometer and Renishaw interferometer. It can be seen that the fluctuating trends of the deviations are almost the same. Figure 11(b) shows the difference of the measurement results between the proposed interferometer and Renishaw interferometer. The maximal value and the standard deviation of the difference are about −55 nm and 27.2 nm, respectively. These results indicate that the results obtained with the proposed interferometer are in good agreement with those obtained with Renishaw interferometer. Therefore, the proposed method can be applied to dynamic displacement measurement in millimeter range.

## 5. Conclusion

In this paper, a precision PGC demodulation for homodyne interferometer modulated with a combined sinusoidal and triangular signal has been proposed to realize nanometer displacement measurement. Introducing an additional triangular signal to drive EOM generates a continuous phase-shifted interference signal for ellipse fitting to correct the AC amplitudes and DC offsets of the quadrature components in PGC demodulation. The signal processing of the demodulation is accomplished in FPGA + ARM in real time. The advantage of the demodulation is not only eliminating the periodic nonlinearity caused by the factors such as the light intensity disturbance, the drift of modulation depth and the carrier phase delay but also being applied in dynamic measurement. Several experiments were carried out for the feasibility verification of the proposed demodulation. The experimental results show that: (1) The proposed PGC demodulation with the combined modulation can achieve precise phase measurement. (2) With the real-time ellipse fitting and correction, the displacement error caused by nonlinear demodulation is reduced from to ± 7.5 nm to ± 1.5 nm for the test linear movement with the speed of 100 nm/s. (3) Standard deviations of less than 0.5 nm in static nanometer displacement measurements are realized. (4) Standard deviation of 27.2 nm in dynamic displacement measurement is realized in the range of 300 mm. Therefore, the proposed demodulation has significant application for dynamic displacement measurement with nanometer accuracy.

## Funding

National Natural Science Foundation of China (NSFC) (51475435 and 51527807); 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

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