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Abstract
To eliminate the nonlinear error of phase generated carrier (PGC) demodulation in sinusoidal phase modulating interferometer (SPMI), an active linearized PGC demodulation with fusion of differential-and-cross-multiplying (PGC-DCM) and the arctangent (PGC-Arctan) schemes is proposed. In this method, the periodic integer multiple of π (π-integer phases) of PGC-Arctan without nonlinear error and the corresponding PGC-DCM results recorded at the same time are fused to obtain a calibration coefficient for PGC-DCM demodulation. Combining the accurate π-integer phases of PGC-Arctan and the calibrated fractional phase in the range of π of PGC-DCM, a linearized PGC demodulation result can be achieved, effectively eliminating the nonlinear error caused by drifts of phase demodulation depth (m) and carrier phase delay (θ). The distinct advantage of the proposed method is that it actively and linearly calibrates the fractional result of PGC-DCM without needing to measure or compensate m and θ. Simulation and displacement measurement experiments with different m and inherent arbitrary θ are performed to validate the proposed method. The experimental results show that nonlinear error of the proposed method can be reduced to about 0.1 nm with real-time linearization.
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1. Introduction
Laser interferometry is a key technology for ultra-precision machining, integrated circuit manufacturing, precision metrology and some scientific researches. In laser interferometry, the phase demodulation technology largely determines the performance of displacement measurement. Among various phase demodulation technologies, the phase generated carrier (PGC) technology [1–3] has been widely used in optical interferometers due to its high sensitivity, wide dynamic range and strong anti-interference. In PGC demodulation, the measured phase signal is up-converted onto the harmonic components of the high-frequency carrier phase signal, which is generated by modulating the laser wavelength with a current modulated laser diode (called laser wavelength sinusoidal modulation: LWSM) [4] or modulating the phase of interferometer (sinusoidal phase modulating interferometer: SPMI) with an electro-optic phase modulator (EOPM) [5]. By extracting the first and second harmonics of the interference signal, a pair of in-phase and quadrature components with respect to the phase signal to be measured can be obtained. The differential-and-cross-multiplying (PGC-DCM) [6] and the arctangent (PGC-Arctan) [7,8] are two typical PGC demodulation schemes used to recover the phase signal from the in-phase and quadrature components. In PGC-DCM scheme, the in-phase and quadrature components are cross-multiplied with each other’s derivatives, and the multiplication results are subtracted and integrated to obtain the phase signal. In PGC-Arctan scheme, the in-phase and quadrature components are divided and arctangent calculated to obtain the phase signal.
Generally, there are three main factors affecting the demodulated result: drift of phase modulation depth (m), carrier phase delay (θ) not being kπ (k=±1, ±2…) and laser intensity disturbance (LID). The PGC-DCM scheme has relatively high linearity and low harmonic distortion, while its accuracy is susceptible to LID. And m deviating greatly from the ideal value (2.37 rad) [9] or θ not being kπ will lead to a large linear error. The PGC-Arctan scheme is inherently immune to LID, but its output result will contain a large harmonic distortion (nonlinear error) when m deviating from the ideal value (2.63 rad) or θ not being kπ. Especially, when θ is close to or at certain values (kπ+π/4, kπ+π/2; k = 0, ±1, …), both demodulation schemes cannot work properly. In most cases, it is hard to maintain m at a fixed value because it varies with the measured displacement in LWSM and the environmental temperature in EOPM-based SPMI. And θ caused by optical path, circuit transmission and photoelectric conversion could be arbitrary [10]. Therefore, to obtain a correct demodulation result, the linear coefficient of PGC-DCM scheme should be calibrated in time and the harmonic distortion elimination should be considered in PGC-Arctan scheme.
Kinds of improved PGC demodulation schemes have been proposed to overcome the effects of m and θ. For example, to eliminate the drift effect of m, differential-and-self-multiplying (DSM) and Arctan scheme [11], PGC-DCM-Arctan scheme [12], J1…J4 scheme [13–15], etc. were proposed. The DSM and Arctan scheme uses the in-phase and quadrature components self-multiplied with their own derivatives to obtain the coefficient related to m. The PGC-DCM-Arctan scheme compensates the coefficient related to m by dividing the two cross-multiplication results of DCM. The J1…J4 scheme extracts four harmonics of the interference signal to directly measure and correct m. To compensate for θ, some schemes synchronize the phase of the local carrier with the phase of the carrier in the interference signal [16–18]. Other schemes adopt the orthogonal detection method to calculate the arctangent of the AC components of orthogonal signals, to recover θ from the interference signal and compensate for it [19–23].
Usually, accurate calculation and compensation of m requires that θ should be equal to kπ. However, in actual applications, the influences of m and θ always exist at the same time. Therefore, the effects of m and θ should be compensated simultaneously for precision PGC demodulation. The ellipse fitting method [24,25] is a recognized effective scheme to realize simultaneous compensation. Due to being time-consuming and needing a large amount of pre-obtained data, the ellipse fitting approach is always realized off-line. In our previous research, we proposed the real-time ellipse fitting method with a combined sinusoidal and triangular modulation, and achieved a nonlinear error of displacement measurement better 0.1nm [26]. Later, by directly extracting θ, we proposed improved J1…J4 schemes to eliminate the nonlinear error introduced by m and θ simultaneously in real time [27,28]. In the above motioned schemes, the PGC demodulated results can be improved a lot. All these schemes are indirect compensation methods, requiring calculation of θ or normalizing the in-phase and quadrature components (affected by m and θ) accurately before arctangent operation. However, θ cannot be calculated correctly when the phase to be measured is close to kπ, and the ellipse fitting method also fails when θ is close to or at certain special values (kπ+π/4, kπ+π/2; k = 0, ±1, …). Therefore, it is necessary to seek a novel PGC demodulation method that actively and directly eliminates nonlinear error caused by m and θ.
Inspired by the facts that the nonlinear error of PGC-Arctan demodulation is periodic and its demodulated phase is accurate at certain values such as Nπ (N = 0, ±1, ±2…) and the demodulated result of PGC-DCM has high linearity and lower harmonic distortion, we proposed an active linearized PGC demodulation with fusion of PGC-Arctan and PGC-DCM schemes for nonlinear error elimination. In this method, the coefficient of PGC-DCM is calibrated with two adjacent π-integer phases Nπ and (N+1)π obtained with PGC-Arctan. Therefore, the fractional phase with linear error correction can be obtained in PGC-DCM. Combining the accurate π-integer phases, the nonlinear error caused by m and θ can be actively eliminated, and the demodulated result with lower harmonic distortion can be obtained. In Section 2, the principle and implementation of the proposed method are presented. In Section 3 and Section 4, the simulation and displacement measurement experiments are performed and the results are analyzed.
2. Principle and implementation of proposed method
2.1 Influences of m and θ in PGC-Arctan and PGC-DCM schemes
In a SPMI, when the optical path difference of interferometer is modulated and a high-frequency phase carrier signal ${V_1}(t) = {A_c}\cos ({\omega _c}t)$ is introduced to the interference signal [17], the interference signal can be expressed as
where S0 and S1 are the amplitudes of the DC and AC component of the interference signal, respectively, m is the phase modulation depth of sinusoidal modulation, ωc is the frequency of the sinusoidal signal (also called as carrier frequency), θ is the carrier phase delay, and φ(t) is the phase signal to be measured.This interference signal I(t) is multiplied with the reference carrier signal and its second harmonic. After low-pass filtering, a pair of in-phase and quadrature signals I1(t) and I2(t) containing φ(t) can be obtained by
From Eq. (3) and (4), the drifts of m and θ will affect the accuracy of coefficient $2{v_1}{v_2}G{({{S_1}B} )^2}$ in PGC-DCM scheme, making the demodulated result φdcm(t) deviate linearly from the actual phase φ(t). While in PGC-Arctan scheme, the demodulated result φatan(t) will have nonlinear error with a periodicity of π. The amplitude of nonlinear error depends on the actual values of m and θ. However, when the phase φ(t) to be measured is equal to the integer multiple of π, i.e., φ(t) = Nπ (N = 0, ±1, ±2…), the demodulated result φatan(t) agrees with φ(t) and the nonlinear error is zero.
As we all know, the linearity error can be corrected easily with a linear fitting. Here, inspired by the demodulation features of PGC-DCM and PGC-Arctan schemes, we utilize two adjacent integer phases Nπ and (N+1)π obtained with PGC-Arctan to calibrate the result of PGC-DCM in proportion, so as to obtain an accurate fractional phase with linear error correction in PGC-DCM. Combining the accurate integer phases obtained with PGC-Arctan, the nonlinear error caused by m and θ in the demodulated result can be actively eliminated.
2.2 Active linearized PGC demodulation for nonlinear error elimination
The principle and implementation of active linearized PGC demodulation by fusing PGC-Arctan and PGC-DCM schemes is shown in Fig. 1. PGC-Arctan and PGC-DCM demodulations are performed in parallel. To make sure that the variation of phase to be demodulated exceed π, an extra triangular phase modulation is introduced, thus two adjacent integer phases Nπ and (N+1)π can be obtained regardless of the static or dynamic measurement. As shown in Fig. 1, the triangular signal generated by a triangular modulation generator (TMG) is superimposed with a sinusoidal signal generated by a direct digital frequency synthesizer (DDS), forming a mixed phase modulation signal. When the mixed signal $V(t) = {A_c}\cos ({\omega _c}t) + {A_t}Tri({\omega _t}t)$ is applied to SPMI, the interference signal I(t) with mixed phase modulation is re-expressed as
Fig. 1. Scheme of PGC demodulation with active harmonic distortion elimination. (TMG: triangular modulation generator, DDS: direct digital frequency synthesizer, LPF: low-pass filter, PSM: phase synchronization matching, Arctan: arctangent calculation, Unwrap: phase unwrapping, MF: mean filter.)
The interference signal I(t) is processed by PGC-DCM and PGC-Arctan schemes simultaneously, the demodulated phases can be re-obtained as
From Eqs. (6) and (7), for a specific phase φ(t) to be measured, the demodulated results φ'dcm(t) and φ'atan(t) change periodically with the triangular modulation. These simultaneously demodulated signals are input to the proposed linearized PGC fusing scheme to eliminate the nonlinear error of φ(t). Besides, after removing the triangular modulation phase by mean filter (MF), the demodulated phase φdcm(t) and φatan(t) can obtained.
Figure 2 shows the idea of active linear correction for demodulated phase with fusion of the PGC-Arctan and PGC-DCM. In the phase synchronization matching (PSM), two adjacent integer phases Nπ and (N+1)π of φ'atan(t) at times t1 and t2 are recorded as φ'atan1(t) and φ'atan2(t). And the phases of φ'dcm(t) are recorded simultaneously as φ'dcm1(t) and φ'dcm2(t). Then, a calibration coefficient can be obtained by
Therefore, at any time t, combining the accurate integer phase of PGC-Arctan and the linearly calibrated fractional phase of PGC-DCM, a corrected phase φ'(t) without nonlinear error can be obtained by
It should be noted that, although the real-time correction is realized with the calibration coefficient of the latest cycle, it is still very effective for eliminating nonlinear error. This is because m and θ drift slowly and the calibration coefficient is updated in each cycle.
Finally, after removing the triangular modulation phase by MF, the demodulated phase φ(t) without nonlinear error can be achieved.
3. Simulative evaluation experiment
To evaluate the feasibility of the proposed PGC demodulation, a simulation experiment was carried out. A signal processing unit with a high-performance FPGA (StratkII EP2S90F 1020I4, ALTERA), four-channel ADCs (AD9246-125, ADI) and four-channel DACs (AD9744-210, ADI) was used to perform the demodulation scheme shown in Fig. 1. According to Eq. (5), a simulative interference signal with mixed sinusoidal and triangular phase modulation was generated by FPGA, converted to analog signal by DAC, and then sampled by ADC for phase demodulation. The modulation frequencies ωc and ωt were 200kHz and 200Hz, respectively. The phase modulation depth of triangular signal was 3.5 rad. The phase modulation depth of sinusoidal modulation m was set to 1.5, 2.0, 2.5 and 2.63 rad respectively. And the carrier phase delay θ was naturally introduced by the circuit transmission (DAC, ADC, etc.), which might be an arbitrary value.
Firstly, the phase signal φ(t) was set as 0°. Without removing the phase introduced by triangular modulation, the demodulated signal of PGC-Arctan before linearization (φ'atan(t)) and the phase signal of proposed PGC demodulation after linearization (φ'(t)) are compared and shown in Fig. 3. The address of triangular signal linearly corresponds to the modulation phase generated by TMG. To evaluate the nonlinear error of φ'atan(t) and φ'(t), linear fittings are performed. The maximum residuals of the linear fit of the demodulated results before and after linearization are summarized in Table 1. We can see that, the curves of φ'atan(t) and φ'(t) are basically triangular waveforms. However, the curves of φ'atan(t) exhibit significant nonlinearity. For example, when m is 1.5 rad, the maximum nonlinear error even reaches about 30°. In addition, due to the arbitrary inherent θ, the curve of φ'atan(t) exhibits nonlinearity about 7° even m is equal to the ideal value of 2.63 rad. In contrast, the curves of φ'(t) exhibit good linearity under different m, with a maximum error less than 0.11°. These comparison results demonstrated that a linearized demodulated result can be achieved with the proposed PGC demodulation.
Fig. 3. The demodulated phase changes caused by triangular modulation before and after linearization with different m and inherent θ.
Table 1. Nonlinear errors of the demodulated results before and after linearization.
Secondly, the phase signal φ(t) was changed from 0° to 720° in increment of 1°. The real-time linearization correction is performed after φ(t) is larger than 360°. After removing the triangular phase modulation, the demodulated phase results are compared with the preset phase values. The experimental results before and after real-time linearization are shown in Fig. 4 and summarized in Table 2. We can see that, for different m, the obtained phase signal before linearization has obvious periodic nonlinearity with a periodicity of π. This result is consistent with Eq. (4). Even when m is 2.63 rad, the nonlinear error is also up to 0.63° due to the inherent θ. On the contrary, no obvious nonlinearities can be observed in the demodulated phase signals after active linearization, and the maximum deviation between the demodulated results and the preset values is only 0.05°.
Table 2. Maximum deviations of the demodulated results from the preset value.
Here, it should be noted that, the deviations before linearization in Table 2 are much smaller than the nonlinear error in Table 1. This is because the mean filtering used to remove the triangular modulation phase will also average partial nonlinear error. Therefore, the proposed active linearized PGC demodulation can effectively suppress the periodic nonlinear error and accurately obtain the phase signal to be measured.
4. Displacement measurement experiment and results
4.1 SPMI with triangular phase modulation
To validate the effectiveness of the proposed PGC demodulation in practical applications, an experimental setup of SPMI was constructed according to the optical configuration shown in Fig. 5(a). This SPMI is composed of a frequency stabilized He-Ne laser (HRS015B, Thorlabs), an isolator (IA), a beam splitter (BS), a polarizer (P), a reference reflector (RR), and a measuring reflector (MR). An electro-optic phase modulator (EOPM, EO-PMNR-C1, Thorlabs) was placed in the reference arm of interferometer. The polarizer aligns the polarization direction of the laser beam from BS to RR with the EOPM crystal axis. The interference signal is detected by a photodetector (PD) and then transmitted to the signal processing unit for displacement measurement.
Fig. 5. Schematic and experimental setup of SPMI with mixed sinusoidal and triangular phase modulation. (a) Optical configuration. (b) Constructed experimental setup.
A mixed sinusoidal and triangular signal generated by the signal processing unit is amplified by a high-voltage amplifier (HVA, ATA-2082, Aigtek) and then applied to EOPM. The modulation frequencies of sinusoidal and triangular signal are also 200kHz and 200Hz, respectively. The amplitudes of sinusoidal and triangular signals are ±80V and ±240V, respectively, and the amplitude of the mixed modulation signal is ±320V. As the half-wave voltage of EOPM is Vπ = 135V, the modulation depths m and z are about 1.86 rad and 5.59 rad, respectively. The relationship between the moved displacement of MR and the demodulated phase change can be expressed as $d = {{\Delta \varphi \times \lambda } / {4\pi {n_\textrm{a}}}}$ (λ is the wavelength of the laser, na is the refractive index of air).
4.2 Nonlinear error elimination verification for nanometer displacement measurement
This experiment was carried out to evaluate the performance of the proposed linearized PGC demodulation for nonlinear error elimination in nanometer displacement measurement. In the experiment, the measured displacement of MR was provided by a nano-positioning stage (P-753.1CD, Physik Instrument) which moved with a speed of 1 µm/s and a displacement step of 10 nm for 512 times. The closed-loop travel range of P-753.1CD stage is 12 µm and its bidirectional repeatability is ±1 nm. For each step, the displacement obtained by the PGC-Arctan scheme, the displacement obtained by the proposed active linearized PGC scheme, and the stage position provided by P-753.1CD were recorded simultaneously until the stage stopped completely. In addition, another similar experiment was performed, and the displacement obtained by the ellipse fitting PGC-Arctan demodulation [24] was recorded for comparison.
Figure 6 shows the demodulated triangular phases obtained by the PGC-Arctan scheme and the proposed linearized PGC scheme when the P-753.1CD stage is stationary. It can be seen that the real-time linearization works well in actual measurement, and the amplitude of φ'(t) is consistent with the modulation depth z.
Fig. 6. Demodulated triangular modulation phases with PGC-Arctan scheme and proposed linearized PGC scheme.
Figure 7(a) shows the displacement measurement results obtained by the PGC-Arctan scheme (before linearization), the proposed linearized PGC scheme (after linearization) and the ellipse fitting scheme, respectively. Taking the stage position as the reference, the maximum displacement deviations of PGC-Arctan, proposed linearized PGC and ellipse fitting schemes are about 7.18 nm, 3.97 nm and 4.70 nm, respectively. Figure 7(b) shows the corresponding fast Fourier transform (FFT) analysis results of the displacement deviations. Theoretically, if nonlinear errors with a displacement period of λ/4 or λ/2 occurs in the phase demodulation, there will be peaks around the sequence number of [512 × 10 nm/(633 nm/4)] ≈32 or [512 × 10 nm/(633 nm/2)] ≈16 in Fig. 7(b). As shown in Fig. 7(b), before linearization, the nonlinear error of PGC-Arctan with a displacement period of λ/4 (sequence number 32) reaches about 3.65 nm. However, after linearization with the proposed PGC demodulation, the error can be reduced to about 0.10 nm. And the nonlinear error of ellipse fitting at the first-order harmonic component of fringe is about 0.24 nm. It should be noted that, due to slow drifts of environmental parameters, FFT results also include some low-frequency components. These experimental results demonstrate that the proposed linearized PGC demodulation can actively eliminate the nonlinear error effectively.
Fig. 7. Nanometer displacement measurement results and FFT analysis. (a) Measurement results with the step of 10 nm for 512 times. (b) FFT analysis results of the displacement deviations with a zoomed inset near the period of λ/4.
4.3 Experiment verification of dynamic measurement in millimeter range
This experiment was carried out to demonstrate the applicability of the proposed demodulation in millimeter range dynamic measurement. In this experiment, the measured displacement of MR was provided by high-precision linear stage (XMS160-S, Newport) with bi-directional repeatability of ± 0.04 µm and movement range of 160 mm. And a commercial interferometer (XL80, Renishaw) with a linear measurement resolution of 1 nm and an accuracy of ± 0.5 ppm was used for comparison. SPMI and XL80 were installed on either side of the XMS160-S stage. MR and the measuring reflector of XL80 were mounted back-to-back on the stage, and moved 80 steps with 2 mm increments and 2 mm/s speed following the stage. The displacements measured by SPMI with the proposed linearized PGC demodulation and measured by the XL80 interferometer were recorded simultaneously, and the stage positions were also recorded. The environmental parameters were obtained with air temperature sensor 81P72 and XC-80 compensator 75T460, and na was calculated by Edlén equations with uncertainty about 4.7×10−8 [29]. The displacements of proposed SPMI and XL80 interferometers are compensated with the same na. The experimental results are shown in Fig. 8. In Fig. 8(a), the deviations are the difference between the positions of the XMS160-S stage and the measurement results of SPMI and XL80, respectively. For clear display, the displacement measurement curve of XL80 is shifted up by 10 mm, and its trend agrees well with the trend of the displacement measurement curve of SPMI. Figure 8(b) shows the deviations of the measurement results between SPMI and XL80, the maximum deviation and standard deviations are 0.213 µm and 0.095 µm, respectively. It should be noted that, the deviations of the measurement results of SPMI and XL80 were mainly caused by the fact that the measurement reflectors of the two interferometers were mounted back-to-back, and the influence of the tilting angle of the stage on the measurements of SPMI and XL80 was opposite. These experimental results indicate the displacements measured by SPMI with the proposed linearized PGC demodulation are consistent with those measured by XL80. Therefore, the proposed linearized PGC demodulation can be applied to SPMI for precision dynamic displacement measurements in the millimeter range.
Fig. 8. Comparison experiment of dynamic displacement measurement. (a) Measurement results of SPMI with proposed PGC demodulation and XL80. (b) Deviations of the measurement results between SPMI and XL80.
5. Conclusion
In this paper, we proposed an active linearized PGC demodulation by fusing the simultaneously demodulated results of PGC-Arctan scheme and PGC-DCM scheme. Using the accurate π-integer phases of PGC-Arctan and the corresponding PGC-DCM results recorded at the same time, a calibration coefficient for PGC-DCM demodulation can be obtained. Combining the accurate π-integer phases and the calibrated fractional phase, a real-time linearized PGC demodulation result is obtained, actively eliminating the nonlinear error introduced by phase demodulation depth m and carrier phase delay θ. Different from current methods of calculating or compensating θ and m, the proposed method actively corrects the demodulated results. Without needing pre-obtained data or complicated calculation, it is simple and feasible, easy to implement in real time. Thus, the proposed method has wider applicability. The simulation and displacement measurement experiment have verified the proposed PGC demodulation method. The experimental results show that: (1) Most of the periodic nonlinear phase error can be eliminated, and the residual is only about 0.11°after linearization; (2) With the real-time active linearization, the displacement error caused by nonlinear demodulation is reduced from 3.65 nm to 0.10 nm; (3) Standard deviation of 0.095 µm in dynamic displacement measurement is achieved in the range of 160 mm. Therefore, the proposed PGC demodulation can realize nanometer displacement measurement and has a great potential in the applications such as the ultra-precision machining, integrated circuit manufacturing and precision metrology. Furthermore, to solve the influence of mean filtering introduced by triangular modulation and improve the dynamic measurement performance, our next work is to increase the modulation frequencies of sinusoidal and triangular modulation of SPMI (e.g., using a high-frequency EOPM).
Funding
National Natural Science Foundation of China (52005449, 52035015, 51875530); Zhejiang Provincial Postdoctoral Science Foundation (ZJ2021122); Natural Science Foundation of Zhejiang Province (LQ20E050002, LZ18E050003).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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