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Abstract
When the microprobe sensor is faced with the demand of high-speed biaxial displacement measurement, due to the characteristics of phase generated carrier (PGC) technology, accompanying optical intensity modulation (AOIM) and unfavorable phase modulation depth (PMD) will bring about the tens of nanometer cyclic nonlinear errors, further hindering high-speed and high-precision measurement. Herein, a light source intensity stabilization system based on semiconductor optical amplifier (SOA) feedback control is achieved to eliminate the error caused by AOIM in the presence of high-frequency and large-amplitude laser modulation. Based on this, the reasons for large nonlinear errors in biaxial measurements and the inability to ensure the stability of the accuracy of multiple measurement axes are methodically examined, and an effective nonlinear error elimination methodology based on the normalized amplitude correction of active temperature scanning is proposed. The continuity and linearity of the temperature scanning are also discussed. The performed experiments show that the above approach is capable of reducing the displacement demodulation error from the nanometer scale to the sub-nanometer scale. Further, the nonlinear error is reduced to within 0.1 nm for both measurement axes and the performance becomes consistent. The dual-axis measurement resolution of the microprobe sensor reaches 0.4 nm and the measurement speed is better than 1.2 m/s with the standard deviation of lower than 0.5 nm.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
To date, the laser interferometer, due to its unique measurement principle of using laser as a carrier and light wavelength as a length reference, is considered one of the most accurate and sensitive measurement tools, which is extensively employed in the field of highly accurate displacement measurement [1–3]. However, in advanced measurement pieces of equipment, there is an increasing demand for equipment miniaturization and simplification, which makes it important to investigate fiber-optic microprobe laser interferometers. This instrument is easy to install and adjust, has a very small probe size, can be isolated from thermal contamination, and can be employed as an embedded device in multi-axis measurements [4–6].
Phase demodulation technology basically determines the measurement accuracy of laser interferometer [7–9], among which phase generated carrier (PGC) demodulation technology is a passive zero-difference demodulation technology. The basic principle of the algorithm is to first combine the modulated measurement signal and the reference signal, then go through the low-pass filter operation, the mathematical calculation of the trigonometric function, and other operations, and finally obtain the measured phase. With the advantages of high sensitivity, excellent linearity, and high dynamic range, it is extensively implemented in the phase demodulation technology of fiber-optic sensors [10–12]. The PGC technique utilizes modulation to convert the measured phase shift into a change in light intensity, which is divided into external and internal modulation. The former generally uses a piezoelectric transducer (PZT) or electro-optical modulator (EOM) as a phase modulator to directly alter the phase of the laser on the reference arm in the optical path of the interferometer, which suffers from the drawbacks of a complex structure, high cost, and large size [13–15,14]. Compared to the former approach, internal modulation uses a cosine modulator to directly modulate the drive current of the semiconductor laser source. This methodology does not need to add other components to the whole fiber system and realize the idea more simply. Additionally, this is highly sensitive in signal monitoring and capable of achieving more accurate measurements, so internal modulation has been mostly employed in biaxial fiber-optic microprobe laser interferometers [16–18].
However, the direct internal modulation of the laser beam is associated with the inevitable problems of carrier phase delay (CPD), accompanying optical intensity modulation (AOIM), and phase modulation depth (PMD) deviation, which cause nonlinear errors that seriously affect the measurement accuracy of the interferometer's displacement. Among these glitches, the CPD problem is better examined and investigated compared to the more serious problems pertinent to the AOIM and PMD. Especially when considering the requirements of multi-axis measurement and high-speed measurement, it is difficult for the present study to solve this problem effectively. For AOIM, researchers such as Tian and Wang [19] employed the look-up table approach to compensate for the effect of light intensity modulation on the interference model but did not take into account the phase difference between the laser frequency modulation and the light intensity modulation. Wang et al. [20] proposed an approach to eliminate the light intensity modulation effect, which utilizes the principle of dual photoelectric detection and signal phase division, and realizes the dual channels by splitting the beam via a beam splitter, and then eliminates the AOIM by the division operation. However, Hu et al. [21] found that it is difficult to achieve dual-channel synchronization via this method, especially when the laser modulation frequency reaches megahertz levels, and the phase difference delay is large. In terms of PMD correction, Volkov et al. [22] proposed a phase modulation depth calculation and control scheme based on the fourth harmonic solution. Chen et al. [23] established a methodology to eliminate the depth effect of active phase modulation based on the orthogonal multi-harmonic frequency reduction. The methods mentioned above are shown to solve PMD error problems. However, for each measurement axis, the former uses four sets of phase-locked amplification units and four sets of digital filters, whereas the latter exploits twice these units and filters, thus occupying a large amount of digital processor resources. It is therefore difficult to adapt this approach to the measurement requirements of more than one measurement axis. In the meantime, although the above methodologies have been proven better at eliminating the PMD effect, they are not able to remove the effects of AOIM, and the effectiveness of the PMD elimination methods is noticeably affected.
At present, the main method that is capable of eliminating the influence of multiple error sources at the same time is the nonlinear error correction method. Inherently, this approach is based on the idea of ellipse fitting, but most of this method is offline processing, so it is difficult to meet the requirements of real-time measurement. At the same time, the elliptical fitting is not able to effectively correct the nonlinear error in the period of less than half the wavelength, even though Wang et al. [24] subsequently proposed an approach to eliminate the periodic nonlinear error of the laser interferometer based on the liquid crystal shifter. Nevertheless, the proposed method required adding a supplementary device in the measurement sensor, which did not meet the requirements of the small size of the microprobe.
In summary, in confronting the demand for high-speed measurement of more than one measurement axis, the existing literature lacks a suitable AOIM elimination method as well as a PMD deviation suppression method that does not consume a lot of data resources and can be processed in real-time, so as to correct the nonlinear error of tens of nanometers to the sub-nanometer level and realize ultra-precision displacement measurement.
In the present work, we design a light source intensity stabilization control system based on a semiconductor optical amplifier (SOA) that solves the problem of nanoscale errors in high-speed measurements by compensating fluctuations in the output intensity of the fiber-optic microprobe sensor laser interferometer in real-time. Subsequently, we proceed with examining the causes of large nonlinear errors in biaxial measurements performed by fiber-optic microprobe laser interferometers and developing a nonlinear error elimination approach based on the normalized amplitude correction. Finally, the designed light source intensity stabilization control system and the nonlinear error correction method are applied to the biaxial fiber-optic microprobe laser interferometer. The experimental results reveal that the interferometer is able to achieve sub-nanometer displacement demodulation for biaxial measurements under the demand of high-speed measurements, and the dual-axis performance is consistent.
2. Displacement measurement principle for biaxial fiber-optic microprobe interferometers
2.1 Error analysis of AOIM of laser under high-speed measurements and large modulation
Figure 1 illustrates a block diagram of the principle of the biaxial microprobe laser interferometer based on the PGC technique. In this article, a biaxial fiber-optic microprobe interferometer is proposed, which possesses the characteristics of small size, high accuracy, and simultaneous biaxial measurement.
Fig. 1. Schematic representation of the measurement structure of the high-speed dual-axis fiber-optic microprobe interferometer (note: DFB: Distributed Feedback Laser; Couple, fiber-optic beam splitter; FC: Fiber Circulator; SMF: Single-Mode Fiber; M1 and M2: two mirrors in the direction of the x-axis; WD1: working distance in the x-axis; M3 and M4: two mirrors in y-axis direction; WD2: working distance in the y-axis; APD: Avalanche Photodetector; ADC1/2: Analog-to-Digital Converter; DAC: Digital-to-Analog Converter; DDS: Direct Digital Synthesizer; LPF: Low Pass Filter; Atan: the inverse tangent algorithm).
The DDS module in the signal processing unit outputs a sinusoidal signal that modulates the laser frequency, and the output light enters the measurement optical path through the fiber-optic beam splitter in the x-axis and y-axis directions, respectively. The two light paths have the same interference principle, which both occur in the spatial cavity consisting of M1 (M3) and M2 (M4). The two interference signals are again returned to the optical fiber circulator and processed by the photodetector and analog-to-digital converter before being fed into the signal demodulation unit. The phase demodulation is performed via the same PGC-Atan algorithm for both channels, and the target phase φ0(t) is finally calculated. The expression for the interferometric signal after APD detection is given by:
where A represents the DC component of the current, B denotes the amplitude of the AC component of the current, C is the phase modulation depth, which is proportional to the working distance L0, and ω0 represents the modulation frequency.In terms of the nature of motion, the reason for the change in the phase to be measured due to the motion of the object is the Doppler frequency shift effect of the laser, so the form of the effect of the motion velocity (v) on the interference signal can be reflected in the motion frequency (f). From the signal analysis, the motion frequency (f) shifts the spectrum of the interference signal, and the maximum motion speed of the target object is limited by the value of the modulation frequency to avoid spectral aliasing. Therefore, if the measurement speed of the system is to be improved, the modulation frequency of the laser should be increased first. Further, to realize different range measurements of the laser interferometer, the laser modulation amplitude is also required assuming that the value of phase modulation depth (C) is 2.63 [25]. This causes the laser frequency modulation process to simultaneously produce light intensity modulation, the so-called phenomenon of accompanying optical intensity modulation (AOIM). At this point, Eq. (1) for the interference signal detected by the APD is rewritten as:
For the traditional PGC demodulation algorithm, the presence of the AOIM term brings a large nonlinear error to the demodulation result, which seriously affects the accuracy of the laser interferometer. Two interferometric signals are modulated and both have optical intensity modulation conditions, and they enter the signal processing unit to realize simultaneous demodulation through the same PGC algorithm for both channels. As illustrated in Fig. 1, the interference signal is divided into two paths, which are multiplied by the one-octave reference signal and the two-octave reference signal produced by the DDS. In addition, the two orthogonal signals after low-pass filtering, u1 and u2, can be evaluated as follows:
To ensure that the amplitudes of u1 and u2 are equal, the initial working distance L0 and the modulation range are generally set reasonably so that C = 2.63 rad, thus having J1(C) =J2(C). The computed value of the target phase, φ*(t), is obtained by the inverse tangent Atan algorithm after dividing u1 and u2. Therefore, the target phase reads:
As shown in Fig. 2, the effect of the AOIM phenomenon on the interferometric signal is simulated according to Eq. (5). The ideal measurement situation is shown in Fig. 2(b), where the blue curve represents the ideal interference signal, and its amplitude should not change with time, so the envelope line of the interference signal should be straight, as shown in the red dashed line. However, as shown in Fig. 2(a), the blue curve represents the actual interference signal, because the laser output frequency is modulated by the sine signal as shown in the red curve, distorting the actual interference signal intensity, so its amplitude is sinusoidal regular change as shown in the yellow envelope. Therefore, the error caused by this phenomenon cannot be ignored.
Fig. 2. Effect of the AOIM on the demodulation results. The red curve in (a) represents the modulation signal, the blue curve signifies the interfering signal distorted by the AOIM, the yellow curve presents the amplitude change curve of the interfering signal, and the blue curve in (b) denotes the ideal interfering signal under the same conditions as (a).
2.2 Analysis of the nonlinear error due to unequal working distances based on the biaxial measurement
It has already been shown that there are specific requirements for the value of the phase modulation depth C in the PGC demodulation algorithm. Further, when the interferometer is applied to multi-axis micro displacement measurements, the value of C only takes into account the effect of the initial working distance because the measurement beams of both measurement axes are emitted by the same laser source. The measuring requirements for dual-axis work may be different in practical applications, and if two measuring axes are used, it is difficult to ensure that the initial working distances L of the main axis and L’ of the secondary axis are exactly the same. Assuming that the working distance of the main axis satisfies the ideal condition of J1(C) =J2(C), and the generated interference signal still satisfies the error model of Eq. (5). Further, according to the proportionality relationship, the expression of the phase modulation depth C’ of the secondary axis at this time is given by C′=L′/L·C; hence, the phase demodulated by the secondary axis is rewritten as follows:
Therefore, according to the demodulation result pertinent to the secondary axis, it can be seen that ${\varphi ^ \oplus }(t ) $ is affected by the working distance of the secondary axis (L′), and the AOIM coefficient (m), and the system is unable to demodulate the real value of the phase to be measured φ0(t). Figure 3 demonstrates the demodulation error simulation based on Eq. (6), presenting the demodulation error over one complete cycle of variation.
Fig. 3. Simulation analysis of the phase demodulation error caused by the AOIM in biaxial measurement: (a) and (b)The displacement demodulation errors at different values of m for a main axis working distance of 4 cm and secondary axis working distances of 3.5 cm and 3.7 cm, respectively.
By extending the analysis of Fig. 3, the following results can be obtained:
- (1) When the value of m is not equal to zero (see the main axes of Figs. 3 (a) and (b)), the plotted results reveal that the error introduced by the AOIM is a periodic error of about φ0. Since there are additional cosine, DC, and sine terms in the numerator and denominator, it is not a sinusoidal error and the error amplitude increases with m.
- (2) By observing the secondary axes of Figs. 3 (a) and (b) when m takes the value of zero, it can be seen that the error introduced by the biaxial measurements presents a sinusoidal periodic error around φ0, which is mainly caused by the fact that there are no additive errors in the numerator and denominator terms. By continuing to compare the error curves of the two secondary axes when their values of m are the same and not zero, it can be seen that the law of changing the magnitude of the error amplitude with L′/L is not certain. This is essentially attributed to the nature of the inverse tangent function and the Bessel function. Because the Bessel function is not linear, there are multiple intersections between different orders of the Bessel function, so the error caused by the biaxiality does not exhibit a simple linear relationship in terms of L′/L.
3. Error suppression method in biaxial high-speed and high-precision fiber-optic microprobe laser interferometer measurements
To solve the above problems caused by AOIM and biaxial measurements, an SOA-based light intensity stabilization module (LISM) and a nonlinear error correction module (NECM) based on the normalized amplitude correction are designed here. Figure 4 illustrates the schematic of the dual-axis fiber-optic microprobe laser interferometer measurement at high speed and large range after adding the error suppression unit.
Fig. 4. Measurement schematic representation of the dual-axis fiber-optic microprobe interferometer at high speeds and large ranges after compensation (note: LISM: Light Intensity Stabilization Module; NECM: Nonlinear Error Correction Module; SOA: Semiconductor Optical Amplifier; LPF: Low-Pass Filter; Gain, P-I controller; PVD: Peak Value Detector).
3.1 Design of the light intensity stabilization control module based on the SOA
This study presents an efficient way to stabilize the final light intensity by modifying the output light intensity gain in real-time in response to the sine law variation phenomenon of the light intensity amplitude generated by AOIM. An intensity stabilization module is incorporated between Couple 1 and the DFB laser to eliminate the AOIM influence, as shown in Fig. 4.
The system control model is shown in Fig. 5, and the working process is as follows: The photodetector detects the output light intensity in real time and converts it into voltage. The preset amplitude is subtracted from it and the error value is input to the controller. In order to modify the gain of the SOA and achieve the goal of modifying the light's intensity, the controller produces the control voltage through the control algorithm, and the driver creates the corresponding current based on the voltage value. Operational amplifiers, capacitors, resistors, and other components are used in the construction of the controller. A constant current chip, whose output current is only correlated with the input voltage and exhibits strong linearity, is used in the driver.
Fig. 5. Block diagram of light intensity stabilization control (note: GC(s): transfer function of the controller; GD(s): transfer function of the driver; GP(s): transfer function of the SOA; H(s): transfer function of the APD).
Prior to designing the control system, SOA devices with a response frequency higher than 5 MHz, along with the appropriate driver circuits and photodetectors, should be chosen because the maximum frequency of the output light intensity change is 5 MHz. Next, according to the response speed as well as the gain in the product datasheet, the controlled object is identified and modeled to yield the transfer function G1(s)=GD(s)GP(s) for the SOA and its driving circuit. The G1(s) expression obtained is as follows:
For the controller, the PID control algorithm is selected. The control circuit uses a P-I control rather than a derivative part to prevent abnormal variations in light intensity during operation from causing excessive control of the controller, which in turn causes damage to the semiconductor diode by transiently excessive output current of the driver circuit. GC(s) can be expressed as follows:
where KP and KI denote the proportional and integral parameters, respectively.The analysis is subsequently carried out by the simulation software and the preliminary tuning of the P-I parameters is carried out using the critical proportional band method. Based on this, the proper KP and KI parameters are determined by fine-tuning, and the final parameters can be achieved by adjusting the circuit's resistance and capacitance values.
The design results are as follows: the system overshoot is 11.7%, the steady-state error is 0, the rise time is 4.11e-08 s, the stabilization time is 1.64e-07 s, and the parameters of the P-I controller are KP = 0.1539, KI = 5252837. Finally, the frequency response of the system is then analyzed to obtain the Bode diagram of the closed-loop system as shown in Fig. 6, it can be inferred that the −3 dB cutoff frequency is 7.56 MHz, which meets the 5 MHz bandwidth requirement.
3.2 Nonlinear error correction for biaxial microprobe laser interferometers
The value of the modulation depth C is no longer 2.63 due to the unequal working distances of both the main and secondary axes in the actual biaxial measurement process, which eventually results in a nonlinear error. To conquer this problematic issue, this paper proposes an effective approach of amplitude normalization via linear transformation to solve the problem of unequal amplitude of two orthogonal signals of any single axis. As presented in Fig. 4, the nonlinear error correction module (NECM) is added to the PGC-Atan algorithm. Taking the secondary axis as an example, the two orthogonal signals of the secondary axis are fed into the peak value detection unit (PVD), and by detecting a complete cycle of the signal, its maximum and minimum values are extracted. After performing the relevant mathematical operations, a pair of orthogonal sine and cosine signals is obtained with equal amplitude normalization.
However, as shown in Fig. 7(a), it should be noted that when the range of motion of the object in a certain measurement axis is less than half of the laser wavelength, then the peak detection module cannot detect the maximum and minimum values of a complete signal cycle, and the calculation results are not trustable at all. Therefore, when the object is in motion, it must first be determined whether the phase shift due to the motion of the object is less than 2π or not. If this is the case, phase compensation is required so that the orthogonal signal can form a complete Lissajous graph, and thus further obtain the correction parameters for amplitude correction.
Fig. 7. Using active temperature scanning to ensure the completeness of the Lissajous graph: (a) The incomplete Lissajous graph when the object motion results in a phase change of less than 2π, (b) The complete Lissajous elliptic graph formed by simulating the object motion using temperature scanning, (c) The Lissajous graph of interference signal generated by temperature scanning changing wavelength in the presence of constant driving current, (d) The Lissajous graph of interference signal generated by driving current scanning changing wavelength subjected to constant temperature, and the blue to red color indicates a gradual increase in the light intensity.
To solve the problem that a complete signal cycle cannot be formed due to the small distance of the actual object, we compensate for the phase integrity by actively changing the laser wavelength instead of moving the object based on the tunable laser wavelength feature. When the interferometer works normally, the phase change of interferometric demodulation can be given by:
where λ represents the laser wavelength, n is the refractive index of air, and L(t) denotes the displacement change.When the wavelength is continuously changed from λ1 to λ2 process, where the corresponding time changes from t1 to t2, the phase difference of the laser interferometer can be stated in the following form:
Assuming that there is no change or a small range of change in the displacement of the object to be measured during this time from the change of t1 to t2. Therefore, Eq. (10) can be reasonably simplified as follows:
Equation (11) clearly displays that by changing the wavelength of the laser, a complete Lissajous graph can be obtained, that is, the phase change Δφ ≥ 2π. Therefore, it is possible to completely simulate the interference phenomenon produced by the movement of the object, which thus realizes the active change of the phase difference of the interferometer as well as the compensation.
With further analysis, we know that DFB lasers are capable of altering their output wavelength by the current tuning and the temperature tuning. Both approaches are discussed below, where the blue curve in Fig. 7(c) presents the Lissajous graph of the orthogonal signal obtained by changing the output wavelength through temperature scanning in the presence of constant current conditions, and the green curve represents the Lissajous graph of the orthogonal signal obtained by changing the output wavelength linearly through simulation. Figure 7(d) illustrates the Lissajous graph of the orthogonal signal obtained by demodulating the system during continuously varying drive currents at constant temperature. By comparing the two, it can be seen that the method of changing the wavelength of the laser output by tuning the temperature does not have the problem of changing the intensity of the output light, so the temperature scanning methodology should be chosen.
Although temperature tuning is capable of continuously altering the laser output wavelength and thus the equivalent phase, the wavelength change process by temperature scanning is not linear. Therefore, in the next step, it is necessary to analyze whether the motion of the simulated object needs to change phase continuously as well as linearly. First, let us analyze the continuity; if the phase change is not continuous, the results also do not form a complete Lissajous graph, and cannot play a compensatory role. In the next step, the linear nature is methodically examined by changing the phase linearly from A-B-C-D shown in Fig. 7(c) and viewing the Lissajous graph equivalently. Where A and C represent the two extreme values of S2(t) respectively, while B and D represent the two extreme values of S1(t) respectively, and the sequence from A-B-C-D is represented as the phase change. As shown in Fig. 7(c), the green curve is shifted to distinguish it from the blue curve and this is consistent with the blue curve, indicating that only continuity is required to perform active phase change of phase, and it is not necessary to be linear. In summary, the Lissajous diagram produced by the temperature scanning is consistent with the Lissajous diagram produced by the motion of an actual object and therefore can be effectively employed as a substitute for the motion of an actual object.
Therefore, as shown in Fig. 7(b), when the range of motion of the object is very small, we choose to use the active temperature scanning unit to realize the change of the laser wavelength shift, thus providing sufficient phase change, which finally guarantees that the cyclic nonlinear error correction unit can work properly.
After obtaining a more ideal Lissajous graph, the two signals are linearized with the following equations to eliminate DC bias and unequal amplitude errors, and finally, the ideal sine and cosine signals are obtained:
Therefore, as long as the amplitude of the signals before and after two cycles do not alter much, the amplitude detected in the previous cycle can be employed to normalize the signal of the current cycle, and then two orthogonal normalized signals are passed through the Atan's algorithm to achieve accurate target phase.
4. Experimental validation
Finally, an experimental test system is appropriately constructed to validate the accuracy and effectiveness of the proposed approach. The laser source used in the experiments is a distributed feedback laser (DFB PRO BFY, Toptica, Germany), which is modulated by a high-speed sinusoidal signal generated by the DDS and a 14-bit digital-to-analog converter chip (DAC). The test object is the P-733.3DD three-dimensional piezoelectric ceramic nano displacement stage from Physik Instrumente, Germany, which has a final displacement resolution of 0.4 nm, and a closed-loop motion range of 30 µm in each of the three axes. Such an apparatus is suitably programable to realize various motion shapes. The photoelectric conversion of the interference signals of the two measurement axes is realized via an APD (APD430C, Thorlab, USA), which is converted to a digital signal by a 16-bit analog-to-digital converter (ADC). The final demodulated phase and displacement are sent to a personal computer for display via a universal serial bus (USB). All operations of the above biaxial modulation and demodulation module are implemented in a field-programmable gate array (FPGA). Additionally, it is necessary to use a Tektronix AFG3252C signal generator (dual-channel output, 1.2 GS/s sampling rate) to generate high-speed modulation signals during testing. All experiments are performed on an air-floating table in an ultra-precision clean laboratory (ΔT = 0.01°C/10 min).
4.1 Performance testing experiment of the LISM
First, the stability test of the selected SOA device should be performed. The PID parameters are then adjusted and the LISM is tested for performance. As presented in Fig. 8, in the test session, a signal generator is employed to generate a sinusoidal signal for optical frequency modulation, and the output light is first attenuated by a variable optical attenuator (VOA) to prevent the APD from being damaged by too much light intensity. Thereafter, 50% light beam is split by a fiber beam splitter for comparison, and the other 50% light is passed through the LISM for compensation testing. Finally, after the two light beams are photoelectrically converted through the APD, their light intensity changes can be observed by an oscilloscope.
Fig. 8. Schematic representation of the performance test experiment of the LISM (note: VOA: Variable Optical Attenuator; SG: Signal Generator; DSO: Digital Storage Oscilloscope).
In the experiment, the stability of the SOA gain is first observed in 60 min, when the test current is 300 mA and the test temperature is 25°C, and the obtained results are displayed in Fig. 9(a). The graphed results indicate that after the gain unit, the output optical power variation is not more than 0.05 dBm, the gain module works stably and can be implemented for the LISM.
Fig. 9. Optical intensity stabilization unit test results: (a) The gain stability of the SOA over a long period, (b) The light intensity modulation coefficient in terms of the modulation amplitude before and after the addition of the LISM (note: The blue lines display the case without adding the LISM, the red lines demonstrate the case with adding the LISM, and the green lines illustrate the ratio of m before and after the addition of the unit mentioned above).
The performance of the actual light intensity stabilization unit was then tested. Using a signal generator, the DC and AC components of the modulating signal are measured before and after stabilization, set the modulation frequency of the modulated signal to 5 MHz according to the requirements of the measurement speed, and the modulation amplitude is continuously increased from 0.5 V to 3 V. The values of AOIM coefficient m before and after stabilization and the ratio of the two were calculated and the results are plotted in Fig. 9(b). From the experimental results, it can be seen that after the LISM, the AOIM coefficient of the output light intensity is reduced to 30% of the original. At the actual operating point, that is, at the output amplitude of 1.194 V, the coefficient after stabilization becomes 0.017, and the maximum displacement demodulation error corresponding to this is obtained as 0.1 nm. This effectively improves the measurement accuracy.
4.2 Performance test experiment of the NECM
Then, we test the working effect of the NECM in the dual-axis measurement system. The signal generator is appropriately implemented to produce two signals to simulate the optical interference phenomenon caused by the uniform movement of the actual object, set the value of the phase modulation depth C of the main axis as 2.63, and the value of C of the secondary axis are taken as different values and inputted to the dual-axis demodulation board respectively for experiments. The results of the final displacement demodulation are analyzed to compare the error magnitudes of the displacement demodulation before and after the correction of the two measurement axes (see Fig. 10), where axis 1 is set as the main axis and axis 2 is set as the secondary axis.
Fig. 10. Nonlinear error correction module test results: (a) and (b) The orthogonal signals, their Lissajous graphs, and the corresponding displacement demodulation errors for the measurement axis 1 with a phase modulation depth of C = 2.63; (c) and (d) The orthogonal signals, their Lissajous graphs, and the corresponding displacement demodulation errors before and after correction for the measurement axis 2 with a phase modulation depth of C = 2.80 rad, respectively.
Due to the continuous change of the phase modulation depth C in the measurement axis 2, the amplitudes of the two orthogonal signals after PGC demodulation are not equal, which is consistent with the conclusion of the nonlinear error analysis above. It can be seen that the peak-peak value of the demodulation error of the displacement of measurement axis 1 is 88 pm, whereas the value of the displacement demodulation error of measurement axis 2 is 18 nm and its error after the NECM is less than 0.1 nm. The result is in line with the accuracy of measurement axis 1. Therefore, the experimental results reveal that the NECM is able to effectively reduce the nonlinear error due to the change of phase modulation depth during the multi-axis measurement.
4.3 Experimental validation of the sub-nanometer resolution and high-speed measurements of a biaxial fiber-optic microprobe laser interferometer
Based on Fig. 4, the overall laser interferometry system is constructed to perform demodulation performance tests of the actual fiber-optic microprobe laser interferometer, examining the demodulation results and errors. In the actual test, the LISM and the NECM are added to enhance the measurement accuracy.
First, we test the measurement speed of the biaxial fiber-optic microprobe laser interferometer. According to the previous analysis, it has been found that the spectral shift of the interference signal is directly proportional to the speed of the moving object. Hence, it is possible to simulate the phenomenon of interference in the presence of various speeds of the moving object by generating the equivalent spectral shift via a programmable signal generator. By continuously changing the movement speed, we can obtain a series of interference signals, which are received and processed by the signal acquisition board to measure the displacement. Using the least squares algorithm to fit it, we can obtain the displacement-time fitting curve results shown in Fig. 11.
Fig. 11. Displacement fitting curves at various motion speeds (1381.50, 1151.25, 767.50, and 383.75 mm/s): (a) Measurement axis 1, (b) Measurement axis 2.
Using the Bessel function formula, the standard deviation of the displacement measurement can be appropriately obtained to measure the error (see Fig. 12). The results indicate that the standard deviation of the displacement measurement is less than 0.25 nm and is basically unchanged at low object movement speeds, but with the growth of the movement speed, the displacement standard deviation becomes extremely large.
Fig. 12. The relationships between the standard deviation of displacements of two measuring axes and the velocity of the moving object.
Additionally, at a speed of 1.2 m/s, the standard deviation of both axes is less than 0.5 nm, which meets the system design requirements. Therefore, the conducted experiment validates that the measurement speed of this biaxial demodulation system is better than 1.2 m/s.
After that, the resolution test of the laser interferometric displacement measurement system is performed. The interferometric optical path is first installed and adjusted so that the stresses coupled to each part of the loop are naturally released. Then by controlling the PI displacement stage to move forward and backward, the resolution test of 1 nm as well as 0.4 nm is performed, respectively, as illustrated in Fig. 13.
Fig. 13. Actual displacement resolution test results of biaxial laser interferometer: (a) and (b) The results of displacement demodulation for measuring axes 1 and 2 at a resolution of 1 nm, respectively, (c) and (d) The results of displacement demodulation for measuring axes 1 and 2 at a resolution of 0.4 nm, respectively.
From the analysis of this figure, in the displacement stage to perform reciprocating movement of 0.4 nm, the results of the demodulation system can still be clearly distinguished. Due to the accuracy limitations of the displacement stage itself, as well as the effect of photoelectric conversion noise and environmental interference, the test results shown here will be worse than those of the ideal time, so the test results exhibit that the resolution of the biaxial fiber-optic microprobe laser interferometry system is better than 0.4 nm, and the biaxial performance is consistent.
Although there is a slow drift in the results, it is clear from the analysis that it is caused by the change in the refractive index of the air due to the change in environmental factors such as temperature and air pressure. Further, the drift direction is consistent in both axes, so it can be reduced later by optimizing the environmental stability.
5. Conclusions
In summary, the nonlinear errors of the biaxial fiber-optic microprobe laser interferometer for high-speed and long-range measurements are analyzed in this paper. The high-frequency and large-amplitude modulation used for measurement needs can lead to the AOIM phenomenon. The C values of the two axes are difficult to equal because they have different working distances. Both of these factors can result in a decrease in the demodulation accuracy of the PGC, which reduces the precision of the displacement measurement.
Then, a set of light source intensity stabilization control system was designed for the AOIM phenomenon, and the nonlinear error of uniaxial measurement was reduced to the subnanometer level. In addition, a nonlinear error elimination method based on normalized amplitude correction and active temperature scanning was proposed for multi-axis measurement. The experimental results showed that the light source intensity stabilization system based on SOA can effectively ensure the stability of the laser light source, and was suitable for two-axis or multi-axis measurement. Compared with the traditional least square ellipse fitting method, the dynamic compensation technology based on normalized amplitude correction can realize real-time fine correction. Using a signal generator, the measurement speed was tested using the electrical signal test method, and the results showed that the measurement speed was better than 1.2 m/s with the standard deviation of lower than 0.5 nm. Then the static resolution was tested and the results showed that both axes of the interferometer reached the limit resolution of 0.4 nm of the micro-displacement stage used.
In the near future, we will eagerly attempt to explore more axes and higher accuracy of displacement measurement by frequency division multiplexing approach using fiber-optic microprobe laser interferometers.
Funding
National Key Research and Development Program of China (2022YFF0705801); National Natural Science Foundation of China (62305090); China Postdoctoral Science Foundation (2023M730883); National Postdoctoral Program for Innovative Talents (BX20230478).
Acknowledgments
The authors sincerely thank Profs. Hongxing Yang, Haijin Fu, and Ruitao Yang for their helpful assistance and valuable suggestions.
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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