1. Introduction

In recent years, fiber optic interferometer has been developed rapidly, because of its simple structure, small size, low cost, and multiaxial measurement capability; thus, it is extensively used in the field for displacement measurement [1,2]. Phase-generated-carrier (PGC) technology is used in signal demodulation of fiber-optic interferometers. The advantages of the latter are high sensitivity, high-dynamic range, and good linearity [3,4]. PGC technology requires a high-frequency carrier phase signal, which can be generated by external or internal modulation. The former is realized with the use of a piezoelectric (PZT) phase modulator or an electro-optic modulator (EOM) to modulate the phase of the reference light in the Michelson interferometer reference arm [5–8]. However, these methods can yield an optical measurement structure that is too large to meet some measurement conditions involving narrow spaces. In comparison, the internal modulation is realized by directly modulating the laser frequency, and this method increases the size of the laser module components, but does not affect the measurement of the optical path [9–11]. Therefore, fiber-optic interferometers with laser frequency modulation are widely used in applications where small volume is required

In the PGC demodulation system, a high-frequency carrier phase signal is used to convert the measured phase signal into a carrier-frequency side band. By extracting the first and second harmonics of the interference signal, the orthogonal components (including the phase shift signal) can be obtained. To extract phase shift from the obtained orthogonal components, the differential-and-cross-multiplying approach (DCM) [3] or the arctangent approach (Arctan) [12] are proposed and developed [13,14]. However, in fiber-optic interferometers with laser frequency modulation, the carrier phase delay and accompanied optical intensity modulation (AOIM) in PGC demodulation cause nonlinear errors, which can seriously restrict the displacement measurement accuracy.

Owing to these problems, various improved PGC schemes have been proposed; these will be discussed.

Some schemes only compensate for the carrier phase delay. A feedback method to indirectly measure the phase delay is proposed and it automatically regulate the phase offset of the carrier signal [15]. However, this scheme is time-consuming because the compensating phase is determined by successively maximizing the amplitudes of low-pass filters. And the result of phase delay is incorrect when the interference signal phase is changing; For the orthogonal detection method [16], the result of phase delay is incorrect when the interference phase is close to an integer multiple of π and this scheme only works when the carrier phase delay is at [0, π]. Recently, a real-time phase delay compensation method has been proposed that takes the sum of squares of the quadrature components to remove the phase delay factors [17]. However, four-channel carriers and more symbol discrimination are used, which takes up more memories. Despite the efficacy of the above methods was verified through experiment, none of those methods can be applied to phase delay compensation in PGC demodulation units affected by AOIM.

Other schemes only compensate for the AOIM. Reference [18] proposed to build a look-up table to compensate the AOIM. However, this method neglects the phase difference between COFM and AOIM in the interference model. And the method of double photoelectric detection and signal division is adopted to eliminate AOIM [19], however the double channel synchronization is difficult to realize, which may cause error.

To date, the ellipse fitting method is the only effective scheme that can eliminate these two effects. The “Heydemann correction” was proposed by Heydemann and is based on least-squares ellitic fitting [20]. To accurately evaluate the fitting parameters, this method requires a target amplitude that is ≥ λ_laser/4 (where λ is the wavelength), which can hardly be achieved in nanoscale displacement measurement. In addition, this complex fitting algorithm is time-consuming and, thus, is mainly used offline [21]. Although the enhanced ellipse-fitting method presented by Požar et al. [22] can improve the computational efficiency and reduce the measurable amplitude to λ_laser/8, it is still insufficient for nanometer vibration measurement. Dai et al. [23] have developed a method called the “gain/offset correction method,” in which the peak values of the quadrature signals are used to correct the direct current (DC) offsets and the unequal amplitudes. As complex least-squares fitting is avoided, this method can be applied to real-time measurement; however, the displacement must still be ≥ λ_laser/4 to facilitate peak value extraction. Reference [7] introduced an additional PZT to drive the reference corner cube to generate periodic interference signals for real-time normalization. However, because another monitor interferometer is required, this method cannot used in applications where small volume is required. Reference [8] used using a triangular signal as additional modulation to generate continuous phase shifted interference signal for ellipse fitting. In addition, the aforementioned correction algorithms also fail when the PGC carrier phase delay is close to or at certain special values (e.g., nπ +π/4, nπ +π/2). In such cases, signal blanking occurs.

In this study, a novel correction method is proposed for the application to nonlinear errors from PGC carrier phase delay and AOIM. Active laser-wavelength scanning by constant variation of the laser drive temperature is used to replace the target movement. The remainder of the paper is organized as follows. In Section 2, the nonlinear errors due to carrier phase delay and AOIM in PGC demodulation are analyzed. In Section 3, the novel nonlinear-error correction method based on pre-compensation and fine correction is described in detail. The experiment setup and the validation experiments are discussed in Section 4.

2. Effects of carrier phase delay and AOIM in PGC demodulation

In fiber-optic interferometers with laser frequency modulation, the interferometer optical path difference is modulated. Under ideal conditions, the interference signal, which encodes the measured displacement information, is given by

To facilitate processing, the formula for the interference signal $S(t)$ is converted to

It is apparent from Eq. (3) that the modulated interference signal contains the DC component, fundamental frequency component, and multiple harmonic components of the modulated frequency. The fundamental frequency component and the second harmonic component can be expressed as

 

Fig. 1. Schematic of fiber-optic Michelson interferometer with laser frequency modulation: FC: fiber optic circulator; SMF: single mode fiber; BS: beam splitter; M1, M2: mirrors 1 and 2, respectively; APD: avalanche photodetector; DDS1, DDS2: direct digital synthesizer 1 and 2, respectively; LPF: low-pass filter; DFB: distributed feedback laser; GRIN: gradient-index lens; ADC: analog-to-digital converter; DAC: digital-to- analog converter.

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However, in real conditions, because the laser wavelength modulation is achieved via direct current modulation, the light intensity is inevitably modulated and therefore AOIM is introduced. Additionally, unwanted phase delay between the carrier component of the detected interference signal and the carrier has adverse effects on PGC demodulation in the fiber-optic interferometric sensor.

The modified interference model with AOIM in PGC demodulation is given by

Referring to Eq. (10), according to the traditional PGC-Arctan demodulation algorithm, the interference signal is multiplied by $\cos ({\omega _0}t)$ and $\cos (2{\omega _0}t)$, respectively, and is passed through the LPF to obtain the following signals,

Then, comparative simulation experiments were performed. In this study, for a fiber-optic Michelson interferometer, the carrier phase delays were assumed to be 0, π/4, π/2, and 3π/4, the carrier frequency was 1 MHz, and the measured target moved 3750 nm (5 interference fringes) at a constant velocity of 0.75 mm/s. Furthermore, the movement time was 5 ms, the sampling rate was 20 MHz, and the modulation depth was set to C = 2.63. The AOIM depth was set to m = 0.1. Figs.  2(a)–2(d) shows the effects of the carrier phase delay and AOIM in PGC demodulation for the two output signals from an LPF when the carrier phase delay was set to 0, π/4, π/2, and 3π/4, respectively.

It is apparent from Fig.  2(a) that, when the carrier phase delay approaches zero, the AOIM-induced nonlinear error dominates and the Lissajous figure is an ellipse. Thus, the nonlinear error can be corrected via ellipse fitting and the gain/offset correction method. However, it is apparent from Figs.  2(b)–2(d) that, when the carrier phase delay is at certain values (such as nπ +π/4 and nπ +π/2), the nonlinear error due to the carrier phase delay dominates. Thus, the Lissajous figure is a line and either one or both signals are mostly attenuated. In that case, neither the demodulation nor nonlinear correction is possible in the system.

 

Fig. 2. Effects of carrier phase delay and AOIM in PGC demodulation: (a)–(d) two output signals (red and green traces) from a LPF and Lissajous figure (blue traces) of S₁(t) and S₂(t) for the carrier phase delays of 0, π/4, π/2, and 3π/4, respectively..

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3. Correction method for nonlinear errors from carrier phase delay and AOIM

As mentioned above, carrier phase delay and AOIM seriously affect the accuracy of the demodulation results in PGC. Therefore, it is necessary to correct the associated nonlinear errors. However, typical nonlinear correction methods require target movements λ_laser/4; these methods fail when the PGC carrier phase delay is proximate to certain values (e.g., nπ +π/4, nπ +π/2). To solve these problems, active laser-wavelength scanning by constantly varying the laser drive temperature is conducted to replace the target movement. Further, a modified nonlinear-error correction method for errors due to PGC carrier phase delay and AOIM is proposed. The corresponding theoretical analysis and correction steps are discussed below in detail.

3.1 Theoretical analysis of target equivalent movement

In Eq. (11), $\varphi (t)$ consists of the initial phase ${\varphi _0}$ and the phase shift ${\varphi _s}(t)$ caused by the measured displacement $x(t)$; that is,

Normally, the temperature tuning coefficient (typical value 0.1nm/°C) is an order of magnitude larger than the current tuning coefficient (typical value 0.01nm/mA). Further, the variation in the current will result in considerable variation in the light power compared with the temperature tuning. In other words, when tuning the same wavelength range, the light intensity will considerably fluctuate during the current tuning process. According to Eqs. (10)–(17), the phenomena that the variation in light intensity will lead to a variation in m and interference signal amplitude can be found, which will further vary the nonlinear parameters. The test parameters of the factory manual of laser in this study are used for relevant simulation, and the results are as follows:

 

Fig. 3. Lissajous figures (different laser drive temperature and current): (a) driving current is constant and driving temperature is variable, and (b) driving current is variable and driving temperature is constant

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The simulation results demonstrate that to vary the same wavelength, adjusting the driving current will vary the Lissajous figures; that is, the nonlinear characteristic parameters are different. Thus, the wavelength scanning realized by current scanning cannot replace the object motion. In contrast, by adjusting the driving temperature, the Lissajous figures coincide and nonlinear characteristic parameters remain constant, which can be used to realize wavelength scanning instead of object motion.

3.2 Correction method of nonlinear-error

Step 1 precompensation:

The main purpose of this step is to prevent the carrier phase delay from being close to the previously mentioned special values (e.g., n +π/4 and n +π/2) and the output signal of the LPF from being seriously attenuated, as these scenarios can cause PGC demodulation and nonlinear correction failure. Therefore, during this step, coarse compensation of the carrier phase delay is used to provide conditions for nonlinear correction. As shown in Fig.  4, the details of the carrier phase delay precompensation method are given below.

 

Fig. 4. Schematic of nonlinearity correction in fiber-optic Michelson interferometers with laser frequency modulation. M1, M2: mirrors 1 and 2, respectively; APD: avalanche photodetector; BS: beam splitter; SMF: single mode fiber; FC: fiber optic circulator; DDS1, DDS2: direct digital synthesizer 1 and 2, respectively; ADC: analog-to-digital converter; LPF: low-pass filter; DAC: digital-to- analog converter; DFB: distributed feedback laser; GRIN: gradient-index lens; EN: enable end of a nonlinearity correction unit; INT: integer processing unit; Pick: optimal value picking.

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By increasing the laser drive temperature, the laser wavelength is scanned for the first time. In this process, the outputs of the direct digital synthesizer DDS1 and DDS2 are set to $\sin ({\omega _0}t)$ and $\cos ({\omega _0}t)$, respectively. The nonlinearity correction module is disabled on the enable side. Referring to Eq. (11), the interference signal collected by the ADC is multiplied by $\sin ({\omega _0}t)$ and $\cos ({\omega _0}t)$ and then passed through the LPF to obtain the two signals. Under normal conditions, as m ≤ 0.1, the two signals can be approximately equal to the following formula in the precompensation process:

Then, the phase delay $\varphi _{_c}^ \ast $ is approximately determined and compensated back into the carrier to realize carrier phase delay error precompensation. If ${\varphi _c} \in [0,\pi )$, the obtained phase delay is correct, and if ${\varphi _c} \in [\pi ,2\pi )$, the obtained delay differs from the true value by π. Because this difference will only lead to a variation in the direction of the demodulation result, after the correction is completed, the laser wavelength is scanned again, and the direction is discriminated and determined using the known wavelength scanning direction.

Step 2 fine correction:

In this step, the residual nonlinear error is mainly corrected. After the carrier phase delay precompensation is applied in the previous step, the output signals from the LPF are guaranteed not to be greatly attenuated, thereby satisfying the conditions of nonlinear correction.

When the target displacement exceeds λ/4, the Lissajous trajectory of the quadrature signals, as shown in Fig.  5(a), becomes larger than a full ellipse. In that case, the four peaks of the two quadrature signals can be extracted, and the parameters for nonlinear error correction can be calculated. However, as shown in Fig.  5(b), if the displacement is less than λ/4, the extraction of the four peaks is not feasible because of the insufficiency of the Lissajous pattern. Hence, correction cannot be performed. To solve this problem, the active temperature scanning unit is used again to perform an active scan of the laser wavelength. This is equivalent to the target movement and is used instead. Its function is to actively offer sufficient displacement to predetermine the parameters for nonlinear error correction when the target displacement is less than a quarter of the wavelength. Details of the nonlinear fine correction are provided below.

 

Fig. 5. Lissajous trajectories with target displacement of (a) < λ/4 and (b) ≥λ/4. (c) Schematic of gain/offset correction method based on peak value extraction. PVD: peak value detector.

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By reducing the laser drive temperature to its initial value, the laser wavelength is scanned for the second time. In this process, the DDS1 output is set to $\cos (2{\omega _0}t)$, while the DDS2 output remains unchanged. The nonlinearity correction module is thus enabled. At this stage, the two filter output signals are close to those given by Eqs. (11) and (12). Equations (11) and (12) incorporate DC offset, unequal amplitude, and non-orthogonal errors. As shown in Fig.  5, the nonlinearity correction module uses two peak value detectors (PVDs) to evaluate the error parameters.

First, the DC offsets and unequal amplitudes can be corrected using the four peaks of the two signals; that is,

4. Experimental validation

The experimental setup of the fiber-optic Michelson interferometer with laser frequency modulation was constructed as shown in Fig.  3. The laser source was a distributed feedback (DFB) laser (DFB PRO BFY, Toptica, Germany) with a DLC laser driver (DLC PRO, Toptica, Germany). The Michelson interferometer was established with the use of a fiber optic sensor (PS-SH-C01-BSR80, Smaract, Germany), and the interference signal was detected by a photodetector (APD430C, Thorlab, USA). Interference signal processing and modulation signal generation was performed by a signal processing board, based on which the interference signals from the photoelectric detector were converted into digital signals by a 16-bit analogue-to-digital converter (ADC). Sinusoidal modulation signals were created by a 14-bit digital-to-analog converter (DAC). The sinusoidal modulation signal had a 1 MHz frequency. The laser frequency sinusoidal modulation was introduced by the alternating current (AC) modulation port of the DFB laser. The working distance of the Michelson interferometer was 40 mm, and the phase modulation depth C in the PGC was 2.63. The measured displacement was provided by the nanoposition stage (P-733, Physik Instrument, Germany) that had a movement range of 30 µm and a bidirectional repeatability of ± 1 nm. The method described in Section 3 was realized in the field programmable gate array (FPGA), and the results were sent to a personal computer through a universal serial bus (USB).

4.1 Validation of hypothesis

On the basis of the preceding nonlinear precompensation, fine compensation of the nonlinear error was performed. In Section 3, we introduced the fine correction method of the nonlinear error. The core hypothesis of this method is that active laser wavelength scanning through active temperature scanning is equivalent to target movement, if the nonlinear parameters obtained using active laser temperature scanning and those obtained when the target is moving are consistent. Using the proposed method, the nonlinear parameters were obtained in advance when the target was stationary. This allowed the achievement of nonlinear error correction when the target motion range was less than λ/4.

To validate the above hypothesis, the active temperature scanning unit was used to drive the laser temperature from 24°C to 25°C, and the output signals from the filters in the PGC demodulation system were collected. The corresponding Lissajous patterns are shown in Fig.  6(a). For the comparison experiment, the laser temperature was set to 25°C and the PI nanometer displacement platform was set to move at a speed of 1 mm/s with a 5-µm motion range. Again, the output signals from the filters in the PGC demodulation system were collected. The corresponding Lissajous figures are shown in Fig.  6(b).

 

Fig. 6. Lissajous figures with characteristic parameters obtained during (a) temperature scanning and (b) target motion. A, B, S_1c, S_2c, α: characteristic parameters of ellipse.

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The labels A, B, S_1c, S_2c, and α in the figure are the characteristic parameters of the ellipse, and are obtained via the ellipse fitting method. The two sets of parameters are listed in Table  1 for comparison purposes. The differences are very small and the maximal discrepancy is -0.00081. We speculate that this deviation was caused by a slight change in the light power during the experimental process. Thus, the above hypothesis is validated.

Table 1. Comparison of elliptic characteristic parameters obtained based on elliptic fitting.

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4.2 Feasibility verification of nonlinear error precompensation

To prove the effectiveness of the nonlinear error precompensation method, the phase delay was set to π/4 and π/2. The measured target remained stationary. The active temperature scanning drove the laser temperature from 25 to 26°C. Using the nonlinear precompensation method described in Section 3, two output signals from filters and phase delay calculations are plotted in Fig.  7. Figs.  7(a) and 7(b) show that the phase delay was calculated and was equal to 45° and 91° when the phase delay was set to π/4 and π/2, respectively. Thus, the calculated and set delays were approximately the same. The obtained rough carrier phase delay was compensated by the carrier to complete the nonlinear precompensation. Before and after precompensation, the nanoposition stage was set to move at a speed of 1 mm/s within a range of 5 µm. The output signals from the filters in the PGC demodulation system were collected at this time. The corresponding Lissajous patterns are shown in Figs.  7(b) and 7(d). Comparison of the figures pre- and postcompensation reveals that after precompensation, the Lissajous figures changed from lines to ellipses. Although the nonlinear errors (the DC offsets and unequal amplitudes) remained, the signals satisfied the conditions for fine compensation for nonlinear errors. Thus, the results of this experiment prove that the developed module is effective.

 

Fig. 7. Nonlinear error precompensation results: (a, c) approximate calculation of carrier phase delays, where S₃(t) (red traces) and S₄(t) (pink traces) are correspond to the left ordinate and calculated carrier phase delays are φ_m(t) (blue traces) correspond to the right ordinate, and (b, d) comparison of Lissajous figures of S₁(t) and S₂(t) before and after precompensation when the phase delay was set to π/4 and π/2, respectively.

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4.3 Feasibility verification of fine correction of nonlinear error

To prove the effectiveness of the fine correction of the nonlinear error, the nanometer displacement platform was set to move at a speed of 0.75 mm/s within a range of 4.5 µm. The nonlinear fine correction was initially deactivated. When the target had moved for a certain period of time, the nonlinear fine correction was activated. The phase demodulation results of the entire process were collected, and the corresponding nonlinear errors were calculated. These data are plotted in Figs.  8(a) and 8(b) for carrier phase delays of π/4 and π/2, respectively. It is apparent from the figure that after nonlinear precompensation, there was a residual nonlinear error of approximately 10 nm. However, after fine correction, the maximum nonlinear error was less than 1 nm. Thus, the nonlinear error correction method was proven to be effective.

4.4 Nanometer displacement measurement

Nanometer displacement measurement was performed to verify the correction method for the nonlinear errors from the PGC carrier phase delay and AOIM. The laser drive temperature was changed from 25 to 26°C and nonlinear error precorrection was performed. The laser drive temperature was changed from 26 to 25°C again and the nonlinear-error fine correction was applied. Nanometer displacement measurements were conducted by moving the nanoposition stage at 5 nm steps at 600 times (i.e., for 3 µm). The measured results are shown in Fig.  8. The stage displacement, the demodulated displacement, and the deviation between them, are shown in Fig.  9(a); the average deviation was almost zero. The maximum error was 1.21 nm and the standard deviation was 0.67 nm. Figure  9(b) shows the fast Fourier transform (FFT) analysis of the displacement deviation depicted in Fig.  9(a). As shown in Fig.  7(b), no apparent peak emerged at the second harmonic components of the fringe, and displacement measurements with nanometer accuracy could be realized.

 

Fig. 8. Experimental results of nonlinear error fine correction for carrier phase delays of (a) π/4 and (b) π/2. Note that demodulation phase values (blue traces) are correspond to the left ordinate and nonlinear errors (red traces) are correspond to the right ordinate, nonlinear error fine correction is enable at 3 ms.

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Fig. 9. Experimental results of displacement measurements at the nanoscale. (a) Measured results for 5 nm steps conducted 600 times (total range = 3 µm). Note that the red line is shifted by 0.5 µm to allow plot visibility, demodulation displacement (black traces) and displacement of the stage (red traces) are correspond to the left ordinate, deviation (blue traces) are correspond to the right ordinate. (b) FFT analysis of displacement deviation.

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5. Conclusion

The carrier phase delay and AOIM have always caused nonlinear errors in fiber optic interferometers with laser frequency modulation. These errors can ultimately affect the PGC demodulation. In this study, the effects of the carrier phase delay and AOIM on PGC demodulation with the PGC-Arctan approach were analyzed, and a nonlinear error correction method was realized. This method included two parts: precompensation and fine correction. These can overcome the problems owing to which the traditional nonlinear correction method fails when the carrier phase delay is at certain values, or the problems of nonlinear correction for motion associated with less than half the laser wavelength. In experiments using fiber-optic Michelson interferometers, the precompensation was validated, along with the hypothesis that active laser wavelength scanning through active temperature scanning was equivalent to target movement. The nonlinear error was less than 1 nm for applications of the proposed nonlinear error fine correction. Hence, the nanometer displacement measurement was realized using the proposed nonlinear error fine correction method for PGC demodulation. Owing to the limitation of PGC phase modulation depth, the fiber-optic Michelson interferometer proposed in this study was only suitable for small range displacement measurements. In the future, we will focus on the measurement of large-range displacements and realize PGC phase modulation depth control.