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Embedded micro-probe fiber optic interferometer with low nonlinearity against light intensity disturbance

Chen Zhang, Yisi Dong, Pengcheng Hu, Haijin Fu, Hongxing Yang, Ruitao Yang, and Limin Zou

Open Access Open Access

Abstract

A novel low-nonlinearity Michelson microprobe fiber interferometer against light intensity disturbance for high-precision embedded displacement measurements is introduced. To analyze the influence of light intensity disturbance on the microprobe and measurement accuracy of the integrated fiber interferometer, an equivalent model of micro-probe sensing with the tilted target is established. The proposed PGC demodulation and nonlinearity correction method with simple principle helps avoid DC component varying with light intensity. The experiments show that residual displacement errors of the micro-probe fiber interferometer are reduced from 4.36 nm to 0.46 nm, thus allowing embedded displacement detection with sub-nanometer accuracy under low frequency light intensity disturbance.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The rapid development of process manufacturing and scientific research fields requires displacement metrology equipment that would yield an accuracy in the range of a few nanometers. Such rapid growth would demand excellent characteristics such as integration, robustness, and size scale. With the mature application of optical fiber communication technology, fiber optic interferometry technology has developed rapidly since the 1970s. Compared to the traditional block prism interferometric system, the fiber-optic interferometer has several advantages, including convenient optical path adjustment, small space requirement, and suitability for extreme environment measurement, and it is widely used in the field of position and displacement metrology.

The fiber optic microsensing probe is the core component of an integrated fiber-optic interferometric system with a compact structure and flexible use, and is available for monitoring micro-displacement and micro-vibration in confined spaces. The fiber optic micro-sensors use a single-mode fiber (SMF) as the signal arm to transmit the beam, whose end face and the target reflective surface form the Fabry-Perot (F-P) interferometric cavity. A few examples of sensors include the F-P interferometer for micro-displacement reconstruction proposed by Liang et al. [1] and the sinusoidal phase-modulated interferometer [2] built by Jin et al. Although the sensing probe with SMF as the signal arm satisfies the miniaturization of the interferometer, the small mode field diameter [3] of the SMF and the unconstrained output beam limit the widespread use. To overcome this issue, Acharya et al. introduced an aspheric lens to collimate the light at the tip of the fiber core in the extrinsic F-P interference optical path [4]. However, the F-P interferometer with the fiber end face directly as a cavity mirror is sensitive to fiber perturbation, thereby causing interference signal instability. In addition, all-fiber sensors prepared by optical fiber melting and etching processes combined with new materials have been reported successively. A few examples of such sensors include micro-Michelson interferometer based on end-collapsed [5] and hollow core fiber for displacement sensing and a spherical fiber F-P interferometer [6] for greater tolerance of target reflectivity. The complex fabrication can increase the cost of fiber-optic sensors and limit large-scale promotion. The gradient index (GRIN) lenses are widely used in the preparation of tiny optical sensing probes owing to their small size as well as the good self-focusing, collimation, and imaging characteristics [7–10]. For instance, an F-P interferometer model based on the GRIN all-fiber probe [3] investigated by Wang et al. improved the output signal strength and dynamic measurement range, as compared to the SMF interferometer. Du et al. proposed a compact F-P interferometric sensor based on graded index fiber for high-sensitivity displacement and vibration sensing [10]. To the best of our knowledge, the integrated displacement measurement system based on GRIN sensor probes mostly adopts a low-finesse optical F-P interference structure [11], which approximates multi-beam interference as a two-beam interference. Thus, there are high-order nonlinear errors in this principle.

The phase-generated carrier (PGC) technology is a widely used demodulation algorithm for optical fiber interferometers mainly because of properties such as high sensitivity, high dynamic range, and good linearity. Considering the integration of the fiber optic interferometer, the internal modulation performed by the circuit to generate the phase carrier is desirable. However, it also introduces the accompanying optical intensity modulation [12], thus causing nonlinear errors. Moreover, owing to the reduced light-receiving area of the receiving signal, the fiber microsensing probe is more sensitive to the tilt angle of the moving target, which indicates that the intensity of the collected interference signal changes with the change in the target tilt angle because of the pitch and yaw in non-ideal movements. Therefore, the realization of a high-precision and integrated interferometric displacement sensing system with a micro-probe based on the light intensity disturbance would improve the robustness of the interferometer. The nonlinearity correction methods with real-time monitoring and dynamic compensation have been reported for nonlinear measurement errors that are introduced by the disturbance of the interference signal intensity. Dai et al. [13] proposed a correction method based on the peak detection to obtain the DC offset and AC unequal amplitude error parameters in real time. Hu et al. realized online correction of non-orthogonal errors by simple addition and subtraction operations on the corrected signal based on this theory [14]. These methods are only feasible when the nonlinear characteristic parameters remain unchanged or are slightly changed during the measurement. In addition, some improved PGC demodulation algorithms have been suggested to eliminate the influence of light intensity disturbances. He et al. [15] improved the PGC demodulation algorithm based on arctangent and differential self-multiplication (PGC-DSM-Arctan) [16] and achieved high stability under light intensity fluctuations. Zhang et al. introduced a changed reference signal envelope to eliminate the influence of light intensity disturbance on demodulation results based on the differential cross multiplication (PGC-DCM) demodulation [17]. However, the use of piezoelectric ceramics to generate the reference compensation signal results in an increase in the complexity of the interference sensing optical path, and this cannot be easily implemented in a structurally compact micro-probe fiber interferometer.

This paper presents a novel integrated fiber optic interferometer based on a Michelson microprobe with a single-arm common optical path for high-precision displacement measurements. An equivalent model of micro-probe interferometric sensing containing a tilted target is established based on coordinate transformation, and the influence of interference signal intensity disturbance on the microprobe and signal demodulation accuracy is theoretically analyzed according to the equivalent model. With the aim of reducing the nonlinear errors of displacement measurement induced by light intensity disturbance, a low-nonlinearity PGC demodulation and error correction method against the light intensity disturbance is proposed. This approach with simple principle for easy realization has no attached devices, and helps reduce the measurement deviation introduced by the interference signal intensity change. Combined with a flexible fiber optic microprobe, the fiber-optic interferometer can conduct embedded precision displacement measurements in a narrow space. The rest of this article is organized as follows: In Section 2, an equivalent model of the fiber microprobe based on the tilted target is first established, and the effect of interference signal intensity disturbance on the coupling performance of the microsensing probe and demodulation accuracy is studied. Then, an improved PGC demodulation and nonlinearity correction method against light intensity disturbance is introduced. Section 3 presents the experimental verification of the microprobe equivalent model and the proposed PGC demodulation and correction method, and the experimental results based on the microprobe fiber optic interferometer.

2. Theoretical analysis and operation principle

2.1 Effects of interference light intensity disturbance on the microprobe

The proposed Michelson micro-probe (see Fig. 1) is composed of a single-mode fiber pigtail, an air gap, a GRIN lens, and a non-polarizing cube beam splitter (NPBS Cube). This micro-probe is implemented through a single-arm common optical path structure, which improves the disturbance immunity of the fiber-optic interferometer. However, the microprobe is extremely sensitive to the variation of the interference signal intensity, due to its millimeter-scale clear aperture. To analyze the effects of this interference light intensity disturbance on the microprobe, the constructed equivalent model of the fiber microprobe with a tilted target is shown in Fig. 1. The Gaussian beams [18,19] output from the fiber collimated by the GRIN lens is split equally by the NPBS; one beam is reflected back to the GRIN lens by the reflecting surface at the top of the NPBS as the reference beam, and the other beam propagates to the mirror and reflects back to the microprobe, wherein it interacts with the reference beam to obtain the interference signal. After the fiber light source with the beam waist w at the end face is transformed by the microprobe, the output beam waist w1 is located l away from the probe. The distance Zwd between the reflection target and probe is called the working distance of the fiber microprobe. The beam reflected by the target with a tilt angle θ becomes a tilted-axis Gaussian beam with an angle 2θ with the optical axis. This makes it difficult to simulate the propagation path of Gaussian beams. The process wherein the Gaussian beam reflected back to the microprobe through the mirror is equivalent to the transmission of the Gaussian beam twice the working distance 2Zwd, with unchanged beam characteristics. Therefore, the reflection target is equivalent to the same optical fiber receiving probe placed symmetrically with the reflection target as the axis, and the interference of the two beams can be regarded as coupling at the end face of the equivalent probe between the signal beam emitted by the original probe and the intrinsic reference beam output from the equivalent probe, as shown in Fig. 1(a). The reflection target with tilt angle θ is equivalent to the receiving fiber probe with an angle of 2θ along the optical axis, as shown in Fig. 1(b). Here, the mirror still serves as the symmetry axis of the sensing optical path with the unaltered transmission characteristics of the tilted-axis beams. The proposed equivalent model based on the tilted reflection target, by means of the equivalence of the incidence of off-axis Gaussian beams to the fiber probe and the incidence of Gaussian beams to the tilted fiber probe, simplifies the construction of the interference optical path of the micro-probe with tilted elements. It also avoids the complicated solution of the wave equation in the misaligned optical system. Thus, the relationship between the coupling performance of the sensing probe and the target inclination angle can be characterized using the established equivalent model.

 figure: Fig. 1.

Fig. 1. Equivalent model of fiber optic micro-probe (a) vertical to the optical axis and (b) with the angle of 2θ with the optical axis. w: radius of output beam from SMF; w1: beam waist radius of micro-probe; l: beam waist position of micro-probe; Zwd: working distance of micro-probe.

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In Fig. 2(a), the coordinate system is established with the origin at the center of the beam waist position outputted from the micro-probe, where the thickness of the NPBS is ignored, and it only involves the light splitting of Gaussian beams, which can be expressed as:

$$E(x,y,z) = {E_0}{\alpha _1}\frac{{{w_1}}}{{w(z)}}\exp ( - \frac{{{x^2} + {y^2}}}{{{w^2}(z)}} - i[k(z + \frac{{{x^2} + {y^2}}}{{2R(z)}}) - arctg\frac{z}{f}]),$$
where E0 is a constant, α1 is the scale factor of the NPBS, k is the wave number, w(z) is the curvature radius of the equiphase surface of Gaussian beams at z, R(z) is the spot radius of the equiphase surface at z, and f is the confocal parameter of Gaussian beams.

 figure: Fig. 2.

Fig. 2. Coordinate system of micro-probe equivalent model (a) with the beam waist position as the origin and (b) with the tilted target mirror.

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The coordinate system is established with the exit surface of the probe as the origin of the coordinate, x-axis along the probe exit surface, the optical axis as the z-axis, and the y-axis perpendicular to the paper. This corresponds to a shift in the coordinate system at the beam waist by a distance of l to the left along the optical axis, thus yielding the expression of the output light field in the new coordinate system:

$$\begin{array}{l} {E_s}(x,y,z) = E(x,y,z - l) = {E_0}{\alpha _1}\frac{{{w_1}}}{{w(z - l)}}\\ \times \exp \left( { - \frac{{{x^2} + {y^2}}}{{{w^2}(z - l)}} - i[k((z - l) + \frac{{{x^2} + {y^2}}}{{2R(z - l)}}) - arctg\frac{{z - l}}{f}]} \right). \end{array}$$

Similarly, a coordinate system represented by (x’, y’, z’) is established at the end face of the equivalent probe, NPBS scale factor α = α1 = α2 = 0.5, and the corresponding effective received light field [20] of the equivalent probe in this coordinate system is expressed as

$${E_r}(x',y',z') = {E_0}{\alpha _2}\frac{{{w_1}}}{{w(z' + l)}}\textrm {exp} \left( { - \frac{{x{'^2} + y{'^2}}}{{{w^2}(z' + l)}} - i[k((z' + l) + \frac{{x{'^2} + y{'^2}}}{{2R(z' + l)}}) - arctg\frac{{z' + l}}{f}]} \right).$$

The coupling efficiency of the optical fiber sensing probe is given by the overlapping integral of the output photoelectric field and the eigenmode field of the equivalent probe at the end face of the equivalent probe, as follows:

$$\eta = \frac{{{{\left|{\int {{{\int { {({E_s}E_r^\ast )} |} }_{z^{\prime} = 0}}dx^{\prime}dy^{\prime}} } \right|}^2}}}{{\int {{{\int { {({E_s}E_s^\ast )} |} }_{z^{\prime} = 0}}dx^{\prime}dy^{\prime}\int {{{\int { {({E_r}E_r^\ast )} |} }_{z^{\prime} = 0}}dx^{\prime}dy^{\prime}} } }}.$$

When the working distance of the probe is fixed and the target mirror is tilted at an angle θ (see Fig. 2(b)), the equivalent probe is also tilted at an angle of 2θ with the optical axis. According to the transformation of the coordinate systems at the two probe end faces (x = x'cos2θ - z'sin2θ - Zwdsin2θ, y = y’, z = x'sin2θ + z'cos2θ + Zwd + Zwdcos2θ), and assuming that the mirror is located beam waist (Zwd = l), Eq. (2) and Eq. (3) are substituted into Eq. (4) with the small-angle approximation [20] in the derivation. This yields the expression between the power coupling efficiency of the microprobe and the target tilt angle:

$$\eta = \exp [ - {\sin ^2}\theta (\frac{{{l^2}{f^2}}}{{w_1^2({l^2} + {f^2})}} + \frac{{w_1^2{k^2}{f^2}}}{{4({l^2} + {f^2})}})].$$

The simulation curve of the power coupling efficiency with the target located at the beam waist is shown in Fig. 3(a). The optical power captured by the micro-probe decreases rapidly with an increase in the deflection angle between the normal of the reflecting surface and the incident beam. This is reflected in the power coupling efficiency, which is approximately 19% when the target tilt angle is ±0.05°. Meanwhile, compared to the high sensitivity of coupling efficiency, the fringe contrast [21] changes smoothly with the tilt angle of the mirror, thus maintaining a relatively high contrast within a range of angles. The optical power coupling efficiency physically reflects the degree of overlap between the reference and measurement spots, as shown in the inset of Fig. 3(b). As the angle of the reflected light increased, the reference spot and the measurement spot gradually dispersed from the initial coincidence, and when the angle increased to 0.10°, there was almost no overlap between the two spots; that is, no signal light was coupled into the fiber probe to interfere with the reference light.

 figure: Fig. 3.

Fig. 3. Simulation verification of the equivalent model of the micro-probe based on the tilted target with w = 5.2, λ = 1.550 µm. (a) Optical power coupling efficiency (red line) and fringe contrast (blue line) and (b) the overlap between the reference and the measurement spot related to the target tilt angle.

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2.2 Effects of interference light intensity disturbance on demodulation accuracy

The effect of light intensity disturbance caused by the target tilt misalignment on the microprobe coupling efficiency is investigated in Section 2.1. The following further constructs the relationship between this interference light intensity disturbance and displacement demodulation accuracy. The photoelectric signal collected by the detector can be expressed as

$$s(t )= A + B\cos ({C\cos ({{w_0}t} )+ \varphi (t )} ),$$
where C is the carrier phase modulation depth, w0 is the carrier angular frequency, φ(t) is the phase of the moving target, and A and B are the DC and AC terms of the interference signal, respectively, expressed as
$$A = {k_v}({{I_r} + {I_s}} )= {k_v}({{\alpha^2}{R_1}{I_0} + {\alpha^2}{R_2}{I_0}\eta } ),$$
$$B = 2{k_v}\sqrt {{I_r}{I_s}} = 2{k_v}\sqrt {({{\alpha^2}{R_1}{I_0}} )({{\alpha^2}{R_2}{I_0}\eta } )} ,$$
where kv represents the detector conversion coefficient, I0, Ir, and Is are, the output intensity of the laser light, the reference light intensity inside the probe, the received signal light intensity, R1 and R2 are the reflectivity of the NPBS reflecting surface and the target mirror in the measurement arm, and η denotes the coupling efficiency of the micro-probe, respectively. From Eq. (8), the interference signal intensity is related to I0 and η when the detector, reflection target, and NPBS are fixed. Thus, the interference light intensity disturbance is mainly caused by the fluctuation of the light source and the instability of the received signal light intensity. To ensure the miniaturization of the optical path, a carrier signal is usually generated inside the interference system to modulate the frequency of the laser light source, which is accompanied by the modulation of the light source intensity (AOIM) [22], which is expressed as
$$s(t )= A[{1 + m\cos ({{w_0}t + {\varphi_m}} )} ][{1 + {B / A}\cos ({C\cos ({{w_0}t} )+ \varphi (t)} )} ],$$
where the first square bracket represents AOIM, which is the light source intensity disturbance related to the modulation frequency, and the second square bracket represents the ideal interference signal collected by the detector, m is the light source amplitude modulation depth, and φm is the phase difference [23] between the frequency modulation and the accompanying intensity modulation of the light source.

According to the PGC-Arctan demodulation algorithm, the interference signal s(t) containing the target motion phase after ADC conversion is mixed with the carrier fundamental frequency cos(w0t) and double frequency cos(2w0t) generated inside the module, respectively. The mixing result passes through the low-pass filter with a cut-off frequency w0/2 and yields two output signals, which are recorded as

$${s_1}(t )={-} B\sqrt {{b^2} + J_1^2(C )} \sin ({\varphi (t )- {\theta_1}} )+ mA\frac{1}{2}\cos {\varphi _m},$$
$${s_2}(t )={-} B\sqrt {{a^2} + J_2^2(C )} \cos ({\varphi (t )- {\theta_2}} ),$$
where a, b, θ1, and θ2 are, respectively, expressed as (Jn(C) is the n-order Bessel function of C):
$$a = \frac{m}{2}({{J_1}(C )\cos {\varphi_m} - {J_3}(C )\cos {\varphi_m}} ),$$
$$b = \frac{m}{2}({{J_0}(C )\cos {\varphi_m} - {J_2}(C )\cos {\varphi_m}} ),$$
$${\theta _2} = \arctan \frac{a}{{{J_2}(C )}},$$
$${\theta _1} = \arctan \frac{b}{{{J_1}(C )}}.$$

From Eqs. (10) and (11), it can be seen that the AOIM demodulation signals s1(t) and s2(t) after PGC demodulation are a group of non-orthogonal signals with DC offsets and unequal amplitudes. If the two output signals are directly subjected to arctangent and phase unwrapping to solve the target phase φ(t), a periodic nonlinear phase error φe is generated, and it is expressed as

$${\varphi _e} = \textrm {Phauw} \left[ {\arctan \frac{{{s_1}(t )}}{{{s_2}(t )}}} \right] - \arctan \frac{{\sin \varphi (t)}}{{\cos \varphi (t)}}.$$

Assuming that the target motion frequency is 1 kHz, the spectrum analysis of the demodulated signal under different conditions is shown in Fig. 4(a). The ideal demodulation signal spectrum has an amplitude at 1 kHz, indicating that the PGC demodulation algorithm filters out high-frequency components and retains the target motion frequency. Additionally, a DC component is introduced by the light source intensity modulation in the AOIM spectrum, consistent with Eq. (10). To realize real-time correction of nonlinear errors, the extremum method [23,24] based on peak detection is usually applied to correct the nonlinearity online. Using this method, the DC offset and AC unequal amplitude correction parameters of the signal are calculated by evaluating the extreme value of the interference period. However, for the optical fiber interferometer, the interference signal intensity instability is not only caused by the AOIM induced by the internal frequency modulation of the light source, but is also reflected in the signal light intensity disturbance (SID) affected by external factors such as the measurement environment and the target spatial position. Therefore, the micro-probe coupling efficiency η changes owing to the target tilt angle and the unstable measurement environment, thus resulting in time-varying disturbance generation of the detection signal intensity coupled to the probe in the actual measurement process. The signal light intensity is a disturbance that affects the stability of the DC offset and AC amplitude [25]; the DC term in Eq. (10) varies with the change in the measurement beam intensity, causing deviations in the DC offset characteristic parameters that are extracted by the above extremum method. This makes it impossible to completely correct the DC errors in the signal. In Fig. 4(a), the spectrum presents a shift centered on the target motion frequency and DC component and stepped at a 20-Hz disturbance frequency, when the signal light intensity changes sinusoidally at a frequency of 20 Hz.

 figure: Fig. 4.

Fig. 4. (a) Frequency spectrum of ideal, AOIM and AOIM&SID demodulation signal. (b) Nonlinear displacement errors under AOIM and AOIM&SID with conventional nonlinearity correction (m = 0.1, ms = 0.1, C = 2.63 (J1(C) = J2(C)), φm= 0 rad). Inset: Amplification of nonlinear errors under AOIM.

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With the target motion frequency of 1 kHz, the picometer displacement errors after conventional nonlinearity correction are shown in Fig. 4(b), and this indicates that the extremum method is capable of nonlinearity compensation caused by the AOIM. If the signal light is disturbed at t0 in the form Is = Is0 × (1 + mscos (2π × 20 Hz × t)) (Is0 is the average intensity of the signal light, ms is the disturbance depth of the signal light), the demodulated signal amplitude is modulated, which has a subsequent impact on the correction accuracy. Thus, the nonlinear error increases from 24 pm to 2.8 nm. This take into account the effect of external factors on signal beam transmission, different from previous studies wherein the signal beam coupling efficiency is assumed to be 100%. The simulation prove that the conventional extremum method is not ideal for signal light intensity disturbances, wherein the extraction of nonlinear characteristic parameters, especially the DC offset, produces serious deviations and additional errors, thus decreasing the displacement measurement accuracy of the optical fiber interferometer. Therefore, a PGC demodulation and nonlinearity correction method that reduces the influence of signal light intensity disturbance is significant in high-precision fiber microprobe interferometers.

2.3 PGC demodulation and error correction method against interference light intensity disturbance

It has been illustrated above that when the signal light intensity fluctuates owing to the measurement environment or target motion, the AC amplitude and DC offset terms in Eqs. (10) and (11) are time-dependent. The former is simultaneously affected by the light intensity fluctuation, and its common-mode error can be eliminated by division; the latter is continuously varying, and it is difficult to accurately obtain the offset parameter corresponding to the correction moment. This is usually adopted as a constant value independent of time or a lag of one period [14]. A novel higher harmonic demodulation-amplitude normalization correction method for eliminating the time-varying DC offset term, is proposed in combination with a simple improvement of the PGC algorithm and nonlinear error correction. As illustrated in Fig. 5, the interference signal converted by the ADC is not mixed with the carrier fundamental frequency and the double frequency generated by the DDS in the system, but is multiplied by the double frequency and triple frequency of the carrier, respectively, and then passed through a low-pass filter with a cut-off frequency of w0/2. The signal mixed with the second harmonic and filtered can be expressed by Eq. (11), and the other output signal is denoted as:

$${s_3}(t )= B\sqrt {{c^2} + J_3^2(C )} \sin ({\varphi (t )- {\theta_3}} ),$$
where the expressions for c and θ3 are:
$$c = \frac{m}{2}({{J_2}(C )\cos {\varphi_m} - {J_4}(C )\cos {\varphi_m}} ),$$
$${\theta _3} = \arctan \frac{c}{{{J_3}(C )}}.$$

 figure: Fig. 5.

Fig. 5. Proposed PGC demodulation and nonlinearity correction method schematic diagram. ADC: analog-to-digital converter; DDS: direct digital synthesizer; LPF: low-pass filter; Arctan: arctangent operation; Phauw: Phase unwrapping; Max: Maximum; Min: Minimum.

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At this time, the carrier phase modulation depth C should satisfy J2(C) = J3(C), and according to the Bessel function, C = 3.77. Comparing Eq. (17) with Eq. (11), it can be seen that the two equations have similar forms, but s3(t) is the result of extracting the third harmonic demodulation of the interference signal, which effectively avoids the DC term induced by AOIM, thus improving the insensitivity to the detected light intensity fluctuation. Because of the opposite sign between s3(t) and s2(t), the s3(t) signal needs to be inverted before entering the residual error correction module in order to obtain a pair of non-orthogonal signals with the same sign. The AC amplitude is affected by the same light intensity disturbance and has no DC offset.

The maximum and minimum values of the above signals within one measurement period were extracted and calculated to obtain the AC gain amplitude. The two signals were normalized separately according to the obtained AC amplitude, and the results were expressed as follows:

$${s_2}^{\prime}(t )= \cos ({\varphi (t )- {\theta_2}} ),$$
$${s_3}^{\prime}(t )= \sin ({\varphi (t )- {\theta_3}} ),$$

It is shown that the second- and third-harmonic PGC demodulation avoids the DC offset error term that dithers with the signal light intensity, but there are still non-orthogonal errors independent of the signal light intensity disturbance. To eliminate non-orthogonal errors, the two normalized signals are subtracted and added to construct a new pair of completely orthogonal signals, denoted as

$${s_2}^ \ast (t )= 2\cos \left( {\frac{{{\theta_3} - {\theta_2}}}{2} + \frac{\pi }{4}} \right)\cos \left[ {\varphi (t )- \left( {\frac{{{\theta_3} + {\theta_2}}}{2} + \frac{\pi }{4}} \right)} \right],$$
$${s_3}^ \ast (t )= 2\sin \left( {\frac{{{\theta_3} - {\theta_2}}}{2} + \frac{\pi }{4}} \right)\sin \left[ {\varphi (t )- \left( {\frac{{{\theta_3} + {\theta_2}}}{2} + \frac{\pi }{4}} \right)} \right].$$

Equations (22) and (23) demonstrate that the non-orthogonal errors have been well compensated, but the constructed orthogonal signals with the target displacements introduce an additional unequal amplitude. Therefore, it is necessary to perform the normalization correction of the AC amplitude again by extracting and calculating the characteristic parameters. Finally, the two signals are obtained as

$${s_2}^{\prime\prime}(t )= \cos \left[ {\varphi (t )- \left( {\frac{{{\theta_3} + {\theta_2}}}{2} + \frac{\pi }{4}} \right)} \right],$$
$${s_3}^{\prime\prime}(t )= \sin \left[ {\varphi (t )- \left( {\frac{{{\theta_3} + {\theta_2}}}{2} + \frac{\pi }{4}} \right)} \right].$$

The AOIM&SID signal with the same parameters as in Section 2.2 is utilized to demodulate and correct the target displacement, and yield the displacement error (see Fig. 6(a)), which is reduced by about an order of magnitude compared to the conventional extremum method. Figure 6(b) shows the nonlinear displacement errors under different signal light intensity disturbance frequencies at the target motion frequency of 1 kHz. The nonlinear error of the conventional correction method fluctuates between 2 nm and 3 nm at low frequency disturbance, and increases continuously with the disturbance frequency greater than 150 Hz. The displacement error with the proposed higher harmonic demodulation-amplitude normalization method varies gradually with frequency, less than 1 nm within the 100-Hz disturbance frequency. The simulation illustrates that the proposed PGC demodulation and correction method can effectively decrease the nonlinear errors, thus suitable for the case of signal optical intensity instability during micro-probe fiber interferometer measurement. This approach with simple principle for realization has no additional devices, which maintains the integration of the micro-probe fiber interferometer.

 figure: Fig. 6.

Fig. 6. Nonlinear displacement errors under AOIM&SID (a) at a 20-Hz disturbance frequency and (b) at different signal light intensity disturbance frequencies, with the proposed PGC demodulation and nonlinearity correction method and conventional correction method.

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3. Experimental validation

An experimental validation setup was built to verify the feasibility of the constructed micro-probe equivalent model with a tilted target, as shown in Fig. 7(a). The light source was a distributed feedback (DFB) laser (DFB PRO, Toptica, Germany) driven by a DLC light source driver (DLC PRO, Toptica, Germany). The internal frequency modulation of the light source is implemented by the 1M-Hz sinusoidal current signal. The optical fiber micro-Michelson probe was applied to collect the signal light reflected back from the reflection target at different tilt angles, which is realized by the angle rotating stage. Figure 7(b) is used to verify the feasibility of the proposed PGC demodulation and correction method for higher harmonic demodulation-amplitude normalization. The signal light with the target displacement generated by the micro-displacement stage (P-733, Physik Instrument, Germany) synthesizes the interference signal with the reference light inside the probe, which is fed to the signal acquisition and processing board (in the red wireframe) via the photoelectric conversion. A continuous variable neutral density filter (NDF) is placed between the fiber microprobe and the target to be measured, and this can equivalently change the power coupling efficiency of the micro-probe by rotation, thus changing the signal light intensity that is coupled back to the microprobe.

 figure: Fig. 7.

Fig. 7. Micro-probe equivalent model (a) and the proposed PGC demodulation and nonlinearity correction method (b) verification experimental setup. DFB Laser: distributed feedback laser; Driver: light source driver module; FC: fiber optic circulator; PIN: PIN photodetector; NDF: neutral density filter; DAC: digital to analog converter; Red wireframe: the signal acquisition and processing module; Photo: physical picture of fiber micro-Michelson interference probe.

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3.1 Feasibility verification of the equivalent model of micro-probe based on the tilted target

The fiber micro-Michaelson probe is shown in the photograph of Fig. 7(b), with an overall length of less than 2 cm and a maximum diameter of 4.7 mm, and its highly integrated miniature structure reflects the applicability of embedded measurements in confined spaces. Because the microprobe is sensitive to the angle change of the motion target, the following experiments verify the feasibility of the micro-probe equivalent model with the tilted target and the effect of the target tilt angle on the coupling performance and interference signal. First, the model of the target tilt angle and the microprobe power coupling efficiency was verified by replacing the NPBS with a glass cube of equal size without the reference light. The tilt angle of the target mirror is changed by the angle rotating stage shown in Fig. 7(a), and the coupling efficiency of the fiber microprobe is characterized by the ratio of the coupling optical power to the total power emitted from the probe. The normalized coupling efficiency at different angular positions is fitted nonlinearly, and the experimental fitting curve and simulation results are shown in Fig. 8(a). Each data point was calculated as the average over five repeated measurements and the error bar represents the standard deviation of each measurement set. These results match with each other, thus proving the feasibility of the equivalent model of the microprobe. The fringe contrast, which is a measure of the interference signal quality, was also tested experimentally. The interference signal is generated by the complete microMichelson probe and the DFB laser with wavelength scanning by the driving current, and the experimental results of the interference fringe contrast are shown in Fig. 8(b). Owing to the loss of light in the experimental process, the experimental value has a loss of approximately 0.32 dB from the theoretical value, and the experimental data have been scaled up accordingly for better comparison. Figure 8(b) indicates that the contrast variation trend with the target mirror reflection angle is consistent with that simulated by the equivalent model in Section 2.1, and the slight deviation is caused by the fluctuation of the experimental environment and the measurement error.

 figure: Fig. 8.

Fig. 8. Experimental results of micro-probe equivalent model validation based on tilted target. (a) Fiber micro-probe coupling efficiency as function of the target mirror tilt angle. (b) Interference fringe contrast as function of the target mirror tilt angle.

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The interference signal amplitude at different optical power coupling efficiencies of the microprobe was further characterized (Fig. 9). A comparison of the curve fitted to the experimental data points and the equivalent model simulation curve revealed that as the micro-probe coupling efficiency increases, the amplitude of the interference signal collected by the detector is larger, and the signal-to-noise ratio is better in the case of a certain noise, which is more conducive to the accurate demodulation of the subsequent interference phase. The interference signal intensity of simulation and fitting has the same variation trend as the micro-probe coupling performance, which verifies the effectiveness of the equivalent model of the micro-probe based on the tilted target.

 figure: Fig. 9.

Fig. 9. Interference signal amplitude as function of fiber micro-probe coupling efficiency.

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3.2 Feasibility verification of the proposed PGC demodulation and nonlinearity correction method

The feasibility of the proposed PGC demodulation and correction method for higher harmonic demodulation-amplitude normalization against light intensity disturbance was verified using the experimental setup shown in Fig. 7(b). When the target mirror is mounted on the micro-displacement stage shifts, the phase of the interference signal changes correspondingly, and this is collected by the microsensing probe and transmitted to the PGC demodulation and correction module for processing. The NDF is placed in the sensing optical path between the microprobe and the mirror to continuously adjust the signal light intensity reflected back from the mirror to simulate signal light intensity disturbance [26].

The signal light intensity disturbance, which is generated by the continuously rotating NDF, is superimposed on AOIM and will cause instability of the interference signal intensity. The proposed PGC demodulation and nonlinearity correction method aims to reduce the effect of interference light intensity disturbance on the phase demodulation accuracy of the fiber interferometric system. At the beginning, the output signals s2(t), s3(t) after higher harmonic demodulation and amplitude normalization correction are acquired under AOIM, AOIM&SID conditions, respectively, and their Lissajous trajectories are shown in Fig. 10. The characteristic parameters of the interference signal are obtained by ellipse fitting, which are considered as actual nonlinear error parameters, as shown in Table 1. After a higher harmonic demodulation of the interference signal, the center of the Lissajous trajectory is located near the origin (Fig. 10(a)), and only a small DC component exists, which may be caused by factors such as unsatisfactory optical components and electronic component parameter deviations. This indicates that the higher harmonic demodulation effectively reduces the DC offset component that is susceptible to light intensity disturbance. Further, through non-orthogonal compensation and AC amplitude normalization correction, the Lissajous trajectory is transformed from elliptical to nearly circular. It can also be intuitively seen from Table 1 that the corrected signal is approximately regarded as a set of orthogonal signals with no DC offset and equal AC amplitude, thus verifying the feasibility of the proposed method for nonlinear error correction under interference light intensity disturbance. Moreover, the output signal under SID is modulated by the low-frequency disturbance of the signal light, which shows that the Lissajous trajectory (see Fig. 10(b)) is no longer the coincidence of multiple cycles, but a multi-layer elliptical spiral trajectory [27], wherein the number of layers is related to the period ratio of signal light disturbance and target movement.

 figure: Fig. 10.

Fig. 10. Lissajous trajectories of the actual output signal after PGC demodulation and nonlinearity correction in the case of (a) AOIM and (b) AOIM&SID.

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Tables Icon

Table 1. Characteristic parameters after PGC demodulation and nonlinearity correction

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The established micro-probe fiber optic interferometer was applied to measure the displacement stage motion, where the measurement was affected by a variable signal intensity produced through continuous adjustment of NDF. In the case of AOIM&SID, the displacement stage is set to move at a speed of 1.5 µm/ms, and the solved displacement is shown in Fig. 11(a). Note that the uncorrected displacement and the demodulated displacement with the conventional method are shifted by 1 µm and 0.5 µm for clarity of presentation, respectively. Within the target displacement range of 5 µm, the uncorrected demodulated displacement shows a superimposed periodic nonlinear variation centered on the actual displacement, which is inconsistent with the target motion state; there is still significant nonlinearities in the result processed with conventional extremum correction method; the demodulated displacement after correction with the proposed method is correlated linearly with the actual displacement of the target, with only a small displacement deviation. The peak-to-peak value of the displacement deviation is 0.46 nm observed in Fig. 11(b) and the standard deviation of the error is 0.14nm, while the displacement deviation corrected with the conventional method is 4.36 nm with a 1.42-nm standard deviation. The experimental results proves that the microprobe fiber optic interferometer with the proposed PGC demodulation and nonlinearity correction method can achieve displacement measurements with sub-nanometer accuracy under low frequency interference light intensity disturbance.

 figure: Fig. 11.

Fig. 11. Experimental results of displacement measurement in the micrometer range. (a) The displacement curve demodulated by the micro-probe fiber interferometer before and after correction as function of the actual displacement of the micro-displacement stage. (b) The deviation of the corrected displacement with conventional and proposed method from the actual displacement.

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4. Conclusion

In this study, an embedded microprobe fiber optic interferometer with low nonlinearity against light intensity disturbance was presented for precision position monitoring and displacement measurements. First, an equivalent analysis model of the microsensing probe based on a tilted target is developed, which equates the tilted-axis Gaussian beam incident micro-probe to parallel Gaussian beam incident tilted micro-probe. This simplifies the construction of the interference model containing a tilted reflection target. The model analysis demonstrated that the conventional extremum method cannot accurately compensate for the light intensity disturbance related to the target motion, resulting in a larger correction deviation. Then in response to this issue, a correction method for higher harmonic demodulation-amplitude normalization is proposed. This method uses a higher harmonic and signal mixing on the basis of PGC demodulation to avoid the DC offset varying with light intensity disturbance. It further realizes the nonlinear error correction under interference light intensity disturbance by amplitude normalization. The experiments showed that the proposed PGC demodulation and correction method could effectively reduce the DC term in the presence of signal light intensity disturbance, thereby improving the robustness of the microprobe fiber-optic interferometric system. The residual error was less than 1 nm in the micrometer displacement, which achieved displacement sensing with sub-nanometer accuracy in the microprobe fiber optic interferometer under light intensity disturbance.

This paper is a preliminary study on the resistance of micro-probe fiber optic interferometer to light intensity disturbance, and the challenges and limitations include: (1) fiber microprobes with various split ratios should be further prepared for different target surface reflectivities. (2) In the experimental verification, the signal light intensity disturbance is simulated by the continuous adjustment of the NDF, and the optical modulator can be further introduced to quantitatively characterize the signal light intensity disturbance at different frequencies. (3) In the future, an air refractive index measurement system will be added to compensate for environmental drift in real time, and this can further improve the displacement measurement accuracy of the microprobe fiber optic interferometer under interference light intensity disturbance.

Funding

National Natural Science Foundation of China (51875140, 52061135114, 52175500); National Major Science and Technology Projects of China (2017ZX02101006-005).

Acknowledgments

C. Zhang would thank the National Natural Science Foundation for this work.

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.

References

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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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Figures (11)
Fig. 1.
Fig. 1. Equivalent model of fiber optic micro-probe (a) vertical to the optical axis and (b) with the angle of 2θ with the optical axis. w: radius of output beam from SMF; w1: beam waist radius of micro-probe; l: beam waist position of micro-probe; Zwd: working distance of micro-probe.

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Fig. 2.
Fig. 2. Coordinate system of micro-probe equivalent model (a) with the beam waist position as the origin and (b) with the tilted target mirror.

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Fig. 3.
Fig. 3. Simulation verification of the equivalent model of the micro-probe based on the tilted target with w = 5.2, λ = 1.550 µm. (a) Optical power coupling efficiency (red line) and fringe contrast (blue line) and (b) the overlap between the reference and the measurement spot related to the target tilt angle.

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Fig. 4.
Fig. 4. (a) Frequency spectrum of ideal, AOIM and AOIM&SID demodulation signal. (b) Nonlinear displacement errors under AOIM and AOIM&SID with conventional nonlinearity correction (m = 0.1, ms = 0.1, C = 2.63 (J1(C) = J2(C)), φm= 0 rad). Inset: Amplification of nonlinear errors under AOIM.

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Fig. 5.
Fig. 5. Proposed PGC demodulation and nonlinearity correction method schematic diagram. ADC: analog-to-digital converter; DDS: direct digital synthesizer; LPF: low-pass filter; Arctan: arctangent operation; Phauw: Phase unwrapping; Max: Maximum; Min: Minimum.

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Fig. 6.
Fig. 6. Nonlinear displacement errors under AOIM&SID (a) at a 20-Hz disturbance frequency and (b) at different signal light intensity disturbance frequencies, with the proposed PGC demodulation and nonlinearity correction method and conventional correction method.

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Fig. 7.
Fig. 7. Micro-probe equivalent model (a) and the proposed PGC demodulation and nonlinearity correction method (b) verification experimental setup. DFB Laser: distributed feedback laser; Driver: light source driver module; FC: fiber optic circulator; PIN: PIN photodetector; NDF: neutral density filter; DAC: digital to analog converter; Red wireframe: the signal acquisition and processing module; Photo: physical picture of fiber micro-Michelson interference probe.

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Fig. 8.
Fig. 8. Experimental results of micro-probe equivalent model validation based on tilted target. (a) Fiber micro-probe coupling efficiency as function of the target mirror tilt angle. (b) Interference fringe contrast as function of the target mirror tilt angle.

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Fig. 9.
Fig. 9. Interference signal amplitude as function of fiber micro-probe coupling efficiency.

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Fig. 10.
Fig. 10. Lissajous trajectories of the actual output signal after PGC demodulation and nonlinearity correction in the case of (a) AOIM and (b) AOIM&SID.

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Fig. 11.
Fig. 11. Experimental results of displacement measurement in the micrometer range. (a) The displacement curve demodulated by the micro-probe fiber interferometer before and after correction as function of the actual displacement of the micro-displacement stage. (b) The deviation of the corrected displacement with conventional and proposed method from the actual displacement.

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Tables (1)

Tables Icon

Table 1. Characteristic parameters after PGC demodulation and nonlinearity correction

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Equations (25)

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E ( x , y , z ) = E 0 α 1 w 1 w ( z ) exp ⁡ ( − x 2 + y 2 w 2 ( z ) − i [ k ( z + x 2 + y 2 2 R ( z ) ) − a r c t g z f ] ) ,
E s ( x , y , z ) = E ( x , y , z − l ) = E 0 α 1 w 1 w ( z − l ) × exp ⁡ ( − x 2 + y 2 w 2 ( z − l ) − i [ k ( ( z − l ) + x 2 + y 2 2 R ( z − l ) ) − a r c t g z − l f ] ) .
E r ( x ′ , y ′ , z ′ ) = E 0 α 2 w 1 w ( z ′ + l ) exp ( − x ′ 2 + y ′ 2 w 2 ( z ′ + l ) − i [ k ( ( z ′ + l ) + x ′ 2 + y ′ 2 2 R ( z ′ + l ) ) − a r c t g z ′ + l f ] ) .
η = | ∫ ∫ ( E s E r ∗ ) | z ′ = 0 d x ′ d y ′ | 2 ∫ ∫ ( E s E s ∗ ) | z ′ = 0 d x ′ d y ′ ∫ ∫ ( E r E r ∗ ) | z ′ = 0 d x ′ d y ′ .
η = exp ⁡ [ − sin 2 θ ( l 2 f 2 w 1 2 ( l 2 + f 2 ) + w 1 2 k 2 f 2 4 ( l 2 + f 2 ) ) ] .
s ( t ) = A + B cos ⁡ ( C cos ⁡ ( w 0 t ) + φ ( t ) ) ,
A = k v ( I r + I s ) = k v ( α 2 R 1 I 0 + α 2 R 2 I 0 η ) ,
B = 2 k v I r I s = 2 k v ( α 2 R 1 I 0 ) ( α 2 R 2 I 0 η ) ,
s ( t ) = A [ 1 + m cos ⁡ ( w 0 t + φ m ) ] [ 1 + B / A cos ⁡ ( C cos ⁡ ( w 0 t ) + φ ( t ) ) ] ,
s 1 ( t ) = − B b 2 + J 1 2 ( C ) sin ⁡ ( φ ( t ) − θ 1 ) + m A 1 2 cos ⁡ φ m ,
s 2 ( t ) = − B a 2 + J 2 2 ( C ) cos ⁡ ( φ ( t ) − θ 2 ) ,
a = m 2 ( J 1 ( C ) cos ⁡ φ m − J 3 ( C ) cos ⁡ φ m ) ,
b = m 2 ( J 0 ( C ) cos ⁡ φ m − J 2 ( C ) cos ⁡ φ m ) ,
θ 2 = arctan ⁡ a J 2 ( C ) ,
θ 1 = arctan ⁡ b J 1 ( C ) .
φ e = Phauw [ arctan ⁡ s 1 ( t ) s 2 ( t ) ] − arctan ⁡ sin ⁡ φ ( t ) cos ⁡ φ ( t ) .
s 3 ( t ) = B c 2 + J 3 2 ( C ) sin ⁡ ( φ ( t ) − θ 3 ) ,
c = m 2 ( J 2 ( C ) cos ⁡ φ m − J 4 ( C ) cos ⁡ φ m ) ,
θ 3 = arctan ⁡ c J 3 ( C ) .
s 2 ′ ( t ) = cos ⁡ ( φ ( t ) − θ 2 ) ,
s 3 ′ ( t ) = sin ⁡ ( φ ( t ) − θ 3 ) ,
s 2 ∗ ( t ) = 2 cos ⁡ ( θ 3 − θ 2 2 + π 4 ) cos ⁡ [ φ ( t ) − ( θ 3 + θ 2 2 + π 4 ) ] ,
s 3 ∗ ( t ) = 2 sin ⁡ ( θ 3 − θ 2 2 + π 4 ) sin ⁡ [ φ ( t ) − ( θ 3 + θ 2 2 + π 4 ) ] .
s 2 ′ ′ ( t ) = cos ⁡ [ φ ( t ) − ( θ 3 + θ 2 2 + π 4 ) ] ,
s 3 ′ ′ ( t ) = sin ⁡ [ φ ( t ) − ( θ 3 + θ 2 2 + π 4 ) ] .
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