The development of advanced equipment manufacturing, semiconductor industry, and scientific research fields has put forward an urgent demand for large-range displacement measurements with high accuracy in narrow spaces [1–3]. The laser interference displacement sensor, an essential approach for high-precision displacement measurement, also shows promising features in terms of large range and miniaturization. In recent years, fiber-optic miniature sensors have been actively researched due to their compact volume, convenient mounting, and embeddable measurements. These probes have already been applied in various cases, such as trace detection, precision positioning, and structural health monitoring [4–6].

Initial research on fiber microprobe sensor focused on low-frequency vibration measurements such as fiber-optic accelerometers and fiber-optic hydrophones [7,8]. With the rapid advancement of fiber-optic sensing technology, these microprobes have gradually been applied in the realm of high-accuracy displacement measurements [9,10]. The Fabry–Perot structured microprobe, composed of a single-mode fiber and a target, enables micrometer-scale displacement sensing with nanometer-level accuracy [11,12]. Liang et al. reconstructed a target 10 μm displacement with a maximum relative error of 4.9% by performing the Hilbert transform to the low-fineness Fabry–Perot cavity interference signal [13]. The microprobe sensor detectable range was broadened to tens of millimeters by introducing a collimating lens to shape the fiber-optic output beam [14,15]. Thurner et al. accounted for return light fading in the millimeter range by adding the coupling efficiency of the measurement light into the Fabry–Perot cavity microprobe interference-sensing model [16,17]. This further improved the theoretical analysis of the fiber-optic microprobe large-range sensing and realized a maximum measurement range of 100 mm. Most fiber microprobes employ a compact Fabry–Perot interference structure and offer the potential for a measurement range of hundred-millimeter scale. However, the sensing mechanism of the Fabry–Perot cavity microprobe approximates multibeam interference to two-beam interference due to the low reflectivity of the cavity surfaces, resulting in higher-order principal errors. Furthermore, as the working distance increases, the measurement accuracy becomes more sensitive to the alignment deviation between the miniature sensor elements. Hence, a comprehensive microprobe interference-sensing model is needed to support high-accuracy displacement measurements over hundreds of millimeters.

The signal demodulation technique is a key factor affecting the measurement accuracy of fiber microprobe interferometers. As a common signal processing means for microprobe sensors, white light interferometry based on a broadband light source or tunable laser was mostly applicable to static distance measurement [18,19]. Quadrature intensity detection could realize dynamic high-accuracy signal demodulation by adjusting the working point position [20,21]. However, this method only works over a quarter of the laser wavelength, restricting the measurement range of the microprobe sensor. The phase-generated carrier (PGC) demodulation was proposed to improve the sensing range [22–24]. Through processing the extracted orthogonal signals, it can efficiently suppress noise jamming, providing high research value in the realm of high-accuracy optical measurements. However, in practice, microprobe large-range sensing inevitably causes variations in the interference light intensity. This seriously affects the phase demodulation accuracy of the PGC algorithm. Therefore, current PGC demodulation methods fail to support the microprobe sensor for high-accuracy displacement measurements in the hundreds of millimeters range.

This paper proposes a novel common-path Michelson microprobe interference-sensing method for high-accuracy, large-range displacement measurements. This method establishes the model relationship between spatial alignment of the collimating lens in microprobe and interference signal, utilizing tilted-axis Gaussian optical field diffraction. In the model, we analyze the effect of different assembly errors in the microprobe element on the measurement accuracy for large-range displacements. Furthermore, a high-accuracy PGC signal processing method based on high-order carrier demodulation is proposed to improve the resistance of microprobe sensors to interference intensity variations. The proposed Michelson microprobe interference-sensing method can effectively reduce nonlinear errors arising from microprobe element misalignment and interference intensity fluctuations in large-range displacement sensing.

The fiber-optic Michelson microprobe, shown in Fig. 1, functions as follows. A Gaussian beam from the single-mode fiber (SMF) is collimated by a gradient refractive index lens (GRIN) after passing through an air gap. It is then divided into two beams by a non-polarizing cube beam splitter (NPBS). The transmitted measurement light is reflected back into the microprobe by the target after covering a spatial transmission distance. The reference light is reflected directly back to the microprobe by the NPBS inner surface, which is coated with a highly reflective film. These two beams then couple and interfere at the input fiber end face of the microprobe. The surfaces of NPBS along its transmission axis and the fiber end are coated with anti-reflection films to eliminate parasitic reflections. The coupling efficiency of the reference light is related to the alignment of the microprobe elements, assuming fixed values for the microprobe beam waist radius $w_t$ and beam waist position $l$. The coupling efficiency of the measurement light is not only affected by element alignment but also sensitive to variations in the microprobe working distance. The coupling efficiencies of both beams represent the light signal intensity entering the SMF, which in turn affect the quality of the interference signal. Assembly alignment errors of the microprobe elements with millimeter-size dimensions can degrade the coupling effect and affect the large-range sensing performance of the integrated microprobe. Thus, it is crucial to establish a model relationship among the microprobe element alignment deviation, spatial-fiber coupling efficiency, and interference signal quality over a large range of detection distances.

Microprobe interference-sensing models in previous reports primarily focused on the transmission and variation of Gaussian optical fields within ideal coaxial optical systems. However, such coaxial conditions are often not satisfied in practice due to mounting adjustment errors in the microprobe elements, resulting in an out-of-alignment state for the optical system. The transmission characteristics of the optical field in such a misaligned system depend not only on those in a coaxial condition but also on the system’s misalignment. We propose a microprobe global interference-sensing model that accounts for element misalignment for large-range, high-accuracy displacement detection. The term “global” refers to the whole process of transmission and coupling interference of a Gaussian beam in multiple media such as optical fibers, space, and elements with gradient refractive index. The model is based on tilted-axis Gaussian optical field diffraction and describes the optical field distribution in microprobe sensors containing the misaligned GRIN. The major idea is to regard the misaligned optical axis as a rotated version of the ideal optical axis, occurring when an optical element is tilted and dislocated due to alignment deviations. The transmission of the optical field along this optical axis can be represented by the optical field diffraction integral and the transmission matrix transformation. We then obtain the optical field distribution along the ideal axis after accounting for the tilted and dislocated element via coordinate transformation.

Compared with the misalignment of the NPBS, the alignment deviation between the GRIN and SMF is more influential on the microprobe interference-sensing signal. Thus, this paper focuses on the influence of tilt and dislocation between the GRIN and SMF on microprobe displacement sensing. By mirror-symmetrically unfolding the reflection path centered on the target mirror M1, the measurement light returning to the microprobe can be viewed as the continuous transmission of the same measurement light over an equivalent working distance as seen in Fig. 2(a). To investigate the Gaussian optical field transmission in the microprobes, the surface of each element is defined as a reference plane (${\mathrm{RP}}_i$), characterized by the Gaussian beam complex parameter $q_i$ (where $i=0$, $1,\dots ,10$). This parameter reflects the Gaussian beam spot size $w_i$ and the wavefront curvature radius $R_i$ at each ${\mathrm{RP}}_i$. The GRIN is rotated centered on $\mathrm{O}$ point of the incident plane by a $\theta$ angle relative to ${\mathrm{RP}}_1$ and shifted upward by a distance $d$, indicating the generation of tilt and dislocation with respect to the SMF. Subsequent studies define the reference planes $1^{*}$, $2^{*}$ and $8^{*}$, $9^{*}$ on the surface of the misaligned GRIN. Coordinate systems are established at each reference plane for measurement optical field transmission. The Gaussian optical field output from the fiber end face is expressed as $$E(x_0,y_0)=E_0\exp (-\frac{x_0^2+y_0^2}{w_0^2}),$$(1)where $E_0$ is the light source amplitude, and $w_0$ is the beam waist radius of the light source. The measurement optical field distribution on different planes is obtained using the Collins diffraction integral formula combined with near-axis optical system transmission matrices [25,26].

The Gaussian optical field output from the fiber is transmitted over a distance $L_0$ and expressed at ${\mathrm{RP}}_1$ as $$E(x_1,y_1)=E_0\frac{w_0}{w_1}\exp (-(x_1^2+y_1^2)(\frac{1}{w_1^2}+\frac{ik}{2R_1})+i\text{arctan}\left(\frac{\lambda L_0}{\pi w_0^2}\right)-ik{L}_0),$$(2)where $\lambda$ is the light source wavelength, and $k=2\pi /\lambda$ is the wave number. Assuming that $\theta$ and $d$ are the misalignment amounts in the $x$-axis direction satisfying the small angle approximation, the coordinate transformations ($x_1=x_1^{*}\cos \theta -{\mathit{z}}_1^{*}\sin \theta +d$, $y_1=y_1^{*}$, $z_1=L_0+z_1^{*}\cos \theta +x_1^{*}\sin \theta$) are applied to obtain the expression for the optical field at ${\mathrm{RP}}_1^{*}$ ($z_1^{*}=0$) on the misaligned optical axis: $$E(x_1^{*},y_1^{*})=E_0\frac{w_0}{w_1}\exp (-({(x_1^{*}+d)}^2+y_1^{*2})(\frac{1}{w_1^2}+\frac{ik}{2R_1})+i{\phi }_1-ik(L_0+x_1^{*}\sin \theta )),$$(3)where attachment phase shift ${\phi }_1=\text{arctan}\left(\frac{\lambda L_0}{\pi w_0^2}\right)$ and since $w_1^2(L_0+x_1^{*}\sin \theta )\approx w_1^2(L_0)$ and $R_1(x_1^{*}\sin \theta +L_0)\approx R_1(L_0)$, it follows that $q_1^{*}\approx q_1$. Then, the optical field on ${\mathrm{RP}}_2^{*}$ is yielded by the diffraction integral and the GRIN transmission matrix ${\mathit{M}}_{21}$: $$E(x_2^{*},y_2^{*})=E_0\frac{w_0}{w_2^{*}}\exp (-({(x_2^{*}-b_g\sin \theta +d{a}_g)}^2+y_2^{*2})(\frac{1}{w_2^{*2}}+\frac{ik}{2R_2^{*}})+i({\phi }_1+{\phi }_2))\times \exp (-ik(L_0+n_0{L}_g))\times \exp (\frac{ik}{2}(b_g{d}_g{\sin }^2\theta -2d_g\sin \theta x_2^{*}+2d{c}_g{x}_2^{*}+d^2{a}_g{c}_g-2b_g{c}_{g}d\sin \theta )),$$(4)where ${\phi }_2=\text{arctan}\left(\frac{\lambda b_g{R}_1}{\pi (a_g{w}_1^2{R}_1+b_g{w}_1^2)}\right)$, and $a_g$, $b_g$, $c_g$, and $d_g$ are the matrix elements sequentially corresponding to ${\mathit{M}}_{21}$, indicated as $$M_{21}=\left(\begin{array}{cc}{a}_g& b_g\\ c_g& d_g\end{array}\right)=\left(\begin{array}{cc}\cos (g{L}_g)& \sin (g{L}_g)/g{n}_0\\ -g{n}_0\sin (g{L}_g)& \cos (g{L}_g)\end{array}\right),$$(5)where $g$ is the self-focusing constant of the GRIN, and $n_0$ is the refractive index on the central axis of the GRIN. The optical field distribution on the misaligned optical axis is transformed to ${\mathrm{RP}}_2(z_2=0)$ on the ideal optical axis by the coordinate transformation ($x_2^{*}=x_2\cos \theta +d_1\cos \theta +z_2\sin \theta ,y_2^{*}=y_2$, $z_2^{*}=-x_2\sin \theta -d_1\sin \theta +z_2\cos \theta$), denoted as $$E(x_2,y_2)=E_0\frac{w_0}{w_2}\exp (-({(x_2+d_1-b_g\sin \theta +d{a}_g)}^2+y_2^2)(\frac{1}{w_2^2}+\frac{ik}{2R_2})+i({\phi }_1+{\phi }_2))\times \exp (-ik(L_0+n_0{L}_g)+ik{x}_2(\sin \theta -d_g\sin \theta +d{c}_g)+ik{d}_1\sin \theta )\times \exp \left(\frac{ik}{2}{K}_1\right),$$(6)where $d_1$ is the dislocation of ${\mathrm{O}}_1$ point on the output plane ${\mathrm{RP}}_2^{*}$ relative to the ideal optical axis, and according to the geometrical relation, $d_1=L_g\sin \theta -d$. Similarly, $w_2^{*}\approx w_2$, $R_2^{*}\approx R_2$, and $K_1=b_g{d}_g{\sin }^2\theta -2d_g{d}_1\sin \theta +2d{c}_g{d}_1+d^2{a}_g{c}_g-2b_g{c}_{g}d\sin \theta$. As seen from Eq. (6), the Gaussian beam output from the fiber is transformed into a tilted-axis Gaussian beam through the misaligned GRIN. The bias axis is located on the $xoz$ plane, where the barycentric spatial position relates to the misalignment amounts $\theta$ and $d$. Additionally, the spot radius and wavefront curvature radius of the tilted-axis Gaussian beam are approximately the same as those of the ideal optical axis and follow the same transmission law.

The tilted-axis Gaussian optical field at ${\mathrm{RP}}_2$ is continuously transmitted along the ideal optical axis, yielding the optical field distribution $E(x_8,y_8)$ at ${\mathrm{RP}}_8$: $$E(x_8,y_8)=E_0\frac{w_0}{w_8}\exp (-({(x_8-b_s\sin {\mathrm{\Phi }}_1+d^{\prime })}^2+y_8^2)(\frac{1}{w_8^2}+\frac{ik}{2R_8})+i({\phi }_1+{\phi }_2+{\phi }_8))\times \exp (-ik(L_0+n_0{L}_g+2p+2n_1{d}_t+2Z_{wd})+ik{d}_1\sin \theta )\times \exp (\frac{ik}{2}(K_1+b_s{\sin }^2{\mathrm{\Phi }}_1-2\sin {\mathrm{\Phi }}_1{x}_8)),$$(7)where ${\phi }_8=\text{arctan}\left(\frac{\lambda b_s{R}_2}{\pi (w_2^2{R}_2+b_s{w}_2^2)}\right)$ and $b_s=2p+2d_t/n_1+2Z_{wd}$ are secondary elements in the optical transmission matrix from ${\mathrm{RP}}_2$ to ${\mathrm{RP}}_8$, with $n_1$ representing the refractive index of the NPBS. Analogous to Eq. (4), $\sin {\mathrm{\Phi }}_1=d_g\sin \theta -d{c}_g-\sin \theta$ and $d^{\prime }=d_1-b_g\sin \theta +d{a}_g$ are regarded as equivalent misalignment amounts. Adopting the tilted-axis Gaussian optical field diffraction proposed above, the same process is applied to the optical field transmitted through the mirror-symmetric GRIN, ultimately yielding the distribution of the measurement optical field at ${\mathrm{RP}}_{10}$: $$\begin{gathered} E_S(x_s,y_s)=E_0\frac{w_0}{w_{10}}\exp (-({(x_s-L_0\sin {\mathrm{\Phi }}_3+d_s^{*})}^2+y_s^2)(\frac{1}{w_{10}^2}+\frac{\mathrm{ik}}{2R_{10}})+i{\phi }_s) \\ \times \exp (-ikL+ik(d_1+d)\sin \theta )\times \exp (\frac{ik}{2}(K_1+K_2+K_3+L_0{\sin }^2{\mathrm{\Phi }}_3-2\sin {\mathrm{\Phi }}_3{x}_s)), \end{gathered}$$(8)where $L=2L_0+2n_0{L}_g+2p+2n_1{d}_t+2Z_{wd}$ is the optical length of the measurement light along the optical axis and ${\phi }_s={\phi }_1+{\phi }_2+{\phi }_8+{\phi }_9+{\phi }_{10}$ is the sum of the attachment phase shifts. The equivalent misalignment amounts $\sin {\mathrm{\Phi }}_3$ and $d_s^{*}$, which affect the transmission direction of the measurement optical field, are denoted as $$\sin {\mathrm{\Phi }}_3=\sin \theta +d_g\sin {\mathrm{\Phi }}_2-c_g{\tilde{d}}_s,$$(9)$$d_s^{*}=a_g{\tilde{d}}_s-d-b_g\sin {\mathrm{\Phi }}_2,$$(10)where $\sin {\mathrm{\Phi }}_2=\sin {\mathrm{\Phi }}_1-\sin \theta$, ${\tilde{d}}_s=d^{\prime }-d_1-b_s\sin {\mathrm{\Phi }}_1$, and phase items $K_2=b_s{\sin }^2{\mathrm{\Phi }}_1+2d_1\sin {\mathrm{\Phi }}_1$, $K_3=b_g{d}_g{\sin }^2{\mathrm{\Phi }}_2+2d_{g}d\sin {\mathrm{\Phi }}_2-2{\tilde{d}}_s{c}_{g}d+{\tilde{d}}_s^2{a}_g{c}_g-2b_g{c}_g{\tilde{d}}_s\sin {\mathrm{\Phi }}_2$.

The NPBS built-in reflective surface M2 is equivalent to NPBS exit surface, and the reference light reflection path is symmetrically unfolded centered on the equivalent reflective surface ${\mathrm{M}}_2^{\prime }$. The reference light transmission coordinate system is constructed in Fig. 2(b), and the distribution of the reference optical field at ${\mathrm{RP}}_{10}^{\prime }$ is expressed as $$E_R(x_r,y_r)=E_0\frac{w_0}{w_{10}^{\prime }}\exp (-({(x_r-L_0\sin {\mathrm{\Phi }}_3^{\prime }+d_r^{*})}^2+y_r^2)(\frac{1}{w_{10}^{\prime 2}}+\frac{\mathit{ik}}{2R_{10}^{\prime }})+i{\phi }_r)\times \exp (-ik{L}^{\prime }+ik(d_1+d)\sin \theta )\times \exp (\frac{ik}{2}(K_1+K_2^{\prime }+K_3^{\prime }+L_0{\sin }^2{\mathrm{\Phi }}_3^{\prime }-2\sin {\mathrm{\Phi }}_3^{\prime }{x}_r)),$$(11)where $L^{\prime }=2L_0+2n_0{L}_g+2p+2n_1{d}_t$ is the optical length of the reference optical field along the optical axis. Equivalent misalignment amounts for the reference optical field $\sin {\mathrm{\Phi }}_3^{\prime }=\sin \theta +d_g\sin {\mathrm{\Phi }}_2-c_g(d^{\prime }-d_1-b_r\sin {\mathrm{\Phi }}_1)$, $d_r^{*}=a_g(d^{\prime }-d_1-b_r\sin {\mathrm{\Phi }}_1)-d-b_g\sin {\mathrm{\Phi }}_2$, where $b_r=2p+2d_t/n_1$ is the second element of the transmission matrix between ${\mathrm{RP}}_2$ and ${\mathrm{RP}}_8^{\prime }$. Further, ${\phi }_r={\phi }_1+{\phi }_2+{\phi }_8^{\prime }+{\phi }_9^{\prime }+{\phi }_{10}^{\prime }$ is the sum of the attachment phase shifts of the reference optical field at ${\mathrm{RP}}_{10}^{\prime }$. Phase items $K_3^{\prime }=b_g{d}_g{\sin }^2{\mathrm{\Phi }}_2+2d_{g}d\sin {\mathrm{\Phi }}_2-2{\tilde{d}}_r{c}_{g}d+{\tilde{d}}_r^2{a}_g{c}_g-2b_g{c}_g{\tilde{d}}_r\sin {\mathrm{\Phi }}_2$, $K_2^{\prime }=b_r{\sin }^2{\mathrm{\Phi }}_1+2d_1\sin {\mathrm{\Phi }}_1$. According to Eqs. (1), (8), and (11), the measurement light-coupling efficiency and the reference light-coupling efficiency are derived using a spatial-fiber coupling efficiency analysis method based on mode-field matching [27]: $${\eta }_s={\left|\frac{2}{w_0{w}_{10}{Q}_s}\exp (i{\psi }_s)\exp \left(\frac{P_s^2-4Q_s{O}_s}{4Q_s}\right)\right|}^2,$$(12)$${\eta }_r={\left|\frac{2}{w_0{w}_{10}^{\prime }{Q}_r}\exp (i{\psi }_r)\exp \left(\frac{P_r^2-4Q_r{O}_r}{4Q_r}\right)\right|}^2,$$(13)where parameters of the measurement light-coupling efficiency $Q_s=\frac{1}{w_{10}^2}+\frac{1}{w_0^2}+\frac{ik}{2R_{10}}$, $O_s={(d_s^{*}-L_0\sin {\mathrm{\Phi }}_3)}^2(\frac{1}{w_{10}^2}+\frac{ik}{2R_{10}})$, ${\psi }_s={\phi }_s-kL+k(d_1+d)\sin \theta +\frac{k}{2}(K_1+K_2+K_3+L_0{\sin }^2{\mathrm{\Phi }}_3)$, and $P_s=2(d_s^{*}-L_0\sin {\mathrm{\Phi }}_3)(\frac{1}{w_{10}^2}+\frac{\mathit{ik}}{2R_{10}})+ik\sin {\mathrm{\Phi }}_3$. Parameters of the reference light-coupling efficiency ${\psi }_r={\phi }_r-k{L}^{\prime }+k(d_1+d)\sin \theta +\frac{k}{2}(K_1+K_2^{\prime }+K_3^{\prime }+L_0{\sin }^2{\mathrm{\Phi }}_3^{\prime })$, $Q_r=\frac{1}{{w^{\prime }}_{10}^2}+\frac{1}{w_0^2}+\frac{ik}{2{R^{\prime }}_{10}}$, $P_r=2(d_r^{*}-L_0\sin {\mathrm{\Phi }}_3^{\prime })(\frac{1}{{w^{\prime }}_{10}^2}+\frac{ik}{2R_{10}^{\prime }})+ik\sin {\mathrm{\Phi }}_3^{\prime }$, and $O_r={(d_r^{*}-L_0\sin {\mathrm{\Phi }}_3^{\prime })}^2(\frac{1}{{w^{\prime }}_{10}^2}+\frac{ik}{2R_{10}^{\prime }})$.

The measurement and reference optical fields coupled into an SMF produce an interference signal, expressed as $$I=I_s+I_r+2\sqrt{I_s{I}_r}\cos (\frac{2\pi }{\lambda }(L-L^{\prime })+{\phi }_0),$$(14)where $I_s={|E_S(x_s,y_s)|}^2{\delta }_s^2{R}_s{\eta }_s$ is the measurement light intensity coupled into the fiber, $I_r={|E_R(x_r,y_r)|}^2{\delta }_r^2{R}_r{\eta }_r$ is the reference light intensity coupled into the fiber, and ${\phi }_0$ is the initial phase difference between the two beams. ${\delta }_s$ and ${\delta }_r$ are the transmissivity and reflectivity of NPBS, and $R_r$ and $R_s$ are the reflectivities of the built-in reflective film of NPBS in the reference arm and target in the measurement arm, respectively. The interference contrast is commonly recognized as a measure of the interference signal quality related to displacement sensing sensitivity and is denoted as $$v=\frac{2\sqrt{I_s{I}_r}}{I_s+I_r}=\frac{2\sqrt{{|E_S|}^2{\delta }_s^2{R}_s{\eta }_s{|E_R|}^2{\delta }_r^2{R}_r{\eta }_r}}{{|E_S|}^2{\delta }_s^2{R}_s{\eta }_s+{|E_R|}^2{\delta }_r^2{R}_r{\eta }_r}.$$(15)

During the microprobe assembly process, the mounting misalignment of millimeter-sized elements can attenuate the coupling efficiency over long working distances, thereby degrading the sensing signal quality of the integrated microprobe for large-range displacement detection. The effects of dislocation and tilt of the GRIN on the spatial-fiber coupling efficiencies are shown in Figs. 3(a)–3(c). Here, the laser wavelength $\lambda =1.5328\mathrm{\mu m}$, the beam waist radius $w_0=5.2\mathrm{\mu m}$, the GRIN self-focusing constant $g=0.188{\mathrm{mm}}^{-1}$, the length $L_g=0.23$$P$ (where pitch $P=2\pi /g$), the spacing from the SMF $L_0=0.55\mathrm{mm}$, and the NPBS of 3 mm size equally splits beams, corresponding to the microprobe beam waist position $l=60\mathrm{mm}$. As seen in Fig. 3(a), the measurement light-coupling efficiency ${\eta }_s$ has a maximum value of 100% when the reflection target is located at the beam waist position outside the microprobe end face. This value decreases rapidly with an increase in the radial displacement of the GRIN by micrometer magnitudes. The reference light-coupling efficiency ${\eta }_r$ with a maximum of 96% shows similar sensitivity to the dislocation of the GRIN to that of the measurement light-coupling efficiency. The misalignment angle tolerance of the GRIN is within $\pm 0.1°$; i.e., the absolute value of the tilt angle cannot exceed 1.75 mrad, as illustrated in Fig. 3(b). Beyond this, spatial-fiber coupling fails to occur. Figure 3(c) demonstrates the relationship between ${\eta }_s$ and the microprobe working distance at different misalignment amounts. In the presence of a single misalignment, ${\eta }_s$ is attenuated over the entire working range. When two misalignment amounts coexist, ${\eta }_s$ attenuation is different depending on their directions. When the GRIN dislocation is in the same direction as tilt, there may be an anomalous enhancement of coupling efficiency versus the working distance curve compared to that for a single misalignment amount. This is explained in terms of the model relationship between two misalignment amounts as depicted in Eqs. (7) and (8). When dislocation is consistent with the tilt direction, this relationship partially counteracts the effect of the GRIN misalignment on the optical field transmission and demodulation results. If the misalignment amounts are in the opposite directions, the measurement light-coupling efficiency is greatly attenuated over a working distance of 0–750 mm due to the superimposed effects of the misalignments.

Figure 3(d) illustrates the role of spatial-fiber coupling efficiency on interference contrast. In ideal spatial-fiber coupling, both the reference and measurement lights are fully coupled into the fiber, resulting in an interference pattern with a bright spot at the center. At this point, the interference signal contrast reaches its maximum output from the photodetector due to the superimposed enhancement of the interference signals at each point in the pattern. When the measurement light is tilted concerning the reference light, the interference pattern transforms from a circular spot to a striped distribution. This is due to the different phase differences between the points on the receiving surface. Consequently, the interference signal intensity received by the photodetector decreases due to the counteraction of dark and light stripes, leading to a reduction in interference contrast.

The relationship curves between the GRIN misalignment and the interference contrast are shown in Figs. 3(e) and 3(f). The contrast remains close to 1 within the range of $|d|<6\mathrm{\mu m}$, $|\theta |<0.1°$ at the working distance of the beam waist position. This is related to the almost synchronous changes in ${\eta }_s$ and ${\eta }_r$ within this range [see Figs. 3(a) and 3(b)], making the system insensitive to element misalignment amounts. When $|d|>6\mathrm{\mu m}$ and $|\theta |>0.1°$, the interference signal strength decreases compared to the intrinsic noise, with reductions in the intensities of the reference and measurement lights coupled into the SMF. The contrast attenuates correspondingly as misalignment increases. If there exists another fixed, non-zero misalignment at this point, it is reflected as a shift in the response curve for contrast. To ensure the spatial-fiber coupling efficiency and interference signal quality of the microprobe, the radial displacement of the GRIN concerning the SMF should be confined within $\pm 6\mathrm{\mu m}$ and the tilt angle should be within $\pm 0.1°$ during the microprobe element alignment assembly.

In the model, we investigated the effects of misalignment of the GRIN with respect to the SMF on the resulting interference signals and sensing performance. This serves to guide the microprobe assembly for reducing the effects of element misalignment on the measurement accuracy over a large range. Although we based the model development work on the Michelson interferometer, the model could be employed in other interferometer configurations with partial adjustments. Moreover, in the scenario where a mirror is not available due to space limitation, the NPBS splitting ratio can be adjusted according to the target’s actual reflectivity to match the reference light and measurement light intensities in wide-range displacement sensing.

The interference light produced by the microprobe-coupled reference light and measurement light is converted by the photodetector and is denoted as $$s(t)=(1+m\cos (w_{c}t)+{\phi }_m)\times (\mathit{A}+B\cos (Z\cos (w_{c}t)+\phi (t))).$$(16)

The first term represents the accompanying optical intensity modulation (AOIM), introduced by the internal modulation of the light source frequency. The second term is the ideal interference signal, occurring under the phase modulation of the sinusoidal carrier signal. Here, $m$ is the depth of the accompanied modulation, and ${\phi }_m$ is the phase difference between the frequency modulation of the light source and the accompanying modulation of the light source intensity. $Z=\frac{4\pi n{Z}_{wd}}{c}{f}_m$ is the carrier phase modulation depth, where $n$ is the air refractive index, $c$ is the speed of light in a vacuum, and $f_m$ is the frequency modulation amplitude. $w_c$ is the carrier signal angular frequency, and $\phi (t)$ is the phase signal of the moving target. $A=k_v(I_s+I_r)=k_v({|E_S|}^2{\delta }_s^2{R}_s{\eta }_s+{|E_R|}^2{\delta }_r^2{R}_r{\eta }_r)$ and $B=2k_v\sqrt{I_s{I}_r}=2k_v|E_S||E_R|{\delta }_s{\delta }_r\sqrt{R_r{\eta }_r{R}_s{\eta }_s}$ are the DC and AC amplitudes of the interference signal. $k_v$ represents the optoelectronic conversion factor of the photodetector.

In the conventional PGC demodulation algorithm, the interference signal $s(t)$ is mixed with the carrier signal $\cos (w_{c}t)$ and the dual-frequency signal $\cos (2w_{c}t)$ [28]. The results are low-pass filtered with a cutoff frequency of $w_c/2$, yielding two output signals: $$s_1(t)=-B\sqrt{\frac{m^2}{4}{\cos }^2{\phi }_m{(J_0(Z)-J_2(Z))}^2+J_1^2(Z)}\sin (\phi (t)-{\theta }_1)+mA\frac{1}{2}\cos {\phi }_m,$$(17)$$s_2(t)=-B\sqrt{\frac{m^2}{4}{\cos }^2{\phi }_m{(J_1(Z)-J_3(Z))}^2+J_2^2(Z)}\cos (\phi (t)-{\theta }_2),$$(18)where ${\theta }_1=\arctan \frac{m\cos {\phi }_m(J_0(Z)-J_2(Z))}{2J_1(Z)}$, ${\theta }_2=\arctan \frac{m\cos {\phi }_m(J_1(Z)-J_3(Z))}{2J_2(Z)}$. The phase signal $\phi (t)$ of the moving target is solved by the inverse tangent operation of the two output signals, where $J_i(x)$ (where $i=0$, 1, 2, 3) are the $i$-th order Bessel functions and $Z$ satisfies $J_1(Z)=J_2(Z)$, with an approximate value of 2.63. Equations (17) and (18) show that the interference signal with AOIM is demodulated by conventional PGC, yielding a set of non-orthogonal signals with DC bias and unequal amplitude. The demodulation result is as follows: $${\phi }^{\prime }(t)=\text{Phauw}\left(\arctan \frac{s_1(t)}{s_2(t)}\right)=\text{Phauw}\left(\arctan \frac{\sin \phi (t)}{\cos \phi (t)}\right)+{\phi }_{\mathrm{error}},$$(19)where Phauw represents the phase unwrapping operation and ${\phi }_{\mathrm{error}}$ is the periodic nonlinear phase solution error, which is generally compensated in real time with the nonlinearity correction method based on characteristic parameter extraction [29,30]. However, in large-range displacement sensing, the interference light intensity varies with the target motion state, leading to continuous variation in the DC bias. It prevents the nonlinear correction method from accurately obtaining the characteristic bias parameter corresponding to the correction moment and even generates the extra phase demodulation error.

We propose a high-accuracy PGC demodulation method removing the time-varying DC term. The interference signal from the two light beams is mixed with the carrier dual-frequency cos($2w_{c}t$) and triple-frequency cos($3w_{c}t$) and low-pass filtered. The filtered signal mixed with the second harmonic can be expressed in Eq. (18), and the other output signal is indicated as $$s_3(t)=B\sqrt{\frac{m^2}{4}{\cos }^2{\phi }_m{(J_2(Z)-J_4(Z))}^2+J_3^2(Z)}\sin (\phi (t)-{\theta }_3),$$(20)where ${\theta }_3=\arctan \frac{m\cos {\phi }_m(J_2(Z)-J_4(Z))}{2J_3(Z)}$. The carrier phase modulation depth $Z$ satisfies $J_2(Z)=J_3(Z)$, with $Z=3.77$ according to the Bessel function. $s_3(t)$ effectively avoids the DC term from conventional PGC demodulation, improving the resistance to variations in interference light intensity. The two signals are normalized and then subtracted and added to construct a set of completely orthogonal output signals, denoted as $$s_2^{\prime }(t)=2\cos (\frac{{\theta }_3-{\theta }_2}{2}+\frac{\pi }{4})\cos (\phi (t)-(\frac{{\theta }_3+{\theta }_2}{2}+\frac{\pi }{4})),$$(21)$$s_3^{\prime }(t)=2\sin (\frac{{\theta }_3-{\theta }_2}{2}+\frac{\pi }{4})\sin (\phi (t)-(\frac{{\theta }_3+{\theta }_2}{2}+\frac{\pi }{4})).$$(22)

Finally, the amplitude normalization correction is performed by extracting the extremum values within one interference period, and the two signals are derived as $$s_2^{\prime \prime }(t)=\cos (\phi (t)-(\frac{{\theta }_3+{\theta }_2}{2}+\frac{\pi }{4})),$$(23)$$s_3^{\prime \prime }(t)=\sin (\phi (t)-(\frac{{\theta }_3+{\theta }_2}{2}+\frac{\pi }{4})).$$(24)

Figure 4(a) illustrates the nonlinear correction results with the proposed PGC and conventional PGC demodulation under interference light intensity variation, where the working distance is located at the microprobe beam waist. Here the carrier signal frequency $f_c=w_c/2\pi =1\mathrm{MHz}$, AOIM parameters $m=0.1$ and ${\phi }_m=0$, and the target moved with a uniform velocity of 3 mm/s. The measurement light introduced an intensity disturbance of 25 Hz to simulate variations in the interference light intensity. Nonlinear displacement errors were obtained from the demodulation results compared to ideal displacements of the target. The conventional PGC demodulation has a periodic nonlinear error with a 27 nm peak-to-peak value, and there is still a residual error of 3 nm combined with the correction method based on characteristic parameter extraction. With the proposed PGC demodulation method, the theoretical nonlinear error is at the pm level after processing the sensing signals under light intensity variations. This indicates that the proposed PGC method can accurately demodulate interference signals with varying light intensity, ensuring microprobe sensing accuracy over large displacement ranges.

In large-range detection, the effect of alignment deviations between miniature probe elements on displacement measurement accuracy is not negligible. As a measure for assessing demodulation system performance, the signal to noise and distortion (SINAD) is defined as the ratio of the desired signal energy to the sum of the total harmonic energy and noise within the demodulation frequency bandwidth [31]. This ratio is employed in this paper to characterize the effect of GRIN’s misalignment amounts on the demodulation of microprobe sensing signals with the proposed PGC algorithm. The demodulated phase SINAD as a function of the microprobe working distance curves for different alignment deviations between the GRIN and SMF was simulated as shown in Fig. 4(b). When the GRIN is radially displaced by 4 μm relative to the SMF, the microprobe SINAD is reduced by 17 dB over the 0–750 mm range. Additionally, the SINAD is attenuated by 8 dB when the GRIN is tilted by 0.05° from the SMF. In cases where both tilt and dislocation misalignments are present, the SINAD attenuation increases to 46 dB over the entire measurement range. It implies a reduction in the quality of the demodulated signal, with large-range displacement measurement accuracy becoming sensitive to noise. The SINAD of the microprobe sensor, assembled under the guidance of the microprobe interference-sensing model, fluctuated by less than 8 dB over a working distance of 0–750 mm. This provides better resistance to noise and harmonic disturbances. Additionally, the maximum SINAD is less than 70 dB due to the addition of random noise in the simulation and harmonic components generated by the AOIM.

Figures 4(c) and 4(d) demonstrate the demodulation nonlinear errors with respect to the working distance with the conventional PGC and proposed PGC algorithm for different GRIN alignment deviations. As shown in Fig. 4(c), alignment deviations largely impact measurement accuracy at longer distances. Within the 0–200 mm range, microprobes with different alignments have nonlinear errors of approximately 30 nm due to the AOIM. The demodulation errors for misaligned microprobes increase to several tens of or even 100 nanometers as the working distance increases. Residual nonlinear errors span several nanometers in the 0–750 mm range for misaligned microprobes, even after compensation through the proposed PGC demodulation algorithm, as depicted in Fig. 4(d). This is related to the heightened sensitivity of spatial-fiber coupling efficiency to tilts and dislocations of miniature optical elements at longer working distances. Such sensitivity results in reduced contrast and signal-to-noise ratio of interference signals over a wide range of target displacements, limiting the nonlinearity correction effectiveness of the proposed PGC algorithm. The proposed interference-sensing method establishes a model relationship between element misalignment and the interference-sensing signal quality of the microprobe. It guides the alignment and assembly of microprobe elements close to the ideal alignment situation. The nonlinearity of the microprobe in ideal alignment is effectively corrected with errors less than 20 pm across the 0–750 mm working distance range, improving the accuracy of the microprobe displacement measurement. Simulation results demonstrate that the proposed microprobe interference-sensing method can effectively reduce nonlinear measurement errors caused by element alignment deviations in larger ranges. This ensures the high-accuracy displacement measurement of the microprobe within the scope of hundreds of millimeters in narrow spaces.

An experimental setup was established to verify the proposed microprobe interference-sensing method for large-range displacement measurements with high accuracy, as shown in Fig. 5. The light source was a distributed feedback (DFB) semiconductor laser (DFB PRO, Toptica, Germany) with a wavelength of 1532.8 nm, supplied with operating temperature and current by a driver (DLC PRO, Toptica, Germany). The modulation of the light source frequency was realized by modulating the operating current with a 1 MHz sinusoidal signal, generated in a homemade data processing unit via a direct digital frequency synthesizer (DDS). The microprobe fiber pigtail and NPBS were fixed, and the GRIN was placed on a multi-degree-of-freedom adjustable fiber coupling stage (MAX313D, Thorlabs, USA). The fiber output laser was reflected back to the microprobe fiber pigtail through the GRIN in different misalignment states, generating different coupling efficiencies and interference contrasts. Then the assembled microprobe sensor (see inset in Fig. 5) was utilized to verify the feasibility for nonlinear error correction of the proposed fiber microprobe interference-sensing method. The interference signal formed by the measurement light containing the target displacement and reference light inside the microprobe was collected by the detector and transmitted to a homemade data processing module. After analog-to-digital conversion, the signal was demodulated and corrected for nonlinear errors, and the results were displayed on a PC. The reflective target was fixed on a linear displacement stage (A123, Physik Instrument, Germany) to produce a target displacement over a wide range of 0–700 mm with a positioning accuracy of $\pm 0.5\mathrm{\mu m}$.

Before adjusting the GRIN misalignment amounts to verify the feasibility of the interference-sensing model, placing NPBS and SMF in parallel was necessary to be ensured in the following steps. First, the SMF was fixed on the base, and the mirror angles were adjusted to maximize the reflected light intensity, ensuring the mirror was parallel to the SMF. Then coarse alignment between NPBS and SMF was established using an indicating light source and a multi-axis displacement adjustment stage to make the light point reflected by the NPBS inner surface coincide with that reflected by the mirror. Finally, the NPBS angular attitudes were finely adjusted to maximize the light intensity reflected to SMF from the NPBS inner surface, thus ensuring NPBS and SMF in parallel. The relative positions between the GRIN and SMF were adjusted using the fiber coupling platform shown in Fig. 5, which allows for the adjustment of radial displacement with a resolution of 0.1 μm and tilt angle with a resolution of 2.8 milli-degrees. The reference light-coupling efficiency is defined as the ratio of the input optical power to the reference light-coupling power and is measured by blocking the measurement optical path with varying misalignments of the GRIN concerning the fiber. The measurement light-coupling efficiency was obtained by dividing the measurement light-coupling power by the input power, where an identical cube beam splitter prism without a built-in reflective surface was used. Additionally, the interference signals at different misalignments of the GRIN were transmitted to an oscilloscope to obtain the interference contrast at a constant distance from the reflection target.

The radial displacements of the GRIN in both directions concerning the fiber pigtail were adjusted, and the corresponding measurement and reference light-coupling efficiencies were recorded and compared with the simulation of the interference-sensing model, as shown in Fig. 6(a). Here, the GRIN’s focusing constant $g=0.327{\mathrm{mm}}^{-1}$, length $L_g=0.25$$P$, and an NPBS size of 3 mm with equal splitting beams were considered. The reflection target was located at the beam waist position of the microprobe, corresponding to an ideal maximum light-coupling efficiency of 100% and a reference light-coupling efficiency of 96%. For easier visual comparison, the power loss due to the fiber devices and environment was compensated, ensuring that the experimental data maximum is consistent with the simulation. It is seen that GRIN dislocation similarly affects the spatial-fiber coupling efficiencies of both beams, with the coupling efficiency reducing to 60% at a radial displacement of 2 μm and further decaying to 3% at 6 μm. Figure 6(b) presents the measurement results of the interference contrast with different misalignments of the GRIN. The contrast stays above 0.94 between radial displacements of $-6$ and 6 μm. This is because the coupling efficiencies of the two beams vary consistently with the radial displacement, rendering the contrast insensitive to GRIN dislocations in the microprobe fiber. As the dislocation increases continuously, the two beams fail to couple efficiently into the fiber for interference; the interference signal light intensity attenuates greatly compared to the measurement noise, leading to a decrease in contrast.

The model relationships between the microprobe coupling efficiencies and interference contrast as a function of the GRIN tilt angle were also experimentally verified, as shown in Figs. 6(c) and 6(d). The coupling efficiencies of the two beams decrease rapidly as the tilt angle of the GRIN increases, reaching coupling efficiencies of 30% and 1% at tilt angles of 0.1° and 0.2°, respectively. The interference contrast remains greater than 0.96 within the tolerance of misalignment angles that allow for coupling and decreases significantly beyond this range. The experimental results shown in Fig. 6 are consistent with the simulation results of the proposed interference-sensing model, thus indicating the feasibility for handling element misalignment.

According to the proposed microprobe global interference-sensing model, a microprobe alignment assembly system was built to realize the alignment and assembly of microprobe elements. The alignment status of the elements was adjusted by using coupling efficiency and contrast as feedback information for precise microprobe assembly. The fabrication process was divided into four steps.Step 1: Determination of microprobe beam waist position. First, the SMF, NPBS, and mirror were aligned using the method described in Section 3.A. The SMF was embedded into the ferrule for easy handling and fixed with a glass sleeve of matching diameter by gluing. The GRIN was then mounted on a five-axis adjustment stage by a precision mechanical gripper, capable of three degrees of freedom positional and two degrees of freedom angular adjustment. The incident end of the GRIN was placed into the glass sleeve. The axial distance of the GRIN relative to SMF was adjusted to match the beam waist spot observed by the beam analyzer with the simulation results of the proposed sensing model. This ensured the microprobe beam waist position was identical to the design value.Step 2: Alignment of GRIN and SMF. First, the mirror was driven to perform a large-range displacement. With the contrast as feedback, two radial displacements and two rotation angles of the GRIN were adjusted to ensure a smooth variation of the contrast across the entire measurement range. Then the measurement optical path was blocked, and the two rotation angles of the GRIN (yaw and pitch) were adjusted within a small range to maximize the reference light-coupling efficiency.Step 3: Assembly between GRIN and SMF. The aligned GRIN was fixed with the glass sleeve using epoxy adhesive. During the curing process, the reference light-coupling efficiency was monitored in real time, and the two angular attitudes of the GRIN were finely adjusted to ensure the curing quality.Step 4: Assembly between GRIN and NPBS. After completing the assembly between GRIN and SMF, NPBS was moved axially to the nearby end face of GRIN. The assembly of NPBS and GRIN was realized by dropping epoxy adhesive at the edge connection of the two elements while continuously monitoring the feedback signal changes during the curing process to ensure the assembly quality.

The microprobe assembled under the guidance of the proposed sensing model measured $\mathrm{\Phi }$$4.5\mathrm{mm}\times 14\mathrm{mm}$ and was packaged by a metal tube. The feasibility of nonlinearity correction for large-range displacement measurements was verified with the assembled microprobe sensor. The proposed PGC phase demodulation method was employed to solve the target displacements in interference signals. The frequency modulation amplitude of the light source was dynamically adjusted based on the variation in working distance to keep the phase modulation depth at the working point ($Z=3.77$) throughout the measurement process. Displacement nonlinearity errors were obtained by comparing the displacements measured by the sensor with the ones yielded by the recognized effective Heydemann correction [24,32].

The measurement light-coupling efficiency and contrast of the assembled microprobe sensor were first tested over a large range of target displacements. As shown in Fig. 7(a), the microprobe coupling efficiency exceeded 20% over the 0–700 mm range, with the maximum value located near the 60 mm beam waist position. The interference contrast was also above 0.4 over this range. Experimental results of the sensor’s coupling efficiency and contrast are consistent with simulation ones of the proposed microprobe sensing model. This demonstrates the effectiveness of the proposed interference-sensing model in guiding the alignment and assembly of microprobe elements. Figure 7(b) shows the demodulation phase SINAD at different target displacements for the microprobe sensor with model-optimized assembly and element alignment deviations. The SINAD of the microprobe with element misalignment attenuates by 24 dB as the target displacement increases, while the SINAD of the assembly-optimized microprobe exhibited a 10 dB variation over the 0–700 mm measurement range. These experimental results illustrate that the sensing model enhances the SINAD of the integrated microprobe under a large range of target displacements by guiding element alignment, thereby facilitating the realization of high-accuracy displacement measurements.

The feasibility of the proposed fiber microprobe-sensing method for nonlinearity correction of displacement measurements was verified. The target follows a linear displacement stage at a uniform speed of 3 mm/s. Figure 7(c) provides residual nonlinear errors of the microprobe sensor signal under GRIN misalignment demodulated with conventional and proposed PGC algorithms. The conventional PGC demodulation had large nonlinear errors due to interference light intensity fluctuations from AOIM of the light source and target mirror deflection in wide-range motions. The proposed PGC algorithm based on high-order carrier demodulation significantly reduces nonlinearity in the sensor’s large-range displacement measurement by eliminating the DC term that varies with light intensity. The enlargement of the gray shaded area is demonstrated in Fig. 7(d). The measurement nonlinear errors of the sensing-model optimized microprobe with the proposed PGC algorithm over the target motion range of 0–700 mm are compared with those of the microprobe under element misalignment. As the target displacement increases, the nonlinear error under GRIN alignment deviations grows to 3.26 nm, with a standard deviation of 0.93 nm. It indicates that the miniature element misalignment in the sensor limits the nonlinearity correction accuracy of the proposed phase demodulation in a large sensing range. The microprobe with model-optimized assembly combined with this demodulation algorithm shows a maximum residual nonlinear error of 0.82 nm and a standard deviation of 0.20 nm over the entire measurement range. Experimental results demonstrate that the microprobe interference-sensing method reduces the maximum nonlinear error to 0.82 nm over the 0–700 mm range by optimizing alignment assembly of microprobe elements and high-accuracy PGC demodulation, thereby achieving embedded displacement measurement with subnanometer accuracy over hundreds of millimeters range.

Herein, a novel fiber microprobe interference-sensing method was proposed for large-range displacement measurements with high accuracy in narrow spaces. The method established the model relationship between alignment deviations of the microprobe element and interference-sensing signal, utilizing tilted-axis Gaussian optical field diffraction. The effect of these alignment deviations on the accuracy of large-range displacement measurements with the microprobe sensor was analyzed. The findings demonstrated that the proposed interference-sensing model effectively suppresses nonlinear errors caused by the element misalignment over large-range measurements. Moreover, a high-accuracy PGC demodulation algorithm was proposed for interference light intensity variations in large-range detection. With high-order carrier demodulation, the algorithm avoided the time-varying DC term that disturbs nonlinearity correction to improve the resistance of microprobe sensors to interference light intensity fluctuations. Experiments indicated that through assembly optimization of the proposed interference-sensing model, the millimeter-sized microprobe sensor reached a measurement range of 0–700 mm. The maximum nonlinear error was reduced from tens of nanometers to 0.82 nm over the entire range combined with the high-accuracy PGC demodulation, which ensures subnanometer accuracy for embedded displacement measurements over hundreds of millimeters. The findings presented bear considerable relevance for industries seeking breakthroughs in long-range displacement measurement, such as robotics, medical devices, and manufacturing. Currently, the microprobe interference-sensing model has only investigated the effect of misalignment between GRIN and SMF on large-range displacement sensing, not considering assembly deviations of the NPBS. In future research, the model relationship between pose changes in NPBS and microprobe interference-sensing signals and measurement accuracy will be added for further improvement.

Acknowledgment. C. Zhang thanks support from the National Key Research and Development Program of China.