A Fiber-Based Chromatic Dispersion Probe for Simultaneous Measurement of X-Axis and Z-Axis Displacements with Nanometric Resolutions

In this paper, a fiber-based chromatic dispersion probe for simultaneous measurement of X-axis and Z-axis displacements with nanometric resolutions by using the full width at half maxima (FWHM) of the detected spectral signal has been proposed and demonstrated. For X-axis, FWHM is employed for indicating the X-axis displacement based on the fact that the FWHM remains almost constant with the varying Z-axis displacement of the fiber detector and shows a linear relationship with the X-axis displacement within a specific Z-axis displacement range. For the Z-axis, the linear relationship between the centroid wavelength λ of the detected spectral signal and the Z-axis displacement is employed for indicating the Z-axis displacement based on the fact that the sensitivity (slope of the λ-Z curve) is also linear with X-axis displacement within a certain X-axis displacement range. Theoretical and experimental investigations have verified the feasibility of the proposed chromatic dispersion probe, which yields X- and Z-axis measurement ranges of 2.3 μm and 15 μm and X- and Z-axis measurement resolutions of better than 25 nm and 50 nm, respectively. Experiments were further performed to evaluate the basic performance of the prototype probe and the maximum measurement errors were less than 10 nm and 60 nm for X- and Z-axis displacements, respectively.


Introduction
With the miniaturization of equipment and components in the industry, microscale stages have been a powerful tool and widely used in the space-limited and small-cavity fields of micro-electro-mechanical systems (MEMS) [1][2][3], precision optics [4][5][6], microscale mechanical testing [7], semiconductor manufacturing [8], and so on. To ensure the high-accuracy motion of these microscale stages, high-precision, cost-effective, and easymounted displacement measurement sensors are necessary and needed [9][10][11].
To address the issue of the displacement measurement of the micro-stage and fulfill the development of the application of the displacement measurement in the space-limited and small-cavity fields, many displacement measurement sensors and techniques [11][12][13][14][15][16][17][18][19] have been proposed so far, such as optical interferometers [11][12][13], digital image correlation sensors [14,15], thin film displacement sensors [16,17], integrated capacitive displacement sensors [18,19], hall-effect sensors [20], and so on. For the optical interferometers, not only the miniature laser diode interferometer [11], the compact laser interferometer based on the displacement sensing method [12], but also the Fabry-Pérot interferometer [13] can achieve good measurement resolutions with large-stroke measurement ranges. However, due to the large-stroke optical path, these optical interferometers are relatively fragile and Figure 1 depicts the schematic diagram of the established fiber-based chromatic dispersion probe for simultaneous measurement of X-axis and Z-axis displacements in which the object to be measured is the entrance face of the employed optical single-mode fiber. By the detected output spectrum from the fiber detector, within the X-axis, FWHM of the output spectrum is employed for indicating the X-axis displacement based on the fact that FWHM remains almost constant with the varying Z-axis displacement of the fiber detector and shows a linear relationship with the X-axis displacement within a specific Z-axis displacement range. For the Z-axis, the linear relationship between the centroid wavelength λ of the detected spectral signal and the Z-axis displacement is employed for indicating the Z-axis displacement based on the fact that the sensitivity S (slope of the λ-Z curve) is also linear with the X-axis displacement within a certain X-axis displacement range. In the decoding process of the simultaneous measurement of X-axis and Z-axis displacements, firstly, the FWHM is calculated from the detected spectrum and then the displacement of the X-axis is decoded through the FWHM. Secondly, with the obtained X-axis displacement, the sensitivity S (slope of the λ-Z curve) by the S-X curve is decoded and obtained. Finally, by the decoded sensitivity S, the corresponding λ-Z curve can be selected and by the λ-Z curve, the Z-axis displacement can be approached. It should be pointed out that the above operations can only be realized within certain Xand Z-axis displacement ranges, which are associated with the measurement ranges of the Xand Z-axis of the established chromatic dispersion probe. It should also be noted that the dispersive light coupling into the entrance face of the optical fiber core has many continuous focal planes within the fiber detector of the established chromatic dispersion probe, and one of the focal planes shown in Figure 1 is the focal plane corresponding to the central wavelength of the light beam. centroid wavelength can make ensure constant stable measurement results and can be regarded as a sign of the robustness of the proposed probe. Figure 1 depicts the schematic diagram of the established fiber-based chromatic dispersion probe for simultaneous measurement of X-axis and Z-axis displacements in which the object to be measured is the entrance face of the employed optical single-mode fiber. By the detected output spectrum from the fiber detector, within the X-axis, FWHM of the output spectrum is employed for indicating the X-axis displacement based on the fact that FWHM remains almost constant with the varying Z-axis displacement of the fiber detector and shows a linear relationship with the X-axis displacement within a specific Z-axis displacement range. For the Z-axis, the linear relationship between the centroid wavelength λ of the detected spectral signal and the Z-axis displacement is employed for indicating the Z-axis displacement based on the fact that the sensitivity S (slope of the λ-Z curve) is also linear with the X-axis displacement within a certain X-axis displacement range. In the decoding process of the simultaneous measurement of X-axis and Z-axis displacements, firstly, the FWHM is calculated from the detected spectrum and then the displacement of the X-axis is decoded through the FWHM. Secondly, with the obtained X-axis displacement, the sensitivity S (slope of the λ-Z curve) by the S-X curve is decoded and obtained. Finally, by the decoded sensitivity S, the corresponding λ-Z curve can be selected and by the λ-Z curve, the Z-axis displacement can be approached. It should be pointed out that the above operations can only be realized within certain X-and Z-axis displacement ranges, which are associated with the measurement ranges of the X-and Z-axis of the established chromatic dispersion probe. It should also be noted that the dispersive light coupling into the entrance face of the optical fiber core has many continuous focal planes within the fiber detector of the established chromatic dispersion probe, and one of the focal planes shown in Figure 1 is the focal plane corresponding to the central wavelength of the light beam.

Principle and Simulation
Simultaneous measurement of X-axis and Z-axis displacements FWHM remains almost constant with the varying Z-axis displacement of fiber detector Linear relationship between the sensitivity S (λ-Z curve slope) and X-axis displacement

Chromatic Dispersion and Imaging Principle
In the established fiber-based chromatic dispersion probe, the light beam passing through a small pinhole within the 4f optical system filter can be regarded as the linear superposition of the paraxial point light sources, which can be simply signified as the superposition of the impulse functions of ∑δ(rsp). In the chromatic dispersion probe, an Sensors 2023, 23, 51 4 of 15 always coherent optical system is realized by employing the single-mode optical fiber as the signal-transferring media regardless of the detailed optical paths and the fiber dimensions [38,39]. The axial response of the chromatic dispersion probe within the entrance face of the fiber detector can thereby be expressed as: where r sp stands for the distance between the light source center and the superimposed paraxial point light source and h effective is the effective point spread function of the probe system associated with the single-mode fiber and can be expressed as: where h(u, v) is the point spread function of the probe system, e(v) is the eigenfunction of the transverse orthonormal modal field in the entrance face of the single-mode fiber, ⊗ is the convolution operation, and u and v are the optical coordinates related to the Z-axis displacement Zand the X-directional distance X sp . u and v can be expressed as: where sin α is the numerical aperture (NA) of the employed chromatic dispersive lens, ρ is the radial variation of the incident field of the fiber shown in Figure 2, and the Xdirectional distance X sp is the distance between the center of the fiber core and the diffraction field generated by the dispersive lens and can be expressed as the distance between the light source center and the superimposed paraxial point light sources r sp and the X-axis displacement X given by Equation (5): where M is the magnification of the optical system. In the simulation, θ is the angle corresponding to the discrete paraxial point light source that needs to be superimposed in a circle from the center of the light source r sp .
superposition of the paraxial point light sources, which can be simply signified as the superposition of the impulse functions of ∑δ(rsp). In the chromatic dispersion probe, an always coherent optical system is realized by employing the single-mode optical fiber as the signal-transferring media regardless of the detailed optical paths and the fiber dimensions [38,39]. The axial response of the chromatic dispersion probe within the entrance face of the fiber detector can thereby be expressed as: where rsp stands for the distance between the light source center and the superimposed paraxial point light source and heffective is the effective point spread function of the probe system associated with the single-mode fiber and can be expressed as: where h(u, v) is the point spread function of the probe system, e(v) is the eigenfunction of the transverse orthonormal modal field in the entrance face of the single-mode fiber,  is the convolution operation, and u and v are the optical coordinates related to the Z-axis displacement Z-and the X-directional distance Xsp. u and v can be expressed as: where sinα is the numerical aperture (NA) of the employed chromatic dispersive lens, ρ is the radial variation of the incident field of the fiber shown in Figure 2, and the X-directional distance Xsp is the distance between the center of the fiber core and the diffraction field generated by the dispersive lens and can be expressed as the distance between the light source center and the superimposed paraxial point light sources rsp and the X-axis displacement X given by Equation (5): where M is the magnification of the optical system. In the simulation, θ is the angle corresponding to the discrete paraxial point light source that needs to be superimposed in a circle from the center of the light source rsp.   The point spread function for the established optical system is given as: where J 0 represents the zero-order Bessel functions of the first kind. It should be pointed out that the used modal field in Equation (2) is given by Ref. [40].
where ψ n is the normalization coefficient of the mode, ϕ is the azimuthal angle shown in Figure 2 and f n (R) represents the transverse electromagnetic field of the used step-index optical fiber and can be expressed as [40,41]: with R = ρ/R fiber and U is the modal eigenvalue [41], in which U and W are related to the dimensionless parameter V for the core and cladding of the fiber and can be found in Ref. [42] (p. 314). Moreover, n stands for the n th mode of the employed fiber and equals 1 in the single-mode fiber.
The effective point spread function can be expressed by bringing Equations (6) and (7) into Equation (2), which is given by: where v c represents the optical coordinate and is given by 2π R fiber sin α/λ and v diff is the optical coordinate of the diffraction field given by 2π X sp sin α/λ. The axial responses of the measurement axis can be expressed by bringing Equations (8)-(10) into Equation (1): where J 1 (rv c ) is the first-order Bessel function of the first kind. Figure 3 shows a flowchart of the measurement principle and the corresponding decoding process of the proposed fiber-based chromatic dispersion probe for displacement measurement of the X-axis. After passing through the optical path, the output spectrum is obtained and detected by the fiber detector in which the FWHM can be calculated. With different X-axis displacements of the fiber detector, different output spectra are acquired with different FWHM values. FWHM-X decoding can be carried out based on the fact that FWHM remains almost constant with the varying Z-axis displacement and yields a linear relationship with the X-axis displacement within a specific Z-axis displacement.   Figure 4 shows a flowchart of the measurement principle and the corresponding decoding process of the proposed fiber-based chromatic dispersion probe for displacement measurement of the Z-axis. Additionally, with the detected output spectrum, the centroid wavelength λ can be obtained. Thus, decoding the linear relationship between the centroid wavelength λ and Z-axis displacement in different X-axis displacements can be performed, in which different sensitivity S (λ-Z curve slopes) can be observed. With the different sensitivity S (λ-Z curve slopes), the linear relationship between the sensitivity S and X-axis displacement can also be inspected within a specific X-axis displacement range.

Chromatic spectrum and dispersion
Light beam  Figure 4 shows a flowchart of the measurement principle and the corresponding decoding process of the proposed fiber-based chromatic dispersion probe for displacement measurement of the Z-axis. Additionally, with the detected output spectrum, the centroid wavelength λ can be obtained. Thus, decoding the linear relationship between the centroid wavelength λ and Z-axis displacement in different X-axis displacements can be performed, in which different sensitivity S (λ-Z curve slopes) can be observed. With the different sensitivity S (λ-Z curve slopes), the linear relationship between the sensitivity S and X-axis displacement can also be inspected within a specific X-axis displacement range.   Figure 4 shows a flowchart of the measurement principle and the corresponding decoding process of the proposed fiber-based chromatic dispersion probe for displacement measurement of the Z-axis. Additionally, with the detected output spectrum, the centroid wavelength λ can be obtained. Thus, decoding the linear relationship between the centroid wavelength λ and Z-axis displacement in different X-axis displacements can be performed, in which different sensitivity S (λ-Z curve slopes) can be observed. With the different sensitivity S (λ-Z curve slopes), the linear relationship between the sensitivity S and X-axis displacement can also be inspected within a specific X-axis displacement range.

Chromatic spectrum and dispersion
Light beam

Simulation and Discussion
According to Equation (11) in Section 2.1, the axial response curves corresponding to different Xand Zaxis displacements of the established chromatic dispersion probe are calculated and plotted by the software MATLAB. The FWHM of the axial response curves can, thereby, be obtained by finding the half-peak width of these axial response curves. It should be noted that the axial response curves have gradually become spectral bimodal distribution when the X-axis displacement is larger than two microns based on the theory of the employed step-index single-mode fiber, in which the cladding of the step-index single-mode fiber can hardly accept the dispersive light coupling into it. Meanwhile, in practical experiments, not only the core but also the cladding of the stepindex single-mode fiber can receive the dispersive light coupling into it; moreover, the focused beam spot diameter by the dispersive lens L4 in the experiment is larger than that of the focused beam spot diameter in simulation due to the influence of the beam quality and the lens spherical aberration. Consequently, no spectral bimodal distribution is found in the practical experiments and Gaussian fitting is used for the calculation of the FWHM in the simulation. Therefore, there is a difference in the FWHM of the simulation and experimental results. Figure 5a,b show the simulation results of the variation of FWHM within the Z-axis displacement ranging from −5 to 95 µm and Z-axis displacement ranging from 30 to 50 µm at X-axis displacements from 4 to 6.3 µm. As depicted in the figures, the maximum variation of FWHM within the Z-axis displacement ranging from −5 to 95 µm is less than 8 nm and is even less than 0.4 nm within the corresponding ranging from 30 to 50 µm at the X-axis displacements ranging from 4 to 6.3 µm. It is observed that FWHM remains almost constant with the varying of the Z-axis displacement of the fiber detector within Z-axis displacement ranging from 30 to 50 µm and can be used for indicating the X-axis displacement regardless of the simultaneous variation of the Z-axis displacement.

Simulation and Discussion
According to Equation (11) in Section 2.1, the axial response curves corresponding to different X-and Z-axis displacements of the established chromatic dispersion probe are calculated and plotted by the software MATLAB. The FWHM of the axial response curves can, thereby, be obtained by finding the half-peak width of these axial response curves. It should be noted that the axial response curves have gradually become spectral bimodal distribution when the X-axis displacement is larger than two microns based on the theory of the employed step-index single-mode fiber, in which the cladding of the step-index single-mode fiber can hardly accept the dispersive light coupling into it. Meanwhile, in practical experiments, not only the core but also the cladding of the step-index singlemode fiber can receive the dispersive light coupling into it; moreover, the focused beam spot diameter by the dispersive lens L4 in the experiment is larger than that of the focused beam spot diameter in simulation due to the influence of the beam quality and the lens spherical aberration. Consequently, no spectral bimodal distribution is found in the practical experiments and Gaussian fitting is used for the calculation of the FWHM in the simulation. Therefore, there is a difference in the FWHM of the simulation and experimental results. Figure 5a,b show the simulation results of the variation of FWHM within the Zaxis displacement ranging from −5 to 95 μm and Z-axis displacement ranging from 30 to 50 μm at X-axis displacements from 4 to 6.3 μm. As depicted in the figures, the maximum variation of FWHM within the Z-axis displacement ranging from −5 to 95 μm is less than 8 nm and is even less than 0.4 nm within the corresponding ranging from 30 to 50 μm at the X-axis displacements ranging from 4 to 6.3 μm. It is observed that FWHM remains almost constant with the varying of the Z-axis displacement of the fiber detector within Z-axis displacement ranging from 30 to 50 μm and can be used for indicating the X-axis displacement regardless of the simultaneous variation of the Z-axis displacement. Simulation results of (a) the variation of FWHM within Z-axis displacement ranging from −5 to 95 μm; and (b) the corresponding Z-axis displacement ranging from 30 μm to 50 μm at the Xaxis displacements from 4 μm to 6.3 μm. Figure 6a,b show simulation results of the linear relationship between the centroid wavelength λ and Z-axis displacement and the linear relationship between the corresponding sensitivity S (λ-Z curve slope) and X-axis displacement at the X-axis displacements from 4 μm to 6.3 μm. As can be observed in the figures, the centroid wavelength λ of the detected spectral signal can be employed for indicating the Z-axis displacement associated with different X-axis displacements by using the corresponding linear decoding of the S-X curve within a specific Z-axis displacement range of 15 μm.  Figure 6a,b show simulation results of the linear relationship between the centroid wavelength λ and Z-axis displacement and the linear relationship between the corresponding sensitivity S (λ-Z curve slope) and X-axis displacement at the X-axis displacements from 4 µm to 6.3 µm. As can be observed in the figures, the centroid wavelength λ of the detected spectral signal can be employed for indicating the Z-axis displacement associated with different X-axis displacements by using the corresponding linear decoding of the S-X curve within a specific Z-axis displacement range of 15 µm.  Figure 6. Simulation results of (a) the linear relationship between the centroid wavelength λ and Zaxis displacement and (b) the linear relationship between the corresponding sensitivity S (λ-Z curve slope) and X-axis displacement at the X-axis displacements from 4 to 6.3 μm.

Experiments
Experiments on the investigation of the X-and Z-axis measurement ranges and resolutions were carried out first. Experiments were further performed to evaluate the basic performance of the prototype probe for simultaneous measurement of X-axis and Z-axis displacements with nanometric resolutions. Figure 7 shows the schematic of the experimental configuration of the proposed chromatic dispersion probe for simultaneous measurement of the X-and Z-axis displacements with nanometric resolutions. To obtain a wide spectral range for measurement, a chromatic LED illuminator (L1CU-4090, Lumileds Holding B.V., Haarlemmermeer, The Netherlands) was selected as the light source with a band-wide spectrum ranging from 400 nm to 800 nm. Collimated by a collimating lens L1 (AC 254-030-A-ML, Thorlabs Inc., Newton, NJ, USA), the light beam was made to pass through an optical 4f system in which a pinhole with a diameter of 50 μm was placed onto the focal planes of the two achromatic lenses L2 and L3. After that, the light beam was split into two sub-beams called the reference sub-beam and the measurement sub-beam by a non-polarized splitter (BS). To monitor the beam quality, the reference sub-beam was inspected by a beam profiler. The measurement sub-beam was collected by a dispersive lens L4 (GCL-010158A, Daheng New Epoch Technology, Inc., Beijing, China), whose physical parameters are shown in Table 1, and received by a fiber detector which was mainly composed of a single-mode optical fiber

Experiments
Experiments on the investigation of the Xand Z-axis measurement ranges and resolutions were carried out first. Experiments were further performed to evaluate the basic performance of the prototype probe for simultaneous measurement of X-axis and Z-axis displacements with nanometric resolutions. Figure 7 shows the schematic of the experimental configuration of the proposed chromatic dispersion probe for simultaneous measurement of the Xand Z-axis displacements with nanometric resolutions. To obtain a wide spectral range for measurement, a chromatic LED illuminator (L1CU-4090, Lumileds Holding B.V., Haarlemmermeer, The Netherlands) was selected as the light source with a band-wide spectrum ranging from 400 nm to 800 nm. Collimated by a collimating lens L1 (AC 254-030-A-ML, Thorlabs Inc., Newton, NJ, USA), the light beam was made to pass through an optical 4f system in which a pinhole with a diameter of 50 µm was placed onto the focal planes of the two achromatic lenses L2 and L3. After that, the light beam was split into two sub-beams called the reference sub-beam and the measurement sub-beam by a non-polarized splitter (BS). To monitor the beam quality, the reference sub-beam was inspected by a beam profiler. The measurement sub-beam was collected by a dispersive lens L4 (GCL-010158A, Daheng New Epoch Technology, Inc., Beijing, China), whose physical parameters are shown in Table 1, and received by a fiber detector which was mainly composed of a single-mode optical fiber connected to a spectrometer (LEDPRO-50, Ocean Optics, Shanghai, China). A piezo actuator (P843-20, Physik Instrumente (PI) GmbH & Co. KG., Karlsruhe, Germany) with a measurement range of 20 µm and a positioning resolution of 0.6 nm and a linear stage (M-112-1DG, Physik Instrumente (PI) GmbH & Co. KG.) with a measurement range of 25 mm and a positioning resolution of 50 nm were utilized for Xand Z-axis precision movement of the single-mode optical fiber. Both the Xand Z-axis moving distances were calibrated by the capacitance probes (ASP-125M-ILA, MTI Instruments Inc., Albany, NY, USA) with a measurement range and resolution of 125 µm and 7.5 nm, respectively.

Measurement Ranges of the X-and Z-Axis
Experiments were first carried out to investigate the FWHM variation with the varying Z-axis displacement of the fiber detector and the linear relationship between the FWHM and the X-axis displacement within a specific Z-axis displacement range. Based on this, the measurement range of the X-axis could finally be observed. In these experiments, the entrance face of the single-mode optical fiber core of the fiber detector was mounted onto the linear stage and was made to have a Z-directional scanning with a moving step of 1 μm within the Z-axis displacement range from 30 to 45 μm. Then, the entrance face of the single-mode optical fiber of the fiber detector was made to move along

Measurement Ranges of the X-and Z-Axis
Experiments were first carried out to investigate the FWHM variation with the varying Z-axis displacement of the fiber detector and the linear relationship between the FWHM and the X-axis displacement within a specific Z-axis displacement range. Based on this, the measurement range of the X-axis could finally be observed. In these experiments, the entrance face of the single-mode optical fiber core of the fiber detector was mounted onto the linear stage and was made to have a Z-directional scanning with a moving step of 1 µm within the Z-axis displacement range from 30 to 45 µm. Then, the entrance face of the singlemode optical fiber of the fiber detector was made to move along X-axis with a moving interval of 100 nm by the piezo actuator at the Z-axis displacements of 30, 35, 40, and 45 µm, respectively. Both the compact linear stage and the piezo actuator were calibrated by the capacitance probes with a measurement range and a measurement resolution of 125 and 7.5 nm, respectively. Figure 8a shows the schematic of the definition of the Xand Z-axis coordinates in which the focal point of the dispersive lens L4 was set to be the coordinate origin. Figure 8b depicts the detected output spectrum by the fiber detector. Figure 8c,d show the experimental results of the variation of the FWHM within a Z-axis displacement range of 15 µm and the linear decoding of FWHM-X curves within the corresponding Z-axis displacement range. It can be seen from the figures that the change and the standard deviation of the FWHM were less than 0.2 nm and 0.1 nm, respectively, and furthermore, the maximum standard deviation of FWHM at the same X-axis displacement σ max was less than 0.1 nm within the Z-axis displacement range of 15 µm. The experimental results demonstrate that FWHM remained almost constant with the varying of the Z-axis displacement of the fiber detector, and furthermore, the linear decoding of the FWHM-X curve could be used for indicating the X-axis displacement regardless of the simultaneous variation of the Z-axis displacement and the X-axis measurement range was 2.3 µm. It should be noted that Figure 8d was only a rough estimated X-axis displacement measurement range associated with the FWHM variation determined by the Z-axis displacement range shown in Figure 8c. It should also be pointed out that the coefficient of determination R 2 was only utilized for a qualitative evaluation of the performance of data fitting, but not as the indicator for the quantitative evaluation of the measurement linearity of the proposed fiber-based chromatic dispersion probe both in simulation and experimental results. Accurate evaluation of the measurement range and measurement linearity with accurate identification of the measurement accuracy and uncertainties will be carried out in future work. ment range. It can be seen from the figures that the change and the standard deviation of the FWHM were less than 0.2 nm and 0.1 nm, respectively, and furthermore, the maximum standard deviation of FWHM at the same X-axis displacement σmax was less than 0.1 nm within the Z-axis displacement range of 15 μm. The experimental results demonstrate that FWHM remained almost constant with the varying of the Z-axis displacement of the fiber detector, and furthermore, the linear decoding of the FWHM-X curve could be used for indicating the X-axis displacement regardless of the simultaneous variation of the Zaxis displacement and the X-axis measurement range was 2.3 μm. It should be noted that Figure 8d was only a rough estimated X-axis displacement measurement range associated with the FWHM variation determined by the Z-axis displacement range shown in Figure  8c. It should also be pointed out that the coefficient of determination R 2 was only utilized for a qualitative evaluation of the performance of data fitting, but not as the indicator for the quantitative evaluation of the measurement linearity of the proposed fiber-based chromatic dispersion probe both in simulation and experimental results. Accurate evaluation of the measurement range and measurement linearity with accurate identification of the measurement accuracy and uncertainties will be carried out in future work.   Figure 9b shows the linear relationship between the corresponding sensitivity S (λ-Z curve slope) and X-axis displacement within the X-axis displacements from 4 μm to 6.3 μm. As shown in the figures, the centroid wavelength λ of the detected spectral signal could be employed for indicating the Z-axis   Figure 9b shows the linear relationship between the corresponding sensitivity S (λ-Z curve slope) and X-axis displacement within the X-axis displacements from 4 µm to 6.3 µm. As shown in the figures, the centroid wavelength λ of the detected spectral signal could be employed for indicating the Z-axis displacement associated with different X-axis displacements by using the corresponding linear decoding of the S-X curve, and furthermore, the Z-axis measurement range was 15 µm. It should be pointed out that the output spectra have gradually become spectral bimodal distribution when the X-axis displacement was larger than two microns based on the theory of the employed step-index single-mode fiber in which the cladding of the step-index single-mode fiber could hardly accept the dispersive light coupling into it. Based on this, Gaussian fitting was used for the calculation of the centroid wavelength in the simulation. Meanwhile, in practical experiments, not only the core but also the cladding of the step-index single-mode fiber could receive the dispersive light coupling into it; moreover, the focused beam spot diameter by the dispersive lens L4 in the experiment was larger than that of the focused beam spot diameter in simulation due to the influence of the beam quality and the lens spherical aberration. Consequently, no spectral bimodal distribution was found in the practical experiments. Therefore, the simulation and experimental results were different because the output spectrum with Gaussian fitting was used for the calculation of the centroid wavelength in the simulation, which finally resulted in a slight difference in the simulation and experimental results of the equations of the S-X decoding. Despite the slight difference, the same trend of the linear relationship could be observed and the difference was, thereby, within the expected tolerance, which demonstrated that the Z-axis displacement could be measured regardless of the X-axis movement.
step-index single-mode fiber could receive the dispersive light coupling into it; moreover, the focused beam spot diameter by the dispersive lens L4 in the experiment was larger than that of the focused beam spot diameter in simulation due to the influence of the beam quality and the lens spherical aberration. Consequently, no spectral bimodal distribution was found in the practical experiments. Therefore, the simulation and experimental results were different because the output spectrum with Gaussian fitting was used for the calculation of the centroid wavelength in the simulation, which finally resulted in a slight difference in the simulation and experimental results of the equations of the S-X decoding. Despite the slight difference, the same trend of the linear relationship could be observed and the difference was, thereby, within the expected tolerance, which demonstrated that the Z-axis displacement could be measured regardless of the X-axis movement. Figure 9. Experimental results of (a) the linear relationship between the centroid wavelength λ and Z-axis displacement and (b) the linear relationship between the corresponding sensitivity S (λ-Z curve slope) and X-axis displacement.

Measurement resolutions of X-and Z-axis
Experiments were then carried out to investigate the measurement resolutions of the X-and Z-axis of the established chromatic dispersion probe, respectively. Similarly, in these experiments, the entrance face of the optical fiber core in the fiber detector was fixed onto the piezo actuator and the linear stage was firstly made to have a scanning along Xaxis with much smaller moving steps of 100, 50, 25, and 15 nm, respectively. Then, the entrance face of the single-mode optical fiber core of the fiber detector was made to move along Z-axis with moving steps of 200, 100, 50, and 25 nm, respectively by the linear stage. Figure 10 shows the experimental results of the X-axis displacement resolution of the established chromatic dispersion probe with the moving steps of 100, 50, 25, and 15 nm, respectively. As can be seen in the figures, the applied moving steps could be clearly discriminated in the moving steps of 100, 50, and 25 nm, and hence, we can conclude that an X-axis displacement resolution of better than 25 nm could be realized in the established chromatic dispersion probe. Similarly, Figure 11 depicts the experimental results of the Zaxis displacement resolution of the established chromatic dispersion probe with different moving intervals of 200, 100, 50, and 25 nm, respectively, which yielded a Z-axis displacement resolution of better than 50 nm for the established dispersion probe. It should be noted that to improve the measurement resolution of the X-axis displacement, the sum of sines function was used to fit the detected spectra. As can be seen in Figures 8,10,11, although different sensitivities were found, only slight change could be observed and good

Measurement Resolutions of X-and Z-Axis
Experiments were then carried out to investigate the measurement resolutions of the Xand Z-axis of the established chromatic dispersion probe, respectively. Similarly, in these experiments, the entrance face of the optical fiber core in the fiber detector was fixed onto the piezo actuator and the linear stage was firstly made to have a scanning along X-axis with much smaller moving steps of 100, 50, 25, and 15 nm, respectively. Then, the entrance face of the single-mode optical fiber core of the fiber detector was made to move along Z-axis with moving steps of 200, 100, 50, and 25 nm, respectively by the linear stage. Figure 10 shows the experimental results of the X-axis displacement resolution of the established chromatic dispersion probe with the moving steps of 100, 50, 25, and 15 nm, respectively. As can be seen in the figures, the applied moving steps could be clearly discriminated in the moving steps of 100, 50, and 25 nm, and hence, we can conclude that an X-axis displacement resolution of better than 25 nm could be realized in the established chromatic dispersion probe. Similarly, Figure 11 depicts the experimental results of the Z-axis displacement resolution of the established chromatic dispersion probe with different moving intervals of 200, 100, 50, and 25 nm, respectively, which yielded a Z-axis displacement resolution of better than 50 nm for the established dispersion probe. It should be noted that to improve the measurement resolution of the X-axis displacement, the sum of sines function was used to fit the detected spectra. As can be seen in Figures 8, 10 and 11, although different sensitivities were found, only slight change could be observed and good measurement linearity could be inspected both in Xand Z-axis measurement range and resolution experiments. Thus, it could still be considered that the sensitivity was a constant and was independent of the displacement values and measurement linearity within the measurement range of the employed chromatic dispersion probe.

Evaluation of the Basic Performance of the Established Prototype Probe
To verify and confirm the feasibility of the proposed chromatic dispersion probe for the simultaneous measurement of Xand Z-axis displacements, experiments were finally carried out to evaluate the basic performance of the established prototype probe. In the experiments, as shown in Figure 12a, from the coordinate origin defined in Figure 8a, the moving path of the fiber detector was depicted with Xand Z-axis moving steps of 100 nm and 1 µm, respectively, along which five measuring points, A to E, were randomly selected. The actual moving displacement was calibrated by the capacitive probe and was called the calibrated value. The displacement decoded from the established chromatic confocal probe was called the measurement value. Figure 12b depicts the experimentally obtained spectrum at the measuring point C and the sum of sines function was used to fit the detected spectrum for the improvement of the measurement resolution of the X-axis. Figure 12c,d show the calibrated values (red points) and the measurement values (blue points) for the confirmation experiments of the simultaneous measurement of Xand Z-axis displacements measurement with nanometric resolutions. As depicted in Figure 12d,e, the maximum measurement errors of the prototype probe were less than 10 nm and 60 nm for Xand Z-axis displacements measurement, respectively. As can be observed in the experiments, the feasibility of the proposed chromatic dispersion probe for simultaneous measurement of Xand Z-axis displacements with nanometric resolutions could be verified and simultaneous measurement of Xand Z-axis displacements of an optical fiber could be realized regardless of the variation of Xand Z-axis displacements of the optical fiber during the simultaneous measurement process, which greatly fulfills the development of the application of simultaneous measurement of dual-axis displacement in the spacelimited and small-cavity fields with nanometric resolutions, in which the micro-sized optical fiber can easily be fixed on a micro-object to be measured.
Sensors 2022, 22, x FOR PEER REVIEW 12 of 16 measurement linearity could be inspected both in X-and Z-axis measurement range and resolution experiments. Thus, it could still be considered that the sensitivity was a constant and was independent of the displacement values and measurement linearity within the measurement range of the employed chromatic dispersion probe.

Evaluation of the basic performance of the established prototype probe
To verify and confirm the feasibility of the proposed chromatic dispersion probe for the simultaneous measurement of X-and Z-axis displacements, experiments were finally carried out to evaluate the basic performance of the established prototype probe. In the experiments, as shown in Figure 12a, from the coordinate origin defined in Figure 8a, the moving path of the fiber detector was depicted with X-and Z-axis moving steps of 100   measurement linearity could be inspected both in X-and Z-axis measurement range and resolution experiments. Thus, it could still be considered that the sensitivity was a constant and was independent of the displacement values and measurement linearity within the measurement range of the employed chromatic dispersion probe.

Evaluation of the basic performance of the established prototype probe
To verify and confirm the feasibility of the proposed chromatic dispersion probe for the simultaneous measurement of X-and Z-axis displacements, experiments were finally carried out to evaluate the basic performance of the established prototype probe. In the experiments, as shown in Figure 12a, from the coordinate origin defined in Figure 8a, the moving path of the fiber detector was depicted with X-and Z-axis moving steps of 100 neous measurement of X-and Z-axis displacements with nanometric resolutions could be verified and simultaneous measurement of X-and Z-axis displacements of an optical fiber could be realized regardless of the variation of X-and Z-axis displacements of the optical fiber during the simultaneous measurement process, which greatly fulfills the development of the application of simultaneous measurement of dual-axis displacement in the space-limited and small-cavity fields with nanometric resolutions, in which the microsized optical fiber can easily be fixed on a micro-object to be measured. Figure 12. Schematic of (a) the moving path of the fiber detector and five randomly selected measuring points A to E. (b) Experimental results of the obtained output spectrum of the measuring points C and Experimental results of the simultaneous measurement of (c) X-and (d) Z-axis displacements together with the corresponding measurement errors of (e) X-axis and (f) Z-axis for the prototype probe.

Conclusions
In this study, a fiber-based chromatic dispersion probe for the simultaneous measurement of X-and Z-axis displacements with nanometric resolutions is proposed and Figure 12. Schematic of (a) the moving path of the fiber detector and five randomly selected measuring points A to E. (b) Experimental results of the obtained output spectrum of the measuring points C and Experimental results of the simultaneous measurement of (c) Xand (d) Z-axis displacements together with the corresponding measurement errors of (e) X-axis and (f) Z-axis for the prototype probe.

Conclusions
In this study, a fiber-based chromatic dispersion probe for the simultaneous measurement of Xand Z-axis displacements with nanometric resolutions is proposed and demonstrated by using the FWHM of the detected spectral signal, which successfully realized the X-axis measurement range and resolution of 2.3 µm and 25 nm, and furthermore, the Z-axis measurement range and resolution of 15 µm and 50 nm, respectively. Experiments were performed to evaluate the basic performance of the prototype sensor, which verified the feasibility of the proposed chromatic dispersion probe and yielded that the maximum measurement errors of the probe were less than 10 nm and 60 nm for the Xand Z-axis displacements, respectively. Due to the micro-size of the optical fiber, it is expected to extend this probe to online calibration in space-limited situations and to enlighten the application in microscale sensing under small-cavity conditions. It should be pointed out that this paper is an initial report of our research on the proposal of a preliminary idea of a fiber-based chromatic dispersion probe for simultaneous measurement of Xand Z-axis displacements by employing the FWHM of the detected spectrum.
As the next step of our research, studies will be carried out to systematically optimize the proposed fiber-based chromatic dispersion probe and achieve longer measurement ranges together with higher measurement resolutions for both the Xand Z-axis. Moreover, the practical application of the developed fiber-based chromatic dispersion probe to the dual-axis displacement measurement under space-limited and small-cavity conditions in the industry remains a challenge to be addressed and will also be taken into consideration in the next step of our research.