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Abstract
Current fiber chromatic confocal probes suffer from a low axial resolution. This study proposes a fiber chromatic confocal method with a tilt-coupling source module for axial super-resolution. An annular intensity distribution is generated at the fiber port of the probe by adjusting the tilt-coupling angle between the fiber and source module to reduce the width of the confocal axial response characteristic curve at each wavelength and to enhance the intensity contrast of the spectral signal. Compared with the coaxial coupling state, the developed 21° tilt-coupling state can reduce the FWHM by approximately 17%, and the axial resolution increases by 1.7 times.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The chromatic confocal method has the advantages of a large measurement range and high accuracy; moreover, it enables 3D surface measurement, axis positioning, and defect detection. Therefore, it is widely used in chip manufacturing [1], micro-optics and micro-mechanical component processing [2–4], and blade tip clearance measurements for aeroengines [5]. Fiber-type structures [6–8] are among the main forms of chromatic confocal technology. The fiber is not only a medium for transmitting light but also an illumination and detection pinhole. A miniaturized fiber-integrated probe makes the measurement process more flexible and suitable for limited space. However, it has the drawbacks of a small numerical aperture and large pinhole size, which limit the improvement in the axial resolution.
For the spectral signal model, Ruprecht et al. [9] established a chromatic confocal response model with a finite detection pinhole, to explore the influence of detection pinhole and numerical aperture on the full width at half maximum of the confocal signal. Hillenbrand et al. [10] proposed three spectral response models: collinear, geometric, and wave optical models. It considers the impact of the illumination pinhole size on the system. Nizami et al. [11] presented a wave-optical model based on multimode fiber data acquisition. The fiber core is modelled as a point source grid. Incoherent summation of the intensities caused by all point sources is used as the confocal signal. Chen et al. [12] proposed a two-dimensional spectral signal model to describe the signal intensity-wavelength-displacement characteristics, in which the extended illumination source is assumed to be homogeneous. So far, the signal model with a finite illumination pinhole has been widely studied. However, the impact of intensity distribution within the illumination pinhole on the system has not been reported yet.
In order to improve the axial resolution, differential methods are applied in chromatic confocal technology. Wang et al. [13] proposed a virtual double-slit differential dark-field chromatic line confocal imaging technology. By subtracting the signals from two lateral shifted virtual slit, a narrower axial response curve can be obtained, thereby improving the axial resolution. Chen et al. [14] developed a broadband differential confocal method. Using double-slits with different sizes to collect signals, the displacement is measured according to the sum-subtraction ratio curve of the two signals. The above methods have a complex optical path structure and cannot meet the requirements for miniaturization of the probe. The combination of a chromatic confocal system and a phase-sensitive spectral optical coherence tomography is an effective way to improve axial resolution [15,16]. The depth-resolved phase information is decoded from the interference. The position decoded from the confocal signal can eliminate the phase ambiguity. However, the interference method is sensitive to the change in the measurement environment. In traditional confocal microscopy techniques, pupil filtering is used to realize axial or lateral super-resolution, which modulates the distribution of the light field on the pupil plane to modify the point spread function and compress the main lobe width of the confocal axial or lateral response characteristic curve. Based on their structural forms, pupil filters can be divided into three types: amplitude-only [17–21], phase-only [22–25], and complex amplitude types [26]. By comparison, there is a significant difference in applying pupil filters to the multimode fiber chromatic confocal system. The pupil plane contains fields radiated by multiple fiber modes. The axial or lateral intensity response is determined jointly by all modes. Phase-only and complex amplitude pupil filters cannot modulate the intensity distribution in the focus space. The reason is that different modes excited by low spatial coherence source are incoherent. For amplitude-only type, such as annular pupil, it can eliminate the modes with low axial resolution through mode selection, thereby achieving axial super-resolution. However, adding a pupil filter to the probe has the following problems: 1) The use of pupil filters may require adding a collimating optical path into the probe (structure as Ref. [27]) or etching the mask structure onto the objective lens (structure as Ref. [28]). It changes the original structure of the objective lens and increases the complexity of the design. 2) If the pupil is applied to a miniaturized fiber-integrated probe, the size of each annular zone of the mask must be significantly small, which increases the difficulty and cost of the manufacturing process.
At present, few studies have investigated the influence of source intensity distribution on the axial resolution of chromatic confocal systems. This is because the source can be made equivalent to a point through optical path optimization when the probe is not limited by volume. However, a fiber chromatic confocal system uses a multimode fiber to transmit light. The core size is relatively large, and the impact of source intensity distribution cannot be ignored. From this perspective, this study presents a fiber chromatic confocal method with a tilt-coupling source module for axial super-resolution. A confocal response model with a finite illumination-detection pinhole is established such that the intensity distribution in the illumination pinhole is described by a linearly polarized mode. This study investigates the reason higher-order modes with intensities concentrated at the edge of the fiber core possess a relatively high axial resolution. Accordingly, tilted coupling is adopted between the fiber and source module to increase the proportion of skew rays coupled to the fiber. The output port of the fiber produces an annular intensity distribution, thereby improving the axial resolution. Here, the probe is integrated by a gradient-index (GRIN) lens and fiber, therefore exhibiting a relatively small volume. The experiment characterizes the improvement of the proposed method on the sensing performance of this type of miniaturized probes.
2. Proposed method
A chromatic confocal method focuses light of different wavelengths at different positions using a dispersive objective lens. After being reflected by the sample, the light component that forms a focal point on the sample surface carries the highest intensity in the returned spectral signal owing to the filtering effect of the detection pinhole. The axial position of the sample is determined by measuring the peak wavelength. Therefore, the greater the difference in intensity between the peak and adjacent wavelengths in the spectral signal, the higher the axial resolution of the system.
To improve the axial resolution without changing the probe structure, a fiber chromatic confocal method with a tilt-coupling source module for axial super-resolution is proposed in this study, as shown in Fig. 1. In the source module, coaxial coupling is commonly used for ray transmission via the fiber. This ensures that the optical axis of the source module coincides with the central axis of the fiber. In contrast, this study adopts a tilt-coupling approach. An angle θ exists between the optical axis of the source module and the central axis of fiber port P1, that is, the coupling angle. By optimizing θ, the intensity distribution at fiber port P2 can be modulated to reduce the full width at half maximum (FWHM) of the confocal axial response characteristic curve for each wavelength. Furthermore, the intensity contrast between the peak and adjacent wavelengths in the spectral signal at fiber port P3 is enhanced, and the axial resolution is improved.
2.1 Confocal response characteristics of each mode in the step-index multimode fiber
To analyze the effect of intensity distribution in the source plane (i.e., the plane with the illumination pinhole) on the axial resolution of the fiber chromatic confocal system, this section explores the confocal axial response characteristic corresponding to each fiber mode.
In low-coherence systems, there can be no interference between different modes. The intensity distribution within fiber port is obtained by intensity distribution superposition of all modes. The confocal axial response of the system is the sum of confocal axial responses of all modes. Due to the different transmission paths of fields radiated by each mode, the sensitivity of each mode to axial displacement is different. The confocal axial response characteristics of each mode in the step-index multimode fiber is discussed next. A confocal response model with a finite illumination-detection pinhole is established. Assuming a weakly guiding fiber, the field distribution of the step-index multimode fiber mode is described by the theory of linearly polarized modes. By comparing the confocal axial response characteristics, the modes that enable the system to achieve a high axial resolution are selected.
Owing to the broadband and low-coherence illumination, the confocal signal should be described according to the incoherent imaging theory. The monochromatic ray transmission process of the fiber probe is shown in Fig. 2. The confocal response model has three aspects. First, the illumination intensity is determined at each sample point as the convolution of the intensity distribution of the source plane and the point spread function of illumination arm. Second, the intensity response in the pinhole caused by the reflected light of a single sample point is determined by the convolution of the point spread function of the detection arm and the pinhole area. Third, the influence of all points in the illuminating area of the sample surface on the intensity received from the pinhole is considered. The confocal intensity response is modeled by the convolution of the confocal intensity caused by a single sample point and the reflectivity coefficient distribution of the sample surface. The details are expressed in Eq. (1).
Fig. 2. Diagram of the equivalent optical path in the fiber probe. The illumination arm is the optical path from the illumination space to the object space. The detection arm is the optical path from the object space to the detection space.
The intensity distribution of the source plane is determined from the field distribution of each mode in the multimode fiber. Assuming that the system uses a weakly guiding fiber, the field distribution E of each mode can be expressed by the linearly polarized mode LPl,m, as expressed in Eq. (2).
According to Eq. (1), the confocal axial response characteristics of each mode under illumination wavelengths of 450, 550, and 650 nm are analyzed. In the simulation, the core diameter of the step-index multimode fiber is 50 µm, and the numerical aperture is 0.22. Figures 4(a)–(c) compare the confocal axial response characteristic curves of the fundamental and high-order-l LPl,m modes. As l increases, the confocal axial response characteristic curve narrows gradually. Figures 4(e)–(f) compare the confocal axial response characteristic curves of the fundamental and m-increased LP0,m modes. In contrast, the width of the confocal axial response characteristic curve corresponding to the m-increased LP0,m mode is similar to that of the fundamental mode.
Fig. 4. Confocal axial response characteristic curves corresponding to each mode under different illumination wavelengths: (a)/(d) 450 nm; (b)/(e) 550 nm; and (c)/(f) 650 nm.
To explain more intuitively the above phenomenon, the intensity distribution on the detection plane is analyzed next. Considering the illumination wavelength of 550 nm as an example, the analyzed modes include LP0,1, LP0,17, and LP16,1. In Fig. 5, the area inside the red dotted circle represents the fiber core (i.e., the pinhole area). As the defocus value u increases, the spread speed of the fundamental mode LP0,1 decreases. Most of the intensity is always concentrated inside the pinhole, resulting in small attenuation. LP0,17 has a high spread speed. However, because the intensity is mainly concentrated in the center and the pinhole is much larger than the intensity distribution range of LP0,17, within a certain defocus range, most of the intensity remains within the pinhole, even if the intensity distribution of LP0,17 expands. The intensity attenuation caused by the pinhole is smaller. This results in a slower decrease in the confocal axial response characteristic curve during defocusing. The corresponding confocal axial response is similar to that of the fundamental mode. In contrast, the intensity of the LP16,1 mode is concentrated at the pinhole edge. Moreover, the spread speed is high. The detected intensity attenuation is relatively large during defocusing. Therefore, the confocal axial response characteristic curve is relatively narrow. In short, during ray transmission, the spatial filtering effect of the pinhole on the beam of the LP16,1 mode is better than that of the LP0,1 and LP0,17 modes, resulting in a narrower FWHM of the confocal axial response characteristic curve.
Based on the previous analysis, the intensity distribution of the LP0,m mode presents the characteristics of a strong center and weak edge. As m increases, it gradually shrinks towards the center of the fiber core. The pinhole has a weak spatial filtering effect. Therefore, the FWHM of the confocal axial response characteristic curve corresponding to the LP0,m mode is relatively wide. In contrast, with increasing l, the intensity distribution of the LPl,m mode gradually approaches the edge of the fiber core, and the corresponding annular intensity profile narrows gradually. Consequently, the spatial filtering effect of the pinhole on the beam of the high-order-l LPl,m mode is more significant. The FWHM of the confocal axial response characteristic curve corresponding to each wavelength is narrower. The intensity contrast between the peak and other wavelengths in the spectral signal is enhanced. An axial super-resolution is achieved. Therefore, the axial resolution of the fiber chromatic confocal system can be effectively improved by increasing the ratio of the high-order-l LPl,m mode intensity to the total intensity and obtaining an annular source with its intensity concentrated on the edge of the fiber core.
2.2 Modulation of intensity distribution on the fiber port and optimization of tilt-coupling angle
In this section, the light transmission characteristics of each mode in the fiber are analyzed. Accordingly, tilted coupling between the fiber and source module is adopted to increase the ratio of the high-order-l LPl,m mode intensity to the total intensity. An annular intensity distribution is generated at the fiber port. The tilt-coupling angle θ is optimized experimentally to narrow the annular intensity distribution with a certain intensity loss.
In step-index multimode fibers, rays travel along two types of paths: meridional and skew rays. The meridional ray is located in the meridional plane. The propagation trajectory intersects the central axis of the fiber, and its projection on the cross-section of the core is a line, as shown in Fig. 6(a). The skew ray propagates in the form of a spiral line, which is neither parallel to nor intersecting the central axis. The projection of the trajectory on the core cross-section is located in an annular area. Moreover, ψ is the angle between the projection and tangent at the reflection point. The smaller the ψ, the more inclined the skew ray is relative to the meridional plane. The condition that rays travel along the skew path, as shown in Fig. 6(b), requires the ray projection AS1 on the fiber port to be inclined to the radial direction O1B (the angle is φ) and the ray incident position to be off-center. Figure 6(c) depicts the relationship between the ray incident condition and the transmission path; Eq. (4) expresses the relationship between the ray incident position r, φ, and ψ. When r→a and φ→π/2, ψ→0. Accordingly, when the ray incident position is away from the center and its projection approaches perpendicularity to the radial direction, the corresponding skew ray is more inclined.
where a is the core radius.Fig. 6. Diagram of the ray transmission path in the fiber. (a) Projections of meridional and skew rays on the cross-section of the fiber; (b) transmission of skew ray; and (c) relationship between the ray incident condition and the transmission path. In (b), the ray enters the fiber at a position S1, which is at a distance r from the center O1. β is the incident angle. φ is the angle between the radial direction O1B and the projection AS1 of the ray on the incident plane. γ is the angle between the radial direction O2S2 and the projection CS2 of the ray on the cross-section where the reflection point S2 is located. In (c), the ray incident conditions include the incident position r and the angle φ between the ray projection and the radial direction. The transmission path is represented by the angle ψ between the ray projection and the tangent.
The-ray transmission path corresponding to each LPl,m mode is analyzed next. The linearly polarized mode LPl,m is a combination of vector modes, which include the transverse electric mode TE0,m, transverse magnetic mode TM0,m, and hybrid modes HEl,m and EHl,m. Specifically, LP0,m is HE1,m. LP1,m comprises TE0,m, TM0,m, and HE2,m. Moreover, LPl,m (l > 1) comprises HEl + 1,m and EHl-1,m. Based on the discussion on mode parameters and ray invariants in [29], for TE0,m, TM0,m, HEl,m, and EHl,m, the order l, mode parameter Ul,m, and ψ satisfy Eq. (5).
Note that for TE0,m and TM0,m with l = 0, the angle ψ is π/2, which implies that the ray is always located in the meridional plane during propagation. Therefore, TE0,m and TM0,m correspond to the meridional rays. For HEl,m and EHl,m, the rays are transmitted along the skew paths. Since Ul,m varies with l, the influence of the increase in l on ψ cannot be directly judged through Eq. (5). Next, the variation in l/Ul,m with respect to l must be determined. Equations (6) and (7) are the eigenvalue equations of HEl,m and EHl,m, respectively. The corresponding Ul,m can be calculated based on the fiber structure.
Combining Eqs. (6) and (7), the variation in l/Ul,m with l corresponds to HEl,m and EHl,m, as shown in Fig. 7. As l increases, l/Ul,m increases. This indicates that ψ decreases with increasing l and the skew rays corresponding to the high-order-l HEl,m and EHl,m are more inclined. According to the relationship between the LPl,m and vector modes, the ray transmission paths of the low-order-l LPl,m modes inside the fiber are dominated by meridional and skew rays with a small inclination. As l increases, the skew rays of the LPl,m modes become more inclined. Therefore, if the ratio of the high-order-l LPl,m mode intensity to the total intensity in the fiber requires to be increased, the ratio of skew rays with small ψ (i.e., large inclination) to all rays must be increased. Accordingly, based on Eq. (4), the rays should primarily be incident from the edge of the core at the incident port. Furthermore, the ray projection should be close to perpendicular to the radial direction.
Fig. 7. Variation in l/Ul,m with respect to l and m. (a) HEl,m and (b) EHl,m. The structure parameters of the fiber are as follows: the core diameter is 100 µm, and the numerical aperture is 0.22.
To achieve this goal, this study adopts a tilt-coupling method to increase the ratio of skew rays with small ψ to all rays by adjusting the angle between the optical axis of the source module and the center axis of the fiber. According to the total reflection condition, the equivalent numerical aperture of the skew ray in Fig. 6(b) is expressed as follows [30]:
where NA is the numerical aperture of the meridional ray.The NAs of the skew ray is larger than that of the meridional ray. Compared with the meridional ray, the incidence angle of the skew ray is larger, and the ability of the fiber to collect the skew ray is stronger. Furthermore, the NAs rises with increasing r and φ. This indicates that the incident angle β of the skew ray gradually increases from the center to the edge of the core, as shown in Fig. 6(b). At the same incident position, the ray incident angle is maximum when φ equals π/2 (i.e., the ray projection is perpendicular to the radial direction). Accordingly, the range of incident rays that can be received by the fiber on the incident plane is as shown in Fig. 8(a). At the center and off-center positions, the range is depicted as a cone and elliptical cone, respectively. The larger the deviation from the center, the larger the volume of the elliptical cone.
Fig. 8. Diagram of the tilt-coupling principle. (a) Range of incident rays that can be received by the fiber; (b) tilted coupling at the center position O1 and off-center position S1. The green cone represents the range in which the fiber can receive the rays. The red cone represents the range of incident rays from the source. θ is the tilt-coupling angle.
The tilt-coupling principle is shown in Fig. 8(b). After the optical axis of the source module is inclined towards the center axis of the fiber, fewer incident rays enter the fiber near the center position O1. This causes fewer rays to travel along meridional paths and skew paths with large ψ. In contrast, at the off-center position S1, the range of rays that can be received by the fiber is an elliptical cone. The incident angle β is relatively large when the ray projection is nearly perpendicular to the radial direction (i.e., the x-axis), which causes most of the rays to enter the fiber along this path after tilted coupling. The angle between the projection of the received rays on the cross-section and the radial direction (i.e., the positive direction of the y-axis) is in the range from φ to π-φ. As θ increases, φ approaches π/2. Based on these insights and Eq. (4), the rays are mainly transmitted along the skew paths with small ψ after tilted coupling. With increasing θ, the ratio of skew rays with large inclination relative to the meridional plane to all rays increases gradually. Consequently, the ratio of the high-order-l LPl,m mode intensity to the total intensity increases. The annular intensity distribution narrows gradually at the fiber output port. However, an increase in θ inevitably increases the loss of incident rays. To elaborate, more incident rays deviate from the acceptable range of the fiber, reducing the optical coupling efficiency.
Therefore, the tilt-coupling angle θ is an important parameter in fiber chromatic confocal systems. On the one hand, it determines the intensity distribution at the fiber output port, which affects the FWHM of the confocal axial response characteristic curve at each wavelength. On the other hand, it determines the optical coupling efficiency, which, in turn, affects the signal-to-noise ratio of the system. To achieve axial super-resolution, the value of θ must be optimized such that the minimum energy is lost while narrowing the width of the annular intensity distribution.
3. Experiment and discussion
3.1 Optimization of coupling angle θ
The coupling angle θ is optimized through the tilt-coupling experiment. Figure 9 shows diagrams of the optical path and experimental setup.
Fig. 9. Tilt-coupling structure. (a) Diagram of optical path; (b) experimental setup. CL: condenser lens; L1: lens; TS: travel stage; RS: rotation stage; P1/P2: fiber port; and O1/O2: objective.
The source module consists of LED (MNWHL4, Thorlabs Inc.), condenser lens CL (ACL25416U-A, Thorlabs Inc.), and lens L1 (LB1471, Thorlabs Inc.). The coupling angle θ between the fiber port P1 and source module is adjusted by rotating the rotation stage RS. The fiber (M43L02, Thorlabs Inc.) has a core diameter of 105 µm and numerical aperture of 0.22. The fiber output port P2 is connected to an imaging module comprising objective lenses O1 (RMS10×, Olympus Co.) and O2 (RMS4×, Olympus Co.) as well as CCD (CMLN-13S2M, Point Grey Inc.) to observe the intensity profile at the fiber output port P2.
Figure 10(a) shows the intensity distribution at P2 for several values of θ. Note that as θ increases, the intensity distribution gradually shifts from a flat top profile to an annular profile. To show more clearly the influence of θ on the intensity distribution, the intensity distribution at a cross-section of P2 (i.e., the position of the red dotted line in Fig. 10(a)) is extracted for analysis. Figure 10(b) shows the intensity distribution of the central cross-section corresponding to several θ values. The central depressed region of the intensity curve gradually extends with increasing θ. To facilitate the following discussion on the influence of θ on the annular intensity distribution, the proportionality parameter ε is defined as the ratio of the diameter of the central dark-field area to the core diameter of the fiber. The central dark-field area is defined as the area with intensity less than 80% of the maximum intensity. Primarily, ε is used to evaluate the width of the annular intensity distribution.
Fig. 10. Intensity distribution. (a) Intensity distribution of the fiber output port; (b) intensity distribution at the central cross-section of the core.
Figure 11 records the variations in the proportionality parameter ε and the fiber coupling power with respect to θ. The fiber coupling power is measured at P2 using an optical power meter (Thorlabs, Inc. S130C). The results demonstrate that with increasing θ, ε increases gradually and tends to be constant when θ > 21°. This indicates that the annular intensity distribution at P2 is nearly the narrowest after the tilt-coupling angle reaches 21°. Furthermore, increasing the coupling angle reduces the fiber coupling intensity. Therefore, based on the aforementioned two factors, θ should be 21°.
3.2 Performance experiment of the fiber chromatic confocal system
This section discusses the confocal axial response characteristics of each wavelength, measurement repeatability, and axial resolution of the fiber chromatic confocal system under tilt- and coaxial-coupling states.
As shown in Fig. 12, the experimental setup comprises four parts: source, splitting, objective, and spectrometer modules. The source module includes LED (MNWHL4, Thorlabs Inc.), condenser lens CL (ACL25416U-A, Thorlabs Inc.), and biconvex lens L1 (LB1471, Thorlabs Inc.). To realize the tilt-coupling state, the fiber port P1 is fixed on the rotation stage RS. By rotating P1, the coupling angle θ between the fiber and source module is adjusted. The splitting module is a 1 × 2 fiber coupler FC with a core diameter of 105 µm and numerical aperture of 0.22. The objective module includes the fiber port P2 and a GRIN lens with a pitch of 0.76. The distance between P2 and the GRIN lens is 1 mm. In the wavelength range of 580–620 nm, the dispersion amount is 325 µm. The spectrometer module comprises objective lenses O1 (RMS20×, Olympus Co.), O2 (RMS20×, Olympus Co.), and O3 (RMS10×, Olympus Co.); slit S (VA100/M, Thorlabs Inc.); diffraction grating DG (GR25-1205, Thorlabs Inc.); achromatic lens L2 (AC254-050-A-ML, Thorlabs Inc.); and CCD (DCU224M, Thorlabs Inc.). The measured surface is a mirror M fixed on an electric travel stage (LNR502/M, Thorlabs Inc.).
Fig. 12. (a) Schematic diagram of the optical path; (b) experimental setup. CL: condenser lens; L1/L2: lens; TS: travel stage; RS: rotation stage; P1/P2/P3: fiber port; FC: fiber coupler; CO: chromatic objective; M: mirror; O1/O2/O3: objective; S: slit; and DG: diffraction grating.
First, the confocal axial response is obtained at each wavelength. The mirror M moves from the axial distance z (the distance between the measured position and the GRIN lens back surface) of 1.5 to 2.5 mm in steps of 1 µm. The intensities of CCD pixels are recorded at each axial position. The results are presented in Fig. 14. The abscissa of the figure represents the horizontal pixel position N of CCD. Each pixel corresponds to a particular wavelength. The ordinate is the axial distance z. The intensity distribution is normalized. Figures 13(a) and (b) show the measurement results under the coaxial coupling and 21° tilt-coupling states, respectively. In contrast, the confocal axial response corresponding to each pixel in the 21° tilt-coupling state is significantly narrower.
Fig. 13. Confocal axial response corresponding to each wavelength: (a) the coaxial coupling state; (b) the 21° tilt-coupling state.
Fig. 14. Relationship between peak position and displacement. (a) Coaxial coupling state; (b) 21° tilt-coupling state.
By using Gaussian fitting method to extract the peak positions from the collected spectral signal, the relationship between peak position and displacement is obtained as shown in Fig. 14. The linear fitting on the data under the two coupling states is performed. The correlation coefficients R are all close to 1, which means an approximate linear change. The root mean square error (RMSE) is 9.525 pixels under the 21° tilt-coupling state. It is relatively small. The tilt-coupling state enhances the anti-disturbance ability.
To describe more clearly the influence of the tilt-coupling state on the confocal axial response characteristic curve, Fig. 15 compares the FWHM of these curves at the wavelengths corresponding to the pixel positions N = 1, 500, and 900 of CCD. The explored coupling angles θ are 0°, 17°, 19°, and 21°. Notably, the FWHM decreases gradually with increasing θ. Specifically, compared to the coaxial coupling state with θ = 0°, the tilt-coupling state with θ = 21° can reduce the FWHM by 18.1%, 17.9%, and 17.4%, corresponding to N = 1, 500, and 900, respectively.
Fig. 15. Monochromatic confocal axial response characteristic curve. Horizontal pixel position in CCD: (a) N = 1; (b) N = 500; and (c) N = 900.
The aforementioned phenomenon confirms that the tilt-coupling between the fiber and source module can compress the FWHM of the confocal axial response characteristic curve at each wavelength. Based on the variation in the intensity distribution on the fiber port with θ in Fig. 10, the annular source is conducive for reducing the FWHM of the confocal axial response characteristic curve.
Next, the repeatability of measurements is evaluated and compared. The same axial position is measured 50 times, and the spectral signal curves are obtained. Figures 16(a) and (b) show the resulting peak positions of the spectral signals measured repeatedly under the coaxial coupling and 21° tilt-coupling states; their standard deviations are 16 and 3 pixels, respectively. The latter deviation is 19% of the former. The measured values in the 21° tilt-coupling state fluctuate limitedly. This proves that the tilt-coupling state reduces the intensity disturbance caused by adjacent wavelengths on the focus wavelength in the spectral signal.
Finally, the axial resolutions of the fiber chromatic confocal system in the coaxial coupling and tilt-coupling states are compared. Starting at a distance of 2 mm from the back surface of the GRIN lens, the mirror M is moved using the electric travel stage in different steps. The spectral signal at each axial position is measured 30 times. Figure 17 records the peak position results of the spectral signal when moving in steps of 3, 5, and 7 µm. Under the coaxial coupling state, the system can clearly distinguish the signal changes caused by movement with a step greater than 5 µm. In contrast, under the 21° tilt-coupling state, the system can clearly resolve the axial step of 3 µm. Its axial resolution is 1.7 times that of the coaxial coupling state.
Fig. 17. Results of axial resolution test. Coaxial coupling state: (a) 3 µm, (b) 5 µm, and (c) 7 µm steps; 21° tilt-coupling state: (d) 3 µm, (e) 5 µm, and (f) 7 µm steps.
The conducted experiments verify the effectiveness of the tilt-coupling state in improving the axial resolution of the system. Compared with the coaxial coupling state, the results show that the developed 21° tilt-coupling state can reduce the FWHM of the confocal axial response characteristic curve by approximately 17%; the axial resolution increases by 1.7 times. The repeatability standard deviation of spectral peak position result is approximately 19% that of the coaxial coupling state. In summary, by adjusting the tilt-coupling angle θ between the optical axis of the source module and the central axis of the fiber, an annular intensity distribution can be generated on the fiber output port. Furthermore, the FWHM of the confocal axial response characteristic curve at each wavelength is reduced. The measurement repeatability and axial resolution of the system are also improved. Simultaneously, the intensity disturbance caused by the adjacent wavelengths on the focus wavelength is suppressed.
4. Conclusion
In this study, a fiber chromatic confocal method with a tilt-coupling source module is proposed for axial super-resolution. This is the first study to improve the axial resolution of fiber chromatic confocal sensors by adjusting the intensity distribution on the source plane. A confocal axial response model with a finite illumination-detection pinhole is established, in which the intensity distribution of the source is described by a linearly polarized mode. By analyzing the influence of each mode on the confocal axial response, the high-order-l LPl,m modes evidently correspond to a higher axial resolution of the system. Furthermore, by adjusting the tilt-coupling angle between the source and fiber, the ratio of the high-order-l LPl,m mode intensity to the total intensity increases. The fiber port in the probe produces an annular intensity distribution, which compresses the FWHM of the confocal axial response characteristic curve and increases the intensity difference between the focal and adjacent wavelengths in the spectral signal. Consequently, an axial super-resolution system is achieved. The coaxial coupling state and the developed 21° tilt-coupling states are explored experimentally. The results confirm that the tilt-coupling state can reduce the FWHM of the confocal axial response characteristic curve at each wavelength. The intensity disturbance caused by the adjacent wavelengths on the focus wavelength is suppressed. Furthermore, the measurement repeatability and axial resolution of the system are improved. This method does not require a change in the structure of the chromatic probes, and the implementation is simple. The proposed approach helps improving the measurement precision of displacement and clearance for situations with limited measurement space such as the parameters measurement of a small hole with high aspect ratio and the online quality monitoring of work pieces. Currently, this work has the limitation of low coupling efficiency of the fiber. Further research will focus on increasing the coupling energy of the fiber to make the system suitable for low-reflectivity sample surfaces.
Funding
National Natural Science Foundation of China (52275526).
Acknowledgements
The authors appreciate the help provided by the National Natural Science Foundation of China.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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