Influence of Surface Tilt Angle on a Chromatic Confocal Probe with a Femtosecond Laser

Abstract

This paper presents an intentional investigation of the effect of the object tilt angle on the tracking local minimum method (TL method), which is the one for detecting the measurement target position of the object optical axis, in a chromatic confocal probe employing a differential dual-fiber-detector optical system with a mode-locked femtosecond laser as the light source. The effect of the object tilt angle on dual-detector confocal probes, and even chromatic confocal probes, has not been investigated in detail so far, although the effect of object tilt angle on scanning confocal probes has been studied. At first, to examine the influence of the object tilt angle on the TL method, a theoretical model is established, and numerical simulations are performed based on the established theoretical equation. Then, the effect of aberrations in confocal optics on the confocal response curve is investigated in experiments. Finally, investigations on the effect of the object tilt angle on the TL method are demonstrated in experiments.

1. Introduction

For semiconductor devices, MEMS (micro electro mechanical systems) and other precision components with microgeometry, measurement, and evaluation of three-dimensional (3D) surface profiles [1,2] are necessary to maintain their quality. For this purpose, measuring instruments are required to have a resolution and accuracy in the order of sub-μm over a measurement range of several hundred μm [1,2,3]. Confocal probes [4,5,6,7,8,9,10] are non-contact 3D surface profiling instruments that satisfy these requirements. Confocal probes are used in many fields since they are capable of carrying out non-contact and high-resolution 3D shape measurement in the atmosphere. In recent years, chromatic confocal probes using a light source with a spectral width have been studied from the viewpoint of measurement efficiency and measurement range [11,12,13,14,15,16]. Furthermore, a dual-detector confocal optical system has been proposed to achieve higher resolution and a wider measurement range for measuring the optical axis position of the object to be measured [17,18,19,20,21]. Our research group has also developed a dual-detector chromatic confocal probe using a femtosecond laser as a light source [22,23,24,25,26,27]. At first, a new z-position detection method, which is referred to as the tracking local minimum method (TL method) [22], has been proposed. So far, the tracking local minimum method (TL method) [22], which uses the minimum point of the sidelobe of the confocal response curve, has been used to achieve an optical axis position measurement resolution of 30 nm in a measurement range of 40 μm, and then a measurement range of 250 μm [23,27] has been achieved by applying the principle of the sidelobe of the confocal response curve. Here, these achievements in the references [23,27] have focused on the measurement range expansion of z-position based on the TL method by extending the sidelobes of the confocal response curves. After that, as another new z-position measurement method, the tracking intersection method (TI method) [26], which detects the z-position from the intersection of confocal response curves obtained with two detectors, has also been proposed and demonstrated by using the developed dual-detector chromatic confocal probe system. It should be noted that the difference is that the TL method identifies the z-position from the local minima of the confocal responses, while the TI method identifies the z-position from the intersection of the confocal responses. Both the TL method and TI method, however, enable highly accurate detection of the z-position of the measurement object without being affected by the shape of the light source spectrum. Furthermore, several investigations have been carried out to make the confocal optical system robust against temperature variation, and several works have also been carried out to improve the measurement accuracy of the proposed dual-detector chromatic confocal probe by examining the amount of detector offset [24,25].

On the other hand, the influence of the tilt angle of the measurement object on the proposed TL method has not been investigated so far. It should be noted that the influence of the tilt angle of the measurement object on the scanning confocal response curve, from which the optical axis position of the measurement object can be detected, has been investigated by several researchers [28,29,30,31,32,33,34,35,36,37]; for example, it has been found that the presence of a tilt angle of the measurement object changes the shape of the confocal response curve, thereby affecting the detection of the measurement object’s optical axis position [28,29,30,31]. However, these investigations have been conducted on scanning confocal probes with monochromatic light sources, and no investigations have been carried out on chromatic confocal probes. Furthermore, in view of the scanning confocal probe’s principle of detecting the z-position of the object to be measured, the focus of most of these studies has been on the effect of the confocal response on peak detection. Here, our confocal probe is not a scanning confocal probe but a chromatic confocal probe. Furthermore, the TL method decodes the z-position of the measurement object not from the peak wavelength of the confocal response curve but from the wavelength of the sidelobe minima of the confocal response curve.

In this study, theoretical and experimental investigations on the influence of the object tilt angle on the TL method are demonstrated. At first, a general numerical model for the chromatic confocal probe is constructed, and theoretical equations for numerical calculations are discussed. Based on the theoretical equation, numerical calculations are then performed and compared with the results of scanning confocal probes in previous studies to discuss the effect of the object tilt angle on the TL method. Finally, experiments are conducted to verify the theoretical calculations and to discuss the effect of the object tilt angle on the TL method. Here, our previous studies have mainly focused on proposing a z-position detection method [22,26], extending the z-position measurement range [23,27], and optimizing the optical system [24,25]. On the other hand, this report differs from our previous studies in that it aims to report the results of an investigation into the effect of the tilt angle of the measurement object on the proposed TL method.

2. Theoretical Equations for the Tracking Local Minimum Method of Mode-Locked Femtosecond Laser Dual-Fiber Differential Detector-Type Chromatic Confocal Probe

2.1. Principle of the Tracking Local Minimum Method

Figure 1 shows a schematic of the z-position detecting algorithm based on the TL method [22,26]. At first, in step 1, the measurement light spectrum I_Mea and the reference light spectrum I_Ref are acquired. Here, the measurement light spectrum I_Mea is acquired without detector optical axis offset, while the reference light spectrum I_Ref is acquired with detector optical axis offset z_d. Therefore, the peak wavelengths of the I_Mea and I_Ref differ by Δλ_d. It should be mentioned that the optical spectra I_Mea and I_Ref are obtained by including the light source spectral shape, the reflectance of the object being measured, and transmittance of the optical element in the measurement light confocal response curve I_M or the reference light confocal response curve I_R [22,23,24,25]. Then, in step 2, the normalized intensity ratio I_TLM is calculated by dividing the obtained measurement light spectrum I_Mea by the reference light spectrum I_Ref. The obtained I_TLM is not affected by the dispersion of the measurement object or optical elements, as well as the light source spectral shape [22,24]. The peak of the I_TLM matches the local minimum point of the reference light spectrum I_Ref. As with the peak wavelength of the confocal spectrum, the wavelength of the minimum point of the reference light spectrum I_R changes with the change in the z-position of the measurement object. This means that it is possible to determine the z-position by identifying the peak wavelength of the normalized spectrum I_TLM. Finally, in step 3, the peak wavelength of the normalized intensity ratio I_TLM, which is referred to as the focal wavelength λ_focused, is calculated based on the center of gravity method [24,26]. The conversion from the focused wavelength λ_focused to the z-position is performed using the z-λ sensitivity (dz/dλ). The z-λ sensitivity (dz/dλ) is required to be calibrated along with the measurement object. In this study, the influence of the object tilt angle on the z-λ sensitivity (dz/dλ) is investigated.

2.2. Modeling for Theoretical Equations

The following is the process of deriving the theoretical equation for the numerical calculations. Figure 2 presents the fiber confocal probe [6]. The light intensity obtained by the fiber-detector of the fiber confocal probe is given by the following equation [4,5,6]:

$$I({\mathit{r}}_s)={\left|U\right|}^2={\left|{\displaystyle {\int }_{-\infty }^{\infty }c(\mathit{m})O(\mathit{m})\exp (2\pi i{\mathit{r}}_s\cdot \mathit{m})d\mathit{m}}\right|}^2$$

(1)

where r_s is the confocal probe scanning position vector (x_s, y_s, z_s) in Figure 2, m is the spatial frequency vector (m, n, s), and c(m) is the optical transfer function. Here, O(m) is obtained by the Fourier transform of the 3D object function o(r) [5,6]. When the object to be measured is an ideal flat mirror with angles θ_X and θ_Y, o(r) is given by the following equation:

$$o(\mathit{r})=\exp \left(j\frac{2\pi }{\lambda }\cdot 2\tan {\theta }_{Y}x\right)\cdot \exp \left(j\frac{2\pi }{\lambda }\cdot 2\tan {\theta }_{X}y\right)\cdot \delta \left(z\right)$$

(2)

where δ is the delta function. By using Equation (1), Equation (2) is transformed into the following equation.

$$\begin{array}{cc}\hfill I({\mathit{r}}_s)& ={\left|{\displaystyle {\int }_{-\infty }^{\infty }c(\mathit{m})O(\mathit{m})\exp (2\pi i{\mathit{r}}_s\cdot \mathit{m})d\mathit{m}}\right|}^2\hfill \\ & ={\left|{\displaystyle {\int }_{-\infty }^{\infty }c\left(\mathit{m}\right)\cdot {\displaystyle {\int }_{-\infty }^{\infty }o(\mathit{r})\exp (-2\pi i\mathit{r}\cdot }\mathit{m})d\mathit{r}\cdot \exp (2\pi i{\mathit{r}}_s\cdot \mathit{m})d\mathit{m}}\right|}^2\hfill \\ & ={\left|{\displaystyle {\int }_{-\infty }^{\infty }c\left(\mathit{m}\right)\delta \left(m-\frac{2\tan {\theta }_Y}{\lambda }\right)\delta \left(n-\frac{2\tan {\theta }_X}{\lambda }\right)\exp (2\pi i{\mathit{r}}_s\cdot \mathit{m})d\mathit{m}}\right|}^2\hfill \\ & ={\left|\exp \left(2\pi i\left(x_s{m}_0+y_s{n}_0\right)\right)\cdot {\displaystyle {\int }_{-\infty }^{\infty }c\left(m_0,n_0,s\right)\exp \left(2\pi i{z}_{s}s\right)ds}\right|}^2\hfill \\ & ={\left|c\left(l_0,u\right)\right|}^2\hfill \end{array}$$

(3)

Here, m₀ = 2tanθ_Y/λ, n₀ = 2tanθ_X/λ, l₀ = (m₀² + n₀²)^1/2, and u is the normalized measurement object z-position, which is expressed by the following equation [5,6]:

$$u(\lambda ,z_s)=\frac{2\pi }{\lambda }(-z_s-f_{\lambda }+f_{{\lambda }_{\mathrm{center}}})\frac{a^2}{f_{\lambda }^2}$$

(4)

where a is the pupil radius, and f_λ is the focal length of the objective lens for the measurement object in each wavelength λ. Each parameter is indicated in Figure 3. Equation (4) means that the origin of the z-position is defined as the focal position of the central wavelength (λ_center). In the case of the fiber confocal probe used in this study, c(l₀, u) is given by the following equation [6,38]:

$$c(l_0,u)=K_r\exp \left(-is0u\right)\exp \left[-\left(A-\frac{iu{l}_0^2}{4}\right)\right]{\displaystyle {\int }_0^{\pi /2}{\displaystyle {\int }_0^{{\rho }_0}\exp \left[-\left(A-iu{{\rho }^{\prime }}^2\right)\right]}{\rho }^{\prime }d{\rho }^{\prime }d{\theta }^{\prime }}$$

(5)

where

$${\rho }_0=-\frac{l_0\left|\cos {\theta }^{\prime }\right|}{2}+\sqrt{1-\frac{l_0^2{\sin }^2{\theta }^{\prime }}{4}}$$

(6)

Here, K_r is a constant factor, s₀ is the spatial frequency offset, and A is the normalized fiber spot size, each given by the following equations [6].

$$s0=\frac{1}{2{\sin }^2\left({\alpha }_0/2\right)}$$

(7)

$$A={\left(\frac{2\pi a{r}_0}{\lambda F}\right)}^2$$

(8)

α₀ in Equation (7) is the numerical aperture of the objective lens for the measurement object, and r₀ in Equation (8) is the fiber spot size, given by the following equation.

$$r_0=\frac{{\rho }_c}{\sqrt{2\ln V_f}}$$

(9)

In Equation (9), ρ_c is the fiber core radius, and V_f is the fiber parameter, given by the following equation using the fiber aperture NA_fiber [6,22].

$$V_f=\frac{2\pi }{\lambda }{\rho }_{c}N{A}_{\mathrm{fiber}}$$

(10)

Using the above equations, it is possible to numerically calculate the effect of the object tilt angle on the TL method. It should be noted that the above equations are theoretical equations for an ideal confocal optical system without the influence of aberrations Φ other than axial chromatic aberration [35,36,37,39,40,41] of the optical system since it is difficult to accurately predict the overall aberrations Φ of the optical system [39]. Therefore, in this paper, numerical calculations are performed without considering the aberrations Φ, and then, the effect of the object tilt angle on the TL method is estimated. Finally, the prediction of experimental results in a confocal optical system including the aberrations Φ is performed to show the validity of the experimental results.

3. Numerical Calculation Results for the Tracking Local Minimum Method

Based on the equations derived in the previous chapter, numerical calculations of the measurement light confocal response curve I_M and the reference light confocal response curve I_R based on our proposed TL method were conducted by using the following Equations (11)–(13):

$$I_{\mathrm{M}}\left(z,\lambda ,{\theta }_X\right)={\left|c\left(l_0,u\right)\right|}^2$$

(11)

$$I_{\mathrm{R}}\left(z,\lambda ,{\theta }_X,z_d\right)={\left|c\left(l_0,u+u_d/2\right)\right|}^2$$

(12)

$$u_d(z,\lambda ,z_d)=-\frac{2\pi }{\lambda }{z}_d\frac{a^2}{F^2}$$

(13)

In Equation (13), z_d is the offset position on the detector optical axis, which was set to 200 μm based on previous experimental results [24], u_d is the normalized detector offset position z_d, and F is the focal length of the objective lens for the fiber-detector. Here, values for lens focal length and fiber were taken from the previous study [22]. In addition, considering the circular symmetry of the optical system, only θ_X was made to change in the calculations.

Numerical calculations were performed using MATLAB R2021a for each light source wavelength λ and z-position. The light source spectral intensity was assumed to be constant regardless of wavelength. Furthermore, the objective lens was assumed to be an optical system aligned under ideal conditions, possessing no aberrations other than an axial chromatic aberration. Due to these assumptions, the effects of the light source spectral shape and the aberrations Φ of the optical system on the confocal response cannot be obtained from the numerical results. It should be noted that these numerical simulations were performed to confirm the effect of the object tilt angle θ_r (=θ_X) on the response curve of an ideal fiber-detector confocal optical system. Figure 4 shows the results of the calculations for the light source whose spectrum is ranging from 1.48 μm to 1.64 μm at θ_X = 0 degree. Figure 4a shows the measurement light confocal response curve I_M and the reference light spectrum I_Ref, and Figure 4b shows the normalized intensity ratio I_TLM calculated by the TL method from the I_M and I_R. It should be noted that the intensities of I_M and I_R, and I_TLM were normalized by the maximum intensity for each wavelength. In addition, Figure 4c,d show cross-sections at z = 16 μm in Figure 4a,b, and at the center wavelength λ = 1.56 μm in Figure 4a,b, respectively. It should also be noted that Figure 4c shows the confocal response curve when the data are observed as a chromatic confocal probe, while Figure 4d shows the confocal response curve when the data are observed as a scanning confocal probe. From Figure 4c,d, it can be confirmed that the minimum point of the reference light confocal response I_R appears as a peak in the normalized intensity ratio I_TLM by the division operation, as shown in Figure 1. Here, for the sake of simplicity, the effect of the object tilt angle on the TL method was discussed with the wavelength fixed as shown in Figure 4d, and then the discussion based on the fixed wavelength extended the considerations to a chromatic confocal probe, as shown in Figure 4c.

Then, the effect of the change in the object tilt angle θ_X on the TL method was estimated with the wavelength λ fixed at 1.56 μm. Figure 5a shows only the reference light confocal response curve I_R for changes in the object tilt angle θ_X, and its contour plot is shown in Figure 5b since the z-position of the target can be obtained by tracking the local minimum point of the I_R. From Figure 5a,b, it can be confirmed that the peak intensity of the reference light confocal response curve I_R decreases and the sidelobe widens as the object tilt angle |θ_X| increases. This is consistent with the trend of numerical results in previous studies [28,29,30,31]. Figure 5c shows a contour plot of the normalized intensity ratio I_TLM. The figure shows that the peak of the I_TLM shifts to the positive direction at the z-position as the hem of the sidelobe of the reference light confocal response curve I_R widens with the object tilt angle θ_X. These phenomena were confirmed over the entire light source wavelength range. Figure 5d shows the results of calculating the peak wavelength λ_focused of the normalized intensity ratio I_TLM at each object tilt angle θ_X in the entire light source wavelength range. Here, the peak wavelength was calculated based on the center-of-gravity method [24]. As can be seen in this figure, it was confirmed that the z-position intercept of the z-λ sensitivity line shifted parallel to the z-position positive direction with the object tilt angle θ_X. This means that in the scanning confocal probe, the peak position of the confocal response does not change, so the object tilt angle θ_X does not affect the z-position detection. On the other hand, in the chromatic confocal probe, the effect of the object tilt angle θ_X cannot be ignored since the wavelength in focus on the object surface for detecting the z-position is changed.

Here, it should be noted that these calculation results have been obtained by examining the effect of the object tilt angle θ_X on the TL method using the theoretical equation for an ideal confocal optical system without the influence of the aberrations Φ of the confocal optical system. A previous study reported that the peak position or sidelobe shape of the confocal response changes when the optical system contains aberrations Φ [31]. Accordingly, in an actual confocal optical system with aberrations Φ, it is expected that the z-position intercept of the z-λ sensitivity line confirmed in Figure 5d will change in the amount of the transition of the z-position intercept translating to the z-position positive direction. Based on the above considerations, we conducted z-λ sensitivity acquisition experiments and discussed the results.

4. Experimental Investigation Results for the Tracking Local Minimum Method

4.1. Experimental Configuration for the Tracking Local Minimum Method

A schematic diagram of the confocal optical system used in the experiment is shown in Figure 6. Supercontinuum light, which is a mode-locked femtosecond laser, emitted from a single-mode fiber was collimated using a collimating lens. After passing through a polarizing beam splitter (PBS) and a quarter-wave plate (QWP), the light was focused onto the object to be measured using a plano-convex lens. Then, the reflected light from the surface of the object was again collimated by a plano-convex lens, passed through the QWP and PBS, and was split into two light beams by a beam splitter (BS). Each beam was treated as a measurement light and a reference light; the two beams were focused onto the fiber end-face using the same objective lens and then analyzed using an optical spectrum analyzer. The reference fiber was given an optical axis offset of z_d = 200 μm. As mentioned above, the normalized intensity ratio I_TLM was obtained and analyzed by dividing the measurement light spectrum I_Mea obtained from the fiber-detector for the measurement light by the reference light spectrum I_Ref obtained from the fiber-detector for the reference light [22,23]. The configuration of this optical system was the same as that in the previous studies [24,27]. The measurement object was fixed to a z-axis stage and a rotation stage. It should be noted that only one rotary stage (θ_r = θ_X) was mounted since the response to the tilt angle of the measurement object was circularly symmetrical with respect to the optical axis.

4.2. Alignment Method of the Optical System

Next, efforts were made to reduce the misalignment y₀ between the center of rotation of the measurement object and the optical axis and the misalignment z₀ between the center of rotation of the measurement object and the surface of the measurement object (Figure 7a). Since the purpose of this study is to investigate the influence of the tilt of the measurement object on the z-position measurement results, it is important to reduce the influence of the misalignment quantities y₀ and z₀ on the confocal responses. As can be seen in Equation (4), the TL method defines the object z-position origin as the position where the light source center wavelength of 1560 nm comes into focus on the surface of the measurement object. Therefore, when the misalignments y₀ and z₀ exist, the light ray with a wavelength of 1560 nm is aligned to focus on the point P(y₀, z₀), as shown in Figure 7a. In this case, when θ_X is given to the mirror using a rotary stage in the presence of y₀ and z₀, the point at which the light of wavelength 1560 nm is focused will change from point P to point P′. Figure 7b shows this situation. The coordinates of point P′(y₀′, z₀′) are given geometrically by the following equation.

$$\{\begin{array}{l}{y_0}^{\prime }=y_0\\ {z_0}^{\prime }=\tan {\theta }_X\cdot \left(y_0+\sin {\theta }_X\cdot z_0\right)+\cos {\theta }_X\cdot z_0\end{array}$$

(14)

From the above equation, the amount of a focusing z-position change (Z) at which the confocal response peaks due to the presence of y₀ and z₀ misalignment between the rotation axis and the optical axis can be expressed as follows.

$$Z={z_0}^{\prime }-z_0+k{\sin }^2{\theta }_X$$

(15)

This leads to the focusing z-position change (Z) due to the existence of misalignment factors y₀ and z₀. The third term, ksin²θ_X, was added to represent the amount of peak shift caused by aberrations Φ in the confocal optics. It should be noted that the reason for adding the term proportional to the square of the sin function is that the results are in good agreement with the experimental results, which will be described later.

Figure 8a,b show the results of calculating the amount of z-position change (Z) at which the confocal response peaks using Equations (14) and (15). Calculations were performed with P(y₀, z₀) = P(0, −200) and P(1050, −200). Here, results in Figure 8a were calculated under the condition of k = 0, while Figure 8b was calculated under the condition of k = 8000. The results for P(0, −200) shown in Figure 8a indicate that the focusing z-position change (Z) where the confocal response peaks show a transition amount of fewer than 1 μm within the range of |θ_X| = 5 degrees. This suggests that the effect on the confocal response becomes small at |z₀| < 200 μm, which is expected from the mechanical installation error. Then, from the results of P(1050, −200) shown in Figure 8a, it can be confirmed that the presence of y₀ induces a linear position change in the z-position where the confocal response peaks with the change in the measurement object angle. This indicates that it is important to set y₀ to be 0 μm to investigate the effect of the object tilt angle on the TL method independently, which is the objective of this study. According to the results of P(0, −200) and P(1050, −200) in Figure 8b, the peak transition of the confocal response should be symmetric about the axis of θ_X = 0 degree to satisfy y₀ = 0 μm.

Based on the above considerations, an experiment was conducted to investigate the effect of the misalignment y₀ between the rotation axis and the optical axis. The measurement light spectrum I_Mea was acquired by scanning the z-stage from −200 μm to +200 μm at 2 μm intervals with y₀ = 0 μm and y₀ = about 1050 μm at each θ_X, respectively. Figure 9a,b shows the measurement light response curve I_M of the measurement light spectrum I_Mea obtained at the source center wavelength of 1560 nm when y₀ = 0 μm, and Figure 9c,d shows that under the condition of y₀ = 1050 μm, respectively. Here, the results obtained for each θ_X in Figure 9a, normalized to 1, are shown in Figure 9b. Figure 9c,d are similar to Figure 9a,b, respectively. The results in Figure 9a,b are in good agreement with those in Figure 8b. Based on the above results, the Figure 9a,b condition was defined as the state in which the measurement object was ideally aligned with the center of rotation of the mirror without any misalignment y₀ and z₀ between the rotation axis of the measurement object and the optical axis.

4.3. Experimental Analysis of the Influence of the Tilt Angle of the Measurement Object on the Tracking Local Minimum Method

Finally, the influences of the tilt angle θ_X of the measurement object on the tracking local minimum method for z-position measuring were evaluated. Figure 10 shows the obtained optical spectra ranging from 1.48 μm to 1.64 μm at θ_X = 0 degree under the above alignment conditions. Figure 10a,b show the results of the measurement light spectrum I_Mea and the reference light spectrum I_Ref, respectively, and the normalized intensity ratio I_TLM calculated from the I_Mea and I_Ref based on the TL method. Note that, unlike Figure 4a,c, Figure 10a,b show the I_Mea and I_Ref including the light source spectrum. Figure 10c shows the cross-sections at z = 30 μm in Figure 10a,b. Figure 10c shows that the measurement light spectrum I_Mea and the reference light spectrum I_Ref are affected by the light source spectral shape, making it difficult to confirm the peak wavelength, while the normalized intensity ratio I_TLM is not affected by the light source spectral shape, and the peak wavelength can be confirmed. Figure 10d shows the cross-sections at the center wavelength λ = 1.56 μm in Figure 10a,b, respectively. The I_Mea and I_Ref in Figure 10d were shown normalized to the maximum intensity. Note that the I_Mea and I_Ref in Figure 10d were equivalent to the measurement light confocal response curve I_M and the reference light confocal response curve I_R. It can be confirmed that Figure 10d differs from Figure 4d in that there is no sidelobe in the negative direction at the z-position. This was due to the effect of aberrations Φ that were not considered in the numerical calculations [31,35,36,37].

Figure 11 shows the results of the experiment conducted to determine the effect of the object tilt angle θ_X on the TL method. Figure 11a shows the reference light spectrum I_Ref at different object tilt angles θ_X, and its contour plot is shown in Figure 11b, corresponding to the numerical calculation results shown in Figure 5. Figure 11a,b show that the peak intensity of the reference light spectrum I_Ref decreased and the hem of the sidelobe widened as the object tilt angle |θ_X| increased. However, unlike Figure 5a,b, the peak position shifted toward the positive z-position as the object tilt angle |θ_X| increased. Note that this can be considered due to the factors expected from Figure 8b and Figure 9. Figure 11c shows a contour plot of the normalized intensity ratio I_TLM. As can be seen in Figure 11c, it can be realized that the peak of the normalized intensity ratio I_TLM also shifted to the positive direction at the z-position in accordance with the shift of the minimum point of the reference light spectrum I_Ref to the positive direction at the z-position due to the tilt angle θ_X of the measurement object. This phenomenon was confirmed over the entire source wavelength band. Figure 11d shows the results of calculating the peak wavelength of the normalized intensity ratio I_TLM at each object tilt angle θ_X in the entire light source wavelength band. It was confirmed that the z-position intercept of the z-λ sensitivity line shifted parallel to the z-position positive direction with the object tilt angle θ_X. This means the effect of the object tilt angle θ_X cannot be ignored in the TL method since the wavelength in focus on the object surface for detecting the z-position has changed.

4.4. Discussion about the Influence of the Tilt Angle of the Measurement Object on the Tracking Local Minimum Method

Based on the above results, the influence of the object tilt angle θ_X on the TL method was discussed by using the following equation shown in Step 3 of Figure 1:

$$z=\frac{dz}{d\lambda }\times {\lambda }_{\mathrm{focused}}+z_0$$

(16)

The z-λ sensitivity (dz/dλ) and z-intercept (z₀) obtained in the experiments were shown in Figure 12a. The calculation was performed by a linear approximation using the least-squares method. The coefficient of determination for the linear approximation R² is shown in Figure 12b. As can be seen in the figures, dz/dλ and z₀ were found to vary in proportion to |θ_X|, except for θ_X = −2 degree, where R² < 0.99. This suggests that the confocal optics should be configured according to the tilt angle θ_X of the object to be measured. In addition, numerical results showed that dz/dλ is constant regardless of θ_X, but experimental results show that it varies. This phenomenon can be expressed by the following equation:

$$z\left({\theta }_X\right)=\frac{dz}{d\lambda }\left({\theta }_X\right)\times {\lambda }_{\mathrm{focused}}+z_0\left({\theta }_X\right)$$

(17)

This means that the z-λ sensitivity (dz/dλ) and z-intercept z₀ change with the object tilt angle θ_X. This factor is thought to be due to the effect of aberrations Φ existing in the confocal optics. In addition to the improvement for a new confocal optics that was not affected by the aberrations Φ, it was also realized that a new z-position detection method can simultaneously measure the angle θ_X information of the measurement point is required.

5. Conclusions

In this study, the effect of the object tilt angle on the TL method, which is a method for detecting the object z-position in our proposed dual-fiber-detection confocal optical system, has been investigated. It should be noted that the proposed TL method uses z-λ sensitivity to convert from the wavelength of the confocal response curve local minimum point to the z-position. At first, a theoretical equation was developed to investigate the effect of the object tilt angle on the confocal response curve. Numerical calculations were then performed based on the constructed equation, and the results have revealed that the hem of the sidelobe of the confocal response curve has widened with the object tilt angle θ_X. In conjunction with this, the z-position intercept of the z-λ sensitivity line, which detects the z-position of the measurement object from the wavelength of the confocal response curve minimum, showed a parallel transition in the z-position positive direction. This indicates that the effect of the object tilt angle θ_X cannot be ignored in the developed chromatic confocal probe employing the TL method that detects the z-position from the wavelength in focusing on the surface of the measurement object. Here, the aberrations Φ in the optical system were not taken into account when examining the theoretical equation. This is the reason why it is difficult to completely estimate the aberrations in the entire optical system. The inclusion of aberrations should be considered in future studies.

In addition, an experimental study was conducted to determine the effect of the object tilt angle on the TL method. At first, experimental investigations realized that the dual-fiber-detector confocal optics contains aberrations, and that the peak wavelength of the confocal response more shifts with the object tilt angle θ_X due to the effect of these aberrations than estimated in numerical calculations. This factor was considered due to the effect of aberrations Φ existing in the confocal optics. These experimental results mean that the z-λ sensitivity (dz/dλ) and z-intercept z₀ change with the object tilt angle θ_X. In the future, new confocal optics that are not affected by the aberrations Φ of the confocal optics will be constructed regarding the aberration correction method described in references [35,39]. For example, aberrations Φ in the optical system can be corrected by changing the optical system tube length [35]. The investigations of the effect of the object tilt angle on the TL method in this paper were conducted without changing the parameters of the lens aperture and beam size of the developed confocal optics in previous works [26,27]. Future work includes optimization of the lens aperture and beam size parameters of the confocal optics. In addition, improving the z-position detection method to a new method that can simultaneously measure the z-position and acquire the angle θ_X information of the measurement point will be carried out as future work.