## 2 Principle of the Extension of the Measurement Range
### Expansion of the Measurement Range
Regarding the non-smoothness of the laser spectrum shown in Figure 2b, axial responses of the fiber-based dual-detector chromatic confocal probe are estimated based on Equations (13) and (14), in  
Figure 4: The main-lobes and the side-lobes extracted from the normalized intensity ratio and the invert normalized intensity ratio obtained at central wavelength \(\lambda\) = 1560 nm, together with side-lobe spacing and main-lobe to side-lobe spacing with target positions ranging from 0 to 250 \(\upmu\)m and defocus \(d\) ranging 50 \(\upmu\)m to 110 \(\upmu\)m. (**a**) Main-lobe and side-lobes obtained from normalized and invert normalized intensity ratio at central wavelength with \(d\) = 50 \(\upmu\)m. (**b**) Main-lobe and side-lobes obtained from normalized and invert normalized intensity ratio at central wavelength with \(d\) = 80 \(\upmu\)m. (**c**) Main-lobe and side-lobes obtained from normalized and invert normalized intensity ratio at central wavelength with \(d\) = 110 \(\upmu\)m. (**d**) The side-lobe spacing and main-lobe to side-lobe spacing at central wavelength with \(d\) = 50 \(\upmu\)m, 80 \(\upmu\)m, and 110 \(\upmu\)m.  
Figure 3: A typical axial response obtained by the dual-detector chromatic confocal probe with a mode-locked femtosecond laser source with axial response of the normalized intensity ratio and axial response of the invert normalized intensity ratio, with target positions ranging from 0 to 120 \(\upmu\)m and a defocus \(d\) of 110 \(\upmu\)m.  
which the proposed axial response is defined as the intensity ratio of the signal 1 and signal 2, not as the single axial response anymore. The distance between the main-lobe and the side-lobe is therefore determined by the relative magnitude of the ratios of signal 1 and signal 2 at the same target position. Furthermore, the relative height of the main-lobe and the side-lobes is also controlled by the relative magnitude of the ratios of signal 1 and signal 2 at the same target position, which is further controlled by the defocus \(d\) that is shown in Equation (11). As illustrated in Equation (11), when we set a defocus \(d\) to be zero, theoretically the values of Equation (13) and Equation (14) always become 1. For the calculations, the parameters summarized in Table 1 are employed to express the effect of the chromatic objective. Figure 5a,b show the axial responses from the normalized intensity ratio and the invert normalized intensity ratio, respectively, calculated at the wavelengths from 1550 nm to 1580 nm. As can be seen in the figures, the main-lobe and the side-lobes of the axial response from the invert normalized intensity ratio are found to be shifted along the axial (\(Z\)-) direction with respect to those from the normalized intensity ratio. In the previous work by the authors [3], only the information from the main-lobe in the normalized intensity ratio has been employed to detect a displacement in the axial direction. As a result, the measurement range has been limited to be approximately 40 \(\upmu\)m. Meanwhile, by using the information from not only the main-lobe but also the side-lobes in the axial responses from both the intensity ratios, the measurement range can be expanded; as can be seen in Figure 5c, the gap of the neighboring lobes (including the main-lobe and side-lobes) in the axial response of \(I_{a}(u,\,u_{d})\) can be covered by the corresponding lobes of the axial response of \(I_{n}(u,\,u_{d})\). Figure 5: The main-lobe and the side-lobes in the axial response obtained by the dual-detector chromatic confocal probe with a mode-locked femtosecond laser source with target positions ranging from 0 to 120 \(\upmu\)m and a defocus \(d\) of 110 \(\upmu\)m. (**a**) Axial response of the normalized intensity ratio with shadow of the main-lobe and side-lobes and the corresponding target position values. (**b**) Axial response of the invert normalized intensity ratio with shadow of the side-lobes and the corresponding target position values. (**c**) The main-lobes and the side-lobes in the axial responses.