## 2 Principle of the Extension of the Measurement Range  
### Imaging Principle  
Figure 1 shows a schematic of the optical configuration for the fiber-based dual-detector chromatic confocal probe with a mode-locked femtosecond laser source [3]. A mode-locked femtosecond laser source is employed, while single-mode step index fibers are used as the signal-propagating media. The \(k\)th optical mode in the mode-locked femtosecond laser has a specific frequency \(v_{k}\), which corresponds to the deterministic working wavelength \(\lambda_{k}\), which can be expressed as follows by using the carrier envelope offset frequency \(v_{\text{ceo}}\) and the pulse repetition rate \(v_{\text{rep}}\)[32; 33; 34]:  
\[v_{k}=v_{\text{ceo}}+k\cdot v_{\text{rep}} \tag{1}\]  
\[\lambda_{k}=\frac{c}{v_{k}} \tag{2}\]  
where \(c\) is the speed of light in vacuum. For the image formation of confocal microscopy, the point spread function (PSF), and the pupil function of imaging lens are critical and are used in the equation of the final axial response of the confocal microscopy. The equation of the PSF, pupil function, and the equation of the effective PSF are given as follows [35; 36; 37; 3]:  
\[h_{i}(u,v)=\int_{0}^{1}P_{i}(u,\rho)J_{0}(\rho v)\rho d\rho,i=1,2, \tag{3}\]  
\[I_{\rm m}(u)=\left|\int_{0}^{1}\exp\left(ju\rho^{2}\right)P_{1\_eff}P_{2\_eff}\rho d \rho\right|^{2} \tag{10}\]  
In order to overcome the non-smoothness of the laser spectrum in the mode-locked femtosecond laser source, a dual-detector differential confocal configuration shown in Figure 1 has been proposed [3]. The light intensity distribution \(I_{\rm r}\) (\(u\), \(u_{d}\)) on the plane with a distance \(d\) from the focal plane of the objective lens is given as follows:  
\[I_{\rm r}(u,u_{d})=\left|\int_{0}^{1}\exp\left[j\Big{(}u+\frac{u_{d}}{2}\Big{)} \rho^{2}\right]P_{1\_eff}P_{2\_eff}\rho d\rho\right|^{2} \tag{11}\]  
where \(u_{d}\) is the optical distance associated with the defocus \(d\) that can be described as follows:  
\[u_{d}=\frac{2\pi}{\lambda}d\mathrm{sin}^{2}\beta \tag{12}\]  
Figure 2a shows typical spectra of the light intensities \(I_{\rm m}\) (\(u\)) and \(I_{\rm r}\) (\(u\), \(u_{d}\)) obtained by the two identical fiber detectors. As can be seen in the figure, it is difficult to distinguish a peak in the spectra, since a mode-locked femtosecond laser source has a non-smooth spectrum, as shown in Figure 2b. The spectrum shown in Figure 2b is from a specific femtosecond laser used in the following experiments and its non-smooth spectrum is not the same as the other generic laser sources. To address the aforementioned problem, the normalized intensity ratio \(I_{\rm a}\) (\(u\), \(u_{d}\)) and the invert normalized intensity ratio \(I_{\rm n}\) (\(u\), \(u_{d}\)) of the obtained signal intensities, determined by the following equations, are employed in the proposed method:  
\[I_{\rm a}(u,u_{d})=\frac{I_{\rm m}(u)}{I_{\rm r}(u,u_{d})} \tag{13}\]  
\[I_{\rm n}(u,u_{d})=\frac{I_{\rm r}(u,u_{d})}{I_{\rm m}(u)} \tag{14}\]  
Figure 2: Typical normalized intensity ratio obtained by the dual-detector chromatic confocal configuration with a mode-locked femtosecond laser source. (**a**) Light intensities obtained by the two identical fiber detectors with a target position of 40 \(\upmu\)m and a defocus \(d\) of 150 \(\upmu\)m; (**b**) Working spectrum of the mode-locked femtosecond laser source; (**c**) Normalized intensity ratio and invert normalized intensity ratio obtained from the two light intensities. Figure 2c shows the normalized intensity ratio and the invert normalized intensity ratio obtained from the spectra of the obtained light intensities shown in Figure 2a, with a target position of 40 \(\upmu\)m and a defocus \(d\) of 150 \(\upmu\)m, in which main-lobes can clearly be distinguished in both the normalized intensity ratio \(I_{\text{a}}\) and the invert normalized intensity ratio \(I_{\text{n}}\), but side-lobes are inconspicuous.