## 2 Principle of the Extension of the Measurement Range
### Expansion of the Measurement Range
Theoretical calculation is expanded to a working range of the mode-locked laser of 1.48 \(\upmu\)m-1.64 \(\upmu\)m and a target position over 250 \(\upmu\)m. Figure 6a,b show the axial responses of the normalized intensity ratio \(I_{\text{a}}\) and the invert normalized intensity ratio \(I_{\text{n}}\). As can be seen in the figure, side-lobes are clearly observed over a target position range of 250 \(\upmu\)m in both of the axial responses. Meanwhile, the existence of a gap between each neighboring lobes is also confirmed in both of the axial responses. In addition, the theoretical calculation results show that the gap becomes wider as the increase of the target position. Figure 6c shows the main-lobes and the side-lobes of the normalized intensity ratio \(I_{\text{a}}\) plotted together with those of the invert normalized intensity ratio \(I_{\text{n}}\). As can be seen in the figure, the two axial responses are almost 180\({}^{o}\) out of phase, and the gaps between each neighboring lobes in the axial response of the normalized intensity ration can be covered by the lobes in the axial response of the invert normalized intensity ratio over a target position range of 150 \(\upmu\)m. \begin{table}
\begin{tabular}{c c c} \hline
**Items** & **Symbol** & **Value** \\ \hline Lens material & - & N-SF11 \\ Surface radii of imaging lens & \(r_{1}\) & 7.85 mm \\  & \(r_{2}\) & \(\infty\) \\  & Sellmeier coefficients & \\ \(B_{1}\) & 1.73759695 & \(C_{1}\) & 0.013188707 \\ \(B_{2}\) & 0.313747346 & \(C_{2}\) & 0.062306814 \\ \(B_{3}\) & 1.89878101 & \(C_{3}\) & 155.23629 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters employed for the theoretical calculation. Figure 6: The main-lobe and the side-lobes in the axial responses obtained by the dual-detector chromatic confocal probe with a mode-locked femtosecond laser source. (**a**) Axial response of the normalized intensity ratio with target positions ranging from 0 to 250 \(\upmu\)m and a defocus \(d\) of 10 \(\upmu\)m. (**b**) Axial response of the invert normalized intensity ratio. (**c**) The main-lobes and the side-lobes in the axial responses. It should be noted that in order to utilize the side-lobes of the axial responses as illustrated in Figure 6c for the expansion of the axial measurement range, calculation of partial derivatives of Equation (13) is necessary:  
\[\frac{\partial I_{\mathrm{a}}}{\partial z}=\frac{\partial I_{\mathrm{a}}}{ \partial u}\frac{\partial u}{\partial z}=\frac{4\pi\mathrm{sin}^{2}\beta\{I_{ \mathrm{r}}\sqrt{I_{\mathrm{m}}}\mathrm{exp}\left(ju\rho^{2}\right)-I_{ \mathrm{m}}\sqrt{I_{\mathrm{r}}}\mathrm{exp}\left[j\left(u+\frac{u_{d}}{2} \right)\rho^{2}\right]\}}{I_{\mathrm{r}}{}^{2}\lambda} \tag{18}\]  
In order to obtain the relative maximum intensity values for wavelength-to-displacement encoding of the side-lobes, let Equation (13) be equal to zero and get the relationship between the target position \(z\) and wavelength \(\lambda\):  
\[\frac{\partial I_{\mathrm{a}}}{\partial z}=\frac{\partial I_{\mathrm{a}}}{ \partial u}\frac{\partial u}{\partial z}=0 \tag{19}\]  
As illustrated in Figure 6c, the main-lobe and side-lobes are employed in the wavelength-to-displacement encoding, but gaps between main-lobe and side-lobes make the final measurement range discontinuous. In order to eliminate the gaps, side-lobes of the axial response of the invert normalized intensity ratio are employed. Therefore, the following relationship should also be satisfied:  
\[\frac{\partial I_{\mathrm{n}}}{\partial z}=\frac{\partial I_{\mathrm{n}}}{ \partial u}\frac{\partial u}{\partial z}=0 \tag{20}\]