## 2 Measurement Principle  
### Image Formation Principle  
Figure 3 shows a schematic diagram of the differential chromatic confocal configuration [27; 28; 29; 30; 31]. A mode-locked femtosecond laser beam from the laser source is made to focus on a target surface by a chromatic objective lens. The reflected laser beam from the target surface is then divided into two sub-beams by a beam splitter, and the sub-beams are then captured by two fiber detector units composed of two identical fibers and two identical collecting lenses, L\({}_{1}\) and L\({}_{2}\). One of the sub-beams is captured by one fiber detector unit (referred to as the measurement detector unit) set at the focal plane of L\({}_{1}\), while the other is captured by another fiber detector unit (referred to as the reference detector unit) with the collecting lens L\({}_{2}\). Figure 2: Reference detector unit with an off-axis detector shift \(d_{\rm m}\) in the newly proposed method. The confocal output \(I_{\rm{1 mea}}(u)\) to be captured by the measurement detector unit, whose optical configuration is shown in Figure 1a, can be expressed by the following equation [27; 35; 36; 37]. \[I_{\rm{1 mea}}(u)=\left|\int_{0}^{1}\exp(ju\rho^{2})P_{\rm{1 eff}}(\rho)P_{\rm{2 eff}}(\rho)\rho d\rho\right|^{2} \tag{1}\]  
where \(\rho\) is the normalized radius of the imaging lens, \(j\) is the imaginary unit and \(P_{\rm{1 eff}}\) and \(P_{\rm{2 eff}}\) stand for the effective pupil functions of the imaging lens (L1 or L2) and illuminating lens (collimating lens), respectively. The parameters \(v\) and \(u\) represent the optical coordinates related to the actual radial and axial coordinates \(r\) and \(z\), respectively, that can be given as follows [27; 35; 36; 37]. It should be noted that only the axial (\(Z\)-) response is considered and measured in the experiments. Therefore, the variable \(\
u\) in Equation (1) is treated to be zero [27] and is omitted in the equation. \[v=\frac{2\pi}{\lambda}r\sin\beta \tag{2}\]  
\[u=\frac{2\pi}{\lambda}z\ \sin^{2}\beta \tag{3}\]  
where \(\sin\beta\) and \(\lambda\) are the numerical aperture of the collecting lens and the light wavelength, respectively. Figure 4 shows schematic diagrams of the reference fiber detector unit by the proposed off-axis differential method and the conventional differential method for the chromatic confocal probe system. As can be seen in Figure 4a, in the reference fiber detector unit based on the conventional differential method [27], the single-mode fiber detector is placed with a defocus \(d\) with respect to the back focal plane of the collecting lens L\({}_{2}\). In this case, the confocal output \(I_{\rm{2 ref}}(u;u_{\rm{d}})\) can be represented as follows:  
\[I_{\rm{2 ref}}(u;u_{d})=\left|\int_{0}^{1}\exp[j(u+u_{d}/2)\rho^{2}]P_{\rm{1 eff }}(\rho)P_{\rm{2 eff}}(\rho)\rho d\rho\right|^{2} \tag{4}\]  
Figure 3: A schematic diagram of the differential chromatic confocal configuration with a mode-locked femtosecond laser source. By using these two confocal outputs, the height of a surface under measurement can be detected through the arithmetic operation based on the algorithm referred to as the tracking local minimum (TLM) method [30]. Meanwhile, in the newly proposed off-axis differential method, a major modification is made to the reference detector unit: the single-mode fiber detector is located on the rear focal plane of the collecting lens L\({}_{2}\) by the optical axis but with an off-axis detector shift \(d_{\text{m}}\), as shown in Figure 4b. In this case, the confocal output \(I_{\text{2 ref\_m}}(u;d_{\text{m}})\) can be represented as follows [38]:  
\[I_{\text{2 ref\_m}}(u,d_{\text{m}})=\left|\int_{0}^{1}J_{0}\left(\frac{d_{\text {m}}}{r_{\text{f}}}\rho v_{\text{f}}\right)\exp\left(ju\rho^{2}\right)P_{\text {1 eff}}P_{\text{2 eff}}\rho d\rho\right|^{2} \tag{5}\]  
where \(r_{\text{f}}\) and \(v_{\text{f}}\) are the radius and the normalized radius of the single-mode fiber, respectively.