## 4 Conclusions  
In this paper, an off-axis differential method for a femtosecond laser differential chromatic confocal probe has been proposed to improve the measurement performance, as well as to simplify the optical alignments and the data processing in the peak wavelength-Z position encoding. A major modification has been made to the conventional differential chromatic confocal setup in such a way that the fiber detector in the reference detector unit has been placed to have an offset with respect to the optical axis of the collecting lens in the detector unit, rather than giving a defocus to the fiber detector. This modification enables the newly proposed off-axis setup to obtain a narrower peak having a profile in the normalized chromatic confocal output, as well as to simplify the optical alignment of the differential confocal setup. With the enhancement of the tracking local minimum method, an improved Z-directional measurement range of 46 \(\upmu\)m has been achieved by the newly proposed off-axis differential method. The possibility of achieving a resolution of 20 nm has also been verified.  
In this paper, attention has been paid to the verification of the feasibility of the newly proposed off-axis method in the femtosecond chromatic confocal probe compared with the conventional setup with the defocus method [27; 28; 29; 30]. Further work contains the investigation on the Z-directional measurement resolution with another signal processing algorithm such as the tracking intersection method [30]. It should also be noted that this paper has focused on the proposal of a preliminary idea of an off-axis differential method for the improvement of the signal-to-noise ratio (SNR), the measurement range and measurement resolution of the chromatic confocal probe. A more detailed investigation of the novelty of the displaced probe, as well as the corresponding application in three-dimensional microstructure imaging, will be carried out in future work. Further optimization and simplification of the established differential chromatic confocal probe system for a higher measurement resolution and investigation on the difference between the results of numerical calculations and experimental results described in this paper, as well as the extension of the measurement range by the optimization of the chromatic objective lens, will also be carried out. The application of the developed mode-locked femtosecond chromatic differential confocal probe to the three-dimensional surface profile measurement also remains as a challenge to be addressed and will be carried out in future work.  
Conceptualization, W.G.; methodology, C.C. and R.S.; software, C.C. and R.S.; validation, C.C., H.M. and R.S.; formal analysis, C.C., R.S. and Y.S.; investigation, C.C. and Y.S.; resources, W.G. and Y.S.; Data curation, C.C.; writing--original draft preparation, C.C.; writing--review and editing, Y.S., C.C.; visualization, W.G. and Y.S.; supervision, W.G. and Y.S.; administration, W.G.; funding acquisition, W.G., Y.S. and H.M. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Japan Society for the Promotion of Science (JSPS) 18H01345.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.