## 1 Introduction  
A confocal probe is an optical displacement sensor often employed in microscopy for form measurement of three-dimensional microstructures for the quality control of ultra-precision machined surfaces [1; 2; 3; 4; 5; 6]. Due to a high lateral resolution and a good depth discrimination performance by the \(Z\)-directional depth-sectioning effect, a confocal probe can also be employed in the surface profilometry of various materials and medical applications in the observation of living cells [7; 8; 9; 10]. Chromatic confocal microscopy [11] has been proposed for the elimination of the axial-scanning of the conventional confocal microscopy. To obtain a better lateral resolution and an in-depth discrimination performance while eliminating the influence of nonuniform surface reflectivity of a target object in the three-dimensional surface topography measurement at nanometric or sub-nanometric level, differential methods and the corresponding signal processing algorithms for a confocal probe [12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26] have been proposed. Most of the differential methods for a confocal probe employ two confocal outputs captured by two detectors with different apertures or focusing conditions. The differential methods can be categorized into two major types. The first type [25] employs two photodetectors having apertures with different sizes. Each of the apertures is placed at the focal plane of the corresponding collecting lens in the confocal setup, and the height information of an object surface under inspection can be decoded from the linear relationship between the axial displacement of the target surface and the normalized intensity ratio of the two confocal outputs. Some of the differential probes of this type can carry out measurement without the scanning along the optical axis. However, the in-depth measurement range is relatively small [26]. The second type employs two detectors having identical confocal apertures [17; 18; 19; 20; 21; 22; 23; 24]. One confocal aperture is located in front of the focal plane of a corresponding collecting lens, while the other is placed behind the focal plane of another corresponding collecting lens to obtain two different confocal outputs. In the case of employing two slits as the apertures for the confocal setups, the in-depth resolution could be degraded [17]. However, this can be overcome by employing two identical pinholes as the apertures for the confocal setup [18; 19; 20; 21; 22; 23; 24]. For this type of differential method, two corresponding signal processing algorithms have been designed. The first signal processing algorithm [21; 22; 23; 24] is designed in such a way that the normalized output is defined as the ratio of intensity subtraction of the two confocal outputs. Meanwhile, this algorithm cannot remove the influence of the nonuniform surface reflectivity of an object under inspection, as well as the influence of the light source. To address these issues, a second signal processing algorithm [18; 19; 20] has been designed. In the algorithm, the ratio of intensity difference and intensity sum (or maximum intensity) of the two confocal outputs are utilized as a confocal output. It should be noted that these two types can be combined with each other [26]. In the previous work by the authors of [27; 28; 29; 30; 31], the dual-detector confocal setup based on the second type, as well as the signal processing algorithm, has been modified for the femtosecond laser differential chromatic confocal probe, where a mode-locked femtosecond laser is employed as the laser source. In the modified differential chromatic confocal setup, the two identical fiber detectors are arranged in such a way that the fiber detector in the measurement detector unit is placed at the focal plane of the collecting lens of the detector unit, as shown in Figure 1a, while the fiber detector in the reference detector unit is placed at a position with a displacement (defocus) \(d\) along with the axial position, as shown in Figure 1b.