## 1 Introduction  
Confocal microscopy [1; 2] with a confocal probe has outstanding image formation properties in depth sectioning imaging and optical tomographic imaging [3; 4; 5], which makes it a potentially powerful tool for image formation in industrial and medical engineering fields, such as three-dimensional (3D) fine structure image formation [4; 5] and living body observation of cells [6; 7], compared with traditional microscopy. It is also expected to measure large-area 3D microstructured surfaces [8; 9]. Meanwhile, the relatively small measurement range of a confocal probe with a monochromatic laser source has prevented the further application of confocal microscopy in these areas. Therefore, the extension of the measurement range of a confocal probe is critical and crucial for expansion of the application range of the confocal microscopy. For the extension of the measurement range of a confocal probe, several efforts have already been made so far [10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20]. Among them, enhancement of the chromatic dispersion has been proven as a way to realize the extension of the measurement range without any axial scanning of a target object or a measuring head of a confocal probe [18; 19; 20]. The employment of a diffractive optical element (DOE) is a candidate method to enhance the chromatic aberration, and five-fold extension of measurement range has been achieved [10]. Other DOEs for enlargement of the chromatic dispersion, such as a micro-lens array [13] and a Fresnel lens [14], have achieved enlarged measurement ranges of 210 \(\upmu\)m and 200 \(\upmu\)m, respectively. With a diffractive lens and objective lenses [11], an expansion of a measurement range has also been achieved [11; 16]. Furthermore, by utilizing a pupil filter technique, such as super-resolution pupil filter [15] or annular pupil filter [17], for the re-shaping of a measuring beam, measurement ranges have also been enlarged to 40 \(\upmu\)m and 14 \(\upmu\)m, respectively. However,the above methods need to employ additional optical components in the optical setup, which could affect the performances of original optical confocal setups. Meanwhile, employing a broadband laser source is another approach to expand the measurement range of a chromatic confocal probe [18; 19], and several methods have been proposed so far. With the employment of a supercontinuum laser source, over 16-fold expanded measurement range has been acquired [19]. However, non-smoothness of the laser spectrum, which means non-uniformity of the spectral intensity distribution, has restricted the full exploration of the whole spectrum of the supercontinuum laser source for a wider axial measurement range in a chromatic confocal probe. In responding to the background described above, a fiber-based dual-detector chromatic confocal probe with a mode-locked femtosecond laser source has been developed [3]. The proposed method has demonstrated that the influence of non-smoothness of the mode-locked femtosecond laser source can be compensated while expanding the measurement range over the whole spectrum of the laser by employing an axial response, defined as the intensity ratio of the two different confocal signals. In this paper, as the second step of research, a new signal processing method for the axial response curve from the fiber-based dual-detector chromatic confocal probe with a mode-locked femtosecond laser source is proposed. The method is designed to further expand the measurement range without modifying the optical setup of the dual-detector chromatic confocal probe. Due to the diffraction effect of a point detector in confocal microscopy [20], there are always a main-lobe and side-lobes in the axial response of a confocal probe, in which main-lobe stands for the highest power intensity distribution curve of the axial response, while side-lobes stand for local maxima power intensity distribution curves of the axial response.