# Long-Distance Measurements Using a Chromatic Confocal Sensor
## 2 Experimental Setup
### Optical Performance  
Several critical parameters for a chromatic confocal sensor include the working distance, the measurement range, the axial and lateral resolution, and the maximum tilt angle of the sample [11]. These parameters collectively determine the sensor's performance, impacting its versatility and suitability for various applications. The working distance of our system corresponds to the effective focal length of the primary aspheric lens, measuring 620 mm for a wavelength of 590 nm. It is worth mentioning that even if the aspherical shape of the main lens is aimed to decrease the spherical aberration it does not remove entirely. Furthermore, the confocal sensor uses a small portion of the center of the lens, as shown in Figure 4, and so the focal lengths for different wavelengths will be larger than those given in the specifications of the lens. In Figure 5, \begin{table}
\begin{tabular}{l c} \hline
**Parameter** & **Value** \\ \hline Curvature & plano-convex \\ \hline Radius & 329.8 mm \\ \hline Diameter & 345 mm \\ \hline Clear aperture & 310 mm \\ \hline Material & N-BK7 \\ \hline Conic & \(-\)0.604 \\ \hline Focal length (590 nm) & 620 mm \\ \hline Central thickness & 60 mm \\ \hline \end{tabular}
\end{table} 
Table 1: Aspheric lens optical parameters. Figure 3: Chromatic confocal sensor render. the focal lengths for the marginal rays of an incident light with a diameter of 25.4 mm are shown. In addition, it is important to note that the central rays of the collimated light incident on the main lens will have even higher focal lengths and extend the working range of the sensor even further, which is why the last optical relay was introduced, to magnify the collimated light after reflection on the sample and to introduce a spatial filter to block these rays (see Figure 6) and maintain the desired working range. Figure 4: Theoretical working range given by the Python simulator. Figure 5: Focal lengths for different wavelengths for the main aspheric lens. Figure 6: Setup plot from the developed Python module. Shorter wavelengths are shown in blue, while longer wavelengths are shown in red. The measurement range is bounded by the extremes of the light source's wavelength spectrum, spanning from 550 nm to 1100 nm. However, our system is tailored to operate within a narrower spectral range, precisely from 550 to 800 nm, because of the antireflective coating used on all of the lenses. Theoretically, if the beam reaching the aspheric lens is perfectly collimated for all wavelengths, the measurement range of the sensor, given by the chromatic dispersion of the optics, will lead to images of the different wavelength components at different longitudinal points along the optical axis of around 15 mm. The exact value of the working range will be delimited by the spatial filter and also by the exact height at which the marginal ray hits the main lens. This value is given by the chromatic aberration of the main lens and also by the height of the marginal ray that comes from the collimator, as seen in Figure 4.  
For the purpose of this work, which is to demonstrate the capability of the chromatic sensor to operate in a long-distance setup, the lateral resolution is not a primary concern. However, it remains influenced by the point spread function of the aspheric lens and the extent of the collimation of the light source. The axial resolution hinges mainly on the spectrometer's resolution and the chromatic aberration inherent in the optical setup. In a focused system, the axial and lateral resolution can be described by the radius of the diffraction of the Airy disk in the lateral (\(x\), \(y\)) image plane, as defined in Equation (1); the equivalent of this in the axial direction is given by Equation (2).