# Precision Auto-Focusing Apparatus Based on Long-Working-Distance, Wide-Spectrum Microscope
## 3 Results  
In this study, imaging characteristics of the focusing system were evaluated by using the method of imaging a 1951 USAF resolution target, as shown in Figure 5. Initially, it is placed near the focal plane of the objective lens. The focusing of the resolution target with the microscope measuring device was realized by the automated focusing method described above.  
At the optimum focusing position, four wavelength LED light sources (\(\lambda\) = 480, 550, 660, and 850 nm) were successively connected by an optical fiber to observe the effects of imaging the resolution target under illumination by different wavelengths, as shown in Figure 6. The minimum line pair position shown in Figure 6 is the group 7 element 6 pattern of 1951 USAF resolution target, which corresponds to 228 line pairs per mm (LP/mm). It can be seen that under illumination by all four wavelengths, the microscope can clearly image the resolution target. The horizontal and vertical intensity profiles of the four images in group 7 element 5 and group 7 element 6 have been presented.  
Figure 5: The 1951 USAF resolution target using for evaluated the microscopic imaging system.  
Figure 6: Imaging results obtained using the 1951 USAF resolution target with different wavelength light sources. (**a**) Image at \(\lambda\) = 480 nm; (**b**) image at \(\lambda\) = 550 nm; (**c**) image at \(\lambda\) = 660 nm; (**d**) image at \(\lambda\) = 850 nm. (**a1**)–(**d1**) are the horizontal and vertical intensity profiles of the four images in group 7 element 5 and group 7 element 6, respectively.  
During the focusing process, a series of images at \(\lambda=480\) nm in different states of defocusing can be obtained by combining the imaging system with displacement driving controls, as illustrated in Figure 7.  
In order to make valid comparisons, the focusing performance under different evaluation functions is analyzed, applying different evaluation functions--the Laplacian algorithm [16], the wavelet algorithm [17], the gradient square [18] algorithm and the improved gradient square algorithm, which are shown in the Equations (1)-(4). The autofocusing algorithm is used in sequence of Figure 7, each focusing algorithm produces a focusing curve as shown in Figure 8, and the extreme point of the curve is the optimal focusing position. Compared with the four focusing algorithms under the experimental conditions in this paper, the improved gradient algorithm has better performance in focusing, and this algorithm is used for fitting in subsequent experiments.  
Figure 7: (a)–(g) Images of resolution target obtained by focusing system with different defocusing states, (a1)–(g1) Intensity profile at group 6 element 1 of each image with different defocusing states.  
In order to verify the accuracy and reliability of our focusing results, repeated focusing experiments were carried out. The resolution target and the microscope measuring device were adjusted to their initial positions, and the focusing process was repeated 25 times. The relative distance between the initial position and the optimal focusing position was obtained, and a comparison of these results for the 25 repeated focusing processes is shown in Figure 9; the peak value and standard deviation of the repeated measurement datasets are 2.74 and 0.75 \(\upmu\)m, respectively, which demonstrates that the experimental equipment constructed in this study can achieve high precision focusing.