# Design of Zoom Optical System from Visible to NIR-II for Vivo Fluorescence Imaging Device
## 2 Design Strategy
### Elimination of Wide Band Chromatic Aberration  
The achromaticity in the wide band range is mainly achieved by the apochromatic model. Each lens needs to satisfy the optical focal equation and the chromaticity equation, and each element satisfies the following equations [9]:  
\[ \begin{split}\varphi=\frac{1}{h_{1}}\sum h_{i}\varphi_{i}\\ L=\frac{1}{h_{1}^{2}\varphi^{2}}\sum h_{i}^{2}C\varphi_{i}\end{split}  \tag{3}\]  
where \(\varphi\) represents the total optical focal power of the optical system, \(\varphi_{i}\) represents the optical focal power of the ith len, h\({}_{i}\) represents the near-axis light object height of the ith len, and C represents the corresponding band dispersion coefficient. This model can solve the problem of chromatic aberration in each band of the optical system to a certain extent. However, for 0.4-1.7 \(\upmu\)m and such a wide band chromaticaberration correction, the dispersion of the optical system in the same image surface position in two or more different bands cannot guarantee good imaging quality. Therefore, the dispersion equation can be extended to solve the dispersion gap between the bands:  
\[P_{12}=\frac{n_{1I}-n_{2r}}{n_{1I}-n_{2r}} \tag{4}\]  
In the material library, we can choose several materials between a high and low P\({}_{12}\) value. The design example in this paper uses the five kinds of glass materials shown in Table 1.  
The chromatic dispersion of different bands in Table 1 is defined by the following equations. \[ \begin{cases}C_{0.4-0.7}=\dfrac{n_{0.4-n_{0.7}}}{n_{0.55-1}}\\ C_{0.7-1.0}=\dfrac{n_{0.7-n_{1.0}}}{n_{0.85-1}}\\ C_{1.0-1.7}=\dfrac{n_{1.0-n_{1.7}}}{n_{1.35-1}}\\ C_{0.4-1.7}=\dfrac{n_{0.4-n_{1.7}}}{n_{1.05-1}}\end{cases}  \tag{5}\]  
In the near-infrared and short-wave infrared bands, the optical glass materials in Table 1 have similar dispersion coefficients, indicating that the dispersion characteristics of these two wavelengths are similar, so we can focus on only one band. When considering the dispersion characteristics over a wide wavelength range, we found that N-SF1 had a relatively large dispersion coefficient and could be used as a color correction element. N-LAF2 and N-LAF34 in the visible and short-wave infrared dispersion degree were different and could be used to complement or to replace according to the chromatic aberration. In the design of fixed-focus optical systems, because the entire optical system components are relatively fixed according to the above-mentioned model approach, through selecting a reasonable structure type, selection of material combinations, initial layout parameters calculation, system optimization, and image quality evaluation, we could usually obtain better design results. However, for the zoom optical system, there are variable elements in the system, and the incident height of the light on the surface of the element changes in the process of field-of-view (FOV) switching, which may lead to large changes in the system's chromatic aberration. In order to avoid mixing, the above achromatic achromatic aberration equation with the initial structural parameters of the zoom optical system solution, which makes the design process complicated, provides a better solution to obtain the optical focal length distribution of each mirror group of the system and then combine the achromatic for each mirror group of the variable focal length system to eliminate chromatic aberration separately. The front fixed group usually bears the main optical focal length and carries out a separate achromatic design; the rear fixed group and other fixed mirror groups mainly play a role in compensating for the remaining aberration of the front group while bearing part of the optical focal length; and the zoom group lenses number is usually less so we cannot design for separate achromatic aberration. At this time, we need to try to choose the