## 2 Theory and Methods  
The transmission, reflection, and absorption spectral properties of the coating surface make the filter achieve its filtering function [14]. By referring the product catalog of the optical giant Edmund, it is known that the types of filters usually include bandpass, longpass, shortpass, and notch filters, etc. An example is shown in Figure 1[15].  
Figure 1: Common filter spectra: (**a**) bandpass; (**b**) longpass; (**c**) shortpass; (**d**) notch filters.  
This article elucidates the design principle behind a multi-layer filter-wheel mechanism, essential for achieving various spectral combinations. Using the double-layer filter wheel as a case study, the mechanism is revealed. Figure 2 illustrates the setup, detailed as follows: the upper gear houses m filters labeled P1, P2...Pm, while the lower gear contains \(n\) filters labeled Q1, Q2...Qn, with m and \(n\) being coprime. A motor drives the double-layered small gear, which in turn rotates the upper and lower gears. Through precise transmission ratio design, a single revolution of the small gear prompts the upper gear to rotate \(1/\text{m}\) of a revolution and the lower gear to rotate \(1/n\) of a revolution, synchronously advancing both gears to the next filter position. This principle is extendable to a multi-layer filter wheel, provided the number of filters in each layer remains coprime.  
Each rotation of the small gear constitutes a single work step, beginning with the initial filter combination of Q1 and P1. As we proceed, after one work step, the combination shifts to Q2 and P2. Upon completing m work steps, the upper gear has made a full revolution, while the lower gear has turned \(\text{m}/n\) of a revolution. Notably, the upper gear resets to P1, but the lower gear does not return to Q1 due to the relative primality of m and \(n\). After the m*\(n\) step, the upper gear rotates \(n\) revolution and the lower gear rotates m revolution, and the upper gear returns to P1 and the lower gear return to Q1; the P1 Q1 combination first overlapped after the first work step, and during this process there are a total of m*\(n\) steps, which proves the existence of m*\(n\) combinations. The operation of this mechanism is illustrated in Figure 3. For practical use, selecting m and \(n\) as consecutive numbers maximizes the number of possible combinations within a given volume, enhancing the versatility of the filter-wheel system.  
In order to meet the above functions, the gear parameters need to satisfy Formula (1).  
\[\begin{gathered}\left(Z_{up1}+Z_{up2}\right)\cdot m_{gear}=\left(Z_ {low1}+Z_{low2}\right)\cdot m_{gear}\\ \hskip 14.226378 ptZ_{up1}/Z_{up2}=1/m\\ \hskip 14.226378 ptZ_{low1}/Z_{low2}=1/n\end{gathered} \tag{1}\]  
where \(Z_{up1}\) is the upper-layer small-gear teeth number, \(Z_{up2}\) is the number of upper-layer big-gear teeth, \(Z_{low1}\) is the number of lower-layer small-gear teeth, \(Z_{low2}\) is the number of lower-layer small-gear teeth, \(m_{gear}\) is the gear module, m is the number of upper layer filters, and \(n\) is the number of lower layer filters. According to Formula (1), the small-gear parameter should satisfy Formula (2).  
Figure 3: Combinations with different work steps of upper gear and lower gear.  
Figure 2: Working diagram of double-layer filter wheel.  
\[Z_{up1}/Z_{low1}=(n+1)/(m+1) \tag{2}\]  
Considering factors such as volume and the number of spectral segments, this article selects m as 5 and \(n\) as 4, resulting in a total of 20 spectral combinations. Based on Formula (2), \(Z_{up1}/Z_{low1}\) = 5/6. However, to prevent root cutting [16], the number of teeth must be greater than 17. Additionally, to reduce manufacturing complexity and control the overall size, it is essential to minimize the number of gear teeth. After careful consideration, the number of teeth on all gears is as follows: \(Z_{up1}\) = 20, \(Z_{low1}\) = 24, \(Z_{up2}\) = 100, \(Z_{low2}\) = 96.