# Optical Design and Stray Light Analysis of Underwater Spectral Radiometer
## 3 Underwater Spectral Radiometer Design
### Spectrometer System Design
If the light emitted from point \(A\) is imaged at point B after passing through point \(P\), then the optical path length of the light ray \(APB\) can be represented by Equation (7) as follows:  
\[F=AP+PB+n(y,z)m\lambda \tag{7}\]  
In Equation (7), m represents the diffraction order of the grating, and \(n(y,z)\) represents the groove number from any point \(P\) on the concave grating to the origin point \(O\). It is determined by two coherent point sources \(C\) and \(D\). \[n(y,z)=\frac{1}{\lambda_{0}}\bigg{[}(\overset{\rightarrow}{C}\overset{ \rightarrow}{D}P)-(\overset{\rightarrow}{C}\overset{\rightarrow}{O} \overset{\rightarrow}{O})\bigg{]} \tag{8}\]  
In Equation (8), \(\lambda_{0}\) represents the wavelength of the recording light source, namely, the wavelength of the two coherent point light sources \(C\) and \(D\). If all optical paths from point \(A\) to point \(B\) through any point of the grating are equal, point \(A\) can be accurately focused on point \(B\). Therefore, by substituting Equation (8) into Equation (7) and expanding the optical path function into a power series, we can eventually obtain:  
\[F=F_{00}+yF_{10}+\frac{1}{2}y^{2}F_{20}+\frac{1}{2}y^{2}F_{02}+\frac{1}{2}y^{3 }F_{30}+\frac{1}{2}zy^{2}F_{12}+\frac{1}{3}y^{4}F_{40}+\frac{1}{4}y^{2}z^{2}F_ {12}+\frac{1}{8}z^{2}F_{04}+\cdots \tag{9}\]  
In Equation (9), \(F_{20}\) represents defocus, \(F_{02}\) represents astigmatism, \(F_{30}\) represents meridional coma, \(F_{12}\) represents sagittal coma, and the remaining terms are higher-order aberrations. Each term \(F_{ij}\) can be expressed in the following form:  
\[F_{ij}=M_{ij}+m\frac{\lambda}{\lambda_{0}}H_{ij} \tag{10}\]  
\(M_{ij}\) is only related to the parameters used in the structure, while \(H_{ij}\) is only related to the parameters used in the manufacture of the structure. Selecting the main aberration coefficients, establish the objective function for minimization:  
\[I_{ij}=\int_{\lambda_{2}}^{\lambda_{1}}\Big{(}F_{20}^{2}+F_{02}^{2}+F_{30}^{2} +F_{12}^{2}\Big{)}d\lambda \tag{11}\]  
Solving the optimal aberration function for the aforementioned flat-field concave holographic grating and transforming it to obtain the optimal parameter values that minimize the aberration expression in Equation (11). According to Formula (11), the design of flat-field concave holographic gratings is the process of solving the optimal solution to minimize the aberration function value of flat-field concave holographic gratings. In this paper, a genetic algorithm is used to  
Figure 4: Flat-field concave holographic grating light path. solve the initial values of the design parameters of the plane-field concave grating. When the solution space of the genetic algorithm is close to the global optimal solution, the convergence speed is slow and the convergence accuracy is decreased, which seriously affects the calculation efficiency. Therefore, when the results approach the global optimal solution, the search is stopped, and the incomplete convergence solution is input into the optical tracking software Zemax 19.4 as the initial structural parameters of the flat-field concave holographic grating. The local optimization algorithm of Zemax optical simulation software is used for secondary optimization, and the optimization is stopped when the design indexes are met and the optimal solution of the grating design parameters is finally obtained. The simulated optical system of the spectrometer's final structure is illustrated in Figure 5, and the specific parameters are listed in Table 2.  
The optical imaging quality of the above design results was evaluated using the optical design software Zemax. The resolution of the spectrometer is closely related to the width of the entrance slit, with a larger slit width resulting in lower spectral resolution [19]. The designed spectrometer in this study uses a slit width of 50 \(\upmu\)m and a height of 1 mm. Nine coordinates representing the center and edges of the optical slit were selected through ray tracing.