# Optical Design and Stray Light Analysis of Underwater Spectral Radiometer
## 3 Underwater Spectral Radiometer Design
### Spectrometer System Design  
The spectrometer system is the core component of the underwater spectral radiometer and is essentially a miniature spectrometer module. This instrument is mainly used for underwater spectral radiation measurements. Considering relevant instruments at home and abroad, as well as the characteristics of underwater \"optical windows\" and signal-to-noise ratio requirements, the performance specifications of the instrument are proposed. The spectral range of this spectrometer is 400-950 nm. For in situ measurement of underwater spectral radiation, a spectral resolution of 5-10 nm is sufficient to capture the  
Figure 2: The overall structure of the underwater spectral radiometer. spectral radiation in water. However, for sharp spectral features, it may not be enough [17]. Therefore, this design specifies a spectral resolution better than 3 nm to address this issue. As shown in Figure 3, there are two commonly used structures for micro-spectrum: one is plane grating C-T (Czerny-Turner) structure, and the other is based on field holographic concave grating structure. This design uses a flat-field concave holographic grating as the dispersive element of the spectrometer. Compared with miniature spectrometers with a C-T structure, the selected structure is easier to assemble, which is advantageous for a more compact and lightweight design. In addition, the former has a smaller F-number, therefore having a larger light-collecting aperture, which is beneficial for improving the system's signal-to-noise ratio. Table 1 presents the main parameters of the miniature spectrometer. As shown in Figure 4, establish a Cartesian coordinate system with \(O\) as the origin. The origin \(O\) coincides with the center of the grating, the \(X\)-axis coincides with the normal to the grating surface, the \(Y\)-axis is perpendicular to the direction of the grating lines, and the \(Z\)-axis is parallel to the direction of the grating lines. Points \(C\) and \(D\) are two recording points located in the \(XOY\) meridian plane, representing the positions of two relevant point sources for producing the grating. Point \(A\) is the position of the light source incidence point, and point \(B\) represents the focal point of wavelength \(\lambda\) after the light from point \(A\) undergoes holographic grating diffraction on the flat-field concave surface. The central main rays \(CO\) and \(DO\) of the recorded beams emitted from points \(C\) and \(D\) are, respectively, denoted as \(r_{C}\) and \(r_{D}\) in terms of their lengths. The angles they form with the \(X\)-axis are, respectively, denoted as \(\gamma\) and \(\delta\). The light beam emitted from \(A\) with a wavelength of \(\lambda\) has a central main ray \(AO\), with a length denoted as \(r_{A}\) and an angle \(\alpha\) with the \(X\)-axis. After diffraction by the grating, the \(m\)-th order diffracted light forms an image at point \(B\) on the detector \(B_{1}B_{2}\) with the image distance from \(OB\) denoted as \(r_{B}\) and the diffraction angle as \(\beta\). \(OH\) is the perpendicular line from \(O\) to the spectrum surface \(B_{1}B_{2}\), with a length denoted as \(r_{H}\) and an angle with the \(X\)-axis as \(\beta_{H}\)[18]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Spectral range/nm & 400–950 \\ Diffraction order & +1 \\ Spectral resolution/nm & \(\leq\)3 \\ Grating density & 217 L/mm \\ Slit width/\(\upmu\)m & 50 \\ F-number & 2.27 \\ Spectrum length/mm & 20.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design specifications for a miniature spectrometer optical system. Figure 3: Common structure of a miniature spectrometer. (**a**) Cross-type C-T structure; (**b**) Flat-field concave holographic grating structure. Let \(P\) be any point on the plane concave holographic grating.