# Optical Design and Stray Light Analysis of Underwater Spectral Radiometer
## 4 Stray Light Suppression and Simulation Analysis
### Analysis and Calculation of Stray Light Suppression Index  
According to the theory of radiation transmission in water, assuming \(E_{S}(\lambda)\) represents the spectral irradiance at the sea surface, and \(T_{s}\) is the transmittance at the sea-air interface, the downward spectral irradiance at a depth \(z\) below the sea surface is given by the following:  
\[E_{d}(\lambda,z)=T_{s}\times E_{S}(\lambda)e^{-K_{d}(\lambda)z} \tag{14}\]  
\(T_{c}\) represents the transmittance of the front optical system of the underwater spectral radiometer, entering the spectral irradiance at the entrance pupil of the spectrometer as follows:  
\[E_{d1}(\lambda,z)=T_{c}\times T_{s}\times E_{S}(\lambda)e^{-K_{d}(\lambda)z} \tag{15}\]  
Figure 8: Distribution of various diffraction orders. (**a**) Distribution of +1 order diffraction light; (**b**) Distribution of 0 order diffraction light; (**c**) Distribution of \(-1\) order diffraction light; (**d**) Distribution of +2 order diffraction light. The entrance radiance of the underwater spectral radiometer would be the following:  
\[L_{s}(\lambda)=E_{d1}(\lambda,z)/\pi \tag{16}\]  
\(F\) represents the \(F\)-number of the optical system, and \(\tau_{o}(\lambda)\) is the grating diffraction efficiency, then the spectral irradiance received by the detector is given by the following:  
\[E_{d}(\lambda)=\frac{\pi\tau_{o}(\lambda)L_{s}(\lambda)}{4F^{2}} \tag{17}\]  
The area of a detector pixel is denoted by \(A_{d}\), during the integration time \(T_{\text{int}}\), the energy received by the detector can be represented as follows:  
\[ \begin{array}{c}Q(\lambda)=E_{d}(\lambda)\times A_{d}\times T_{\text{int}} \times\Delta\lambda\\ =\frac{\pi\tau_{o}(\lambda)L_{s}(\lambda)}{4F^{2}}\times A_{d}\times T_{\text{ int}}\times\Delta\lambda\end{array}  \tag{18}\]  
Considering the quantum efficiency of the detector, the number of photoelectrons generated by the detector during the integration time is given by the following:  
\[S_{e}(\lambda)\ =\eta(\lambda)\times Q(\lambda)/\left(\frac{hc}{\lambda}\right) \tag{19}\]  
Translating the Equations (14)-(18) into Equation (19), the number of photo-generated electrons in the detector is as follows:  
\[S_{e}(\lambda,z)\ =T_{c}\times T_{s}\times\frac{A_{d}}{4F^{2}}\times\frac{ \lambda}{hc}\times T_{\text{int}}\times\tau_{o}(\lambda)\times E_{S}(\lambda) \times\eta(\lambda)\times\Delta\lambda\times e^{-K_{d}(\lambda)z} \tag{20}\]  
where \(\eta(\lambda)\) is the quantum efficiency, and \(\Delta\lambda\) is the spectral resolution, \(h\) is the Planck constant, and \(c\) is the speed of light, the definitions of the remaining quantities are given above. The detector chosen for this system is a silicon photodiode array. Considering only the detector's dark current noise, the signal-to-noise ratio of the underwater spectral radiometer is as follows:  
\[SNR_{1}=\frac{S_{e}}{\sqrt{S_{e}+N_{dark}^{2}}} \tag{21}\]  
Equation (22) gives the specific expression for dark current noise as follows:  
\[N_{dark}=\frac{I_{dark}\times T_{\text{int}}}{q} \tag{22}\]  
Let \(S_{R}\) be the number of electrons generated by stray light. The method for calculating the signal-to-noise ratio of the optical system under the influence of stray light is as follows:  
\[SNR=\frac{S_{e}}{S_{R}+\sqrt{S_{e}+N_{dark}^{2}}} \tag{23}\]  
In Ocean Class I water bodies, the 490 nm wavelength of light holds significant importance in aquatic observations, particularly due to its close correlation with the optical properties and biological relevance of water. Taking the downward spectral irradiance of sunlight at 490 nm in oceanic water as an example, analyzing the relationship between the signal-to-noise ratio of the system and the proportion of stray light, the relevant parameters are as follows:  
* The solar zenith angle is 0 degrees.