# Optical Design and Stray Light Analysis of Underwater Spectral Radiometer
## 2 Theoretical Foundation  
The spectral measurement of upward and downward irradiance and radiance in the ocean at different depths provides one of the best methods for studying the optical characteristics of seawater [10]. Compared to the measurement of spectral radiance on land, there are numerous factors that affect underwater radiance measurements. In the field measurements of water color remote sensing, the key challenges to address include a wide dynamic range, self-shadowing effects, and the signal-to-noise ratio issue [11]. The light entering the water is primarily composed of two parts: direct radiation from the sun and diffuse radiation from the sky. As sunlight passes through the sea surface and enters the seawater, the light undergoes attenuation due to the absorption and scattering within the water. According to the Beer-Lambert Law, light experiences exponential decay as it propagates through water. Taking the downwelling spectral irradiance in the water as an example, let us analyze the irradiance conditions observed by the underwater spectral radiometer. Assuming the water in the ocean is homogeneous, the downward spectral irradiance located z meters below the sea surface can be represented by Equation (1). \[E_{d}(\lambda,z)=E_{d}(\lambda,0^{-})e^{-K_{d}(\lambda)z} \tag{1}\]  
In Equation (1), \(E_{d}(\lambda,z)\) and \(E_{d}(\lambda,0^{-})\), respectively, represent the downward spectral irradiance at a depth of z meters below the sea surface and 0 m below the sea surface. \(K_{d}(\lambda)\) represents the attenuation coefficient of the water for a wavelength \(\lambda\). From Equation (1), it is evident that after passing through a certain depth of seawater, spectral radiation attenuation is quite pronounced. In order to measure underwater profile data at a specific depth, an underwater spectroradiometer should possess the capability of detecting faint oceanic light. In addition, the underwater light field distribution is determined by the inherent optical properties of the water. The presence of underwater optical instruments can interfere with the light field distribution around the instrument, and this effect is known as the instrument's self-shadowing effect. Self-shadowing refers to the shadow impact caused by the external structure of the sensor itself, and underwater optical instruments are generally affected by the self-shadowing effect [12]. Taking the measurement of upward spectral radiance as an example to analyze the impact of the instrument's self-shadowing effect on the measurement results, the shadow situation generated during underwater upward spectral radiance measurement is illustrated in Figure 1.  
Figure 1: Self-shadow in the instrument field of view. An underwater spectral radiometer is typically designed in a cylindrical structure, represented by the disk in Figure 1, which symbolizes the instrument's housing. When sunlight is incident at a specific solar zenith angle, a shadow is cast on the opposite side of the instrument [13]. When an underwater spectral radiometer measures the upward radiance spectrum \(L_{u}(\lambda)\) in the water, the instrument's field of view experiences a reduction in spectral energy due to self-shadowing. Therefore, the presence of shadows within the instrument's field of view inevitably leads to a discrepancy between the measured values and the actual values. The measurement error in the upward spectral irradiance \(E_{d}(\lambda)\) in the water, influenced by the instrument's characteristics, follows mechanisms similar to those described for the measurement of upward radiance \(L_{u}(\lambda)\). The key difference is that the solid angle for the observation field of upward spectral irradiance is \(2\pi\). Consequently, the measurement error caused by shadowing effects in upward spectral irradiance is much smaller than the radiance measurement error [14].