# A Novel Fluorescent-Material-Based Simple Method for Sunscreen Evaluation
## 4 Validation of Proposed Method
### Numerical Analysis  
Although the analytical model established in this study is not necessarily precise, it can verify the feasibility of the developed method and clarify the feasible conditions. The numerical model is composed of a fluorescent layer of a continuous medium with a thickness index of \(\tau_{f}\); the radiation is scattered and/or absorbed in this medium. The refractive index of the fluorescent layer is assumed to be 1.65, which is the same as that of the UV-curable resin used. At the interface between the layer and air (\(n=1.0\)), differences in the indices generate Fresnel reflections at the surface. Although the actual fluorescent layer is sandwiched between glass slides, the effect of the glass slides can be omitted because they have almost the same refractive index as the resin. Propagations of UV radiation and fluorescence within the layer are described by radiation transfer Equations (7) and (8), respectively [27] (Figure S2):  
\[\frac{dI_{e}(\tau,\mathbf{\Omega})}{d\tau}=-I_{e}(\tau,\mathbf{\Omega})+ \frac{\omega_{e}}{4\pi}\int_{4\pi}I_{e}\big{(}\tau,\mathbf{\Omega}^{\prime} \big{)}\Phi\big{(}\mathbf{\Omega}^{\prime}\rightarrow\mathbf{\Omega}\big{)}d \mathbf{\Omega}^{\prime}-(1-\omega_{e})\ Q\cdot I_{f}(\tau,\mathbf{\Omega}) \tag{7}\]  
\[\frac{dI_{f}(\tau,\mathbf{\Omega})}{d\tau}=-I_{f}(\tau,\mathbf{\Omega})+\frac {\omega_{f}}{4\pi}\int_{4\pi}I_{f}\big{(}\tau,\mathbf{\Omega}^{\prime}\big{)} \Phi\big{(}\mathbf{\Omega}^{\prime}\rightarrow\mathbf{\Omega}\big{)}d \mathbf{\Omega}^{\prime} \tag{8}\]  
where \(\mathbf{\Omega}\) is a unit vector representing the propagation direction of the radiation, and \(\tau\) is the coordinate along the direction of the distance index. \(I_{e}\) is the intensity of the UV radiation propagating within the layer. \(I_{f}\) is the intensity of the fluorescent radiation in the \(\mathbf{\Omega}\) direction that is produced by absorbing excitation radiation when \(I_{e}\) passes the microvolume at position \(\tau\) in the \(\mathbf{\Omega}\) direction, and \(\omega\) is the albedo, which is defined as follows. The radiation intensity decreases as the radiation passes through the scattering/absorbing medium according to the extinction coefficient \(\beta\), which can be expressed by Equation (8), where \(\sigma\) is the scattering coefficient and \(\alpha\) is the absorption coefficient. The albedo \(\omega\) is defined by Equation (9):  
\[\beta=\sigma+\alpha \tag{9}\]  
\[\omega=\frac{\sigma}{\beta} \tag{10}\]  
In other words, the albedo \(\omega\), is the fraction of the amount of scattered radiation out of the absorbed excitation radiation as it propagates within the medium. \(\omega_{e}\) is the albedo of the excitation radiation, whereas \(\omega_{f}\) is that of the fluorescent radiation. Fluorescent materials absorb only the UV radiation; the generated fluorescence is not absorbed but scattered. Therefore, the albedo of the fluorescent layer was assumed to be \(\omega_{f}=1\). \(Q\) is the fluorescence yield, which describes the probability of the material emitting fluorescence when an activated atom returns to the ground state. Although the actual fluorescence yield \(Q\) of this fluorescent material is unclear, the ratio of the measured fluorescence intensity is not affected by the value of \(Q\). Therefore, the analysis was performed as \(Q=1\). Generally, fluorescence is assumed to be emitted isotropically. When UV radiation and fluorescence are scattered in the fluorescent layer, they are assumed to be anisotropic scattering and follow the Henyey-Greenstein function \(\Phi(\mathbf{\Omega}^{\prime}\mathbf{\rightarrow}\mathbf{\Omega})\). The scattering phase function depends on the wavelength of the radiation. However, the difference in the scattering phase functions between the excitation radiation and fluorescence is considered negligible because these radiations are scattered many times within the fluorescent layer. Therefore, the same scattering phase function was assigned to both types of radiation in this model.