## 2 Micro-Imaging System  
### Ray Transfer Matrix of the Micro-Imaging System  
Figure 1 presents the optical configuration of the proposed system and shows the components in the optical path. The Z-axis of the coordinate system coincides with the optical axis. d\({}_{0}\) is the distance between a cover glass and a single nanoparticle; and d\({}_{1}\), d\({}_{2}\), and d\({}_{3}\) are the distances between two optical components, which are unknown parameters; and s\({}_{0}\), s\({}_{1}\), and s\({}_{3}\) are the optical component thicknesses, as shown in Figure 1, which are the given parameters. Based on these parameters, the optical matrixes of the proposed system are as follows:  
\[P_{0}=\left[\begin{array}{cccc}1&0&\text{d}_{0}&0\\ 0&1&0&\text{d}_{0}\\ 0&0&1&0\\ 0&0&0&1\end{array}\right]\,P_{1}=\left[\begin{array}{cccc}1&0&\text{d}_{1}&0 \\ 0&1&0&\text{d}_{1}\\ 0&0&1&0\\ 0&0&0&1\end{array}\right],P_{2}=\left[\begin{array}{cccc}1&0&\text{d}_{2}&0 \\ 0&1&0&\text{d}_{2}\\ 0&0&1&0\\ 0&0&0&1\end{array}\right],P_{3}=\left[\begin{array}{cccc}1&0&\text{d}_{3}&0 \\ 0&1&0&\text{d}_{3}\\ 0&0&1&0\\ 0&0&0&1\end{array}\right],P_{3}=\left[\begin{array}{cccc}1&0&\text{d}_{3}&0 \\ 0&1&0&\text{d}_{3}\\ 0&0&1&0\\ 0&0&0&1\end{array}\right],\]  
\[L_{0}=\left[\begin{array}{cccc}1&0&\text{n}_{\text{i}}\,\text{s}_{0}/\text {n}&0\\ 0&1&0&\text{n}_{\text{i}}\,\text{s}_{0}/\text{n}\\ 0&0&\text{n}_{\text{i}}/\text{n}_{\text{a}}&0\\ 0&0&0&\text{n}_{\text{i}}/\text{n}_{\text{a}}\end{array}\right],\]  
\[L_{1}=\left[\begin{array}{cccc}(\text{n}_{\text{a}}-\text{n})/\text{n}&0& \text{n}_{\text{a}}\,\text{s}_{1}/\text{n}&0\\ 0&(\text{n}_{\text{a}}-\text{n})/\text{n}&0&\text{n}_{\text{a}}\,\text{s}_{1}/ \text{n}\\ 4\text{n}_{\text{a}}(\text{n}_{\text{a}}-\text{n})/(\text{n}_{\text{a}}\,\text{ n}\,\text{s}_{1})&0&(\text{n}_{\text{a}}-\text{n})/\text{n}&0\\ 0&4\text{n}_{\text{a}}(\text{n}_{\text{a}}-\text{n})/(\text{n}_{\text{a}}\, \text{n}\,\text{s}_{1})&0&(\text{n}_{\text{a}}-\text{n})/\text{n}\end{array} \right],\]For the optical matrixes above, \(P_{0}\), \(P_{1}\), \(P_{2}\), and \(P_{3}\) are the transfer matrixes for propagation through the distances d\({}_{0}\), d\({}_{1}\), d\({}_{2}\), and d\({}_{3}\), respectively; and \(L_{0}\), \(L_{1}\), and \(L_{2}\) are the transfer matrixes of the cover glass, spherical lens, and cylindrical lens, respectively. Here, R\({}_{\rm c}\) is the radius of curvature of the cylindrical lens, n\({}_{\rm i}\), n\({}_{\rm a}\), and n are the refractive indexes of the measured sample, air, and optical glass, respectively, all of which are given parameters. The integrated ray transfer matrix of this micro-imaging system is as follows:  
\[\mathbf{W}=P3\cdot L2\cdot P2\cdot L1\cdot P1\cdot L0\cdot P0=\left[\begin{array} []{cccc}\mathrm{W}(1,1)&0&\mathrm{W}(1,3)&0\\ 0&\mathrm{W}(2,2)&0&\mathrm{W}(2,4)\\ \mathrm{W}(3,1)&0&\mathrm{W}(3,3)&0\\ 0&\mathrm{W}(4,2)&0&\mathrm{W}(4,4)\end{array}\right] \tag{1}\]  
where \(\mathbf{W}(\textit{row},\textit{col})\) indicates the parametric expression or constant value at row = _row_ and column = _col_ of the matrix \(\mathbf{W}\). Assuming the vector of a light ray from a point light source is \(X_{\rm e}=[x_{\rm e},y_{\rm e},a_{\rm e},b_{\rm e}]^{\rm T}\), where \(x_{\rm e}\) and \(y_{\rm e}\) are the X- and Y-axis coordinates, respectively, and \(a_{\rm e}\) and \(b_{\rm e}\) are the X- and Y-axis slopes of the corresponding ray, respectively, the corresponding vector of the light ray incident on the CCD is \(X_{\rm i}=[x_{\rm i},y_{\rm i},a_{\rm i},b_{\rm i}]^{\rm T}=\mathbf{W}\cdot X_{ \rm e}\).