# Design of 4 \(\times\) 1 Power Beam Combiner Based on MultiCore Photonic Crystal Fiber
## 2 Principle of the Work  
Figure 1 shows a schematic sketch of the multicore PCF 4 \(\times\) 1 power combiner design based on the coupling between five cores (yellow color). The coefficients \(\kappa_{1},\kappa_{2}\) are the coupling coefficients between the cores: \(\kappa_{1}\) denotes the coupling between core 3 and core 1, or between core 2 and core 1. \(\kappa_{2}\) denotes the coupling between core 4 and core 1, or between core 5 and core 1. \(d\) is the diameter of the air-holes (white color), \(d\) is the diameter of the small air-holes (black color) and \(\Lambda\) denotes the pitch--the distance between two air holes. Classical coupled-mode equations can be used to analyze the light coupling between the five cores in the multicore PCF 4 \(\times\) 1 power combiner. In our model, we integrated eight small air holes (black color), which leads to cancellation of other coupling that may occur between closer cores such as core 3-core 5, core 3-core 4, core 2-core 4, and core 2-core 5, and by assuming polarization independence, the solutions of the mode amplitudes can be given as  
\[\mathrm{A}_{r}=\alpha_{n}\mathrm{e}^{\left(-\mathrm{i}\mu\left(\beta+\epsilon \right)\right)} \tag{1}\]  
where \(\mathrm{A}_{r}\) (\(r\) = 1, 2, 3, 4, 5) are the amplitudes of the fundamental mode in core \(r\), \(\beta\) is the propagation constant of the fundamental propagation mode, z is the propagation distance, \(\alpha_{n}\) (\(n=1,2,3,4,5\)) is  
Figure 1: Schematic sketch of the 4 \(\times\) 1 power combiner design at the \(x\)–\(y\) plane. the amplitude constant, and \(\varepsilon\) is an eigenvalue. In our design, there are five cores, which mean that five coupled-mode equations can be used to describe our design. However, the combiner is based on multicore PCF with symmetrical hexagonal structure; in other words, the mode amplitude is equal in core 2-core 3 (A\({}_{2}\) = A\({}_{3}\)) and core 4-core 5 (A\({}_{4}\) = A\({}_{5}\)). Therefore, the five coupled-mode equations can be reduced to three equations which are given by  
\[\frac{d(\alpha_{1}\mathrm{e}^{(-|\mathrm{z}(\beta+\varepsilon))})}{dz}+\mathrm{ j}\beta\alpha_{1}\mathrm{e}^{(-|\mathrm{z}(\beta+\varepsilon))}=-\mathrm{j} \Big{\{}2\alpha_{2}\mathrm{e}^{(-|\mathrm{z}(\beta+\varepsilon))}\kappa_{1}+2 \alpha_{4}\mathrm{e}^{(-|\mathrm{z}(\beta+\varepsilon))}\kappa_{2}\Big{\}} \tag{2}\]  
\[\frac{d(\alpha_{2}\mathrm{e}^{(-|\mathrm{z}(\beta+\varepsilon))})}{dz}+\mathrm{ j}\beta\alpha_{2}\mathrm{e}^{(-|\mathrm{z}(\beta+\varepsilon))}=-\mathrm{j} \alpha_{1}\mathrm{e}^{(-|\mathrm{z}(\beta+\varepsilon))}\kappa_{1} \tag{3}\]  
\[\frac{d(\alpha_{4}\mathrm{e}^{(-|\mathrm{z}(\beta+\varepsilon))})}{dz}+\mathrm{ j}\beta\alpha_{4}\mathrm{e}^{(-|\mathrm{z}(\beta+\varepsilon))}=-\mathrm{j} \alpha_{1}\mathrm{e}^{(-|\mathrm{z}(\beta+\varepsilon))}\kappa_{2} \tag{4}\]  
where the boundary conditions are given by  
\[\mathrm{A}_{1}(\mathrm{z}=0)=0,\mathrm{A}_{2}(\mathrm{z}=0)=\mathrm{A}_{4}( \mathrm{z}=0)=0.25 \tag{5}\]  
The three coupled-mode Equations (2)-(4) can be simplified to three linear equations, which are given by  
\[\varepsilon\alpha_{1}=2\alpha_{2}\kappa_{1}+2\alpha_{4}\kappa_{2} \tag{6}\]  
\[\varepsilon\alpha_{2}=\alpha_{1}\kappa_{1} \tag{7}\]  
\[\varepsilon\alpha_{4}=\alpha_{1}\kappa_{2} \tag{8}\]  
The matrix system can be described as follows:  
\[\left[ \begin{array}{ccc}\varepsilon&-2\kappa_{1}&-2\kappa_{2}\\ -\kappa_{1}&\varepsilon&0\\ -\kappa_{2}&0&\varepsilon\end{array} \right]\left[ \begin{array}{c}\alpha_{1} \\ \alpha_{2}\\ \alpha_{4}\end{array} \right]=\left[ \begin{array}{c}0\\ 0\\ 0\end{array} \right] \tag{9}\]  
The eigenvalues and the eigenvectors can be found by solving the matrix system (Equation (9)). The field E(z) can be represented by a linear combination of the eigenvectors. In a particular solution, where \(\kappa_{1}=\kappa_{2}\), a complete transfer of the energy from cores 2, 3, 4, and 5 to the central core 1, can be obtained.