## 2 Materials and Methods  
As shown in Figure 1, the STMC is made of two identical two-mode fibers (Corning SMF-28e, core/cladding = 8.2/125 \(\upmu\)m, roore = 1.4637, ncladding = 1.457), and completes the excitation of the LP\({}_{11}\) mode in the conical region and the output of the LP\({}_{11}\) mode in the coupling region to realize the conversion of the LP\({}_{01}\) mode to the LP\({}_{11}\) mode. Figure 1: The structure and principle of STMC. The coupling region working mechanism of the STMC conforms to mode-coupling theory, and the coupling equation is [36] as follows:  
\[\frac{dA(z)}{dz}=jk_{21}B(z)e^{i(\beta_{A}-\beta_{B})z} \tag{1}\]  
\[\frac{dB(z)}{dz}=jk_{12}A(z)e^{i(\beta_{B}-\beta_{A})z} \tag{2}\]  
In the equation, \(A(z)\) and \(B(z)\) are the power distributions of the two modes along the axis; \(\beta\) is the propagation constant; \(z\) is the distance that light travels along the coupling region; and \(K\) is the coupling coefficient. If light is incident only from the first two-mode fiber (TMF), the boundary conditions \(A(0)=1\) and \(B(0)=0\) are met. From this, Equation (2) is integrated:  
\[B(L)=jk_{12}\int_{0}^{L}A(z)e^{i(\beta_{B}-\beta_{A})z}dz \tag{3}\]  
It can be seen that when the light transmission distance is \(L\) in the second FMF or TMF, a light wave field with power \(B(L)\) is established. When \(\beta_{A}-\beta_{B}\
eq 0\), Equation (3) is equal to an effective value within the coupling distance \(L\), that is, only if there are two modes or the same mode with similar propagation constants (equal effective refractive index) can a valid value be obtained, in which case effective coupling occurs. In traditional mode selection couplers, direct coupling between LP\({}_{01}\) and LP\({}_{11}\) modes occurs, while in the STMC, taper pulling of the TMF can excite the LP\({}_{11}\) mode in the fiber [37]; therefore, there is a coupling of two modes in the coupling region of the STMC, that is, the coupling between the two LP\({}_{01}\) modes and the coupling between the two LP\({}_{11}\) modes. By controlling the cone size of the TMF, the LP\({}_{01}\) mode can be limited to the fiber core with minimal coupling to the other fiber, and the LP\({}_{11}\) mode can be leaked into the cladding in order to be more coupled; specific size control is shown in the simulation analysis below. We used the finite element method (FEM) to calculate the variation in the effective refractive index of LP\({}_{01}\) and LP\({}_{11}\) modes with the fiber diameter (cladding diameter) as in the SMF-28e fiber; the results are shown in Figure 2. According to the principle of total internal reflection, the condition under which the mode is tethered to the core in the SMF-28e fiber is \(1.457<\) neff \(<\) 1.4637. When the effective refractive index of the mode is less than 1.457, the mode leaks into the cladding. As shown in Figure 2, (The illustration in Figure 2 is a part of the figure in Figure 2. Its actual meaning is the distribution of mode LP\({}_{01}\) and mode LP\({}_{11}\) wh  
Figure 2: The change curve of effective refractive index with fiber diameter. 15-85 \(\upmu\)m, the blue line is the cladding index of TMF.) by controlling the tensile diameter of the fiber, the LP\({}_{01}\) mode in the TMF can be bound in the core, and the LP\({}_{11}\) mode leaks into the cladding, so that the LP\({}_{01}\) mode is separated from the LP\({}_{11}\) mode. According to the simulation results, when the fiber diameter is 75 \(\upmu\)m, the effective refractive index of the LP\({}_{11}\) mode in the TMF is 1.457; at this time, the LP\({}_{11}\) mode begins to leak into the cladding, and LP\({}_{11}\) mode coupling begins to occur in the coupling region of the STMC, while the effective refractive index of the LP\({}_{01}\) mode is 1.457 \(<\) neff \(<\) 1.4637. At this time, the LP\({}_{01}\) mode is bound to the core for transmission; therefore, the coupling of the LP\({}_{01}\) mode occurs only to a minor degree in the coupling region of the STMC.