# Research on Improving Optical Field Distribution of TOSA Based on 3D Defocusing Coupling
## 3 Design of 3D Defocusing Coupling Model
The ideal coupling efficiency of the spherical lens is [30]:  
\[\eta=\frac{\left|\iint\psi_{O}(x,y)\psi_{F}^{*}(x,y)dxdy\right|^{2}}{\iint\psi_{O} (x,y)\psi_{O}^{*}(x,y)dxdy\cdot\iint\psi_{F}(x,y)\psi_{F}^{*}(x,y)dxdy} \tag{10}\]  
\(\psi_{O}\) is the mode field distribution on the principal plane of the light emitted by the laser after passing through the ball lens, and \(\psi_{F}\) is the mode field distribution on the principal plane of the single-mode fiber. \(\psi_{O}^{*}\) and \(\psi_{F}^{*}\) are the complex conjugate distribution of the light source and the single-mode fiber on the principal plane, respectively. For single-mode fiber, the mode field radius of 1310 nm is 4.6 um. The coupling efficiency is the highest when \(\psi_{O}\) is closest to \(\psi_{F}\). Conversely, the highest the coupling efficiency is, the best \(\psi_{O}\) matches \(\psi_{F}\). According to the relationship between the optimal coupling efficiency and the optical field, in order to guarantee the uniform distribution of the optical field in the fiber, the TOSA must be coupled to the maximum optical power on the optical plane, that is, the coupling efficiency must be highest. In this paper, a 3D defocusing coupling model for a TOSA is established. The structure of the TOSA is shown in Figure 7b. Before coupling, the LC receptacle is linked to a power meter using fiber jumpers, and the power value is associated with the coordinate of the end-face of fiber. The end-face of fiber is moved up the X/Y plane and along the Z-axis with the power changing. For the first step, the end-face of fiber is placed near the focal point of the LD TO and moved up with coarse step on the X/Y plane before the power meter reads the optical power through the fiber jumper. Once a little optical power value (P1) is read at some coordinate, this coordinate point (X\({}_{1}\), Y\({}_{1}\)) is as a dot. The second step, the end-face of fiber is move up with fine step within radius 0.5 mm or smaller. The highest optical power (P\({}_{2}\)) and the coordinates (X\({}_{2}\), Y\({}_{2}\)) are obtained in this range. The third step, the end-face of fiber is moved up a fine step along the positive Z-axis and then repeats the first and the second steps. The highest optical power (P\({}_{3}\)) and the coordinates (X\({}_{3}\), Y\({}_{3}\)) are obtained in the plane. If P\({}_{3}\) > P\({}_{2}\), the end-face of fiber is moved up a fine step along the positive Z-axis again and repeated the first step and the second step. Then, the optical power (P\({}_{4}\)) is compared with P\({}_{3}\). If P\({}_{4}\) > P\({}_{3}\), the third step is repeated until the optical power (Pmax) is the highest. When the Pmax is within the specification of optical power, the receptacle is welded with a ring using laser welding, then the ring is welded on the body. If the Pmax is out of specification, the third step is repeated until the optical power is within the specification. Conversely, if P\({}_{3}\) < P\({}_{2}\), the end-face of fiber is moved up along the negative Z-axis, and repeated the first step and the second step. Then, the power (P\({}_{5}\)) is compared with P\({}_{3}\). If P\({}_{5}\) > P\({}_{3}\), the end-face of fiber is moved up a fine step along the positive Z-axis again, and repeated the first step and the second step until the optical power (Pmax\({}^{\prime}\)) is the highest. When the Pmax is within the specification of optical power, the receptacle is welded with a ring by laser welding, then the ring is welded on the body. If the Pmax is out of specification, the step is repeated until the optical power is within the specification. Figure 8 shows an optical simulation of the 3D defocusing coupling. When the end-face of fiber is at the focal point, the optical field is mainly concentrated in and around the core, and there is only little stray light in the cladding, which is caused by the 8\({}^{\circ}\) end-face of fiber.