# Research on Improving Optical Field Distribution of TOSA Based on 3D Defocusing Coupling
## 2 Analysis of the Influence of Optical Field Distribution in TOSA  
### Factors Affecting the Optical Field  
LD beam quality and the TOSA coupling mode are the main factors affecting LD energy distribution in the single-mode fiber. Common parameters for evaluating LD beam quality include far-field the divergence angle, focal spot size, beam quality \(\beta\) factor, Strehle ratio, encircled energy ratio, \(M^{2}\) factor, etc. [9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23]. For a TOSA, the far-field divergence angle, focal spot size, and \(M^{2}\) factor are the main influencing factors. The far-field divergence angle (\(\theta\)) in the characteristic parameters of the Gaussian beam is defined as the angle corresponding to \(1/\mathrm{e}^{2}\) of the maximum intensity of the beam. When the laser transmits along the Z-axis, the far-field divergence angle (\(\theta\)) is:  
\[\theta=\lim_{z\rightarrow\infty}\frac{\omega(z)}{z} \tag{1}\]  
\[\omega_{(z)}=\omega_{0}\sqrt{1+\left(\frac{\lambda z}{\pi\omega_{0}{}^{2}} \right)^{2}} \tag{2}\]  
Equation (2) is substituted into Equation (1):  
\[\theta=\frac{\lambda}{\pi\omega_{0}} \tag{3}\]  
\(\omega(z)\) is the beam width (beam radius) at the Z-axis, and \(\omega_{0}\) is the waist radius. The far-field divergence angle represents the divergence of the beam. The larger the far-field divergence angle, the faster the beam diverges, the worse the quality of the beam, and the shorter the distance that the beam can transmit. Focal spot size (\(\omega f\)) refers to the size of the spot formed on the focal plane of the optical system after the laser beam passes through the focusing optical system. When the focal length of the focusing optical system is constant, the \(\omega f\) is proportional to the far-field divergence angle (\(\theta\)) of the measured beam. In this case, it is equivalent to using the size of the focal spot as the evaluation standard of the beam quality as the far field divergence angle. The larger the focal spot radius is, the larger the far field divergence angle is, and the worse the beam quality is. Taking the model in Figure 1 as a reference, the divergence angles \(\theta\bot\) is set as \(15^{\circ}\), \(20^{\circ}\), \(25^{\circ}\), and \(30^{\circ}\), respectively, and the corresponding RMS radius, geometric radius, and Airy spot radius were obtained by simulation, as shown in Table 1.  
Beam quality \(\beta\) is defined as the ratio of the far-field divergence angle of the actual beam to the ideal far-field divergence angle. \(\beta\) is generally greater than 1. The closer to 1, the higher the beam quality is, and \(\beta=1\) is the maximum. \(\beta\) is mainly used to evaluate the laser beam emitted from the laser resonator, which can reasonably evaluate the near-field beam quality. It is a static performance index to describe the laser beam quality without considering the effects of atmospheric scattering, turbulence, and thermal halo. The laser beam quality factor (\(M^{2}\)) is defined as the product of the waist width of the actual beam and the far-field divergence angle divided by the product of the waist width of the fundamental mode Gaussian beam and the far-field divergence angle. \[M^{2}=\frac{\omega\theta}{\omega_{0}\,\theta_{0}} \tag{4}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\theta_{\bot}\) & **RMS Radius (\(\upmu\))** & \
\begin{tabular}{c} **Geometric Radius** \\ **(\(\upmu\))** \\ \end{tabular} &
\begin{tabular}{c} **Airy Spot Radius** \\ **(\(\upmu\))** \\ \end{tabular} \\ \hline \(15^{\circ}\) & 4.845 & 6.468 & 8.288 \\ \(20^{\circ}\) & 13.305 & 30.01 & 5.762 \\ \(25^{\circ}\) & 35.672 & 80.889 & 4.235 \\ \(30^{\circ}\) & 77.143 & 172.84 & 3.152 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variation of far-field divergence angle and focal spot. Figure 1: Optical path model of TOSA.