## 2 Theory
### M\({}^{2}\)-Factor
\begin{split}\langle r_{\text{sx}}^{2}\rangle=\frac{\pi w_{0}^{2}}{P} \sum\limits_{l_{1}=1}^{N}\sum\limits_{l_{2}=1}^{N}\sum\limits_{l_{1}=-\frac{l_ {1}-1}{2}}^{\frac{l_{1}-1}{2}}\sum\limits_{l_{2}=-\frac{l_{2}-1}{2}}^{\frac{l_ {2}-1}{2}}\sum\limits_{p_{1}=-\frac{p_{1}-1}{2}}^{\frac{p_{1}-1}{2}}\sum\limits _{p_{2}=-\frac{p_{1}-1}{2}}^{\frac{p_{1}-1}{2}}\frac{(-1)^{l_{1}+l_{2}}}{N^{2}} \Bigg{(}\begin{array}{c}N\\ l_{1}\end{array}\Bigg{)}\Bigg{(}\begin{array}{c}N\\ l_{2}\end{array}\Bigg{)}\\ \times\frac{\alpha s(2\pi^{2}TB^{2}z/3-2\alpha_{3}A^{2}-A^{2}\beta_{3}^{2})}{1+ l_{1}+l_{2}}\\ \Big{\langle}r_{\text{sy}}^{2}\Big{\rangle}=\frac{\pi w_{0}^{2}}{P} \sum\limits_{l_{1}=1}^{N}\sum\limits_{l_{2}=1}^{N}\sum\limits_{l_{1}=-\frac{ l_{1}-1}{2}}^{\frac{l_{1}-1}{2}}\sum\limits_{p_{2}=-\frac{p_{1}-1}{2}}^{\frac{l_{2}-1}{2}} \sum\limits_{p_{1}=-\frac{p_{1}-1}{2}}^{\frac{p_{1}-1}{2}}\sum\limits_{p_{2}=- \frac{p_{1}-1}{2}}^{\frac{p_{1}-1}{2}}\frac{(-1)^{l_{1}+l_{2}}}{N^{2}}\Bigg{(} \begin{array}{c}N\\ l_{1}\end{array}\Bigg{)}\Bigg{(}\begin{array}{c}N\\ l_{2}\end{array}\Bigg{)}\\ \times\frac{\alpha s(2\pi^{2}TB^{2}z/3-2\alpha_{3}A^{2}-A^{2}\beta_{4}^{2})}{l_ {1}+l_{2}}\\ \langle\theta_{\text{x}}^{2}\rangle=\frac{\pi w_{0}^{2}}{P} \sum\limits_{l_{1}=1}^{N}\sum\limits_{l_{2}=1}^{N}\sum\limits_{l_{1}=-\frac{ l_{1}-1}{2}}^{\frac{l_{1}-1}{2}}\sum\limits_{l_{2}=-\frac{l_{1}-1}{2}}^{\frac{l_{1}-1}{2}} \sum\limits_{p_{1}=-\frac{p_{1}-1}{2}}^{\frac{p_{1}-1}{2}}\frac{(-1)^{l_{1}+l_ {2}}}{N}\Bigg{(}\begin{array}{c}N\\ l_{1}\end{array}\Bigg{)}\Bigg{(}\begin{array}{c}N\\ l_{2}\end{array}\Bigg{)}\frac{\alpha_{5}}{l_{1}+l_{2}}\\ \Bigg{\{}\frac{2\alpha A_{4}(\alpha_{2}-2A\alpha_{3}\alpha_{4})-2A^{2}\alpha_{3 }\alpha_{4}^{2}-2\alpha_{1}}{k^{2}}+\frac{\pi^{2}Tz}{3}[-2A\alpha_{4}(\frac{2B }{A^{2}k}+\frac{B}{Ak}-\frac{2B^{2}\alpha_{4}}{Ak^{2}})\\ -\frac{2\alpha_{4}^{2}B^{2}}{k^{2}}+(\frac{2}{A^{2}}+\frac{2}{A}+2)]-(\frac{ \beta_{1}-A\alpha_{4}\beta_{3}}{k})^{2}\Bigg{\}}\\ \Big{\langle}\theta_{\text{y}}^{2}\Big{\rangle}=\frac{\pi w_{0}^{2}}{P} \sum\limits_{l_{1}=1}^{N}\sum\limits_{l_{2}=1}^{N}\sum\limits_{l_{1}=-\frac{ l_{1}-1}{2}}^{\frac{l_{1}-1}{2}}\sum\limits_{l_{2}=-\frac{l_{1}-1}{2}}^{\frac{l_{1}-1}{2}} \sum\limits_{p_{1}=-\frac{p_{1}-1}{2}}^{\frac{p_{1}-1}{2}}\frac{(-1)^{l_{1}+l_ {2}}}{N}\Bigg{(}\begin{array}{c}N\\ l_{1}\end{array}\Bigg{)}\Bigg{(}\begin{array}{c}N\\ l_{2}\end{array}\Bigg{)}\frac{\alpha_{5}}{l_{1}+l_{2}}\\ \Bigg{\{}\frac{2A\alpha_{4}(\alpha_{2}-2A\alpha_{3}\alpha_{4})+2A^{2}\alpha_{3 }\alpha_{4}^{2}-2\alpha_{1}-}{k^{2}}+\frac{\pi^{2}Tz}{3}[-2A\alpha_{4}(\frac{2 B}{A^{2}k}+\frac{B}{Ak}-\frac{2B^{2}\alpha_{4}}{Ak^{2}})\\ -\frac{2\alpha_{4}^{2}B^{2}}{k^{2}}+(\frac{2}{A^{2}}+\frac{2}{A}+2)]-(\frac{ \beta_{2}-A\alpha_{4}\beta_{4}}{k})^{2}\Bigg{\}}\\ \langle r_{\text{sx}},\theta_{\text{x}}\rangle=\frac{\pi w_{0}^{2}}{P} \sum\limits_{l_{1}=1}^{N}\sum\limits_{l_{2}=1}^{N}\sum\limits_{l_{1}=-\frac{ l_{1}-1}{2}}^{\frac{l_{1}-1}{2}}\sum\limits_{l_{2}=-\frac{l_{1}-1}{2}}^{\frac{l_{1}-1}{2}} \sum\limits_{p_{1}=-\frac{p_{1}-1}{2}}^{\frac{p_{1}-1}{2}}\frac{(-1)^{l_{1}+l_ {2}}}{N^{2}}\Bigg{(}\begin{array}{c}N\\ l_{1}\end{array}\Bigg{)}\Bigg{(}\begin{array}{c}N\\ l_{2}\end{array}\Bigg{)}\frac{\alpha_{5}}{l_{1}+l_{2}}\\ \times\bigg{[}\frac{A(\alpha_{2}-2A\alpha_{3}\alpha_{4})}{k}+\frac{\pi^{2}Tz}{3} (\frac{2B^{2}\alpha_{4}}{k}-\frac{2B}{A}-B)+\frac{A\beta_{3}}{k}(\beta_{1}-A \alpha_{4}\beta_{3})\bigg{]}\\ \langle r_{\text{sy}},\theta_{\text{y}}\rangle=\frac{\pi w_{0}^{2}}{P} \sum\limits_{l_{1}=1}^{N}\sum\limits_{l_{2}=1}^{N}\sum\limits_{l_{1}=-\frac{l_{1}-1 }{2}}^{\frac{l_{1}-1}{2}}\sum\limits_{l_{2}=-\frac{l_{1}-1}{2}}^{\frac{l_{1}-1}{ 2}}\sum\limits_{p_{1}=-\frac{p_{1}-1}{2}}^{\frac{p_{1}-1}{2}}\frac{(-1)^{l_{1}+l_ {2}}}{N^{2}}\Bigg{(}\begin{array}{c}N\\ l_{1}\end{array}\Bigg{)}\Bigg{(}\begin{array}{c}N\\ l_{2}\end{array}\Bigg{)}\frac{\alpha_{5}}{l_{1}+l_{2}}\\ \times\bigg{[}\frac{A(\alpha_{2}-2A\alpha_{3}\alpha_{4})}{k}+\frac{\pi^{2}Tz}{3} (\frac{2B^{2}\alpha_{4}}{k}-\frac{2B}{A}-B)+\frac{A\beta_{4}}{k}(\beta_{2}-A \alpha_{4}\beta_{4})\bigg{]}\\ \end{split}