# Beam Propagation Factor of a Cosh-Airy Beam
## Appendix A  
\[ \begin{array}{c}T_{0}(a_{+},a_{-})=\int_{-\infty}^{\infty}[F_{1}^{*}(a_{+},\xi)+ F_{1}^{*}(a_{-},\xi)][F_{1}(a_{+},\xi)+F_{1}(a_{-},\xi)]d\xi\\ =\frac{w_{0}}{4\sqrt{2}n^{3/2}}\bigg{[}\sqrt{\frac{1}{a_{+}}}\exp(\frac{2\sigma _{+}^{3}}{3})+\sqrt{\frac{1}{a_{-}}}\exp(\frac{2\sigma_{-}^{3}}{3})+2\sqrt{ \frac{2}{a_{+}+a_{-}}}\exp(\frac{\sigma_{+}^{3}+a_{-}^{3}}{12}+\frac{a_{+} \sigma_{+}^{2}+a_{-}\sigma_{+}^{2}}{4})\bigg{]}\\ \end{array}  \tag{11}\]  
\[ \begin{array}{c}T_{1}(a_{+},a_{-})=\int_{-\infty}^{\infty}[F_{2}^{*}(a_{+}, \xi)+F_{2}^{*}(a_{-},\xi)][F_{1}(a_{+},\xi)+F_{1}(a_{-},\xi)]d\xi\\ =\frac{w_{0}^{2}}{4\sqrt{2}n^{3/2}}\bigg{\{}\sqrt{\frac{1}{a_{+}}}\exp(\frac{2 \sigma_{+}^{3}}{3})\Big{(}\!a_{+}^{2}-\frac{1}{4a_{+}}\!\Big{)}+\sqrt{\frac{ 1}{a_{-}}}\exp(\frac{2\sigma_{-}^{3}}{3})\Big{(}\!a_{-}^{2}-\frac{1}{4a_{-}} \!\Big{)}\\ +\sqrt{\frac{2}{a_{+}+a_{-}}}\exp(\frac{a_{+}^{3}+a_{-}^{3}}{12}+\frac{a_{+} \sigma_{+}^{2}+a_{-}\sigma_{+}^{2}}{4})\bigg{]}\!\Big{[}2a_{+}a_{-}-\frac{2a_{ +}+2a_{-}\sigma_{+}^{4}+2\sigma_{+}^{2}}{2(a_{+}+a_{-})^{2}}\!\Big{]}\!\Big{\}} \end{array}  \tag{12}\]  
\[ \begin{array}{c}T_{2}(a_{+},a_{-})=\int_{-\infty}^{\infty}[F_{2}^{*}(a_{+}, \xi)+F_{2}^{*}(a_{-},\xi)][F_{2}(a_{+},\xi)+F_{2}(a_{-},\xi)]d\xi\\ =\frac{w_{0}^{3}}{4\sqrt{2}n^{3/2}}\bigg{\{}\sqrt{\frac{1}{a_{+}}}\exp(\frac{2 \sigma_{+}^{3}}{3})\Big{(}\!a_{+}^{4}+\frac{a_{+}}{2}+\frac{3}{16a_{+}^{2}} \!\Big{)}+\sqrt{\frac{1}{a_{-}}}\exp(\frac{2\sigma_{-}^{3}}{3})\Big{(}\!a_{-} ^{4}+\frac{a_{-}}{2}+\frac{3}{16a_{-}^{2}}\!\Big{)}\\ +\sqrt{\frac{2}{a_{+}+a_{-}}}\exp(\frac{a_{+}^{3}+a_{-}^{2}}{12}+\frac{a_{+}^ {2}a_{-}^{2}+a_{+}\sigma_{+}^{2}}{4})\Big{(}\!2a_{+}a_{-}(a_{+}-a_{-})^{2}+2a_ {+}^{2}a_{-}^{2}+\frac{1}{8(a_{+},+a_{-})^{2}}\\ \end{array}  \tag{13}\]  
\[ \begin{array}{c}\times\!\!\Big{[}21(a_{+}^{6}+d_{-}^{6})-18(a_{+}^{5}a_{-} +a_{+}a_{-}^{5})-5(a_{+}^{4}a_{-}^{2}+a_{+}^{2}a_{-}^{4})+4a_{+}^{3}a_{-}^{3}- 20(a_{+}^{3}+a_{-}^{3})\\ +36(a_{+}^{2}a_{-}+a_{-}^{2})+12\Big{]}-(a_{+}^{6}-a_{-}^{6}-3a_{+}^{4}a_{-}^{ 2}+3a_{+}^{2}a_{-}^{4}-6a_{+}^{3}+6a_{-}^{3}-6a_{+}^{2}a_{-}+6a_{+}a_{-}^{2}) \frac{(a_{+}-a_{-})}{2(a_{+}+a_{-})^{3}}\!\Big{\}}\end{array}  \tag{14}\]  
\[\begin{array}{c}T_{3}(a_{+},a_{-})=\int_{-\infty}^{\infty}\!\!\Big{|}\frac{ 2\mathcal{E}(w_{0},0)}{\sigma_{0}}\Big{|}^{2}dx_{0}=\frac{1}{w_{0}^{2}}\!\! \int_{-\infty}^{\infty}[a_{+}F_{1}^{*}(a_{+},\xi)+a_{-}F_{1}^{*}(a_{-},\xi)+F_{ 3}^{*}(a_{+},\xi)+F_{3}^{*}(a_{-},\xi)]\\ \times\!\!\big{[}a_{+}F_{1}(a_{+},\xi)+a_{-}F_{1}(a_{-},\xi)+F_{3}(a_{+},\xi)+F _{3}(a_{-},\xi)\big{]}d\xi\\ =\frac{1}{16\sqrt{2}n^{3/2}w_{0}}\bigg{[}\frac{1}{a_{+}}\sqrt{\frac{1}{a_{+}}} \,\exp(\frac{2\sigma_{+}^{2}}{3})+\frac{1}{a_{-}}\,\sqrt{\frac{1}{a_{-}}}\, \exp(\frac{2\sigma_{-}^{2}}{3})+\frac{4a_{+}+4a_{-}-2a_{+}^{4}-2a_{+}^{4}+4a_{ +}^{2}a_{-}^{2}}{(a_{+}+a_{-})^{2}}\\ \times\sqrt{\frac{2}{a_{+}+a_{-}}  
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