## 2 Mathematical Formulation
\tag{31}\]  
Now in view of Equations (16a) and (31), we have  
\[A_{1}(\beta^{{}^{\prime}})=\varkappa_{-}(\beta^{{}^{\prime}},0^{+},\omega^{{}^{ \prime}})+\varkappa_{1}(\beta^{{}^{\prime}},0^{+},\omega^{{}^{\prime}})+ \varkappa_{+}(\beta^{{}^{\prime}},0^{+},\omega^{{}^{\prime}}), \tag{32}\]  
and  
\[-A_{1}(\beta^{{}^{\prime}})=\varkappa_{-}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{ \prime}})+\varkappa_{1}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}})+ \varkappa_{+}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}}). \tag{33}\]For detailed treatment of Equations (32) and (33) we refer the readers to the Appendix A of the paper and the unknown coefficient \(A_{1}\left(\beta^{{}^{\prime}}\right)\) is calculated to be:  
\[A_{1}\left(\beta^{{}^{\prime}}\right)=-\frac{1}{\gamma}\left[\frac{bm\gamma_{ \oplus}\left(-\beta^{{}^{\prime}}\right)\left[\exp-i(\beta^{{}^{\prime}}-k_{1} \cos\theta_{0})l\right]}{\gamma_{\ominus}\left(k_{1}\cos\theta_{0}\right)\left( \beta^{{}^{\prime}}-k_{1}\cos\theta_{0}\right)}\right]-bm\gamma_{\oplus}\left(- \beta^{{}^{\prime}}\right)\widetilde{R}_{2}\left(-\beta^{{}^{\prime}}\right) \exp(-i\beta^{{}^{\prime}}l)\]  
\[+bm\gamma_{\oplus}\left(-\beta^{{}^{\prime}}\right)\gamma_{\oplus}\left(k \right)\widetilde{T}\left(-\beta^{{}^{\prime}}\right)\widetilde{C}_{2}\exp(-i \beta^{{}^{\prime}}l)+\frac{iu_{0}g^{*}(\omega^{{}^{\prime}})\gamma_{\oplus} \left(-\beta^{{}^{\prime}}\right)\exp(-i\beta^{{}^{\prime}}l)}{\beta^{{}^{ \prime}}\gamma_{\oplus}\left(0\right)}\]  
\[+\frac{iu_{0}g^{*}(\omega^{{}^{\prime}})\gamma_{\oplus}\left(k\right)\gamma_{ \oplus}\left(-\beta^{{}^{\prime}}\right)\widetilde{T}\left(-\beta^{{}^{\prime }}\right)\widetilde{T}\left(k\right)\exp(-i\beta^{{}^{\prime}}l)}{k\left(1- \gamma_{\oplus}\left(k\right)\widetilde{T}^{2}\left(k\right)\right)}\left( \frac{1}{\gamma_{\oplus}\left(k\right)}-\frac{1}{\gamma_{\oplus}\left(0 \right)}\right)\]  
\[-\frac{\widetilde{T}\left(-\beta^{{}^{\prime}}\right)\widetilde{T}\left(k \right)\gamma_{\oplus}\left(k\right)\gamma_{\oplus}\left(-\beta^{{}^{\prime} }\right)iu_{0}g^{*}(\omega^{{}^{\prime}})\exp(-i\beta^{{}^{\prime}}l)}{k \left(1-\gamma_{\oplus}\left(k\right)\widetilde{T}^{2}\left(k\right)\right)} -\frac{bm\gamma_{\ominus}\left(-\beta^{{}^{\prime}}\right)}{\gamma_{\oplus} \left(k_{1}\cos\theta_{0}\right)\left(\beta^{{}^{\prime}}-k_{1}\cos\theta_{0} \right)}\]  
\[-bm\gamma_{\oplus}\left(\beta^{{}^{\prime}}\right)e^{ik_{1}l\cos\theta_{0} }\widetilde{R}_{1}\left(\beta^{{}^{\prime}}\right)-\frac{iu_{0}g^{*}(\omega^{ {}^{\prime}})\gamma_{\ominus}\left(\beta^{{}^{\prime}}\right)}{g^{{}^{\prime} }\gamma_{\ominus}\left(0\right)}+\frac{\gamma_{\oplus}\left(k\right)\gamma_{ \ominus}\left(\beta^{{}^{\prime}}\right)in_{0}g^{*}(\omega^{{}^{\prime}}) \widetilde{T}\left(\beta^{{}^{\prime}}\right)\widetilde{T}\left(k\right)}{k \left(1-\gamma_{\oplus}\left(k\right)\widetilde{T}^{2}\left(k\right)\right)} \left(\frac{1}{\gamma_{\ominus}\left(k\right)}-\frac{1}{\gamma_{\oplus}\left(0 \right)}\right)\]  
\[-\frac{\gamma_{\oplus}\left(k\right)\gamma_{\oplus}\left(\beta^{{}^{\prime}} \right)iu_{0}g^{*}(\omega^{{}^{\prime}})\widetilde{T}\left(\beta^{{}^{\prime} }\right)\widetilde{T}\left(k\right)}{k\left(1-\gamma_{\oplus}\left(k\right) \widetilde{T}^{2}\left(k\right)\right)}+bm\gamma_{\oplus}\left(k\right) \gamma_{\oplus}\left(\beta^{{}^{\prime}}\right)\widetilde{T}\left(\beta^{{}^{ \prime}}\right)\widetilde{C}_{1}. \tag{34}\]  
On inverting the Fourier transform over the variable \(\beta^{{}^{\prime}}\) in Equation (16b), the function \(\widetilde{\varphi}(x^{{}^{\prime}},y^{{}^{\prime}},\omega^{{}^{\prime}})\) can be obtained as  
\[\widetilde{\varphi}(x^{{}^{\prime}},y^{{}^{\prime}},\omega^{{}^{\prime}})= \frac{1}{2\pi}\int_{-}\ \varkappa(\beta^{{}^{\prime}},y^{{}^{\prime}},\omega^{{}^{\prime}})\exp \left(-i\beta^{{}^{\prime}}x^{{}^{\prime}}\right)d\beta^{{}^{\prime}}=\frac{1} {2\pi}\int_{-}\ A(\beta^{{}^{\prime}},\omega^{{}^{\prime}})\exp\left(-\gamma \left|y^{{}^{\prime}}\right|-i\beta^{{}^{\prime}}x^{{}^{\prime}}\right)d\beta^ {{}^{\prime}}. \tag{35}\]  
Now \(\widetilde{\varphi}(x^{{}^{\prime}},y^{{}^{\prime}},\omega^{{}^{\prime}})\) can be written as  
\[\widetilde{\varphi}(x^{{}^{\prime}},y^{{}^{\prime}},\omega^{{}^{\prime}})= \widetilde{\varphi}^{separated}(x^{{}^{\prime}},y^{{}^{\prime}},\omega^{{}^{ \prime}})+\widetilde{\varphi}^{interacted}(x^{{}^{\prime}},y^{{}^{\prime}}, \omega^{{}^{\prime}}), \tag{36}\]  
\[-A_{1}(\beta^{{}^{\prime}})=\varkappa_{-}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{ \prime}})+\varkappa_{1}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}})+ \varkappa_{+}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}}). \tag{33}\]0  
\[-A_{1}(\beta^{{}^{\prime}})=\varkappa_{-}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{ \prime}})+\varkappa_{1}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}})+ \varkappa_{+}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}}). \tag{33}\]1  
and  
\[-A_{1}(\beta^{{}^{\prime}})=\varkappa_{-}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{ \prime}})+\varkappa_{1}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}})+ \varkappa_{+}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}}). \tag{33}\]2  
\[-A_{1}(\beta^{{}^{\prime}})=\varkappa_{-}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{ \prime}})+\varkappa_{1}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}})+ \varkappa_{+}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}}). \tag{33}\]3  
\[-A_{1}(\beta^{{}^{\prime}})=\varkappa_{-}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{ \prime}})+\varkappa_{1}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}})+ \varkappa_{+}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}}). \tag{33}\]4  
\[-A_{1}(\beta^{{}^{\prime}})=\varkappa_{-}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{ \prime}})+\varkappa_{1}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}})+ \varkappa_{+}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}}). \tag{33}\]5\[-A_{1}(\beta^{{}^{\prime}})=\varkappa_{-}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{ \prime}})+\varkappa_{1}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}})+ \varkappa_{+}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}}). \tag{33}\]6 \[-A_{1}(\beta^{{}^{\prime}})=\varkappa_{-}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{ \prime}})+\varkappa_{1}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}})+ \varkappa_{+}(\beta^{{}^{\prime}},0^{-},\omega^{{}^{\prime}}). \tag{33}\]7  
Here \(\widetilde{\varphi}^{separated}(x^{{}^{\prime}},y^{{}^{\prime}},\omega^{{}^{ \prime}})\) gives the diffracted field produced by the edges at \(x^{{}^{\prime}}=0\) and \(-l\) and \(\widetilde{\varphi}^{interacted}(x^{{}^{\prime}},y^{{}^{\prime}},\omega^{{}^{ \prime}})\) accounts for an interaction field which is the impact of one edge over the other edge.