## Appendix C First and Second Order Derivatives of \(B\left(\theta_{1},\theta_{2}\right)\)  
\[B\left(\theta_{1},\theta_{2}\right)=\frac{1-\exp\left[j\frac{2\pi}{\lambda} \Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right]}{1-\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)\right]}\equiv \frac{N\left(\theta_{1},\theta_{2}\right)}{D\left(\theta_{1},\theta_{2}\right)} \tag{26}\]  
\[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}}=\left(j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{1}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{27}\]  
\[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}}=\left(-j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{2}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{28}\]  
\[\frac{\partial^{2}N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}^{2}} = \left(-j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{1}\right)M \right)\exp\left(j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1 }\right)M\right) \tag{29}\] \[+\left(\frac{2\pi}{\lambda}\Delta\left(\cos\theta_{1}\right)M \right)^{2}\exp\left(j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin \theta_{1}\right)M\right)\]  
\[\frac{\partial^{2}N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}^{2}} = \left(j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{2}\right)M \right)\exp\left(j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1 }\right)M\right) \tag{30}\] \[+\left(\frac{2\pi}{\lambda}\Delta\left(\cos\theta_{2}\right)M \right)^{2}\exp\left(j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1 }\right)M\right)\]  
\[\frac{\partial^{2}N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}d\theta_{2}}=- \left(\frac{2\pi}{\lambda}\Delta M\right)^{2}\left(\cos\theta_{1}\right) \left(\cos\theta_{2}\right)\exp\left(j\frac{2\pi}{\lambda}\Delta\left(\sin \theta_{2}-\sin\theta_{1}\right)M\right) \tag{31}\]  
\[\frac{\partial D\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}}=\left(j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{1}\right)\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)\right] \tag{32}\]\[\frac{\partial D\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}} = \left(-j\frac{2\pi}{\lambda}\Delta\left(\cos\theta_{2}\right) \right)\exp\left[j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1 }\right)\right]\] (A33) \[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}}=\left(j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{1}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{27}\]0 (A34) \[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}}=\left(j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{1}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{27}\]1  
\[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}}=\left(j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{1}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{27}\]2 (A35) \[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}}=\left(j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{1}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{27}\]3  
\[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}}=\left(j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{1}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{27}\]4 (A36) \[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}}=\left(j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{1}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{27}\]5 (A37)  
\[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}}=\left(j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{1}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{27}\]6 (A38) \[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}}=\left(j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{1}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{27}\]7  
\[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}}=\left(j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{1}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{27}\]8 (A39) \[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}}=\left(j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{1}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{27}\]9  
\[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}}=\left(-j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{2}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{28}\]0 (A40) \[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}}=\left(-j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{2}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{28}\]1 (A41)  
\[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}}=\left(-j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{2}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{28}\]2 (A42) \[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}}=\left(-j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{2}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{28}\]3\[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}}=\left(-j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{2}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{28}\]4 \[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}}=\left(-j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{2}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{28}\]5  
\[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}}=\left(-j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{2}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{28}\]6 \[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}}=\left(-j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{2}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{28}\]7 \[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}}=\left(-j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{2}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{28}\]8  
The first order and the second order derivatives of \(a_{k}\left(\theta_{1}\right)\) and \(a_{k}\left(\theta_{2}\right)\) are given in Appendix D.  
Appendix D First and Second Order Derivatives of \(a_{k}\left(\theta_{1}\right)\) and \(a_{k}\left(\theta_{2}\right)\)  
\[\frac{\partial N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}}=\left(-j \frac{2\pi}{\lambda}\Delta\left(\cos\theta_{2}\right)M\right)\exp\left[j\frac{2 \pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1}\right)M\right] \tag{28}\]9 \[\frac{\partial^{2}N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}^{2}} = \left(-j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{1}\right)M \right)\exp\left(j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1 }\right)M\right) \tag{29}\]0 (107) \[\frac{\partial^{2}N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}^{2}} = \left(-j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{1}\right)M \right)\exp\left(j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1 }\right)M\right) \tag{29}\]1 (108) \[\frac{\partial^{2}N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}^{2}} = \left(-j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{1}\right)M \right)\exp\left(j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1 }\right)M\right) \tag{29}\]2  
Appendix E Derivations of \(f_{kl}\left(\theta_{1},\theta_{2}\right)\) and \(g_{kl}\left(\theta_{1},\theta_{2}\right)\)  
Let \(f_{kl}\left(\theta_{1},\theta_{2}\right)\) and \(g_{kl}\left(\theta_{1},\theta_{2}\right)\) denote the partial derivatives of \(P_{kl}\left(\theta_{1},\theta_{2}\right)\) with respect to \(\theta_{1}\) and \(\theta_{2}\), respectively:  
\[\frac{\partial^{2}N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}^{2}} = \left(-j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{1}\right)M \right)\exp\left(j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1 }\right)M\right) \tag{29}\]3 \[\frac{\partial^{2}N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}^{2}} = \left(-j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{1}\right)M \right)\exp\left(j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1 }\right)M\right) \tag{29}\]4\[\frac{\partial^{2}N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}^{2}} = \left(-j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{1}\right)M \right)\exp\left(j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1 }\right)M\right) \tag{29}\]5 (A50) \[\frac{\partial^{2}N\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}^{2}} = \left(-j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{1}\right)M \right)\exp\left(j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{2}-\sin\theta_{1 }\right)M\right) \tag{29}\]6  
**Appendix F. Derivations of \(f_{kl}\left(\theta_{1},\theta_{2}\right)\) and \(g_{kl}\left(\theta_{1},\theta_{2}\right)\) with the Tayler Series Approximation**  
Applying the Taylor series expansion in (A49) and (A50) yields  
\[f_{kl}\left(\hat{\theta}_{1},\hat{\theta}_{2}\right) = f_{kl}\left(\theta_{1}^{(0)},\theta_{2}^{(0)}\right)+\delta \theta_{1}^{(u=1)}\left(\left.\frac{\partial f_{kl}\left(\theta_{1},\theta_{2 }\right)}{\partial\theta_{1}}\right|_{