Generated by the Physics Derivation Graph. Eq. \ref{eq:4809503} is an initial equation. \begin{equation} Z = |Z| \exp( i \theta ) \label{eq:4809503} \end{equation} Conjugate both sides of Eq. \ref{eq:4809503}; yields Eq. \ref{eq:5663009}. \begin{equation} Z^* = |Z| \exp( -i \theta ) \label{eq:5663009} \end{equation} Multiply Eq. \ref{eq:4809503} by Eq. \ref{eq:5663009}; yields Eq. \ref{eq:4577339}. \begin{equation} Z Z^* = |Z| |Z| \exp( -i \theta ) \exp( i \theta ) \label{eq:4577339} \end{equation} Simplify Eq. \ref{eq:4577339}; yields Eq. \ref{eq:4362190}. \begin{equation} Z Z^* = |Z|^2 \label{eq:4362190} \end{equation} Eq. \ref{eq:6461198} is an initial equation. \begin{equation} I = | A + B |^2 \label{eq:6461198} \end{equation} Change variable \(Z\) to \(A + B\) in Eq. \ref{eq:4362190}; yields Eq. \ref{eq:4137499}. \begin{equation} (A + B)(A + B)^* = |A + B|^2 \label{eq:4137499} \end{equation} Substitute LHS of Eq. \ref{eq:4137499} into Eq. \ref{eq:6461198}; yields Eq. \ref{eq:9192406}. \begin{equation} I = (A + B)(A + B)^* \label{eq:9192406} \end{equation} Distribute conjugate to factors in Eq. \ref{eq:9192406}; yields Eq. \ref{eq:2300056}. \begin{equation} I = (A + B)(A^* + B^*) \label{eq:2300056} \end{equation} Simplify Eq. \ref{eq:2300056}; yields Eq. \ref{eq:9934418}. \begin{equation} I = A A^* + B B^* + A B^* + B A^* \label{eq:9934418} \end{equation} Change variable \(Z\) to \(A\) in Eq. \ref{eq:4362190}; yields Eq. \ref{eq:3404497}. \begin{equation} A A^* = |A|^2 \label{eq:3404497} \end{equation} Change variable \(B\) to \(Z\) in Eq. \ref{eq:4362190}; yields Eq. \ref{eq:2303305}. \begin{equation} B B^* = |B|^2 \label{eq:2303305} \end{equation} Substitute LHS of Eq. \ref{eq:3404497} into Eq. \ref{eq:9934418}; yields Eq. \ref{eq:4729665}. \begin{equation} I = |A|^2 + B B^* + A B^* + B A^* \label{eq:4729665} \end{equation} Substitute LHS of Eq. \ref{eq:2303305} into Eq. \ref{eq:4729665}; yields Eq. \ref{eq:8296872}. \begin{equation} I = |A|^2 + |B|^2 + A B^* + B A^* \label{eq:8296872} \end{equation} Change variable \(Z = |Z| \exp( i \theta )\) to \(Z\) and \(B\) to \(5513927\) in Eq. \ref{eq:7875296}; yields Eq. \ref{eq:#6}. \begin{equation} B = |B| \exp(i \phi) \label{eq:7875296} \end{equation} Change variable \(A\) to \(Z\) in Eq. \ref{eq:4809503}; yields Eq. \ref{eq:2018605}. \begin{equation} A = |A| \exp(i \theta) \label{eq:2018605} \end{equation} Conjugate both sides of Eq. \ref{eq:2018605}; yields Eq. \ref{eq:1584527}. \begin{equation} A^* = |A| \exp(-i \theta) \label{eq:1584527} \end{equation} Conjugate both sides of Eq. \ref{eq:7875296}; yields Eq. \ref{eq:1174231}. \begin{equation} B^* = |B| \exp(-i \phi) \label{eq:1174231} \end{equation} Substitute LHS of Eq. \ref{eq:7875296} and LHS of Eq. \ref{eq:1174231} and LHS of Eq. \ref{eq:2018605} and LHS of Eq. \ref{eq:1405078} into Eq. \ref{eq:#5}; yields Eq. \ref{eq:#6}. \begin{equation} I = |A|^2 + |B|^2 + |A| |B| \exp(-i \theta) \exp(i \phi) + |A| |B| \exp(i \theta) \exp(-i \phi) \label{eq:1405078} \end{equation} Simplify Eq. \ref{eq:1405078}; yields Eq. \ref{eq:5595798}. \begin{equation} I = |A|^2 + |B|^2 + |A| |B| \exp(i (\theta - \phi)) + |A| |B| \exp(-i (\theta - \phi)) \label{eq:5595798} \end{equation} Change variable \(x\) to \(\theta - \phi\) in Eq. \ref{eq:7002927}; yields Eq. \ref{eq:9190817}. \begin{equation} \cos(x) = \frac{1}{2} \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right) \label{eq:9190817} \end{equation} Multiply both sides of Eq. \ref{eq:9190817} by \(2\); yields Eq. \ref{eq:8635275}. \begin{equation} 2 \cos(x) = \left( \exp(i (\theta - \phi)) + \exp(-i (\theta - \phi)) \right) \label{eq:8635275} \end{equation} Substitute LHS of Eq. \ref{eq:8635275} into Eq. \ref{eq:5595798}; yields Eq. \ref{eq:5493675}. \begin{equation} I = |A|^2 + |B|^2 + |A| |B| 2 \cos( \theta - \phi ) \label{eq:5493675} \end{equation} Eq. \ref{eq:7781977} is an initial equation. \begin{equation} \theta = \phi \label{eq:7781977} \end{equation} Substitute LHS of Eq. \ref{eq:7781977} into Eq. \ref{eq:5493675}; yields Eq. \ref{eq:2413866}. \begin{equation} I_{\rm coherent} = |A|^2 + |B|^2 + |A| |B| 2 \cos( 0 ) \label{eq:2413866} \end{equation} Eq. \ref{eq:9739736} is an initial equation. \begin{equation} |A| = |B| \label{eq:9739736} \end{equation} Substitute LHS of Eq. \ref{eq:9739736} into Eq. \ref{eq:2413866}; yields Eq. \ref{eq:2139818}. \begin{equation} I_{\rm coherent} = |A|^2 + |A|^2 + |A| |A| 2 \label{eq:2139818} \end{equation} Simplify Eq. \ref{eq:2139818}; yields Eq. \ref{eq:7442815}. \begin{equation} I_{\rm coherent} = 4 |A|^2 \label{eq:7442815} \end{equation} Eq. \ref{eq:4842351} is an initial equation. \begin{equation} \langle \cos(\theta - \phi) \rangle = 0 \label{eq:4842351} \end{equation} Substitute LHS of Eq. \ref{eq:4842351} into Eq. \ref{eq:5493675}; yields Eq. \ref{eq:8093224}. \begin{equation} I_{\rm incoherent} = |A|^2 + |B|^2 \label{eq:8093224} \end{equation} Substitute LHS of Eq. \ref{eq:9739736} into Eq. \ref{eq:8093224}; yields Eq. \ref{eq:5409843}. \begin{equation} I_{\rm incoherent} = |A|^2 + |A|^2 \label{eq:5409843} \end{equation} Simplify Eq. \ref{eq:5409843}; yields Eq. \ref{eq:3246829}. \begin{equation} I_{\rm incoherent} = 2|A|^2 \label{eq:3246829} \end{equation} Divide Eq. \ref{eq:7442815} by Eq. \ref{eq:3246829}; yields Eq. \ref{eq:6088608}. \begin{equation} \frac{I_{\rm coherent}}{I_{\rm incoherent}} = 2 \label{eq:6088608} \end{equation} Eq. \ref{eq:6088608} is one of the final equations.