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