Optimal Förster distance for two-state contrast ($R_{0,\mathrm{opt}}=\sqrt{AB}$)

For FRET efficiency $E(R)=1/[1+(R/R_0{)}^6]$ and distinct positive reporter distances A and B, the absolute contrast is maximized at the geometric mean Förster distance $R_0=\sqrt{AB}$. Substitution of $x=R_0^6$ makes the derivative have the sign of $A^6{B}^6-x^2$. The maximum is $|B^3-A^3|/(B^3+A^3)$ and the two efficiencies then sum to one. If $A=B$, every choice gives zero contrast and there is no unique optimum.