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The FRET dipole orientation factor describes the geometry of the two transition electric dipoles relative to their separation. Let ${\hat{\mathbf{\mu }}}_D$, ${\hat{\mathbf{\mu }}}_A$ and $\hat{\mathbf{r}}$ be their unit vectors. ThenFor a separation along $z$, parallel electric dipoles along $z$ give ${\kappa }^2=4$, parallel ones along $x$ give $1$, and one along $x$ with the other along $y$ gives $0$. Thus rotating a pair can change FRET efficiency without changing its separation.

The familiar value ${\kappa }^2=2/3$ requires suitable rapid, independent orientation averaging during the FRET donor lifetime. Random orientations across a population of slowly rotating fluorescent proteins do not establish that dynamic condition. ^[ref]

The FRET donor lifetime is the mean time spent in the excited state. For the constant-rate model of FRET efficiency,After a sufficiently brief excitation pulse, the FRET donor's fluorescence follows exponential decay, $I(t)=I(0)e^{-t/\tau }$. With the illustrative values ${\tau }_D=4$ nanoseconds and ${\tau }_{DA}=2$ nanoseconds, the inferred FRET efficiency is $1/2$. Multiplying $I(0)$ changes brightness without changing this decay time.

The lifetime comparison requires the same nontransfer decay rates in the reference and paired samples. Changes in the FRET donor's environment or a different nonradiative loss can also shorten the FRET donor lifetime. Heterogeneous FRET efficiencies or slowly varying FRET dipole orientation factors need an appropriate decay model; substituting any fitted average lifetime into the single-pair formula can mislead. ^[ref]