The angle dependency in FCPM

Assuming the transition dipole moments for absorption $\vec{M}_{abs}$ and emission $\vec{M}_{em}$ are parallel to the long axis of dye molecules and the excitation light $\vec{E}_0$ is linearly polarized, the absorbed light intensity is given by

$$I_{abs}\propto\left(\vec{M}_{abs} \vec{E}_0\right)^2=\left(\left|\vec{M}_{abs}\right|\left|\vec{E}_0\right|cos\left(\zeta\right)\right)^2$$

Assuming further that the rotational relaxation time $\tau_{rot}$ of the dye is slow compared to the lifetime $\tau_{excite}$ of the excited state, the orientation $\zeta'$ of dye during re-emittance of light can be considered identical to the orientation $\zeta$ during absorption.  The detectable light intensity is then given by

$$I_{det}=I_{abs}\left(\vec{M}_{em} P\right)^2=I_{abs}\left(\left|\vec{M}_{abs}\right|\left|\vec{E}\right|\cos^2\left(\zeta'\right)\right)$$

With $\zeta = \zeta'$ follows

$$I_{det}=\left(\left|\vec{M}_{abs}\right|\left|\vec{M}_{em}\right|\left|\vec{E}_0\right|\left|\vec{E}\right|\right)^2\cdot \cos^4\left(\zeta\right)$$

In consequence, the detectable light intensity scales with the angle $\zeta$ between the excitation light and the transition dipole moment $\vec{M}$ over the relation

$$I_{det}\propto \cos^4\left(\zeta\right)$$