The above formula assumes that the receiving antenna is polarization-matched. Normally, there is a polarization mismatch between the transmitting and receiving antennas. In the case of directional transmitting and receiving antennas, Friisâ formula takes the following form:
:<math> \frac{P_R}{P_T} = G_T G_R \left( \frac{\lambda_0}{4\pi R} \right)^2 ( \left| \mathbf{ \hat{u}_T \cdot \hat{u}_R } )\right|^2 </math>
where '''u<sub>T</sub>''' and '''u<sub>R</sub>''' are the unit polarization vectors of the transmitting and receiving antennas, and G<sub>T</sub> and G<sub>R</sub> are their gains, respectively.