Loss occurs when a receiver is not matched
to the polarization of an incident electromagnetic field.

In the case of the polarization of a field emitted by a transmitting
antenna, first, look at the far zone of the transmitting antenna,
as shown in the following figure. At this location―which is
the location of the receiving antenna―the electromagnetic field
is orthogonal to the direction from transmitter to receiver.

You can represent the transmitted electromagnetic field, `fv_tr`,
by the components of a vector with respect to a spherical basis of
the transmitter's local coordinate system. The orientation
of this basis depends on its direction from the origin. The direction
is specified by the azimuth and elevation of the receiving antenna
with respect to the transmitter's local coordinate system.
Then, the transmitter's polarization, in terms of the spherical
basis vectors of the transmitter's local coordinate system,
is

$$E={E}_{H}{\widehat{e}}_{az}+{E}_{V}{\widehat{e}}_{el}={E}_{m}{P}_{i}$$

In the same manner, the receiver's polarization vector, `fv_rcv`,
is defined with respect to a spherical basis in the receiver's
local coordinate system. Now, the azimuth and elevation specify the
transmitter's position with respect to the receiver's
local coordinate system. You can write the receiving antennas polarization
in terms of the spherical basis vectors of the receiver's local
coordinate system:

$$P={P}_{H}{{\widehat{e}}^{\prime}}_{az}+{P}_{V}{{\widehat{e}}^{\prime}}_{el}$$

This figure shows the construction of the different transmitter
and receiver local coordinate systems. It also shows the spherical
basis vectors with which to write the field components.

The polarization loss is the projection (or dot product) of
the normalized transmitted field vector onto the normalized receiver
polarization vector. Notice that the loss occurs because of the mismatch
in direction of the two vectors not in their magnitudes. Because the
vectors are defined in different coordinate systems, they must be
converted to the global coordinate system in order to form the projection.
The polarization loss is defined by:

$$\rho =\frac{|{E}_{i}\cdot P{|}^{2}}{\left|{E}_{i}{|}^{2}\right|P{|}^{2}}$$